TWO DIMENSIONAL TOPOLOGICAL SYSTEMS: NOVEL
PROPERTIES
By
XUELE LIU
Bachelor of Arts/Science in Applied Physics
Wuhan University
Wuhan, Hubei, China
2000
Master of Arts/Science in Theoretical Physics
Beijing University
Beijing, China
2007
Submitted to the Faculty of the
Graduate College of
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
DOCTOR OF PHILOSOPHY
July, 2012
COPYRIGHT c⃝
By
XUELE LIU
July, 2012
TWO DIMENSIONAL TOPOLOGICAL SYSTEMS: NOVEL
PROPERTIES
Dissertation Approved:
Dr. Xincheng Xie
Dissertation Advisor
Dr. Weili Zhang
Dr. John W. Mintmire
Dr. Yin Guo
Dr. Sheryl Tucker
Dean of the Graduate College
iii
TABLE OF CONTENTS
Chapter Page
1 Introduction: Topology in Condense Matter 1
1.1 Topology in Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . 5
1.2 How to Get topology: from Spin Hall Effect to Quantum Spin Hall Effect 9
1.2.1 Rashba Spin-Orbit Interaction . . . . . . . . . . . . . . . . . . 10
1.2.2 Spin Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.3 Quantum Spin Hall Effect . . . . . . . . . . . . . . . . . . . . 15
1.3 Understanding Topology: Physics v.s. Mathematics . . . . . . . . . . 18
1.3.1 Physics: Berry Phase and Hall Conductance . . . . . . . . . . 18
1.3.2 Mathematics: Genus and Winding Number . . . . . . . . . . . 22
1.3.3 Topological Protection . . . . . . . . . . . . . . . . . . . . . . 25
2 Transport Property of Mesosystem 28
2.1 Numerical Calculation of Landau-B¨uttiker Formula . . . . . . . . . . 29
2.1.1 Analysis of the transmission coefficients . . . . . . . . . . . . . 30
2.1.2 Method to the Get Surface Part of Gc . . . . . . . . . . . . . 35
2.1.3 Schedule to Calculate Transmission Coefficients . . . . . . . . 37
2.2 About the Lattice Hamiltonian and the Fourier Transformation . . . 39
2.2.1 Get the Lattice Hamiltonian: Finite Difference Formulation . 40
2.2.2 Get the Hamiltonian of Layers . . . . . . . . . . . . . . . . . . 43
2.2.3 Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . 47
2.3 Application: Spin Nernst effect in the Absence of a Magnetic Field . 51
2.3.1 Motivation and Background . . . . . . . . . . . . . . . . . . . 51
iv
2.3.2 Research Methods and the Definition of Thermal Quantities . 53
2.3.3 Numerical Results and Analysis . . . . . . . . . . . . . . . . . 57
3 The Topological System with a Twisting Edge Band: Position-Dependent
Hall Resistance 63
3.1 Motivation and Background . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Model and Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3 Twisting Edge Band . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4 The Translational Invariance Symmetry Breaking of the Hall Resistance 68
4 Proximity Effect and Majarana Fermion 74
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2.1 A Generalized Haldane Model in Square Lattice . . . . . . . . 77
4.2.2 Winding Numbers . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3 Ground states and anyonic excitations . . . . . . . . . . . . . . . . . 80
4.3.1 The (1,1,-1)-state and abelian anyons . . . . . . . . . . . . . . 80
4.3.2 Weak-strong pairing phase transition and non-abelian anyons 82
4.3.3 Insulator-QAHE transition and non-abelian anyons . . . . . . 83
4.4 Edge states and Majorana fermion modes . . . . . . . . . . . . . . . . 84
4.5 Phase Diagram and Conclusions . . . . . . . . . . . . . . . . . . . . . 86
5 Summary 87
BIBLIOGRAPHY 90
A Method to Calculate the Surface Green Function of leads 96
A.1 Single layer system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
A.1.1 Inverse method (Phys. Rev. B 23, 4997(1981); J. Chem. Phys.
120, 7733 (2004)): . . . . . . . . . . . . . . . . . . . . . . . . . 96
v
A.1.2 Iteration method I (J. Phys. F 14, 1205 (1984)): . . . . . . . . 97
A.1.3 Iteration method II (J. Phys. F 15, 851 (1985)): . . . . . . . . 98
A.2 Multilayer system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
A.2.1 Inverse method for multilayer system . . . . . . . . . . . . . . 100
A.2.2 Iteration methods for multilayer system . . . . . . . . . . . . . 101
vi
LIST OF TABLES
Table Page
vii
LIST OF FIGURES
Figure Page
1.1 Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Quantization of Hall Conductance[15, 16] . . . . . . . . . . . . . . . . 6
1.3 Disorder Effects v.s. Edge states[16] . . . . . . . . . . . . . . . . . . . 7
1.4 Spin Hall effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Potential of 2D System . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Spin Orbital Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 Spin Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.8 Quantum Spin Hall effect, from [27] . . . . . . . . . . . . . . . . . . . 16
1.9 Compare of edge states, from [27] . . . . . . . . . . . . . . . . . . . . 17
1.10 the genus of different 2D surface . . . . . . . . . . . . . . . . . . . . . 22
1.11 the winding number around a given point . . . . . . . . . . . . . . . . 23
2.1 Deal the system by layers . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 The block structure of hamiltonian . . . . . . . . . . . . . . . . . . . 34
2.3 Nernst Effect and Spin Nernst Effect. Red dot denotes hot source and
blue dot denotes cold source . . . . . . . . . . . . . . . . . . . . . . . 52
2.4 Schematic diagram of the four-terminal cross-bar sample. The area
with SOI is marked by gray. A thermal gradient ΔT is applied between
the longitudinal lead-1 and lead-3. . . . . . . . . . . . . . . . . . . . . 54
viii
2.5 Ns (red solid) and S (black dotted) vs. Fermi energy EF for different
Rashba VR. The (scaled) transmission coefficient T1,2 + T1,3 (thin blue
dashed) and spin transmission coefficient ΔT2,3 (thin blue solid) are
also shown. The other parameters are T = 0.01, and L = 19a. . . . . 58
2.6 (a) A simple model: Current J because of voltage gradient(red dashed)
and thermal power S because of thermal gradient (blue dotted) vs.
Fermi energy EF at a two-lead system with Rashba VR = 0. The
(scaled) transmission function T1,3 is also shown (black solid). The
plot in the small box shows fL − fR with temperature difference. (b)
and (c): the eigen energy of the lead En,ky v.s. longitudinal wave vector
ky (units: 1/2a) for different VR . . . . . . . . . . . . . . . . . . . . . 59
2.7 (Ns (red solid) and S (black dashed) vs. Rashba SOI VR for fermi
energy EF = −3. For compare, the blue dotted line shows the seebeck
when lead-2,4 have the same VR as lead-1,3. The other parameters are
T = 0.01, and L = 19a. . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.8 (a). Ns vs. the strength of disorder W for different Fermi level EF =
−3.8(solid blue), EF = −3.6(dashed red), EF = −3.0(dotted black),
EF = −2.2(dotted dashed green); (b). Ns vs. Fermi energy EF = −3.8
for W = 0 (dashed black) and W = 1.4 (solid black).Other parameters
are VR = 0.05, T = 0.01, and L = 19a. . . . . . . . . . . . . . . . . . 61
ix
3.1 (Color online) (A) The energy band structures of zigzag-edge ribbon of
topological system, with the ribbon width W = 50a and a =
√
2a0. We
choose the parameters tab = 10 and ta1 = ta2 = tc +0.1, tb1 = −tc −ts,
tb2 = tc for all the subplots. (A1) indirect semi-metal with tc = 1.4
and ts = −0.4, (A2) the twisting edge band system with tc = 1.4 and
ts = 0.4, and (A3) the normal topological system with tc = 0.7 and
ts = 1. (B) The distribution |ψ|2 of the four edge states of (A2). (C)
The lattice structure of the system. (D) The schematic diagram of the
four edge states of (A2). (E) The twisting edge band of (A2) can be
treated as mix of the topological protected and unprotected systems. 66
3.2 (Color online) For the three sets of parameters used in Fig.??(A), the
corresponding resistances of the system v.s. the Fermi energy: (a) the
longitudinal resistances and (b) the Hall resistances. The wide lines are
for R6,5 and R2,6, the narrow lines are for R2,3 and R3,5. In both figures,
the pair of lines with the broadest quantized plateau (−2 ∼ 2) are for
Fig.??(A3); the pair of lines only have plateau within −0.8 ∼ 0.8 are
for Fig.??(A2); for Fig.??(A1), the pair of lines have no plateau. Other
parameters used for the calculation: the ribbon width W = 50a, the
distance between vertical leads L = 20a. (c) is the schematic diagram
of the 6-lead measurement we used for (a) and (b). . . . . . . . . . . 69
3.3 (Color online) For the parameters used in Fig.??(A2), the resistances
v.s. Fermi energy for sample widthes W = 50a (the broadest black
line), 60a (the red line), and 80a (the thinnest green line). . . . . . . 71
x
3.4 (Color online) (A). For the parameters used in Fig.??(A2), the 4-lead
measurement of Hall resistances v.s. the position to measure at EF = 0.
From top to bottom, the blue, red, green, and black lines are for the
disorder strength Dis = 0, Δ/8, Δ/4, and Δ/2, respectively. Here
the gap is Δ = 0.16|tab|. The results are calculated with the width of
sample W = 70a, by the average of 700 disorder configurations. (B).
The schematic diagram of another method to realize twisting edge bands. 72
4.1 The square lattice model for Hamiltonian (1) Left: The two-sublattice
and hopping. Right: The flux distribution. Hopping along arrowed
vertical links generates a phase π and arrowed horizontal links a phase
π/2. A net flux of −2π (π/2) is accumulated for the dark (light) grey
triangular blocks. The rest of the hopping is real. . . . . . . . . . . . 78
4.2 : The frames and frame change from M < 4t′ to M > 4t′. (a)
(−2tqy, 2tqx,−M − 4t′) at (0, 0). The frame is not changed and is
the right. (b) (−2tqy,−2qx,−M +4t′) at (π, 0). The frame is changed
from the right for M < 4t′ to the left for M > 4t′. . . . . . . . . . . 80
4.3 (color online) The energy dispersion E along kx showing the evolution
of the edge states with proximity induced pairing Δ. Top panel starts
with (1a) the QAHE state (M < 4t′) while the bottom panel starts
with (1b) the insulating state (M > 4t′) when Δ = 0. The gap closes
at Δ = Δ0 (1b and 2b). The chiral edge states located at the y = 0
and y = L boundaries are marked by red dashed lines and, in the case
of degeneracy, the blue solid lines. t = 4t′. . . . . . . . . . . . . . . . 84
4.4 The phase diagram with the Chern numbers, the ground state wave
functions and the gapless edge modes. . . . . . . . . . . . . . . . . . . 85
xi
CHAPTER 1
Introduction: Topology in Condense Matter
In 1972, the famous paper, more is di erent [1] was published. In this paper, Dr.
P.W. Anderson pointed out that the behaviors of complex physical systems could not
be simply understood in terms of elementary particles and the laws governing them.
The subtitle of this paper — ’Broken symmetry and the nature of the hierarchical
structures of science’ — gives the main theme of condensed matter physics in the last
century: discovering and classifying the distinctive phases of matter. The Landau’s
theory [2] was proved to be a powerful tool, which characterizes different states of
matters by the principle of spontaneous breaking of symmetry. In such a theory, the
symmetry breaking can be described by a order parameter, which is only nonzero at
the symmetry breaking state. For example, the superconductor can be described by
using the number of cooper pairs as the order parameter.
In 1980, the quantum hall (QH) effects [3] was experimentally observed. The
theoretical analysis found that the states responsible for QH effects does not break
any symmetry, thus it can not be described by the theory of spontaneous symmetry
breaking. The later study [4, 5] showed that the QH effect should be characterized
by a new paradigm - the topological order. In the QH effect, the electric currents
only go along the edge of the sample; the bulk of the system is in a gap of band
structure, thus is insulated. Because the current only flows on the edge, dissipation
is avoided and a quantized hall conductance is produced. These phenomena give the
fundamental characterization of topological phases: a number of gapless boundary
states and the quantized value of hall conductance.
1
The QH effect is so attracting that many great works have been done by the fol-lowing
years. In 1998, the Nobel Prize in Physics was awarded jointly to Robert B.
Laughlin, Horst L. St¨ormer and Daniel C. Tsui for their discovery of fractional quan-tum
hall effect - ”a new form of quantum fluid with fractionally charged excitations”.
Besides the work on the QH effect itself, Some other scientists were interested by
the topological phases [6]. A general question is, can we find the topological phases in
the system other than QH system? In 1988, Dr. F. D. M. Haldane firstly presented
a two band model of quantum anomalous hall (QAH) effect [7], in which the Landau
level is not needed. The QH effect and QAHE belong to the same topological class
that the time-reversal (TR) symmetry is broken. There are always pure electronic
currents at the edge of this class of materials. Mathematically, they are characterized
by the rst chern number. No new breakthrough was made in the following ten years
until a new topological class is observed. This new topological class conserves the
TR symmetry, in which the spin-orbital interaction plays the key role [8, 9]. In such
a system, the states of two opposite spin counterpropagate at each edge, these so-called
gapless ’helical’ edge states give zero electric current in total at every edge.
Such a topological class is called topological insulator. The topological insulator is
mathematically characterized by a Z2 topological invariant [10], i.e. the second chern
number.
The non-trivial topology was also shown in px +ipy superconductors theoretically
[78]. The zero energy majorana fermions is predicted in such a system with gapless
excitation. Majorana fermion is the particle that is it’s own anti-particle, which
is characterized by non-abelian statistics. If it is observed, zero energy majorana
fermion may greatly advance the quantum computation applications. However the
px + ipy superconductor materials is hard to realized. Many efforts are then focused
on finding other materials instead. It is found that the topological superconductor
can be produced by the proximity effect between a topological insulator and a normal
2
s-wave superconductor [74].
In this thesis, I will discuss some novel properties of 2D topological systems. For
simplicity, I forces on TR symmetry broken system, i.e. the first chern number system.
The outline of the thesis is as follows. In this chapter, I will discuss the connection
between the mathematical concepts of topology and the physical applications. The
topological system can be understood within the band theory, which can be exper-imentally
researched by the transport properties of the sample. In chapter 2, I will
show how to numerically calculate transport properties by Landau-B¨uttiker formula,
with the introduce of lattice green function. By using that, we can calculate the
current or voltage of each lead that connect to the sample, get the hall conductance.
In the end of chapter 2, as an application, we use this method to calculate the spin
nernst effect in the absence of a magnetic field. In chapter 3, I will discuss the topo-logical
system with a curved edge band. Specifically, we study a ν = 1 topological
system with one twisting edge-state band and one normal edge-state band. In that
system, I show that it consists of both topologically protected and unprotected edge
states, and as a consequence, its Hall resistance depends on the location where the
Hall measurement is done even for a translationally invariant system. This unique
property is absent in a normal topological insulator. In chapter 4, I study the QAH
effect described by a class of two-component Haldane models on square lattices. We
show that the proximity effect can also drive a conventional insulator into a QAH
state with a Majorana edge mode and the non-abelian vortex excitations.
In the rest of this chapter, I will introduce the topology in physics by the fol-lowing
ways. In section 1, the direct feeling of the topology phase will be given by
discussing the quantum hall effect. In section 2, spin hall effect (normal system)
and the quantum spin hall effect (topological non-trivial system) are introduced. By
comparing their hamiltonian, the possible condition to get a topology non-trivial two-band
system is analyzed. In section 3, I will show how to understand topology in
3
physics: I will give the definition of Berry phase, and discuss it’s relationship to the
Hall conductance; then I will analyze the relationship between berry phase and the
mathematical topological terms: genus and winding number; Lastly, the topological
protection will be discussed. As the end, I discuss the physical quantities related to
topological invariant — the edge states and the hall conductance — at the system
have the boundary.
4
1.1 Topology in Quantum Hall Effect
Before Quantum Hall effect, we have class hall effect. The classic Hall effect was
discovered in 1879 by Edwin Herbert Hall, in his working on the doctoral degree
at the Johns Hopkins University [13]. Charge accumulations, the property of
Hall Effect was firstly found at that time. It was found that, with a magnetic field
perpendicular to the 2D system, a transverse voltage difference (the Hall voltage)
is induced by the longitudinal electric current (Fig.1.1). The theory for this classic
phenomena is simple: Because of the existence of perpendicular magnetic field, the
charge carrier experience the Lorenz force, which is perpendicular to their motion and
inside the plane. This force curves the charge’s path so that the charge accumulates
at the edge of the 2D sample.
