Study of the Kitaev Models

Project report submitted by

Arnab Barman Ray

3rd Year Int. Msc
Indian Institute Of Technology Kharagpur

May 2016 - July 2016

Supervisor: Dr.Saurabh Basu

Department of Physics
Indian Institute of Technology Guwahati 781 039, India

July, 2016

Abstract
In this project we numerically reproduce some of the results obtained by
Kitaev in two of his models- the Kitaev Chain and the Kitaev honeycomb
lattice. We see how the Kitaev Chain exhibits unpaired Majorana Modes
which gives rise to robust degenerate ground states in a certain part of the
parameter space.These ground states are such that both phase error and
classical error can be avoided thus helping in fault-tolerant quantum computation.We also numerically investigate the bulk spectrum and see how a
phase transition takes place as we move from one part of the parameter space
to another. We generate the Density of states for the Kitaev Honeycomb matrix for 450 lattice sites. We also see how phase transitions take place when
we change the spin-coupling parameters.

Contents
1 The
1.1
1.2
1.3
1.4
1.5

Kitaev Chain
The Hamiltonian And Majorana Operators
Two special cases . . . . . . . . . . . . . .
General conditions for Majorana Operators
The Bulk Spectrum . . . . . . . . . . . . .
Topology . . . . . . . . . . . . . . . . . . .

2 The
2.1
2.2
2.3

Kitaev Honeycomb
14
Lattice and the Hamiltonian . . . . . . . . . . . . . . . . . . . 14
Jordan-Wigner Transformation and related Operators . . . . . 15
The Bulk Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 16

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11
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List of Figures
1.1
1.2
1.3

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6
7

1.4
1.5
1.6
1.7

Coupling between the Majorana Modes[7] . . . . . . . . . .

DOS for the trivial(up) and topological(down) case . . . . .
Energy difference between the two ground states,(left)linear
scale,(right)logarithmic scale . . . . . . . . . . . . . . . . . .
Trivial DOS 1 & 2 . . . . . . . . . . . . . . . . . . . . . . .
Critical Point . . . . . . . . . . . . . . . . . . . . . . . . . .
Topological DOS 1 & 2 . . . . . . . . . . . . . . . . . . . . .
Spectra at different parameter values . . . . . . . . . . . . .

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12
13

2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9

The Honeycomb lattice.[9] . . . . . . .

Plaquette Operator . . . . . . . . . . .
Unit Cell[8] . . . . . . . . . . . . . . .
The Phase Diagram . . . . . . . . . . .
Gapless phase . . . . . . . . . . . . . .
Borderline gapped phase . . . . . . . .
Gapped Phase . . . . . . . . . . . . . .
Phase transitions . . . . . . . . . . . .
DOS for different coupling parameters

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Chapter 1
The Kitaev Chain
1.1

The Hamiltonian And Majorana Operators

In his paper[1], Kitaev introduced a toy model which simulates the properties
of a wire placed on the surface of a p-wave superconductor. The wire has L
fermionic sites which can be occupied or unoccupied by spinless fermions.The
jth state is given by the wavefunction (xj). In basis of these states, Kitaev
introduces the Hamiltonian, in the language of second quantization[5] as:

H=

X
j

1
(w(aj aj+1 + aj+1 aj ) (aj aj ) + aj aj+1 + aj+1 aj ) (1.1)
2

Here, aj and aj follow the usual fermionic anti-commutation relations,

{aj , ak } = 0

{aj , ak } = 0

{aj , ak } = jk

(1.2)

w is the hopping amplitude, is the chemical potential and is the

superconducting gap. We absorb the phase of the superconducting gap while
defining the Majorana Operators:
c2j1 = ei/2 aj + ei/2 aj

c2j = iei/2 aj + iei/2 aj

where is obtained from the superconducting gap , = ||ei

(1.3)

Figure 1.1: Coupling between the Majorana Modes[7]

In terms of these Majorana Operators, the hamiltonian can be rewritten
as:
H = i/2

X
[c2j1 c2j + (w + ||)c2j c2j+1 + (w + ||)c2j1 c2j+2 ] (1.4)
j

As we can clearly see from (1.2),the transformation is orthogonal and

each fermionic site gives rise to two Majorana Modes which are hermitian,
i.e ck = ck . The orthogonal transformation ensures the following properties
for the modes:
{cj , ck } = 2jk and c2j = 1
It is instructive to think of an electron to be made up of two Majorana
quasi-particles but one has to be careful because the Majorana modes themselves do not represent excitations which could physically exist because the
number operator constructed as: cj cj = c2j = 1 and hence makes no sense.
It is more correct to think in terms of combinations of majorana modes
which gives rise to fermionic excitations such as in (1.2),aj = 21 ei/2 (c2j1
ic2j ).
(1+ic2j1 c2j )
The fermionic number operator, aj aj =
. Hence,(ic2j1 c2j ) =
2
1 or 1 depending on whether the fermionic site is occupied or unoccupied.
We define the Parity operator as: P = j (ic2j1 c2j ). P commutes with
the Hamiltonian H in (1.3).The eigenstates of H hence have definite parity
i.e P=-1 or +1, depending on whether they have an odd or even number of
fermions.
A real orthogonal transformation of the Majorana operators given as c~0 =
M~c where M M T = 1, might or might not change the fermionic parity P =
(i)L j cj = det(M )(i)L j c0j of a state.

