theoriesandapplicationsoftopologicalinsulators xu-ganghe(何何...

Theories and Applications of Topological Insulators

A Dissertation presented

by

Xu-Gang He (何何何绪绪绪纲纲纲)

to

The Graduate School

in Partial Fulfillment of the

Requirements

for the Degree of

Doctor of Philosophy

in

Physics

(Theoretic Condensed Matter Physics)

Stony Brook University

May 2016

ii


The Graduate School

Xu-Gang He

We, the dissertation committe for the above candidate for the

Doctor of Philosophy degree, hereby recommend

acceptance of this dissertation

Advisor: Prof. Wei KuPhysics and Astronomy Department

Prof. Derek TeaneyPhysics and Astronomy Department

Prof. Xu DuPhysics and Astronomy Department

Prof. Oleg ViroMathematics Department

This dissertation is accepted by the Graduate School

Charles TaberDean of the Graduate School

iii

Abstract of the Dissertation

Theories and Applications of Topological Insulators

by

Xu-Gang He

Doctor of Philosophy

in

Physics

(Theoretic Condensed Matter Physics)


2016

As a new kind of state of the materials, topological insulators have beenintensively studied by researchers very recently. The concept of topologycoming from the pure mathematics, captures the key aspect of the phaseterm of the wave functions of topological insulators. In other branches ofphysics, the topology properties have been studied for some famous objects–such as magnetic monopoles and quantum Hall effects. Our study in topo-logical insulator, a close cousin of quantum Hall effect, reveals its topologicalcharacteristic and phase transition property from the complex crystal mo-mentum point of view. In fact, using the analytic continuation method, wefound that the effective magnetic monopole distributes in the complex mo-mentum space, and its swapping mechanism during the topological phasetransition, guarantees robust metallic states at the critical point. Severalimportant aspects of topological insulators have been studied–such as intrin-sic instability due to Mexican hat bands structure, higher Chern numbermodel and exact supersymmetry of quantum mechanics. Based on the aboveresults, further possible applications have been proposed. Lastly, I introducesome recently inspiring experiments and the future possibilities.

iv

To My Parents, 何何何龙龙龙and 肖肖肖琳琳琳玲玲玲And My Lovely Wife, 张张张黎黎黎丽丽丽

CONTENTS v

Contents1 Chapter 1 1

1.1 Priory Considerations . . . . . . . . . . . . . . . . . . . . . 1

2 Chapter 2 42.1 Physics Rediscovery of the Mathematical Concepts . . 42.2 Discovery of TI’s in Experiments . . . . . . . . . . . . . . 10

3 Chapter 3 203.1 Hall Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Quantum Hall Effects . . . . . . . . . . . . . . . . . . . . . 233.3 IQHE vs Topology . . . . . . . . . . . . . . . . . . . . . . . 303.4 Monopole in Complex Momentum Space . . . . . . . . . 34

4 Chapter 4 494.1 Higher Chern Number Model . . . . . . . . . . . . . . . . 494.2 Real Space Models . . . . . . . . . . . . . . . . . . . . . . . 554.3 Phase Transition with Fixed Mass Term . . . . . . . . . 59

5 Chapter 5 635.1 Band Structures of TI’s . . . . . . . . . . . . . . . . . . . . 635.2 Symmetry Breaking Instabilities . . . . . . . . . . . . . . 70

6 Chapter 6 776.1 Supersymmetry in Quantum Mechanics . . . . . . . . . 776.2 Supersymmetry in TI . . . . . . . . . . . . . . . . . . . . . 82

7 Chapter 7 867.1 Bulk Signature of the Topological Phase Transition . . 86

8 Chapter 8 988.1 High Resistance of the In-doped Pb1−xSnxTe . . . . . . 98

9 Conclusions 105

Appendices 107

CONTENTS vi

A Landauer-Büttiker Formalism 107A.1 Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . 107A.2 Quantum Spin Hall Effect . . . . . . . . . . . . . . . . . . . . 109

B Invariance of Chern Number Formalism 110

LIST OF FIGURES vii

List of Figures/Tables/Illustrations

List of Figures1 Single monopole at the center of a sphere . . . . . . . . . . . . 72 Quantum spin Hall effect . . . . . . . . . . . . . . . . . . . . . 123 Experiments on Quantum spin Hall effect . . . . . . . . . . . . 164 ARPES results . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Spin texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Classical Hall effect . . . . . . . . . . . . . . . . . . . . . . . . 227 IQHE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 Landau levels . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Laughlin ribbon . . . . . . . . . . . . . . . . . . . . . . . . . . 2910 Monopole in d-space . . . . . . . . . . . . . . . . . . . . . . . 3512 Monopole string in complex k-space . . . . . . . . . . . . . . . 4013 Monopole swapping in topological phase transition . . . . . . 4514 Kink and Non-kink . . . . . . . . . . . . . . . . . . . . . . . . 5415 Higher Chern number . . . . . . . . . . . . . . . . . . . . . . . 5616 Contour Plot of the components of ~d . . . . . . . . . . . . . . 6017 Winding term around the Γ point . . . . . . . . . . . . . . . . 6118 DOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6719 Maxican hat band structure . . . . . . . . . . . . . . . . . . . 6820 Superconducting susceptibility . . . . . . . . . . . . . . . . . . 7121 Lifshitz transition . . . . . . . . . . . . . . . . . . . . . . . . . 7322 Symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . 8123 Band structure . . . . . . . . . . . . . . . . . . . . . . . . . . 9024 Infrared reflectance at room temperature . . . . . . . . . . . . 9125 Infrared reflectance at low temperatures . . . . . . . . . . . . 9226 Resistivity with temperature dependence . . . . . . . . . . . . 9927 Surface conductivity . . . . . . . . . . . . . . . . . . . . . . . 103

LIST OF TABLES viii

List of Tables1 DOS of VHS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

LIST OF TABLES ix

Acknowledgements

I would like to thank my advisor, Professor Wei Ku, for leading me into

the exciting frontiers of topological insulators. I will always remember his

stimulating guidances, invaluable discussions, and his devotion to me. He has

helped me to learn the critical skills necessary as an independent researcher.

I am grateful to Wei and his family, for putting up with the occasional after-

hours discussions, sometimes at midnight. I am fortunate to have all the

former members of the Wei’s group. Weiguo Yin taught me very valuable

analytic methods in Green’s functions and helpful discussions on ab initio cal-

culations. Chia-Hui Lin brought me into the numerics, and especially thank

him for willing to work through some of my crazy ideas in the early stage of

some of these projects. I have spent many wonderful hours in Stony Brook

and at Brookhaven National laboratory. I have learned a lot from many

excellent professors of physics department. During the class period, I have

many happy discussions with Chen Cong, Chia-Yi Ju, Chengjian Wu and

Yanliang Shi, and have shared joyful memories. Studying in the BNL gave

me opportunities to work with other outstanding researchers, post docs and

students. I was lucky to meet Professor Genda Gu and his student Ruidan

Zhong. They introduced me very interesting knowledge about experiments.

I thank my collaborator Xiaoxiang Xi who did great experiments and made

very constructive comments on my paper. Last and the most important, I

dedicate this thesis to my parents and my wife. Without their unconditional

support, this thesis, and my PhD study would not have become possible.

LIST OF TABLES x

Publications

Xu-Gang He and Wei Ku, Model of arbitrary Chern number at a single

Dirac point, (preparing for publishing).

Xu-Gang He and Wei Ku, Hunting down magnetic monopoles in 2D topo-

logical insulators, (preparing for publishing).

Cheng Zhang, Xu-Gang He, Hang Chi, Ruidan Zhong, Genda Gu, Wei

Ku, John Tranquada and Qiang Li1, Introducing Surface State into

Superconducting High Indium-doped SnTe, (preparing for publishing).

Xu-Gang He, Xiaoxiang Xi and Wei Ku, Generic Symmetry Breaking In-

stability of Topological Insulators due to a Novel van Hove Singularity,

(Submitted arXiv:1410.2885).

David M. Fobes, Igor A. Zaliznyak, Zhijun Xu, Genda Gu, Xu-Gang He,

Wei Ku, John M. Tranquada, Yang Zhao, Masaaki Matsuda, V. Ovidu

Garlea, Barry Winn, “Forbidden" phonon: dynamical signature of bond

symmetry breaking in the iron chalcogenides, arXiv:1509.05930.

Ruidan Zhong, Xugang He, J. A. Schneeloch, Cheng Zhang, Tiansheng Liu,

I. Pletikosić, T. Yilmaz, B. Sinkovic, Qiang Li, Wei Ku, T. Valla, J.

M. Tranquada, and Genda Gu, Surface-state-dominated transport in

crystals of the topological crystalline insulator In-doped Pb1−xSnxTe,

Phys. Rev. B 91, 195321 (2015).

LIST OF TABLES xi

Xiaoxiang Xi, Xu-Gang He, Fen Guan, Zhenxian Liu, R.D. Zhong, J.A.

Schneeloch, T.S. Liu, G.D. Gu, X. Du, Z. Chen, X.G. Hong, Wei Ku,

and G.L. Carr, Bulk Signatures of Pressure-Induced Band Inversion

and Topological Phase Transitions in Pb1−xSnxSe, Phys. Rev. Lett.

113, 096401 (2014).

1

1 Chapter 1

1.1 Priory Considerations

Topology is an abstract mathematical concept stemmed from set theory and

geometry, which is an important branch of modern mathematics. Based on

the study of the real line and euclidean space, the topological spaces have

been defined as a collection of subsets (open sets) from a set satisfied certain

rules. Topology focuses on how the elements of a set related to each other

spatially and further classify spaces using some sort of measures. Although

strictly speaking, metric does not needed to be defined in a topological space;

However for real physical considerations, in this thesis, we always take metric

(probably always Euclidean) for granted.

In physics, for theoretical construction, researchers usually start from

Lagrangian or Hamiltonian. Either of them has its own advantages. For

example, Lagrangian would represent (gauge) symmetries more explicitly

as a scalar, while Hamiltonian with energy dimension has been defined by

canonical conjugate variables. Though they are equivalent to each other

by the Legendre transformation, but for specific problem the complexity of

quantization or predicting observables, could be very different. In this thesis,

I use Hamiltonian formalism for topological insulators which is different in

topological field theory, such as Chern-Simons term in the Lagrangian as a

topological indicator.

The goal of physics is almost always the same–describing the movement

of objects. In other words, we have to derive, from either Lagrangian or

1.1 Priory Considerations 2

Hamiltonian formalism, the equation of motion which turns out to be par-

tial differential equations (PDE’s) of time and coordinate variables. This

philosophy of locality leads the evolution of the physical theory intertwined

with its appropriate mathematical representations. Actually, modern physics

started along with the calculus, and contemporary microscopic theories have

been developing with the language of vector space (or field). However, on the

contrary, the global property of physical systems have been underestimated

for a long time.

The pioneer work, considering a global characteristic of a meaningful

physical object, can be tracked back to Dirac’s famous topological argument

for the quantization of electron charge due to the magnetic monopole in

1931 [1]. Based on that, T.T. Wu and C.N. Yang found out the right math-

ematical tool–fiber bundle, to describe the magnetic monopole and further

the non-abelian gauge theories [2]. As C.N.Yang said in his commentary,

“...Wu and I explored these global connotations. We showed that the gauge

phase factor gives an intrinsic and complete description of electromagnetism.

It neither underdescribes nor overdescribes it...". On the other hand, in con-

densed matter physics, the topology concept was introduced by the Quantum

Hall Effect in TKNN formula [3]. Soon later, the integer quantum Hall con-

ductance has been related to the integral Chern number by integrating the

Berry curvature and the winding number of the Berry phase around a closed

loop in the first Brillouin zone. It is worth noting that although they are very

different objects in particle physics and condensed matter, respectively; the

frame work and mathematical entities are the same–topological invariants of

1.1 Priory Considerations 3

the principal bundle. We will see that there are close connections between

two of them which is my standing point to represent my results.

I will arrange the content as following. The Chapter 2 as a general in-

troduction, will briefly present the mathematical concepts used in TI’s and

physical discovery of TI’s. In Chapter 3, starting from the (quantum) Hall

effects, the parallel understanding of the topological insulators can be easily

demonstrated. In last part of Chapter 3, I will show how to use complex

crystal momentum to hunt the effective magnetic monopole that is previ-

ously proposed but has not been fully investigated, especially in the physical

point of view. In Chapter 4, I will show the higher Chern number model

in TI’s with a new type of topological phase transitions. Several interesting

characteristics will be showed in Chapters 5 and 6: the special band structure

of TI’s and its consequences, and exact quantum mechanic supersymmetry

associated with TI’s model. The last two Chapters 7 and 8 will introduce

two recent experiments to fulfill the theoretical and practical goals.