Figure 1.1: Hall Effect
The quantum hall effect was not noticed until one hundred year later. In 1975, the
quantization of the Hall conductance GH was predicted by Ando, Matsumoto,
and Uemura [14], who even did not believe this themselves. Five years later, 1980,
Klaus von Klitzing, who worked on the MOSFETs sample, firstly discovered that
the Hall conductivity was exactly quantized [3]. For this finding, von Klitzing was
awarded the 1985 Nobel Prize in Physics.
The Hall resistance RH (≡ 1/GH) is defined as transverse voltage divided by the
5
Figure 1.2: Quantization of Hall Conductance[15, 16]
longitudinal current (Fig.1.2a). In the quantum hall effect, it has the form:
RH =
VH
I
=
h
ne2 (1.1)
This can be explained by the ballistic transport theory. When the length of the
conductor is much shorter than the mean free path, the conductor is ballistic [17].
Then the current at energy E is carried by numbers of subbands which is cut by
energy E. Each subband can carry current I = −e(μ+ −μ−)/h. Here μ+ and μ− are
the chemical potentials of left lead and right lead contact to the sample (Fig.1.2b);
Obviously μ+ −μ− = eV , where V is the bias voltage applied on the two leads. Thus
we can get I = e2/h for one subband and I = ne2V/h for all the subbands crossing
energy E. In quantum hall effect, because the positive charge moves to the upside
and the negative charge moves to the downside, the upside should have the chemical
potential as μ+ and downside as μ−. thus we can get I = ne2VH/h and the quantized
hall conductance RH.
It should be mentioned that the bias voltage V is only applied on the two leads;
6
when the ballistic transport happens, the transmission is perfect thus the transmission
coefficienttn = 1. Because there is no voltage change at the edge of the sample along
the transport direction, VL in Fig.1.2a should be zero, we should always get R = 0.
Figure 1.3: Disorder Effects v.s. Edge states[16]
However, in Fig.1.2c, while the quantized plateau of RH is changed, we can clearly
see the non-zero peaks of R. To understand it, we need step a little more inside.
The exact solution of an infinite quantum hall system (i.e. a strong magnetic field
perpendicular to the infinite 2D system) is a set of quantized landau levels. When
there exists boundary, the flat landau band is curved, as shown in Fig.1.3a. In that
way, the bulk states of the QH effects are gaped; however in the edge there always
exists states. The number of edge states at the fermi level is same as the number of
filled bulk states. In the ballistic transport theory, I = ne2VH/h, number n is totally
determined by the states at the fermi surface. Thus we can clearly see that the
quantized hall resistance RH are totally determined by the number of edge
states. This number is the so called topological number, the quantity to characterized
the topological phases. In the following of this chapter, We will show the relationship
between the hall resistance (as can be seen later, it exists at all non-trivial first chern
7
number topological system) and the topological quantity. When the fermi surface
passes through the bulk landau level, the bulk states can also afford the transport
channel. As the results, the charges are no more localized on the edge, the current
can directly transport from left to right through the bulk of the material. Thus the
bulk channel can give a nonzero longitudinal resistance R. It is easy to see the when
the fermi surface passes through the bulk landau level, the number of edge states are
also changing. This is the reason the nonzero R only happens when RH is changed.
The existence of gapless edge states at the gapped bulk states makes
another important property of quantum hall effect: it’s difficult to be affected by
the impurities. With the existence of impurity, the bulk landau level is broaden
(Fig.1.3b). When the fermi energy is within the original flat Landau level, the bulk is
still an extended state (corresponding to the bottom of Fig.1.3c-left, the ’hill’ of which
represents the impurity potential); the charges can move in all the samples. However,
because of the impurity diffusion, transmission tn < 1. When the fermi energy is far
away from the original Landau level, the broadened parts of the bulk state is localized
around the impurities; charges can not be transported by the localized state. Thus
the peak of longitudinal resistance R may changes smaller and a little broaden but it
still keep zero at the main parts of the gap of original landau levels. However, as the
edge state is not affected by the impurity diffusion, charges can transport at the edge
with tn = 1. Besides the width of plateau is a little shortened, the hall resistance RH
is almost not affected.
In this section, with a schematic diagram, I will introduce the two important
properties of QH effects: (1) the existence of number of gapless edge states at the
gapped bulk states; (2) the quantized hall resistance. It will be seen later that these
two properties can be used to characterized all non-trivial topological system, and
they are directly related to the topological number of the system. We also show the
topological protection of the system: it is not easy affected by the disorder. People
8
may think the schematic picture (Fig.1.3a) of the the curved bands closing the edge
is kind of weird. Because in quantum mechanics, the eigen energy of the system is
always solved for the whole space, we never get the eigen energy of the whole system
by some special position. However, it can be shown that [16], in QH system, the eigen
states of different energy are centered at different positions, it decays fast when away
from the center. When the eigen energy is higher, the center of eigen states are more
close to the edge. In this way, we can plot the eigen energy as the function of the
center of eigenstates, which gives the good picture of edge states [18]. However, it is
hard to use this to describe the general topological system, as the bulk states are in
general do exist in the whole bulk. Later in the different topological systems, I will
draw the energy bands in k- space, as we usually do. We will see that there are still
gapless states at the bulk gap of energy bands. We can check the eigenstates of the
gapless modes, which are only nonzero at the edge.
1.2 How to Get topology: from Spin Hall Effect to Quantum Spin Hall
Effect
As the quantum hall effect has such the elegant properties: charge accumulation,
quantized conductance and edge states, we may wonder that if we can get the spin
counterpart. If we can get a so-called spin hall effect, i.e. due to some force, the spin-up
moves down and the spin-down moves up (Fig.1.4), it should be very interesting.
We will show that this can be realized by the Rashba spin-orbit interaction (SOI).
Figure 1.4: Spin Hall effects
9
1.2.1 Rashba Spin-Orbit Interaction
Spin-orbit interaction is a relativistic effect. From the low-energy limit of dirac equa-tion,
we can get one term as spin-orbit coupling[24]. This can also be understood in
a classic way: in the relativistic theory, a electron moving in a electric field create a
magnetic field Be = − γ
c2v × E (For a common system, the speed v ≪ c, we have
γ = (1 − v2/c2)−1/2 ≃ 1). The electron with spin moving in this magnetic field will
have the Zeeman-like term
HSOI = −μs · Be = σ ·
[
p
me
× ∇
(
e~
c
V
2mec2
)]
= aσ · (p × ∇V ) (1.2)
Here we use the fact that μs = − e~
2mecσ. The quantum counterpart should consider
the commutation, we have
HSOI =
a
2
[σ · (p × ∇V ) − σ · (∇V × p)] (1.3)
=
a
2
[(σ × p) · ∇V + ∇V · (σ × p)]
Figure 1.5: Potential of 2D System
For a quasi 2D system, assuming we have V (r) = V (z) + V (x, y). Because in
z− direction, the sample is very thin, we will get a sudden change in V (z) (Fig.1.5),
thus ∇V (z) ≫ ∇V (x, y) and ∇V (r) =bz
d
dzV (z). The differential can be treated as
10
constant within the small height of z, by requiring a constant α = a d
dzV , we have
HSOI = (σ × p) · αbz
(1.4)
The above equation is nothing new but the SOI in a 2D system. If we only consider
an pure electron moves in the 2D system, noticing 2mec2 ∼ 1Mev, the coefficient α
thus the SOI should be very small. Dr. E. I. Rashba firstly researched the quasi-free
electrons in solids with the consideration of SOI [19]. It is shown that, the SOI effect
is broaden because of the periodic crystal structure. Such an effect is thus called
Rashba SOI. The Rashba model in solid can be derived based on k · p theory [20] or
a tight binding approximation [21]. The research shows that, in such a system, the
SOI can be rewritten as
HSOI ∼ σ ·
[
p
m∗
× ∇
(
V
Eg
)]
(1.5)
here m∗ is the effective mass, Eg is the energy of band gap between the orbital bands
(without considering the spin and SOI). Generally, we have Eg ∼ 1eV , thus we can
observe a notable Rashba SOI.
The Rashba SOI can be observed on the surface of Au(111) [22]. It is also found
in the semi-conductors which have diamond structure (zinc blende structure). This
type of semiconductors are generally III-V or II-VI compounds, such as GaAs, InSb
and HgxCd1−xTe. GaAs and InSb are the traditional spin hall effect materials. In
HgxCd1−xTe, we can observe the quantized spin hall effect. we will discuss these two
effects separately.
11
1.2.2 Spin Hall Effect
Now, we will give a rough picture to show that Rashba SOI can gives spin hall effect.
The full Hamiltonian of a SOI system can be written as
H =
bp2
2m∗ + σ · Beff =
bp2
2m∗ + (σ ×bp
) · αbz
(1.6)
Let’s solve it for an infinite 2D system. At this situation, px and py can be thought
as good quantum number and we can set p2 = p2
x + p2y
, thus.
H =
p2
2m∗ + α (σxpy − σypx) =
p2
2m∗ α (py + ipx)
α (py − ipx) p2
2m∗
=
p2
2m∗ iαpe−iφ
−iαpeiφ p2
2m∗
=
p2
2m∗ αpei(π/2−φ)
αpe−i(π/2−φ) p2
2m∗
where φ is the direction of the 2D momentum p (p, φ) and cos φ = px/p, sin φ = py/p.
we can easily get the eigenvalue by solving:
p2
2m∗ − E αpei(π/2−φ)
αpe−i(π/2−φ) p2
2m∗ − E
= 0 (1.7)
so that (
p2
2m∗
− E
)2
− α2p2 = 0 (1.8)
The eigenvalue is
Es (p) =
p2
2m∗ + sαp , s = ±1, p ≥ 0 (1.9)
The plot of E v.s. p can be seen at Fig.1.6.a; since p is the magnitude of momentum
p > 0, by rotating Fig.1.6.a about z− axis, we can get the plot of E v.s. px, py. It
can be seen that though HSOI = μ · Be is a Zeeman-like term, it is in fact totally
12
⇒
Figure 1.6: Spin Orbital Interaction
different with Zeeman effect. The breaking of degeneracy by Zeeman effect is split in
up-down ways; but here it is split in left-right ways. From Eq.(A.18), we know that
spin σ is always perpendicular to the momentum p (Fig.1.6.c). In fact this can be
critically solved by the following ways.
The eigenstates of the system can be given as
ψs (p) =
√1
2
ise−iφ
1
φ = arctan
py
px
(1.10)
The eigenstates of two-band system can always be thought as the eigenstates of
σn = σ · n. With n (sin θ cos ϕ, sin θ sin ϕ, cos θ), we have the eigenstates of σn as
ψ+ =
cos θ/2 exp (−iϕ/2)
sin θ/2 exp (iϕ/2)
and ψ− =
−sin θ/2 exp (−iϕ/2)
cos θ/2 exp (iϕ/2)
(1.11)
compare with ψs (p), we know that
ψ+ (p) =
√1
2
ei(π/2−φ)
1
=
cos (π/4) ei(π/2−φ)/2
sin (π/4) e−i(π/2−φ)/2
(1.12)
13
ψ− (p) =
√1
2
−ei(π/2−φ)
1
=
−sin (π/4) ei(π/2−φ)/2
cos (π/4) e−i(π/2−φ)/2
(1.13)
From above, we can get the σn of ψs (p) is at the direction (θ = π/2, ϕ = −π/2+φ),
i.e., spin of ψ+ (p) is at direction (θ = π/2, ϕ = −π/2 + φ); spin of ψ− (p) is at the
direction (θ = π/2, ϕ = π/2 + φ) (Fig.1.6.c). Thus the spin is always perpendicular
to the momentum p(p, π/2, φ).
Now let’s suppose a small longitudinal bias voltage so that the spin carriers can
move; the spin may also change directions, however, it should always be perpendicular
to the momentum p. From the Bloch equation, the time-dependent magnetic field of
Zeeman term can change the direction of spins:
~
dbn
dt
=bn
× B + c~
dbn
dt
×bn
(1.14)
By solving it, we can get [25]
nz,p = s
e~pyEx
αp3 (1.15)
Let’s understand this equation by using the lower band E− (p) as an example, when
we have s = −1. We will have:
• if py > 0, i.e. the electron is move to the up edge, we can get nz,p < 0, the spin
should points to the down direction;
• if py < 0, i.e. the electron is move to the down edge, we can get nz,p < 0, the
spin points to the up direction;
All in one, we showed the a different spin is accumulated at different edge, thus a
spin hall voltage can be archived. In this way, we can have the spin hall effect, which
had already been observed by different groups in 2004 [23].
However, such a spin hall effect is not perfect. Firstly, from Eq.(1.15), for s = 1,
the higher band E+ (p), spin moves to the inverse direction with E− (p). Thus we can
14
only see an obvious spin hall effect when fermi energy is smaller than the minimum
of E+ (p); otherwise, in each edge of the sample, the spin polarization of two bands
is in a different direction and they will cancel each other.
Figure 1.7: Spin Precession
Even only E− (p) is involved, a detailed calculation shows the spin precession
during the transport (Fig.1.7). When the electron transports along the horizonal
direction, it is not localized at the edge. Alternatively, it oscillatesd between the
upper edge and the down edge. During this oscillating motion, the spin rotate itself
at the same time: while spin points to the down direction in the upper edge; when it
reaches to the down edge, the direction has already been changed to the up direction.
Though at the different edge we can observe the different direction of spin, the spin
does not run at the edge. This picture is totally different with our initial idea to get
spin hall effect - where we simply want spin-up and spin-down goes to different edge
with no precession. The spin hall effect we get here is not so well. For it is not due
to the edge state, it is easily affected by the disorder. Though disorder may bring the
positive affects and strengthen the signal at some special situations (See the end part
of chapter 2), the system is not stable.
1.2.3 Quantum Spin Hall Effect
The spin hall effect we introduced above is almost close to what we want, but there is
still something missed. A few years ago, the quantum spin hall effect is predicted in
graphene [10] and the semiconductor materials HgxCd1−xTe [8]. As the spin-orbital
15
(a): MB < 0 (b): MB > 0
Figure 1.8: Quantum Spin Hall effect, from [27]
in graphene is very weak, the experiment of quantum spin hall effect was finally
observed at the HgTe/CdTe quantum well [9]. The four sub-bands involved in the
effective hamiltonian are |s, ↑⟩ , |px + ipy, ↑⟩ and |s, ↓⟩ , |−(px − ipy), ↓⟩. The two
subbands with the same spin form the pseudo-spin orbital system. Detaily
He (k) =
H(k) 0
0 H∗(−k)
(1.16)
H(k) = ε(k) + d · σ (1.17)
We can see that He (k) is block diagonalized, and H∗(−k) is in fact the time-reversal
symmetry part of H(k). Thus we only need discuss H(k), which is a two band
hamiltonian. The parameters are given as
dx = Akx, dy = Aky (1.18)
dz = M − B(k2x
+ k2
y), ε = C − D(k2x
+ k2
y)
Compare with (A.18), we can see that the term dx, dy and ε are same as the pure
Rashba SOI model, besides the difference of sign or additional constant. The biggest
difference between the above equations and (A.18) is that now we have σz term, dz.
16
Besides, quantum spin hall effect also need the parameters dz satisfy MB > 0, so that
dz can change sign when k2 = k2x
+ k2
y is bigger. We can check this from Fig.1.8. In
Fig.1.8a, MB < 0, the system is still a normal insulator. In Fig.1.8b, MB > 0, there
exist two external bands between the gap, which are connected from the conductance
band to the valence band. It can be easily checked that, the corresponding eigenstates
of blue band and red band are separately localized at the upper and down edge of
the sample. Thus we have a well defined edge states, similar as the QH effect, the
quantum spin hall effect is thus obtained.
(a): chiral edge states (b): helical edge states
Figure 1.9: Compare of edge states, from [27]
There are still some difference between the QH effect and the above QSH effect.
In QH effect, the time reversal symmetry is broken, the current of upper edge states
(positive charge) and the current the down edge states (negative charge) are in the
opposite direction (Fig.1.9a), which is the so called chiral edge states. In QSH effect,
the time reversal symmetry is conserved. The H(k) above is corresponding to the
spin up states, the time reversal counterparts H∗(−k) gives another edge states runs
in the opposite direction at the same edge (Fig.1.9b). These types of edge states are
called helical edge states. Topologically, it is said QSH effect are characterized by a
Z2 topological invariant [10]. We will not discuss much about Z2 here, instead, we
are still focus on the origin of non-trivial topology.
