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Topological insulator and the Dirac equation

Article · September 2010

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Shun-Qing Shen, Wen-Yu Shan and Hai-Zhou Lu

Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong
(Dated: September 29, 2010)
We present a general description of topological insulators from the point of view of the Dirac
equation. The Z2 index for the Dirac equation is always zero, and thus the Dirac equation is
arXiv:1009.5502v1 [cond-mat.mes-hall] 28 Sep 2010

topologically trivial. After the quadratic Bp2 term in momentum p is introduced to correct the
band gap mv 2 of the Dirac equation (v has the dimension of speed), the Z2 index is modified as 1
for a dimensionless parameter mB > 0 and 0 for mB < 0. For a fixed B there exists a topological
quantum phase transition from a topologically trivial system to a non-trivial one system when the
sign of the band gap mv 2 changes. A series of solutions near the boundary in the modified Dirac
equation are obtained, which is characteristic of topological insulator. From the solutions of the
bound states and the Z2 index we establish an explicit relation between the Dirac equation and
topological insulators.

PACS numbers: 72.25.-b, 73.20.-r,03.65.Pm

Recent years it was discovered that a class of new ma- where mv 2 is the band gap of particle and m and v have
terials possesses a feature that their bulks are insulating dimensions of mass and speed, respectively. αi and β are
while their surfaces or edges are metallic. This metallic the Dirac matrices. In 3D spatial space, one representa-
behavior is quite robust against impurities and interac- tion of the Dirac matrices in terms of the Pauli matrices
tion, and is protected by the time reversal symmetry of σ i (i = x, y, z) is[4, 5]
the band structures. The materials with this new feature    
is called topological insulator.[1–3] The Dirac equation, 0 σi σ0 0
αi = ,β = (2)
a relativistic quantum mechanical wave one for elemen- σi 0 0 −σ 0
tary spin 1/2 particle,[4, 5] enters the field of topolog-
where σ 0 is a 2 × 2 identity matrix. The general
ical insulators in two aspects. First of all, topological
solutions of the wave functions can be expressed as
insulators possess strong spin-orbit coupling, which is a
Ψν = uv (p)ei(p·r−Ep,ν t)/~ . The dispersion relations
consequence of a non-relativistic approximation to the
of four energy bands are Ep,ν(=1,2) = −Ep,ν(=3,4) =
Dirac equation. It makes the spin, momentum and the p
Coulomb interaction or external electric fields couple to- v 2 p2 + (mv 2 − Bp2 )2 . The four-component spinors
uv (p) can be expressed as uv (p) = Suν (p = 0) with
gether, which produces very profound effect on energy
band structure in solid.[6] For example, it is possible that 
1 0 − pǫzpv − pǫ−pv

the band structures in some materials becomes topolog-
ǫp  0 1 − pǫ+pv pǫzpv 
r
ically non-trivial because of strong spin-orbit coupling. S=  pz v p− v

(3)
2Ep,1  ǫp 1 0

Another aspect is that the effective Hamiltonians to the p+ v
ǫp 
two-dimensional (2D) and three dimensional (3D) topo- ǫp − pǫzpv 0 1
logical insulators have the identical mathematical struc-
where p± = px ±ipy , ǫp = Ep,1 + mv 2 − Bp2 , and uν (0)


ture of the Dirac equation.[7–9] In these effective models
the equations are used to describe the coupling between is one of the four eigen states of β.
electrons in the conduction and valence bands, not be- The topological properties of the modified Dirac equa-
tween the electron and positron as in the original Dirac’s tion can be gained from these solutions of a free particle.
theory. The positive and negative spectra are for the The Dirac equation is invariant under the time reversal
electrons and holes in semiconductors. symmetry, and can be classified according to the Z2 topo-
In this paper we start with the modified Dirac equa- logical classification following Kane and Mele.[10]. In the
tion to provide a unified description for a large family representation for the Dirac matrices in Eq. (2), the time-
of topological insulators. A series of solutions from 1D reversal operator here is defined as[11] Θ ≡ −iαx αz K,
to 3D are presented to demonstrate the existence of the where K the complex conjugate operator that forms the
end, edge and surface states in topological insulators. complex conjugate of any coefficient that multiplies a
We start with a modified Dirac Hamiltonian by intro- ket or wave function (and stands on the right of K).
ducing a quadratic correction −Bp2 in momentum p to Under the time reversal operation, the modified Dirac
the band gap or rest-energy term, equation remains invariant, ΘH(p)Θ−1 = H(−p). Fur-
thermore we have the relations that Θu1 (p) = −iu2 (−p)
and Θu2 (p) = +iu1 (−p), which satisfy the relation of
H = vp · α + mv 2 − Bp2 β. Θ2 = −1. Similarly, Θu3 (p) = −iu4 (−p) and Θu4 (p) =


