Field Mesoscopic Fluctations

A DVANCES IN P HYSICS, 2000, VOL. 49, NO.
3, 321 ± 394
Field theory of mesoscopic ¯ uctuations in superconductor± normal-metal

systems
ALEXANDER A LTLAND{{, B. D. Simons{ and D. TARAS-SEMCHUK{*

{Institut fuÈr Theoretische Physik, UniversitaÈt zu KoÈ ln, ZuÈlpicher Strasse 77, 50937
KoÈ ln, Germany
{ Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE, UK
[Received 9 November 1998 ; accepted in revised form 8 November 1999]
Abstract
Thermodynamic and transport properties of mesoscopic conductors are
strongly in¯ uenced by the proximity of a superconductor: an interplay between
the large scale quantum coherent wave functions in the normal mesoscopic and the
superconducting region, respectively, leads to unusual mechanisms of quantum
interference. These manifest themselves in both the mean and the mesoscopic
¯ uctuation behaviour of superconductor± normal-metal (SN) hybrid systems being
strikingly di erent from those of conventional mesoscopic systems. After review-
ing some established theories of SN quantum interference phenomena, we intro-
duce a new approach to the analysis of SN mesoscopic physics. Essentially, our
formalism represents a uni® cation of the quasi-classical formalism for describing
mean properties of SN systems on the one hand, with more recent ® eld theories of
mesoscopic ¯ uctuations on the other hand. Thus, by its very construction, the new
approach is capable of exploring both averaged and ¯ uctuation properties of SN
systems on the same microscopic footing. As an example, the method is applied to
the study of various characteristics of the single particle spectrum of SNS
structures.
Contents PAGE
1. Introduction 322
2. Andreev re¯ ection and the proximity e ect 326
2.1. Single-particle spectrum 329
2.2. Failure of semiclassics: quantum di raction 330
3. Quasi-classics 332
3.1. The Gorkov equations 332
3.2. Quasi-classical approximation 333
3.3. Dirty limit 333
3.4. Solution of the Usadel equation 334
4. Beyond quasi-classics 335
4.1. Perturbative diagrammatic methods 336
4.2. Multiple scattering formalism 337
5. Field theory for SN systems 339
5.1. Field integral and the ensemble average 340
5.2. Example: a bulk superconductor 343
5.3. Gradient expansion and `medium energy action’ 347
6. Stationary phase analysis 350
* e-mail: jpdt1@ cus.cam.ac.uk

Advances in Physics ISSN 0001± 8732 print/ISSN 1460± 6976 online # 2000 Taylor & Francis Ltd
http://www.tandf.co.uk/journals
322 A. Altland et al.
7. Fluctuations 356
7.1. a-Type ¯ uctuations: quantum corrections to the quasi-classical
theory 359
7.1.1. The C-modes: non-perturbative corrections to quasi-classical
Green functions 360
7.1.2. Perturbative corrections to the quasi-classical Green
functions 366
7.2. b-Type ¯ uctuations: the Goldstone mode 372
7.2.1. Level statistics in SNS structures 374
7.3. c-Type ¯ uctuations: quantum corrections to level statistics 375
8. Discussion 378
Acknowledgements 379
Appendix A: Boundary conditions of the Usadel equation 380
Appendix B: Symmetries of the action and Á-® elds 381
Appendix C: Saddle points and analytic continuation 381
Appendix D: Solutions to the Usadel equation 384
D.1. SN junction 384
D.2. SNS junction with coincident phases 385
Appendix E: E ective action of the Goldstone mode 386
E.1. Time reversal invariant action 386
E.2. Broken time reversal invariance 387
E.3. Boundary conditions 388
Appendix F: Renormalization of the minigap edge 390
References 392
1. Introduction
Physical properties of both superconductors and mesoscopic normal metals are
governed by mechanisms of macroscopic quantum coherence. Their interplay in
superconductor ± normal-metal (SN) systems, i.e. hybrid systems comprised of a
superconductor adjacent to a mesoscopic normal metal, gives rise to qualitatively
new phenomena (see [1] for a review): aspects of the superconducting characteristics
are imparted on the behaviour of electrons in the normal region. This phenomenon,
known as the `proximity e ect’ , leads to both the mean (disorder averaged)
properties of SN systems being substantially di erent from those of normal metals,
and various types of mesoscopic ¯ uctuations.
Although these two classes of phenomena are rooted in the same fundamental
physical mechanismÐ a tendency towards the formation of Cooper pairs in the
normal metal region of a SN systemÐ there are also major di erences. Even so, it is
notable that more than two decades passed between the ® rst analyses of the
manifestations of the proximity e ect in the mean properties of SN systems and
the mere observation that the same e ect may also be exhibited in mesoscopic
¯ uctuations. The intimate connection between mean and ¯ uctuation manifestations
of the proximity e ect will be discussed in some detail below. At this stage we simply
itemize some basic proximity e ect induced phenomenaÐ of both mean and
¯ uctuation typeÐ and brie¯ y comment on the theoretical approaches that have been
applied to their analysis.
Mean properties of SN systems. Superconductors strongly modify the
physical properties of adjacent normal metals. For example, the proximity of
Field theory of mesoscopic ¯ uctuations in SN systems 323
a superconducting condensate tends to induce singular behaviour in the

normal metal density of states (DoS). The properties of these singularities
depend on both the coupling to the superconductor and purely intrinsic
characteristics of the normal mesoscopic component. This complex
behaviour indicates that we are confronted with an interplay between
mechanisms of quantum coherence in the superconductor and in the
mesoscopic N region. DoS singularities are only one of many more examples
of manifestations of the proximity e ect. For example, supercurrents may
¯ ow through normal metal regions and the conductance of the normal metal
may vary with the phase of the superconducting order parameter [2, 3, 4].
Indeed the conductance may, quite counterintuitively, increase as a function
of the impurity concentration , through the phenomenon known as
re¯ ectionless tunnelling [1, 5].
All these phenomena share the common drawback that they are
exceedingly di cult to describe within conventional perturbative techniques
of condensed matter physics. Broadly speaking, the reason for these
problems is that the conventional `reference point’ of perturbative
approaches to dirty metals, i.e. a ® lled Fermi sea of electrons with an
essentially structureless dispersion relation, represents a poor starting point
to the description of SN hybrids. In fact, the proximity of a superconductor
leads to a strong modi® cation of the states in the vicinity of the Fermi
surface, which implies that it is di cult to perturbatively interpolate between
the conventional weakly disordered metal limit and the true state of the N
component of a SN system.
In the late sixties, Eilenberger [6] also introduced a novel approach to the
description of bulk superconductors which subsequently turned out to be
extremely successful in the analysis of SN systems. This so-called quasi-
classical approach provided essentially a controlled coarse-graining procedure
by which the Gorkov equation for the microscopic Green function of
superconductors could be drastically simpli® ed. Since on the one hand the
Green function contains all the information that is needed to describe SN
phenomena, whilst on the other handÐ for the reasons indicated aboveÐ its
computation in proximity e ect in¯ uenced environments is in general
tremendously di cult, it is clear that Eilenberger’s method represented a
breakthrough . Extending Eilenberger’s work, Usadel [7] later derived a
nonlinear, di usion-type equation, known as the Ùsadel equation’ , for the
quasi-classical Green function of dirty metals (for the precise de® nition of the
term `dirty’ , see below). Based largely on the pioneering work of Eilenberger [6]
and Usadel [7], a powerful array of quasi-classical methods has since been
developed to study the mean properties of SN systems.
In view of what has been said above about the di culties encountered in
perturbativ e approaches, it is instructive to re-interpret the solution of the
quasi-classical equations in terms of the language of conventional
diagrammatic perturbation theory. Referring for details to later sections,
we merely notice here that the quasi-classical Green functions actually
represent high-order summations of quantum interference processes caused
by multiple impurity scattering. More precisely, the solution of the Usadel
equation generally sums up in® nitely many so-called di uson diagrams, the
fundamental building blocks of the perturbative approach to dirty metals.
Whereas in normal metals high-order interference contributions to physical

observables are usually small in powers of the parameter g¡1 (g ¾ 1 is the
dimensionless conductance of a weakly disordered metal), here they represent
the leading order contribution even to a single Green function. In passing we
note that, very much as the conductance of normal metals may be expanded
in terms of the weak localization corrections, disorder averaged properties of
SN systems can be systematically expanded beyond the leading quasi-
classical approximation in powers of g¡1 . We will come back to this issue
below. Having made these observations one may anticipate that the tendency
to strong quantum interference in SN systems not only a ects their mean
properties but also leads to the appearance of unusual mesoscopic ¯ uctuation
behaviour.
Mesoscopic ¯ uctuations. It has been shown both experimentally [8, 9] and
theoretically [2, 3, 10± 17] that mesoscopic ¯ uctuations in SN systems not
only tend to be larger than in the pure N case, but also can be of qualitatively
di erent physical origin. After what was said above it should be no surprise
that the pronounced tendency to exhibit ¯ uctuations again ® nds its origin
in an interplay between standard mechanisms of mesoscopic quantum
coherence and the proximity e ect.
A list of examples of such SN-speci® c mesoscopic ¯ uctuation phenomena
includes the following:
(a) As in N systems, sample-speci® c ¯ uctuations of the conductance are
universal. However, due to proximity e ect induced coherence
mechanisms, the ¯ uctuations di er from those of normal systems by
rational numerical factors. Depending on the topology of the device, the
¯ uctuations can be smaller [18] or signi® cantly larger [8, 9] than in the
normal case.
(b) Besides the normal current, the critical Josephson current, Ic , through
SNS junctions exhibits ¯ uctuations [3], which become universal in the
limit of a short …L ½ ¹† junction [19]:
8
< …eEc †2 ; L ¾ ¹;
2
h¯Ic i ¹
:
…eD†2 ; L ½ ¹:
Here ¹ ˆ …D=D†1=2 is the coherence length of dirty superconductors , D

the order parameter, D the di usion constant, Ec ˆ D=L 2 the Thouless
energy and L the system size (note that we set h- ˆ 1 throughout).
(c) Such ¯ uctuations of the supercurrent are relatively robust [3]: for
instance, a relatively strong magnetic ® eld will exponentially suppress the
average supercurrent, but reduces the variance of the supercurrent
¯ uctuations by only a factor of 2.
(d) The single particle spectrum in the vicinity of the DoS anomaly exhibits
characteristic types of statistics [10, 17].
(e) Novel types of universal spectral ¯ uctuations appear [10].
As compared to the mean properties of SN systems, the physics of ¯ uctuation
phenomena is less well understood. First, the quasi-classical approach is not
tailored to an analysis of ¯ uctuations. Although, to compute ¯ uctuations,
one needs to average products of Green functions over disorder, the quasi-
classical equations are derived for single disorder averaged Green functions
and can, to the best of our knowledge, not be extended to the computation of
higher order cumulants{. Secondly, diagrammatic methods, for reasons
similar to those outlined above, are ruled out in cases where the proximity
e ect is fully established.
Important progress has been made by extending the scattering formulation of
transport in N-mesoscopic systems to the SN case [18, 11, 20± 22]. This
approach made possible an e cient calculation of both ¯ uctuation and weak
localization contributions to various global transport properties of SN
systems [1]. Unlike the quasi-classical formalism, however, the scattering
matrix approach is not microscopic. Instead, the di erent components of an
SN system are treated as `black boxes’ which are described in terms of
phenomenological stochastic scattering matrices. This approach, whilst
extremely powerful in the analysis of global transport features (the
conductance say), cannot address problems that necessitate a local and truly
microscopic description. For example, it is not suitable for the calculation of
spectral ¯ uctuations, both global and local, the analysis of local currents, and
so on.
The purpose of this paper is to introduce a theoretical approach to the study of
SN systems which essentially represents a uni® cation of the above quasi-classical
concepts with more recent ® eld theoretical methods developed to study N-meso-
scopic ¯ uctuations. As a result we will obtain a modelling of SN systems that treats
mean and ¯ uctuation manifestations of the proximity e ect on the same footing,
thereby revealing their common physical origin. This work represents the develop-
ment of ideas that we originally presented in a short letter [23].
Our starting point will be a connection, recently identi® ed, between quasi-
classical equations for Green functions on the one hand and supersymmetric
nonlinear ¼-models on the other. As was shown by Muzykhantski i and
Khmelnitskii [24], the former can be regarded as the classical equations of motions
of the latter. In other words, the ¼-model formulation has been shown to provide a
variational principle associated with quasi-classics. So far, these connections have
not been exploited within their natural context, namely, superconductivity . To ® ll
this gap, we will demonstrate here that by embedding concepts of quasi-classics into
a ® eld theoretical framework, one obtains a ¯ exible and fairly general theoretical
tool to the analysis of SN systems. In particular , it will be straightforward to extend
the quasi-classical equations so as to account for the consequences of time-reversal
symmetry, the connections to perturbative diagrammatic approaches will become
clear, andÐ most importantlyÐ the e ective action approach may be straightfor-
wardly extended to the computation of mesoscopic ¯ uctuations. In doing so, it will
become clear in which way both the characteristic features of the Usadel Green
function and SN-mesoscopic ¯ uctuations originate in the same basic mechanisms of
quantum interference.
{ Instead of deriving equations for the Green functions themselves one may attempt to set up a
quasi-classical approximation for their generating functional (U. Eckern, private communication).
Whether or not such an approach has been realized and/or made working in the concrete analysis of
¯ uctuation phenomena is unknown to us.
In this paper the emphasis will be on the construction of the approach, that is,
most of its applications will be deferred to forthcoming publications. However, in
order to demonstrate the practical use of the formalism we will consider at least one
important representative of mesoscopic ¯ uctuation phenomena, namely ¯ uctuations
in the quasi-particl e spectrum, in some detail: the DoS of N-mesoscopic systems
exhibits quantum ¯ uctuations around its disorder averaged mean value which may
be described in terms of various types of universal statistics. The analogous question
for SN systemsÐ What types of statistics govern the disorder induced ¯ uctuation
behaviour of the proximity e ect in¯ uenced DoS?Ð has not been answered so far.
Below we will show the emergence of some kind of modi® ed Wigner± Dyson statistics
[25], within the newly constructed formalism. A concise presentation of both the ® eld
theory and its application to SNS-spectral statistics is contained in [23].
The organization of the paper is as follows. In section 2, we review the basic
microscopic mechanism responsible for SN quantum interference phenomena. In
section 3, we discuss the quasi-classical approach to the computation of single
particle Green functions. In section 4, we brie¯ y review the diagrammati c and
statistical scattering approaches, as the only methods so far developed to compute
mesoscopic ¯ uctuations, as well as a few elements of semiclassics. These sections
have been included because we believe that a certain background knowledge of the
theoretical concepts that have been applied to mesoscopic superconductivity is
helpful for understanding the peculiarities of the problem. It should be emphasized,
though, that the short methodological review of section 4 cannot serve as an
introduction into the respective methods; readers who are not accquainted with
diagrammatics and/or the transfer matrix approach may prefer to skip this section
and to directly proceed to section 5.
In the central sections 5± 7 we introduce the aforementioned ® eld theoretical
framework. In section 5, we derive the e ective action for a di usive SN structure in
the form of a supersymmetric nonlinear ¼-model. In section 6, we obtain the saddle-
point equations of the action and examine their solution for some simple geometries.
As mentioned above, these saddle-point equations, obtained by a stationary phase
analysis of the e ective action, are none other than the quasi-classical equations of
motion. In section 7, we address the central issue of this paper, the behaviour of
¯ uctuations around the saddle-point solutions. The action displays a spontaneous
breaking of symmetry, whose massless, or Goldstone modes are the di usion modes
of the system. The interaction of the di usion modes is incorporated naturally within
this formalism, despite their strong modi® cation due to the proximity e ect, and
leads to mesoscopic ¯ uctuations. We calculate in this section the renormalization of
the spectrum of a quasi-1D SNS junction due to such ¯ uctuations. We also
demonstrate the spectral statistics of the SN structure to be described at low energies
by a modi® ed version of a universal Wigner± Dyson, or random matrix, theory. The
® eld theoretic formalism will also allow us to examine the onset at higher energies of
non-universal corrections which serve to destroy the correlations described by such a
universal model. In section 8, we conclude with a discussion.
2. Andreev re¯ ection and the proximity eŒect

Consider a normal metal at mesoscopic length scales, that is, scales much less
than both L ’ and L T , where L ’ is the dephasing length due to electron± electron
interactions and L T ˆ …D=T †1=2 sets the scale at which the quantum mechanical
coherence is cut o by thermal smearing e ects. The interest of such mesoscopic

materials stems from the fact that their physical behaviour is strongly in¯ uenced by
e ects of large scale quantum interference. Such e ects manifest themselves in both a
variety of ¯ uctuation phenomena and (non-stochastic) quantum corrections to
physical observables.
At the same time, the physics of bulk superconductors is also determined by
mechanisms of macroscopic quantum coherence. For example, the Cooper pairs
forming a superconducting condensate represent two-electron states whose phase
coherence extends over a (possibly macroscopic) scale set by the superconducting
coherence length.
Given that the physics of both mesoscopic metals and bulk superconductors is
in¯ uenced by quantum coherence, it is appropriate to expect that novel interference
mechanisms arise when two systems of this type are combined. This is indeed what
happens and has led to the continued interest in the physics of SN hybrid systems. A
key piece of information required for the understanding of large scale manifestations
of SN quantum coherence is the manner in which normal metals and super-
conductors exchange quantum phase information on a microscopic level. The basic
coupling mechanism between a superconductor and normal metal is a form of
interface scattering, known as Andreev re¯ ection [26, 27]. In this process, depicted in
® gure 1 (a ), an electron at an energy below the superconducting gap, D, strikes the
SN interface. Due to its low excitation energy it represents a forbidden quasi-particle
state and is unable to enter the S region. Instead, however, it may be Andreev
re¯ ected o the boundary as a hole. As a result two excess charges are left at the
interface which disappear into the superconducting condensate as a Cooper pair.
The detailed physics of Andreev scattering and its consequences for SN
structures has been reviewed extensively in the literature (see e.g. [1, 26, 28± 31]).
Here we merely summarize some of its essential features that will be of importance
throughout.
(a) As opposed to ordinary specular re¯ ection, Andreev re¯ ection represents a
process of `retro-re¯ ection’. More precisely, apart from a slight angular
mismatch proportional to the excitation energy, °, of the electron above the
Fermi energy, °F , the hole is re¯ ected back along the trajectory of the
incoming electron.
(b) An electron with excitation energy ° is scattered into a hole with energy ¡°.
a b
Figure 1. (a ) Andreev re¯ ection of an electron at an SN interface and (b ) a typical pair of

Feynman paths that lead to a non-zero value of hÁ y Á y i.
(c) The hole acquires a scattering phase p=2 ¡ ’, where ’ is the phase of the
superconducting order parameter at the interface.
An important consequence of the existence of the Andreev scattering mechanism is
the formation of a Cooper-pair amplitude hhÁ y…x†Á y …x†ii in the normal metal region.
Here hh. . .ii represents not only the quantum mechanical expectation value, but also
a disorder average. The creation of an average local pairing amplitude can be
understood heuristically from a simple semiclassical consideration: consider the
creation of an electron somewhere at a point x inside a disordered metal adjacent to
a superconductor (see ® gure 1 (b )). Due to the presence of disorder, the electron will
propagate di usively and may eventually strike the SN interface and be Andreev
re¯ ected. In general the newly created hole may now di use along its own path.
However, a particularly interesting situation arises if the hole happens to propagate
along the path of the incoming electron back to the point of creation. As a result we
obtain a non-vanishing pairing ® eld amplitude hh Á y …x†Á y …x†ii. The point is that
during their propagation through the disordered background both the incoming
electron and the outgoing hole accumulate a quantum mechanical scattering phase
which depends sensitively on microscopic details of the disorder. However, owing to
the fact that the two particles propagate along the same path, these phases cancel
each other to a large extent. (For an excitation energy ° ˆ 0 the cancellation is, in
fact, perfect. For non-vanishing ° one obtains a phase mismatch ¹ L p °=vF / °L 2 =D,
where L p is the length of the scattering path, D the di usion constant, vF the Fermi
velocity, L the separation of x from the interface and we have used the fact that for
di usive motion L p =vF / L 2 =D.) Of course, while more generic path pairs, where the
electron and hole follow di erent paths, also contribute to hÁ y …x†Á y…x†i , their
contributions vanish upon disorder averaging due to their strong phase dependence.
The non-vanishing of hhÁ y …x†Á y…x†ii is the basic content of the proximity e ect.
Besides its resilience against disorder, the pairing ® eld amplitude possesses a number
of important featuresÐ all of which are related to the phase argument aboveÐ that
will be of importance for all that follows.
(a) hhÁ y…x†Á y …x†ii varies weakly as a function of x. More precisely, it does not
¯ uctuate on atomic scales but rather on scales set by …D=°†1=2 .
(b) hhÁ y…x†Á y …x†ii decays exponentially as a function of °L 2 =D. If either T or the
inverse dephasing time ½’¡1 exceed °, the decay rate is set by these energy
scales.
(c) Quantitative expressions for the di usive pairs of quantum paths entering the
physics of the proximity e ect are provided by so-called di uson modes.
Their meaning in the present context will become clear below.
(d) The pairing ® eld amplitude depends on the phases of the order parameters of
the adjacent superconductors. If only a single superconducting terminal with
constant phase, ’, is present, the phase dependence is simply ¹ exp …i’†. In
this case the phase is inessential and can be eliminated by means of a global
gauge transformation. More interesting situations arise when more than one
superconductor are present, in which case the phase sensitivity of the pairing
amplitude provides the mechanism for the stationary Josephson e ect.
The non-vanishing of the pairing ® eld amplitude heavily in¯ uences the properties of
the normal metal components of SN systems. Widely known examples of proximity
e ect induced phenomena are the DC and AC Josephson e ect, which allow the
Figure 2. The geometry of the SNS junction.
possibility of supercurrent ¯ ows through SNS sandwiches. Another important

phenomenon is the dependence of the N conductance on the phases of adjacent
superconductorsÐ again triggered by the phase sensitivity of the proximity
amplitude [4, 32]. However, as mentioned in the introduction, the emphasis in this
paper will be on a study of the in¯ uence of the proximity e ect on the single particle
spectrum.
2.1. Single-particle spectrum

To understand the basic connection between the proximity e ect and the single
particle spectrum, let us begin by considering the simple geometry of an SNS
sandwich, shown in ® gure 2, where the N layer is of width L , of otherwise in® nite
extent and clean. This system was ® rst considered by Andreev [26] who applied
scattering theory to the electron wavefunction to show that the spectrum in the N
region, for trajectories at a ® xed angle to the interface, is discrete below the
superconducting gap. The Àndreev levels’ correspond to bound states with energies
° given by the quantization rule,
…jv°L j ‡ sgn …v † D 2’† ˆ …D ¡ °2 †1=2