From the above mechanism to get QSH effect, we can easily find that a non-trivial
topology system, which contains gapless edge states, need a sign-changing coefficient
of σz. As the three σi should be physically equivalent, it seems that for a two band
17
system, a non-trivial topology need the coefficients of σx, σy and σz can
change sign when k varies. We will continue the discussion of this in the next
section.
1.3 Understanding Topology: Physics v.s. Mathematics
In this section, we try to give the intuitive view of the topology system. As mentioned
above, QH effects and QSH effects are belong to the different classes of topological
materials. To help understanding this, we choose QH system as an example. Mathe-matically,
this class of matter is distinguished by the rst Chern number n. Here n
is actually the number of edge states at fermi surface, or the number to characterize
the quantized hall conductance σxy = e2
h n. Physically, the rst Chern number n can
be get by Berry phase. We can also understand the Berry phase mathematically from
another two point of views, the genus and the winding number. We will discuss them
in the following of this section.
1.3.1 Physics: Berry Phase and Hall Conductance
The Berry phase is defined by the following ways. Supposing hamiltonian of the
system is depend on a set of parameters R = (R1,R2, . . .), and R can varies at
parameter space, we have H = H (R). The general example of parameters R is the
momentum k, when we describe the hamiltonian in k space, thus we have H = H (k).
In this way, for one energy band |n (R)⟩, we can define Berry connection (or Berry
vector potential ):
A(n) (R) = i ⟨n (R)|∇R |n (R)⟩ (1.19)
Correspondingly, we can define Berry curvature as:
Ω(n) (R) = ∇R × A(n) (R) (1.20)
18
Berry phase (or the so called geometric phase) of energy band |n (R)⟩ is defined as:
γ(n) =
∮
C
dR · A(n) (R) =
∫
S
dS · Ω(n) (R) (1.21)
here S is the whole parameter area. For the Bloch wave k, S is the first Brillouin
zone; C is the boundary of S. If R is in a two dimensional parameter space, we have
Ω(n) (R) = i
[
⟨∂Rxn
∂Ryn
⟩
−
⟨
∂Ryn |∂Rxn⟩
]
(1.22)
In the review of method introducing by TKNN [4], we can check that the hal-l
conductance is quantized by the berry phase. From the linear response theory,
the response current in the vertical direction while applied electric field along the
longitudinal direction can be described by the Hall conductance
σxy (EF ) =
e2~
i
Σ
E <EF<E
(vy)αβ (vx)βα
− (vx)αβ (vy)βα
(Eα − Eβ)2 (1.23)
here k is the Bloch vector, i.e. the crystal momentum, and we have
(vx)αβ =
1
~
⟨α| ∂H
∂kx
|β⟩ , (vy)αβ =
1
~
⟨α| ∂H
∂ky
|β⟩ (1.24)
|α⟩ is the state below fermi energy, |β⟩ is the state above fermi energy. Fermi energy
is at the band gap. From
H (k) |β (k)⟩ = Eβ (k) |β (k)⟩ (1.25)
make a derivation ∂k on both sides
(
∂k H
)
|β⟩ + H
∂k β
⟩
=
(
∂k Eβ
)
|β⟩ + Eβ
∂k β
⟩
(1.26)
19
left multiply by ⟨α|
⟨α|
(
∂k H
)
|β⟩ + ⟨α|H
∂k β
⟩
= ⟨α|
(
∂k Eβ
)
|β⟩ + ⟨α|Eβ
∂k β
⟩
(1.27)
⇒
⟨α|
(
∂k H
)
|β⟩ + Eα ⟨α
∂k β
⟩
=
(
∂k Eβ
)
⟨α |β⟩ + Eβ ⟨α
∂k β
⟩
(1.28)
thus
⟨α|
(
∂k H
)
|β⟩ =
(
∂k Eβ
)
δαβ + (Eβ − Eα) ⟨α
∂k β
⟩
(1.29)
we have
⟨α
∂k β
⟩
=
⟨α|
(
∂k H
)
|β⟩
(Eβ − Eα)
−
(
∂k Eβ
)
(Eβ − Eα)
δαβ =
⟨α|
(
∂k H
)
|β⟩
(Eβ − Eα)
(1.30)
here we using that fact Eα ̸= Eβ thus⟨α |β⟩ = 0. Based on this, we have ∂k [⟨α |β⟩] =
∂k (δαβ) = 0, thus ⟨α
∂k β
⟩
= −
⟨
∂k α |β⟩, we can get
⟨α| ∂H
∂kμ
|β⟩ = (Eβ − Eα) ⟨α
∂k β
⟩
= −(Eβ − Eα)
⟨
∂k α |β⟩ (1.31)
the hall conductance can be rewritten as
σxy (EF ) =
e2
i~
Σ
E <EF<E
[⟨
∂kyα |β⟩ ⟨β |∂kxα⟩ − ⟨∂kxα |β⟩ ⟨β| ∂kyα
⟩]
(1.32)
as the competence of the eigenstates need
Σ
α,β (|α⟩ ⟨α| + |β⟩ ⟨β|) = 1. Thus
σxy (EF ) =
e2
i~
Σ
E <EF
[⟨
∂kyα |∂kxα⟩ − ⟨∂kxα ∂kyα
⟩]
−
[⟨
∂kyα |α⟩ ⟨α |∂kxα⟩ − ⟨∂kxα |α⟩ ⟨α| ∂kyα
⟩]
(1.33)
noticing the normalization of eigenvector, ⟨α |α⟩ = c, which gives
⟨
∂kyα |α⟩ = −⟨α
∂kyα
⟩
,
the second part of above equation is zero. We can define σxy (EF ) =
Σ
E <EF
σ(α)
xy ,
20
with σ(α)
xy the hall conductance for the αth band. We can get σ(α)
xy by the sum of all
energy belong to the αth band and below the fermi surface. It should be mentioned
that the αth band may not be completely filled when the fermi energy lies within
the band. However, we can expand the sum to the whole αth band by multiply a
factor f(EF ). We can ask f(EF ) = 0 if the corresponding Eα > EF or else we have
f(EF ) = 1.
σ(α)
xy (EF ) =
e2
h
1
2πi
∫
d2k
[⟨
∂kyα |∂kxα⟩ − ⟨∂kxα ∂kyα
⟩]
f(EF )
=
e2
h
1
2π
∫
d2kΩ(α)
xy f(EF ) =
e2
h
γ(α)(EF )
2π
(1.34)
It is easy to see that f(EF ) is in fact the Fermi-Dirac distribution function at zero
temperature. It is natural to expand the definition of f(EF ) to the finite temperature
and we have the above form of σ(α)
xy (EF ) unchanged.
σxy =
Σ
α
σ(α)
xy =
Σ
α
e2
h
γ(α)
2π
(1.35)
The hall conductance of some band may not be an integer when the fermi surface is
within the band. However, for the completed filled band, the integral should gives the
quantized number. We can understand this from the following ways. We already know
that if the fermi surface is within the gap, we have the quantized hall conductance
σxy =
Σ
α σ(α)
xy = e2
h n. If the fermi surface is between the first band and the second
band, we already know n = 1, correspondingly, sigma(1)
xy is quantized by 1. We can
easily see that all the completed band sigma(1)
xy are quantized by moving the fermi
surface to each band gap. From above, we know that n =
Σ
α
γ( )
2π , which is in fact
the quantization of berry phase.
For the completely filled band, n = γ( )
2π is in fact the first chern number. It reflects
the topological properties of the gapped band structure. While the gap is not closing,
21
it should not change. We will come back to this in the last subsection.
1.3.2 Mathematics: Genus and Winding Number
In mathematics, a 2D surface can be topological characterized by their genus g, which
counts the number of holes in the surface. As shown in Fig.1.10, though have the
different shape, the basketball and the football has the same genus g = 0; For donuts
and the cup with the handle, there is one hole, thus the genus is g = 1.
(a) (b)
(I). genus g = 0, the n = 0 QH system
(c) (d)
(II). g = 1, the n = 1 QH system
Figure 1.10: the genus of different 2D surface
In physics, the berry phase has the form γ ∼
∮
C
A · dl. As we know, this contour
integration is nonzero only when there are singularities within the contour. When
one band is completed filled, all the singularities are involved, γ/2π is quantized,
these singularities can be treated as a set of singularized monopoles, and the integral
counts the number of monopoles. It is similar to the genus, which measures the holes
on the 2D surfaces. In this way, the quantized berry phase can be understand as
genus, which gives number of holes (monopoles) of Berry connection A at the first
Brillouin zone, which is a 2D surfaces. Thus Fig.1.10(I) describe a system has on hall
conductance; Fig.1.10(II) describe a QH system have hall conductance e2/h.
Another mathematic term involved is the winding number. In the 2D plane, the
winding number is defined as the times one closed curve travels counterclockwise
22
Figure 1.11: the winding number around a given point
around a given point. As shown in Fig. 1.11. For a two band system, the berry phase
can be understood as a winding number.
The hamiltonian of a general two-level system is
H1 = px (k) σx + py (k) σy + pz (k) σz + h0 (k) I (1.36)
The vector p (k) can be parameterized as p = p (sin θ cos ϕ, sin θ sin ϕ, cos θ), it is easy
to check, the eigenvalue of the system is
λ± = h0 ± p (1.37)
The corresponding eigenstate is independent of h0 (k), we have
ψ+ =
cos θ
2e−iϕ
sin θ
2
, ψ− =
sin θ
2e−iϕ
−cos θ
2
(1.38)
For the lower band, we can calculate the Berry connection. noticing
∇ =
∂
∂r
er +
1
r
∂
∂θ
eθ +
1
r sin θ
∂
∂ϕ
eϕ (1.39)
23
as ⟨−| ∂r |−⟩ ≡ 0 and we also have
⟨−| ∂θ |−⟩ = i
[
sin θ
2eiϕ −cos θ
2
]
1
2 cos θ
2e−iϕ
1
2 sin θ
2
= 0 (1.40)
i ⟨−| ∂ϕ |−⟩ = i
[
sin θ
2eiϕ −cos θ
2
]
−i sin θ
2e−iϕ
0
= i
(
−i sin2 θ
2
)
= sin2 θ
2
(1.41)
The only nonzero term of A(−) is A(−)
ϕ . Noticing r in equation (1.39) is in fact p here,
we have
A(−)
ϕ =
i
r sin θ
⟨−| ∂ϕ |−⟩ =
sin2 θ
2
r sin θ
=
1
2r
tan
θ
2
=
1
2p
tan
θ
2
(1.42)
we can get the Berry curvature
Ω(−) (k) = ∇R × A(−) (k) =
1
r sin θ
∂
∂θ
(
sin θA(−)
ϕ
)
er − 1
r
∂
∂r
(
rA(−)
ϕ
)
eθ
=
1
r2 sin θ
∂
∂θ
(
sin2 θ
2
)
er =
er
2r2 =
p (k)
2p (k)3 (1.43)
When the k goes through the whole space, p may not arrive at some directions, it
may also point the same direction for more than one time. Because of this, we can
define Ω(−) (k) in p space by (θ, φ) of the whole sphere by ask
Ω(−) (k) = f (θ, φ)
p
2p3 (1.44)
here f (θ, φ) counts the times of p point at (θ, φ) when k goes through the whole
parameter space. f (θ, φ) = 0 means p can never point at such a direction. In this
way, the Berry phase of the system defined in the 2D k space is mapped to the berry
phase on the whole sphere of p space:
γ(−) =
∫
S
dS · Ω(−) (R) =
1
2
∫
S
f (θ, φ) dω (1.45)
24
here dω is the solid angle, it is chosen for the whole sphere.
For the quantized hall effect or other n = 1 topological systems, p can point to
any direction one time, thus γ(−) = 1
2
∫
S dω = 2π; for a general n topological systems,
p points to all direction n times, we have f (θ, φ) ≡ n, thus γ(−) = 2πn. In this way,
the topological number n can be viewed as the times p can cover the whole sphere
when k goes through the whole parameter space. It is same as the winding number
defined above in a plane.
1.3.3 Topological Protection
Now, let’s talk about the topological protection. Mathematically, topological invari-ance
is used to classify different geometry objects. The objects belong to the same
class can be smoothly deformed into each other without change topological proper-ties.
For example, in Fig.1.10, we can see that the donuts and the cup with handle
are belong to the same class with g = 1, which have only one hole on the surface.
They can smoothly deform into one another without creating any new hole. We can
understand the winding number in the same way. The path of the closed curve have
the fixed winding number about a given point can varies the shape without creating
new circles around the point.
Thus, the key point in topological invariance is the smooth deformation. Physi-cally,
the solution of the many body systems are always a energy gap separating the
ground states from the excited states. Thus we can consider the smooth deformation
in physics by continuously change hamiltonian without closing the gap. Once the gap
is closed and reopen, the phase transition is happened. The states before gap closing
and after gap closing are said they can not be smoothly deformed into each other. In
this way, we have the topological protected phases. The topological property of the
system can not changed while any disorder deform the hamiltonian unless they close
the band gap. Thus the physical quantities related to the topological number is very
25
stable, they are not easily affected by the disorder.
The QH effect happens at 2D system. For a quasi 2D materials, the band structure
can be treated as a mapping from the crystal momentum k to the Bloch Hamiltonian
H(k). As H(k) is periodic in k space, we can only discussed it in the first Brillouin
zone. The pair of the edge of the first Brillouin zone are equivalent for H(k), for
examples H(kx,−π) = H(kx, π). Thus the first Brillouin zone can be treated as
a torus by connecting the corresponding opposite edges. For the two-band system,
as we shown before, H(k) is defined on a 2D sphere. The map from the torus to
the sphere, also maps the genus on the surface of torus to the winding number of
a ball. All of these topological terms can be understand by the physical quantity
Berry phase. The quantized Berry phase classifies the topology of the gapped band
structures. The topological equivalent H(k) can smoothly deform into one another,
without closing the band gap. Mathematically, the topological invariant n is called
the first chern number.
Before the end of this chapter, let’s discuss more about the relationship between
topological invariant and the measurable physical quantity: the number of edge states
at fermi surface and the hall conductance. By smoothly changing the hamiltonian
without closing the gap, the dispersion of edge states can changes. Specifically, the
edge modes may curves thus one edge mode can pass though the fermi surface many
times (see chapter 3). Supposing originally the edge mode runs along right direction at
one edge. The curve of this edge mode now gives many edge states at fermi surface,
we may have NR of them running right while NL of them running left. However,
the difference of the right moving states and the left moving states is fixed by the
topological number, we have
NR − NL ≡ n (1.46)
Thus the difference is topological protected. The hall conductance of the above are
get from the spacial-infinite systems, and it should not be affected by above smooth
26
deformation. However, a real system is always finite. In chapter 3 we will show, in
such a situation, the hall conductance may be affected by the distortion of the edge
states.
27
CHAPTER 2
Transport Property of Mesosystem
The topological system can be well understood by the band theory [28]. The hall
conductance, the measurable physical quantity directly related to topological number,
can be calculated by the transport theory of mesosystem. At the low temperature, the
transport properties of the sample can be described by the Landauer-Buttiker (LB)
formula. The LB formula can be deducted by the non-equilibrium Greens function
method [29] or by the intuitive point view of quantized transport [17], I will not discuss
the details here. Instead, we will discuss the tricks to get the transmission coefficients,
which lies in the center of LB formula. The convenient way to get the transmission
coefficients is based on the lattice model by the tight-binding approximation. In the
first section, we will show how to get the lattice hamiltonian, and then discuss how
to calculate the quantities used to get the transmission coefficients.
The research on transport properties of the system is to deal with the hamiltonian
of real sample, which only have the finite size. It is known that the quantized transport
conductance is consistent with the band theory. Generally, the band structure shows
the relationship between the energy E and the momentum vector ki in some special
direction. It need ki to be a good quantum number, thus the system should be
translational invariant along this special direction, or the system is infinite at this
direction. To discuss the topological property of the system, we need calculated
the chern number. As shown in the last chapter, the calculation should be done at
the whole first Brillouin zone. Thus we need k is a good quantum number in any
direction, we should deal with the system is translational invariant at any direction,
28
i.e. we need consider the system are totally infinite. Thus, to totally discuss the
topological system, we need the hamiltonian of finite sample, the hamiltonian for
the system is infinite in the special direction and the hamiltonian of infinite systems.