Figure 1.2: DOS for the trivial(up) and topological(down) case

1.2

Two special cases

Now we look at two special cases of the Hamiltonian:

(a)The trivial case: = w = 0 and < 0 we get the hamiltonian:
Ha =

X
j

X
i
1
c2j1 c2j
(aj aj ) = ()
2
2
j

(1.5)

So, the Majorana operators from a single site are paired together to give rise
to the original electrons.The ground state has occupation number 0.
(b)The topological case: = w > 0 and = 0.
Hb = iw

X
j

csj c2j+1

L1
X
1
= 2w
(
aj a
j )
2
j=1

(1.6)

As we can see the Majorana modes from neighbouring sites are coupled.The
operators c1 and c2l do not appear in the Hamiltonian.New fermionic modes
constructed as a
j = 12 (c2j + ic2j+1 ) are the eigenmodes of this phase.
Now, we can construct a non-local fermionic mode from the two Majorana
operators that are left out from the hamiltonian, f = 12 (c1 ic2L ) is a w
energy fermionic eigen-mode of the Hamiltonian. The ground state of (1.5)
must satisfy a
j |0 i = 0 j. Hence the parity operator P reduces to the
form, ic1 c2L . Now, both c1 and c2L commutes with Hb .(this implies that
the ground state of the Hamiltonian is an eigenstate of both the Majorana
operators).So does P . If |0 i is the ground state, it can be shown that
7

both c1 |0 i and c2L |0 i cannot be equal to |0 i.Hence, a degeneracy in the

ground state exists.Infact the two ground states |0 i and |1 i represent nonoccupation and occupation of the non-local fermionic mode f .
l1
|0 i = |0if l1
j=1 |0ij and |1 i = |1if j=1 |0ij

(1.7)

The two modes can be swapped between each other by an operation

of c1 ,c2L or f . Also, P |0 i = +1 and P |1 i = 1, representing odd and
even fermionic states.Since they involve a non-local mode, local perturbations
cannot lift the degeneracy as they cannot contain both of the two majorana
modes. Hence these states can be used in non-local storage of information.

1.3

General conditions for Majorana Operators

The Hamiltonian in (1.3) can be written in a matrix form in the Majorana

matrix,
i
(1.8)
H = C HC where C = [c1 , c2 , . . . , c2L ]
4
where the matrix H has the form:

0
(w + ||)
0
0...

0
(w + ||)
0
0
0...

0
(w + ||)
0

0
(w + ||) . . .
H=

(w ||)

0
(w
+
||)
0
.
.
.

..
..
..
..
..
..
.
.
.
.
.
.

H is a skew-symmetric matrix. This means that it has purely imaginary

eigenvalues of the form: im . We try to find solutions for one of the eigenvalues to be zero.
H can be reduced to the block-diagonal form by a real orthogonal matrix
W like:

A = W HW T

0 1
0
0...
1 0

0
0...

0
0 2 . . .
=

0 2 0 . . .

..
..
..
..
.. . .
.
.
.
.
.
.

So, now the Hamiltonian is of the form H =

8

0
m m bm bm .

(1.9)

We try to find the conditions on W so that one of the eigenvalues is zero,

Any skew-symmetric matrix H can be block-diagonalized by utilizing its
eigenvectors.
Hu~m = +im = H~u = im We define new real vectors m = i(um
um )/2 and m = (um + um )/2 = Hm = m m and Hm = m m .
The transformation matrix W is defined as[6]:
W2j1 = [jT ] and W2j = [jT ]

(1.10)

It can be shown that W is orthogonal. In his paper, Kitaev employs an

ansatz of the form:
X
X
j
0 j
0 j
b0 =
(+
x+ +
x )c2j1 , b =
(+ xj
(1.11)
+ + x )c2j1
j

Using the form of W , it can be shown in the limit

L , that the conditions

2 4w2 +4||2

, in the parameter
are perfectly satisfied with m = 0 if x =
2(w+||)
space 2w > , || =
6 0.
For finite L however, the two ground states are separated by an energy
difference of 2 eL/l0 .
This has been verified for chains upto 80 sites long in Figure 1.3. The
linear fitting gives the power of the exponential to be proportional to
0.4568, which is not very close to the theoretical value of 0.5039(an error of
9.35 %). This may be attributed to numerical inaccuracy for large matrices.
We verify the division of parameter space of the Hamiltonian into two
parts one of which exhibits Majorana Zero Modes i.e when 2w > ||. When
|2w| < ||, the Hamiltonian does not contain unpaired Majorana fermions
and zero modes are non-existent.
We start with the values = 10, w = 4 and || = 3 and serially increase
the value of w by 0.5 and plot the resulting five histograms which depict the
density of states.
As expected, there are no Majorana Zero Modes when = 10, w = 4(left)
and || = 3(right). A similar case is exhibited by = 10, w = 4.5 and
|| = 3. Both the cases are gapped.