2
tan x ; …1†
x °
where D ’ is the phase di erence across the junction and vx is the component of the
electron velocity normal to the interface. Equation (1) then indicates an average
Andreev level spacing of the order of the inverse ¯ ight time, jvxj=L , across the
normal region. A non-zero phase di erence, D ’, leads to a shift in the levels so as to
produce two separate branches of the spectrum, for electrons and holes respectively.
To determine the observed spectrum, it is necessary to sum the pole contributions
to the DoS from bound states which arise from all possible velocity directions,
according to equation (1). Note that non-zero contributions survive down to
arbitrarily small energies due to trajectories travelling close to parallel along the
interface. The energy dependence of the weights of these poles is so as to produce a
total DoS which is linear in energy °, at energies ° ½ D and in three dimensions.
The introduction of a ® nite concentration of impurities leads, upon disorder
averaging, to a smearing of the formerly sharp pole structure. A less obvious
outcome of this process is the appearance of a sharp cut-o in the spectrum, the
`minigap’ , below which the DoS vanishes entirely (as ® rst analysed , in a dirty system,
by Golubov et al. [33]). The minigap, Eg , is smaller than the superconducting gap, D,
and depends on D ’, attaining its maximum for D ’ ˆ 0 and shrinking to zero as D ’
approaches p. In general Eg depends on D ’ in a non-sinusoidal fashion [34],

although in the limit of a short, di usive junction [35], L ½ ¹, the dependence
becomes sinusoidal, Eg ˆ D cos …D ’=2†.
In the general case, the formation of a minigap in the metallic DoS represents a
highly non-trivial phenomenon. For example, it has been shown [36] that the
presence or absence of a gap depends on the classical dynamical features of the
metallic probe contacting the superconductor: for samples with integrable classical
dynamics (such as the cubic system described above) there is no gap but rather a DoS
that vanishes linearly at the Fermi energy. The existence of states all the way down to
zero energy may be understood by means of Bohr± Sommerfeld quantization
arguments. By contrast, in systems with chaotic classical dynamics, a gap opens
whose magnitude may depend on both the coupling strength to the superconductor
and the intrinsic classical transport time through the metal region.
2.2. Failure of semiclassics: quantum di raction

The prediction of a minigap provides a useful test for any analytic approach to
the physics of SN systems: for example, an application of standard approximation
schemes to semiclassical formulae for the the DoS fails to predict correctly a gap.
To explain this point, let us consider the simple case of a di usive metallic cube of
linear extension L attached to a superconductor . In this case the DoS gap is of width
’ Ec . In order to embed the previous heuristic path arguments regarding the
proximity e ect into a quantitative calculation of the proximity e ect in¯ uenced
DoS one might apply the semiclassical Gutzwiller trace formula [37], a powerful
computational tool often used in the analysis of the DoS and correlations thereof{.
The Gutzwiller trace formula essentially states that the DoS can be obtained as a
sum over periodic orbits,
X
¸ ¹ - †;
A¬ exp …ihS ¬
¬
where ¬ denotes the orbit, S ¬ its classical action and A¬ is a stability factor which
will not be essential throughout. Averaging the DoS leads to a reduction to a sum
over those periodic orbits whose action is small, of the order of h- (cf. e.g. [40] for a
review.) In the pure N case no such orbits can be found, implying that the average
N DoS is structureless. In the presence of a superconductor , however, the situation is
di erent and low action periodic orbits exist. A representative of the most apparent
category of such orbits is depicted in ® gure 3 (a ). The smallness of the action of these
paths follows from the fact that if the action of the electronic segment of the path
(solid line) has an energy dependent action S…°†, the action of the hole segment
(dashed line) will be ¡S …¡°†. The two contributions nearly cancel each other. Notice
that the structure of these orbits bears resemblance with the so-called `diagonal
approximation’ commonly employed semiclassical analyses: in general, each periodic
orbit ¬ contributing to the Gutzwiller formula for the SN DoS decomposes into a
particle segment ¬p and a hole segment ¬h , ¬ ˆ ¬p [ ¬h . Prior to demanding that the
action of the orbit be small, the particle and the hole segment can be chosen largely
{ Although the trace formula is most often applied to purely normal systems, the generalization
to superconducting systems follows straightforwardly by a doubling of the space to include particle and
hole sectors. See e.g. [38, 39] for application of such semiclassical formulae to clean SN systems.
a b
Figure 3. Trajectories that are (a ) included and (b ) not included within a semiclassical
treatment.
P -
independently, i.e. ¸ ¹ ¬p ;¬ h A¬p [¬h exp …ihS ¬p [¬h † takes the form of a coherent
double sum over particle and hole segments. To a ® rst approximation
P , the projection
onto
P orbits with a small action leads to a collapse of ¬p ;¬h to the `diagonal’ sum
¬p over orbits with ˆ
¬p ¬h , i.e. orbits like the one depicted in ® gure 3 (a ). This
selection rule parallels the diagonal approximation scheme commonly applied to the
analysis of pairs of Green functions in N systems [40]. Yet whereas in N physics
corrections to the diagonal scheme are generally associated with small quantum
coherence corrections, the diagonal approximation in the SN case fails to correctly
predict even coarse structures of the average DoS. Even if complex paths with
multiple Andreev scattering are taken into account, the diagonal sum does not, for
example, produce a real gap in the DoS.
In order to understand this failure we ® rst have to notice that small-action path
pairs exist which do not fall into the scope of the diagonal approximation (by which
we mean that they cannot be obtained as a superposition of two identical segments,
one electron- and one hole-like). A common feature of these `non-diagonal’ path
con® gurations is that they contain `junction points’ where the paths of electrons and
holes split (cf. ® gure 3 (b )). In order to understand the existence of these splittings
one has to keep in mind that the paths entering the semiclassical picture do not
correspond to rigorously de® ned solutions of classical equations of motion but
should rather be thought of as objects that are smeared out (in con® guration space)
over scales comparable with the Fermi wavelength [41]. As a result two classically
ìdentical’ paths may split and recombine at some later stage, a process which is not
accounted for by the diagonal approximation. This splitting, as it is caused by the
wave nature of the electrons, is sometimes referred to as a quantum di raction
phenomenon.
Note that the junctions appearing in ® gure 3 (b ) are reminiscent of similar
processes needed to generate weak localization corrections to N-mesoscopic
observables [41], that is, corrections to the `standard’ diagonal approximation
scheme. However, whereas weak localization corrections represent a correction of
O…g¡1 † (g ¾ 1 is the dimensionless conductance) , the di raction contributions
appearing in the present context can by no means be regarded as small. In fact they
are as important as the leading order diagonal contributions , which implies that
processes with up to an in® nite number of `junction points’ have to be taken into
account. This fact not only explains the failure of the diagonal approximation but
also the di culties encountered in diagrammati c analyses of the proximity e ect.
The point is that the diagrammatic counterpart of each of the `legs’ appearing in
® gure 3 (b ) is a Cooperon mode. The perturbative summation of in® nitely many
Cooperons represents a di cult problem, in particular in cases where the system is
truly extended in the sense that it cannot be treated within an ergodic or zero mode
approximation (for a perturbativ e analysis of the zero-mode scenario, see Brouwer
et al. [42]). Fortunately there is an alternative approach, the quasi-classical method
reviewed below, which provides a highly e cient tool for the e ective summation of
all interference corrections contributing to the DoS and other physical observables.
3. Quasi-classics
In order to prepare the discussion of quasi-classics we ® rst need to introduce its
microscopic basis in general, and the Gorkov Green function in particular.
3.1. T he Gorkov equations

As long as interaction e ects are neglected{ the complete information on any SN
system is encoded in its single particle Gorkov Green function, Gr;a . The latter is
de® ned by the Gorkov equations [43], whose matrix representation reads
0 1 e 2 1
· ‡ °§ ¡ ^
p ¡ A…r1 † ¡V …r1 † D…r1 †
B 2m c C r;a
B CG …r1 ;r2 †
@ A
¤ 1 e 2
¡D…r1 † · ¡ °§ ¡ ^
p ‡ A…r1 † ¡V …r1 †
2m c
ˆ 1¯d …r1 ¡ r2 †; …2†
where
Gr;a F r;a
Gr;a ˆ …F y r;a
Gy
r;a †: …3†
Here Gr;a and F r;a represent the normal and anomalous Green function, respectively,
D…r† is the superconducting order parameter (which, in principle, has to be
determined self-consistently), °§ ˆ ° § i0, and V is the impurity potential.
The equation above may be represented in a more convenient way by introducing
Pauli matrices, ¼ph ±
i , operating in the two-component particle hole space{. Separating
the order parameter into its modulus, jD…r†j, and phase, ’…r†, equation (2) takes the
form
1 e 2
°F ¡ p ¡ A…r†¼ph
^ 3 ¡V …r† ‡ ¼ph
3 D ^ …r1 † ‡ °§ Gr;a …r1 ;r2 † ˆ ¯d …r1 ¡ r2 †; …4†
2m c
^ ˆ ¼ph
where D ph
1 jDj exp ‰¡i’…r1 †¼3 Š. The presence of an impurity potential , V …r†,
makes the solution of equation (4) di cult. However, as pointed out by Eilenberger
{ In this paper theÐ importantÐ role Coulomb interactions may play in SN physics will not be
discussed.
{ Recall the general de® nition of the Pauli matrices,
¼ˆ … 01 10 †; ¼2 ˆ … 1i ¡i
0 †
; ¼3 ˆ … 10 0
¡1†:
[6] and Larkin and Ovchinnikov [44], a crucial simpli® cation applies in the case
where the wavelength of the electrons, ¶F , is small as compared to the characteristic
scales over which the order parameter D…r† and vector potential A…r† vary. Under
this condition one may resort to the `quasi-classical approximation’ (which, in the
context of mesoscopic superconductivity , should not be confused with the `semi-
classical approximation’ ).
3.2. Quasi-classical approximation

The starting point of the quasi-classical approach is the observation that the
spatial structure of the Green function is comprised of rapid oscillations, over a
spatial scale of the Fermi wavelength, modulated by a slowly ¯ uctuating background
over longer scales. A quasi-classical analysis of the proximity e ect involves an
averaging over the rapid variations of the Green function. At the same time,
su cient information is retained within the slower modes of the Green function to
provide a useful approximation to the full consequences of the proximity e ect. The
advantage is a great simpli® cation of the corresponding kinetic equations. As the
derivation of the quasi-classical equations has been reviewed extensively in the
literature (e.g. [31, 45, 46]), we restrict ourselves here to a brief summary of the main
results and ideas of the approach.
The quasi-classical , or Eilenberger, Green function, g r;a …n; r†, is obtained from
the Gorkov Green function by (a) a Wigner transform , (b) an impurity average and
(c) an integral over the kinetic energy variable. The precise de® nition reads
… …
i
r;a
g …n;r† ˆ d¹p d…r1 ¡ r2 †Gr;a …r1 ;r2 † exp …¡ip ¢ …r1 ¡ r2 ††; …5†
p
where r ˆ …r1 ‡ r2 †=2, ¹p ˆ vF …p ¡ pF †, n ˆ p=p and pF ˆ mvF is the Fermi momen-
tum. The application of this approximation to the Gorkov equation, equation (4),
leads to the Èilenberger equation’ [6]:
i r;a 0
vF n ¢ rr gr;a …n;r† ˆ i ¼ph ^
3 …°§ ‡ D…r†† ‡
hg …n ;r†in 0 ;gr;a …n; r† ; …6†
2½
where ½ is the elastic scattering time due to impurities. This equation essentially
represents an expansion to leading order in the ratio of ¶F to the scale of spatial
variation of the slow modes of the Gorkov Green function. It can be shown that the
Eilenberger Green function obeys the nonlinear normalization condition (see, for
example, the discussions in [47, 48]),
gr;a …n;r†2 ˆ 1: …7†
Equation (6) represents an equation of Boltzmann type which is much simpler than
the original Gorkov equation, but may still be di cult to solve in general. However,
signi® cant further simpli® cations are possible in the `dirty limit’.
3.3. Dirty limit

The `dirty limit’ is speci® ed by the conditions ` ˆ vF ½ ½ ¹ (implying that the
dominant transport mechanism on the scales of the coherence length is di usion)
and ° < ½ ¡1 (implying that `time scales’ °¡1 much longer than the scattering time are
explored). Under these conditions, the dependence of the Green function on the
angular direction (represented by n) is weak and one may expand in its ® rst two
spherical harmonics:
gr;a …vF ;r† ˆ gr;a r;a

0 …r† ‡ n ¢ g1 …r† ‡ ¢ ¢ ¢ ; …8†
where gr;a r;a
0 …r† ¾ n ¢ g1 …r†. A systematic expansion of the Eilenberger equation in
terms of g1 then leads to a nonlinear and second-order equation for the isotropic
component,
0 …r†rg0 …r†† ‡ i‰¼3 …°§ ‡ D …r††;g0 …r†Š

ph
D r…gr;a r;a ^ r;a
ˆ 0; gr;a
0 …r†
2
ˆ 1; …9†
known as the Ùsadel equation’ [7]. In order to specify a solution, one has to
supplement the equation with appropriate boundary conditions. The analysis of the
boundary behaviour of the equation becomes somewhat technical. For this reason a
more detailed discussion of the boundary conditions,
¼…¡†g0 @r g0 …¡† ˆ ¼…‡†g0 @r g0 …‡†; …10†
GT
¼…§†g0 @r g0 …§† ˆ ‰g0 …‡†;g0 …¡†Š; T ½ 1; …11†
2
g0 …‡† ˆ g0 …¡†; T ’ 1; …12†
is the subject of Appendix A. Here T 2 ‰0;1Š is a measure for the transparency of the
SN interface, g0 …§† denotes the Green functions in® nitesimally to the left, respect-
ively right, of the junction, ¼…§† is the metallic conductivity on either side of the
junction, and GT (cf. equation (A 7)) is the tunnel conductance of the interface.
3.4. Solution of the Usadel equation

Solutions of the Usadel equation with appropriate boundary conditions have
been derived for a vast number of geometries. Indeed a systematic and general
solution scheme, based on an e ective circuit theory, has been constructed by
Nazarov [49]. Furthermore, a number of quasi-classical predictions seem to be
borne out well experimentally (see e.g. [32, 50± 54]). In the ® eld theoretic context
introduced below, the Usadel equation and its boundary condition will reappear on
the level of the mean ® eld analysis in section 6. Explicit solutions for some simple
geometries will be discussed in that section.
At this stage the question might arise of how the comparatively simple quasi-
classical approach manages to produce correct results for physical observables like
the mean DoS whilst the methods alluded to earlier, semiclassics and diagrammatic
perturbative methods, fail to do so. This point can be understood by qualitatively
constructing a connection between the Usadel approach and the real space di usion
picture discussed in section 2.2. To this end we focus on the solution of the Usadel
equation in the N region of a some larger SN structure{. Since, by de® nition, the N
region does not support an order parameter, we are led to consider an Usadel
equation where D appears as a boundary operator , Db , at the interface to the
S regions. To acquire some understanding of the basic structure of the solution of the
{ In principle one would have to solve the equation in both the N and the S region, followed by a
connection of the solutions according to the boundary conditions discussed above. In practice, however,
it often su ces to restrict attention to a particular region of interest and to treat the complementary
region as a boundary perturbation. This strategy is frequently adopted in mesoscopic superconductivity
where the interest is primarily on proximity in¯ uenced N regions. For explicit realizations of this
scheme, see below.
equation, one may attempt to expand in powers of the boundary operator Db , i.e. in
the number of Andreev scattering events. In cases where the coupling to the
superconductor is signi® cant, the outcome of this scheme will have little in common
with the `real’ solution (which often enough is even non-analytic in the strength of
Db ). It has, however, the pedagogical advantage of making explicit the connection to
the approaches discussed above: by explicitly carrying out the perturbative expan-
sion, one ® nds that each power of Db is accompanied by a di usion pole structure
¹…D@ 2 ‡ i!†¡1 . These poles are the Cooperon modes which we saw above to
originate from points of Andreev scattering (cf. ® gure 3 (a )). For higher orders of
Db , `branching’ processes come into play, with the formation of structures such as
the one depicted in ® gure 3 (b ). (More precisely, the perturbative solution contains
terms where di usion poles originate at the boundaries and terminate at common
end points, which are located in the interior of the N region and are integrated over.
Within a diagrammatic approach, each branching point is represented by a Hikami
box. Physically, these points represent the separation of two trajectories that are
classically identical, but quantum mechanically smeared over a wavelength [41].)
Except for cases where the strength of the boundary scattering is asymptotically
weak (equivalently, a regime in which the Usadel solution may be linearized in its
o -diagonal components in particle± hole space), an in® nite order expansion in the
number of branching points is needed to correctly describe the proximity e ect or,
worse, the solution is not perturbatively accessible at all. This qualitative picture
explains both the strength of approaches based on a direct solution of the Usadel
di erential equation (as opposed to the arti® cial `perturbative’ scheme considered
here) and the inadequacy of diagrammati c perturbative expansions which, in
practice, are limited to contributions with a low number of branching points.
4. Beyond quasi-classics
The quasi-classical approach allows for the e cient calculation of a wide
collection of physical observables. In general, any observable that may be expressed
in terms of a single disorder averaged Green function is a candidate for quasi-
classical analysis. Note that, by extending the formalism so as to include Keldysh±
Green functions [55], observables that are commonly expressed in terms of two-
particle Green functions also become accessible. Important examples are the
conductance and other transport quantities. However, there are important classes
of observables which do not fall into the above category, thereby falling beyond the
scope of quasi-classics. The list of inaccessible quantities may be grouped roughly
into four di erent categories.
(a) Physical observables which, by de® nition, are not expressible in terms of
single particle Green functions. An example is given by the magnetic ® eld
dependence of the London penetration depth for a bulk superconductor , as
studied by Larkin and Ovchinnikov [44]. Its analysis requires the
computation of the average of four momentum operators, hp…0†p…t†
p…0†p…t†i, a quantity that involves two- rather than one-particle Green
functions.
(b) Higher order quantum interference corrections to single particle Green
functions. Very much like weak localization corrections of O…g¡1 † to the
classical Drude conductance, the quasi-classical Green function represents
the leading order term of a series expansion in powers of g¡1 . The next to
leading order contributions become important in cases where one is

interested in quantum corrections of weak localization type or strong
localization e ects.
(c) The quasi-classical approach (in its extension to include a Keldysh
component) does not account for the corrections to two-particle Green
functions due to the interference of mutually time-reversed trajectories. An
example property that is a ected in this way is the conductivity, as we see
below.
(d) Most importantly, the quasi-classical approach does not allow for the study
of mesoscopic ¯ uctuation phenomena. The analysis of ¯ uctuations requires
the computation of disorder averages of two or more Green functions. Due
to the impurity induced interference between di erent Green functions,
quasi-classical techniques are inapplicable to these problems.
Given that these classes of problems cannot be addressed within quasi-classics, it
becomes necessary to seek some alternative approach. Here, we brie¯ y review two of
perhaps the most important theoretical tools currently established, namely, dia-
grammatics and the scattering matrix approach.
4.1. Perturbative diagrammatic methods

Microscopic diagrammatic methods have been applied to the study of various SN
phenomena. The list of diagrammatic analyses includes computations of Josephson
current ¯ uctuations through SNS junctions [3], investigations of the phase sensitivity
of the N conductance [2], computations of universal conductance ¯ uctuations of SN
systems [11], and more.
The basic building block of diagrammatic analyses are the di usion modes of
both di uson and Cooperon type. What makes these modes di erent from their
counterparts in pure N systems is that they now include Andreev scattering events, as
represented in ® gure 4. In the presence of the proximity e ect, any of the Andreev
scattering vertices appearing in these modes is in turn to be renormalized by further
a b
Figure 4. (a ) shows the trajectories whose interference leads to the ® rst weak localization
correction to the conductivity: (b ) shows the corresponding diagrams, where the
curvy line is the Cooperon and the triangles represent Andreev re¯ ection processes.
Figure 5. First order renormalization of the Andreev scattering process necessary to include
the proximity e ect.
di usion modes, as indicated in ® gure 5. Note that these diagrams are the formal
representation of what in real space are the `legs’ of the star® sh-like structures
appearing in ® gure 3. The problem with the diagrammati c approach is that in
situations where the proximity e ect is fully established, the Andreev vertices
renormalize heavily, i.e. one has to sum self-consistently nested series of the diagrams
depicted in ® gure 5. Another way of putting this is to say that one has to
perturbatively reconstruct the solution of the Usadel equation, a di cult if not
impossible task. In fact, an incomplete account of the proximity-induced renorma-
lization processes may lead to incorrect results: for example, diagrammatic analyses
of universal conductance ¯ uctuations by Takane and Ebisawa [11] failed to
reproduce correctly their surprising insensitivity to external magnetic ® elds, as later
demonstrated by Brouwer and co-worker [1, 15]. However, in cases where the
proximity e ect is either suppressed or of secondary importance, diagrammatic tools
can be applied successfully to the study of SN systems.
In summary, it can be said that diagrammatic s is applicable to the perturbative
analysis of SN phenomena in cases with a weakly pronounced proximity e ect. As is
usual with diagrammatic methods, non-perturbativ e problems, such as localization,
® ne structure level statistics, and so on, cannot be addressed.
4.2. Multiple scattering formalism

Scattering theory provides a powerful theoretical tool for the analysis of quantum
transport through mesoscopic systems in general, and SN systems in particular. The
reason for the e ciency of the scattering theoretical formalism in the study of SN
systems is not only its relative simplicity, but also the fact that the proximity e ect
does not seem to lead to essential complications. Due to this latter advantage ,
scattering theory has for a long time existed as the only tool for computing
mesoscopic ¯ uctuations under the in¯ uence of the proximity e ect.
The starting points of the scattering approach are generalizations of the standard
multi-channel Landauer formulae for N-mesoscopic systems to the SN case. For
example, in the case of a single N sample attached to a superconductor the
conductance may be expressed as [11, 20, 21]
2e2 y 4e2 y
GSN ˆ tr …1 ¡ S ee S ee
y
‡ S he S he † ˆ tr S he S he ; …13†
h h
where e is the electron charge and S ee …S eh † are the matrices describing the scattering
of electrons incoming from the normal metal to electrons (holes). In a next step, the
scattering matrices are expressed in terms of (a) the transmission matrices of the
normal metal compound (which are known in terms of their transmission eigenvalue
distribution functions [1, 56]), and (b) matrices describing the scattering o the
superconductor.
The scattering theoretical approach is particularly powerful if the observables of
interest take the form of `linear statistics’ , i.e. quantities, X, that can be represented as
X
Xˆ f …T n †;
n
where f is some function and T n is related to the nth normal transmission matrix
eigenvalue. This is often but not always the case: for example, in time reversal
invariant cases, the conductance of the above SN system may be formulated as (see
Beenakker [22]),
N
4e2 X T n2
GSN ˆ : …14†
h nˆ1 …2 ¡ T n †2
However, if time reversal invariance is broken, such a simple representation is no

longer possible and expressions involving not only eigenvalues but also the
diagonalizing matrices appear. It has been shown by Brouwer and Beenakker [42]
that, even under these more complicated circumstances, scattering theory remains
applicable, although at the expense of introducing further series of diagrams{. Due
to the complex structure of the diagrammatic series appearing in the SN problem,
only quantum dots (i.e. ergodic systems that can be modelled in terms of a scattering
matrix distributed through a single Haar measure) rather than arrays thereof could
be analysed in this way: we again encounter the notorious di culties accompanying
perturbative approaches to SN systems.
In passing we note that in some cases SN quantum dots, chaotic or disordered,
can be modelled in terms of simple random matrix theory. More precisely, random
matrix techniques become applicable if the proximity e ect is suppressed. This
happens if the system is subject to a magnetic ® eld (of the order of a few ¯ ux quanta
through the system), or if the phases of the adjacent superconductor s average to
zero{. In spite of the fact that the proximity e ect is suppressed, the mechanism of
Andreev scattering remains active and manifests itself in the SN random matrix
ensembles having symmetry properties that di er substantially from the standard
Wigner± Dyson ensembles.
To summarize, statistical scattering theory represents a powerful tool for the
analysis of both mean and ¯ uctuation characteristics of global transport quantities.
Clearly, observables belonging to this category are of outstanding importance from
the experimental point of view. Nevertheless, problems remain for which one is
interested in observables that are local and/or microscopically de® ned. This
{ What becomes necessary is to supplement the conventional transfer matrix approach (by which
we mean the derivation and solution of a Fokker± Planck equation for the eigenvalues) by diagrammatic
methods accounting for the presence of the diagonalizing matrices.
{ The latter mechanism is rather subtle and it is not clear whether it can be realized in practice.
The reason is that even minute phase ¯ uctuations of O…g¡1 † invalidate the applicability of random
matrix theory. However, systems with a phase-suppressed proximity e ect are realized in nature as
vortices in type II superconductors, see e.g. Caroli et al. [57]. Here the vortex centre has a non-vanishing
metallic DoS, an indication of proximity e ect suppression.
complementary class of quantities is inaccessible through phenomenological scatter-

ing analyses. Thus at least one alternative theoretical tool for the analysis of SN
systems is called for.
5. Field theory for SN systems