These three type of hamiltonian are related by fourier transformation. We will discuss
more details on it in the second section. As we use the lattice model for the finite
sample, the fourier transformation need k as the reduced momentum vector and
actually constrained in the first Brillouin zone.
In the last section, as the application of using LB formula, we calculate the spin
nernst effect in the absence of a magnetic field, which is the extension of spin hall
effect. We will see that, as the system is not protected by the topology, the results
are easy affect by the disorder. Fortunately, sometimes, the affects may be positive
as it can strengthen the signal at some special situations
2.1 Numerical Calculation of Landau-B¨uttiker Formula
Experimentally, we will research the transport properties of the sample by connecting
leads. For each lead, we will apply either voltage V or electronic current J, and then
to measure the one are not given. For example, we will apply a small longitudinal
voltage gradients, thus the voltage of the left lead and the right lead are known.
We can then measure either the current of the voltage of the rest leads. When
we measure the current, one side of the the current meter is connect the sample,
another side is connect to the earth, neglecting the effects of the current meter, we
should have the voltage of the lead V = 0. This is the so-called close boundary
condition; when we measure the voltage of the lead, the voltage meter does not allow
any electronic current, we should have J = 0, this is the so called open boundary
condition. Numerically, this process can be modeled by the Landau-B¨uttiker (LB)
formula. The LB formula gives the particle current Ipσ in lead-p with spin σ equals
29
to either ↑ or ↓:
Ipσ =
1
~
Σ
q̸=p
∫
dE Tpσ,q(E)[fp(E) − fq(E)] p = 1, 2, . . . (2.1)
The charge current in lead-p can be written as Jpe = e(Ip↑+Ip↓). fp(E) is the electric
Fermi-Dirac distribution of lead-p
fp(E) =
1
exp
[
E−eVp−EF
kBTp
]
+ 1
(2.2)
Through this, the LB formula related the charge current of lead-p to the voltage Vq
and temperature Tq of each leads.
The key step of LB formula is to get the transmission coefficients Tpσ,q(E). We
can get the voltage and currents of each leads by solve the set of LB formula. Thus
we can also get the different type of conductance or resistance. Though this way, we
can totally research on the transport properties of the sample.
As the transmission coefficients is so important, in the following subsection we
will discuss how to calculate it.
2.1.1 Analysis of the transmission coefficients
The transmission coefficients Tpσ,q(E) can be given by the following set of equations
[29]:
Σr
pσ = H+
pσ,C
Gr
pσHpσ,C (2.3)
Gr
C (E) = [Ga
C (E)]
†
=
[
E − HC −
Σ
pσ
(
Σr
pσ
)
]−1
(2.4)
Γpσ (E) = i
(
Σr
pσ
− Σr†
pσ
)
, Γq = Γq↑ + Γq↓ (2.5)
Tpσ,q (E) = Tr[ΓpσGrc
ΓqGac
], Tp,q = Tp↑,q + Tp↓,q (2.6)
30
Let’s simply explain the above equations. Supposing the system is connected by
only one lead, hamiltonian of the whole system can be written as
H =
HC HC,p
Hp,C Hp
(2.7)
here HC is the hamiltonian of the sample, Hp is the hamiltonian of the leads. Hp,C =
H
†
C,p is the interaction between the leads and the sample. We always have the green
function of the whole system satisfy
(
ε† − H
)
Gr = I. In details, we have
ε† − HC −HC,p
−Hp,C ε† − Hp
Gr
C Gr
C,p
Gr
p,C Gr
p
= I (2.8)
here Gr
C is the one we defined in (2.4), it is the center part (sample part) of the retard
green function of the whole system. We can easily get (2.4) by the following ways:
(
ε† − HC
)
Gr
C
− HC,pGr
p,C = I
(
ε† − Hp
)
Gr
p,C
− Hp,CGr
C = 0 ⇒ Gr
p,C =
(
ε† − Hp
)−1
Hp,CGr
C = Gr
pHp,CGr
C
(2.9)
⇒
(
ε† − HC
)
Gr
C
− HC,pGr
pHp,CGr
C = I (2.10)
thus we can get
Gr
C =
1
(ε† − HC) − Σr
p
with Σr
p = H
†
p,C
Gr
pHp,C and Gr
p =
(
ε† − Hp
)−1
(2.11)
If we consider the spin, obviously, we can get equation (2.3), Σr
p =
Σ
σ Σpσ =
Σ
σ H
†
pσ,C
Gr
pσHpσ,C . Here Σr
p reflect the affect of the lead-p to the center part, it
is so called the self energy due to lead-p. In which Gr
p =
(
ε† − Hp
)−1
is the free re-tarded
green function of the pure lead, which does not consider the interaction with
31
others. When the system is connected to many leads, there is no directly interaction
among these leads, the total self energy of all leads should be the sum of the self
energy of each one. In this way, we get (2.4). (2.5) is the definition of level-width
function. When we get the self energy Σpσ and the center green function Gr
C, It is
natural to get the transmission coefficients through the last equation (2.6).
However, a deep analysis shows that, if without any approximation, above set of
equations are impossible to solve. Firstly, to avoid the affects of reservoir, the lead are
chosen to be semi-infinite. Consequently, we have infinite size of Hp, it is impossible
to do the numerical calculation. Secondly, even ignore the self energy, the size of the
center sample are always very huge, if directly use formula (2.4) to calculate Gr
C, we
should get the results by a very long time.
A good approximation is based on layers. The sample and the lead of the system
are treated as many layers (see Fig.2.1). Based on tight-binding model, it is supposed
the interaction is only exists between the nearest layers. In this way, the calculation
can be simplified. Firstly, the hamiltonian of the center (sample) HC and the leads
Hp can be written as triangular blocks; secondly, the interaction between the center
and the leads is only non-zero between the contacted layers. In detail, this can be
seen from Fig.2.2, where A denotes
(
ε† − HC
)
and corresponding GC, which is for
center area; D denotes
(
ε† − Hp
)
and Gp, which is for lead; C denotes −Hp,C and
Gp,C , means the interaction between center area and lead; D denotes −HC,p and GC,p.
Because of the nearest neighbor approximation, the interaction between center area
and lead only exists at the connecting layers: i.e. a and d, thus c and b is in fact the
non-zero interaction. Without the approximation, the self energy ΣD should have the
same size of A; now because only a, b, c, d are involved, the nonzero part of ΣD is in
fact s, which is at the corresponding position of a to A. Because of this, when we
calculating Tpσ,q (E) =Tr[ΓpσGrc
ΓqGac
], we only need the ga part of Grc
. By the same
method we get Grc
, we can have ga = [a − s − AbA−1
a Ac]
−1.
32
... 4 3 2 1 1 2 3 4 ... ... 4 3 2 1 1 2 3 4 ...
Lead 1 The Sample Lead 2
(a): Two leads: the layers of the sample are pair of lines
... 4 3 2 1 1 2 3 4 ...
Lead 1 The Sample Lead 2
1
2
3
4
...
Lead 3
1
2
3
(b): Three leads: the layers of the sample are square
Figure 2.1: Deal the system by layers
Based on this, we can rewrite the transmission coefficients Tpσ,q(E) as
sr
pσ = h+
pσ,Cgpσhpσ,C (2.12)
gr (E) = [ga (E)]
†
=
[
(
ε† − hC
)
−
Σ
pσ
(
sr
pσ
)
− AbA−1
a Ac
]−1
(2.13)
Γpσ (E) = i
(
sr
pσ
− sr†
pσ
)
, Γq = Γq↑ + Γq↓ (2.14)
Tpσ,q (E) = Tr[ΓpσgrΓqga], Tp,q = Tp↑,q + Tp↓,q (2.15)
Here in equation (2.13), hC denotes the layers of center area, who are connect to the
leads. We should have more than one leads in general, thus we may have more than
33
Figure 2.2: The block structure of hamiltonian
one hC in the total HC. However, we can always arrange the matrix, or by appropriate
choose the layers of the sample, so that all the surface layers of the sample is at the
a place of Fig.. For example, as in Fig.2.1a, the sample is connected to two leads, we
can choose the layer of sample as the pair of lines, the first layer is actually the two
end of the sample connected to lead; in Fig.2.1b, the sample is connected to three
leads, we can choose all the boundary of the sample as the first layer. Of course, the
calculation based on Fig.2.1a should be faster than Fig.2.1b, as it has the smaller
matrix size. correspondingly, the place and size of gc at GC is same place as hC at
HC.
In equation (2.12), gp is the surface part of the Gp =
(
ε† − Hp
)−1
, corresponding
to the part of Hp that connecting to the center. h+
pσ,C and hpσ,C are the nozero part
of Hp,C and HC,p. denotes the interaction between the edge layers of center area and
leads. gc is surface part of GC, has the same size of hC.
Now the problem is changed to the calculation of the surface green function of
center gr
c and the surface green function of leads grp
. We will discuss gr
c in the following
section. To calculate the surface green function of semi-infinite leads, we need firstly
34
supposing the leads are translational invariant, the hamiltonian of each layer are
same, so as the interaction between any two layers. In the appendix, I will introduce
two methods to calculate grp
.
2.1.2 Method to the Get Surface Part of Gc
Supposing we have n layers with nearest neighbor interaction, the interaction between
ith layer and (i + 1)th layer is (h10)i. we want to get gnn (Here we use a different
notation as Fig, the surface layer of Gc is denoted by n, the most inside layer is
denoted by 1).
As the hamiltonian of the sample is triangular blockaded, we can write ε − H as
ε − H =
ε − (h00)1
−(h01)1
−(h10)1 ε − (h00)2
−(h01)2
−(h10)2 ε − (h00)3
. . .
. . . . . . −(h01)n−1
−(h10)n−1 ε − (h00)n
− Σ
=
D1 C
†
1
C1 D2 C
†
2
C2 D3
. . .
. . . . . . C
†
n−1
Cn−1 Dn
(2.16)
35
The general form of green function (ε − H)Gc = I gives
D1 C
†
1
C1 D2 C
†
2
C2 D3
. . .
. . . . . . C
†
n−1
Cn−1 Dn
g11 . . . g1n
g21 . . . g2n
. . . . . . . . .
gn1 . . . gnn
= I (2.17)
Now let’s deal with it. Firstly, ask 1st line × nth column
D1g1,n + C
†
1g2,n = 0 (2.18)
set F1 = D1
C
†
1g2,n = −D1g1,n = −F1g1,n (2.19)
we have
g1,n = −F−1
1 C
†
1g2,n with F1 = D1 (2.20)
Then, ask jth line × nth column (1 < j < n)
Cj−1gj−1,n + Djgj,n + C
†
j gj+1,n = 0 (2.21)
we have
C
†
2g3,n = −(C1g1,n + D2g2,n) = −
(
D2 − C1D−1
1 F1
)
g2,n = −F2g2,n (2.22)
g2,n = −F−1
2 C
†
2g3,n (2.23)
C
†
j gj+1,n = −(Djgj,n + Cj−1gj−1,n) = −
(
Dj − Cj−1F−1
j−1C
†
j−1
)
gj,n = −Fjgj,n (2.24)
36
i.e.
gj,n = −F−1
j C
†
j gj+1,n 1 < j < n) (2.25)
with
Fj = Dj − Cj−1F−1
j−1C
†
j−1 (2.26)
Lastly, ask nth line × nth column
Cn−1gn−1,n + Dngn,n = I (2.27)
thus
I = Dngn,n + Cn−1gn−1,n = Dngn,n − Cn−1F−1
n−1C
†
n−1gn,n (2.28)
i.e
gn,n =
(
Dn − Cn−1F−1
n−1C
†
n−1
)−1
(2.29)
In summary, to get gnn, we can define a set of Fj , j choose from 1 to n, corre-sponding
to the most inside layer to the layer near to the surface layer.
F1 = D1
Fj = Dj − Cj−1F−1
j−1C
†
j−1 1 < j < n
(2.30)
we thus have
gn,n =
(
Dn − Cn−1F−1
n−1C
†
n−1
)−1
(2.31)
2.1.3 Schedule to Calculate Transmission Coefficients
Now let’s review the schedule to calculate transmission coefficients.
1. get the surface green function of the lead: gpσ
supposing the lead has the periodic structure, i.e. it is repeat of basic cells. Each
cell may have more than one different layers. Thus we have the inlayer hamilto-
37
nian {(h00)i , i = 1, . . . , k} and the interlayer hamiltonian {(h10)i , i = 1, . . . , k},
here (h01)k is the interlayer hamiltonian between the kth layer of the nth cell
and the 1st layer of the (n + 1)th cell.
we can using several methods to get gpσ (see appendix);
2. using sr
pσ = h+
pσ,Cgpσhpσ,C , get the surface part of self energy of leads;
3. get the surface part of GC, i.e. gr (E) (more details see the notes and program
in folder: surface of center)
supposing the center area have n layers,
ε† − HC − Σ =
ε − (h00)1
−(h01)1
−(h10)1 ε − (h00)2
−(h01)2
−(h10)2 ε − (h00)3
. . .
. . . . . . −(h01)n−1
−(h10)n−1 ε − (h00)n
−
Σ
sr
pσ
=
D1 C
†
1
C1 D2 C
†
2
C2 D3
. . .
. . . . . . C
†
n−1
Cn−1 Dn
(2.32)
here (h00)n denotes hC,
Σ
sr
pσ is the self energy. we can get
(a) from
F1 = D1 (2.33)
get
Fj = Dj − Cj−1F−1
j−1C
†
j−1 1 < j < n (2.34)
38
(b) we have
gr (E) = gn,n =
(
Dn − Cn−1F−1
n−1C
†
n−1
)−1
(2.35)
4. calculating Γpσ (E) = i
(
sr
pσ
− sr†
pσ
)
and Γq = Γq↑ + Γq↓
5. get Tpσ,q (E) =Tr[ΓpσgrΓqga]
(a) gr can be written as
gr =
g11 g12 . . . g1p g1R
g21 g22 . . . g2p g2R
. . . . . . . . . . . . . . .
gp1 gp2 . . . gpp gpR
gR,1 gR,2 . . . gR,p gR,R
(2.36)
here lead j connects sample at the position of the of gjj (1 ≤ j ≤ p) . gR,R
means the part of surface does not connect any leads. We have
Tpσ,q (E) = Tr[ΓpσgrΓqga] = Tr[Γpσgr
pσ,qΓq (gr)
†
q,pσ] = Tr[Γpσgr
pσ,qΓq
(
gr
pσ,q
)†
]
(2.37)
i.e.
Tpσ,q (E) = Tr[ΓpσbΓqb†] with b = gr
pσ,q (2.38)
2.2 About the Lattice Hamiltonian and the Fourier Transformation
The above schedule is based on the tight-binding hamiltonian. It is easily to get
the tight-binding hamiltonian based on lattice model. In lattice model, the complete
basis is chosen as a set of function that is only defined at the lattice points. If
the system have inner freedom, each point may have more than one expanding basis,
corresponding the inner freedom. The little dots in Fig. are corresponding to the unit
39
cell of system. the unit cell is the basic unit to form the whole system. Each unit cell
may contains more than one points, corresponding to the system have the sublattice
structure. If we are researching on the free electrons without considering any crystal
structure, the lattice hamiltonian can be given from finite difference formulation. In
the first subsection, we will introduce this method. In such a situation, the calculation
may become accuracy by thicken the density of points. If the crystal structure is
considered, the lattice is in fact the crystal lattice. The lattice model is in fact the
approximation of wannier function, more inner freedom such as higher orbitals may
be considered. We can see this in chapter 3. In the second subsection, I will get the
layer hamiltonian based on the basic hamiltonian of unit cells. In the third subsection,
I will discuss the fourier transformation form of lattice hamiltonian, we will focus on
the situation that each unit cell have more than one points.
As the system are defined only at the lattice, the fourier transformation corre-sponding
the translational invariant is in fact based on bloch momentum vector k
and bloch wave function. As only the unit cell are translational invariant, k is corre-sponding
to the unit cell other than the basic lattice points.
2.2.1 Get the Lattice Hamiltonian: Finite Difference Formulation
In this subsection, using the system have SOI and in the magnetic field as example,
we will show how to get the lattice hamiltonian by the finite difference formulation.