(1)
 2

+iu3 (−p). Thus the solutions of {u1 (p), u2 (−p)} and at x = 0. The stationary Dirac equation has the form
{u3 (p), u4 (−p)} are two degenerate Kramer pairs of pos-
mv 2 + B~2 ∂x2
    
itive and negative energies, respectively. The matrix of −iv~∂x ϕ ϕ
=E .
overlap {huµ (p)| Θ |uν (p)i} has the form −iv~∂x −mv 2 − B~2 ∂x2 χ χ
(6)
2 2
p− v
We may have a series of extended solutions which spread
i mvE−Bp i Epzp,1
v
 
0 p,1
−i E p,1 in the whole space, but are not what we are interested
2 2
p+ v
 −i mvE−Bp 0 i Epzp,1
v
iE here. To find solutions of bound states near the end of the
 

 p,1
2
p,1
2
. chain, we also requires that the wave function vanishes
p− v
 iE −i Epzp,1
v
0 i mvE−Bp 
 p,1
2 2
p,1
 at x = +∞. Following Zhou et al,[14] we take the trial
p+ v
−i Epzp,1
v
−i E p,1
−i mvE−Bp
p,1
0 wave function ϕ, χ ∝ e−x/ξ . The secular equation and
(4) boundary conditions give a solution of the bound state
For the two negative energy bands u3 (p) and u4 (p), the with zero energy
submatrix of overlap can be expressed in terms of a single    
number as ǫµν P (p), ϕ C sgn(B)
= √ (e−x/ξ+ − e−x/ξ− ) (7)
χ 2 i
mv 2 − Bp2 √
where ξ −1 v