4

2 Chapter 2

2.1 Physics Rediscovery of the Mathematical Concepts

In the original work of P.A.M. Dirac, the magnetic monopole is associated

with an artificial (gauge field) divergent Dirac string without any real physical

meanings. In 1975, one far-reaching paper considered the suitable mathemat-

ical language of gauge theory, starting by resolving this Dirac string problem.

It turned out that if one accept the concept of fiber bundle, then the artificial

Dirac string is not necessary [2]. Moreover, the fiber bundle theory provides

rich insights on the physics. As our concerns here, the global guage can be

classified by various topological invariants, and further, the topological in-

variant would be observables, such as charges of the magnetic monopoles and

quantum Hall conductances. The topological invariant associated with the

global gauge type (principal fiber bundle) does not change with any global

gauge transformations. Two of topological invariants are important in our

discussions:

A. The winding number;

B. The first Chern number.

The former is defined by loop integral of the local gauge transformation over

the Berry phases; and the latter is the integral of the Berry curvature on the

first Brillouin zone (base space).

Let’s check out the idea. We use the simplest 2D Chern insulator model

which is the prototype of other models, such as quantum Hall spin effect and

2.1 Physics Rediscovery of the Mathematical Concepts 5

crystalline topological insulators,

H = di(k)σi, (1)

where σi’s are the Pauli matrices; and the ~d is

~d (kx, ky) = (sin (kx) , sin (ky) ,M − 2 + cos (kx) + cos (ky)) , (2)

for the square lattices. This is a two-band model which shows the topological

phase transition at the critical point kx = ky = M = 0 [4]

For simplicity but without losing too much generality, we can consider

the low-energy limit model with the infinite k-space domain,

~d (kx, ky) =(kx, ky,M −

k2x

2 −k2y

2

). (3)

This model has a topological defect analogous to the magnetic monopole

in the topological non-trivial phase whereM > 0 and ~d can be represented in

spherical coordinate system, ~d =∣∣∣~d∣∣∣ (sin (θ) cos (ϕ) , sin (θ) sin (ϕ) , cos (ϕ)).

Solving the Hamiltonian (Eq. 1), one get the lower energy eigenstate as

∣∣∣u−⟩ =

sin(θ2

)e−iϕ

− cos(θ2

) , (4)

with the eigen-energy E = −√∣∣∣~d∣∣∣2. The relevant gauge field is the Berry


connection:~An(k) = i

⟨un (k)

∣∣∣~∇∣∣∣un (k)⟩, (5)

where un is a Bloch function with band index n. Thus we have

~A− = i⟨u−∣∣∣~∇∣∣∣u−⟩ = 1− cos (θ)

2∣∣∣~d∣∣∣ sin (θ)

φ. (6)

It easy to get, except the south point, we have

~∇× ~A− =~d

2∣∣∣~d∣∣∣3 , (7)

which means there is a monopole with a half unit of magnetic charge setting

at the center of the sphere.

With a topological defect, one has to choose a suitable set of open cov-

ering Ui over the S2 in the d space. One can fix a gauge on each covering,

and during the overlap, one need local gauge transformation from one cov-

ering to another one (FIG. 1). Another way to see the necessity of several

pieces manifolds to cover the sphere is that we lose the Stokes’ theorem here.

Because if one calculates the flux out the sphere by Stokes’ theorem, one get

vanishing result since there is no boundary of the sphere. However, it is,

of course, not true; since the gauge field is always divergent somewhere no

matter which guage we choose.

Under the setup of the principal U(1) fiber bundle, let’s calculate the

monopole charge. Choosing different gauges, we obtain different solutions of

model (Eq. 1), with the same energy E = −√∣∣∣~d∣∣∣2, related with local gauge


A2

A1

B

Figure 1: One magnetic monopole (red point) resides at the center of a sphereon the left hand-side picture. On the right hand-side, two different gauge fieldsbelonging to the same energy branch, are defined on the hemispheres. The Stockes’theorem can not be applied on the sphere, but is available for the two-hemispherecase. Most important, the physically unnecessary Dirac string is not needed.


transformation.

ψA−1 = 1√

2d(d−d3)

d3 − d

d1 + id2

, divergent at north point;

ψA−2 = 1√

2d(d+d3)

− (d1 − id2)

d3 + d

, divergent at south point.

(8)

⇒

A−1 = id1∇d2−id2∇d1

2d(d−d3) ,

A−2 = id1∇d2−id2∇d12d(d+d3) ,

(9)

where d =√d2

1 + d22 + d2

3. Then the gauge transformation A−2 → A−1 − i∇f−

is

i∇f− = id1∇d2 − id2∇d1

d21 + d2

2= ∇ ln

d1 + id2√d2

1 + d22

, (10)

which is corresponding to the azimuth angle of our model on the sphere. It

would be more clear from the Hamiltonian (Eq. 1),

H =

d3 d1 − id2

d1 + id2 −d3

=

d3

√d2

1 + d22e−iφ√

d21 + d2

2eiφ −d3

. (11)

In winding term d1 + id2, the azimuth angle is

iφ = ln d1 + id2√

d21 + d2

2

. (12)


Then the difference of the Berry phase is

γ−2 − γ−1 =∮A−2 · dl −

∮A−1 · dl

= −i∮∇f− · dl = −i

∮∇φ · dl = N−W ,

(13)

where N−W is the winding number of the Berry phase of the lower-energy

wavefunction, which is a topological invariant belonging to the homotopy

group π1 (U (1)) = Z. By the same method, one can find that winding

number of the upper-energy wavefunction is in the same homotopy group,

with N+W = −N−W .

Then what is the relation between winding number and the Chern num-

ber? The Chern number is the integral of the third direction component of

the magnetic field,

C =∫∫©Fzds =

∫∫© (∇× A)z ds (14)

Then as for winding number, we may use the Stokes’ theorem but for multiply

pieces of the covering as before.

⇒ C =∮A2 · dl −

∮A1 · dl = NA

W , (15)

where the gauge field has been chosen such that A2 is smooth on the upper

hemisphere and A1 is smooth on the lower one. Then the Chern number is

identical to the winding number NAW related to the gauge field.

In the following chapters, we will see that with this concepts of topological

2.2 Discovery of TI’s in Experiments 10

invariants, one can easily construct the model for higher Chern numbers.

Actually, one may construct the real space higher Chern number model that

can be realized in the experiments.

Our formal discussion here provides limited insights in the physical con-

siderations. Especially, we are working in the artificial 3D d space rather

than physical crystal momentum space. Later, we will show that in QHE,

the magnetic monopole helps to quantize the electron charge and induces

quantum hall conductances in momentum space. The topological arguments

are sort of static statements about the topological classification of differ-

ent insulating phases. Then what does happen during the phase transition?

With the analytic continuation of the crystal momentum, we can draw the

intuitive picture for the change of the monopole charges in the next chapter.

2.2 Discovery of TI’s in Experiments

The topological insulators have been found in a kind of semiconductors shar-

ing similar properties, such as narrow band gaps and strong spin-orbital

couplings. We have to note that in theory, the existence of the topological

insulating states does not need the extra conditions other than the winding

term and nontrivial radial part. However, as far as we know, the materials

which can realize Chern insulator model indeed need some more constraints.

To study the topological phase transition, one also need consider how to drive

the system from the topological trivial phase to the nontrivial phase for com-

parisons. As we will discuss in the third chapter, this process is unavoidably


experiencing a band gap closing and reopening transition which swaps the

effective magnetic monopoles with corresponding magnetic charges. That is

why almost all TI’s studied are semiconductors with very narrow band gaps

near the critical points. However, on the other hand, if the band gap is too

small, then the thermal effects become an important obstacle to prob the

desired surface state from the bulk conductivity due to thermal excitations.

Accordingly, the topological insulators have to be verified at low tempera-

tures, sometimes even lower than the boiling point of Helium.

Many other methods attempt to overcome these adverse circumstances. I

have involved two of these kinds of exciting experiments. One is to use high

pressure to drive some material going through topological phase transition,

which may not happen only by decreasing the temperature. Another way is

to add the special impurity which may stabilize the Fermi level with extra

localized states which may increase the bulk resistance by several orders. I

will present the results in the later chapters.

At this stage, all TI’s need help of local interactions that couple both the

spin and the orbital degrees of freedom. These interactions require electrons

to flip their spin as they hop from one orbital to another on the neighbor-

ing lattice site, to minimize (at least partially) the local energy. This local

spin-orbital entanglement may have a global effect on the wavefunction sym-

metry and low-energy dynamics. As we will see that this coupling usually

corresponds to the winding term in the model, which initiates the nontrivial

Berry phase around a closed loop in the k-space. We know that in chemical

compound composed of elements with small atomic numbers Z’s, sometimes,


Figure 2: The spin transportation profile. With the help of the time-reversalsymmetry, there is no back scattering at the edge of the TI’s. Then although theconductance in the bulk is vanishing, there are two conducting channels on theedge for different spin polarizations, which are the so-called “topological protected”surface states. Adopted from M. König et al. Science 318, 5851, 766-770 (2007).


one may even ignore the spin-orbital coupling in real experiments. However,

the spin-orbital coupling increases as Z4 and could no longer be neglected for

heavier atoms. In the case of topological insulators, the spin-orbit coupling

is responsible for band inversion in the known topological phase transitions.

Also the spin-orbit coupling profoundly determines the unconventional spin

polarization of the topological surface state which is one of the key features

to be verified.

The first experimental realization of TI is the 2D quantum spin Hall effect

(QSHE) in Hg(Cd)Te quantum wells that had been predict in 2006 [4] and

verified in 2007 (FIG. 2) [5]. In the experiment, the topological insulator

HgTe had been sandwiched by CdTe which is trivial. On the edge of HgTe,

there is a pair of conducting states counterpropagating with opposite spin

polarizations. The total charge current is always vanishing but the spin

current conductance is quantized as 2e2/h. In the topological nontrivial

phase, QSHE disgonalized with two blocks in the form of the Chern insulator

model but opposite spin polarizations.

Heff (kx, ky) =

H (k) 0

0 H∗ (−k)

, (16)

where the diagonal Hamiltonian is

H (k) = ε (k) + di (k)σi, (17)

which is the Chern insulator model plus an energy shift term depneding on


the real materials but with no topological meaning. It is not hard to see that

the surface states, if they exist, have to propagate in the opposite direction

with locked spin polarization. The time-reversal symmetry guarantees that

there is no back scattering between the chiral currents, and thus the currents

transport without dissipations.

Since the quantum Hall conductance is zero, one needs some sophisticated

method to do the measurements (FIG. 3). First, the temperature has to be as

low as 30mK to avoid any bulk thermal excitations. Second, in the quantum

Hall regime, the nonlocal transport is described by a quantum transport

theory based on the Landauer-Büttiker formalism (Appendix A),

Ii = e2

h

∑j

(TjiVi − TijVj), (18)

where Ii is the current flowing out from the ith electrode to the sample, Tij

is the transmission probability from the jth electrode to the ith electrode,

and Vi is the voltage on the ith electrode.

General speaking, the number of transmission channels of 2D sample

scales with its width, and thus the transmission matrix could be complicated

and nonuniversal. But if the quantum transport is entirely dominated by the

edge states, then the the transmission matrix can be simplified enormously.

For example, if there is only quantum Hall state, then T (QH)i+1,i = 1, for

i = 1, ..., N with periodic “boundary" condiion–N + 1 ∼ 1 and the rest of el-

ements are vanishing. However, for QSHE, since the time reversal symmetry

guarantees two copies of chiral state, the nonvanishing transmission matrix


elements are

T (QSH)i+1,i = T (QSH)i,i+1 = 1. (19)

For multi-terminal device, the resistance defined as

Rij,lk = Vi − VjIi − Ij

. (20)

Then from Landauer-Büttiker equation, one expects that for QSHE, four-

terminal resistance of R14,23 = h/2e2 and two-terminal resistance of R14,14 =

3h/2e2 (FIG. 3). This result is dramatically different from the quantum Hall

effect: R14,14 = h/e2 and R14,23 = 0. The experimental results confirmed the

QSHE in this 2D system.

For 3D TI, the conclusive experimental results came from the surface sen-

sitive instruments–such as Angle-resolved photoemission spectroscopy (ARPES)

or Scanning Tunneling Microscopy (STM) which verified the Dirac point lin-

ear dispersions in the gap regime on the 3D TI’s (FIG. 4) [6]. Furthermore,

one has to consider the spin-polarization in TI’s. With the time-reversal

symmetry, Chern number has to be vanishing but topological invariant of

the QSHE may not. More general argument, the surface state no matter

in 2D or 3D TI, which behaves as massless relativistic particles with an in-

trinsic angular momentum (spin) locked to its translational momentum, is

considered to be the key to verify the existence of TI.