In the following central part of the paper we are going to introduce a novel
approach to the analysis of SN systems which is based on ® eld theoretical and, by
construction, microscopic concepts. The formalism will be applicable to observables
that can be expressed in terms of one or products of single particle Green functions.
If this criterion is met, both mean values (including quantum corrections to quasi-
classical results) and mesoscopic ¯ uctuations can be computed. In a few exceptional
cases, distribution functions can be obtained. It will quickly turn out that the
formalism is intimately related to each of the approaches reviewed above: on the
mean ® eld level it reproduces quasi-classics , perturbative ¯ uctuations around the
mean ® eld can be interpreted diagrammatically , and the connection to scattering
theory is established through general parallels between ¼-models and the transfer
matrix approach [58].
Prior to embarking on any kind of detailed discussion, let us brie¯ y outline the
main conceptual steps of the construction of the ® eld theory.
(1) Starting from the microscopic Gorkov Hamiltonian of an SN system we will
construct a generating functional for the disorder average of the product of a
retarded and an advanced Green function. (Generalizations to products of
more than two Green functions are straightforward.) The functional will be
of a nonlinear ¼-model type. Essentially it represents a supersymmetric
extension of earlier (replicated) ® eld theoretical approaches to bulk
superconductors [59]. In the present formulation, the order-parameter ® eld
will be imposed and not computed self-consistently (a common and mostly
inessential simpli® cation in the ® eld of SN systems{).
(2) It will then turn out that the spatial inhomogeneity of the order parameter
® eld poses a substantial problem: a straightforward perturbative evaluation
of the ® eld theory, by which we mean a perturbative expansion around any
spatially homogeneous reference ® eld con® guration, is impossible. It goes
without saying that this is nothing but the manifestation, in a ® eld theoretical
context, of the general perturbative di culties characteristic to SN systems.
(3) The way out will be to subject the ® eld theory, prior to any perturbative
manipulations , to a stationary phase analysis. Given what has been said
under the previous item, it is clear that the saddle-point con® gurations of the
theory must be spatially inhomogeneous. More speci® cally it will turn out
that the stationary phase equation of the theory is simply the Usadel
equation. In other words, the quasi-classical approach to SN systems turns
out to be equivalent to the mean ® eld level of the ® eld theoretical formalism.
{ Self-consistent calculations of D…r† in mesoscopic SN junctions, as achieved analytically by

Zaikin [31] in the clean case and numerically by several authors (e.g. [60 ± 62]) in the dirty case show that
D is suppressed in the S region near the interface. It may also become non-vanishing in the N region if
an electron± electron interaction is included there. We will neglect these e ects as we do not anticipate
that they would have a signi® cant impact on our results.
(4) We then turn to the issue of ¯ uctuations. Broadly speaking, two qualitatively
di erent types of ¯ uctuations will be encountered: massive ¯ uctuations
around the mean ® eld (giving rise to quantum corrections to quasi-classics)
and a Goldstone mode. The latter will induce correlations between retarded
and advanced Green functions and thereby mesoscopic ¯ uctuations.
To keep the discussion of the above hierarchy of construction steps from being
too abstract, the computation of correlations in the single particle spectrum will
serve as a concrete example of an application of the theory.
Before turning to the actual construction of the ® eld theory, it should be noted
that essential components of the machinery we are going to discuss are not original,
but have been introduced earlier: the general supersymmetric ® eld theoretical
approach to N-mesoscopic systems has been constructed by Efetov [63]. Oppermann
[59], and later Kravtsov and Oppermann [64], introduced a fermion-replicated ¼-
model description of bulk disordered superconductors. As far as technical aspects
are concerned, the formalism we are deriving represents a supersymmetric extension
of Oppermann’ s model, tailored to the description of spatially inhomogeneous
structures. In its early stages, the construction of the model follows a by now
absolutely standard strategy. Essentially, this requires an adaptation of Efetov’s
model to allow for the particular structure of Gorkov Green functions. For this
reason our presentation of the early construction steps will be concise, but never-
theless self-contained.
5.1. Field integral and the ensemble average

As is usual in the construction of ® eld theories of mesoscopic systems, the ® rst
construction step is to represent products of matrix elements of Gorkov Green
functions in terms of supersymmetric Gaussian ® eld integrals. To keep the discussion
comparatively simple, we focus on the case of two-point correlation functions of the
general type hGr …° ‡ !‡ =2†Ga …° ¡ !‡ =2†i and choose, as a speci® c example, the
quantity
htrph;r …¼ph r ph a
3 G …° ‡ !‡ =2†† trph;r …¼ 3 G …° ¡ !‡ =2†† ;
i …15†
which appears in the computation of the ¯ uctuations of the DoS{,

1
R…°;!‡ † ˆ h¯¸…° ‡ !‡ =2†¯¸…° ¡ !‡ =2†i: …16†
h¸…°†i2
Here ¯¸ ˆ ¸ ¡ h¸ i, ¸…°† ˆ ¡…1=2p† Im tr ph;r …¼ph r

3 G …°†† and trph;r denotes a trace with
respect to both position and particle hole index.
In order to represent objects of this kind in terms of Gaussian ® eld integrals, we
® rst introduce a 16-component vector ® eld, Á ˆ fÁ¶;s;¬;t …r†g ; ¶;s;¬;t ˆ 1;2, with
{ Note that the correlation function R…°; !‡ † di ers in two respects from the analogous quantity
in N-mesoscopic systems: (i) the mean DoS, h¸…°†i, will not, in general, be constant. Hence, R…°;!‡ †
may explicitly depend on the centre coordinate °, a fact that can be remedied by an unfolding procedure
(see e.g. [65]). (ii) As opposed to N systems, correlation functions such as hG r G r i are non-trivial in the
sense that they do not equal the product of averages. Both aspects (i) and (ii) will be commented on
later in more detail.
Table 1. The signi® cance of the two-valued

indices ¶; s; ¬; t
index signi® cance abbreviation
¶ advanced/retarded ar
s particle/hole ph
¬ boson/fermion bf
t time reversal tr
complex commuting (anticommuting) components ¬ ˆ 1 …¬ ˆ 2†. The signi® cance

of the two-valued indices ¶; s;¬;t is summarized in table 1. Apart from the index s
accounting for the 2 £ 2 matrix structure of the Gorkov Green function, all other
indices are standard in supersymmetric approaches to disordered systems. For a
discussion of their signi® cance we refer to Efetov’s book [63]. We next introduce the
action
…
- 1 e 2 !‡ ar
S ¡i Á · ¡
ˆ p ¡ A¼ph
^ « ¼tr3 ¡V ‡ ¼ph ~
3 « D‡°‡ ¼ Á; …17†
2m c 3 2 3
where
~
D…r† ˆ D ¼ph tr ph tr
1 « ¼ 3 exp …¡i’…r†¼3 « ¼3 †;
…18†
ph bf
and the Pauli matrices ¼ar tr
i ;¼ i ;¼i , and ¼i ; i
ˆ 1;2;3 operate in the two-dimen-
sional spaces of ¶;¬; s; t indices respectively. Like with the ¼-model of conventional
-
disordered systems, the ® elds Á and Á appearing in the above action are not
independent but rather related to each other by certain symmetry operations. For
the sake of completeness a discussion of these symmetries (which will not play an
explicit role throughout but do determine the structure of the Q-® elds to be discussed
later on) has been included in Appendix B.
As in analogous theories of N systems, the action, S , can be employed to
represent correlation functions in terms of Gaussian ® eld integrals. Speci® cally, the
expression (15) takes the form
3 G …° ‡ !‡ =2†† trph;r …¼3 G …° ¡ !‡ =2††
i
… … …
ˆ 16 D…Á; Á† exp …¡S‰Á; ÁŠ† Á1 ¼3 « ¼3 Á1 Á- 2 ¼bf
1 - - - bf ph ph
3 « ¼3 Á2 ; …19†
where the indices on the Á-® elds refer to the ar space, and all other indices are
summed over.
After this preparationÐ which essentially has comprised of an extension of the
existing supersymmetric framework to account for the additional ph structureÐ we
may proceed in strict analogy to standard procedures.
(a) First, averaging over Gaussian-distribute d disorder, with the correlation
function
1
hV …r†V …r 0 †i ˆ ¯…r ¡ r 0 †;
2p¸n ½
where ¸n denotes the DoS of a bulk N system, generates the quartic

contribution to the action,
…
1 -
S int ˆ …ÁÁ†2 :
4p¸n ½
(b) Next, S int is decoupled by introducing a 16 £ 16 Hubbard ± Stratonovich
matrix ® eld,
… …
1 - p¸n
exp …¡S int † ˆ DQ exp ¡ ÁQÁ ¡ str Q 2 ;
2½ 4
where `str’ denotes the supersymmetric extension of a matrix trace{.
(c) In a third step we integrate over the Á-® elds to arrive at the Q-represented
action,
…
p¸n
S ‰QŠ ˆ 12 strr ln G¡1 ¡ str Q 2 ; …20†
8½
where
1 e 2 !‡ ar i
G¡1 ˆ · ¡ p ¡ A¼ph
^ « ¼tr
3 ‡¼ph ~
3 « D…r† ‡ ° ‡ ¼ ‡ Q …21†
2m c 3 2 3 2½
and strr denotes a trace extending over both internal and spatial degrees of freedom.
The next step in the standard construction of the ¼-model is to subject the action
S‰QŠ to a saddle-point analysis. As we will see shortly, the presence of a super-
conductor with a spatially inhomogeneous order parameter will necessitate sub-
stantial modi® cations to the standard mean ® eld scenario. In order to gain some
insight into the structure of the mean ® eld equations, the next section will be devoted
to the study of the comparatively simple case of a bulk disordered superconductor.
However, prior to specializing the discussion, let us make some general remarks as to
the structure of the stationary phase equations.
Varying the action (20) with respect to Q generates the stationary phase equation,
- i
Q…r† ˆ G…r;r†: …22†
p¸n
Before embarking on the explicit computation of solutions to equation (22)Ð which
represents a 16-dimensional matrix equationÐ it is convenient to elucidate further its
structure. In fact, a striking simpli® cation arises from the fact that the Q-
independent part of the kernel G¡1 is diagonal in all indices save the ph indices.
In the standard case, that is, no superconductivity and hence no ph indices, the
-
diagonality is complete and one may start out from an ansatz for Q which is fully
diagonal. Exploiting the fact that the energy-di erence between the Green functions,
!‡ , is typically small in comparison with all other energy scales of the system, one
might be tempted to argue that G¡1 is not only diagonal but even approximately
proportional to the unit matrix in the internal indices. As a consequence one might
-
assume that Q is proportional to the unit matrix as well. This, however, is wrong.
The in® nitesimal imaginary increment contained in !‡ gives rise to a phenomenon of
spontaneou s symmetry breaking in the sense that the saddle-point solution in the
{ We use the convention that strA ˆ trAbb ¡ trAff .

retarded sector di ers from the one in the advanced sector (see e.g. [66, 63]). More
precisely, the saddle-point solution of the ® eld theory for N systems reads{
-
Q N ˆ ¼ar3 :
What kind of modi® cations arise in the presence of superconductivity ? First, the
kernel G¡1 is no longer fully diagonal. It contains a non-trivial matrix structure in
ph space{. Thus, the simplest ansatz for a saddle-point solution is diagonal in all
indices save the ph indices. Secondly, we may expect that, as in the N case, the
structure of the solution depends on the in® nitesimal increments added to !‡ . An
inspection of equation (17) tells us that the role played by the matrix ¼ar 3 in the
standard case will now be taken over by ¼ph 3 « ¼ar
3 . Finally, the solution in the N case,
-
Q N ˆ ¼ar
3 , was fully universal in the sense that it did not depend on any energy scale.
Since the `perturbations’ arising in the action due to the presence of the super-
conductorÐ most notably the order parameterÐ are weak in comparison with the
dominant energy scale, ·, it is sensible to assume that the eigenvalues of the saddle-
point solution will still be §1.
Starting from the comparatively simple example of a bulk superconductor [59] we
will next con® rm these suppositions by explicit calculation.
5.2. Example: a bulk superconducto r

In this section we assume the order parameter to be spatially constant. The
resulting saddle-point equation has previously been discussed in [59]. Speci® cally, we
~
assume D…r† ² ¡jDj¼ph
2 , corresponding to a constant gauge ’
ˆ p=2, and assume
that the vector potential vanishes, A ˆ 0. Homogeneity of the gap function implies
- -
homogeneity of the saddle-point solution Q…r† ˆ Q. We next introduce the ansatz
0 ph 1
¼
B 1 C
- B C
Q ˆ q ¢ rph ; rph ˆ B ¼ph C;
@ 2 A
ph
¼3
where r is a vector of Pauli matrices operating in ph space. The components of the
vector q are diagonal matrices which are trivial in all but the ar space,
q‡ r
qˆ … q¡
†a …23†
and normalized to unity, qT§ ¢ q§ ˆ 1. To proceed it is convenient to adopt an elegant

parametrization for the Green function suggested by Eilenberger [6]. First note that
in momentum representation G¡1 can be written as
G¡1 …p† ˆ ¡¹p ‡ iw ¢ rph ;
where
{ Note that ¼ar

3 is commonly denoted by L in the literature.
{ One might argue that the o -diagonality may be removed via a global unitary transformation,
at least in the case of a spatially homogeneous order parameter. This, however, would contravene the
conditions enforced by analyticity on the structure of the imaginary increments contained in the action,
which, as we saw in the N case, play a crucial role in determining the structure of the solution.
p2
¹p ˆ ¡ ·;
2m
q
wˆ ¡ ir; r ˆ …° ‡ i0¼ar ^3 ‡ iDe^1 ;
3 †e
2½
and !‡ has been set to zero. Using the fact that …w ¢ rph †…w ¢ rph † ˆ w ¢ w ² jwj2 , it is
a straightforward matter to show that
- 1 X 1ph ‡ sw^ ¢ rph
G…p† ˆ ; …24†
2 sˆ§1 ¡¹p ‡ isjwj
where w ^ ˆ w=jwj. Performing the trace over momenta, and making use of the
relations
X …
p2
p
f
2m
… †
¡ · ’ Vol ¢ ¸n d¹p f …¹p †; …25†
where `Vol’ is the system volume, as well as

… (
X sn 0; n ˆ 0;
d¹p ˆ …26†
sˆ§1
¡¹p ‡ isjwj ¡2pi ‡ O…max …D;°†=·†; n ˆ 1;
the saddle-point equation (22) takes the simple form
i i X
G…r; r† ˆ G…p† ˆ w
^¢r
p¸n p¸n Vol p
,qˆw
^;
which is solved by
ir
qˆ ; …27†
R1=2
R ˆ jDj2 ¡ …° ‡ i0¼ar 2
3 † : …28†
The appearance of the square root in the solution for q requires us to make a choice
of branch cut in the plane of R. As discussed in [64], the correct choice is to take the
branch cut along the negative real axis in R. In this way it is guranteed that all saddle
points are accessible to the integration variables, without crossing the branch cut of
the logarithmic part of the e ective action. This leads to the solution for q being
ar
proportional to ¼ar
3 (1 ) in ar space for j°j > jDj (j°j < jDj). Indeed, introducing the
parametrization
q ˆ …sin ³s ;0;cos ³s †; …29†
we ® nd
(
j°j ¼ar
3 ; j°j > jDj;
cos ³s ˆ 1=2
jRj ¡i sgn …°†; j°j < jDj;
(
ijDj ¼ar
3 ; j°j > jDj;
sin ³s ˆ 1=2
…30†
jRj ¡i sgn…°†; j°j < jDj:
In the particular limit of zero order parameter, the solution collapses to a ph diagonal
- ˆ ph
one, q ˆ e^3 ¼ar3 ?Q ¼3 « ¼ar
3 . This result is consistent with the conventional
saddle-point equation of the normal conductor (cf. the remarks made towards the
end of the preceding section): for vanishing order parameter, the particle± hole
extension simply generates two copies of the normal Hamiltonian. The e ect of a
non-vanishing value of D is to induce a rotation of the saddle point in the ph sector.
In the extreme limit of ° ! 0 with a ® nite gap, the saddle point rotates as far as
q ˆ sgn …°†e^1 , or ³s ˆ sgn …°†p=2, while remaining proportional to 1ar .
The di erence in the ar structure of the solutions for q in the cases of j°j > jDj and
j°j < jDj, while relying upon our choice of the branch cut in the plane of R, ® nds a
natural interpretation in the form of the local DoS, which can be computed from q
as{
1 ph
h¸…r†i ˆ ¡ Im trph h¼3 Gr …°; r; r†i
2p
-
¸n ph Q!Q
ˆ Re hstr …Q¼bf ar
3 « ¼ 3 « E11 †iQ ¡! ¸n Re ‰q‡ …r†Š 3 : …31†
8
ar
(For a de® nition of the matrix E11 , see Appendix B.) Here h. . .iQ denotes the
functional expectation value of the ® eld Q and the ® nal expression is obtained by
evaluating the functional integral at its saddle-point value. For the bulk case, this
gives a superconducting DoS, ¸s , of ¸s ˆ ¸n Re cos ³s , which leads by equation (30)
to the familiar BCS form,
j°j
¸s ˆ Y…j°j ¡ jDj† :
…jDj ¡ °2 †1=2
2
Here we see that the result of the 1ar structure of q below the gap, j°j < jDj, is a
vanishing DoS.
We may next ask whether the saddle-point solution, q, is unique. Anticipating
-
that all saddle-point con® gurations must share the eigenvalue structure of Q, a
general ansatz probing the existence of alternative solutions reads
- -
Q ! T QT ¡1 ; …32†
where T is some rotation matrix. In order to understand the structure of the resulting
saddle-point manifold, it is essential to appreciate that there are three parametrically
di erent energy scales in the problem.
(a) The asymptoticall y largest scale in the problem is the chemical potential. The
existence of the large parameter ·=E, where E may be any other scale
involved, stabilizes the eigenvalue structure of the matrix Q. (This follows
ultimately from the structure of the pole integral (26)Ð see also the
corresponding discussion in [6].) Thus, as long as we are not interested in
{ To derive equation (31) one starts out from the functional representation of the local DoS,
…
- - -
trph h¼ph
3 G r
…°; r; r†i ˆ 1
4
D…Á; Á† exp …¡S‰Á; ÁŠ†Á1 …r†¼bf ph
3 « ¼3 Á1 …r†:
After the Hubbard± Stratonovich transformation the pre-exponential terms take the form of a functional
ph
expectation value ¹ hstr …Q¼bf ar
3 « ¼3 « E 11 †iQ which, upon evaluation in the saddle-point approximation
leads to equation (31).
corrections of O…E=·†, con® gurations of the type of equation (32) exhaust

the ® eld integration domain of interest.
(b) The next largest scales are D and/or °. Amongst the con® gurations
parametrized by equation (32), there are some that are massive in these
parameters and some that are not.
(c) The smallest scale is !‡ . Its physical signi® cance is that of an inverse of the
time scales at which we are probing correlations. With regard to correlation
functions in !‡ , ® eld ¯ uctuations that are massive in the intermediate
parameters ° and D are clearly of little if any relevance.
After these preparatory remarks, it should be clear that we will be concerned mainly
with ¯ uctuation matrices T that still lead to solutions of the saddle-point equation
up to corrections / !‡ . An inspection of the action (20) tells us that such T have to
ful® ll the condition ‰T ;rph Š ˆ 0. On the other hand the matrices T must not
-
commute with Q, as otherwise they would be ine ective. Taking these two facts
-
together, we see that the most relevant ¯ uctuations, T , around Q, are those that
generate rotations in ar space: for !‡ ˆ 0 any con® gurations ful® lling the above
conditions again represent solutions of the saddle-point equation. In other words,
the T ’ s generating these con® gurations are Goldstone modes.
Before extending the discussion to spatially inhomogeneous problems and the
impact of the existence of Goldstone modes, let us comment on a mathematical
aspect of the construction of the theory. We have seen that, for D 6ˆ 0, the saddle-
-
point solution Q di ered substantially from the standard saddle-point ¼ph ar
3 « ¼ 3 of
the bulk metallic phase. This raises the question of whether the superconducting
saddle point, q ¢ rph , and ¼ph ar
3 « ¼3 are both contained in the ® eld manifold of the
nonlinear ¼-model. Clearly this question will be of concern as soon as we deal with
SN hybrid systems, and, in fact, the answer is negative. However, it turns out that the
problem can be surmounted by analytic continuation of the parameter space
spanning the Q-® eld manifold. Since the discussion of the manipulations needed
to access both saddle points is inevitably somewhat technical, it has been deferred to
Appendix C and may be skipped by readers who are not interested in details of the
formalism.
Our ® nal objective will be to describe SN systems rather than bulk super-
conductors. What makes the analysis of SN systems technically more involved is
that the action is manifestly inhomogeneous: in our comparatively coarse modelling,
where the order parameter is externally imposed, the N component of an SN system
will be characterized by a sudden vanishing of the order parameter. Within a more
accurate description, based on a self-consistently determined order parameter ® eld,
the situation would be even more complicated. As in the preceding section, the SN
action may also be subjected to a mean ® eld approach. However, due to the imposed
inhomogeneities in the order parameter, the stationary phase con® gurations will in
general no longer be spatially uniform. At ® rst sight it may not be obvious how
solutions to the spatially inhomogeneous stationary phase problem may be found.
The correct strategy for this problem is again prescribed by the existence of the
threefold hierarchy of energy scales, discussed above. Before going into details, let us
give a brief account of the forthcoming construction steps.
(1) We will ® rst employ the general ansatz
Q…r† ˆ T …r†¼ph ar ¡1
3 « ¼3 T …r† …33†
akin to the one used in the bulk case. Equation (33) implies that the Q
matrices have an eigenvalue structure set by the matrix ¼ph ar
3 « ¼ 3 thereby
automatically solving the saddle-point problem with regard to the highest
energy scale · (cf. the corresponding remarks made above).
(2) In a second step we substitute the above ansatz into (20) and derive a
`medium-energy’ e ective action that contains no energies higher than ° and/
or D. Thirdly we will perform a second stationary phase analysis thereby
determining those ® eld con® gurations (33) that extremise the medium energy
action.
(3) By accounting for ¯ uctuations around these con® gurations, we will ® nally be
able to explore the low energy physics on scales !‡ .
Beginning with the derivation of the `medium energy action’ , we now formulate this
program in more detail.
5.3. Gradient expansion and `medium energy action’