Figure 1.3: Energy difference between the two ground states,(left)linear

scale,(right)logarithmic scale

10

Figure 1.4: Trivial DOS 1 & 2

Figure 1.5: Critical Point

At critical point, where 2w = || = 10 does exhibit Majorana zero modes
as is found, but the spectrum is gapless and hence the states are not robust.(Figure 1.5)
As expected we can clearly see the zero modes when w = 5.5 and =
10(left) as well as when w = 5.5 and = 10(right). Both these spectra are
gapped and support robust ground states(Figure 1.6).

1.4

The Bulk Spectrum

In this section, we investigate the properties of the bulk spectrum for the
chain.We write the original fermionic position operators terms of the creation
operators, ak of the momentum states 1L eikx .
1 X
aj =
hk|(x j)iak
L kF BZ
11

(1.12)

Figure 1.6: Topological DOS 1 & 2

The Hamiltonian(1.1) is then rewritten in the Bogoliubov-deGennes form
using contributions from both +k an -k operators as:

 

 2w cos k 2 i sin k
ak
(1.13)
H = a ak

2 i sin k
2w cos k + ak
Diagonalizing this we end up with the spectrum:
q
(k) = (2w cos k + )2 + 4||2 sin2 k
The diagonalized Hamiltonian is: H =
operators a
k = uk ak + vk ak and
uk =

k a
k +
k k a

(1.14)

constant, where the

k k + (2w cos k + ) 1/2

k (2w cos k + )
(
) and vk = (
)uk (1.15)
|k |
2k
k

, where k = 2 i sin k.
The spectrum is shown below for different values of the parameters. we
see the phase is gapless at the critical point: = 2w.

12

Figure 1.7: Spectra at different parameter values

1.5

Topology

13

Chapter 2
The Kitaev Honeycomb
2.1

Lattice and the Hamiltonian

In his paper[2] Kitaev introduced the Honeycomb lattice as two triangular

lattices meshed together:

Figure 2.1: The Honeycomb lattice.[9]

The two triangular lattices are respectively termed odd(blue circles) or
even(red circles).He also defines three types of links that exist in the model
as x-links,y-links and z-links as shown in the picture.
The spin Hamiltonian is of the form:
X
X
X y y
H = Jx
jx kx Jy
j k Jx
jz kz
(2.1)
xlinks

ylinks

. Jx ,Jy and Jz are the coupling parameters.

14

zlinks

2.2

Jordan-Wigner Transformation and related

Operators

Before we proceed to the fermionization of the spin-system, we first look at

a conserved quantity called the plaquette operator,Wp = 1x 2y 3z 4x 1 5y 6z ,
where the lattice points are numbered as: Clearly,[Wp , H] = 0. Hence, Wp is

Figure 2.2: Plaquette Operator

a conserved quantity of the eigenstates.
Now,we move on to the fermionization of the system. We introduce
new fermionic states for each lattice site j, a1j and a2j . The occupation
states,|1, 1i and |0, 0i represent the up and down spin states of the site respectively.The occupation states,|1, 0i and |0, 1i represent unphysical states
which we weed out later. Hence, the effect is to extend the space from 2N dimensions to 22N dimensions, where N is the number of lattice points. We define four Majorana operators for each site given be: bx = (a1j + a1j )/2 by =
(a1j a1j )/2i bz = (a2j + a2j )/2 c = (a2j a2j )/2i. They follow the usual
ant-commutation rules (1.4).
Since, we have extended the space, we need to project the eigenstates
onto the physical space. This is achieved by the operator Dj = ibx by bz c. The
(1+D )
projection operator is defined as D = j 2 j . Iff D|i = |i, |i is a
physical wavefunction.
=
To work in the extended space, new Pauli operators are defined as alpha
ib c. Within the physical space, they have all the properties of the original
pauli matrices, as indicated by Dj j = j .
Th resulting Hamiltonian is quartic in operators and has the form:
iX