In another experiment, the chiral property of bismuth based class of ma-

terial has been investigated (FIG. 5). It has been revealed that a spin-

momentum locked Dirac cone carrying a nontrivial Berry phase that is spin-


(A)

(B)

Figure 3: Two transportation experimental results on the 2D quantum wells.(A) The geometry of the sample and prob position.(B) The resistance R14,23 with different sample sizes (d× L×W ).(I): 5.5nm×20.0µm×13.3µm; (II): 7.3nm×20.0µm×13.3µm; (III): 7.3nm×1.0µm×1.0µm; (IV): 7.3nm×1.0µm×0.5µm.Adopted from M. König et al. Science 318, 5851, 766-770 (2007) and AndreasRoth et al. Science 325, 294 (2009).


polarized, which exhibits a tunable topological fermion density in the vicin-

ity of the Kramers point and can be driven to the topological spin transport

regime. It even claimed that the topological nodal state is shown to be

protected even up to 300 K. Researchers believe that this experiment with

many similar ones make the applications of topological insulating states in

spintronic and quantum computing technologies promising at room temper-

atures.


Figure 4: Adopted from D. Hsieh et al. Nature 452, 970-974 (2008).


Figure 5: Adopted from D. Hsieh et al. Nature 460, 1101-1105 (2009).

20

3 Chapter 3

3.1 Hall Effects

The applications of topological theory on condensed matter physics started

from quantum Hall effects. The Hall Effect has been found by Edwin Hall

in 1879, 18 years before the electron was discovered. Hall voltage describes

a voltage difference across an electrical conductor under an external mag-

netic field, transverse to an electric current in the conductor (FIG. 6). In

conductor, electric current consists of the same charged particles, typically

electrons, holes, ions. When a magnetic field applied on the charge carri-

ers, they experience a force–so-called Lorentz force, besides the electric field

force:~F = q( ~E + ~v × ~B) (21)

Under the Lorentz force, the charged carriers have to be deviated from

its path and trend to accumulate on edges, which induce the Hall voltage.

Hall effect as a generic phenomena in conducting materials, has been used

as a standard method in the transportation measurements. For example,

measuring the carrier density by Hall coefficient which is defined as the ratio

of the induced electric field to the product of the current density and the

external magnetic field.

RH = EyjxB

= VHallt

IB= − 1

ne, (22)

3.1 Hall Effects 21

where n is the electron concentration and e is the elementary charge. In

semiconductors, it can also be used to determine the carrier types of the

current by measuring the Hall coefficient:

RH = pµ2h − nµ2

e

e (pµh + nµe), (23)

where p is the hole concentration and n is the electron concentration; µe is

the electron mobility and µh is the hole mobility.

Actually, the classic Hall effects can be well explained in the framework

of elementary electromagnetism and have been tested for long time. What

was really surprising to physicists was the discovery of Integer Quantum Hall

Effects (IQHE) in 1980.

3.1 Hall Effects 22

VHall

I

B

x

yz

d

W

L

he-

Ey

Figure 6: Classical Hall effect in demonstration. The current is tending to deviatetransversally by the Lorentz force due to the magnetic field perpendicular to the2D plane, described by Eq.( 43). The value of the Hall voltage is continuous,showing a 2D bulk effect.

3.2 Quantum Hall Effects 23

3.2 Quantum Hall Effects

In two-dimensional electronic systems, at low temperatures a series of steps

appear in the Hall resistance as a function of magnetic field instead of the

monotonic increase. K. von Klitzing and Th. Englert had found this kind of

flat Hall plateaus in 1978. However, the precise quantized value of the Hall

resistance in units of h/e2 = 25812.807557(18)Ω was not recognized until

February of 1980. Five years later, in 1985, K. von Klitzing was awarded

Nobel Prize in Physics for the discovery of Quantum Hall effect.

Let me list the most remarkable features in the IQHE experiments:

A. No symmetry or symmetry-breaking is needed.

B. The quantum value of the Hall conductance is extremely accurate to a

few parts per billion (ppb).

C. The longitudinal resistivity (equivalently conductivity) vanishes, then the

bulk of sample is a perfect insulator; but electrons can transport with-

out dissipation along the edges of the sample.

The IQHE is very generic, observed in wide range of semiconductors,

which leads physicists believe that it would not have anything to do with

the microscopic structures of the samples. Furthermore, these Hall plateaus

occur at incredibly precise values of resistance which are the same no matter

what sample is investigated. This is very rare in physical experiments be-

cause, in real systems, we would always expect corrections of various sorts,

due to, for instance, electron-electron interactions, impurities, substrate po-


Figure 7: Experimental measurements of the Hall resistance RH and of the lon-gitudinal resistance Rxx for a GaAs/AlGaAs heterostructure at a temperature of0.1K. The Quantum Hall resistance and the longitudinal resistance in demonstra-tion. The plateaus of the Hall resistance represent their quantum essence. At thesame time, the longitudinal conductivity vanishes that means the sample is a per-fect insulator in bulk. Adopted from B. Jeckelmann and B. Jeanneret. SéminairePoincaré 2, 39-51 (2004).


tentials, finite size effects, etc. Actually, the Klitzing constant has been used

as electrical resistance standard in modern experimental physics.

The first attempt of explanation came from the Landau quantization

which modified the canonical momentum with the electromagnetic gauge

field:

H = 12m∗ (p− eA)2, (24)

where p is the momentum operator and A is the vector field. To solve it, one

needs fix the gauge; one of the valid ways is the Landau gauge: ~A = (By, 0, 0).

Assume the wavefunctions with the form of Ψ (x, y) = eikxxϕ (y), one can

obtain the familiar quantum harmonic oscillator:

H =p2y

2m∗ + 12m

∗ω2c (y − y0)2 , (25)

where the cyclotron frequency ωc is ωc = eB/m∗, and y0 = ~kx

eB. Solving this

Hamiltonian gives us the famous Landau quantization,

En = ~ωc(n+ 1

2

), n ≥ 0. (26)

Thus there are discrete Landau levels, and the allowed quantum states

accumulate on them with finite density of states (DOS); Between the Lan-

dau levels, system becomes perfectly insulating because of no extended state

propagating in the sample (FIG. 8 (A)). However, it is far from the complete

story.

First, one needs explain the Hall plateau in σxy and its coincidence with


(A)

(B)

Extended States

Localized States

Figure 8: The density of states at Landau levels of pure sample (A) or with disor-ders (B).(A) Landau level itself does not support the plateau; neither available states dis-tributed around the Landau level.(B) However, with the help of the disorder, the extended states spread aroundthe Landau level, corresponding to the phase transition. Between the extendedstates are the localized states which are responsible for insulating bulk and theHall plateaus.


vanishing of σxx. It turns out that the disorder plays the trick. When the

filling value is slightly away from a special filling, says ν0 at which Fermi

level resides on one of the Landau level, the quantum state might change

to another Landau levels. However, with disorders, one has in reality the

mobility gap between Landau levels, which still does not support propagating

states in the bulk, but never prohibits the localized states with a finite DOS.

Thus the added states created by increasing magnetic field, are localized by

the disorder and accordingly, do not contribute to transport, keeping σxx

vanishing (FIG. 8 (B)).

But why does σxy maintain a finite constant? We adopt the “gedanken"

cylindric geometry to show the accuracy of quantization of Hall conductivity

in two-dimensional metals which was introduced by R.B. Laughlin [7]. The

external magnetic field applies on the ribbon as showed in (FIG. 9). Without

electric field in y direction in Hamiltonian (Eq. 24), there is no Hall effect

because:

ρxy = −Eyjx. (27)

Thus Laughlin proposed the Hamiltonian by adding an extra term to the

Landau Hamiltonian,

H = 12m∗ (p− eA)2 + eE0y. (28)

It can be solved by the same way–transformed to a quantum harmonic os-

cillator, but with y0 = ~kx

eB− m∗E0

eB2 . Then one can find out the eigenstate


as,

ψ = eipxx/~√Lx

(m∗ω

π~

)1/4e−

m∗ω2~ (y−y0)2

. (29)

And the velocity is,

vx = 12m∗ (ψ∗ (px − eBy)ψ + h.c.) ,

⇒ vx =√

m∗ωπ~

1Lx

eBm∗

(y0 − y + m∗E0

eB2

)e−

m∗ω~ (y−y0)2

.(30)

Thus the electric current is,

Ix =∫ Ly

0evx (y) dy = eE0

BLx. (31)

Then if the Fermi energy resides between two Landau levels with n sub-bands

occupied, one get the total electric current,

Itot = nNIx = n eBhLxLy

eE0BLx

= n e2

hE0Ly.

⇒ σxy = n e2

h.

(32)

where N is the Landau degeneracy Φ/Φ0 with Φ the total flux going through

the ribbon and Φ0 is the flux quantum h/e.


I

x

y

B

V

Figure 9: The cylindric geometry of the Laughlin ribbon to show the QuantumHall conductivity in two-dimensional metals. The external magnetic field is normalto the x and y directions locally.

3.3 IQHE vs Topology 30

3.3 IQHE vs Topology

But why is the Hall conductance so precisely and robustly quantized? The

answer lies in its topological property associated with the gauge symmetry.

We have to emphasize at the beginning of this discussion, that the field

strength Fµν of electromagnetism, in quantum theory, does not completely

describe all electromagnetic effects on the wave function of the electron. The

missing part is the non-integrable phase factor manifesting the gauge symme-

try in the language of fiber bundle; otherwise one must not void the physical

artificial divergent points of gauge field, such as Dirac string. As previ-

ous authors correctly mentioned, “...the gauge phase factor gives an intrinsic

and complete description of electromagnetism. It neither underdescribes nor

overdescribes it..." [2]

Now we show the intrinsic topological property of IQHE following M.

Kohmoto [8]. For simplicity, we assume that there is one quantum magnetic

flux in the unit cell which means the translation symmetry holds. TKNN

first intorduced a way to calculate the quantum Hall conductances, based on

the Kubo formula [3], which can be modified as

σH = e2

h

12π

∫MBZ

d2k (∇k × A (k))3, (33)

where the gauge field is defined in (Eq. 5) and the integral area is the magnetic

Brilouin zone (here 1BZ). Different from the monopole at the center of the

d-sphere S2 in last chapter, here we consider the 2D 1BZ as torus T 2. Again,

since a torus does not have a boundary, if the Berry connection can be defined


uniquely over the entir torus, then Stokes’ theorem would give us σH = 0.

Then it is natural to consider a principal U(1) bundle over T 2. A torus

has a set of four coverings which are contractible–Hi, i = 1, ..., 4 that overlap

with their neighbors and are slight larger than the following coverings:

H ′1 = (kx, ky) |0 < kx < π, 0 < ky < π ,

H ′2 = (kx, ky) |π < kx < 2π, 0 < ky < π ,

H ′3 = (kx, ky) |0 < kx < π, π < ky < 2π ,

H ′4 = (kx, ky) |π < kx < 2π, π < ky < 2π .

(34)

Thus it is possible to choose a phase convention with the smooth phase

factor on each covering, exp [iθi (kx, ky)] = u (kx, ky)/|u (kx, ky)|. On the

overlap between two coverings Hi ∩ Hj, we have a non-singular transition

function Φij,

Φij = exp i[θi (kx, ky)− θj (kx, ky)

]= exp

[if ij (kx, ky)

], (35)

which is a map Φij : U (1) → U (1). Based on this transition function, a

principal U(1) bundle over T 2 is specified. To find out more topological

meaning, one may try to write down the connection 1-form as

ω = g−1Ag + g−1dg = A+ idχ, (36)

where g = eiχ ∈ U (1) is a fiber. The transition function act on fibers by left

multiplication, which relates the local fiber coordinate g and g′ in Hj and Hi


as

g′ = Φg. (37)

Then the transformation on the gauge field (Berry connection) is

A′ = ΦAΦ−1 + ΦdΦ−1 = A− i ∂f∂kµ

dkµ, (38)

which is the familiar gauge transformation or so-called compatibility condi-

tion. One can prove that ω is invariant under the transformation,

ω → g−1Φ−1(ΦAΦ−1 + ΦdΦ−1

)Φg + g−1Φ−1d (Φg)

= g−1Ag + g−1dΦ−1Φg + g−1Φ−1dΦg + g−1dg

= g−1Ag + g−1dg.

(39)

So this is indeed a legitimate connection 1-form with a choice of guage field

A. Then we have a differential geometry on the topological space. The

curvature is obtained by covariant derivative,

F = Dω = dA = ∂Aµ∂kν

dkν ∧ dkµ. (40)

In above calculation, we have used the fact that the structure constant is 0 of

Abelian group. Then the Chern number is an integral of the Berry curvature

over the T 2 manifold,

C1 = 12π

∫F = 1

2π

∫ ∂Aµ∂kν

dkν ∧ dkµ. (41)


This number is an integer independent of a particular connection chosen.

Comparing with (Eq. 33), one finally has

σH = e2

hC1, (42)

A contribution to the Hall conductance from a single band in unit of e2/h,

is given by the first Chern number.