In constructing the e ective medium energy action, it is again crucial to exploit
the existence of a scale separation in energy. Anticipating that the relevant ® eld
con® gurations T …r† ¯ uctuate weakly as a function of r, we ® rst borrow a
parametrization of the kinetic energy operator that has previously been used in
constructing the quasi-classical equations of superconductivity [6]
1 e 2 1 2 1
p ¡ A¼ph
^ « ¼tr3 ’ p ‡ p¢^
q; …34†
2m c 3 2m m
where
eph
q^ ˆ ¡i@ ¡ A¼3 « ¼tr
3:
c
The idea behind equation (34) is that the slowly ¯ uctuating entities in the action,
most notably Q, e ectively do not vary on scales of the Fermi wavelength. Thus, it
makes sense to decompose the momentum operator into two parts, ^ pˆ p‡^q , where
the `fast’ component, p, has eigenvalues of order of the Fermi momentum, pF , and so
can be treated as a c-number with regard to slowly varying structures. The `slow’
component, ^ q, accounts for both slow spatial variations and the magnetic ® eld. For a
more substantial discussion of (34) we refer to the original literature [6].
We next substitute (33) and (34) into the action (20) to obtain
2 3
1 1 2 i ph ar
S ‰QŠ 2 strr;p ln 6· ¡
ˆ p ‡ ¼3 « ¼3 ‡V 1 ‡ V 2 ‡ T ‰V 1 ‡ V 2 ; T Š7 ;
¡1
2m 2½
6|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚} 7
4 5
¡1
…G0 †
where
1 !‡ ar
V1 ˆ ¡ p ¢ q; V 2 ˆ ¼ph ~
3 « D…r† ‡ ° ‡ ¼
m 2 3
and strr;p denotes a trace of internal indices, the `fast’ p’ s and the spatial coordinate.
We next expand to lowest non-vanishing order in the `slow’ operators V i . As will
become clear from the structure of the resulting series, the small parameters of the
expansion are …l=L †2 , for V 1 , and °½; D ½ for V 2 . Here L denotes the typical scale at
which the matrices T ¯ uctuate. To lowest order we obtain
2
S‰QŠ ! 12 strr;p G0 T ¡1 ‰V 2 ;T Š ¡ 14 strr;p G0 …V 1 ‡ T ¡1 ‰V 1 ; T Š† :
Note that there is no contribution at ® rst order in V 1 . The reason is that V 1 is linear
in the vectorial fast momentum p, whilst G0 is even in p. Thus, the trace over fast
momenta annihilates this contribution.
To prepare the tracing out of the fast momenta, we next formulate some useful
identities describing the behaviour of the `fast’ Green function G0 . The following
relations can be proved straightforwardly by explicitly performing the momentum
integrations (cf. equation (25)) and using some Pauli matrix algebra.
(a) G0 in its momentum representation may be written as (cf. [6] and equation
(24))
X 1 ‡ s¼ph « ¼ar
3 3
G0 …p† ˆ 12 i
:
sˆ§1
¡¹ p ‡ s 2½
(b) The momentum trace over a single Green function becomes

X
G0 …p† ˆ const: ¢ 1 ¡ ip¸n …Vol†¼ph ar
3 « ¼3 :
p
(c) Further, if operators A ^ and B^ vary slowly in space, then

X h i 2 X h i
str G0 …p†p ¢ AG ^ ˆ m …Vol†2pD¸n
^ 0 …p†p ¢ B str …1 ‡ s¼ph
3 † ^ ¢ …1 ¡ s¼ph †B
A 3
^ :
p
4 s
An application of these identities to the e ective action above leads to

S ! S1 ‡ S2 ;
where
… h
p¸n !‡ ar i
3 « D …r† ‡ ° ‡
str Q¼ph
S 1 ˆ ¡i ~ ¼ ;
2 2 3
and, setting O^ ˆ ^ q ‡ T ¡1 ‰^ qT ,
q ;T Š ˆ T ¡1 ^
… h i
S 2 ˆ ¡14pD¸n str …1 ‡ ¼ph 3 « ¼ ar ^
3 † O…1 ¡ ¼
ph
3 « ¼ar ^
3 † O :
By using the identity

str ‰…1 ‡ ¼ph ar ^ ph ar ^ ˆ
3 « ¼3 †O…1 ¡ ¼ 3 « ¼ 3 † OŠ ¡ 12 str ‰‰q^; QŠ2 Š;
we obtain
… h i
pD¸n ~ @Q
~ ;
S2 ˆ ¡ str @Q
8
where
e ph
@~ ˆ @ ¡ i A‰¼3 « ¼tr
3 ;Š …35†
c
represents the covariant derivative. Putting everything together we obtain our ® nal
result for the `medium energy action’
… h i
p¸n e 2 ‡ 4iQ¼ph « D ^ ‡ ° ‡ !‡ ¼ar :
S‰QŠ ¡
ˆ str D…@Q† 3 3 …36†
8 2
Correlation functions are now obtained by substituting the action (36) into a
functional integral over all Q-® elds subject to the constraint Q 2 ˆ 1:
…
DQ…¢ ¢ ¢† exp …¡S ‰QŠ†; …37†
where DQ denotes the invariant measure on the manifold of matrices Q 2 ˆ 1. For

instance, the correlation function (15) takes the form (cf. equation (19))
3 G …° ‡ !‡ =2†† trph;r …¼3 G …° ¡ !‡ =2††
i
2
… … …
p¸n ar bf ph ar bf ph
ˆ¡ DQ str QE11 ¼3 « ¼3 str QE22 ¼3 « ¼3 exp …¡S‰QŠ†: …38†
4
In the limit D ! 0, the functional integral represents a superposition of two
independent copies of normal metal ¼-models, one corresponding to the particle,
the other to the hole sector. Due to the decoupling of these two components, the
ph structure becomes meaningless. For D ˆ 6 0, the situation is more interesting.
Given the spatially inhomogeneous structure of the action, an exact computation of
correlation functionsÐ in the sense of a complete integration over the nonlinear ® eld
manifoldÐ is in general not feasible. Under these circumstances, the ® rst and
seemingly straightforward approach one might try is a perturbative one. Yet, as
usual with perturbative approaches in SN physics, straightforward perturbation
theory does not work here.
To understand the origin of the di culties let us introduce the parametrization
Q ˆ exp …W †…¼ph ar
3 « ¼3 † exp …¡W †; …39†
where ‰W ;¼ph ar ‡ ˆ
3 « ¼3 Š 0. The parametrization (39) is frequently used in perturba-
tive analyses of the ¼-model. In standard (N) applications of the ¼-model, its
substitution into the action leads to a series
S‰QŠ ! S ‰W Š ² S …2† ‰W Š ‡ S …4† ‰W Š ‡ S …6† ‰W Š ‡ ¢ ¢ ¢ ; …40†
where S ‰W Š denotes the contribution of 2nth order in W . The functional can then
…2n†
be evaluated by expanding perturbatively around the second order contribution

exp …¡S …2† ‰W Š† and applying Wick’s theorem. The resulting Taylor series converges
rapidly due to the fact that contributions S …2n† ‰W Š are multiplied by large coupling
constants gn ¾ 1 (all of which are parametrically of the same order).
In the case D 6ˆ 0 the situation is more complicated. The point is that
contributions S …m† ‰W Š, m being odd, arise from the perturbative expansion of the
vertex ¹ str …Q¼3 D ^ †. In particular, a non-vanishing contribution of ® rst order in W
ph
emerges. The presence of this term invalidates perturbation theory. (This can be seen
formally by means of a simple power counting argument: in the expansion of
exp …¡S …1† ‰W Š†, each W is multiplied by a large coupling constant g1 . On the other
hand, the Wick contraction of two W ’s, gives a factor g¡1 2 . Thus, the perturbative
series expansion of exp …¡S …1† ‰W Š† diverges in the parameter g21 =g2 ¾ 1).
To get some idea of how these problems can be overcome, it is helpful to
understand the physical origin of the divergences arising in perturbation theory. To
this end let us consider the disorder average of the ph block of the Gorkov Green
function, G12 …°†, as a simple example of a quantity that strongly couples to the
divergence of the W -perturbation series. When expressed in terms of the functional
integral, the average hG12 …°†i takes the form
ar ph
hGr;12 …°; r;r†i ¹ hstr …Q…r†E11 « E12 « ¼bf
3 † Q ¹ str …W …r†X† W ;
i h i
where h. . .iQ=W stands for functional integration in the Q- respectively W -representa-

ar ph
tion of the theory and X is the ® xed matrix, X ˆ E11 « E12 « ¼bf
3 .
Suppose now, we intended to evaluate this functional expectation value
perturbatively. To lowest order in W we would obtain
… …
hGr;12 …°; r; r†i ¹ str …W …r†X† str dr 0 …W …r 0†D…r
^ 0 †† ¹ dr 0 K…r;r 0 †D…r 0 †;
0
where h. . .„i0 stands for functional integration weighted by the quadratic action
S …2† ‰W Š ¹ str …W K¡1 W †. The kernel, K, governing S …2† ‰W Š is the familiar di usion
pole K¡1 ¹ D@ 2 ‡ i°. Thus we see that the ® rst correction to the ànomalous’ Green
function G12 …°† is proportional to the order parameter andÐ owing to the spatially
long ranged behaviour of the di usion poleÐ stretches far into the normal metal.
Moreover, since the characteristic energy scale of the di usion pole is max …°;Ec †, we
see that the correction is of O…D=max …°;Ec ††, which, for su ciently strong order
parameter/coupling between N and S, exceeds unity. Remembering that the character-
istic scale of the dimensionless quasi-classical Green function is unity we have to
conclude that the perturbation series resulting from a naive W -expansion of the
functional integral does not converge. To understand both the reason for this failure
and the particular form of the ® rst order correction, it is instructive to compare with
the type of divergencies that appear within diagrammatic perturbation theory. In
diagrammatic analyses, the correction to ® rst order in D to the anomalous Green
function is indeed given by a single di usion mode. The real-space representation of
this term has already appeared in ® gure 1 (b ), while the corresponding diagrammatic
representation has also appeared as ® gure 5. Noting that this correction is only the ® rst
contribution to what becomes, upon summation, a full representation of the proximity
e ect, the origin of the problem becomes clear: by perturbatively expanding around
ph
¼3 « ¼ar 3 , we have chosen the metallic limit of the Gorkov Green function,
Ga;r ˆ § i¼ph 3 as a reference point. The superconductor, however, drives the adjacent
normal metal region to a state that is far from conventionally metallic. In order to
force a description of the system in terms of a perturbation theory around the metallic
limit, we have to pay the price of an in® nite perturbation series. Even worse, due to the
e ective spatial inhomogeneity of each perturbative contribution , arising from the
space dependence of the di uson, summation of the series becomes impossible. To
summarize, the considerations above tell us that perturbative approaches based on
spatially constant reference con® gurations are doomed to fail and that the origin of the
problem is the spatially inhomogeneous manifestation of the proximity e ect.
6. Stationary phase analysis

Given what has been said at the end of the previous section, the correct strategy
for overcoming the problems arising in perturbation theory becomes apparent: prior
to any perturbativ e attempts, it is preferable to seek a solution to the stationary
- -
phase equation ¯S‰QŠ=¯ Q ˆ 0. Due to the spatial inhomogeneity of the problem, no
-ˆ -
uniform solutions Q const: will be found. Once a solution Q has been obtained,
both perturbative and non-perturbative evaluation schemes may be safely super-
imposed. The reason is that, by construction, no linear terms appear when the action
-
is expanded around Q.
We ® nd it convenient to formulate the stationary phase analysis in a gauge where

the phase dependence of the order parameters at the SN boundaries has been
eliminated, at the expense of introducing a vector potential in the bulk N region. To
be speci® c, we perform the gauge transformation
i p i p
Q…r† ! exp ¡ ‡ ’…r† ¼ph tr
3 « ¼ 3 Q…r† exp ¡ ¡ ‡ ’…r† ¼ph tr
3 « ¼3 ; …41†
2 2 2 2
where, within the superconducting region, ’…r† is equal to the phases of the order
parameter, as in equation (18), and in the normal region can be chosen arbitrarily.
Inserting the gauge transformed ® eld into the action we obtain
… h i
p¸n e 2 ‡ 4iQY ¢ rph ;
S‰QŠ ˆ ¡ str D…@Q† …42†
8
where
Y ˆ iD…r†e^1 ‡ …° ‡ !‡ ¼ar
3 =2†e
^3 ; …43†
and the vector potential entering the covariant derivative has been transformed by
e e 1
A ! A ¡ @’: …44†
c c 2
Note that the right-hand side of equation (44) may be interpreted as …¡2m† times the
super¯ uid velocity. To ® nd the stationary phase equation, we introduce a small
variation
Q ! exp …¯W †Q exp …¡¯W † ’ Q ‡ ‰¯W ;QŠ
into the action and demand vanishing of the contribution at ® rst order in ¯W . A
straightforward calculation then yields the equation
- - -
D @~i …Q @~i Q† ¡ i Q ;Y ¢ rph ˆ 0: …45†
The ® rst step to analysing the general set of solutions of this equation is to specify
that the solution is as simple as possible, i.e. as diagonal as possible. Noting that the
equation is diagonal in ar, tr and bf space (in bf it is even trivial), we see that, as in
the case of a bulk superconductor , a su ciently general ansatz reads
q§ ˆ q§1 E11
tr
‡ q§2 E22
tr
; q ¢ q ˆ 1bf ;ar;tr ; …46†
where q§ refers to the retarded/advanced blocks de® ned in equation (23) and q§1;2
are
vectors of complex numbers (i.e. structureless in bf-space). The restriction of the
saddle-point equation to the blocks q§ ² q§ ¢ rph now reads
D @~i …q§ @~i q§ † ¡ i q§ ;Y§ ¢ rph ˆ 0; …47†
where
Y§ ˆ iD…r†e^1 ‡ …° § !‡ =2†e^3 :
Comparing equation (47) with equation (9) and identifying q§ with g§ 0 , we identify
the stationary phase equation of the nonlinear ¼-model as the Usadel equation. One
consequence is that we are immediately able to write down the boundary conditions
at a (perfectly transmitting) SN interface, by direct analogy with the Kuprianov and
Lukichev relations, equations (10) and (11):
¼q§ @? q§ ˆ ¼q§ @? q§ ; …48†

x‡ x¡
q§ ˆ q§ ; …49†
x‡ x¡
the former of which implies current conservation at the interface. Here @? is the
normal derivative across the (planar) boundary , and x§ denotes a space point
in® nitesimally to the left/right of a boundary point x. Note that the normal state
conductivities, ¼, in the left and right region may di er.
In passing we note that, although we have stated above and will use further the
relations for a perfectly transmitting interface, in general we need not keep to such a
restriction within this formalism. For instance, we could have equally well employed
the following conditions in the limit of small transparency, again by analogy with
equations (10) and (11):
¼q§ @? q§ ˆ ¼q§ @? q§ ; …50†

x‡ x¡
GT
ˆ q§ ;q§ ; …51†
2 x‡ x¡
where GT is the tunnel conductance of the junction, as given by equation (A 7). By

modelling the tunnel barriers microscopically (as was done, e.g. in [67]) these
boundary conditions can in fact be rederived within the ¼-model formalism.
The coincidence of the stationary phase equation of the ¼-model with the Usadel
equation, which, as mentioned in the introduction, was ® rst observed by Muzy-
kantskii and Khmelnitskii [24] in a di erent context, has fundamental consequences
for all that follows.
(a) Already on the level of the saddle-point equation, the ¼-model contains all
the information that is otherwise obtained quasi-classically. In particular,
solutions of the equation can in most cases of interest be imported from the
extensive literature on Usadel equations for SN systems.
(b) The facts that (i) the solutions of the Usadel equation for the retarded and
the advanced Green function are di erent and (ii) the !‡ ˆ 0 action is
isotropic in ar space, imply that we encounter a situation of spontaneous
symmetry breaking: the mean ® eld does not share the symmetries of the
action and a Goldstone mode, operating in ar space, will appear.
These two observations su ce to dictate a further strategy: one ® rst has to solve or
-
import a solution of equation (45). Then the solution Q ˆ diag …q‡ ;q¡ † is substituted
back into the action and ¯ uctuations around the block diagonal solution are
- -
introduced, via Q ! T QT ¡1 . In analysing ¯ uctuations, the main emphasis will be
on exploring the role of the Goldstone mode. However, before we proceed to the
actual formulation of this program, it is worthwhile to stay for a moment at the
mean ® eld level and to acquire some familiarity with the Usadel equation and
the structure of its solution.
We ® rst note that the di erent `sectors’ , q§1;2 , of the solution vector are not
independent but rather connected to each other via symmetry relations.
(1) The general relation (cf. equation (2))
GA …r; r 0 † ˆ ¼ph R 0 y ph
3 …G …r ;r†† ¼ 3
…52†
implies
q¡ ˆ diag …1; 1; ¡1†…q‡ †¤ : …53†
(2) Taking the transpose of the Usadel equation in the tr-sector 1, we obtain
q2 ˆ diag …1;¡1;1†q1 :
As for the spatial behaviour of the solution, some remarks may be made in general.
Deep in a superconducting region, the large ¼ph 2 -component of Y enforces that
q ’ e^1 . Conversely, deep in a normal metal, q will be aligned with e^3 . The Usadel
equation describes a smooth interpolation between these two limits, where the
gradient term inhibits strong spatial ¯ uctuations (cf. ® gure 6).
In order to say more about the spatial structure of the solution to the Usadel
equation, we have to restrict the discussion to speci® c examples. Here we will
consider two simple prototype systems, representative of the wide classes of systems
with (a) quasi in® nite and (b) compact normal metal region. Since we consider a
quasi-1D geometry in each case, we denote by x the position variable perpendicular
to the interface.
In® nite SN junction. Consider the model system shown in ® gure 7. The
normal metal and superconductor regions occupy x > 0 and x < 0
respectively, so that the gap function is modelled by D…x† ˆ jDjY…¡x†. The
system is quasi one-dimensional in the sense that its constant width is
comparable with the elastic mean free path (i.e. there are many conducting
channels but no di usive motion in the transverse direction.) We assume that
no external magnetic ® eld is present. Furthermore, since there is only one
Figure 6. Schematic plot of the retarded component (real part) of the saddle-point solution,
q‡ , for an SNS junction with coincident phases of the order parameters.
Figure 7. The geometry of the in® nite, quasi-1D SN junction.

superconducting terminal, an elimination of the phase of the order parameter

does not induce a gauge potential and we can globally set A ˆ 0.
The analytic solution of the corresponding Usadel equation is reviewed in
Appendix D.1. Due to the global absence of a vector potential, the spatial
rotation of the vector q takes place in the 1± 3 plane only. Hence, it can be
parametrized as (cf. the analogous form for a bulk superconductor ,
equation (29))
q…x† ˆ …sin ³…x†;0;cos ³…x††:
In ® gure 8 we have plotted the curve in the complex plane that is traced out
by ³…x† upon variation of x.
Figure 9 shows the local DoS ¸…x† ˆ ¸n Re cos ³…x† (cf. equation (31))
obtained for a particular value of the material parameter ® ˆ
1=2 1=2
¸n D n =…¸s D s †, where D n;s are the di usion constants in the N,S region.
Note that the deeper one proceeds into the S region, the closer the DoS
Figure 8. Dotted line: the trajectory of the retarded component of ³ as a function of

position for an in® nite SN junction with coincident phases and °=D ˆ 0:6 and
® ˆ 0:1. The trajectory starts at x ˆ 0 and approaches the origin as x ! 1. Solid
line: the same for an SNS system, with L =¹ ˆ 5, starting at x ˆ ¡L =2 and ending at
x ˆ L =2. The trajectory reverses direction at the centre of the N region, x ˆ 0.
Figure 9. Local DoS of the in® nite SN junction as a function of both energy and position
for ® ˆ 0:1.
Figure 10. The e ect of ® on the energy dependence of the DoS, at x ˆ 1:5¹.
approaches the characteristic BCS form. Due to the proximity e ect, the
structure of the subgap (° < jDj) DoS in the N region remains non-trivial.
Only in the asymptotic limit x ! 1, the region of suppressed DoS shrinks to
zero and `normal’ behaviour is restored. More precisely, substantial
alteration of the DoS induced by the proximity e ect is restricted spatially to
a region of several di usion lengths from the interface into the normal metal,
or several (dirty) superconducting coherence lengths into the superconductor.
Figure 10 shows how variation in ® a ects the DoS. In particular, we may
take the `rigid’ limit ® ! 0, for which the bulk superconducting value of the
Usadel angle is imposed at the interface, and retain a non-trivial structure in
the spectrum.
For a more comprehensive discussion of the local DoS of the system above
we refer the reader to e.g. [61].
SNS junction. As an example representative for the class of SN systems with
compact metallic region we next discuss the quasi one-dimensional SNS
junction displayed in ® gure 2. As opposed to the SN system, the physics of
the SNS system does depend on the phases of the order parameters in the
superconducting terminals. For this reason SNS systems with an in-built
possibility to tune the phases of the superconductors are sometimes referred
to as Andreev interferometers. In Appendix D.1 the quantitative solution of
the Usadel equation is discussed explicitly for the comparatively simple case
D ’ ² ’1 ¡ ’2 ˆ 0.
An analytic solution to the general case, D ’ 6ˆ 0, is also possible, although
cumbersome. In the limits of a very short (L ½ L ° ) and very long (L ¾ L ° )
junction, separate approximation schemes to the general solution may be
employed: the short junction case has been treated by Kulik and
Omelyanchuk [35] (for L ½ ¹;L ° ) and Likharev [30, 68] (for ¹ ½ L ½ L ° ),
while the long junction case (L ¾ ¹;L ° ) has been treated by Zaikin and
Zarkov [31, 69]. At this point we restrict ourselves to a discussion of a few
qualitative characteristics of the solutions obtained in various regimes.
The most conspicuous feature of the system is the appearance of a
minigap, Eg . As mentioned in section , the precise form of the gap depends on
Figure 11. The local DoS for the SNS junction as a function of both energy and position,
for L =¹ ˆ 5, ® ˆ 0:1 and ’ ˆ 0.
the phase di erence D ’. Its maximum width is taken for ¯’ ˆ 0, while this
width [33] tends to D and Ec for short and long (as de® ned above) junctions,
respectively. With growing phase di erence, Eg shrinks until, at D ’ ˆ p, it
closes altogether [34] (apart from a `microgap’ of the width of the single
particle level spacing [10]Ð the latter is induced by the general phenomenon
of level repulsion in disordered metals).
We take as illustrative the case of an SNS junction with coincident phases,
and of arbitrary width. In this case, the solution for q again lies in the 1± 3
plane and is parametrized, as before, by the angle cos ³ ˆ q ¢ e^3 . Figure 8
shows the trajectory of the retarded ³ in the complex plane as a function of
position, and ® gure 11 shows the local DoS as a function of both energy and
position. Note, as compared to the in® nite SN case (® gure 9), the appearance
of the minigap, Eg , which is below the superconducting gap and independent
of position.
Figures 12 and 13 show how variation of the material parameter ® a ects
the local DoS. Further, ® gure 13 shows how the local DoS at the SN
interface decreases as ® is reduced (e.g. as the ratio of the disorder
concentrations in the S and the N region is lowered.)
7. Fluctuations
Having discussed the mean ® eld content of the theory, we now turn to the central
issue of this paper, that of mesoscopic ¯ uctuations. The generality of the ® eld
theoretic machinery we have been discussing is such that it may be employed to
analyse the majority of ¯ uctuating observables in SN systems. Our strategy of
extending the conventional ¼-model formalism allows us to take full advantage
of the versatility of an approach that has been greatly developed in the study of
¯ uctuation phenomena in purely N-mesoscopic systems. Yet in order not to diversify
too much, we focus here on the discussion of the speci® c example of spectral
¯ uctuations. In fact, the status of spectral ¯ uctuations is slightly higher than that of
an ordinary example, since a vast number of ¯ uctuation phenomena are directly or
indirectly related to ¯ uctuations in the single particle spectrum (see [63, 70] for
review).
Figure 12. The e ect of ® on the energy dependence of the local DoS at the centre of the
junction. Here L =¹ ˆ 3.
Figure 13. The e ect of ® on the position dependence of the DoS for the SNS junction, with
°=D ˆ 0:5 and L =¹ ˆ 3.
A speci® c issue is the nature of ¯ uctuations in the local DoS around the mean
values (displayed in ® gures 9 and 11). To characterize these ¯ uctuations quantita-
tively, we employ the correlation function (16). Representing the correlation function
in terms of Green functions (cf. equation (15)), we see that we need to compute
functional expectation values of the type speci® ed in equation (38). An evaluation of
equation (38) on the mean ® eld level discussed previously leads to a vanishing
¯ uctuation component, h¼ph r ph a i h ph r ih¼ph
3 G ¼3 G ¡ ¼ 3 G
ai
3 G , due to the fact that the mean
-
® eld con® guration, Q, is diagonal in the ar indices. In other words, there is no
connection between advanced and retarded components, and the functional evalua-
tion of the product of Green functions equals the product of the averages.
Consequently the physics of mesoscopic ¯ uctuations is contained entirely in
- -
¯ uctuations around the block diagonal mean ® eld con® guration: Q ! T QT ¡1 . At
Table 2. Summary of the three ¯ uctuation types.
type structure in ar space ph space leads to
a diagonal non-diagonal corrections to Usadel Green function

b non-diagonal / 1ph mesoscopic ¯ uctuations
c non-diagonal non-diagonal destruction of b-type ¯ uctuations
® rst sight the analysis of the ¯ uctuation degrees of freedom, T Ð after all a 16-
dimensional matrixÐ seems to be a complicated task. Fortunately the totality of
¯ uctuations may be organized into three separate types, each with its own physical
signi® cance. Such a classi® cation scheme leads to a substantial simpli® cation of the
analysis. To be speci® c, we distinguish between ¯ uctuation matrices, T , that
(a) are diagonal in the space of advanced and retarded components. These types of
¯ uctuations still do not give rise to correlations between retarded and
advanced Green functions. None the less, they are of physical signi® cance:
quantum corrections to the Usadel solution are described by ¯ uctuations of
this type. We elaborate on these e ects in section 7.1.
(b) are non-diagona l in ar space but are proportiona l to unity in ph space,
‰T ; rph Š ˆ 0. Fluctuations of this type do induce correlations between
di erent Green functions and thereby mesoscopic ¯ uctuations. They will be
discussed in detail in section 7.2.
(c) ful® ll neither of the conditions (a) and (b). Whereas the physical signi® cance of
these ¯ uctuations is less clearly identi® able than one of the a- or b-type
¯ uctuations, they are nevertheless of importance. The reason is that the c-
type ¯ uctuations tend to destroy at su ciently high energies the correlations
that derive from mesoscopic b-type ¯ uctuations.
The three ¯ uctuation types are summarized in table 2. Besides the criteria (a) ± (c),
further restrictions to be imposed on the ¯ uctuation matrices follow from two
fundamental symmetries of the model: ® rst, general convergence criteria (see
Appendix B and [63]) enforce the condition
T y ˆ ²T ¡1 ²¡1 ; …54†
where
bf
² ˆ E11 « ¼ph ar bf
3 « ¼3 ‡ E22 :
Secondly, the tr-space structure of the model implies (cf. equation (B 2) of