The hamiltonian of system can be written as
H =
1
2μ
(
P − q
c
A
)2
+ α (σxPy − σyPx) + U (2.39)
Let’s firstly consider the situation that A = 0. Thus we have
H = −~2
2μ
∇2 − iα~ (σx∂y − σy∂x) + U (2.40)
40
Supposing the whole space is in the lattice structure, the lattice point can be
written as i (x, y) = i (na, mb), here a and b are the unit length in x and y di-rection.
The value of any function at the lattice points can be written as Fn,m =
F (x = na, y = mb). Based on the idea of finite difference formulation, the rst order
derivative at lattice point (n,m) can be described as
(∂xF)n,m =
1
2a
(Fn+1,m − Fn−1,m) (2.41)
(∂yF)n,m =
1
2a
(Fn,m+1 − Fn,m−1) (2.42)
the rst order derivative at lattice point (n,m) can be get from the follows
(∂xF)n+1
2,m =
1
a
(Fn+1,m − Fn,m) =⇒
(
∂2
xF
)
n,m =
1
a
(
(∂xF)n+1
2,m
− (∂xF)n−1
2,m
)
(2.43)
1. (a) thus
(
∂2
xF
)
n,m =
1
a2 (Fn+1,m − 2Fn,m + Fn−1,m) (2.44)
(
∂2
yF
)
n,m
=
1
a2 (Fn,m+1 − 2Fn,m + Fn,m−1) (2.45)
Based on this, we can have the hamiltonian as
H = −~2
2μ
∇2 − iα~ (σx∂y − σy∂x) + U
=
Σ
i
− ~2
2μa2
[(
c
†
i+δxci − 2c
†
i ci + c
†
i−δxci
)
+
(
c
†
i+δyci − 2c
†
i ci + c
†
i−δyci
)]
+ U
Σ
i
c
†
i ci+
+
Σ
i
−i
α~
2a
[(
c
†
i+δyσxci − c
†
i−δyσxci
)
−
(
c
†
i+δxσyci − c
†
i−δxσyci
)]
(2.46)
41
Because
Σ
i
− ~2
2μa2
(
c
†
i+δxci + c
†
i−δxci
)
=
Σ
i
− ~2
2μa2
(
c
†
i+δxci + c
†
i ci+δx
)
=
Σ
i
− ~2
2μa2
(
c
†
i+δxci + H.c.
)
(2.47)
noticing
(
ic
†
i+δyσxci
)†
= −iciσxc+
i+δy, we have
Σ
i
−α~
2a
[
c
†
i+δy (iσx) ci − c
†
i−δy (iσx) ci
]
=
Σ
i
−α~
2a
[
c
†
i+δy (iσx) ci − c
†
i (iσx) ci+δy
]
= −α~
2a
Σ
i
[
c
†
i+δy (iσx) ci + H.c.
]
(2.48)
For σy, we have
(
ic
†
i+δxσyci
)†
= −iciσyc+
i+δx. Define
t =
~2
2μa2 , ε = 4t + U, VR =
α~
2a
(2.49)
the hamiltonian can be written as
H = −t
Σ
i
[
c
†
i+δxci + c
†
i+δyci + H.c.
]
+ ε
Σ
i
c
†
i ci
− VR
Σ
i
[
c
†
i+δy (iσx) ci − c
†
i+δx (iσy) ci + H.c.
]
(2.50)
Now let’s discuss the situation A ̸= 0, i.e. the lattice form of
H =
1
2μ
(
P − q
c
A
)2
+ α (σxPy − σyPx) + U (2.51)
The theory of Feyman path integral gives
⟨a|b⟩
A,ϕ̸=0 = ⟨a|b⟩
A,ϕ=0
· exp
[
iq
~c
(∫ b
a
A · ds −
∫ tb
ta
ϕdt
)]
(2.52)
42
where |a⟩ and |b⟩ are the wavefunction at position a and b. This formula shows
the effect of vector potential A can be expressed by an external phase. Using this
formula, we have
H = −t
Σ
i
[
c
†
i+δxci exp
[
iq
~c
∫ i+δx
i
(A · ds)
]
+ c
†
i+δyci exp
[
iq
~c
∫ i+δy
i
(A · ds)
]
+ H.c.
]
+
Σ
i
εic
†
i ci
− VR
Σ
i
[
c
†
i+δy (iσx) ci exp
[
iq
~c
∫ i+δy
i
(A · ds)
]
− c
†
i+δx (iσy) ci exp
[
iq
~c
∫ i+δx
i
(A · ds)
]
+ H.c.
]
(2.53)
Use the landau gauge, choose A along x direction, we have
Ax = −By , Ay = Az = 0 (2.54)
The phase of the lattice i (n,m) is
∫ i+δx
i
(A · ds) =
∫ i+δx
i
Axdx ≃ Ax
(
i +
1
2
δx
)
· a = −Bmab (2.55)
∫ i+δy
i
(A · ds) =
∫ i+δy
i
Aydy = 0 (2.56)
Supposing θ = qabB
~c , we have
H =
Σ
i
εic
†
i ci − t
Σ
i
[
c
†
i+δxcie−imθ + c
†
i+δyci + H.c.
]
−VR
Σ
i
[
c
†
i+δy (iσx) ci − c
†
i+δx (iσy) cie−imθ + H.c.
]
(2.57)
2.2.2 Get the Hamiltonian of Layers
After we get the lattice hamiltonian, we can have the hamiltonian of the unit cell
h0, and the hamiltonian between the unit cell. In the following, we will get the
hamiltonian of layer for different situations by the basic hamiltonian. The next nearest
neighbor interaction of unit cell are considered.
43
The general hamiltonian can be written as
H =
Σ
i
[
h0c
†
i ci +
(
hxc
†
i+δxci + hyc
†
i+δyci + h.c.
)]
+
Σ
i
(
h1xyc
†
i+(δx,δy)ci + h2xyc
†
i+(δx,−δy)ci + h.c.
)
(2.58)
the matrix form of H can be written as
H =
H00 H
†
10
H10 H00 H
†
10
. . . . . . . . .
H10 H00 H
†
10
H10 H00
(2.59)
here H00 is the in-layer hamiltonian, H10 is the interaction between layers. The actual
form of H00 and H10 are determined by the way we choose layers. They are formed
by h0 and hx, hy, h1xy, h2xy.
h0 and hx, hy, h1xy, h2xy are the hamiltonian elements based on the structure of
smallest unit cells and the interactions among them. Because the unit cell has inner
freedom or inner lattices, these elements are generally the matrix. The actual form
of them are determined by how we labeling the layers, the way to label unit cells
at each layer, and the way to label the inner freedom of the unit cell. In the real
calculation, we may firstly already have a set of hamiltonian elements based on unit
cells: the in-cell hamiltonian eh0; and the interaction between cells ehx, ehy, eh1xy, eh2xy.
When they are given, the way to label the inner freedom of the unit cell is fixed, so
as the positive direction of ehx, ehy, eh1xy, eh2xy (see the following figures). However, the
positive direction thus the actual form of hx, hy, h1xy, h2xy may differs from those
of ehx, ehy, eh1xy, eh2xy. Because their positive directions are determined by the way to
label the layers, and the way to label unit cells at each layer, which are varies when
44
calculate the different parts of the system (different leads or the center sample). Thus
though eh0 ≡ h0, we may have hx, hy, h1xy, h2xy different with ehx, ehy, eh1xy, eh2xy.
1. we can choose the coloum as the basic layer, the matrix form is (for left & right
lead, and the center area), we have:
(H00)x =
h0 h†
y
hy h0 h†
y
. . . . . . . . .
hy h0 h†
y
hy h0
, (H10)x =
hx h2xy
h1xy hx h2xy
. . . . . . . . .
h1xy hx h2xy
h1xy hx
(2.60)
(a) if count from positive directions of both ehx and ehy (here from left to right,
and down to up),
I II III IV
4
3
2
1
hy
hx
h1xy
h2xy
hx
hy
h1xy
h2xy
I, 1 ! II, 1
I, 1 ! I, 2
I, 1 ! II, 2
I, 2 ! II, 1
(1). Based on column, from I ! II
we have:
h0 = eh0 , hx = ehx , hy = ehy , h1xy = eh1xy , h2xy = eh2xy (2.61)
(b) if count from positive direction of ehx, and negative direction of ehy ((here
is left to right and up to down),
we have:
h0 = eh0 , hx = ehx , hy = eh†
y , h1xy = eh2xy , h2xy = eh1xy (2.62)
45
I II III IV
1
2
3
4
hy
hx
h1xy
h2xy
hx
hy
h1xy
h2xy
I, 1 ! II, 1
I, 2 ! I, 1
I, 2 ! II, 1
I, 1 ! II, 2
(4). Based on column, from I ! II
2. if we choose the row as the basic cell,
(H00)y =
h0 h†
x
hx h0 h†
x
. . . . . . . . .
hx h0 h†
x
hx h0
, (H10)y =
hy h
†
2xy
h1xy hy h
†
2xy
. . . . . . . . .
h1xy hy h
†
2xy
h1xy hy
,
(Hlead,C)y =
h1xy hy h
†
2xy
h1xy hy h
†
2xy
. . . . . . . . .
h1xy hy h
†
2xy
h1xy hy h
†
2xy
(2.63)
(a) if count from negative direction of ehx, and positive direction of ehy (here is
from down to up, and right to left) (for upper lead),
IV III II I
4
3
2
1
hy
hx
h1xy
h2xy
hx
hy
h1xy
h2xy
1, II ! 1, I
I, 1 ! I, 2
II, 1 ! I, 2
II, 2 ! I, 1
,
,
,
hy
h1xy
h
†
2xy
1, I ! 2, I
1, II ! 2, I
1, I ! 2, II
(3). Based on row, from 1 ! 2
46
we have:
h0 = eh0 , hx = eh†
x , hy = ehy , h1xy = eh
†
2xy , h2xy = eh
†
1xy (2.64)
(b) if count from positive direction of ehx, and negative direction of ehy (here
from up to down, and left to right) (for down lead),
I II III IV
1
2
3
4
hy
hx
h1xy
h2xy
hx
hy
h1xy
h2xy
1, I ! 1, II
I, 2 ! I, 1
I, 2 ! II, 1
I, 1 ! II, 2
,
,
,
h†
y
h
†
1xy
h2xy
1, I ! 2, I
1, II ! 2, I
1, I ! 2, II
(2). Based on row, from 1 ! 2
we have:
h0 = eh0 , hx = ehx , hy = eh†
y , h1xy = eh2xy , h2xy = eh1xy (2.65)
2.2.3 Fourier Transformation
Now, lets discuss the fourier transformation form based on the hamiltonian of layers
we get above.
1. If the inner freedom is sth. like spin, does not depend on position, for hamilto-nian
H =
Σ
i
[
h0c
†
i ci +
(
hxc
†
i+δxci + hyc
†
i+δyci + h.c.
)]
+
Σ
i
(
h1xyc
†
i+(δx,δy)ci + h2xyc
†
i+(δx,−δy)ci + h.c.
)
(2.66)
47
(a) if kx is a good quantum number
H =
Σ
kx,iy
[
h0 +
(
hxe−ikxa + h†
xeikxa)]
c
†
kx,iyckx,iy+
Σ
kx,iy
[(
hy + e−ikxah1xy + eikxah
†
2xy
)
c
†
kx,iy+δyckx,iy + h.c.
]
(2.67)
we have the matrix form
H = (H00)x + e−ikxa (H10)x + eikxa (H10)
†
x (2.68)
here (H00)x and (H10)x are determined by 1a, based on how we counting
the basic cell of each coloumn and each row.
(b) if ky is a good quantum number
H =
Σ
ix,ky
[
h0 +
(
hye−ikya + h†
yeikya)]
c
†
ix,kycix,ky+
Σ
ix,ky
[(
hx + e−ikyah1xy + eikyah2xy
)
c
†
ix+δx,kycix,ky + h.c.
]
(2.69)
we have the matrix form
H = (H00)y + e−ikya (H10)y + eikya (H10)
†
y (2.70)
here (H00)y and (H10)y are determined by 1b, based on how we counting
the basic cell of each row and each coloumn.
48
(c) both are good quantum number
H =
Σ
k
[
h0 +
(
hxe−ikxa + h†
xeikxa)
+
(
hye−ikya + h†
yeikya)]
+
Σ
k
[(
e−ikxae−ikyah1xy + eikxaeikyah
†
1xy
)
+
(
e−ikxaeikyah2xy + eikxae−ikyah
†
2xy
)]
c
†
kck
(2.71)
2. If the inner freedom is also depends on position, the halmitonian is only dif-
fered by a unitary transformation, which does not change the eigenstate and
eigenvalue.
we can rewrite H as
H =
Σ
i,α1α2
[
(h0)α1,α2
c
†
i,α1ci,α2 +
(
(hx)α1,α2
c
†
i+δx,α1ci,α2 + (hy)α1,α2
c
†
i+δy,α1ci,α2 + h.c.
)]
+
Σ
i,α1α2
(
(h1xy)α1,α2
c
†
i+(δx,δy),α1
ci,α2 + (h2xy)α1,α2
c
†
i+(δx,−δy),α1
ci,α2 + h.c.
)
(2.72)
here the index α1α2 means the inner freedom depends on position, i.e. the
different atom in the cell; because there still have inner freedom depends on
spin, (h0)α1,α2
is still a matrix, it is a block elements at the matrix h0. Supposing
the coordinates of the αj atom is rj = αj (xj , yj); define a new unitary matrix
[
e T (k)
]
ij
= δije−ik·ri ,
[
e T (kx)
]
ij
= δije−ikxxj by the coordinates of inner atoms
within one cell;
e T (k) =
e−ik·r1
e−ik·r2
. . .
e−ik·rn
(2.73)
49
define unitary matrix eT
(k) as the repeat of diagonal block e T (k), so that eT
(k)
has the same dimension as H00 and H10
eT
(k) =
e T (k)
e T (k)
. . .
e T (k)
(2.74)
Noticing e T (k) has the same dimension as h0 and hx, hy, h1xy, h2xy, the
matrix form of above discussion di ers from the system that has no location-
depending inner freedom by ONLY the unitary transformation eT
(k), which dose
not change the eigen value and eigen vector:
(a) kx is a good quantum number
H =
Σ
kx,iy
[
e T (kx)
(
h0 + e−ikxaxhx + eikxaxh†
x
) e T† (kx)
]
c
†
kx,iyckx,iy+ (2.75)
Σ
kx,iy
[[
e T† (kx)
(
hy + e−ikxaxh1xy + eikxaxh
†
2xy
)
e T† (kx)
]
c
†
kx,iy+δyckx,iy + h.c.
]
the matrix form is
H = eT
(kx)
[
(H00)y + e−ikxa (H10)y + eikxa (H10)y
] [
eT
(kx)
]†
(2.76)
(b) ky is a good quantum number
H =
Σ
ix,ky
[
e T (ky)
(
h0 + e−ikyayhy + eikyayh†
y
) e T† (ky)
]
c
†
ix,kycix,ky+
Σ
ix,ky
[
e T (ky)
(
hx + e−ikyayh1xy + eikyayh2xy
) e T† (ky) c
†
ix+δx,kycix,ky + h.c.
]
(2.77)
50
the matrix form is
H = eT
(ky)
[
(H00)x + e−ikya (H10)x + eikya (H10)x
] [
eT
(ky)
]†
(2.78)
(c) both are good quantum number,
H =
Σ
k
e T (k)
[
h0 +
(
hye−ikyay + h†
yeikyay
)
+
(
hxe−ikxa + h†
xeikxa)] e T† (k) c
†
kck+
Σ
k
e T (k)
(
e−ikxae−ikyah1xy + eikxaeikyah
†
1xy
)
e T† (k) c
†
kck+
+
Σ
k
e T (k)
(
eikxae−ikyah
†
2xy + e−ikxaeikyah2xy
)
e T† (k) c
†
kck+ (2.79)
the matrix form is
H = e T (k)
[
h0 +
(
hye−ikyay + h†
yeikyay
)
+
(
hxe−ikxa + h†
xeikxa)] e T† (k)+
e T (k)
[(
e−ikxae−ikyah1xy + eikxaeikyah
†
1xy
)
+
(
eikxae−ikyah
†
2xy + e−ikxaeikyah2xy
)]
e T† (k)
(2.80)
2.3 Application: Spin Nernst effect in the Absence of a Magnetic Field
2.3.1 Motivation and Background
With the development of the micro-fabrication technology and the low-temperature
measurement technology, a great amount of efforts have been paid for the research
of the thermoelectric properties in the last two decades [31, 32]. Comparing to the
conductance, the thermoelectric coefficients of electronic systems are more sensitive to
the details of the density of states[33, 34, 35], which is very important for the design
of the electronic devices. The thermopower (seebeck coefficient) of the quantum
dot was measured in the last few years[31]. Recently, the Nernst effect, a Hall-like
thermal effect, has been theoretically studied [36] and had been detected, for
51
example, in bismuth[37] in which, with the existence of a perpendicular magnetic
field, a transverse current is induced by the longitudinal thermal gradient (Fig.2.3).