3.4 Monopole in Complex Momentum Space 34

3.4 Monopole in Complex Momentum Space

The magnetic monopole is one of the most puzzling particles in the fundamen-

tal physics. It stems from the very original study of electromagnetism [1],

then revives with modern interests of the gauge theory [2], and more re-

cently, grand unified and superstring theories [9]. In Maxwell equations,

without magnetic source particles, the magnetic field lines never have source

or sink, which is different from electric field. Thus the duality of electric

and magnetic fields only exists without sources. In 1931, P.A.M. Dirac ar-

gued that the strict quantized value of elementary electric charge could be

explained if there was a monopole [1]. After that, in 1975, T.T. Wu and

C.N. Yang put forward the appropriate mathematical tool–fiber bundle in

which no non-physical Dirac string is needed. The importance is that the in-

tegral formalism and the topology of gauge field specify the intrinsic meaning

unrelated to the irrelevant details used to describe the system. However, al-

though a lot of promising programs are in progress [10, 11], as a fundamental

particle, it is still undiscovered in nature.


?

?

B

d-sphere

MM

(a)

(b)

B

B B

Z

Z

ky

kx

kx

kx

ky

ky

Figure 10: Arrows show the normal component of effective magnetic field. The 2Dreal k-plane stretches and bends down to cover the half lower space. With periodicboundary condition, the wrapped k-space has the same topology as torus. Whatis the effective monopole we expected?


The effective magnetic monopole is also extremely interesting, such as

in the Quantum Hall effect [8, 12] when the topological concept came into

the condensed matter physics, and now in Topological Insulators (TI’s) [14,

13, 15] and spin ice experiments [16, 17, 18, 19]. In 2D TI’s, the anomalous

quantum hall effect is related to the effective magnetic monopole described

by the Berry curvature (effective magnetic field) on the artificial sphere in

3D d-bspace. Its effective magnetic charge is associated with the topolog-

ical invariants–first Chern number and winding number which classify the

different global gauge type.

However, it is not surprise that this kind of static topological statements

does not provide any information about the phase transition in the physi-

cal related momentum space. Furthermore, the topological phase transition

always happens at certain singular k-point–so-called Dirac point. why does

the change of the local k-point alter the global property? Under the d-space

considerations, the phase transition behaves un-physically due to the sud-

den jump of the value of the magnetic charge at the origin of the sphere in

d-space. It fails to explain the appearance of the Dirac point.

In this paper, we will address the above questions by studying the mag-

netic monopole in k-space, describing the distribution of its charge and in-

vestigating what happens during topological phase transition. It turns out

that analytic continuation of the momentum to complex space is the most

prominent way. The effective magnetic monopoles reside on the branch point

which connect two successive bands but in the complex k-space except the

critical point.


M

kx

ky +

C = 1/ 2

C = 1/ 2 C = 1

C = 1/ 2 C = 0

C = 1/ 2

C = 0

C = 1

+

(a)

(b)

1/ 2

1 / 2

1 /2

1 / 2

0

0

0

Figure 11: Fig. (A) shows the necessity of the artifical external field besides thepoint monopole. Fig. (B) describes the failure of the intuitive propose to wrapreal k-space to serve the desired result required by the magnetic Coulomb law ifthere was monopole. Dashed line separates the magnetic field in such a way thatinside (ouside) it would not contribute the total fluxe through the k-plane on upper(lower) panel in Fig. (B).


However, let’s show that the most natural way does not work. For a 2D

TI, adding an artificial real z direction and wrapping the real k-space in this

kx, ky, z 3-dimensional space (FIG. 10). Then if there is a monopole inside

the torus, the total flux should be a quanta obeying Coulomb-type law; if

pure magnetic charge inside is zero, then the total flux also vanishes. Further-

more, if the point monopole is right on the real k-plane, the system is going

through the phase transition. Unfortunately, this intuitive understanding is

not accurate.

First, for the k-plane, one has to have extra external field rather than sin-

gle monopole to realize the quanta flux as showed in (FIG. 11 (A)). In other

words, one need certain artificial extra magnetic charge source at infinity

which has not physical meaning in the current problem. Second, because of

the boundary condition, one needs stretch and wrap the k-space to a compact

torus. However, this process fails to provide us the desired results whether

the torus includes or excludes the point monopole. Because, in the wrapped

space, all magnetic field emitted from single monopole has to sink into the

infinity point which inside the wrapped space. However, this situation is

essentially different from the case that the magnetic field of a monopole goes

through a torus but in a flat space. So one can not have an easy way to wrap

the k-space to satisfy the magnetic Coulomb law without artificial external

field.

The reason is as simple as that the z direction used to construct 3D space

in which the 2D k-space is embedded, is not arbitrary. Following the classic

electromagnetism, if the monopole exists, the source and/or sink of the field


behaves as a pole of the derived field. Then the natural choice of the third di-

mension is the imaginary part of the crystal momentum that guarantees the

Berry curvature has divergent point. In the following, without lossing gener-

ality, we choose complex momtum ky and real kx: kx,Re ky , Im ky =

x, y, z. In this coordinates, we have ~∇ = _x ∂∂x

+ _y ∂∂y

+ _z ∂∂z. Then our

formulae for the vector field and the field stength are modified, and both of

them are regular 3D vectors, based on the Berry’s definitions [20]:

~A =⟨u(~k)

∣∣∣~∇∣∣∣u(~k)⟩,

~F = ~∇× ~A.

(43)

Then from the Maxwell theories, we know that ~F is absence of pole except

at source charge. Obviously, the third component of ~F on the real k-plane

is equivalent to the Berry curvature, and the Chern number can be obtained

by

C = 12πi

∫1BZ

Imky=0

~F · zdkxdRe ky. (44)


++

+ +

+

+++

+

+++

+ +

++

+++

+++

+++++

+++

++

+ +++++

+++

kx

0

+ +

+

+

+

+

++

+

+

++

+

+

+

2=

=

0

B

ky

kyky

kx

ky

Figure 12: The top fig. shows the monopole string in 3D k-space with imaginaryky as the third axis. There is a mirror symmetry between the real k-plane but withopposite charge signs. Due to the symmetry, the upper and lower side monopolestring contribute the exactly same flux through the plane. Thus, wrapping aroundeither side monopole string into the torus gives the desired result.


In fact, the similar method has been introduced several decades ago,

first by H. A. Kramers [21]; and based on that, W. Kohn proposed a com-

prehensive description of analytic properties of band structures in the one-

dimensional lattice with periodic potential [22]. The basic idea behind their

work is that to pursue the complete solution of the Schrödinger equation,

one has to analytically extend the crystal momentum to the complex space

in order to allow the electronic energy covering the band gap parts. Thus

both of wavefunctions φn,k and eigen-energy are multivalued analytic func-

tions, and for each band index n, En,k represents a Riemann sheet, which has

to connect to the next one through the branch point–the doubly degenerate

point belonging to the both Riemann sheets. Considering complex k-space

is not just an expedient mathematical treatment, but the complete physical

information of the electronic bands. Based on the position of the branch

point, analytic method provides convergent properties of φn,k, En,k and the

localization properties of Wannier functions [22]; decaying characteristic of

impurity states [23] and boundary effects [24]. For insulators, on the tradi-

tional band part, there are extended states which propagate in the system;

while in the band gap, at most there are localized states due to the non-

vanishing imaginary part of the crystal momentum. Thus if the two relevant

bands are tending to touch each other, the branch point must run down in

the complex momentum space and finally arrive at the real axis; meanwhile,

the localized in-gap states become metallic.

From the effective monopole point of view, the magnetic charge has to

reside on the branch points which can be showed easily by the definition of


the Berry curvature [3]:

Fn,kxky = i∑n′ 6=n

⟨n∣∣∣∂H(k)∂kx

∣∣∣n′⟩ ⟨n′ ∣∣∣∂H(k)∂ky

∣∣∣n⟩− (kx ↔ ky)(En − En′)2 . (45)

Only at the branch point (the degenerate point), the Berry curvature di-

verges. It turns out that the total magnetic charge can cancel each other in

the trivial phase or sum up to quanta in the topological phase.

To show the our idea, we use the well-known BHZ model [4]:

H = di(k)σi, (46)

where σi’s are the Pauli matrices; and the vector ~d (kx, ky) =(kx, ky,M − k2

x

2 −k2

y

2

).

This is the simple two-band model which shows the topological phase tran-

sition at the critical point kx = ky = M = 0 [4]. Under the analytic continu-

ation, our field strength vector reads as:

Fi = i

2εijkd ·(∂j d× ∂kd

), (47)

which d = ~d/∣∣∣~d∣∣∣. It is easy to verify that ~∇ · ~F = 0 everywhere except ~d = 0

where is the monopole of the magnetic charge.

As we mentioned, monopole has to be at the branch point. In this model,

energies for the two bands are E± = ±√|~d|2, and the branch points can be

found out by solving E± = 0. Analytic continuing the momentum to the

complex space, one can obtain the branch points by solving the following


equation: ∣∣∣~d∣∣∣ = 0⇒ k2x + k2

y +(M − k2

x

2 −k2y

2

)2

= 0. (48)

For M = −0.2, the system is in the topological trivial phase. The mag-

netic charge, not a single point but forming a closed ring (ended at infinity),

is represented as (FIG. 12). The shape of the monopole string is mirror

symmetric about the real momentum plane, but with the opposite charge

sign on the both sides. Then the field contributions on the real k-plane from

upper and lower charge sources are the same, which means no matter which

direction one wrap the real k-space, the total flux along Im ky is invariant.

In this picture, no external magnetic field is needed, and the charge density

follows the magnetic Coulomb law. Furthermore, one has to note that no

matter how to wrap the real k-plane, there is always monopole inside (and

outside) of the compact real k-space. What different between different phases

is that the total magnetic charge inside is different.

Then the natural next step is investigating the topological phase tran-

sition. Our results are summarized in the FIG. 13. Because of the mirror

symmetry about the real k-plane, we then only consider the upper space.

As the mass term is approaching the critical point, the bottom of the upper

monopole string is reaching the real k-plane and becoming linear gradually.

More importantly, the range of negative magnetic charge shrinks, and more

and more charge accumulates around the tip kx, ky = 0, although the total

charge is still vanishing. However, it can change dramatically only through

the critical point: the upper side tip branch point with cumulative magnetic


charge (one half unit) swaps with the lower one. In other words, the total

charge has to be altered by the tip branch point with half quantum charge go-

ing through the real axis, which in turn means that the topological invariant

can be changed only through a metallic state with one branch point touch-

ing the real k-space. After the phase transition, the monopole string departs

from the real k-plane, with quantum magnetic charge but never comes back

in our model. This whole process is as robust as the existence of the branch

points, because the branch points are the only possible point at which the

monopole can reside. In fact, we know that such as single impurity would not

alter the property of the branch points. It is also clear now that swapping

certain branch points across the real k-plane describes the essentiality of the

bands inversion process that is necessary to the topological phase transition.

Furthermore, our results also help to answer certain important question.

Topology describes the global property of the system, but the topological

phase transition usually happens around local k-point, such as Dirac point

in TI’s. This connection is clarified by our results: the total effective mag-

netic charge is invariant in the whole complex 3D space. Then as long as

the monopoles do not cross over the real k-plane, the global property is un-

changed, although the monopole string may change its shape and its charge

distribution may also alters as well. As a result, only two branch points

(kx, ky = 0) that go through the real k-plane, can give rise to the topological

phase transition.

More interesting, although we are dealing with the complex variable which

may not have clear physical meaning in its form, things would be different if


kx

M = 0.4 M = 0.4M = 0.01

C =1

2

C =1

2

magnetic

charge

M = 0.01

C =1

2

C =1

2

M = 0

ky

0

Figure 13: The monopole string changes along the topological phase transitionfrom the topological trivial phase (left) to the non-trivial one (right). The greencolor is the positive magnetic charge part; while red is negative. Near the criticalpoint, charge is cumulative around the tip point (half quantum charge) but withthe total charge unchanged. During the phase transition, the conjugate tip branchpoints swap each other with their magnetic charge. After that, the total charge onboth sides change a unit magnetic charge but with different sign, and the monopolestrings depart from the real k-plane.


one accept the complex representation of field strength composed of electric

and magnetic fields as components. In other words, one can consider the

Riemann-Silberstein (RS) vector representation [25]:

~F = ~E + i ~B, (49)

where both of ~E and ~B are real field. In our model, the ~E and ~B are

perpendicular to each other, ~E · ~B = 0. This representation has been used

in photon wave function and quantization of the electromagnetic field [26];

Then the Maxwell equtions without sources,

i∂t ~F = ~∇× ~F ,

~∇ · ~F = 0,(50)

become,i(∂t ~F + 4π~j

)= ~∇× ~F ,

~∇ · ~F = 4πρ,(51)

where ~j = ~je + i~jm and ρ = ρe + iρm.