Appendix B)
T T ˆ ½ T ¡1 ½ ¡1 ; …55†
where
bf
½ ˆ E22 « i¼tr bf tr
2 ‡ E11 « ¼ 1 :
For future reference we note that it is often convenient to represent both the matrices
T and the above symmetries in terms of the generators of the ¯ uctuations:
T ˆ exp …W †; …56†
where the generators W are subject to the constraints

W y ˆ ¡²W ²¡1 …57†
and
W T ˆ ¡½W ½ ¡1 : …58†
The above scheme generally classi® es the various types of ¯ uctuation corrections to
the quasi-classical picture of dirty superconductivity. Beginning with the a-modes,
we now turn to a more comprehensive discussion of these ¯ uctuations. Although we
do not provide by any means a comprehensive survey of the full diversity of e ects,
we will ® nd even in simple geometries of SNS structures a signi® cant range of
¯ uctuation types and associated phenomena. In order to prevent the subdivision of
these phenomena into excessively many classes, we assume throughout that the
spatial extent of the N region is su ciently large that Ec ½ D. In this case, the
minigap of the SNS junction with zero phase di erence, D ’ ˆ 0, lies at energy Ec .
The signi® cance of this restriction will be discussed further below, in section 8.
7.1. a-T ype ¯ uctuations: quantum corrections to the quasi-classical theory

In this section, ¯ uctuations of type (a) around the Usadel saddle point will be
considered. After specifying the general structure of these ¯ uctuations, we will
exemplify their e ect by discussing the quantum corrections to the quasi-classical
picture of single particle properties of SNS structures. Other types of SN structures
can straightforwardly be subjected to an analogous analysis.
-
Fluctuations of type (a) are diagonal in ar space. Since the saddle point Q is also
ar diagonal , the full e ect of the a-type ¯ uctuations may be studied by considering
just one of the ar sectors of the model action (36). For example, we may consider the
retarded sector describing the behaviour of a single, disorder-average d retarded
Green function. The restricted action is given by
… h i
p¸n e 11 †2 ‡ 4iQ 11 ijDj¼ph ‡ °¼ph ;
S ret ‰Q 11 Š ˆ ¡ str D…@Q 1 3 …59†
8
where the eight-dimensional matrix ® eld Q 11 denotes the rr block of Q and the
parameter ! (which is meaningless for a single Green function) has been dropped. To
keep the notation simple, we will henceforth (until the end of the section) denote Q 11
again by Q.
Following the general philosophy of our classi® cation, we organize the ® eld Q
-
into a saddle-point contribution Q (which is given by the solution of the retarded
Usadel equation) and a-type ¯ uctuations around it:
Q ˆ RQ f R¡1 ; Q f ˆ T ¼ph
3 T
¡1
; …60†
where R represents the inhomogeneous rotation parametrizing the saddle point,

- ph
Q ˆ q ¢ rph ˆ R¼3 R¡1 ; …61†
and the rotation matrices T generate the a-¯ uctuations. More precisely, T 2 G=H,
where G is the group of eight-dimensional matrices subject to the constraints (see
equation (54)),
T y ˆ ²a T ¡1 ²a¡1 ; bf
²a ˆ E11 « ¼ph bf
3 ‡ E22 ;
…62†
and equation (55). The subgroup H»G is de® ned through

H ˆ fh 2 Gj‰h;¼ph3 Š ˆ 0 g {. Note that these symmetry relations imply that the
matrices T span Efetov’s eight dimensional coset space of orthogonal symmetry.
In other words, save for the presence of the order parameter (and the di erent
coupling of magnetic ® elds), the action (59) is identical with that of a conventional ¼-
model for an advanced and a retarded normal metal Green function.
In order to understand more fully the e ect of the ¯ uctuation matrices T , we ® rst
have to analyse their commutation behaviour with the di erent contributions to the
action (59), i.e. the magnetic ® eld, the energy term and the order parameter.
Surprisingly, it turns out that there is a subset of T ’ s which not only commute
through the order parameter (no matter what its phase) but also are insensitive to
magnetic ® elds. For reasons that will become clear below, we will call these matrices
the C-modes. In the limit ° ! 0, the C-modes become completely massless. This
implies that for low energies these modes need a special, or non-perturbative ,
treatment.
7.1.1. T he C-modes: non-perturbativ e corrections to quasi-classical Green functions

Amongst the set of a-type matrices, T , consider the subset T C ² exp …W C †,
subject to the additional constraints ,
(1) ‰¼ph
1 ;W C Š
ˆ 0 (order parameter 1-component commutes through), and
ph
(2) ‰¼3 « ¼tr3 ; W C Š ˆ 0 (no coupling to magnetic ® eld).
In combination with equations (57) and (58), this gives altogether four constraints
and the non-trivial statement is that a set of generators W C obeying all of them
actually exists. These are the C-modes. Before turning to the actual construction of
these modes, let us qualitatively discuss some of their general properties.
First note that the conditions (1) and (2) above imply that the C-modes further
commute with the 2-component of the order parameter. This follows from the
observation that the phase twist needed to interpolate between the 1- and the 2-
component of the order parameter is equivalent to the appearance of a magnetic
gauge ® eld which (see condition (2)) is invisible to the W C ’s. Being insensitive to both
magnetic ® elds and order parameters with arbitrary phase positioning, the C-modes
{ Note that there is some freedom in parametrizing ¯ uctuating ® eld con® gurations. For example,
as an alternative to equation (60), one might let the ¯ uctuation matrices T act from the òutside’
-
Q ˆ T QT ¡1 :
As we will see shortly, there are situations where this parametrization is advantageous. In general,
however, it creates unwanted problems. To see this, interpret the T ’s as `rotations’ acting on the unit
-
vector q appearing in Q ˆ q ¢ ¼ph . Clearly, there are rotations that are ine ective (namely those that
rotate q around itself) and should be excluded from the con® guration space of the T ’s. In practice,
however, it is di cult to disentangle these rotations from the relevant ones. For example, parametrizing
the matrices T in terms of some kind of spatially ® xed coordinate systems, T ˆ T …³1 ;. . .†; where ³i are
rotation angles around certain ® xed axes, one ® nds that the e ective action S‰³1 ; . . .Š contains unphysical
divergencies. These are due to the fact that some `directions’ in the parameter space spanned by the ³’s
correspond to ine ective rotations, thereby being energetically costless. The way to remove these
spurious degrees of freedom is to introduce a `moving’ coordinate frame which, by construction, only
parametrizes rotations around axes perpendicular to q. This is exactly what the representation (60)
achieves. Due to T 2 G=H, the T ’s contain only degrees of freedom that e ectively modify the matrix
ph
¼3 :
merely couple to the gradients and the energy term in the action (59). In the limit of
small °, they become completely massless. More precisely, for a spatially constant
T C0 2 GC , where GC is the subgroup of G ful® lling the extra constraints (1) and (2),
- ¡1 °!0 -
S‰T C0 Q … T C0 Š ˆ S‰QŠ.
Physically, the C-modes represent modes of quantum interference in SN systems
which survive magnetic ® elds. Within a di erent formalism, these modes have for the
® rst time been noticed in [71]. Subsequently various physical phenomena caused by
their presence have been discussed in the literature:
(a) In the initial paper, [71] mentioned above, it was observed that in SN systems
weak localization corrections to the conductance may survive magnetic ® elds.
The quantum interference process responsible for that e ect is associated
with the C-modes.
(b) In [10], an SN system subject to a magnetic ® eld, but not exhibiting a minigap
(due to suppression of the proximity e ect by the magnetic ® eld), was
considered. In this case the single-particle DoS vanishes at the chemical
potential on a scale set by the mean level spacing. The existence of this
`micro-gap’ is also an e ect caused by the C-modes.
(c) Feigl’man and Skvortsov [72] discuss the e ect of C-modes on the transport
behaviour of vortices in moderately clean type II superconductors.
(d) The above phenomena relate to mean single particle properties. For the sake
of completeness, we here mention some manifestations of C-mode
¯ uctuations in two-particle properties: the level statistics of SN quantum
dots in a magnetic ® eld falls into a symmetry class that is di erent from any
of the standard Wigner± Dyson classes. Referring to a classi® cation scheme
due to Cartan, the SN± magnetic ® eld symmetry class has been termed
`class C’ [10]. As in standard mesosocopic systems, these level ¯ uctuations
can also be associated with channels of microscopic quantum interference.
Whereas Wigner± Dyson ¯ uctuations in di usive N systems are caused by
`di uson’ and `Cooperon’ modes, the class C ¯ uctuations are connected to
the modes speci® ed above, and hence the name `C-modes’. A ¼-model
formulation of the C-mode spectral statistics of random matrix ensembles
was presented in [17].
C-type level statistics in vortices has recently been microscopically derived
by Skvortsov et al. [73]. Thermal transport carried by C-modes through the
core of superconductor vortices is considered in a recent paper by Bundschuh
et al. [74].
Being e ective already on the level of single particle properties, the C-modes must
originate from interference processes between particles and holes. However, they
cannot be identical with the modes displayed in ® gure 1 (b ), since the latter are ® eld
sensitive. A typical type-C path con® guration is displayed in ® gure 14. In the analysis
below we will derive quantitative expressions for processes of this type.
After these general remarks, we next turn to the analysis of the C-mode
contribution within the ¼-model formalism. Speci® cally, we will discuss the e ect
of these modes on the mean density of states of the SNS geometry displayed in
® gure 2.
Before turning to the core of the discussion, let us make a technical remark
which will have some impact on the organization of the remainder of the section:
the invariance of the action under spatially constant C-transformations,
Figure 14. Semiclassical illustration of an interfering path con® guration contributing to the
C-mode corrections to the DoS. Compare the relative orientation of the arrows with
those appearing in ® gure 1 (b).
- ¡1 °!0 -
S‰T C0 Q … T C0 Š ˆ S‰QŠ suggests an interpretation of these modes as a global
-
symmetry of the action. In particular, it is more natural to let them act on Q from
-
the outside: Q ˆ T C QT C¡1 (compare with the inside representation of general a-
¯ uctuations, equation (60), and the footnote on p. 360). Of course it is possible to
forcefully contrive an inside parametrization for the C-¯ uctuations, via
-
T C QT C¡1 ˆ RT~C ¼3 T~C¡1 R¡1 , where the unitarily transformed T~ ˆ R¡1 T C R. In prac-
tice, however, this representation is inconvenient and, more seriously, makes it
di cult to separate the C-mode ¯ uctuations from general ¯ uctuations around the
saddle point. These considerations imply that, in general, it is di cult to treat the C-
modes and the rest of the a-type ¯ uctuations simultaneously. Physically, however,
these problems are of little signi® cance: below it will be shown that the minimum
price in energy associated with a non-C-¯ uctuation is of O…Ec †. This implies that two
regimes with qualitatively di erent ¯ uctuation behaviour exist:
(a) low energies, ° ½ Ec , where the C-modes are relevant, whereas the other a-
¯ uctuations can safely be ignored, and
(b) high energies, ° > Ec , where all ¯ uctuations have a comparable action of
O…Ec † and it is pointless to carefully distinguish between the di erent types
(C or other).
Below we will discuss these cases separately. Although our analysis is not applicable
to the crossover regime of intermediate energies, we do not expect qualitatively
remarkable phenomena to arise there.
L ow energies, ° ½ Ec . To quantitatively analyse the ¯ uctuation physics in this
regime, we need ® rst to derive an e ective action for the C-modes.
Fluctuations other than C are ignored.
To this end, we use that ‰T C ; ¼ph

1 Š
ˆ ‰T C ;¼ph tr ˆ
2 « ¼3 Š 0 and represent the
Q-® eld as
Q ˆ q3 Q C ; Q C ˆ T C¼ph
3 TC :
¡1
…63†
Substituting this parametrization into equation (59) , we obtain the desired
action
… h i
p¸n
S C ‰Q C Š ¡
ˆ str Dq23 …@Q C †2 ‡ 4iq3 °Q C ¼ph
3 : …64†
8
Among the general set of C-type ® elds Q C , there is a particular mode
Q 0C ˆ T C0 ¼ph 0 ¡1
3 …T C † which not only has C-symmetries but also is spatially
constant. Substituting this `zero-mode’ into (64), we obtain the action
i~
s h i
ph
S C0 ‰Q 0C Š ˆ ¡ str Q 0C ¼3 ; …65†
2
where
…
s~ ˆ p°¸n q3 : …66†
-
Note that deep in the normal region, where q3 ˆ 1, we have s~ = p°=d (where
-
d is the mean level spacing), so that s~ coincides with the standard parameter s
[63] commonly employed in the literature on spectral correlations in metals.
In order to give the above zero mode action some physical signi® cance, it
has to be shown that it is energetically gapped against the action of the higher
(spatially ¯ uctuating) ® eld con® gurations of C-symmetry. The spectrum of
¯ uctuating C-modes can be determined at various levels of accuracy. For our
purposes, it su ces (a) to demonstrate that a ground state gap exists and (b)
to coarsely estimate its magnitude. To do so, we ® rst note that the mode
spectrum is essentially determined by the gradient operator appearing in
equation (64). Integrating by parts, the latter can be rewritten as
…
¹ q23 str QDq¡2 2
3 @…q3 @†Q : …67†
2
The rationale behind this reformulation is that Dq¡23 @…q3 @† may be regarded
as a di erential „operator which is Hermitian with respect to the scalar
product hf ;gi ² q23 fg. Being Hermitian with compact support (taking
q3 ! 0 in S), the spectrum of the di erential operator is discrete. To estimate
the spacing D E between the zero eigenvalue of the spatially constant
eigenmode and the ® rst excited eigenvalue, we use the fact that the range of
support of the operator is set by L , the extension of the normal region.
Standard reasoning for the eigenvalue structure of one-dimensional
Hermitian di erential operators with compact support then leads to the
estimate D E ¹ D=L 2 ˆ Ec . Notice that the actual spatial structure of the
excited eigenfunctions may be complicated. For example, unlike with
standard applications of ¼-models to N systems, typical eigenmodes of the
action obey neither Neumann nor Dirichlet boundary conditions, but rather
exhibit more complicated edge behaviour which, in principle, may be derived
from equation (67) once q3 is known. For our purposes, entering this
discussion will turn out to be unnecessary.
The considerations above show that for energies ° ½ Ec , the action is

governed by the spatially constant C-mode{. It is reasonable to ask for which
physical applications such low energies may be expected to play a role. For
zero phase di erence between the superconductor terminals, the minigap is of
O…Ec † and, with regard to the DoS, the C-mode ¯ uctuations are expected to
be of little importance. To actually make visible the impact of these
¯ uctuations on the DoS, we concentrate here on the case of a junction close
to p-phase di erence, i.e. a junction where the gap is nearly but not
completely closed: Eg ½ Ec . Note that previous studies of the C-type
¯ uctuations have concentrated on the limiting case where the proximity e ect
is totally suppressed (Eg ! 0). It is worth remarking that, while we have
limited here our comments to issues surrounding the DoS, subgap properties
of SNS junctions, such as the Josephson coupling, may well be a ected by the
C-¯ uctuations, even under the broader conditions of a fully-established
proximity e ect. However, the analysis of these phenomena lies beyond the
scope of this paper.
Focusing on the range of applicability of the action (65), we notice that for
very small energy parameters, so that s~ ˆ O…1†, the zero mode must be
treated in a non-perturbativ e manner. Very much as in the study of normal-
metal ¼-models at low energy scales, the ¯ uctuations become unbounded as
s~ ! 0. Rather than perturbatively expanding in terms of the generators W C ,
it then becomes necessary to integrate over the entire manifold of matrices
T C. Non-perturbativ e analyses of this type have previously been applied to
the study of a random matrix ensemble [17] of non-proximity e ect SN
structures and of normal core excitations of vortices in superconductors [73].
Here we discuss how the C-¯ uctuations a ect the low energy behaviour of the
Gorkov Green function in SNS junctions.
To this end, we ® rst need to specify a global parametrization of the
matrices T C . It is a straightforward (if lengthy) matter to show that a general
8 £ 8 matrix T C ˆ exp …W C †, subject to both the general constraints (57) and
(58) and (1) and (2), can be parametrized as
T C ˆ vua; …68†
a ˆ exp …i³2 E bf
22 « ¼ph tr
1 « ¼1 ;†
bf
u ˆ exp … iyE22 « ¼tr
3 « 1ph ;
¶ ¡ ·¼tr
v ˆ exp … · ‡ ¶¼ tr
3
3
† «1 ph ;
bf
where ¶ and · are Grassmann variables. For the (invariant) measure

associated to the integration over the matrix Q C we obtain (cf. the analogous
but more di cult calculations of integration measures in [63])
{ A closer analysis, similar to that presented in section 7.3 with regard to the destruction of
correlations by c-type ¯ uctuations, shows that the range of stability of the C-zero mode is in fact limited
-
by ° ¹ dg1=2 ½ Ec .
… … 2p …p …
dy d³ sin ³
dQ C …. . .† ˆ d¶ d·…. . .†:
0 2p 0 2 sin2 …³=2†
Substituting the parametrizatio n (68) into the action (65) we obtain
S C0 ‰Q 0C Š ˆ ¡2i~
s…cos ³ ¡ 1†: …69†
As an example, we apply this action to a calculation of the C-mode
corrections to the local DoS. Substituting the zero mode integration over Q C
for the functional expectation value in equation (31) and performing the
(trivial) integrations over Grassmann variables and y, we obtain
…
1 p
¸ ¸n Re q3 1 ¡
ˆ … 2 0
d³ sin ³ exp ‰2i~
s…cos ³ ¡ 1†Š †
1 ¡ exp …¡4i~
s†
ˆ ¸n Re q3 1 ¡ … 4i~
s
†
sin …4~s† …1 ¡ cos …4~
s††
ˆ ¸n Re q3 1 ¡ … 4~s
¡
4i~
s
† :
The last line tells us that for small energies, s~ ! 0, the DoS always (i.e.
including the case of a p-junction) vanishes on a scale set by the mean level
spacing. This is the DoS `micro-gap’ that has been discussed previously in
[10, 17, 72, 73]. Moreover, for general s~, possessing real and imaginary
components, the DoS is not only determined by Re q3 but also by the
imaginary component, Im q3 , of the Usadel solution. Finally, s~ contains the
Usadel solution in an integrated form, that is, the C-mode introduces some
non-local in¯ uence on the local DoS by the Usadel solution at di erent
points of the system. The corrections vanish algebraically as s~¡1 . For
Im s~ > 1 the (oscillatory) factors containing the exponentiated parameter s~
can be neglected and we obtain the simpli® ed result
q3
¸ ’ ¸n Re q3 ¡ 14 Im : …70†
s~
„
Note that the de® nition (66) implies that for the global DoS ( ¹ ¸) the
algebraic corrections vanish. For a diagrammatic interpretation of this
correction to the Usadel DoS, see [10].
For D ’ ˆ p ¡ 0:0025 and g ˆ 5, the quasi-classical DoS and the
corrections to it are displayed in ® gure 15. We display here the DoS in the
vicinity of the minigap, at an energy regime similar to that considered by
Zhou et al. [62]. For this set of parameters, the quasi-classical DoS displays a
very strong peak, which dies down for …° ¡ Eg †=Eg ˆ O…1†.
L arge energies, ° > Ec . For large energies, the isolated 0-mode action is no
longer of signi® cance. The energy of all modes, C or other, is larger than Ec .
In particular, spatially ¯ uctuating con® gurations (with a `kinetic’ energy cost
of O…D=L 2 ˆ Ec †) need to be taken into account, too. Since the energy
associated with all these ¯ uctuations is parametrically of the same order,
separating the C-modes from the rest becomes pointless. In the next section
we discuss the corrections to quasi-classics at energies larger or comparable
to Ec arising from a perturbative treatment of all a-type ¯ uctuations.
Figure 15 (a ). Quasi-classical DoS and (b ) the C-mode correction for D ’ ˆ º ¡ 0:0025 and
g ˆ 5, at the centre of the junction and in the vicinity of the minigap. The dotted line
represents the non-oscillatory part of the correction, as given by equation (70).
7.1.2. Perturbative corrections to the quasi-classical Green functions