H=
Ajk cj ck where Ajk = 2Jjk ujk and ujk = ibj jk bk jk
(2.2)
4
hj,ki

Since, ujk (link operator) commute with each other and the Hamiltonian, the
Hilbert space can be broken up into direct sums of Hu where u is the ordered
15

set of all values of ujk .However, since for an assumed physical eigenstate of
the link operator,
{Dj , ujk } = 0 = Dj ujk |i =
ujk Dj |i
= Dj ujk |i = ujk |i = ujk |i = ujk |i
,the subspaces are not invariant under the projection operator and hence are
unphysical. Acting D on a physical state flips the sign all ujk .Now, we can use
the plaquette operator Wp = (j,k)boundary(p) ujk . This operator commutes
with the hamiltonian and D which can be seen from the fact the action of
D flips the signs of the ujk however, since there are six of them in Wp , its
value remains unchanged. Also, since Wp2 = 1, the eigenvalues are 1 for
each plaquette.
As it turns out, the lowest energy ground state of the configuration is
when
Wp = 1 p.
(2.3)

2.3

The Bulk Spectrum

[h] We take a triangular lattice with an associated basis to describe the

honeycomb as shown: Using this basis, we write the Hamiltonian as H =

Figure 2.3: Unit Cell[8]

i
4

s,,t, As,t cs ct , where the second indices indicate positions inside the
unit cell while the first indicates the lattice site.
We fourier transform the Majorana operators in the case when(2.3) when
all ujk = 1, getting the form:

H=

iX
A, (q)aq, aq, .
4 q,s,t

16

(2.4)

Figure 2.4: The Phase Diagram

Diagonalizing the resulting matrix, we arrive at the spectrum,
(q) = 2|Jx ei(~q.n~1 ) + Jy ei(~q.n~2 ) + Jz |.
(2.5)

n1 and n2 are the basis vectors given by: (1/2, 3/2) and (1/2, 3/2).
~q is in the reciprocal space of the honeycomb lattice. The three terms of
the equation are floating vectors in the 2-D complex space.They can always
have a solution if they obey the triangle inequality and vice-versa(due to the
triangle law of addition of vectors).
The different phases of the honeycomb can be represented by a slice of
the plane Jx +Jy +Jz = 1 in the parameter space.This results in the following
triangle: Finding accurately if a spectrum is gapless from the graph of its
excitation spectrum is not possible, because even the minimum values(as
found from the vectors in matlab) only tend to zero as we decrease the stepsize. Hence, say an apparently gapped spectrum with a minimum energy
value of 0.05 might in reality be gapless in the exact limit.(Though it is
easy to see if the expression has a solution using software) We present the
excitation spectrum for a gapless phase where Jx = Jy = Jz = 10 in figure
(2.5).But, due to the limits of numerical computation, we see that the two
minima dont actually touch. Now, we give the excitation spectrum for a
borderline, gapped phase where Jx = Jy = 6 and Jz = 12.5 in Figure(2.6),
in this case too, the minima are almost as distant as in the gapless phase.

17

Figure 2.5: Gapless phase

Figure 2.6: Borderline gapped phase

However, in a gapped phase far from the gapless phase, given by Jx =
Jy = 6 and Jz = 16 in Figure (2.7), The minima are very well-separated.

Figure 2.7: Gapped Phase

We can observe the changes in the minimum energy values as we move
along the three different rays, OX,OY and OZ in the triangle(Figure 2.4).
The three graphs are shown in Figure (2.8). A marked change in the minimum values can be observed in the final graphs which indicates a phase
transition. And this occurs exactly where we would like it to be, at the point
when the parameters invalidate the triangle inequality.
The Density of states(for 450 sites) for the vortex free case is given for
different parameter values, as we go along the ray OZ in Figure(2.4) is given
in Figure (2.9). As expected, the spectrum is gapped when the couplings
dont satisfy triangle inequality and gapless otherwise.

18

Figure 2.8: Phase transitions

Figure 2.9: DOS for different coupling parameters

19

Bibliography
[1]

Unpaired Majorana Fermions in Quantum Wires, A. Kitaev,2001

[2]

Anyons in an exactly
yond,A.Kitaev,2008

[3]

Majorana Modes and Topological States in Realistic DrivenDissipative Quantum Systems,Charles Edouard Bardyn

[4]

Kitaev model on the honeycomb and diamond lattices,Ludo

Nieuwenhuizen

[5]

Introduction to Many Body Quantum Theory in Condensed

Matter Physics,Henrik Bruus et.al

[6]

http://physics.stackexchange.com/questions/261788/calculatingthe-boundary-modes-in-kitaev-chain

[7]

http://physics.stackexchange.com/questions/199647/unpairedmajoranas-in-the-kitaev-chain

[8]

Group theoretical and topological analysis of the quantum

spin Hall effect in silicene,F,Geissler et.al.

[9]

Floquet Majorana Edge Mode and Non-Abelian Anyons in

a Driven Kitaev Model,Masahiro Sato et.al.

20

Solved

Model

And

Be-