In our consideration, we do not take into account the dynamic part, but

the second equation is easy to be verify. If the Maxwell equations were

satisfied, it is trivial to see that they would be invariant here, under the

electromagnetic duality:

(~E, ~B

)→(~B,− ~E

),

(ρe, ρm)→ (ρm,−ρe) ;(~je,~jm

)→(~jm,−~je

).

(52)


And the charge conservation law holds in the compact form:

∂tρ+ ~∇ ·~j = 0. (53)

In summary, we resolved the problem about how to define effective monopole

in TI’s in a consistent way by introducing the complex crystal momentum.

It turns out that monopole string resides at the branch points which connect

successive bands in analytic continuation. The total charge of the monopole

determines the topological phase of the insulating system. The topological

phase transition happens when the tip of the monopole string goes through

the real k-plane. Each tip monopole point of the complex conjugate pair

brings half quantum magnetic charge with opposite sign between them, and

gives rise to the global change of the topological invariant. We have showed

our results by investigate a two-band low energy model, but the conclu-

sion is general, especially the periodic BHZ model has almost exactly the

same property. However, this non-analytic transition may not necessarily

leads to certain topological phase transition. For example, if under some

circumstances, two opposite charged pairs of branch points go through the

real k-plane; then the transition does not change the total Chern number,

though it is still possible that different topological invariant may be defined

as in QSH effect [4]. Actually, in our opinion, whether a non-analytic tran-

sition always induce a topological phase transition is an open question. But

all topological phase transitions have to go through the critical metallic state

as branch point approaches the real k-plane. The complexity of the effective


field is not totally unphysical as it looks like; actually, if one adopt the RS

vector representation, the inseparableness of the electromagnetic field comes

back in more satisfied form.

49

4 Chapter 4

4.1 Higher Chern Number Model

As we mentioned in the previous chapters, unlike IQHE, TI’s do not need

the external magnetic field or Landau quantization. The mystery is encoded

in the nontrivial topological property of the Berry phase of the Bloch wave

function under the condensed matter circumstance. Thus, going through

the band inversion process and catching Dirac point(s) with non-vanishing

topological invariant are crucial for an electronic system being a topological

nontrivial phase. It should not be overemphasized that the problems which

Chern number a Dirac point has and what the origin of the topological

invariant is, occupy the central position of researches at this stage.

For a single Dirac point with Chern number C = 1, many beautiful

and excited works have been done, most of them related to the linear Dirac

cones. Some efforts have been put into creating higher Chern number in a

single band, especially expecting the fractional Chern insulator. One may

introduce multi-orbital hopping by arranging multiple layer topological flat

bands, which may realize a quasi-2D system possesses an arbitrary integer

Chern number single band. The idea is to enlarge the Brillouin zone by

enhancing the translational symmetry. As a result, it can be understood

that during the topological phase transition from a normal insulating phase,

the bands has to close at N Dirac points at different k points with each of

them contributing an unit Chern number.

Contrary to the above idea, we may also construct a generic model with

4.1 Higher Chern Number Model 50

arbitrary Chern number associated to a single gapless state, without any help

of symmetry enhancement. In such model, single (generalized) Dirac point

may has arbitrary Chern number depending on the winding term (it will be

made more clear soon) in the model. Through the topological phase transi-

tion, the system may change by any topological invariant but corresponding

to the band gap closing at one k point in the first Brillouin zone.

The Chern insulator model is the building block of almost all other TI

related materials: Two identical copies of Chern insulator for opposite spin

polarizations can recover the time reversal symmetry and lead to the Quan-

tum Spin Hall effect. Through an adiabatic change of parameter(s) (keeping

topological invariant), simple Chern number model can connect to the Topo-

logical Kondo insulators. Time reversal symmetry would be replaced by lat-

tice symmetry, and mirror Chern number can be defined almost without any

essential modification from usual one, except that only certain lattice surface

may show the topological properties in the so-called topological crystalline

insulators. 2D Chern insulator model also can embedded into a 3D system,

interesting results come out such as Weyl semimetals. Furthermore, with

close analog with Bogoliubov de Gennes (BdG) formalism, especially the

similar Hamiltonian structure, topological superconductor (TS) developed

parallelly to Chern insulator. There is no doubt that the listed theoretical

studies can be pushed much further directly if the C = N minimum Chern

insulator model works, and future research agenda can be made immediately.


So our model starts from generalizing the Chern insulator model (Eq. 1):

H (k) =

M − 12kx −

12ky (kx − iky)N

(kx + iky)N −(M − 1

2kx −12ky

) . (54)

To show the higher winding property clearly, we use the cylindric coordi-

nate system,ρ =

√k2x + k2

y,

φ = tan−1(kykx

),

(55)

and the z direction is arbitrary. Then our model (Eq. 54) becomes

H (ρ, φ) =

M − 12ρ

2 ρNe−iΘ(φ)

ρNeiΘ(φ) −(M − 1

2ρ2) , (56)

where Θ (φ) = Nφ in our model.

Let’s find out the Berry curvature for this model. Choosing a gauge

arbitrarily (Eq. 8), one may modify it as

|ψ〉 = 1√2d (d− d3)

d3 − d√d2

1 + d22eiΘ

= d√2d (d− d3)

(d3−d)d√

d2−d23

deiΘ

= 1√2

−√

1− d3√1 + d3e

iΘ

,

(57)


where d3 = d3/d is the normalized third component of the d vector.

Then the Berry curvature which is independent of the gauge choice, is

F = i (∇〈ψ|)× (∇ |ψ〉) = 12∇Θ×∇d3, (58)

where Del in the cylindric coordinate is

∇ = ρ∂

∂ρ+ φ

1ρ

∂

∂φ+ z

∂

∂z. (59)

One obtains the Berry curvature as

F = z12∇φΘ∇ρd3

= z12N ·

M + ρ2

2(ρ2 +

(M − ρ2

2

)2) 3

2.

(60)

To calculate the Chern number, one has to integrate the Berry curvature


over the 1BZ.

C = 12π

∫1BZ

dk2Fz

=1

2π

∫ 2π

0

∫ ∞0

NM + ρ2

2

2(ρ2 +

(M − ρ2

2

)2) 3

2ρdφdρ

= Nρ2

2 −M

2(ρ2 +

(M − ρ2

2

)2) 1

2

∣∣∣∣∣∣∣∣∣∞

0

=

0, M < 0,

N, M > 0.

(61)

From Eq.( 60) and ( 61), we can see that the Berry curvature and the

topological invariant in 2D system can be divided into two parts: the winding

part and the kink-like radial part. Thus in the following, we will call the term

d1 + id2 as winding term. The radial terms behaves like 1D kink: although

both of cases belong to the lower energy band, namely the ground states,

by fixing the ending points, they can not be related by smooth deformation

from one to the another (FIG. 14).

Then one of the available ways to obtain the arbitrary Chern number

model, is to multiply the phase of the winding term with arbitrary integers

and at the same time keep the radial term in the topological non-trivial

phase, but other than those, any smooth deformations of the Hamiltonian

do not change the topological property of the system. Then it is suggested

that in momentum space, we may easily get the arbitrary Chern number


2 4 6 8 10

-1.0

-0.5

0.5

1.0

2 4 6 8 10

0.90

0.95

1.00kink non-kink

(A) (B)

Figure 14: The comparison of kink and non-kink configuraions in semi-infinite 1Dspace.(A) Fixing the ending points, the kink corresponds to the non-vanishing integralin Eq.( 61).(B) The non-kink configuration is equivalent, or adiabatically connected to thetotal flat case with vanishing integral in Eq.( 61).

4.2 Real Space Models 55

model in periodic cases by multiply the winding terms (off-diagonal terms in

Hamiltonian) with arbitrary integers. The Hamiltonian then is written as,

Hperiodic (kx, ky) = di · σi

=

M − 2 + cos kx + cos ky (sin kx − i sin ky)N

(sin kx + i sin ky)N − (M − 2 + cos kx + cos ky)

.(62)

Although we may not have the clean winding and radial kink parts in

this model, the Chern number can be calculated numerically by the following

formula that is equivalent to Eq.( 14),

C = 14π

∫1BZ

dkxdkyd ·

∂d

∂kx× ∂d

∂ky

, (63)

where d = ~d/∣∣∣~d∣∣∣

Then we can show the different topological phases with the various values

of the mass term M . Examples have been summarized in the figures 15.

4.2 Real Space Models

It is not hard to understand the higher Chern number model from the momen-

tum space, but can it be realized in real experiments? So one need construct

reasonable higher Chern model in the real space for further researches. For

the C = 1 Chern insulator, one may only need the hopping term on the near-

est lattice neighbor sites that effectively sit at the off-diagonal places of the

Hamiltonian. Thus it was thought that to create the higher Chern number,

one might have to consider far neighbors’ contributions. However, it turns


-1 0 1 2 3 4 5

0

1

2

3

-1

-2

-3

M

C

-1 0 1 2 3 4 5

0

1

2

3

-1

-2

-3

M

C

(B)

(A)

Figure 15: Figure (A) & (B) represent the complete topoloigcal phases accordingto the mass term.(A) The maximum Chern number is C = 1 corresponding to N = 1 in Eq.(62)(B) If N = 2, then C = 2 case can be achieved.


out that we only need the help from the second neighbors in the simplest

case.

H =∑n,m

tnmC†n↑Cm↑ +

∑n,m

−tnmC†n↓Cm↓ +(∑n,m

unmC†n↑Cm↓ + c.c.

), (64)

where C† and C are creation and annihilation operators, respectively; n and

m are the site numbers; ↑ and ↓ represent the spin (or pseudo-spin) degrees

of freedom.

unm = |unm| ~Rnm∣∣∣~Rnm

∣∣∣p eiφnm , (65)

represents the spin-flip interaction that could be complex due to various

possible orbital interactions.

Before any further discussions, we have to claim that the real space model

is general in the sense that not only for the simple square lattice (we will

discuss later), but also available for any kind of 2D Bravais lattices. In fact,

the form of Eq. ( 63) to calculate topological invariant, is invariant in any

coordinate system (Appendix B).

The Fourier transform reads as,

Cjσ = ∑

qeiqajCqσ,

C†jσ = ∑qe−iqajC†qσ.

(66)

Consider t-terms of the first and second nearest neighbors:

Ht1 = 2t1 (cos kx + cos ky)(C†k↑Ck↑ − C

†k↓Ck↓

), (67)


and

Ht2 = 4t2 cos kx cos ky(C†k↑Ck↑ − C

†k↓Ck↓

). (68)

It can be verified directly that only with first neighbor interaction, we can

not construct higher Chern number for single Dirac point no matter what

power term we use. With difference choice of the power term and relative

phase in unm, one may have various topological phases. But to our best

interests, we consider the power p = 2 for both first and second neighbors,

Hu1 = 2u1 (cos kx − cos ky)C†k↑Ck↓ + c.c., (69)

and

Hu1 = −4iu2 sin kx sin kyC†k↑Ck↓ + c.c.. (70)

Then we have,

Hh = M + 2t1 (cos kx + cos ky) + 4t2 cos kx cos ky 2u1 (cos kx − cos ky) + 4iu2 sin kx sin ky

2u1 (cos kx − cos ky)− 4iu2 sin kx sin ky − [M + 2t1 (cos kx + cos ky) + 4t2 cos kx cos ky]

(71)

In Eq. 71, if there is only one of two terms at off-diagonal places, then

the Chern number is vanishing due to no winding term. But here, we can

combine them together with no relative phase in u1 and u2–such as φ’s = 0.

By Eq. 63, it is easy to verify that the Dirac point at the Γ point, has

Chern number C = 2. However, at the critical point, the band structure has

4.3 Phase Transition with Fixed Mass Term 59

quadratic dispersion in all directions comparing to the linear dispersion of

the usual Chern insulator model. Then a finite effective mass of the Dirac

point is expected.

To view the phase transition, we draw the contour plots of all the com-

ponents the ~d, which has been determined by kx, ky,M. The system goes

through the first topological phase transition at dx = dy = dz = 0 that cor-

responds to the Γ point of the first Brillouin zone (FIG. 16). Then to show

the winding term intuitively, we consider the winding term in the model

dx + idy ∼ eiΘ(kx,ky) (FIG. 17). Then we can see that along a closed loop

(anticlockwise) in the k-space around the Γ point, the phase rotates 2 · 2π.

The winding number (positive or negative) at other topological critical points

can be investigated in the same way.

4.3 Phase Transition with Fixed Mass Term

The real space higher Chern number model suggests another possibility of

the topological phase transition: reversing the winding direction. As we men-

tioned before, both of nontrivial winding term and kink-like radial part are

necessary conditions of the system in the topological nontrivial phase. The

various topological phase are classified by values of the topological invari-

ant with sign. Thus one may reverse direction of the winding to change the

topological invariant by sign, but with fixed mass term.