In the following we consider the impact of a-type ¯ uctuations on the `high’
energy (° ¹ Ec ) behaviour of the average Gorkov Green function. To keep our
discussion simple, we limit consideration in this section to a SNS geometry where
time-reversal symmetry is maintained (i.e. where the phase di erence between the
two superconductors , D ’, is zero). The violation of time-reversal symmetry through
the variation of the phase of the order parameter across the junction will not change
our discussion qualitatively.
Speci® cally, the questions we are going to address are
(a) do quantum corrections lead to the suppression of the minigap in the normal
region, and
(b) if so, does the minigap edge remain sharp, or are states introduced at all
energy scales below the gap?
As in the previous sections, the theory developed in this section may also be
straightforwardly generalized to other types of geometries and observables.
For large energies ° ¹ Ec , it is convenient to parametrize the totality of a-type
¯ uctuations as in equation (60). Due to the comparatively large energy cost
associated with ¯ uctuations around the Usadel saddle point, it is su cient to expand
the action to low orders in terms of the generators W of the ¯ uctuation matrices T .
Substituting the parametrization (60) into the action (59) , we obtain
… h i
p¸n e f †2 ‡ 4i°R¡1 ¼ph RQ f ;
S ‰QŠ ¡
ˆ str D…@Q 3 …71†
8
where @e ˆ @ ‡ ‰R¡1 @R; Š. Note that the above action does not contain the super-
conducting order parameter. This is accomplished by demanding that the non-C-
type ¯ uctuations obey Dirichlet boundary conditions at the NS interface. As a
consequence these modes are spatially varying with a minimum ¯ uctuation energy of
O…Ec †, which justi® es their perturbative treatment. As for the C-modes, these do not
couple to the order parameter anyway. In principle, their treatment is di cult
because, as mentioned above, they (a) ful® ll mixed boundary conditions di erent
from Dirichlet or Neumann and (b) are di cult to separate from the complementary
set of a-type ¯ uctuations. However, for large energies we believe these complications
to be physically irrelevant: the spectrum of all ¯ uctuating modes is discrete with a
typical spacing of O…Ec †. For energies ° comparable with Ec , all modes need to be
summed over. Under these conditions, the detailed structure of boundary conditions
and/or eigenvalues of individual modes becomes largely inessential; what matter are
the global features of the energy spectrum associated with the ¯ uctuations, most
importantly, the typical mean energy spacing. For this reason, we feel justi® ed in
ignoring the di erent boundary behaviour of the C-modes and to globally impose
Dirichlet boundary conditions (thereby correctly modelling the typical spacing
between consecutive eigenmodes). We believe that this simpli® cation does not lead
to qualitative errors.
To obtain the perturbative expansion of the action, we employ the exponential
parametrization T ˆ exp …W † and expand the generators W in terms of ph Pauli
ph ph ph
matrices, W ˆ w1 ¼1 ‡ w2 ¼2 (so that ‰W ;¼3 Š‡ ˆ 0). For zero phase di erence, the
Usadel solution encoded in the rotation matrices R can be parametrized in terms of a
single angle ³ (cf. equation (D 1)). The rotation matrices R mediating between ¼ph 3
and the Usadel saddle point then take the simple form
³ i
R ˆ exp ¡i ¼ph ; …@R†R¡1 ˆ ¡ ¼ph @³: …72†
2 2 2 2
Substituting these expressions into the action and expanding up to second order in
wi , it is a straightforward matter to show that
…
S‰W Š ˆ p¸n str 0 ‰D……@w1 †2 ‡ …@w2 †2 ¡ …@³†2 w21 †
¡ 2i° cos ³…w21 ‡ w22 †Š ‡ O…W 3 †: …73†

Here and in the following, the notation `str0 ’ represents a supertrace over all degrees
of freedom except for the ph components, which have been traced over. The absence
of terms at ® rst order in W is assured by the expansion around the saddle-point
con® guration of Q.
To eliminate the term in ¹…@³†2 , we make use of the fact that the Usadel
equation (D 2) possesses the ® rst integral,
D…@³†2 ¡ 4i°…cos ³ ¡ cos ³…0†† ˆ 0; …74†
where we have used the fact that in the middle of the junction, @³ ˆ 0. Substituting
this result into equation (73), we obtain
…
S‰W Š ˆ p¸n str0 ‰D……@w1 †2 ‡ …@w2 †2 †
‡ 2i°w21 …2 cos ³…0† ¡ 3 cos ³† ¡ 2i°w22 cos ³Š: …75†

To compute corrections to the DoS, we substitute the exponential parametrization
into the functional representation (31) to ® nd
¸n ph ¡1
¸…x† ˆ Rehstr…R¼bf 3 « ¼3 R Q† Q
i
8
ˆ ¸n Re cos ³‰1 ‡ 12 hstr0 …¼bf 2 2 i
3 …w1 ‡ w2 †† W Š
1
ˆ ¸n Re cos ³ 1 ‡ …P 1 …x;x† ¡ P2 …x;x†† ; …76†
p¸n L d¡1
?
where x is the coordinate along the junction and L ? is the spatial extent of the N
region in all other directions. The last line in equation (76) is obtained by an
application of Wick’s theorem [63] to the Gaussian expectation values h¼bf 2i
3 wi W . The
`propagators’ Pi play the role of generalized di usion poles. They are de® ned
through
‰¡D@x2 ‡ 2i°…2 cos ³…0† ¡ 3 cos ³…x††ŠP 1 …x;y† ˆ ¯…x ¡ y†;
…77†
‰¡D@x2 ¡ 2i° cos ³…x†ŠP 2 …x;y† ˆ ¯…x ¡ y†:
Without going into details we remark that the relative minus sign between P1 and P 2
in (76) derives from the di erent symmetries of the matrices w1 and w2 under matrix
transposition (cf. equation (58)).
We will not proceed any further analytically. In order to quantitatively evaluate
the P-dependent corrections to the DoS, one would have to compute the generalized
di usion poles (77). Due to the presence of the spatially varying terms ¹cos ³, a
general solution of the di erential equations is di cult{. None the less, quite a few
characteristic properties of the DoS corrections can be deduced from (76) simply by
inspection.
{ In fact, relatively standard techniques [75] may be employed for the solution of equation (77).
This follows from the fact that, upon substitution of the Usadel solution, equation (D 12), these
equations are classi® ed as `LameÂ ’ equations [76]. In comparatively simply situations, such as
asymptotically large energies ° ¾ Ec , in® nite SN rather than ® nite SN systems, and so on, analytical
solutions are available. However, in order not to diversify the discussion unnecessarily we do not
elaborate on these cases.
For asymptotically large energies ° ¾ Ec , one expects no in¯ uence of the

superconductor on the normal metal. Indeed, in that limit, cos ³ ! 1 implying that
(a) the Usadel DoS becomes metallic and (b) P 1 ¡ P2 ! 0, i.e. no quantum
corrections to the DoS.
For intermediate energies just above the minigap edge Eg , cos ³ varies smoothly
as a function of position. In this regime equation (76) gives corrections of O…g¡1 † to
the DoS whose quantitative evaluation is di cult.
Finally, let us consider subgap energies, ° < Eg (remaining of course outside the
regime ° ½ Ec ). Here, according to the quasi-classical analysis, the DoS vanishes,
implying that cos ³ is purely imaginary and the e ective action (75) purely real. As a
consequence , the propagators P i are real, too, and the DoS, as computed according
to (76) , vanishes identically below the quasi-classical minigap edge. In other words,
the perturbative inclusion of ® rst order quantum corrections does not give rise to the
appearance of states below the quasi-classical edge. The vanishing of Re …cos ³†
actually su ces to demonstrate that the robustness of the gap pertains to all orders
of perturbation theory.
This conclusion presents something of a puzzle: taking into account quantum
corrections, the above result indicates that the minigap remains ® xed at energy Eg .
The latter is determined by the bare value of the di usion constant, D ˆ vF l=d.
However, the intuition a orded by the one-paramete r scaling theory of
localization [77] suggests that observables such as the tunnelling DoS should depend
only on the value of D renormalized by weak localization corrections. In bulk
normal metallic samples, weak localization corrections (to two-particle properties)
stem from mechanisms of quantum interference between trajectories connected
through a time reversal operation (see ® gure 16 (a )). In the present case, weak
localization e ects can arise due to the interference of particles and holes (see
® gure 16 (b )). Since holes bear similarity with time reversed particles, there is no
conceptual di erence to the above N-interference mechanism, and one expects a
standard renormalization of the di usion constant (albeit already on the level of one-
particle properties). Yet, according to the analysis above, the minigap edge, a
function of the unrenormalized D, is robust against perturbative quantum correc-
tions in the particle± hole channel.
In fact, the absence of weak localization corrections to the minigap edge signals
the failure of the perturbation theory. To properly identify quantum weak
localization corrections to the di usion constant, and therefore the minigap edge,
it is necessary to renormalize the saddle-point equation itself. This situation parallels
that encountered in the study of the renormalization of the gap in a dirty bulk
superconductor where quantum corrections (in the Cooper channel) lead to a
renormalization of the gap equation (see e.g. [78]). In the present case, weak
localization corrections to the minigap edge are obtained within a renormalization
group procedure. Since, operationally , this procedure is somewhat technical, its
description has been made the subject of Appendix F.
The renormalization group procedure described in Appendix F may be employed
safely down to energy scales in excess of Ec . However, at energy scales in the vicinity
of the minigap, the Cooperon propagator depends sensitively on the geometry and it
becomes necessary to include the additional ¯ ow in R (the matrix rotating to the
RG-a ected saddle point of the theory), coupled to that in D, as the cut-o is
lowered towards Eg . Although the manner in which such renormalization processes
are included self-consistently lies beyond the scope of this paper, the outcome of the
Figure 16. Renormalization of the di usion constant, D, by interference of (a ) trajectories

with their time-reversed counterparts and (b ) particles and holes.
RG procedure can be summarized as follows. Treating quantum interference

correction within a RG scheme leads to a shift of the minigap edge. The overall
structure of the gap edge (e.g. the non-analytic behaviour of the DoS in the vicinity
of the DoS) is maintained. In particular , no states are found below the (renorma-
lized) gap edge.
These ® ndings leave us with the question whether indeed, no states exist below
the (renormalized) gap edge or whether the computation simply has not been
accurate enough. Although we will not provide here a quantitative analysis, we
Figure 17. Schematic indication of the role of the various DoS corrections. Eg¤ denotes the
renormalized gap edge.
believe that the second option is the correct one: to ® nd states below the minigap,
one must account for contributions to the action which cannot be accessed by a
perturbative shift of the inhomogeneous saddle point. Contributions of this kind
have been identi® ed in bulk normal conductors as soliton-like con® gurations of the
Q-matrix ® elds, and have been associated with a rare class of states which are
described as ànomalously’ or `nearly’ localized within the metallic phase [79± 81].
Poorly contacted to the superconductor , these states are able to exist at energies
below the minigap and generate contributions to the average DoS exponentially
small in g. Although we see the SN system as a useful and challenging arena in which
to investigate the localization properties of such rare states, their consideration lies
beyond the scope of this paper.
Before leaving this section, let us in summary list theÐ admittedly diverseÐ set of
a-type ¯ uctuation mechanisms renormalizing the single-particle properties of meso-
scopic SN structures (see ® gure 17).
(a) For energies ° ¾ Ec , the N-component behaviour is largely metallic.
However, the presence of the superconductor is exerted in terms of massive
quantum corrections to the DoS and other single particle properties. The
larger ° is, the smaller are the corrections.
(b) Energies just above the minigap edge Eg are the most di cult to analyse.
Quantum corrections to quasi-classics are carried by di usion type modes
whichÐ due to the pronounced energetic and spatial inhomogeneity of the
DoSÐ are di cult to treat analytically. By perturbativel y including such
corrections, one obtains corrections to the DoS above the gap. Both the
position of the gap and the vanishing of the subgap density of states remain
unchanged.
(c) By embedding the di usive modes into an iteration of RG analyses and
solutions of renormalized Usadel-type mean ® eld equations, one arrives at a
shifted minigap edge. The non-analyticity of the gap is maintainedÐ that is,
there are no smooth DoS `tails’ leaking downwards out of the sharp edge.
(d) Presumably, `nearly localized’ subgap states can be found with a probability
that is exponentially small in the metallic conductance g.
(e) Eventually, for energies ° ½ Ec , the ¯ uctuation physics is governed by the C-
mode whose impact on various physical observables (for non-proximity e ect
SN structures) has already been discussed in the literature.
It is important to question whether the above corrections can be made
experimentally visible. As far as the DoS is concerned, the answer must be a
conservative one: the chances are that it will be impossible to separate the high
energy 1=g corrections from the Usadel background. Furthermore, for good metals
(g ¾ 1) , ® nding nearly localized subgap states will also be di cult, since, as shown in
[79± 81], disorder con® gurations leading to nearly localized states are exponentially
rare. Thus, as far as the mean DoS is concerned, the above ¯ uctuation contributions
will probably be hard to detect. However, the primary purpose of this section has
been to demonstrate that a variety of interference mechanisms adding to the
standard quasi-classical picture exist in principle. If and to what extent these
¯ uctuations give rise to observable changes in single particle properties other than
the DoS (e.g. the Josephson coupling characteristics) represents a subject of future
research.
We now leave the issue of the renormalization of single particle properties and
turn to the discussion of correlations between more than one Green function, as
described by the b- and c-type ¯ uctuations.
7.2. b-T ype ¯ uctuations: the Goldstone mode

In this section we discuss the class of ¯ uctuations around the Usadel saddle point
which has been denoted above by `type (b)’. Unlike the a-¯ uctuations , ¯ uctuations of
type (b) induce correlations between di erent Green functions. What makes the b-
¯ uctuations particularly important is their Goldstone mode character: in the limit of
vanishing energy di erence between the considered Green functions, these modes
become truly massless, a signature for the presence of pronounced mesoscopic
¯ uctuations.
Consider the action (36) in the simple case !‡ ˆ A ˆ 0. Obviously, any
- -
transformation Q ! T 0 QT 0¡1 leaves the action invariant provided that
‰T ;¼ph
i Š
ˆ 0. Among the group of matrices G0 ² fT ˆ T 0 « 1ph g , there is a subgroup
H0 » G0 , ‰H0 ;¼ar
3 Š-
ˆ 0 which leaves invariant not only the action, but also the saddle-
point solution Q itself. As a consequence, ¯ uctuation matrices contained in the
subgroup H0 are completely ine ective and do not couple to the theory. However,
the elements T 0 of the coset space G0 =H0 do generate non-trivial transformations of
the diagonal saddle point. Moreover, in the limit T 0 ˆ const:, these transformations
do not alter the actionÐ they are Goldstone modes.
Being Goldstone modes, the e ective action of the T 0 ’s can only contain gradient
terms and mass terms induced by sources of symmetry breaking, such as ® nite !‡
and A. The actual structure of the action depends crucially on its behaviour under
time reversal. For the sake of simplicity, we focus here on the two pure symmetry
cases:
~ for the Goldstone modes in an SN junction

Figure 18. The e ective di usion constant, D,
as a function of both energy and position, for ® ˆ 0:1.
(i) Orthogonal symmetry: The action is time reversal invariant, A ˆ 0.

(ii) Unitary symmetry: Time reversal invariance is broken, jAjL F¡1
0 ¾ g
¡1=2 .
Here, F0 is the ¯ ux quantum. Note that A denotes the vector potential with account
for the phase di erence between the superconducting terminals, so that phase
di erences D ’F¡10 ¾ g
¡1=2 su ce to drive the system into the unitary symmetry
class. The reason that g ¡1=2 appears as a measure for the strength of the perturbation
is that, for jAjL F¡1
0 ¾ g¡1=2 , the dimensionless coupling constant of the symmetry
breaking operator in the action exceeds unity (see Appendix E). Alternatively, one
may say that under these conditions, the mass of the `Cooperon’ greatly exceeds the
level spacing.
The derivation of the e ective action, S 0 , of the Goldstone modes is somewhat
technical and has been deferred to Appendix E. Here we merely state the result,
…
p
S 0 ‰Q 0 Š ˆ ¡csym ‰¸n D~ str0 …@Q 0 @Q 0 † ‡ 2i!‡ ¸ str0 … Q 0 ¼ar
3 †Š; …78†
4
where D~ ˆ …D=2†…1 ¡ q‡ ¢ q¡ † plays the role of a space dependent di usion

coe cient. The variation of D~ with both position and energy is shown in ® gure 18
for an SN junction with a typical choice of material parameter. In addition, ¸ is the
space dependent, local DoS, as displayed already in ® gure 9. (Notice that in the bulk
of S both the di usion constant and ¸ vanish. Hence, the support of the action of Q 0
is restricted to the N region.)
Further, in the case of
(i) orthogona l symmetry: csym ˆ 1, the matrices T 0 are eight-dimensional and
obey the time reversal symmetry relation (55), and of
(ii) unitary symmetry: csym ˆ 2, the matrices T 0 are four-dimensional (i.e. they do
not carry a tr-index structure) and equation (55) is meaningless.
In either case the matrices T 0 obey the restricted version of the symmetry relation
(54){:
y
T 0 ˆ ²0 T 0¡1 ²0¡1 ; …79†
where
bf
²0 ˆ E11 « ¼ar bf
3 ‡ E22 :
In summary, we see that the symmetry of the Goldstone ® elds is identical to those of
the standard Efetov Q-matrix manifolds [63]. In other words, by freezing out the
ph degrees of freedom, the large 16-dimensional ¼-model manifold collapses to
smaller ones of dimensionality 8 (4) which are symmetrically identical to those
encountered in orthogonal (unitary) applications of the standard ¼-model.
Besides the general symmetry relations, a further condition to be imposed on the
® elds is that they obey Neumann boundary conditions @? T 0 …x† ˆ 0 at all SN
interfaces. The derivation of these boundary conditions is discussed in Appendix E.
So far the discussion has been for a general SN geometry. In order to actually
demonstrate how the b-modes generate mesoscopic ¯ uctuations we next consider a
speci® c example, namely, the problem of DoS ¯ uctuations above the minigap edge in
an SNS structure.
7.2.1. L evel statistics in SNS structures

It is well known that the single particle spectrum of mesoscopic, purely normal
systems is governed by various types of mesoscopic ¯ uctuations (see e.g. [63] for a
review). The ¯ uctuation behaviour can be characterized conveniently in terms of
correlations ¹h¸…° ¡ !=2†¸…° ‡ !=2†i between the DoS’s at di erent energies.
Extensive analyses of correlation functions of this type have shown that the DoS
correlations become increasingly universal in character, the lower the energy
separation ! (a fact that follows heuristically from the interpretation of ! as an
inverse time scale). In particular, for energies ° < Ec the correlations become fully
universal in the sense that they depend on nothing more than the mean (and
-
constant) level spacing d and the fundamental symmetries of the system. This is
the regime of Wigner± Dyson statistics. For larger energies, the Wigner± Dyson
behaviour crosses over to other and less universal types of statistics. Nevertheless,
the correlations remain energetically long ranged in the sense that they decay
algebraically as a function of !.
Here we ask to what extent this behaviour carries over to the ¯ uctuation
behaviour of the SNS DoS in the vicinity of the minigap. As compared to normal
metals, the situation is more intricate in that already the mean DoS is a ected by
mechanisms of quantum coherence. A conceivable situation is that the (strong)
modes of quantum interference, giving rise to the particular structure in the mean
DoS, decouple entirely from the modes responsible for DoS ¯ uctuations. Another
{ At ® rst sight it seems like we are facing a problem here: a rotation matrix cannot be of the form
T ˆ T 0 « 1ph and simultaneously obey the general relation (54). The reason that matrices of b-type are
nonetheless permitted is that relation (54) is in fact too strict. What matters is the restriction of the
symmetry to the boson± boson and the fermion± fermion block of the matrices T (see the corresponding
discussion in [82].) With regard to these bf-diagonal blocks, matrices T 0 obeying equation (79) are
compatible with equation (54).
possibility is that one might end up with some kind of inseparable conglomerate of
modes of interference, and thereby fundamentally non-universal types of spectral
statistics. Here we demonstrate that the true picture lies somewhere halfway between
these two extreme options: it is still possible to identify a regime of universal Wigner±
Dyson statistics, albeit superimposed on an energetically non-uniform DoS. How-
ever, its range of validity shrinks down to a small energy window beyond which the
correlations do become entirely non-universal.
To obtain speci® c information about spectral ¯ uctuations in an SNS structure,
we apply the action (78) to the analysis of the SNS geometry, shown in ® gure 2.
The set of ® eld con® gurations obeying Neumann boundary conditions at the
SN interfaces obviously contains a subset with trivial spatial dependence,
T 0 ˆ const:Ð the `zero mode’. The zero mode action reads:
p !
S 0 ‰Q 0 ˆ const:Š ˆ ¡i - ‡ str0 ‰Q 0 ¼ar3 Š: …80†
2 d…°†
- „
Here d…°† ˆ … ¸…°††¡1 denotes the average level spacing, which, in contrast to the
purely normal case, is now energy dependent. Note the similarity with the action,
(65), of the C-zero mode. The di erence is in the pre-factor and in the physical spaces
in which the matrices Q and Q 0C , respectively, operate.
In order to demonstrate any signi® cance of the isolated zero-mode action, we
have to show that it is separated by an energy gap from the action of all other ® eld
con® gurations. At this point, the boundary conditions begin to play a crucial role.
Expanding the ® elds in terms of cosines, that is, a complete set of functions
compatible with Neumann boundary conditions, we see that, next to the zero mode,
the ® eld with least curvature varies as ¹ cos …2px=L †. Due to the presence of the
-
gradient term, the action associated with this con® guration is of O…D=…L 2 d†. Thus,
for !‡ ½ Ec , the zero-mode action is energetically gapped against all ¯ uctuating
contributions and plays a dominant role.
We will see in the next section that, due to the presence of the c-type ¯ uctuations,
the range of stability of the zero mode is actually much smaller than !‡ < Ec . Yet,
restricting ourselves for a moment to the consideration of the zero mode action, we
can, without any further calculation, draw immediate conclusions about the level
statistics in SNS systems over small correlation intervals, !‡ . In fact, actions of the
structure (80) are standard in applications of the ¼-model in N-mesoscopic physics:
they appear (a) whenever a model may be subjected to a zero-mode approximation
or (b) when one is dealing with ¼-model analyses of a single random matrix
ensemble. With regard to spectral statistics, the existence of the zero-mode action
(80) implies that level correlations for small energies are of W igner± Dyson type.
Furthermore, a comparison of action (80) with the analogous action for
N systems [63] shows the correlations to depend on an average level spacing that
is e ectively halved. This is due to the strong `hybridization’ of levels at energies
¹ °F § ° induced by Andreev scattering at the SN interface.
A more comprehensive discussion of level ¯ uctuations, including the di erences
to the types of spectral statistics found in N materials, will be given after the c-type
¯ uctuations have been incorporated in our analysis.
7.3. c-T ype ¯ uctuations: quantum corrections to level statistics

In the previous section, an e ective action for b-type ¯ uctuations was derived,
and the latter were shown to be Goldstone modes of the theory. Furthermore, for the
SNS geometry at energy scales !‡ ½ Ec , the e ective action was shown to be

dominated by a zero mode which established universality of level statistics within the
ergodic regime. Higher modes give rise to non-universal corrections on energy scales
!‡ ¹ Ec . At the same time, c-type ¯ uctuations, that is, ¯ uctuations that commute
with neither rph nor rar , also become important.
The aim of this section is to examine the role played by c-type ¯ uctuations in
limiting the regime over which universal correlations persist{. Since, for states above
-
the minigap, c-type ¯ uctuations incur a mass which is of order °=d > g ¾ 1,
¯ uctuation corrections to the universal level statistics can be treated within a
perturbative manner. Our approach will be based on the perturbative treatment
introduced by Kravtsov and Mirlin [83] in studying similar corrections in normal
disordered conductors.
We begin by employing the general parametrization
" #
3
X
ph ph
Q ˆ T 0 T q ¢ r T ¡1 T 0¡1 ; T ˆ exp W · ¼· ; …81†
·ˆ0
where Q 0 ˆ T 0 q ¢ rph T 0¡1 represents the spatially homogeneous zero mode. Here we
have applied the notation ¼0 ² 1, and, separating the „zero mode, we impose the
requirement that the ¯ uctuations obey the constraint W 0 ˆ 0. Here, as before,
q ˆ q ¢ rph represents the saddle-point, or Usadel, solution.
Applying this parametrization , and expanding to quadratic order in the ® elds W ,
the total correction to the zero mode action (80) takes the form
…
p¸n e ;qŠ2 ‡ 2‰@W
e ;W Šq@q†
e
¯S ‰Q 0 ;W Š ˆ ¡ str fD…‰@W
8
¡ 2i°q‰W ;‰W ;¼ph