Figure 2.3: Nernst Effect and Spin Nernst Effect. Red dot denotes hot source and
blue dot denotes cold source
In the spintronics area, the spin thermal coefficients are also of focus recently[33,
34, 35]. In a recent paper, by considering a system with a spin-orbit interaction
(SOI), the Nernst effect and a novel thermal effect, the spin Nernst effect, have been
fully studied in a two-dimensional electron gas[38]. It is found that, because of a
perpendicular magnetic field B, the Nernst signal exhibits a series of peaks. When
the SOI exists, the peaks split and the spin Nernst effect appears. With a small B or
a large SOI, the spin Nernst effect becomes more pronounced. It also shows that the
spin Nernst effect is easier to be affected by disorder than the Nernst effect.
There is no doubt that a perpendicular magnetic field B is essential for the exis-tence
of the Nernst effect. However, in the spin Hall effect, the transverse spin current
is due to a SOI rather than a perpendicular magnetic field. Similarly, for the spin
Nernst effect, B may not be needed either. One may suspect that the spin Nernst
effect is in fact the combination of the existence of thermopower and a SOI. Thus,
the focus of the current work is to study the spin Nernst effect in the absence of a
perpendicular magnetic field, and its interplay with the thermopower.
52
2.3.2 Research Methods and the Definition of Thermal Quantities
In this work, the property of spin Nernst effect is developed in a two-dimensional
electron gas system with a Rashba SOI but without a perpendicular magnetic field
B. For this set-up, the Nernst effect disappears thus we focus on the spin Nernst
effect – a transverse spin current induced by a longitudinal thermal gradient ΔT . A
traditional way to analyze such a Hall-like system is to add vertical probes to detect
the transverse properties. Thus we set a four-terminal cross-bar sample, as shown in
Fig.1[38]. A longitudinal thermal gradient ΔT is added between the leads 1 and 3.
This thermal gradient induces a transverse spin current Js in the closed boundary
condition with a SOI, which can be measured at leads 2 and 4. The seebeck coefficient
of such a system can be directly measured at leads 1 and 3.
By using a tight-binding model and the Landauer-Buttiker (LB) formula, the spin
Nernst coefficient Ns (Ns ≡ Js/ΔT ) and the seebeck coefficient S (S ≡ −ΔV/ΔT )
are calculated. The Rashba SOI used in our calculations covers a wide range with
some beyond the accessibility of today’s sample. The seebeck coefficient S shows a
few peaks consequently when the fermi energy EF goes through the energy band. Due
to the interface of our setting (zero Rashba SOI at lead 2,4), we find a negative S. It is
confirmed that spin Nernst effect can not be simply thought as the combination of the
seebeck coefficient and the Spin hall effect [38]. A big spin Nernst coefficient Ns can be
found with a zero seebeck coefficient S. However, when the peaks of seebeck coefficient
occur with a non-zero Rashba SOI, the spin Nernst effect exhibits big amplitude or
sometimes also peaks. The Fermi energy EF also affects Ns. When the Fermi energy
EF is close to the bottom of the energy band (−4t), the oscillatory amplitude of
Ns becomes more pronounced. The effect of disorder on Ns is also investigated.
When EF = −3.8t, we can see a large increase of Ns with increasing of the strength
of disorder. Its value at the peak is about three-fold of that without disorder. In
addition, we find that the strength of disorder when Ns vanishes, indicating that the
53
Figure 2.4: Schematic diagram of the four-terminal cross-bar sample. The area with
SOI is marked by gray. A thermal gradient ΔT is applied between the longitudinal
lead-1 and lead-3.
system goes into an insulating regime, is independent of the Fermi energy.
In the tight-binding representation, the Hamiltonian with SOI can be written
as:[39],
H =
Σ
iσ
εic
†
iσciσ +
Σ
iσσ′
[c
†
i+δy,σ(−tI − iσxVR)σσ′ciσ′
+c
†
i+δx,σ(−tI + iσyVR)σσ′ciσ′ + H.c.] (2.81)
where c
†
iσ(ciσ) is the creation (annihilation) operator of electrons in the site i = (n,m)
with spin σ, and δx and δy are the unit vectors along the x and y directions. εi is the
on-site energy, which is set to 0 everywhere for the clean system. When the center
region is a disorder system, εi is set by a uniform random distribution [-W/2,W/2].
Here t = ~2/(2m∗a2) is the hopping matrix element with the lattice constant a, I
is a two-dimensional identity matrix. The strength of Rashba SOI is represents by
VR = α~/2a, where α is the Rashba spin-orbital coupling. VR is set to zero in the
lead-2 and lead-4.
Considering a small temperature gradient ΔT on the longitudinal lead-1,3, we
can set the temperatures T1 = T + ΔT /2, T3 = T − ΔT /2, T2 = T4 = T . The
charge current in lead-p can be written as Jpe = e(Ip↑ + Ip↓) and the spin current is
54
Jps = (~/2)(Ip↑ −Ip↓). Here Ipσ is the particle current in lead-p with σ equals to ↑ or
↓. Ipσ can be obtained by the LB formula:[38, 39]
Ipσ =
1
~
Σ
q̸=p
∫
dE Tpσ,q(E)[fp(E) − fq(E)] (2.82)
where Tpσ,q(E) is the transmission coefficient from the lead-q to the lead-p with spin σ
and E is the energy of the incident electron. fp(E) is the electronic Fermi distribution
function of the lead-p,
fp(E) =
1
exp
[
E−EF−Vp
kBTp
]
+ 1
(2.83)
when change of temperature Tp is small or we have a small bias Vp, the equation
above can be Taylor expanded. Using
g (T, V ) = g (T0, V0) +
∂g
∂T
T=T0
(T − T0) +
∂g
∂V
V =V0
(V − V0) + ... (2.84)
define f as the zero order of Taylor expansion of the Fermi distribution function, it
is the same for all four leads, f = 1/{exp[(E − EF )/kBT ] + 1}. we have
fp (Tp, Vp) =
1
exp
[
E−EF−Vp
kBTp
]
+ 1
= f + A
Tp − T0
T0
+ A
(Vp − V0)
ε
(2.85)
with
A = f2exp
[
ε
kBT
]
ε
kBT = f (1 − f)
ε
kBT (2.86)
The spin Hall current in lead-2 and lead-4 can be calculated with the closed
boundary condition in both lead-1,3 and lead-2,4, i.e. V1 = V3 = 0 and V2 = V4 = 0.
From symmetry of the system, we know that J2s = −J4s[38]. After the Taylor
expansion, the spin Nernst coefficient Ns ≡ J2s/ΔT can be reduced to:
Ns =
1
4π
∫
dE(ΔT23 − ΔT21)
E − EF
kBT 2 f(1 − f), (2.87)
55
here ΔT2p = T2↑,p − T2↓,p.
For the calculation of the longitudinal seebeck coefficient S, we need the open
boundary condition at lead-1,3, i.e. J1e = J3e = 0 to find the difference ΔV = V1−V3.
Different from a quasi-one-dimensional 2-leads system[40], the extra leads-2,4 also
affects the longitudinal seebeck coefficient S of the entire system. For example, with
a perpendicular magnetic field B, the longitudinal seebeck coefficient S is affected by
the bias in leads-2,4, V2 and V4. However, without B, the sample’s symmetry increases
from C2 symmetry to D2 symmetry, i.e., we have T1,2 = T1,4, Here T1,2 = T1↑,2+
T1↓,2. After the Taylor expansion, we can get the longitudinal seebeck coefficient
S ≡ −ΔV/ΔT as:
S =
1
T
∫
dE (T1,2 + T1,4 + 2T1,3) (E − EF )f (1 − ∫ f)
dE (T1,2 + T1,4 + 2T1,3) f (1 − f)
. (2.88)
The equation above shows that, even with a higher symmetry, the longitudinal seebeck
coefficient S is still affected by the transport properties from lead-2 and lead-4.
With the D2 symmetry, the relationship between S and Ns can be further derived.
In fact, we can rewritten S =
(
A↓ + A↑) /[∫
dEF (ε)
(
a↑ + a↓)]
. The D2 symmetry
gives ΔT23 = −ΔT21. Noticing T3↑,1 = T3↓,1, the spin Nernst coefficient can be
simplified as Ns =
(
A↓ − A↑)/
(2πkBT ). Here ε = E − EF and F (ε) = f (1 − f),
a↑ denotes the spin up term: a↑ = T2↑,1 + T3↑,1, and a↓ the spin down term a↓ =
T2↓,1+T3↓,1, we also use the notation of the integral term A↑ =
∫
dEεF (ε) a↑/
T and
A↓ =
∫
dEεF (ε) a↓/
T . Because of the symmetry, only leads-1,2,3 are used in the
simplified expression of S and Ns, we only need the upper half of the sample for our
investigation. In fact, a↑ (a↓) and A↑ (A↓) reflects transport properties of spin-up
(spin-down) electrons in the upper half of the sample. Roughly speaking, S can be
seen as the sum of spin-up and spin-down terms, while Ns as the difference of them.
56
2.3.3 Numerical Results and Analysis
In the numerical calculations, t = ~2/(2m∗a2) is set as the energy unit. If taking
the effective electron mass m∗ = 0.05me and the lattice constant a = 12.5nm, t is
about 5meV . Temperature is fixed by kBT = 0.01t, which is about 1K. The size
of center region is L = 19a, about 237nm. In a reasonable experimental range thus
far VR ∈ [0, 0.1][41]. However, in order to thoroughly study the relationship between
the spin Nernst coefficient Ns and the seebeck coefficient S, we extend the range of
VR up to [0, 1] in our calculation.
Fig.2.5 shows the spin Nernst coefficient Ns and the seebeck coefficient S versus
the Fermi Energy EF in the clean system (W = 0). It is clearly seen that the seebeck
coefficient S peaks at the positions where there are step-changes of transmission
function T1,2+T1,4. These peaks can be explained by a simple model only with a 2-lead
system without the Rashba SOI, shown in Fig.2.6(a). The transmission coefficient
T1,3 is a step function (solid-black curve). The reason is as follows. The sample can
be considered as a multi-channel system at a low temperature (here T ∼ 1K). When
fermi energy increases, more channels in the lead are used to transport current. Thus,
ΔV of two leads as well as the current increase with increasing of fermi energy (red-dashed
curve). However, the S (blue-dotted curve) can not accumulates while EF
increases, it only peaks while the channel number changes and S is close to zero with
a fixed channel number. This can be seen from the LB formula (2.82), if lead-p and
lead-q have different temperatures, fp(E, T +ΔT)−fq(E, T−ΔT) is an antisymmetry
function of E − EF (see plot in small box of Fig.2.6(a)): when E < EF , fp < fq,
current flows from lower temperature lead to higher temperature one; when E > EF ,
current flows in the opposite direction. Only when the two flows are not equal, i.e.
Tpσ,q has an antisymmetry part, we can have a nonzero current. Thus for Fig.3(a),
only when T1,3 is at the step-change point, it has antisymmetry part and can give a
non-zero S.
57
0
0.1
(a) V
R
=0.02
N
s
& S
0
0.1
(b) V
R
=0.05
−4 −3.5 −3 −2.5
0
0.1
(c) V
R
=0.1
E
F
N
s
& S
−3.5 −3 −2.5
0
0.1
(d) V
R
=0.45
E
F
Figure 2.5: Ns (red solid) and S (black dotted) vs. Fermi energy EF for different
Rashba VR. The (scaled) transmission coefficient T1,2+T1,3 (thin blue dashed) and spin
transmission coefficient ΔT2,3 (thin blue solid) are also shown. The other parameters
are T = 0.01, and L = 19a.
58
(b) V
R
=0.1
k
y
(c) V
R
=0.45
k
y
E
n,k
y
(a) V
R
=0
J & S
−3.9 −3.8 −3.7
E
F
E
F
0
Figure 2.6: (a) A simple model: Current J because of voltage gradient(red dashed)
and thermal power S because of thermal gradient (blue dotted) vs. Fermi energy EF
at a two-lead system with Rashba VR = 0. The (scaled) transmission function T1,3 is
also shown (black solid). The plot in the small box shows fL − fR with temperature
difference. (b) and (c): the eigen energy of the lead En,ky v.s. longitudinal wave
vector ky (units: 1/2a) for different VR
This conclusion can also be used to analyze spin-involved quantities. From Fig.2.6,
we can see that Ns (red solid line) shows an oscillatory structure. Besides the peaks
at VR = 0 (Ns is zero at this point), the magnitude of Ns oscillation is also large at the
peaks of S; but at the exact maximum point of S, where VR is quite small (VR . 0.1),
Ns is generally close to zero. This is because the spin transmission coefficient ΔT2,3
generally has an extreme value when the transmission coefficient jumps at a step.
Around an extreme value, any function is almost symmetry, thus one only can get
a low value of Ns. While at both sides of the extreme value, ΔT2,3 monotonically
increases or decreases, we can get a local maximum magnitude of Ns. Now why ΔT2,3
has an extreme value at a peak of S for a small VR. Due to the Rashba SOI, each
eigen-energy band splits into two sub-bands with opposite spin directions. These two
sub-bands degenerate at ky = 0, and the lower sub-band has two valleys below this
degenerate point. The two sub-bands are very close to each other when VR is small.
If the lower sub-band of high level (for example, E1,ky ) has the similar spin direction
59
with the upper sub-band of low level energy (for example, E0,ky ), ΔT2,3 continually
increases when EF goes from the upper band of E0,ky to the two valleys of lower
sub-band of E1,ky , and than rapidly decreases when EF goes through the degenerate
point (ky = 0) of E1,ky , thus we get a peak in ΔT2,3; otherwise we get a valley in
ΔT2,3.
When VR is very big, we can see the external peaks for both Ns and S. For
example VR = 0.45 in Fig.2.5(d), close to the 2nd and 3rd main peaks of S, we can see
a very sharp sub-peak of Ns. In fact, these are also the small peaks of S, though not
very big. This is because for these two band (see Fig.2.6(c)), the two valleys of the
lower sub-band is far from the degenerate point at ky = 0, the two channel of these
two sub-bands is separated. Thus we can see two peaks. At this time, the change of
spin transmission coefficients can be roughly thought as the change of transmission
coefficients, thus we can see Ns peaks at the S’s peak.
0 0.2 0.4 0.6 0.8 1
−0.04
0
0.04
E
F
=−3
V
R
N
s
& S
Figure 2.7: (Ns (red solid) and S (black dashed) vs. Rashba SOI VR for fermi energy
EF = −3. For compare, the blue dotted line shows the seebeck when lead-2,4 have
the same VR as lead-1,3. The other parameters are T = 0.01, and L = 19a.
In Fig.refcII6, we show the spin Nernst coefficient Ns and the seebeck coefficient
S versus the Rashba SOI VR in the clean system (W = 0) for EF = −3. The seebeck
coefficient S decreases and maintains for a small value for quite a while before shows
60
−0.005
0
0.01
0.02
0.025
N
s
0 1 2
W
(a)
−0.05
0
0.03
0.05
0.1
.15
N
s
−3.8 −3.6 −3.4 −3.2 −3
E
F
(b)
Figure 2.8: (a). Ns vs. the strength of disorder W for different Fermi lev-el
EF = −3.8(solid blue), EF = −3.6(dashed red), EF = −3.0(dotted black),
EF = −2.2(dotted dashed green); (b). Ns vs. Fermi energy EF = −3.8 for W = 0
(dashed black) and W = 1.4 (solid black).Other parameters are VR = 0.05, T = 0.01,
and L = 19a.
another peak. This is because increasing VR moves the energy bands and makes them
go through the fermi energy. It should be mentioned that we found the negative
seebeck coefficient S (see also Fig.2.6(d)), which means a longitudinal current occurs
in the opposite direction of the temperature gradient ΔT . This is due to the boundary
conditions VR = 0 at leads-2,4. As a compare, we also show S for a uniform system,
i.e. leads-2,4 having the same strength of VR as in the sample. For this situation, the
seebeck coefficient S is no longer negative. In fact, when the Rashba SOI is absent in
the leads-2,4, an interface between VR = 0 and VR ̸= 0 ocurrs[38], this interface causes
additional scattering for an incident electron. In some special case like EF = −3, this
may make the electrons below EF easier to transport than the electrons above EF ,
thus a negative S.