In Hamiltonian ( 71), provided both u1 and u2 are real, if one keeps

u1 invariant but u2 changes its sign, then the Chern number changes its


0

0

kx

ky

1 0 -1 1 0 -1

(A)

0

kx

ky

0

(B)

0

kx

ky

0

(C)

-2 -5 -8 -11

Figure 16: Picutres (A), (B) and (C) are the contour plots for dx, dy and dzrespectively. The first topological phase transition happens at the Γ point, when(dx, dy, dz) = (0, 0, 0).


0

kx

ky

0

(1,0)

(0,1)

(-1,0)

(0,-1)

(1,0)

(0,1)

(-1,0)

(0,-1)

Figure 17: The red points indicate the values of (dx, dy). The closed loop (anti-clockwise) at least goes through the phase 2 · 2π, indicating the winding numberequal to 2.


sign through a topological phase transition at u2 = 0. Actually, it also can

be understand from the phase of the wavefunction Eq. ( 57). The spinor

represents the direction of its spin polarization. So changing the sign of u2

is equivalent to change the sign of Θ(kx, ky); and then going around a closed

loop in the k-space results the same number of winding but with the opposite

sign.

Note that this kind of topological phase transition can not be realized

in the simplest case with only the nearest neighbor interaction, because of

no intrinsic relative phase in the winding term–∆φ = φ1 − φ2 from Eq. 65.

There is another interesting feature of this sort topological phase transition.

Due to the periodic symmetry, the Dirac points in all the known cases occur

in the higher symmetric points of the first Brillouin zone. However, with the

help the intrinsic phase difference, the topological phase transition due to

the winding direction reversing, can happen along the 0 contour curves (ac-

cording the value of the mass term) in FIG. 16, either (A) or (B), depending

on which u interaction is being tuned.

63

5 Chapter 5

5.1 Band Structures of TI’s

Interestingly, topological insulators are sometimes found accompanied with

additional symmetry breaking phases. For examples, the thin film of Cr-

doped Bi2(SexTe1−x)3 is found to enter a magnetic phase [27], allowing the

realization of the long-sought quantum anomalous Hall effect [30]. Another

example, the charge density wave instability by the chiral symmetry breaking

in the 3D Weyl semimetals, made by the topological insulator multilayer, has

been proposed to form axion insulators, with the dissipationless transport on

the axion strings [31, 32]. One thus wonders “Is there a generic reason for the

strong tendency toward symmetry-breaking instabilities in the topological

insulating phase?" and “How should one engineer it to realize new quantum

phenomena and to tailor their unique functionalities?"

One of the the most exciting possibilities is to realize the topological

superconductivity. The edge state of a topological superconductor has a

peculiar nature that it is its own anti-particle, a special particle named Ma-

jorana fermion. The Majorana fermion has some exotic properties that make

them scientifically interesting, such as the non-Abelian statistic rather than

the Bose-Einstein statistic of the bosons or Fermi-Dirac statistic of the nor-

mal fermions. This also allows them to be used for practical applications,

such as to create Majorana qubits and to realize the topological quantum

computation. So far, the main thinking of the field is to utilize the supercon-

ducting proximity effect to create topological superconducting state at the

5.1 Band Structures of TI’s 64

interface between a topological insulator and a fully gapped superconduc-

tor [28]. However, it would be highly desirable to also explore the intrinsic

bulk superconducting instability of doped topological insulators.

Another fascinating possibility is the topological ferromagnetism. For a

while it has been postulated but was demonstrated only very recently [29, 30]

that a novel kind of Hall effect, the quantum anomalous Hall effect (QAH),

can be realized in a ferromagnetic topological systems in the absence of an

external magnetic field. Such a quantum anomalous Hall effect is unique in

hosting a dissipationless charge- and spin-current in the edge, contrary to

the spin-only current by the quantum spin-Hall effect of typical topological

insulators, and the charge-only current in the regular Hall of typical met-

als. However, the current prevailing method (doping magnetic elements into

TI’s thin films) suffers from the serious issue, namely the loss of topological

properties at high concentration of magnetic impurities necessary to achieve

strong enough magnetic order. Thus, it would be very interesting to make

use (or at least complement with) intrinsic bulk magnetic instability of topo-

logical systems as a cleaner, more effective approach.

We will point out a generic feature in the topological insulating phase that

renders the electronic system vulnerable against symmetry-breaking instabil-

ities. For present TI’s, the topological phase transition has to go through the

band gap closing and reopening process–so-called bands inversion process,

swapping the special branch point with half unit of topological invriant be-

tween the real k-axis. Deep into the topological phase, the inverted band

unavoidably develops a Mexican-hat dispersion that gives rise to a novel van


Hove singularity (VHS) [33] at the band edge in both 2D and 3D systems.

In essence, the geometry of the Mexican-hat dispersion hosts a singular den-

sity of states (DOS) with a 1D-like divergent exponent. In doped systems,

this may also cause a Lifshitz transition–a change of Fermi surface topology,

involving appearance of additional disconnected Fermi sheets with charac-

teristic shapes. In the absence of Fermi surface nesting, the divergent DOS

would particularly enlarge the phase space of the zero-momentum channel

and favor the superconductivity or ferromagnetism. These generic features,

which we will demonstrate with prototypical 2D and 3D models, not only

explains the observed broken symmetry states in many topological systems,

but also suggest a clear route to activate additional functionalities via tun-

ing chemical potential by slight doping or gating, such as the long-sought

topological superconductivity and quantum anomalous hall effect.

We start by examining the evolution of the electronic bands structure

across the topological phase transition via a band inversion. For a 2D system,

we use the Chern insulator model with low energy limit (Eq. 3). Then the

dispersions are,

E(2)± (k) = ±

√k2 +

(12k

2 −M)2, (72)

where k2 = k2x+k2

y. Note that this is a rescaled generic low-energy renormal-

ized Hamiltonian of many well-studied models, including the BHZ model [4].

In the realistic regime with the band width much larger than the spin-orbit

coupling( B 1): dx(k) = sin kx, dy(k) = sin ky, and dz = B(2 − cos kx −

cos ky)−M . For generic 3D cases, we use the simplified model of the topolog-


ical insulator family including such as Bi2Se3 crystal, which had been realized

in the experiments [34]:

H(2)3D (k) =

M − 12k

2 kz 0 k−

kz −(M − 12k

2) k− 0

0 k+ M − 12k

2 −kz

k+ 0 −kz −(M − 1

2k2)

, (73)

where k± = kx ± iky. The resulting dispersions have the same form as

(Eq. 72), but with k2 = k2x + k2

y + k2z .

Figure 18 summarizes the evolution of the band structure and the corre-

sponding DOS’s, from the topologically trivial phase (M < 0) to the topo-

logically non-trivial phase (M > 0) in both 2D and 3D. As expected, at the

phase boundary (M = 0) a metallic state is guaranteed. Near the Dirac

point, where two bands coincide in energy, the dispersion is linear and the

DOS approaches zero.

Notice that deep into the topological phase (M > 1), the system develops

a Mexican-hat band dispersion. This development of band structure is easily

understood from Fig. 19. When the band inversion is stronger than the gap

opening between the two bands (2M > Egap = 2 ·min|E(k)|), the disper-

sion unavoidably evolves into a Mexican hat. Obviously, the development of

such a feature is generic in all band-inversion scenarios.

The appearance of the Mexican-hat dispersion has important physical


M=0M=-0.5

Egap

=2|M|

(a)

(c)

M=-1.8

E

k-4

-2

2

4

0

Egap

=2|M|

k

M=0.5 M=0.75 M=1 M=1.8

∆E(k=0)=2M

DO

S (

3D

)

Egap

=0 Egap

=2M

DO

S (

2D

)

(b)

Egap

=2M Egap

=2√2M-1Egap

=2M =2√2M-1

γ=1/2

0

1

3

4

2

E E E E E E E

γ=1/2

0

5

10

γ=1/4

0

1

3

4

2

E E E E E E E-1.6 -1.8 -2.0

γ=1/2

0

5

10

Ban

d S

tru

ctu

re

0 -1 -3 -4-2 0 -1 -3 -4-2 0 -1 -3 -4-2 0 -1 -3 -4-2 0 -1 -3 -4-2 0 -1 -3 -4-2

-4

-2

2

4

0

k k k k k

Figure 18: (color online; arb. unit.) (a) Evolution of the band dispersion accordingto the Eq.( 74), (b) The change of the DOS for 2D system and (c) for 3D system,from a normal insulator (M < 0) to a topological insulator (M > 0). Each columnis labelled by the mass term at the top, and the length of the double-headedarrows describe the band gaps given at the bottom. When the band demonstratesMexican-hat dispersion at M > 1, the DOS diverges at the band edge ∼ |ω|−γwith a 1D-like divergent exponent γ = 1/2. The regular VHS corresponding to thetip of the Mexican hat (an additional step function in 2D and square root functionin 3D), can also be observed at |ω| = 1.8 in the right most panels.


Degenerate

MinimaE

k

Eedge

EF

kF10

2MEgap

KM

Non-degenerate

Maximum

kF2

0

Figure 19: (color online) Formation of the Mexican-hat dispersion deep in theband-inverted state. Dashed lines shows the inverted bands without the inter-bandcoupling. Introduction of the inter-band coupling makes the system insulating viaa gap opening. As long as the band-inversion is stronger than the gap opening, aMexican-hat dispersion is unavoidable. KM denotes the radius of the bottom ofthe Mexican hat.


g (ω) M 0 = 0 = 1 > 12D 1

π|M |

1−M + 1π

1−2M(1−M)3 |ω| |ω|

π1

2π

(Egap

|ω|

) 12 1

π

(Egap

|ω|

) 12

3D√

2π2

(|M |

1−M

) 32√|ω| ω2

π21π2

(Egap

|ω|

) 14 KM

π2

(Egap

|ω|

) 12

Table 1: Different analytical limit of the DOS at the band edge from a normalinsulator (M < 0) to a topological insulator (M > 0). With a Mexican hat band,the DOS diverges at the band edge with a 1D-like exponent, for both 2D and 3Dsystems. The 3D DOS is also proportional to the radius–KM of the bottom sphereof the Mexican hat dispersion (cf. Fig. 19).

consequences. For example, the DOS,

gD (ω) = 2∫ dDk

(2π)Dδ(|ED,± (k) | −

(|ω|+ Egap

2

))(74)

becomes divergent at the band edge. Here, ω is the energy measured from the

band edge, the factor 2 accounts for the spin degree of freedom, and D = 2, 3

for 2D and 3D cases, respectively. Indeed, Fig. 18(b)(c) and Table 1 show that

the DOS’s for both 2D and 3D topological systems diverge at the Mexican

hat band edges with a divergent exponent, γ = 1/2, same as that found in

the band edge of a regular 1D system. In fact, these 1D-like divergent DOS’s

are consistent with several recent experimental observations [35, 36, 37].

5.2 Symmetry Breaking Instabilities 70

5.2 Symmetry Breaking Instabilities

The singular DOS’s suggest that deep into the topological insulating phase,

systems with band inversion are intrinsically vulnerable against symmetry

breaking instabilities. This is because of the proximity of the chemical po-

tential to the DOS singularity due to the smallness of the band gap or the

intrinsic doping of the systems. For systems with near nested band structure,

this may lead to charge density wave or spin density wave states. Otherwise,

more generally, this would enhance the instability in the q = 0 channel, such

as ferromagnetism and superconductivity. The Pauli paramagnetic suscepti-

bility is:

χPauli = µ2Bg (µ)→∞, if g (µ)→∞., (75)

where µB is the Bohr magneton.

The bare pairing susceptibility χ0 = (|q| → 0+, ω = 0) are:

χ02D = 1

2g (µ) ln(qvF4ωD

), (76)

and

χ03D = g (µ) ln

(qvF2ωD

). (77)

Both are proportional to the density of state at the chemical potential. Here

vF is the Fermi velocity and ωD is the Debye frequency. Since their bare

susceptibility are proportional to the DOS at the chemical potential, they

are divergent when the DOS becomes singular. This offers the most natural

explanation of the recent observation of additional superconducting instabil-


0 10 20

0.4

0.6

0.8

30

T(K)

χ(meV-1)

χµ2

χ µ1

χ(0)

µ2

χ(0)µ1

Tc1 Tc2

1/Vpp

/4/4

Figure 20: (color online) Illustration of enhanced superconducting instability viadivergent DOS. Bare pairing susceptibility χ(0) (dashed lines) grow as chemcalpotential moves from µ1 = 0.6meV (in green) to µ2 = 0.3meV (in red), closer tothe DOS singularity. Consequently, the RPA-dressed susceptibility χ (solid lines)diverge at higher tansition temperature Tc2.

ity in the topological phase [38, 39]. Note the very similar phenomena also

occur in Rashba gases [40] and Iron-based superconductor [41].