3 ŠŠ
‡ 4i!‡ U0 ¼ph 3
3 ‰W ;qŠ ‡ O…W †;
g …82†
where U0 ˆ T 0¡1 ¼ar

3 T 0 . In contrast to a normal conductor, inhomogeneity of the
saddle-point solution q allows a term linear in W to survive in the action. The
presence of this term has important consequences on the range over which level
correlations are universal.
To proceed, it is convenient to further separate c-type ¯ uctuations into two
classes, W ˆ W A ‡ W D :
(a) Modes diagonal in ar space (‰W ·D ;¼ar
3 Š
ˆ 0) , but o -diagonal in ph space, are
termed D modes; W D ˆ W 1D « ¼ph D
1 ‡ W 2 « ¼2 .
ph
A ar
(b) Modes o -diagonal in ar space (‰W · ;¼3 Š‡ ˆ 0) , but diagonal in ph space,
are termed A modes{; W A ˆ W 0A « 1ph ‡ W 3A « ¼ph 3 .
{ It is conceivable that the unfolding procedure of the previous section, that allows for an
energetically inhomogeneous mean DoS, may itself be a source of decorrelation on energy scales
!‡ ¹ Ec . Such a mechanism for non-universal corrections to the level statistics would be separate to that
described here for the c-type ¯ uctuations and so is not contained within our present analysis.
{ The denotation À’ respectively `D’ modes is again motivated by Cartan’s classi® cation scheme
of symmetric spaces (see [10] for a discussion of the scheme focusing on its application to the symmetry
classi® cation of SN systems).
On the level of the quadratic action, no mixing between these modes occurs.
Finally, considerations analogous to those presented in connection with the a-type
¯ uctuations show the spectrum of these modes to be discretely spaced, where the
typical `level distance’ is of O…Ec †. In passing we note that the c-modes, as introduced
above, are in fact not complementary to the a-modes. For example, the above W D ’s
contain modes / 1ar , which, by de® nition, belong to type (a). However, the present
analysis, regarding the impact of non-universal ¯ uctuations on the b-type Goldstone
mode, does not require a separation of the a- and c-modes and it is su cient to
continue with the present de® nition of the c-modes.
Once again, to keep our discussion simple, we limit consideration to pure
symmetry classes of either orthogonal or unitary type. This leads to a simpli® cation
of the e ective action (82) allowing an explicit integration over the ¯ uctuations W .
Speci® cally, for pure symmetry, the action takes the form ¯S ˆ S A ‡ S D , where
…
S A p¸n str ‰W 0A …¡@ D@†W
ˆ ~ A A ar
0 ‡ i!‡ W 0 ¼ 3 …‰q3 Š ‡ ¡ ‰q3 Š¡ †U0 Š; …83†
…
S D p¸n str ‰W 2D …¡D@ 2 ¡ 2i°q3 †W 2D ‡ 2!‡ W 2D q1 U0 Š
ˆ …84†
and D~ is the space and energy dependent di usion constant that has been introduced
in section 7.2 for the Goldstone modes. Note that the W 3 ¯ uctuations do not couple
linearly to U0 in the pure symmetry case and so may be dropped.
As can be seen from the general structure of the action, c-type ¯ uctuations in the
vicinity of the minigap are generally `massive’ , that is, governed by an action which is
-
at least of order °=d > g ¾ 1. It is thus permissible to treat these ¯ uctuations in a
simple Gaussian approximation. Applying the shift operations
i!‡ ^
W 0A ! W 0A ¡ ~ ¡1 ¼ar …‰q3 Š ¡ ‰q3 Š †U0 ;
P0 …¡@ D@† 3 …85†
2 ‡ ¡
W 2D ! W 2D ¡ !‡ P^2 …¡D@ 2 †¡1 q1 U0 ; …86†

where P^· represents a projector onto the ® eld space of W · , then performing the
Gaussian integral, we obtain the renormalized zero-mode action
2
S ‰Q 0 Š ˆ S 0 ‰Q 0 Š ¡
g
… † str ‰¼
µ…°† !‡
-
d…°†
ar 2
3 ;Q 0 Š ;
…87†
where µ…°† ¹ O…1† represents a constant that depends on the sample geometry.
Equation (87) has a structure equivalent to that found in the study of universal
parametric correlation functions and explicit expressions for the two-point correlator
of DoS ¯ uctuations for both orthogonal and unitary ensembles can be deduced from
[84]{. Qualitatively, the additional contribution in (87) counteracts the zero-mode
¯ uctuations for non-vanishing frequencies !‡ .
Furthermore, we ® nd a marked di erence in the manner in which level
correlations are suppressed as compared to the purely N case. Already for energy
-
separations !‡ =d…°† ¹ g1=2 , the zero-mode integration is largely suppressed which
{ With reference to the speci® c correlation function R2 , we remark that only massive ¯ uctuations
in the ph sector contribute to connected correlators of the form hGA A
°1 G°2 i allowing such terms to be
neglected.
manifests in an exponential vanishing of the level correlations on these scales. This

represents a qualitatively smaller energy scale than that for the purely N case, for
which Wigner± Dyson statistics prevail all the way up to frequencies !‡ ’ Ec . In
addition, the exponential suppression of correlations in the SN case di ers from the
purely N case, for which the Wigner± Dyson statistics are succeeded by other forms
of algebraically decaying spectral statistics in the high frequency domain !‡ > Ec
[85]. Note that a similar phenomenon of zero-mode suppression has recently been
observed by Skvortsov et al. [73] in their analysis of the level statistics of normal core
excitations in type II superconductor vortices.
8. Discussion
In conclusion a general framework has been developed in which the interplay of
mesoscopic quantum coherence phenomena and the proximity e ect can be
explored. The connection between the conventional quasi-classical approach and
the ® eld theoretic approach adopted here has been emphasized. In applying the
e ective action we have introduced a classi® cation of di erent modes of ¯ uctuations.
To keep our analysis simple we have focused on a regime in which the contact
between the superconductor and normal regions is metallic, and where D ¾ Ec .
Experimental analyses are often carried out in the complementary regime where
tunnel barriers separate S and N and/or D < Ec . It is straightforward to modify the
theory so as to accommodate tunnel barriers, small order parameters and, in fact,
altogether di erent sample layouts. However, in order not to diversify the present
exposition of the formalism even further, we have restricted ourselves to the analysis
of the relatively simple systems discussed above. Whereas certain of our conclusions
(e.g. the existence of a Wigner± Dyson regime of spectral correlations) carry over to
cases of di ering sample layouts, others do not. More speci® cally, reducing the
strength of the order parameter below Ec a ects both the behaviour of certain of the
¯ uctuation classes discussed above and the spatial structure of the solutions of the
Usadel equation [35]. Rather than attempting to set up a most general `phase
diagram’ of mean ® eld and ¯ uctuation regimesÐ given the diversity of SN systems
with qualitatively di erent physical behaviour, certainly a fruitless taskÐ it is more
e cient to treat di erent problems individually, i.e. to start out from the most
general form of the action (36) and to restrict the analysis to those ¯ uctuation modes
that encompass the physics particular to the problem under consideration (see, e.g.
[86] and [74] for recent examples.) Whether or not a certain type of ¯ uctuation
around the Usadel saddle point is `relevant’ or not can be deduced from the way it
couples to the di erent contributions to the action.
Finally, we note that in this paper we have focused on the in¯ uence of
mesoscopic ¯ uctuations on the proximity e ect in disordered SN structures.
However, mechanisms of quantum interference analogous to those discussed here
also induce mesoscopic ¯ uctuations in irregular clean or `quantum chaotic’
structures. Moreover, the proximity e ect is strongly in¯ uenced by such coherence
phenomena allowing them to be employed as a potential probe of chaotic
behaviour [36, 87]. The question then arises as to whether the framework developed
above for disordered SN structures may be generalized to account for chaotic or
ballistic SN structures.
To address this question, it is instructive to focus ® rst on the properties of purely
normal chaotic structures. In fact the connection between the statistical ® eld theory
of normal disordered conductors and ballistic chaotic structures was motivated by

the quasi-classical approach of Eilenberger discussed previously. Recognizing that
the Usadel equation could be associated with the equation of motion corresponding
to the saddle point of the action of the di usive nonlinear ¼-model [79], Muzy-
kantskii and Khmel’nitskii proposed that the Eilenberger equation could be
identi® ed with a ballistic analogue of the nonlinear ¼-model action [24]. In this case,
the di usive character of the action was replaced by a kinetic operator. Their work
found support in subsequent investigations based on the study of energy averaged
properties of (again normal) chaotic structures which led to a microscopic derivation
of the ballistic action [88]. Taken together, these studies showed that, while density
relaxation in disordered conductors is di usive, in general chaotic structures it is
governed by modes of the irreversible classical evolution operator.
The generalization of the ballistic ® eld theory to encompass the proximity e ect
follows naturally from the ideas presented in this paper. Expanding the ® eld space of
Q to accommodate particle± hole degrees of freedom and, as in the di usive model,
introducing the inhomogeneous order parameter D ^ , the e ective ballistic action
takes the form
… h
p !‡ ar i
S‰QŠ ˆ i - str 2iT ¼ar 3 « ¼
ph
fH ;T ¡1 g
¡ ¼
ph
« D ^ ‡ ° ‡ ¼ Q ; …88†
2d 3 3
2 3
where fH ; g represents the Poisson bracket of the classical Hamiltonian, and the
ph ¡1
supermatrix ® elds Q ˆ T ¼ar 3 « ¼3 T depend on the 2d ¡ 1 phase space coordinates
parametrizing the constant energy shell, xk ˆ„ …r;p† „ 2d¡1 . (With this de® nition the
integration measure is normalized such that ² dxk ˆ 1.) In the presence of a
Gaussian distributed ¯-correlated impurity potential, the ballistic action is supple-
mented by a further term corresponding to a collision integral [24]
…
p dr dn dn 0
S coll ˆ - str ‰Q…n;r†Q…n 0 ;r†Š; …89†
4d½ L d S d2
„
where n ˆ p=jpj and S d ˆ dn. Indeed, for a strong enough impurity potential,
-
d½ ½ 1, a moment expansion of the action recovers the di usive action. Varying the
action with respect to Q, and applying the identi® cation g…n;r† $ Q…n;r†, the saddle-
point equation of motion coincides with the Eilenberger equation of transport,
equation (6).
Although, in principle the ballistic action represents a complete theory of
statistical correlations in chaotic SN structures, an analytical description of the
modes of the classical evolution operator has proved di cult to construct. In
particular, the sensitivity of weak localization corrections to mechanisms of
`quantum di raction’ and ìrreversibility’ in normal clean chaotic structures has
proved di cult to quantify [41]. In the SN geometry the same mechanisms have a
dramatic e ect on the single-particle properties of the device such as the minigap
structure in the local DoS [36] (see the discussion in section 2.2). For this reason, we
believe that SN structures may provide a versatile arena in which properties of
quantum chaotic systems can be explored.
Acknowledgements
We are indebted to Boris Altshuler, Anton Andreev, Dima Khmel’nitskii,
Vladimir Fal’ko, Alex Kamenev, Vladimir Kravtsov, Igor Lerner and Martin
Zirnbauer for useful discussions. One of us (DT-S) acknowledges the ® nancial

support of the EPSRC and Trinity College.
Appendix A: Boundary conditions of the Usadel equation

In addition to the transport equations provided by quasi-classics, it is necessary
to specify boundary conditions at the SN interfaces. For the Eilenberger equation, at
a planar SN interface with an arbitrary transmission coe cient, T , these conditions
have been derived by Zaitsev [47]. Note that these boundary conditions cannot be
obtained using the standard quasi-classical Green function (5) alone. Instead, one
must go back to a more microscopic formulation.
Following the general philosophy of this section we shall not review the
(somewhat technical) derivation of the boundary conditions but merely formulate
the main results. The Eilenberger Green function, g…n;r†, may be separated into
symmetric and antisymmetric parts, g ˆ gs ‡ ga , with respect to the operation
vF ! ¡vF . The antisymmetric part, ga , is continuous across the interface. We note
that this results in the conservation of the supercurrent density,
p2 D E
j ˆ ¡ F n tr¼ph r
3 g …n;r† ; …A 1†
4p n
across the interface. In contrast, the symmetric part experiences a jump depending
on the transmission coe cient. The resulting conditions are:
ga …‡† ˆ ga …¡† ² ga ; …A 2†
T T
ga R…1 ¡ ga ga † ‡ …gs …‡† ¡ gs …¡††2 ˆ ‰gs …¡†; gs …‡†Š; …A 3†
4 4
where R ˆ 1 ¡ T is the re¯ ection coe cient, the r,a superscripts have been dropped,
and gs …§† denotes the Green function in® nitesimally to the left respectively right of
the junction.
For a perfectly transparent …T ˆ 1† interface, both parts of g are continuous. In
the low transparency limit, T ½ 1, we have ga ¹ T and (A 3) reduces to
T
ga ˆ ‰gs …¡†;gs …‡†Š: …A 4†
4R
The above boundary conditions simplify further in the dirty limit. As shown by
Kuprianov and Lukichev [89], the reformulation of (A 2) and (A 4) in terms of the
Usadel Green function leads to the pair of conditions,
¼…¡†g0 @r g0 …¡† ˆ ¼…‡†g0 @r g0 …‡† …A 5†
T ½1 GT
ˆ ‰g0 …‡†;g0 …¡†Š; …A 6†
2
where ¼…§† is the metallic conductance on either side of the interface, and
…
e2 ¸n vF 1 T
GT ˆ d…cos ¬† cos ¬; …A 7†
2 0 R
is the tunnel conductance of the junction, ¬ is the angle between n and r, and ¸n is the
bulk, normal-metallic DoS. Note that T may depend here on ¬. Note also that the
second condition of equation (11) applies only in the limit T ½ 1. Lambert et al. [90]
have recently examined this restriction and how it may be relaxed. In the following,
we exclusively consider the opposite case of perfect transmittance , T ˆ 1, for which

the second condition, equation (11), should be replaced by
g0 …‡† ˆ g0 …¡†: …A 8†
Appendix B: Symmetries of the action and Á-® elds

In this appendix we elucidate the symmetries of the action (17) and the Á-® elds.
The minimal form of a Gaussian action suitable for the computation of pair-
correlation functions is given by
… h i
- !‡ ph
S ˆ ¡i ¿ H0 ‡ ¼3 « ¼ar3 ¿;
2
where H is the Gorkov Hamilton operator in the Eilenberger representation and ¿
an 8-component super vector ® eld, containing ar, ph and bf components. Con-
vergence of the Gaussian integral over the bosonic components of this ® eld requires
that
-y
¿b ˆ ¼ar bf
3 « ¼ 3 ¿b : …B 1†
-
The fermionic components of the ® elds ¿ and ¿ are independent of each other.
To incorporate the full symmetry of the problem (essentially time reversal and
particle ± hole symmetry) it is necessary to double the ® eld space. Like with the
analogous ® eld integral for normal compounds, this is conveniently done by
employing the identity
…
i n -h !‡ ph i h !‡ ph i o
ar -T
S ˆ¡ ¿ H‡ ¼3 « ¼ar
3 ¿ ‡ ¿T bf
¼ 3 H T
‡ ¼ « ¼3 ¿ ;
2 2 2 3
where the factor of ¼bf
3 arises from the de® nition of the supertransposition operator.
De® ning an enlarged ® eld through
Áˆ
1
21=2 … ¿-¿ †;
T
-
2
1 -
Á ˆ 1=2 … ¿;¿T ¼bf
3 ;
we obtain
… " H #
S ˆ ¡i Á
-
… HT
† ‡
!‡ ph ar
¼ « ¼3 Á:
2 3
Noticing that H T ˆ HgjA!¡A this action is identi® ed with (17). One veri® es that the
-
® elds Á and Á ® elds are related by the symmetry
-
Á ˆ ……¼tr1 « E11
bf
‡ i¼tr2 « E22
bf
†Á†T ; …B 2†
where the matrices Eijx , x ˆ ar;ph;bf ;tr, are de® ned as …Eijx †i 0 j 0 ˆ ¯ii 0 ¯jj 0 , and the
indices i 0 ;j 0 refer to the space x.
Appendix C: Saddle points and analytic continuation

This appendix is devoted to a discussion of the question of how the Q-saddle-
point con® gurations that appear in applications with superconductivity may be
accessed from the starting point of the diagonal saddle-point con® guration,
ph
¼3 « ¼ar3 , characteristic for bulk metallic phases. In order to specify what we mean
by àccess’ , we ® rst have to summarize some facts regarding the structure of the ® eld
manifold of the ¼-model. In the polar representation of Efetov [63], a general Q-
matrix is parametrized as
Q ˆ T …¼ph ar
3 « ¼3 †T
¡1
; …C 1†
ph
where ¼3« ¼ar
3 is the generalization of the standard matrix, L ˆ ¼ar
3 ,
to applications
with a ph substructure. The rotation matrices obey T 2 G=H, where G is a group of
matrices that ful® ll various symmetry conditions, dictated by both the internal
symmetries of the model under consideration and convergence criteria. The group
H » G is determined by the condition ‰H; ¼ph ar ˆ
3 « ¼3 Š 0. The most important
constraint for the present discussion has to do with convergence and reads,
T y ˆ ²T ¡1 ²¡1 ; …C 2†
where
bf
² ˆ E11 « ¼ph ar bf
3 « ¼3 ‡ E22 :
„
The functional integration dQ extends over the coset space G=H. A key question to
address is whether or not all stationary phase points of the action are accessible
within the integration domain speci® ed by (C 1).
To analyse this issue, we ® rst focus on the simple case of a bulk superconductor
in the regime jDj ¾ °. In this case, the saddle point is unique and reads (cf.
section 5.2)
-
Q ˆ ¼ph
1 :
The above question reduces in this case to whether or not there exists a solution to
the equation,
ph ph
¼1 « 1ar ˆ T …¼3 « ¼ar
!
3 †T
¡1
…C 3†
Strictly speaking, no such solution exists. To demonstrate this point, we ® rst note
that, since the rhs of equation (C 3) is trivial in ar, bf and tr space, it is su cient to
focus on each sector of these spaces separately. Without loss of generality, we focus
on the retarded± retarded (rr) sector, where equation (C 3) takes the form,
ph ! ph
¼1 ˆ T ¼3 T ¡1 …C 4†
and we have, for reasons of notational simplicity, denoted the rr-restricted rotation
matrices again by T . The analogous equation for the advanced± advanced block
carries an overall minus sign on the right-hand side. Specializing the discussion
further to the the fermion± fermion ( ) sector, we ® nd that, in this sector, no problem
arises in ® nding a solution to equation (C 4) with the correct symmetries. When
restricted to the block, the symmetry relation (C 2) takes the form
y
T ff ˆ T ff¡1 ;
i.e. the matrices are unitary. At the same time, the restriction of equation (C 4) to
the sector reads
ph ! ph
¼1 ˆ T ff ¼3 T ff¡1 ;
which may be solved by a unitary rotation matrix. For future reference, we explicitly
write the solution as
Figure 19. Visualization of the deformed integration contour.
T ff ˆ exp …¡i³ff ¼ph

2 † 2 G=H:
³ff ˆp4
In the boson ± boson (bb) sector, the situation is more problematic: we fail to ® nd a
solution to (C 4) with the correct symmetries. The restriction of the symmetry
criterion (C 2) on the rr block reads
y
T bb ˆ ¼ph ¡1 ph
3 T bb ¼ 3 ;
which fails to include any T bb that further ful® l

ph ! ph ¡1
¼1 ˆ T bb ¼3 T bb :
The resolution of this problem is provided by analytic continuation. In the
derivation of the ¼-model, the symmetry condition (C 2) is enforced by convergence
requirements. As long as no singularities are encountered, the condition may be
relaxed, in the sense that the integration contours may be analytically continued to
regions where the symmetry criterion is no longer ful® lled. Supposing now we are
integrating over the subset (cf. ® gure 19)
fT ˆ exp …¡i³bb ¼ph
2 †j³bb 2 iR » G=H;
g
the saddle point we wish to access is reached by distorting the integration contour so
as to cross the point ³bb ˆ p=4. In lifting the integration path o the imaginary axis,
no singularities are encountered. Moreover, a closer analysis shows that, in
accordance with the basic conditions to be imposed on saddle-point integrals, the
direction of steepest descent is parallel to the imaginary axis. Analogous arguments
may be applied to the aa sector. Consequently we conclude that q does represent a
proper saddle point of the Q-integration.
The question remains as to what happens if we encounter more complicated
saddle-point con® gurations, such as those with a ® nite value of °=D or even with
spatial variation. Although no mathematical proof has been given, we expect that by
analytic continuation such con® gurations remain accessible. Independent evidence
for the validity of such an assumption is provided by physical criteria: the disordered
mean DoS may be calculated within the framework of the quasi-classical approach,
while the solution to the quasi-classical equations coincides with the solution of the
¼-model mean ® eld equations. Suppose now that the solution was inaccessible in
either block, bb or , or both. In this case the functional integral would exhibit
supersymmetry breaking on the mean ® eld level and it would be obscure how to
reproduce the correct results of quasi-classics. Given such sources of evidence, we
adopt a pragmatic point of view and take for granted the accessibility of the Usadel
saddle points.
Appendix D: Solutions to the Usadel equation

In this section we provide explicit solutions of the Usadel equation for two
simple, quasi-1D geometries: an SN junction and an SNS junction with coincident
phases of the order parameters of the S regions ( D ’ ˆ 0).
D.1. SN junction
We begin with the SN geometry. As with the case of the bulk superconductor , we
introduce an angular parametrization for q…x†:
q…x† ˆ …sin ³…x†;0;cos ³…x††: …D 1†
Since in this geometry the phase of the order parameter is spatially constant, the
gauge transformation of section 6 allows us to set the second component of q to
zero. The saddle-point equation for q, equation (47) , becomes a sine± Gordon
equation,
D 2
@ ³ ‡ i° sin ³ ‡ D …x† cos ³ ˆ 0; …D 2†
2 x
while the boundary condition at the interface, equation (49) , becomes
¼s @x ³…0¡ † ˆ ¼n @x ³…0‡ †: …D 3†
Also, the symmetry relation, equation (53), becomes
³¡ …x† ˆ p ¡ …³‡ †¤ : …D 4†
In addition, there are further boundary conditions at in® nity, at which the bulk
values of the angle are approached , so that
(
³s ; x ! ¡1;
³…x† ! …D 5†
³n ; x ! 1;
where ³s is de® ned by the equations for the bulk order parameter, equations (30),
and …³n †‡;¡ ˆ 0;p. Note that, although the conditions (D 5) takes di erent forms in
the ar sectors, the existence of the relation (D 4) means that we need solve only for
the retarded component, ³‡ , and in the following we drop the ‡ subscript.
The solution of the sine± Gordon equation, equation (D 2) , with boundary
condition (D 5) , is of the following (solitonic) form:
8 " #
> 1=2 1=2
¡1 exp ¡ 2R
>
>
>
>
>
<
³s ‡ 4 tan ……
Ds
† x† tan
³…0† ¡ ³s
4
; x < 0;
³…x† ˆ " # …D 6†
>
> 1=2
>
>
>
> 4 tan¡1 exp ¡
: … … D † x† tan ³…0†4 ;
¡2i°
n
x > 0;
where R ˆ jDj2 ¡ °2 as before.