Finally we discuss the disorder effect on the spin Nernst effect. Fig.2.8 shows Ns
versus disorder strength W for different Fermi energies. The calculations are averaged
61
over 500 disorder configurations. Around W < 1.7, Ns shows an oscillatory structure.
Ns changes sign with increasing of the disorder strength (see EF = −3.6 and −3.0
in Fig.5). It is interesting to see that, comparing to a clean system (W = 0), Ns
can be unexpectedly increased by disorder W. This is because the disorder changes
the oscillating structure of Ns (see Fig.2.6b). As expected, the disorder decreases the
strength of oscillating, however, it also shifts the peak positions of Ns. It is possible
to have a peak in Ns at finite disorder while it is almost zero initially at clean limit.
In Fig.5a, around W = 1 ∼ 1.5, for the Fermi level EF = −3.8, −2.2, we can see that
Ns is up to about three times of Ns at W = 0. The behavior of Ns v.s. W is very
apparent when the Fermi level EF is close to the bottom of energy band (EF = −4).
For EF = −3.8, Ns is much bigger than those at other Fermi levels, and we can see
a very remarkable peak at about W = 1.4. For EF = −2.2, Ns begins from −0.005,
changes its sign at about W ∼ 1.25 and than increases, again reaches to 0.005 at
about W ∼ 1.75. With a very big disorder, Ns should go to zero as system enters
into an insulating regime. We find that the zero of Ns occurs at W = 3 for VR = 0.05.
This is roughly independent of the locations of the Fermi energy.
In summary, in the absence of a perpendicular magnetic field, the interplay be-tween
the spin Nernst effect and the seebeck effect is investigated in a two-dimensional
cross-bar with a spin-orbit interaction. The spin Nernst effect exhibits an oscillatory
structure for a wide range of the Rashba SOI. With a large Rashba SOI, the Ns os-cillation
has a peak when the seebeck coefficient possesses one. However, the inverse
condition is not always satisfied, namely, the seebeck coefficient can be almost zero
while Ns has a peak. The disorder effect on the spin Nernst effects is also studied.
We find that disorder can enhance Ns up to three times for some Fermi levels. In
addition, the disorder can also change the sign of spin Nernst effect. Moreover, the
limit of disorder where Ns goes to zero is independent of the Fermi energy.
62
CHAPTER 3
The Topological System with a Twisting Edge Band: Position-Dependent
Hall Resistance
3.1 Motivation and Background
The topological system has attracted much attention in recent years [28, 43]. About
twenty years ago, by proposing the quantum anomalous Hall effect (QAHE) in graphene
[7], Haldane gave a simple two-band model to study a topological system. Recently,
the topological insulator material is first predicted and then experimentally observed
in some two-dimensional (2D) systems [10, 8, 9]. The three-dimension topological
materials are also discovered soon after [44].
In research of the robustness of topological system, the analysis of edge states is
to be an effective approach [45]. The helical edge states for 2D topological systems
are shown to have the topological protection of Z2 [46], and the scattering between
them is prohibited without breaking time reversal symmetry. While with edge bands
distortion, they may cross the Fermi surface more than one time, which may also give
rise to some extra edge states [28]. However, these extra edge states can not bring
new topological phases, and are not protected by the topology [47]. They are thought
easy to be affected and are treated as unimportant in the earlier studies.
In this chapter, we show a nontrivial effect from the topological unprotected edge
states. While a system is with both the topological protected and unprotected edge
states, the Hall conductance depends on the measurement location even for a transla-tionally
invariant system. This novel property survives at a finite disorder, however,
it is absent in both topological trivial systems and normal topological systems. Thus,
63
this unique property is the hallmark of a topological system with a twisting edge
band.
3.2 Model and Hamiltonian
The band structure of our system is shown in Fig.4.1(A). Below we provide one
example of how to achieve this band structure. Without loss of generality, we take
the simple ν = 1 topological system as an example, which consists of one pair of
topological protected edge states. The AB-stacked square lattice QAHE system [48]
is chosen, in which the two type of atoms are needed. As shown in Fig. 1(C), we
can assume atom A at s level and atom B at the lowest p level [49]. Generally, this
p-orbital may not along the direction of lattice structure, here we choose it along
±⃗e1-direction. The check board magnetic field is also applied by the Peierls phase
ϕ0 = π/2 when an electron jumps from A to B along ±⃗ey-direction. Supposing the
on-site energy of A and B are the same, set to be the zero energy point. The tight-binding
Hamiltonian can thus be written as H = H1 +H2, with H1 (H2) the nearest
(next-nearest) hopping Hamiltonian:
H1 = −tab
Σ
i
[
b
†
i+δxai + eiϕ0b
†
i+δyai + h.c.
]
+tab
Σ
i
[
a
†
i+δxbi + e−iϕ0a
†
i+δybi + h.c.
]
(3.1)
H2 = −
Σ
i
[
ta1a
†
i+δe1ai + ta2a
†
i+δe2ai + h.c.
]
−
Σ
i
[
tb1b
†
i+δe1bi + tb2b
†
i+δe2bi + h.c.
]
(3.2)
The sign of hopping energies are determined by the sign of overlap integrals of two
atomic wave functions centered at different sites [50]. In Fig. 4.1(B), the shape and
sign of atomic wave functions are shown. In the first line of Hamiltonian (1), tab is
the hopping term from A to B along the +x and +y-directions, i.e. from s-orbital of
64
A to the positive part of p-orbital of B, set to be positive. While in the second line
of (1), the hopping from B to A along the same direction is from the negative part
of p-orbital of B to s-orbital of A, so it gets a negative sign. ta1 and ta2 are the next
nearest neighbor hopping at A sublattice along ⃗e1 and ⃗e2, respectively. tb1 and tb2
are the counterparts for the B sublattice. One can check ta1, ta2, tb2 > 0 and tb1 < 0.
Besides, we also have ta1 = ta2 and |tb1| ̸= |tb2|, due to the anisotropy of the p level.
It is easy to discuss this tight-binding Hamiltonian in k-space. Because the sys-tem
is translationally invariant, we have H(k) = h0(k) + σ · p(k). Here px(k) =
2tab sin(kya0), py(k) = −2tab sin(kxa0). The next nearest hopping gives pz(k) =
−[(ta1−tb1) cos(kxa0+kya0)+(ta2−tb2) cos(kxa0−kya0)] and a nonconstant h0(k) =
−[(ta1 + tb1) cos(kxa0 + kya0) + (ta2 + tb2) cos(kxa0 − kya0)]. Here a0 is the distance
between the nearest neighbor atoms A and B. The Chern number of the system can
be calculated in k-space [28, 4] by ν =
∫
d2kF/2π. For our system, when there exists
a real gap, the Chern number of the lower band gives ν = 1.
3.3 Twisting Edge Band
The coexistence of distorted edge band and normal edge band originates from the
symmetry breaking of the eigenvalue λ± = h0±|p|. These two eigenvalue correspond
separately to the upper and down bands. Because of the next nearest hopping, the
symmetry of λ± reduces from C4 to C2. The Dirac points at (0, 0) and (±π,±π)
have different energy values as the Dirac points at (0,±π) and (±π, 0). If the system
is constrained at ⃗ex or ⃗ey direction, each projected Dirac point in fact contains two
type of Dirac points, so the projected Dirac points remains the same. However, if
the system is constrained at ⃗e1-direction [see Fig.4.1(C)], i.e., if with the zigzag edge,
each projected Dirac point contains only one type of Dirac point, the two projected
Dirac points are different with each other, as shown in Fig.4.1(A). Consequently, the
two edge bands may have different group velocities |∂ε(k)/∂k|. In this way, the edge
65
0 0.5 1
−0.5
0
0.5
(A1)
0 0.5 1
a c
b d
(A2)
K (:/a)
0 0.5 1
(A3)
Energy (|tab|)
5 8 47 50
0.4
0.8
|A|2
Position
(B)
a,
b
c
d
(D)
c
b
a
d
~ex
~ey
~e1
~e2
+
(E)
Figure 3.1: (Color online) (A) The energy band structures of zigzag-edge ribbon of
topological system, with the ribbon width W = 50a and a =
√
2a0. We choose the
parameters tab = 10 and ta1 = ta2 = tc + 0.1, tb1 = −tc − ts, tb2 = tc for all the
subplots. (A1) indirect semi-metal with tc = 1.4 and ts = −0.4, (A2) the twisting
edge band system with tc = 1.4 and ts = 0.4, and (A3) the normal topological system
with tc = 0.7 and ts = 1. (B) The distribution |ψ|2 of the four edge states of (A2).
(C) The lattice structure of the system. (D) The schematic diagram of the four edge
states of (A2). (E) The twisting edge band of (A2) can be treated as mix of the
topological protected and unprotected systems.
66
bands are distorted.
To get a twisting edge band, we need a little more effort. Define At = (ta1+ta2) and
Bt = (tb1+tb2), we can get the bulk gap of the system as Δ = 2(|At−Bt|−|At+Bt|).
While AtBt > 0 gives an indirect negative gap Δ. In this case, although the system
has a twisting edge band [see red curve in Fig.4.1(A1)], but it is without a bulk gap,
which creates an indirect semi-metal. The bulk insulator needs Δ > 0 thus AtBt < 0.
When gap Δ is large, the system may only have a distorted edge band but no twisting
edge band [see Fig.4.1(A3)], which is the normal 2D topological insulator. When gap
Δ is positive but small, we may have a twisting edge band [Fig.4.1(A2)]. We also have
another bigger ‘gap’ Δ2 = 2(|At −Bt|+|At +Bt|), corresponding to the normal edge
band [see the blue curve in Fig.4.1(A)]. In our system At > 0, and it’s no harm to set
At > |Bt|, then we can get Δ = −4Bt and Δ2 = 4At. This means that the twisting
and normal edge bands are independently determined by Bt and At, respectively. If
we choose tb1 = −tc − ts, tb2 = tc with tc, ts > 0, the gap Δ is simplified to Δ = 4ts.
Due to the edge band being twisted, it can cross Fermi surface EF three times,
marked by a, c and d [see Fig.4.1(A2)]. The other normal edge band meets Fermi
surface at b. Fig.4.1(B) shows the distribution |ψ|2 v.s. location for these four s-tates.
We can see that, all four states are localized on the edges of the sample, the
distribution is almost zero inside the bulk. Among them, the three states a, c and d
are localized on the upper edge, while state b is localized on the lower edge [see also
Fig.4.1(D)]. As the Chern number of the system is ν = 1, only one pair of the edge
states are protected by the topology, while another two are not protected. There is no
doubt that b is protected by topology since it is the only one edge state on the lower
edge. The other topology protected state is a mixture of these three degenerated edge
states a, c and d on the upper edge. Here we notice that the present system can be
treated as the combination of the normal topological system plus a topological trivial
system with one pair of unprotected edge states, as shown in Fig.4.1(E).
67
3.4 The Translational Invariance Symmetry Breaking of the Hall
Resistance
Now, let us study the transport property of the system using the 6-lead set-up. As
shown in Fig.4.2(c), lead-1 and lead-4 are made by the same materials of the sample,
which can support the well-defined edge states inside the gap of sample. The vertical
leads 2, 3, 5, 6 are made of a metal, which can afford as much modes as possible. A
small longitudinal voltage gradient is applied by setting the lead-1 at V/2 and the
lead-4 at −V/2, providing the longitudinal current I1. We use the zero temperature
Landauer-B¨uttiker formula Ip = e2
h
Σ
q̸=p(Vp − Vq)Tp,q, with Tp,q the transmission
coefficient from the lead q to p [51]. The vertical voltage Vp can thus be obtained by
using the open boundary condition, i.e. by letting the corresponding leads to have
zero current: Ip = 0 with p = 2, 3, 5, 6. Finally, the Hall and longitudinal resistances
can be obtained from Rp,q ≡ (Vp − Vq)/I1.
For the three sets of parameters used in Fig.4.1(A), by changing the Fermi energy,
in Fig.4.2(b) we plot the Hall resistance R2,6, measured on the left side of the sample,
and R3,5, on the right side. We also draw in Fig.4.2(a) the longitudinal resistance
R2,3 for the upper edge, and R6,5 for the bottom edge. For the parameters used
in Fig.4.1(A1) with an indirect negative gap, the coexistence of twisting edge band
and bulk band does not directly show a topological property. Two Hall (longitudinal)
resistances are very small and almost equal, because that the system is translationally
invariant. For the parameters used in Fig.4.1(A3), though the edge band is already
somewhat distorted with two edge currents having different speeds, the measurement
can give no new information other than the normal topological insulator. The Hall
(longitudinal) resistances measured at different place (edge) are the same. Within
the gap, the Hall resistances give a quantized plateau (h/e2) characterized by the
topological number ν = 1, and two longitudinal resistances are zero, because of the
absence of back scattering.
68
0
0.1
0.2
0.3
−2 −0.8 0.8 2
−1
−0.5
0
R2,3 (h/e2)
R6,5 (h/e2)
R2,6 (h/e2)
R3,5 (h/e2) Fermi Energy (|tab|/10)
(a)
(b)
1
2 3
4
6 5
(c)
Figure 3.2: (Color online) For the three sets of parameters used in Fig.4.1(A), the
corresponding resistances of the system v.s. the Fermi energy: (a) the longitudinal
resistances and (b) the Hall resistances. The wide lines are for R6,5 and R2,6, the
narrow lines are for R2,3 and R3,5. In both figures, the pair of lines with the broadest
quantized plateau (−2 ∼ 2) are for Fig.4.1(A3); the pair of lines only have plateau
within −0.8 ∼ 0.8 are for Fig.4.1(A2); for Fig.4.1(A1), the pair of lines have no
plateau. Other parameters used for the calculation: the ribbon width W = 50a, the
distance between vertical leads L = 20a. (c) is the schematic diagram of the 6-lead
measurement we used for (a) and (b).
69
For the parameters used in Fig.4.1(A2), the twisting edge band case, the results are
very different and interesting. When the Fermi energy EF is within the gap but out of
the range of the twisting of edge band, all measurements still show normal topological
property by giving the plateau. When the Fermi energy goes within the twisting area,
the situation is totally changed. Let us first look at the longitudinal resistance. We
still have R6,5 = 0, because there is only one edge state b on the bottom edge of the
sample, no back scattering is allowed there, the voltage drop is zero with V6 = V5.
However, on the upper edge, R2,3 is nonzero and it is about 0.25h/e2. This is because
we have three edge states on the upper edge, two of them move to the right and the
other one moves to the left. As one pair of them moves in the opposite directions, not
topologically protected, the back scattering is allowed. Thus, the voltage may drop,
V2 ̸= V3 to give a nonzero resistance R2,3 on the upper edge. Specifically, as lead-2 is
on the left side of the lead-3, we have V/2 = V1 > V2 > V3 > V4 = −V/2. The two
Hall resistances also change and they are no longer equal to the value h/νe2, although
the Chern number of the system is still ν = 1. In particular, as V2 ̸= V3 and V6 = V5,
we can see that the left side Hall resistance R2,6 = (V2 − V6)/I1 is no longer same as
the right side Hall resistance R3,5 = (V3−V5)/I1: |R2,6| is decreased to about 0.7h/e2
within the twisting-edge-band region but |R3,5| is larger than 0.9h/e2. It should be
emphasized again that the present system is translationally invariant. However, from
the results above, the Hall resistance does break the translational invariance. This
novel phenomenon, the breaking of the translational invariance of the Hall resistance
in a translationally invariant system, origins from the twisting edge band and the
combination of the topological protected and unprotected edge states. This property
is unique to the topological system with a twisting edge band and can not be observed
in either normal topological insulators or non-topological systems. In addition, we
also witnessed the oscillation of R3,5 and R2,3 for the parameters used in Fig.4.1(A2).
This is because of the Fabry-Perot interference between the lead-2 and lead-3. The
70
number of oscillation are determined by the distance between them.