Fig. 20 illustrates an example of utilizing the singularly large DOS at the

band edge to activate the additional order and new functionality via tuning

the chemical potential. Taking the realistic parameters from layered topo-

logical insulators [34], we calculate (Eq. 77) the T -dependent bare pairing

susceptibilities χ(0)’s (dashed lines) and the RPA-dressed pairing susceptibil-

ities χ’s (solid lines) with a reasonable effective electron-electron attacking


Vpp = 5meV:

χ (T ) = χ0 (T )(1− Vppχ0 (T )) . (78)

At chemical potential µ1 = 0.6meV, corresponding to a higher doping

of 6 × 1019cm−3, χ diverges at Tc1 = 2.3K, indicating a superconducting

transition. Upon lowering the doping to 4 × 1019cm−3, with µ2 = 0.3meV

closer to the DOS singularity, χ(0) (dashed red line) increases due to the larger

DOS, and consequently χ (solid red line) diverges at a higher Tc2 = 11.3K.

Compared to the recent proposal of interface superconductivity in topo-

logical crystalline insulators with flat bands [42], the bulk superconductivity

created via the Mexican hat in doped topological insulators has its advan-

tage. While it does suffer from low carrier density, the potential drawback

of smaller phase stiffness should be compensated by the large kinetic energy

(the relative big band width) and the bulk nature of the superconductivity.

In essence, in terms of the superfluid behavior, it would be in the same regime

as the underdoped high-Tc cuprates.

Fundamentally, notice that the VHS created by the Mexican hat is a

qualitatively new class of VHS on its own, different from the known ones.

Originally, based on the Morse theory and the quadratic dispersion of the mo-

menta, the singular DOS is characterized by the non-degenerate extremum

or saddle point, which at most, can lead to a kink in 3D or a logarithmic

divergence in 2D in general. Only in the rare case of the flat bands (here as

M = 1), it is possible to realize more singular DOS caused by the quartic

dispersion. On the other hand, the Mexican-hat dispersion hosts the degen-


kF

ky

0

0

0 0 kx

(a) (b) (c)

kF2kF1

Figure 21: (color online) Demonstration of the Lifshitz transition occurring indoped topological insulators: a single sheet of Fermi surface (a) would turn intotwo sheets of Fermi surface (b) due to formation of the Mexican-hat dispersion(cf.Fig. 19). Inclusion of strong anisotropy might even split the Fermi surface intomore sheets (c).

erate extrema at the bottom of the hat band, in addition to the common

non-degenerate extremum at the tip of the Mexican hat. The former gives

rise to the 1D-like divergent behavior of the DOS and the latter produces

the regular VHS, an additional step function in 2D and square root function

in 3D (cf. the right most panels of Fig. 18). Although at the band edge the

dispersion relations can still be approximated by the quadratic momenta, the

degenerate extrema (a ring in 2D and a sphere in 3D) have 1D codimension,

and consequently the DOS diverges at the band edge like a 1D system [43].

Interestingly, the appearance of the Mexican hat dispersion may also give

rise to a Lifshitz transition in a doped system. For example, observing the

Fermi surface evolution of a doped systems in Fig. 19 and 21 (a)&(b), one

finds that the number of Fermi surface grows from one to two per Dirac point.


Across the Lifshitz transition, the non-analytical change of the correspond-

ing DOS is known to lead to salient effects on thermodynamic, transport or

magnetic properties [44]. One thus expects clear signatures of such a transi-

tion in most measurements. This could also be another way to qualitatively

understand the enhancement of superconductivity discussed above.

It might also be instructive to make a connection to the similar singu-

lar DOS at the gap edge of a fully gapped superconductor with weak gap

anisotropy, eg: s-wave, px+ipy or dx2+y2 +idxy. Notice that in these systems,

when the superconducting gap on the Fermi surface is smaller than the Fermi

energy, the band would demonstrate effectively a Mexican hat dispersion as

well, just with a reduced spectral weight. Therefore, it is trivial to under-

stand that other than an overall 1/2 factor related to the weight reduction,

the resulting DOS has the same 1D-like divergent behavior at the gap edge.

Thus it would not be very unusual that in these systems, superconductivity

can coexist with other symmetry breaking phases.

Finally, for completeness, it is necessary to consider the effect of anisotropy

of the dispersion around the Mexican hat. Such an anisotropy can in prin-

ciple lift the strict degeneracy at the band edge, effectively recovering the

higher-dimensional behavior with less singular DOS. However, this is obvi-

ously a very small energy scale, especially when the radius of the Mexican

hat is small. Above this small energy range, the tendency toward a divergent

behavior would still be present, so our above discussion remains valid. Of

course, if one drives the system into the much deeper band inversion phase

where the radius of the Mexican hat becomes larger, the anisotropy can be


more effective in lifting the degeneracy. In that case, the system might go

through another Lifshitz transition at low doping, from two sheets of Fermi

surface to possible multiple pockets. Fig. 21 (c) demonstrates such a possi-

bility, corresponding to adding anisotropic 3rd order terms in the dispersion

relation. Furthermore, the anisotropy might deform the Fermi surface in

ways that would improve the nesting condition for the charge density wave

or spin density wave states. All these interesting possibilities allow further

tunability of the bulk physical properties of the generic band-inverted sys-

tems.

In summary, we point out that deep into the band inverted state, the

topological insulators are generically vulnerable against symmetry breaking

instability, due to the novel van Hove singularity near the chemical poten-

tial. This new class of VHS is caused by the characteristic Mexican-hat

dispersion at the band edge, which effectively reduces the codimension of the

degenerate extrema to one, and guarantees the divergent DOS with a 1D-like

exponent for both 2D and 3D cases. This singular DOS can boost up the

instability of the system toward superconductivity or ferromagnetism, which

can be effectively tuned via chemical potential through doping or gating. In

addition, associated with the formation of the Mexican-hat-like dispersion, a

doped system would experience a Lifshitz transition that may multiply the

number of Fermi surfaces with modified shapes. Our study not only explains

the existing experimental observations, but also suggests a specific route to

activate novel functionalities via additional symmetry breaking phases in the

topological insulators, particularly for the long-sought quantum anomalous


hall effect and topological superconductivity.

77

6 Chapter 6

6.1 Supersymmetry in Quantum Mechanics

Although in Chern insulator, one need break time-reversal symmetry to get

non-vanishing Chern number in condensed matter, the topological property

itself does not depend on the symmetry. It has been argued that the topo-

logical phase transition does not relate to any kind of spontaneous symmetry

breaking process.

From the high energy physics, we know that there exists transformation

which convert bosons into fermions and vice versa–so-called Supersymmetry

(SUSY). Although this kind of symmetry has not been discovered in nature,

it has been widely and intensively considered and believed that it has to

play some important role in the grand unified theory. One of the motiva-

tions for SUSY is to stabilize the Higgs mass to radiative corrections that

are quadratically divergent. The interactions involving Higgs boson causes

a large renormalization of the Higgs mass and unless there is an accidental

cancellation (probably from its superpartner), the natural size of the Higgs

mass is the greatest scale possible (the hierarchy problem). Another consid-

eration comes from the discrepancy between the gravity and quantum theory.

The powerful Coleman-Mandula theorem says that within the framework of

Lie algebras, there is no way to unify gravity with the gauge symmetries,

because there is no allowable Lie algebra mixing of Poincaré group and an

internal group. The SUSY may be a possible “loophole" of the theorem,

since it contains additional generators (supercharges) of a Lie superalgebra

6.1 Supersymmetry in Quantum Mechanics 78

or “graded" Lie algebra, not a Lie algebra.

However, what we will discuss here is not the SUSY from the particle

physics point of view on the second quantization level; namely there is no

creation or annihilation operators of particles and its superpartners. We

consider the SUSY on the quantum mechanics level, as different type of

symmetry from such as translational or rotational symmetries represented

by Lie algebras.

Actually, SUSY is not totally strange to condensed matter physics. One

of the pioneer papers of SUSY in condensed matter can be tracked to 1980’s

on ferroelectric semiconductors Pb1−xSnxTe. Now we know that a large range

of IV-VI semiconductors belonging to the topological (crystalline) insulators.

Some of them have been verified recently, including Pb1−xSnxTe [45].

Let me list the basic properties of SUSY: The physical states involving

both bosonic and fermionic degrees of freedom,

|nB, nF 〉 , nB = 0, 1, 2, ...∞, nF = 0, 1. (79)

Again for quantum mechanics, we consider eigen levels of a Hamiltonian not

Fock states of second-quantization.

The creation and annihilation operators are

[b−, b+

]= 1,

f−, f+

= f−f+ + f+f− = 1,(80)


and (f−)2

=(f+)2

= 0, (81)

which is called nilpotency.

Between bosonic and fermionic operators,

[b, f ] = 0. (82)

Then the transformation operators are,

Q+ = qb−f+, Q− = qb+f− . (83)

The Q’s operators are also nilpotent inherited from the fermionic operators

f ’s. For N = 2 (two Q operators), we can also construct

Q1 = Q+ +Q−, Q2 = i (Q− −Q+) . (84)

The the simplest SUSY Hamiltonian with a single boson and a single fermion

degree of freedom, can be written as

H = Q21 = Q2

2 = Q+, Q− . (85)


⇒ H = q2(b+b− + f+f−

)= q2

(b+b− + 1

2

)+ q2

(f+f− − 1

2

)= Hb +Hf .

(86)

This Hamiltonian describes a supersymmetric oscillator. First, from Eq. 85,

we note that the energy spectrum is non-negative. Except from the zero

energy E = 0, there are always twofold degeneracy (N = 2) of the energy

levels. For zero energy state, we further have

H |0〉 = 0⇔ Qi |0〉 = 0, ∀i . (87)

Second, in this SUSY harmonic oscillator model, the vacuum has a zero

energy because the energy of the boson zero-point vibration is canceled ex-

actly by the negative energy of the fermion zero-point energy. This famous

cancellation is a manifestation of reduction of the infinite energy of the zero-

point energy in supersymmetry field theory where there are infinite degrees

of freedom. So from the point of supersymmetry theories, infinite energies of

boson and fermion vacuums (positive and negative energies, respectively) are

simply a consequence of the artificial breaking up of the zero energy of the

vacuum of the “unified" theory (including both bosons and fermions) into

positive and negative (both infinite) terms [46]. It is important that this

property is a property of not only the “free" theories of such simple harmonic

oscillator but also a property in problems incorporating an interaction, and

this is true outside the framework of perturbation theory, if the interaction


satisfies certain requirements.

If there exists a supersymmetrically invariant state, meaning taht it is

annihilated by the Q, then it is automatically the true vacuum state, since

it has zero energy and any state that is not invariant under supersymmetry

has positive energy [47]. Thus, if one supersymmetric state exists, it has to

be the ground state and supersymmetry is not spontaneously broken. Only

if there is no supersymmetrically invariant state, the SUSY spontaneously

broken. In one sentence, SUSY is unbroken if and only if the energy of the

vacuum is exactly zero.

SUSY exact ⇔ H |GS〉 = 0. (88)

Figure 22: A classical illustration of the differences between supersymmetry andglobal symmetries. In (a), the expectation value of the scalar field breaks aninternal symmetry, but does not break supersymmetry, because the vacuum energyis zero. In (b), supersymmetry is spontaneously broken. Adopted from E. Witten,Nuclear Physics B185, 513-554 (1981)

6.2 Supersymmetry in TI 82

If the supersymmetry is exact, for each particle, there is a superpartner

with the same mass. In nature, no superpartner of an fundamental particle

even has been found, if supersymmetry exists, it has to be broken in our

energy level. To investigate the supersymmetry and its breaking mechanics,

E. Witten first proposed simple quantum mechanics model. In the following,

we will see that our topological insulator model fits into the Witten’s model

in sort of surprising but reasonable way and the surface state is identical to

the zero energy ground state. So the TCI realizes the exact SUSY and in

simplest way, the Chern number is equal to the Witen’s index.

6.2 Supersymmetry in TI

Witten’s N = 2 SUSY quantum mechanics model:

Q1 = 12 (σ1p+ σ2W (x)) ,

Q2 = 12 (σ2p− σ1W (x)) .

(89)

But they are not independent in the simple model,

Q2 = −iσ3Q1. (90)

Then the Hamiltonian is

H = 12

(p2 +W 2 (x) + ~σ3

dW

dx

). (91)

At the tree level, the ground state energy is the minimum of W 2 and the


number of sypersymmetrically invariant, zero-energy state is equal to the

number of solutions of the equation W (x) = 0. In 1985, B. A. Volkov and

O. A. Pankratov introduced a model to describe the band inversion in semi-

conductors Pb1−xSnxTe (or Se):

H =

0 −i εg(z)2 + ~σ · ~p

i εg(z)2 + ~σ · ~p 0

, (92)

where εg is the energy gap between the conduction and valence bands; for

the reason which will be clear soon, we define M (z) = εg (z)/2.