The integration constant ³…0† is ® xed by imposing the condition (D 3) at the
interface, to give
…³…0† ¡ ³s † ° 1=2 ³…0†
sin ˆ® sin ; …D 7†
2 iR1=2 2
where ® is a parameter representing the mismatch between the two materials:
¼n =¼ s
®ˆ : …D 8†
…D n =D s †1=2
In the limit ® ! 0, the bulk value of the angle, ³s , is imposed asymptotically at the
interface, ³…0† ! ³s Ð the boundary condition becomes `rigid’ .
By equation (31), the local DoS is obtained from the relation,
¸…x† ˆ ¸n Re cos ³…x†: …D 9†
D.2. SNS junction with coincident phases

We turn now to the geometry of an SNS junction, of width L , and with
coincident phases of the order parameters in the S regions. Overall symmetry about
the origin leads to the condition,
@x ³…0† ˆ 0: …D 10†
Since an identical condition holds at a normal-insulator interface, the solution here
also applies to an SNI junction of width L =2. There are further conditions at in® nity,
³…x† ! ³s ; jxj ! 1: …D 11†
The solution of equation (D 2) for ³…x†, incorporating equations (D 10) and (D 11), is
as follows:
8
> 1=2 1=2
¡1 exp ¡ 2R
>
>
>
>
<
³ s ‡ 4 tan … …… Ds
†…jxj ¡ L =2† tan
4 †
³…L =2† ¡ ³s
†
; jxj > L =2;
³…x† ˆ
>
> 1=2
>
>
>
: 2 sin …
¡1
sin
³…0†
2
sn i …… †
¡2i°
Dn
x ‡ K sin
³…0†
2
…;sin
³…0†
2
† ††
; jxj < L =2:
…D 12†
Here K and sn are the complete elliptic integral of the ® rst kind and the Jacobi
elliptic function, respectively (see [91]). The two integration constants, ³…0† and
³…L =2†, are related by
1=2
L
sin
³…L =2†
2
ˆ sin
³…0†
2
sn i…… †
¡2i°
Dn 2
‡ K sin …
³…0†
2
;sin
³…0†
2
; † † …D 13†
and the conditions at the interfaces, corresponding to equation (D 3) , give the further
relation
1=2 1=2
…³…L =2† ¡ ³s †
sin
2
ˆ®
°
iR1=2
…sin 2 ³…L =2†
2
¡ sin 2
³…0†
2
† ; …D 14†
where the parameter ® is de® ned by equation (D 8) as before. The integration

constants may then be determined by numerical solution of equations (D 13) and
(D 14).
Appendix E: EŒective action of the Goldstone mode

The subject of this appendix is a derivation of the e ective action for the
Goldstone modes, represented by rotation matrices T 0 such that ‰T 0 ;rph Š ˆ 0. We
consider separately the cases of orthogonal and unitary symmetry.
E.1. T ime reversal invariant action

We begin with the case of orthogonal symmetry. Substituting the ansatz
-
Q ² T 0 QT 0¡1 into the e ective action, equation (36), we notice that we obtain two
T 0 dependent terms: a gradient term, and a term proportional to !‡ . Note that the
two remaining vertices, proportional to ° and D, ^ commute through and do not
couple to T 0 . Focusing on the gradient term ® rst, we obtain
… …
pD¸n pD¸n - -
¡ str…@Q@Q † ¡ ˆ str… …@ ‡ ‰T 0¡1 …@T 0 †; : Š†Q…@ ‡ ‰T 0¡1 …@T 0 †; : Š†Q
8 8
… 0 1
pD¸n - 2 - -
ˆ¡ str@… ‰T 0¡1 …@T 0 †; QŠ ‡2 ‰T 0¡1 …@T 0 †; QŠ@ Q A ‡ T 0 ¡ independent: …E 1†
8 |‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚} |‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚}
a† b†
We ® rst observe that the contribution b) vanishes, through the use of the condition
-
Q 2 ˆ 1. Turning to the a)-term, we write
- Ï @Q
Ï
a† ˆ str ‰T 0¡1 …@T 0 †; QŠ2 ˆ str … @Q ;
where @Ï is de® ned to be a derivative that acts only on T 0 . Making use of the
ph commutativity of the T 0 ’ s, we next introduce
-
Q ˆ T 0 QT 0¡1 ˆ 12D q ¢ rph Q 0 ;
where
Q 0 ² T 0 ¼ar
3 T0
¡1
and D q ² q‡ ¡ q¡ . Tracing out the ph indices, we now obtain
Ï @Q
Ï D q¢D q
a† ˆ str… @Q ˆ str0 …@Q 0 @Q 0 †:
2
Substituting this expression into equation (E 1) and using the fact that
D q ¢ D q ˆ 2…1 ¡ q‡ ¢ q¡ †, we ® nally arrive at
… …
pD¸n p¸n ~
¡ str …@Q@Q † ¡
ˆ D str0 …@Q 0 @Q 0 † …E 2†
8 4
as the gradient term of the Goldstone action. Here,
D
D~ ² …1 ¡ q‡ ¢ q¡ † …E 3†
2
plays the role of an e ective (and generally space dependent) di usion coe cient.
Note that the symmetry relation (53) implies that this di usion coe cient is real.
We next turn to the computation of the !‡ -dependent vertex:
… …
ip¸n !‡ ip¸n !‡
¡ str Q¼ph
3 « ¼ ar ˆ
3 ¡ D q3 str0 … Q 0 ¼ar
3
4 2
…
ip!‡
ˆ¡ ¸ str0 … Q 0 ¼ar
3 ; …E 4†
2
where D q3 is the 3-component of the vector D q and ¸ ˆ ¸n D q3 , the space dependent
local DoS (cf. equation (31)). Combining equations (E 2) and (53), we ® nally obtain
…
p¸n ~ ip!‡ ¸
S 0 ‰Q 0 Š ¡
ˆ D str0 …@Q 0 @Q 0 † ‡ str0 … Q 0 ¼ar
3 …E 5†
4 2
as the ® nal expression for the time reversal invariant Goldstone mode action, as
~ and
included in equation (78). We emphasize that both the di usion coe cient, D,
the local DoS, ¸, are space dependent.
E.2. Broken time reversal invariance

If the invariance under time reversal is broken by an external magnetic ® eld and/
or signi® cant phase di erences between the adjacent superconductors , rotations
operating non-trivially in tr space are frozen out. More precisely, if the total ¯ ux
threading the system, F=F0 , exceeds 1=g1=2 , symmetry breaking contributions to the
action with coupling constants greater than unity appear{. In the infrared limit,
¯ uctuations coupling to these vertices become inessential and only T matrices with
tr block-diagonal structure survive, through their commutativity with the tr
symmetry-breakin g operators:
tr tr
T 0 ˆ T 1 « E11 ‡ T 2 « E22 :
The symmetry relation (55) implies that the two blocks are related to each other by
T 1T ˆ T 2 : …E 6†
Substituting the block diagonal form into the action of equation (36), all T -
invariance breaking operators drop out. Thus, similarly to the preceding subsection,
we again arrive at equation (E 5) as an e ective Goldstone action. Exploiting the tr
block structure of the T matrices, the action may be written as the sum of two
contributions, S ˆ S 1 ‡ S 2 , where the subscript refers to the tr index. Due to
equation (E 6) and the invariance of the `str’ under matrix transposition , the two
contributions are identical and we obtain
{ The fact that 1=g1=2 is the relevant scale follows simply from gauge invariance: in the presence
of ® elds, the gradient operator appearing in (E 2) generalizes to the gauge invariant form
D@ 2 ! ¡D…i@ ¡ eA=c†2 . For a static ® eld of strength F=Area, the vector potential A ¹ F=L and the
`diamagnetic term’ is of order D…eA=c†2 ¹ Ec …F=F0 †2 . Since the basic energy unit of the action is the
- -
mean level space, d , the dimensionless coupling strength is of O……F=F0 †2 Ec =d† ˆ O……F=F0 †2 g†. For
1=2
F=F0 > g the coupling strength exceeds unity.
…h i
p¸n ~
S 0 ‰Q 0 Š ˆ ¡ D str0 …@Q 0 @Q 0 † ‡ ip!‡ ¸ str0 … Q 0 ¼ar
3 …E 7†
2
as the ® nal result for the T non-invariant Goldstone mode action, as also included in
equation (78). In equation (E 7) , we have used
-
Q 0 ˆ T 0 QT 0¡1 ;
whereÐ for the sake of a homogeneous notationÐ we have denoted T 1 again by T 0

- -
and the tr block Q 11 by Q. Note, however, that the matrix dimension of the ® elds in
(E 7) is twice as small as the one in (E 5) , as the tr index structure is missing, and that
there is no symmetry relation such as equation (55).
E.3. Boundary conditions

In order to make the gradient terms appearing in equations (E 5) and (E 7) well
de® ned, boundary conditions at all interfaces between the N region and external
regions need to be speci® ed. Whereas the boundary conditions to be imposed at
interfaces to insulators (@T ˆ 0) and idealized leads (T ˆ 1) have been derived
previously [63], the interfacial behaviour at SN boundaries has so far not been
analysed. Note that the present analysis applies to the boundary condition of the
Goldstone mode as opposed to the behaviour of a single-particle Green function (the
ar diagonal blocks of the Q matrices)Ð the latter has been summarized already in
Appendix A.
Some insight into the structure of the boundary conditions may be gained from
the fact that two types of currents across the SN boundary may be identi® ed.
(a) A potentially non-vanishing electric current. (Two elementary charges ¯ ow
across the interface whenever Andreev scattering takes place.)
(b) A quasiparticle current that vanishes (even in the case of a non-zero re¯ ection
coe cient at the interface). The quasiparticle current democratically counts
the ¯ ow of electrons and holes. Since incoming electrons are either re¯ ected
or Andreev converted into holes, the normal component of the boundary
quasiparticle current vanishes.
Taking into account the fact that the Goldstone mode does not distinguish between
particles and holes, one may anticipate that the boundary condition reads @? T 0 ˆ 0,
corresponding to zero quasiparticle current ¯ ow. We con® rm this supposition below.
First it is necessary to decide on the location of the NS boundary. Since
superconductive behaviour penetrates, in the sense of the proximity e ect, into the
normal region (and vice versa), the position of the boundary is to some extent
arbitrary and need not necessarily coincide with the material boundary, at which the
jump in D appears. However within the superconductor both the e ective di usion
constant and the DoS vanish on a scale of the order of ¹ ˆ …D=D†1=2 , which is much
smaller than the di usion length, L ° (for ° ½ D). With a type of coarse graining in
mind for which the details of variation over scales of ¹ in the S region becomes
irrelevant, we make a simple choice of the physical SN interface as the e ective one
for the Goldstone action.
To derive boundary conditions for the Goldstone mode, we employ the method
of boundary Ward identities, as previously used in [92]. As is usual with Ward
identities, the scheme is to subject the Q degrees of freedom to an in® nitesimal gauge
transformation and to exploit the fact that physical expectation values ought to be
invariant, whilst the action need not.
Speci® cally, we perform the in® nitesimal rotation
Q 0 ! exp …¡R†Q 0 exp …R† ’ Q 0 ¡ ‰R;Q 0 Š; …E 8†
where
R‡¡ …x†
R…x† ˆ …R ¡‡ …x†
†« 1 tr;bf ;ph ; …E 9†
and the matrix structure refers to the ar indices. A straightforward calculation then
shows that the e ective action (E 5) transforms as
S 0 ‰Q 0 Š ! S 0 ‰Q 0 Š ‡ ¯S N ‰Q 0 ;RŠ ‡ ¯S S=N ‰Q 0 ;RŠ;
…
ip!‡ ¸
¯S N ‰Q 0 ;RŠ ˆ p¸n D~ str0 …R@ …Q 0 @Q 0 †† ‡ str0 … R‰Q 0 ;¼ar
3 Š ;
2
…
p¸n
¯S S=N ‰Q 0 ;RŠ ˆ ¡ dS D S=N str0 ‰RQ 0 @? Q 0 Š; …E 10†
2 S=N
„
where S=N dS denotes a surface integral over the SN boundary (induced by an
integration by parts necessary to shuƒ e all derivatives from R to Q 0 ) and D S=N is the
local di usion coe cient at the boundary. In order to use (E 10) to construct a
boundary condition, we consider the functional expectation value
X ² hF…Q 0 …y††iQ0 ; …E 11†
where F may be an arbitrary function of the matrix Q 0 …y† at point y 2 N. Whereas

both the action and F…Q 0 …y†† need not be invariant under the transformation (E 8) ,
the expectation value, X, must be. Expanding the expression (E 11) to ® rst order in R
and omitting the matrix arguments in the notation, we obtain
X ! X ‡ h¯F ‡ F…¯S N ‡ ¯S S=N †i
and hence
h¯F ‡ F…¯S N ‡ ¯S S=N †i ˆ! 0:
Since the action ¯S S=N is singular at the boundary , its contribution to the above
expression must vanish individually, that is, we have to demand
* … +
F…Q 0 …y†† dSD S=N str0 ‰RQ 0 @? Q 0 Š ˆ! 0
S=N
Q0
for any function F. As R is arbitrary, this can be generally true only if

hF…Q 0 …y†† str0 ‰RQ 0 …x†@? Q 0 …x†Ši ˆ 0; 8y 2 N; x 2 S=N ; …E 12†
where R may be any matrix of the structure (E 9). In order to transform
equation (E 12) into a more practical form, by which we mean an e ective condition
to be imposed on the di erential operator governing the action, we subject both the
action and the expectation value in equation (E 12) to a perturbative expansion.
Introducing
B
Q 0 ˆ exp …W †¼ar
3 exp …¡W †; W ˆ … B- †; …E 13†
and expanding the action to lowest order in B, we obtain

…
-
S 0 ‰Q 0 Š ! str0 … BP ¡1 B ‡ ¢ ¢ ¢ ;
where P is a shorthand for the di usion type operator governing the quadratic
action. Note that for the present discussion the detailed structure of P is of no
concern. Further, choosing
ar
F…Q 0 …y†† ˆ str0 … Q 0 …y†E21 « ¼bf
3 ;
equation (E 12) takes the form

-
hF…Q 0 …y†† str0 ‰RQ 0 …x†@? Q 0 …x†Ši ! str0 …B…y†¼bf
3 † str0 …@? B…x†† -
B;B
ˆ 0; 8y 2 N; x 2 S=N ;
where h. . .iB;B- stands for a functional average with respect to the above quadratic
action. Computing the expectation value by means of Wick’ s theorem, we obtain
® nally
@? P…x;y† ˆ 0; 8y 2 N; x 2 S=N:
This is the required boundary condition. It implies that the eigenfunctions of the
operator P must be drawn from the set of functions obeying Neumann boundary
conditions at the SN interface. In order to install this condition generally, we restrict
the functional integration to the set of ® eld con® gurations, B, that obey the same
boundary condition, @? B ˆ 0. Since T ˆ exp …W †, where W is given in
equation (E 13), an alternative formulation reads @? T ˆ 0.
Appendix F: Renormalization of the minigap edge

In section 7.1.2 the role of a-type ¯ uctuations on the single-particle properties
was investigated within the framework of a perturbative expansion around the
inhomogeneous saddle-point solution of the Usadel equation. There it was shown
that, in the SNS geometry, the minigap induced by the proximity e ect is not
destroyed by quantum ¯ uctuations. However, this calculation failed to account for
the shift of the minigap edge resulting from the quantum renormalization of the
di usion constant due to mechanisms of weak localization. This phenomenon is
described below.
To take into account weak localization corrections, we apply a conventional
momentum shell renormalization group (RG) procedure to the e ective action
similar to the one detailed, e.g. in [63]. Albeit straightforward in principle, the actual
technical implementation of the RG for the superconductor problem comes with
certain di culties. The point is that, of the loop corrections integrated out in each
RG step, two di erent categories exist: `normal’ diagrams akin to the weak
localization type loops appearing in the theory of normal metals, and the above
discussed proximity type diagrams coupling to inhomogeneities of the order par-
ameter. Technically, the latter are induced by linear ® eld vertices in the e ective
action which cannot be brought under control perturbatively. To circumvent this
problem, we do not superimpose the RG on the trivial saddle point of a purely

normal problem but rather on the spatially inhomogeneous Usadel saddle point.
This procedure removes the anomalous diagrams. On the other hand, the spatial
inhomogeneity of the è ective’ vacuum gives rise to some other problems which, as
will become clear below, prevent us from renormalizing the theory down to
arbitrarily large scales. E ectively we will account for the renormalization of the
action due to quantum interference corrections on scales small in comparison with
the spatial variation of the saddle point. We do not believe that this limitation has
qualitative consequences for the outcome of the analysis.
Beginning with the parametrization de® ned in equation (60), we factorize
rotations T ˆ T > T < into fast T > and slow T < degrees of freedom. Here rotations
T > (T < ) involve spatial ¯ uctuations on scales shorter (longer) than b=L, where L
represents an ultraviolet cut-o and 0 < b < 1. Applying the parametrization
Q ˆ UQ > U¡1 ; Q > ˆ T > ¼ph

3 T> ;
¡1
U ˆ RT < ; …F 1†
where the rotation R de® nes the saddle point (60), we obtain
…
p¸n
Sˆ¡ str ‰D……@Q > †2 ‡ 4Q > @Q > ¢ U ‡ ‰U ;Q > Š2 † ‡ 4i°Q > U ¡1 ¼ph
3 UŠ; …F 2†
8
where U ˆ U ¡1 @U. Setting T > ˆ exp …W †, and expanding Q > to quadratic order in
in the generators of rotations W , the action separates into three contributions :
S ˆ S S ‡ S SF ‡ S F ; …F 3†
where, de® ning Q < ˆ U¼ph3 U

¡1 and Q¶< ˆ U ¡1 ¼ph U,
3
…
p¸n
SS ˆ ¡ str ‰D…@Q < †2 ‡ 4i°¼ph
3 Q < Š; …F 4†
8
…
S SF ˆ ¡p¸n str ‰D f‰W ;@W Š ¢ U ‡ …U ¼ph 2 ph 2
3 W † ‡ …U ¼3 † W
2
¡ @W ¢ U ‡ W …U ¼ph3 †
2
g ‡ i°…W ¼ph ¶ 2 ph ¶
3 Q < ‡ W ¼3 Q < †Š; …F 5†
…
p¸n L2
SF ˆ Dstr …@W †2 ‡ 2 W 2 : …F 6†
2 b
We now integrate over the fast ¯ uctuations, at the one-loop level, to obtain a new
e ective action, of the form S ˆ S S ‡ hS SF iF . Taking the energy cut-o L=b to be far
above Ec , it is su cient to neglect the linear terms in W . (More precisely, the ® elds
W ¯ ucutate rapidly in comparison with both the slow ® elds and the saddle-point
® elds R, implying that contributions linear in W e ectively average to zero.) We ® nd
…
p¸n
hS SF iF ˆ dr DP…r;r† str …@Q < †2 ; …F 7†
4
where P…r; r 0 † represents the di usion propagator
L2
…
2p¸n D ¡@ 2 ‡
b2
†
P…r;r 0 † ˆ ¯…r ¡ r 0 †: …F 8†
Altogether, applying the rescaling, at one-loop we obtain the renormalized action

… h i
p¸n
S0 ˆ ¡ str D eff …@Q†2 ‡ 4i°¼ph
3 Q ; …F 9†
8
where D eff ˆ D‰1 ¡ 2P…0;0†Š denotes the renormalized di usion constant. From this
result, we see that the bare di usion constant is subject to a standard weak
localization correction [77], albeit derived from purely within the particle± hole
sector.
As a result of the renormalization procedure, no new terms are generated in the
e ective action. Instead, we obtain the usual kinetic term but with a renormalized
di usion coe cient. However, as a consequence of the renormalization ,
q ¢ ¼ph
3
ˆ R¡1 ¼ph
3 R no longer represents the saddle point of the theory. Accordingly,
it is necessary to recalculate the saddle-point solution in the presence of the
renormalized di usion constant. The result is a corresponding renormalization of
the minigap edge discussed in the text.
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A DVANCES IN P HYSICS, 2000, VOL. 49, NO.

Field theory of mesoscopic ¯ uctuations in superconductor± normal-metal

ALEXANDER A LTLAND{{, B. D. Simons{ and D. TARAS-SEMCHUK{*

[Received 9 November 1998 ; accepted in revised form 8 November 1999]

* e-mail: jpdt1@ cus.cam.ac.uk

a superconducting condensate tends to induce singular behaviour in the

Whereas in normal metals high-order interference contributions to physical

Here ¹ ˆ …D=D†1=2 is the coherence length of dirty superconductors , D

2. Andreev re¯ ection and the proximity eŒect

coherence is cut o by thermal smearing e ects. The interest of such mesoscopic

Figure 1. (a ) Andreev re¯ ection of an electron at an SN interface and (b ) a typical pair of

Figure 2. The geometry of the SNS junction.

possibility of supercurrent ¯ ows through SNS sandwiches. Another important

2.1. Single-particle spectrum

…jv°L j ‡ sgn …v † D 2’† ˆ …D ¡ °2 †1=2

approaches p. In general Eg depends on D ’ in a non-sinusoidal fashion [34],

2.2. Failure of semiclassics: quantum di raction

3.1. T he Gorkov equations

3.2. Quasi-classical approximation

3.3. Dirty limit

gr;a …vF ;r† ˆ gr;a r;a

0 …r†rg0 …r†† ‡ i‰¼3 …°§ ‡ D …r††;g0 …r†Š

3.4. Solution of the Usadel equation

leading order contributions become important in cases where one is

4.1. Perturbative diagrammatic methods

4.2. Multiple scattering formalism

However, if time reversal invariance is broken, such a simple representation is no

complementary class of quantities is inaccessible through phenomenological scatter-

5. Field theory for SN systems

{ Self-consistent calculations of D…r† in mesoscopic SN junctions, as achieved analytically by

5.1. Field integral and the ensemble average

which appears in the computation of the ¯ uctuations of the DoS{,

Here ¯¸ ˆ ¸ ¡ h¸ i, ¸…°† ˆ ¡…1=2p† Im tr ph;r …¼ph r

Table 1. The signi® cance of the two-valued

index signi® cance abbreviation

complex commuting (anticommuting) components ¬ ˆ 1 …¬ ˆ 2†. The signi® cance

where ¸n denotes the DoS of a bulk N system, generates the quartic

{ We use the convention that strA ˆ trAbb ¡ trAff .

5.2. Example: a bulk superconducto r

and normalized to unity, qT§ ¢ q§ ˆ 1. To proceed it is convenient to adopt an elegant

{ Note that ¼ar

where `Vol’ is the system volume, as well as

corrections of O…E=·†, con® gurations of the type of equation (32) exhaust

5.3. Gradient expansion and `medium energy action’

(b) The momentum trace over a single Green function becomes

(c) Further, if operators A ^ and B^ vary slowly in space, then

An application of these identities to the e ective action above leads to

By using the identity

where DQ denotes the invariant measure on the manifold of matrices Q 2 ˆ 1. For

be evaluated by expanding perturbatively around the second order contribution

where h. . .iQ=W stands for functional integration in the Q- respectively W -representa-

6. Stationary phase analysis

We ® nd it convenient to formulate the stationary phase analysis in a gauge where

¼q§ @? q§ ˆ ¼q§ @? q§ ; …48†

¼q§ @? q§ ˆ ¼q§ @? q§ ; …50†

where GT is the tunnel conductance of the junction, as given by equation (A 7). By

q¡ ˆ diag …1; 1; ¡1†…q‡ †¤ : …53†

Figure 7. The geometry of the in® nite, quasi-1D SN junction.

superconducting terminal, an elimination of the phase of the order parameter

coherence is cut o by thermal smearing e ects. The interest of such mesoscopic

2.2. Failure of semiclassics: quantum di raction

An application of these identities to the e ective action above leads to

Figure 16. Renormalization of the di usion constant, D, by interference of (a ) trajectories

where D~ ˆ …D=2†…1 ¡ q‡ ¢ q¡ † plays the role of a space dependent di usion

SNS geometry at energy scales !‡ ½ Ec , the e ective action was shown to be