0
0.1
0.2
0.3 (a)
0
0.1
0.2
(b) 0.3
−0.8 0.8
−1
−0.5
0 (c)
−0.8 0.8
−1
−0.5
(d) 0
R2,3 (h/e2)
R6,5 (h/e2)
R2,6 (h/e2)
R3,5 (h/e2)
Fermi Energy (|tab|/10)
Figure 3.3: (Color online) For the parameters used in Fig.4.1(A2), the resistances
v.s. Fermi energy for sample widthes W = 50a (the broadest black line), 60a (the
red line), and 80a (the thinnest green line).
In order to confirm that the breaking of the translational invariance of Hall resis-tance
is due to the edge states, we show the Hall and longitudinal resistances versus
the width of the sample in Fig.4.3. The change of width only has the effect on the
bulk bands and should not affect the edge bands when the sample is wide enough.
From Fig.4.3, it can be seen that outside the gap, all the four resistances are changed
when the width changes. However, within the gap, the resistances maintain the same
for different widthes. It clearly shows that the breaking of the translational invariance
of the Hall resistance does come from the twisting edge band.
One may argue that the 6-lead measurement itself already breaks the translational
invariance, as the left Hall bar is close to the higher voltage side and the right Hall
bar is close to the lower voltage side [52]. Following we consider the 4-lead set-up of
Hall resistance [see the inset in Fig.4.4(A)] and vary the measurement position. In
addition, disorder effect is also studied. Let us suppose the system having a uniform
distributed Anderson disorder, that does not break the translational invariance. In
the presence of disorder, the Hall resistance |R2′,4′ | increases with the measure position
71
moving from the left to the right [see Fig.4.4(A)]. This clearly implies that the Hall
resistance depends on the measure position, breaking the translational invariance. In
addition, on the left edge of the sample, the Hall resistance is almost not affected by
the disorder. When the sample is long enough, the Hall resistance measured on the
right edge is close to |R2′,4′ | = h/νe2, quantized by the topological number ν = 1.
50 100 150 200
−1
−0.9
−0.8
−0.7
R2′ ,4′ (h/e2)
Position of lead-2′, 4′
(A)
1′
2′
3′
4′
(a) (b) (c)
Antiband E=0
Crossing S.O.I
(B)
Figure 3.4: (Color online) (A). For the parameters used in Fig.4.1(A2), the 4-lead
measurement of Hall resistances v.s. the position to measure at EF = 0. From
top to bottom, the blue, red, green, and black lines are for the disorder strength
Dis = 0, Δ/8, Δ/4, and Δ/2, respectively. Here the gap is Δ = 0.16|tab|. The results
are calculated with the width of sample W = 70a, by the average of 700 disorder
configurations. (B). The schematic diagram of another method to realize twisting
edge bands.
Finally, we should point out that though our results are obtained from an ideal
model, the twisting edge bands can be found in some real systems. For example,
supposing we initially have the two band system as shown in Fig.4.4(B-a), whose
symmetry axis of upper band is shifted from that of the lower band. Then with the
anti-band crossing [Fig.4.4(B-b)], the pseudo spin-orbital interaction may open a gap
and leads to a twisting edge band [Fig.4.4(B-c)].
In conclusion, we have shown that, with a twisting edge bands, the system has
both the topological protected and unprotected edge states. In such a system, the
72
Hall resistance is not determined by the topological number alone. In particular,
the Hall resistance depends the measure position even for a translationally invariant
system.
73
CHAPTER 4
Proximity Effect and Majarana Fermion
In this chapter, I will study the quantum anomalous Hall effect described by a class
of two-component Haldane models on square lattices. We show that the latter can be
transformed into a pseudospin triplet p+ip-wave paired superfluid. In the long wave
length limit, the ground state wave function is described by Halperin’s (1, 1,−1) state
of neutral fermions analogous to the double layer quantum Hall effect. The vortex
excitations are charge e/2 abelian anyons which carry a neutral Dirac fermion zero
mode. The superconducting proximity effect induces ‘tunneling’ between ‘layers’
which leads to topological phase transitions whereby the Dirac fermion zero mode
fractionalizes and Majorana fermions emerge in the edge states. The charge e/2 vortex
excitation carrying a Majorana zero mode is a non-abelian anyon. The proximity
effect can also drive a conventional insulator into a quantum anomalous Hall effect
state with a Majorana edge mode and the non-abelian vortex excitations.
4.1 Introduction
The discovery of the quantum Hall effect (QHE) [53] opened an era for studying
topological quantum phases [4]. Some twenty years ago, Haldane [7] proposed the
quantum anomalous Hall effect (QAHE) for electrons on a two-dimensional lattice.
This is generalized to the time reversal invariant topological insulators[54] in two
dimensions [55, 56, 57, 58] and three dimensions [59, 60, 61, 62, 63]. With the band
inversion in these system, it raises the hope for realizing the QAHE in a two-band
model of two-dimensional magnetic insulators [64]. Candidate materials for this effect,
74
HgTe doped with Mn [65] and a tetradymite semiconductors doped with transition
metal elements [66], have been predicted. Other proposals are also made in condensed
matter systems recently[67, 68]. The QAHE has also been proposed for cold atom
systems [69, 70, 71].
Another advance is the search for topological phases with non-abelian anyons [72]
that have potential applications for quantum computing [73]. In addition to the
ν = 5/2 fractional QHE, it has been shown theoretically that the superconducting
proximity effect on the surface state of topological insulator [74] and on semiconduc-tors
with strong spin-orbit coupling and Zeeman splitting [75] provides a new avenue
for generating the Majorana zero mode and non-abelian vortex excitations.
In this work, we study the QAHE for a class of two-component Haldane models
on a square lattice. The physical degrees of freedom represented by the components
depend on the microscopic details: the real spins of electrons, band indices [64], the
top-bottom surface states of three-dimensional topological insulator [66], as well as
the sublattice indices as exemplified below. We show that, in a pseudospin represen-tation,
the QAHE system can be transformed into a chiral p + ip-wave pairing state
involving both pseudospin components. The ground state wave function is given by a
determinant of the pairing functions whose long wave length limit is a charge neutral
Halperin (1, 1,−1) state analogous to the double layer QHE [76]. There are abelian
anyon excitations with charge e/2, despite that the present model describes an integer
QAHE system.
When s-wave superconductor develops in the QAHE system due to the proximity
effect, tunneling between the different isospin components takes place. We find that
the system displays continuous transitions between topological phases with abelian
and non-abelian anyon excitations. Specifically, a topological phase transition from
the Hall conductance 2e2/h to e2/h happens for sufficiently strong proximity induced
pairing because one of the pseudospin components is driven to a strong pairing s-
75
tate analogous to that in the ν = 1/2 double-layer fractional QHE discussed by Ho
[77] and Read and Green [78]. The remaining pseudospin component is in the weak
pairing state described by a Moore-Read Pfaffian [72]. A Majorana zero mode ap-pears
in the edge state and the vortex excitation carrying this Majorana mode is a
non-abelian anyon. Interestingly, the model also exhibits a topological trivial phase
without the QAHE when the triplet p + ip-wave pairing is in a topologically unpro-
tected weak pairing state. We find that the proximity effect can drive one of the
pseudospin components into a topologically protected weak pairing state with quan-tized
Hall conductance −e2/h, edge Majorana zero mode, and non-abelian anyon
vortex excitations (see also Ref.[79]).
4.2 Models
The Hamiltonian of the two-component model in the lattice momentum space is given
by
H0 =
Σ
k
[(px + ipy)c
†
akcbk + h.c. (4.1)
+ hz(k)(c
†
akcak − c
†
bkcbk) + h0(k)(c
†
akcak + c
†
bkcbk)]
where ca(b)k annihilate an electron of component a(b) with momentum k and h0(k) is
the dispersion due to hopping among electrons of the same component. The physical
origin of the terms proportional to px +ipy and hz depends on the system of interest.
If (a, b) label the electron spin, px + ipy → kx + iky arises from Rashba spin-orbit
coupling [64] and hz is the magnetization. If (a, b) are the orbital indices, px + ipy
describes the orbital hybridyzation [64, 65] and hz is the crystal field splitting. For the
doped tetradymite semiconductors, they constitute the spin-orbit coupling associated
with the three-dimensional topological insulator [66].
76
4.2.1 A Generalized Haldane Model in Square Lattice
In Fig. 4.1, we give an explicit realization of the Hamiltonian (4.1) for spinless fermions
on a square lattice where (a, b) label the A and B sublattices. This turns out to be a
modified Haldane’s model where the complex hopping induces ±π staggered plaquette
flux with the link phase distribution shown in Fig.4.1 (right panel). The corresponding
Halimtonian with this setup is given by
H0 = −t
Σ
ia
[c
†
ia+δxcia + ic
†
ia+δycia + h.c.]
+ t
Σ
ib
[c
†
ib+δxcib
− ic
†
ib+δycib + h.c.]
− t′Σ
ia
[c
†
ia+δaa1cia + c
†
ia+δaa2cia + h.c.] (4.2)
+ t′Σ
ib
[c
†
ib+δbb1
cib + c
†
ib+δbb2
cib + h.c.]
− M
Σ
i
(c
†
iacia
− c
†
ibcib)
where c
†
ia;b is the creation operator of the electron at the site ia,b on the A or B
sublattice. The coordinates and vectors are figured in the left panel of Fig. 4.1. The
lattice spacing a is set to be unit. t and t′ are the nearest neighbor hopping amplitude
and the next nearest neighbor’s. ±M is the on-site energy on ia and ib, respectively.
It can also be realized in cold atom context with a simpler staggered flux while the
atom orbital is different on the A and B sublattice [70].
Making the Fourier transformation, the Hamiltonian is exactly given by Eq. (4.1)
with px + ipy = −2t(sin ky + i sin kx), hz(k) = −M − 4t′ cos kx cos ky, and h0 = 0,
where t and t′ are the hopping amplitudes between nearest and next nearest neighbors
indicated in Fig.4.1 and ±M is the on-site energy for the A and B sublattices. Note
that the ‘magnetization’ contains cos kx cos ky and is different from that used in [64].
A similar model can also be realized in cold atom systems [70]. For h0 = 0, there is
77
an important O(2) symmetry associated with the U(1) particle number conservation
and a Z2 under c
†
a,k
→ icb,−k, i.e., a C · S (particle-hole · sublattice) symmetry. If
h0 ̸= 0 but can be adiabatically driven to zero, we think the results of this work are
also valid.
We now study Hamiltonian (4.1) in the A-B sublattice where k is confined to the
reduced first Brillouin zone bounded by kx±ky = ±π due to the A-B sublattices. Our
results can be extended directly to other relevant cases discussed above with lattice
translation symmetry.
The eigen-energy of (4.1) is given by E0 = ±
√
h2z
+ |p|2. There are two inde-pendent
Dirac points at (0, 0) and (π, 0) in the reduced zone. When an extended
s-wave pairing is induced by proximity effect [74] on the QAHE system, the total
Hamiltonian is given by H = H0 + Hsc, where
Hsc =
Σ
k
f(k)(ca−kcbk + c
†
bkc
†
a−k), (4.3)
and f(k) = 2Δ(cos kx + cos ky). The eigen-energy of the total Hamiltonian is given
by E = ±
√
(hz ± f)2 + |p|2. The energy gap of the QAHE at the Dirac point (0, 0)
closes when Δ = 1
4 (M + 4t′), whereas the QAHE gap at (π, 0) is unperturbed since
f(k) = 0 for kx ± ky = ±π.
y
x
A
B
y
x
aa1 bb1
bb2 aa2
2
2
2
2
2
2
2
−2
−2
2
Figure 4.1: The square lattice model for Hamiltonian (1) Left: The two-sublattice
and hopping. Right: The flux distribution. Hopping along arrowed vertical links
generates a phase π and arrowed horizontal links a phase π/2. A net flux of −2π
(π/2) is accumulated for the dark (light) grey triangular blocks. The rest of the
hopping is real.
78
To unveil the ground state wavefunction and the topological properties of E-q.
(4.1), we introduce a pseudospin representation by mixing the electron and hole
of different component (which in fact indicates a particle-hole transformation of b-component)
c↑k = (cak + c
†
b−k)/
√
2, c↓k = i(cak − c
†
b−k)/
√
2. (4.4)
Under this unitary transformation, the Hamiltonian in terms of fermions carrying the
pseudospin becomes
H0 =
Σ
k;s=↑,↓
[hzc
†
skcsk − ((px − ipy)cskcs−k + h.c.)/2]. (4.5)
Both the pseudospin-↑ and ↓ fermions, having a band dispersion hz(k), are in the
p+ip-wave paired states. Hamiltonian (4.5) is closely related to the ν = 1/2 double-layer
fractional QHE if we identify the pseudospins with the even/odd states of the
isospin (layer index) in the context of triplet chiral p-wave pairing [77, 78].
4.2.2 Winding Numbers
We now calculate the winding number, which describes the mapping from the reduced
zone (a torus) to a target sphere specified by the unit vector n = (px,py,hz)
E0
. The
winding number is given by C = C↑ + C↓ with Cs = 1
4π
∫
d2kn · ∂kxn × ∂kyn in the
continuum limit. A direct calculation yields C↑ = C↓ = 1 and C = 2 which reflects
the fact that the whole first Brillouin zone covers twice of the sphere with such a map
n. This result can be understood intuitively by considering the vector components of
n near the two Dirac points (0, 0) and (π, 0). For a small deviation q ∼ 0, they are
(−2tqy, 2tqx,−M−4t′)/E0 near (0, 0) and (−2tqy,−2qx,−M+4t′)/E0 near (π, 0) (See
Fig. 4.2). Therefore, the Dirac points are mapped to the north and south poles which
are covered once. A semi-sphere including a pole contributes ±1/2 to the winding
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number depending on the pole’s frame. When M < 4t′, both poles are in the right
hand frame and each semi-sphere contributes 1/2 which gives C↑,↓ = 1 and C = 2.
Thus, this QAHE has Hall conductance 2e2
h .
(a) (b)
qx
-qy
-z
Figure 4.2: : The frames and frame change from M < 4t′ to M > 4t′. (a)
(−2tqy, 2tqx,−M − 4t′) at (0, 0). The frame is not changed and is the right. (b)
(−2tqy,−2qx,−M + 4t′) at (π, 0). The frame is changed from the right for M < 4t′
to the left for M > 4t′.
4.3 Ground states and anyonic excitations
4.3.1 The (1,1,-1)-state and abelian anyons
The paired state has BCS coherence factors |us,k|2 = 1
2(1 + hz
E0
), |vs,k|2 = 1
2(1 − hz
E0
)
and pairing functions
gs(k) = vs,k/us,k = g(k) = −(E0 − hz)/p. (4.6)
For M < 4t′, the effective chemical potential at the Dirac point (π, 0) μ(π,0) =
−hz,(π,0) = M − 4t′ < 0 such that us ∼ 1 and gs ∼ vs ∼ qy + iqx. This is in the
strong pairing regime [78] and can be thought as the ‘infinity’ point in the continuum
theory. On the other hand, the effective chemical potential at the Dirac point (0, 0)
μ(0,0) = −hz;(0,0) = M + 4t′ > 0 which leads to vs ∼ 1 and gs ∼ 1/us ∼ 1/(qy − iqx).
This singular pairing function in the long wavelength limit is the hallmark of the
topologically nontrivial p + ip weak pairing phase [78]. Therefore, the long distance,
low energy physics is determined by the weak pairing of the qusiparticles carrying
80
both pseudospins near the Dirac point (0, 0). The ground state is given by
|Gs⟩ ∝ exp[
1
2
Σ
k
g(k)(c
†
↑kc
†
↑−k + c
†
↓kc
†
↓−k)]|0s⟩, (4.7)
where |0s⟩ is the vacuum for the fermions carrying the pseudospin, csk|0s⟩ = 0. From
the transformation (4.4), it is clear that this vacuum is empty of the a-electrons but
filled with the b-electrons. The ground state in Eq. (4.7) resembles the neutral part
of the (3, 3, 1)-state in ν = 1/2 double layer fractional QHE [76, 78]. Going back to
the original a-b component electrons, the ground state wave function of Eq. (4.1) is
of the form of a determinant of the pairing function g(r), the Fourier image of g(k),
Ψ({rai
, rbi
}) = ⟨0|
Π
i
ca(rai
)cb(rbi
)|G⟩ ∝ det[g(rai
− rbj
)],
where ra,b
i are the coordinates of the two-component electrons in the ground state |G⟩
out of the vacuum ca(b)k|0⟩ = 0. The ground state |G⟩ and the vacuum |0⟩ of the origi-nal
Hamiltoni