P (p) = i p . (5) ± = 2|B|~ 1 ± 1 − 4mB and C =
(mv 2 − Bp2 )2 + v 2 p2 p  
2(ξ + + ξ − )/ ξ − − ξ +  is the normalization constant.
The main feature of this solution is that the wave func-
which is the Pfaffian for the 2 × 2 matrix. According to tion dominantly distributes near the boundary. ξ − > ξ +
Kane and Mele,[10] the even or odd number of the ze- and decides the spatial distribution of the wave func-
ros in P (p) defines the Z2 topological invariant. Here we tion. These two length scales characterize the bound
want to emphasize that the sign of a dimensionless pa- state. When B → 0, ξ + → |B| ~/v and ξ − = ~/mv.
rameter mB will determine the Z2 invariant of the mod- In this limit we notice that ξ + approaches to zero while
ified Dirac equation. Since P (p) is always non-zero for ξ − becomes a finite constant in this limit. In the four-
mB ≤ 0 and there exists no zero in the Pfaffian, we con- component form to Eq.(1), the two degenerate solutions
clude immediately that the modified Dirac Hamiltonian have the forms,
for mB ≤ 0 including the conventional Dirac Hamilto-  
nian (B = 0) is topologically trivial. sgn(B)
C  0  −x/ξ
+ − e−x/ξ − )
For mB > 0 the case is different. At p = 0 and Ψ1 = √   (e (8a)
2 0 
p = +∞, P (0) = isgn(m) and P (+∞) = −isgn(B).
i
In this case P (p) = 0 at p2 = mv 2 /B. p = 0 is always  
one of the time reversal invariant momenta (TRIM). As 0
a result of an isotropic model in the momentum space, C  sgn(B)   (e−x/ξ+ − e−x/ξ− ) (8b)
Ψ2 = √ 
we can think all points of p = +∞ shrink into one point 2 i 
if we think the continuous model as a limit of the lattice 0
model by taking the lattice space a → 0 and the recip-
rocal lattice vector G = 2π/a → +∞. In this sense This set of solutions can be used as a basis to study the
as a limit of a square lattice other three TRIM have higher dimensional equation.
P (0, G/2) = P (G/2, 0) = P (G/2, G/2) = P (+∞) which In 2D, the equation is also decoupled into two
has an opposite sign of P (0) if mB > 0. Similarly for independent
equations, H2D,± = vpx σ x ± vpy σ y +
a cubic lattice P (p) of other seven TRIM have opposite mv 2 − Bp2 σ z . These two subsets of equations breaks
sign of P (0). Following Fu, Kane and Mele[12, 13], we the ”time” reversal symmetry under the transformation
conclude that the modified Dirac Hamiltonian is topolog- of σ i → −σ i and pi → −pi in the subspace. The pos-
ically non-trivial only if mB > 0. itive and negative sign before vpy σ y determine the dif-
ferent chirality in H2D,± . In this case, the Chern num-
In the following we shall focus on the topologically non- ber or Thouless-Kohmoto-Nightingale-Nijs integer can be
trivial case with mB > 0, and present a series of topo- used to characterize whether the system in the subspace
logical solutions of edge or surface states in the Dirac is topologically trivial or non-trivial.[15] For these two
equation with an open boundary. The existence of these equations the Chern number has the form [16, 17]
edge and surface states also further demonstrates that
the modified Dirac model can describe a family of topo- n± = ±(sgn(m) + sgn(B))/2. (9)
logical insulators very well. In 1D, the equation can be
decoupled into two sets of independent
equations in the which gives the Hall conductance σ ± = n± e2 /h. When
form H1D (x) = vpx σ x + mv 2 − Bp2x σ z . For a semi- m and B have the same sign, n± becomes ±1, and the
infinite chain, we consider an open boundary condition systems are topologically non-trivial. But if m and B
 3

have different signs, n± = 0. Strictly speaking, the Chern Under a unitary transformation, Φ1 = √1 (|Ψ1 i − i |Ψ2 i)
2
number for the modified Dirac equation in Eq. (1) is al- and Φ2 = −i
√ (|Ψ1 i + i |Ψ2 i),
we can have a gapless Dirac
2
ways equal to zero although they are not zero in the sub- equation for the surface states
spaces of H2D,± , which reflects the ”spin” Chern number
is topologically non-trivial.[18]
To find a solution of an edge state in 2D, we consider Hef f = vsgn(B)(py σ y + pz σ z ). (14)
a semi-infinite plane with the boundary at x = 0. py is a
good quantum number. At py = 0, the 2D equation has The dispersion relations become Epy = ±vpy . In this way
the same form as the 1D equation. The x dependent part we have an effective model for a single Dirac cone of the
of the solutions of the bound states has the identical form surface states.
as in 1D. Thus we use the two 1D solutions {Ψ1 , Ψ2 } as To solve the 3D equation explicitly, the exact solutions
the basis. The y dependent part ∆H2D = vpy αy − Bp2y β of the surface states with the boundary are
is regarded as the perturbation to the 1D Hamiltonian.
In this way, we have a 1D effective model for the helical Ψ± = CΨ0± (e−x/ξ+ − e−x/ξ− ) exp[+i (py y + pz z) /~]
edge states (15)
  where
|Ψ1 i
Hef f = (hΨ1 | , hΨ2 |)∆H2D = vpy sgn(B)σ z
cos 2θ sgn(B) sin θ2 sgn(B)
   
|Ψ2 i
(10)  −i sin θ sgn(B)  0 θ
Ψ0+ =   ; Ψ =  i cos 2 sgn(B)
 
2 
The dispersion relations for the bound states at the θ − θ
 sin 2   − cos 2 
boundary are ǫp = ±vpy . Electrons will have positive i cos θ2 i sin 2θ
and negative velocity in two different states, respectively, (16)
and form form a pair of helical edge states. Thus the 2D with the dispersion relation E ± = ±vpsgn(B) and p =
equation can describe a system of quantum spin Hall sys-
q
p2y + p2z . The penetration depth becomes p dependent,
tem. In the condition of mB > 0, the exact solutions of
the edge states to this 2D equation have the similar form v  
ξ −1
p
of 1D ± = 1 ± 1 − 4mB + 4B 2 p2 /~2 . (17)
  2 |B| ~
sgn(B)
C  0  −x/ξ Again the existence condition of the solution is in a good
Ψ1 = √   (e + − e−x/ξ− )e+ipy y/~
(11a)
0 agreement with the topologically non-trivial condition
2  
i based on the analysis of the Pfaffian.
  Now we address relevance of these solutions to real
0 materials. The first example is the complex p-wave spin-
C  sgn(B)   (e−x/ξ+ − e−x/ξ− )e+ipy y/~ less superconductor, which has two topologically dis-
Ψ2 = √  (11b)
2 i 
tinct phases, i.e., weak and strong pairing phases.[7] The
0 weak pairing phase is identical to the Read-Moore quan-
with the dispersion relation Epy = ±vpy sgn(B)σ z . The tum Hall states. The system can be described by the
penetration depth becomes py dependent, Bogoliubov-de Gennes equation becomes

ξ k −∆·k
    
v  q  ∂ uk uk
ξ −1
± = 1 ± 1 − 4mB + 4B 2 p2y /v 2 . (12) i~ = (18)
2 |B| ~ ∂t vk −∆k −ξ k vk

The existence condition of edge state solution is in agree- where the spinor of (uk , vk ) are the parameters in
ment with the topologically non-trivial condition. This
Q
the BCS variational ground state, |Ωi = k k +
(u
reflects the relation of the Chern number of the number † †
vk ck c−k ) |0i, where |0i is the vacuum state. For com-
of edge state solution.[19, 20] plex p-wave pairing, we take ∆k at small k
In 3D, we consider an y-z plane at x = 0. We can
have an effective model for the surface states by means k2
∆k = ∆(kx − iky ); ξ k = −µ (19)
of the 1D solutions of the bound states. Consider py 2m
and pz dependent part as a perturbation to 1D H1D (x),
∆H3D = vpy αy + vpz αz − B(p2y + p2z )β.The solutions at In this way the Bogoliubov-de Gennes equation has the
py = pz = 0 are identical to the two 1D solutions. A exact form of 2D modified Dirac equation as in H2D,± .
straightforward calculation as in the 2D case gives For a positive chemical potential µ < 0, this is a weak
  pairing phase. Its Chern number is 1, and the phase
|Ψ1 i is topologically non-trivial. Thus we can find the edge
Hef f = (hΨ1 | , hΨ2 |)∆H3D = vsgn(B)(p × σ)x .
|Ψ2 i state solution near the boundary of a complex p-wave
(13) superconductor. Here we want to emphasize that the
 4
2
k
kinetic energy 2m plays a role as a quadratic correction cal edge states near the boundary; (3) in 3D, there exists
to the band gap of Dirac equation. the solution of the surface states near the surface; (4)
The second example is the quantum spin Hall sys- in higher dimensions, there always exists the solution of
tem. Haldane proposed a spinless fermion model for higher dimension surface.
IQHE without Landau levels, in which an effective Hamil-
tonian with the same form of 2D Dirac equation was In short we conclude that the modified Dirac equation
obtained.[21] The Haldane’s model was generalized to can provide a description of a large families of topological
the graphene lattice model of spin 1/2 electrons, which insulators from one to higher dimension.
exhibits quantum spin Hall effect.[22] Bernevig, Hughes
I would like to thank Jian Li, Victor Law and T. K. Ng
and Zhang predicted that quantum spin Hall effect can
for discussions. This work was supported by the Research
be realized in HgTe/CdTe quantum well and proposed
Grant Council of Hong Kong under Grant No.: HKU
an effective model by introducing a quadratic correc-
7051/10P and HKUST3/CRF/09.
tion to the band gap, which is identical to the 2D
case in Eq.(1) besides an kinetic term breaking the
particle-hole symmetry.[8] The prediction was confirmed
by measuring the non-zero conductance in the band gap
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