To see the topology more explicitly, we apply a unitary transformation

U ,

U =

iσz 0

0 1

, (93)

on the Hamiltonian:

H → UHU † = H ′. (94)

⇒ H ′ =

0 M (z)σz + σxpy − σypx + ipz

M (z)σz + σxpy − σypx − ipz 0

=

0 H

H† 0

,(95)


where H is

H = ipz1 +

M (z) i (px − ipy)

−i (px + ipy) −M (z)

(96)

We immediately recognize that the second term in H is the familiar Chern

insulator model. This 3D material can be considered, composed with multiple

layers with difference value of the mass term M(z) along the z direction.

Actually, in the experiment, to investigate the inverted band, the sample

had been synthesized by changing the composition during the growth of the

crystal [48]. Then the Witten’s mode reads as,

H ′ = − (τ2 ⊗ 1pz − τ1 ⊗W (z, px, py)) ∼ −Q2, (97)

with the superpotential in the form of the Chern insulator W = ~σ · ~d (z, p) .

Now the superpotential W is a matrix, then the number of the zero modes

is the number of the zero eigenvalues of matrix W . Thus it is clear that the

topological nontrivial superpotential has to have zero mode to be the exact

SUSY which in turn means that one need zero eigenvalue solution of the

chern insulator model that is the surface state. The surface state only exists

when the system is in the topological insulating state. So in this case, the

Chern insulator model is the superpotential (neither the supercharge nor the

complete Hamiltonian) but fits into the supercharge Q.

Note that the SUSY based on the topological insulating state is dynam-

ical stable. As argued by Witten, SUSY at tree level could be broken by

dynamical mechanics, but keeping the Witten index unchanged. Witten in-


dex is ∆n = nB − nF which is invariant. Since the Witten index is an odd

number, in the exact spectrum, the number of supersymmetric states would

not be vanishing which means the SUSY is still exact. And in this simple

case, the Chern number is essentially the same of the Witten’s index.

Then how to break the SUSY? For symmetries other than SUSY, it would

be rigorously broken in inifinite systems. By contrast, supersymmetry can

perfectly well be spontaneously broken in a finite volume [49]. SUSY break-

ing just means that the ground-state energy is positive, which is possible

for supersymmetric theories in a finite volume or even for supersymmetric

theories with only a finite number of degrees of freedom–such as in quan-

tum mechanics. As we mentioned, the breaking of SUSY results from the

ground state with finite energy. In our model, after the topological phase

transition from topological nontrivial state to trivial state, the system loses

the protected surface state (or zero mode) which in turn, means the SUSY

broken. In this sense, we conclude that at least in this kind of TI’s, the phase

transition related to the SUSY breaking or recovery.

86

7 Chapter 7

7.1 Bulk Signature of the Topological Phase Transition

Verifying the TI is challenging. First, unlike the symmetry breaking phase

transition, there is no direct measurable signal (order parameter) for the

topological order. Second, although there is topological protected surface

state propagating on the edge, it would be challenging for bulk measurements,

especially obvious and essential signals. Now, the most reliable method is

the ARPES that is sensitive to the surface states. However, as we mentioned

before, the surface state is the result caused by the global topology which

belongs to the bulk properties. It is desirable to investigate the possible

method to test the topological phases. Third, at present, the TI’s materials

are all semiconductors or even bad metals (with very high carrier density),

which have pretty good conductance and easily wash out the signal from

surface states.

An effective approach for establishing the bulk signatures of TIs is to

follow the evolution of characteristic features, starting from the trivial insu-

lating state through the topological phase transition (TPT) and into the TI

phase. Applying pressure offers a particularly attractive method for control-

lably driving a material through such a transition. Generally, a hallmark of a

TPT in a non-interacting system is band inversion: the bulk band gap closes

at the phase transition and reopens afterwards, inverting the characters of

the bottom conduction band and top valence band. Although the resulting

change in the bulk band structure is expected to be dramatic, detecting and

7.1 Bulk Signature of the Topological Phase Transition 87

understanding the associated experimental signatures are surprisingly chal-

lenging. For example, APRES is not compatible with pressure tuning nor is

it sensitive to the bulk. Previously, two groups [50, 51] reported investiga-

tions for a pressure-induced TPT in BiTeI, but reached different and actually

contradictory conclusions.

In one case, an observed maximum in the free carrier spectral weight was

interpreted as strong evidence for a TPT. In the other, a monotonic redshift

of the interband absorption edge was interpreted to indicate the absence of

such a transition. Obviously, resolving this contradiction is necessary for our

understanding of TIs to move forward.

Here we clarify this current controversy by demonstrating bulk signa-

tures of a pressure-induced band inversion and thus a TPT in Pb1−xSnxSe

(x = 0.00, 0.15, and 0.23). A maximum in the free carrier spectral weight

is reconfirmed in this system and is possibly a generic feature of pressure-

induced TPTs when bulk free carriers are present. The absorption edge

initially redshifts and then blueshifts under pressure, but only when its over-

lap with the intraband transition is suppressed. Extra evidence for the TPT

is uncovered, including a steeper absorption edge in the topological phase

compared to the trivial phase and a maximum in the pressure dependence

of the Fermi level. The TPTs in Pb1−xSnxSe imply the creation of 3D Dirac

semimetals at the critical pressure, serving as a route for pursuing Weyl

semimetals. The robust bulk signatures of TPTs identified here are expected

to be useful for exploring a variety of candidate pressure-induced TI’s.

Lead chalcogenides are candidate topological crystalline insulators (TCIs)


under pressure, with the role of time-reversal symmetry in TIs replaced by

crystal symmetries. In TCI, the topological invariant of Chern number of

TI has been replaced by the mirror Chern number. Similar to TIs, TCIs’

nontrivial band topology is associated with an inverted band structure below

100 K. These narrow-gap semiconductors crystallize in the rock salt struc-

ture and share simple band structures (only one fundamental gap) ideal for

investigating bulk characteristic of TPTs. At ambient condition, PbX (X =

S, Se, or Te) has a direct band gap at the L point of the Brillouin zone, with

the L−6 (L+6 ) character for the bottom conduction band (top valence band).

Band inversion is known to be induced in Pb1−xSnxX (X = Se and Te) by

doping or, in the series of Pb-rich alloys, by cooling. Pressure-induced band

inversion in PbSe and PbTe has been proposed on theoretical grounds [52]

especially in the context of TPTs, but experimentally it has not been firmly

established.

In this work we present a systematic infrared study of pressure-induced

band inversion in Pb1−xSnxSe. Samples with nominal x = 0.00, 0.15, and 0.23

were synthesized by a modified floating zone method. Hall effect measure-

ments determined a hole density of roughly 1018 cm−3 for PbSe at room tem-

perature. High-pressure experiments were performed using diamond anvil

cells at Beamline U2A of National Synchrotron Light Source, Brookhaven

National Laboratory. Samples in the form of thin flakes (≤ 5 µm) were

measured in transmission while thicker (> 10 µm) pieces were measured in

reflection.

For comparison with the experiment, an ab initio method based on the


WIEN2k package was used to simulate the pressure effects on PbSe. We

combined the local spin density approximation and spin-orbit coupling for

the self-consistent field calculations, adjusting only the lattice parameter a

to mimic pressure effects. The mesh was set to 46×46×46 k-points and

RMT×KMAX = 7, where RMT is the smallest muffin tin radius and KMAX the

plane wave cutoff. The effect of doping was considered in the self-consistent

calculation according to the experimental hole density. While the calculation

does not yield the exact lattice parameter at which the TPT occurs, we found

the results to be in qualitative agreement with our experimental observations.

We determined that, at room temperature, Pb1−xSnxSe maintains the

ambient-pressure structure up to 5.1, 4.0, and 2.9 GPa for x = 0.00, 0.15,

and 0.23, respectively. In the following we focus on the ambient-pressure

structure and investigate pressure-induced TPTs.

We began by revealing a maximum in the pressure-dependence of the

bulk free carrier spectral weight. This serves as a signature of band inversion.

Though the free carrier response of the bulk has made analysis of the surface

behavior challenging, it actually provides a sensitive and convenient probe

of band inversion and thus TPTs. For doped semiconductors with simple

bands, the low-frequency dielectric function for intraband transitions can be

described by ε(ω) = ε∞ − ω2p/(ω2 + iωγ), where ω is the photon frequency,

ε∞ the high-energy dielectric constant, ω2p/8 the Drude spectral weight (ωp

the bare plasma frequency), and γ the electronic scattering rate. The Drude

weight ω2p/8 connects with the band dispersion through the Fermi velocity


/

← →

∼ +← ←

/

← →

/

← ←

+← ←

/

Figure 23: (color online). (a,b) Electronic band structure of PbSe at various latticeparameter ratio a/a0 along the L-Γ and L-W directions of the Brillouin zone. a0is the experimental lattice constant in the zero temperature limit and at ambientpressure. The TPT occurs at a/a0 ≈ 1.0255. The dashed lines indicate the Fermilevel EF for a hole density of N = 1018 cm−3. (c) The direct band gap Eg at the Lpoint, absolute value of the Fermi level |EF | relative to the top valence band, andEg + 2|EF | [roughly the energy threshold for direct interband transitions in thepresence of free carriers, as indicated by the arrow in (a)] as a function of a/a0.(d) Fermi velocity vF along the L-Γ and L-W directions as a function of a/a0 forN = 1018 cm−3.


#!&

#

"$

%

#!&

#

%

#!&

#

%

#!&

ħ

!""$!

Figure 24: (color online). (a–c) Pressure-dependent mid-infrared reflectance ofPb1−xSnxSe measured at the diamond-sample interface and at room temperature.The dashed lines are guides to the eye for the shift of the plasma minimum. (d)Example fits (solid lines) to the experimental data of x = 0.23 (dots). (e) Pressuredependence of ωp extracted from the fit.

vF , ωp ∝ vF , because ω2p (in general as a tensor) for a single band,

ω2p,αβ = ~2e2

π

∫dk vα(k)vβ(k) δ (E(k)− EF ) ,

where k is the crystal momentum, E(k) the band dispersion,

vα (k) = ∂E (k)~∂kα

, (98)

the αth component of the Bloch electron mean velocity, and EF the Fermi

energy. The measured reflectance has a minimum near the zero crossing in the

real part of ε(ω), called the plasma minimum, located at ω ∼ ωp/√ε∞. The

plasma minimum is universally observed in Pb1−xSnxSe for x = 0.00, 0.15,

and 0.23 (see FIG. 24). Upon increasing pressure, it initially blueshifts and


" #

!

! "

" #

!

" #

−

⋅

−

⋅

Figure 25: (color online). Pressure-dependent mid-infrared absorbance of PbSemeasured at (a) 298 K and (b) 70 K. Data in the blank region between 200–300meV are not shown because of unreliability caused by diamond absorption. (c) Realpart of the interband optical conductivity σ1 of intrinsic PbSe at various a/a0 fromfirst-principles calculations. The dashed line shows the result including holes witha density of 1018 cm−3. (d)[inset to (b)] A diagram illustrating hybridization opensup a band gap (in the bands shown as solid lines) when the conduction band andvalence band cross (dashed lines).

then redshifts, indicated by the dashed lines in FIG. 24. Since the phase space

for intraband transitions reaches a minimum at the gap-closing pressure and

ε∞ increases monotonically under pressure, the maximum in ωp/√ε∞ must

be attributed to vF going through a maximum near the critical pressure of

the TPT (FIG. 23(d)).

The expression for ε(ω) provides excellent fits to our data, exemplified for

Pb0.77Sn0.23Se in FIG. 24(d) and quantifying the maximum in the pressure

dependence of ωp [FIG. 24(e)]. Such a maximum was also observed in BiTeI

and is likely generic in pressure-induced TPTs when a significant carrier

density exists.

Having established the TPTs, we now turn to the interband transitions


to address the controversy over the absorption edge. FIG. 25(a) shows the

absorbance [defined as −log(transmittance)] of PbSe at 298 K for pressures

up to ∼5.1 GPa, at which point a structural phase transition occurs. To as-

sist the discussion, we roughly define three photon energy regions, illustrated

in FIG. 25(a). Region I is dominated by the intraband transition, but is out-

side the spectral range for the instrument used in the measurement. Region

III hosts the majority of the absorption edge, defined as the steep rising

part due to the onset of interband transitions and indicated by the arrow in

FIG. 25(a), which is expected to redshift and then blueshift across the TPT.

The absorption edge shown in Region III of FIG. 25(a) redshifts monotoni-

cally under pressure, indicating band gap closing, but not reopening. Above

1.3 GPa, the initial rising part of the absorption edge which determines the

band gap moves into Regions II and I, overlapping significantly with the

intraband transition.

Close inspection of Region II in FIG. 25(a) reveals band gap reopening.

Despite of the overlap with the conspicuous tail of the intraband transition,

the interband absorption edge in Region II shows a clear change of slope:

it becomes steeper as the pressure is increased to 5.1 GPa, possibly due to

the band gap reopening. For a more conclusive observation of the band gap

reopening, we cooled the sample to 70 K in order to reduce the electronic

scattering rate γ, so that the intraband transition peak became narrower and

overlapped less with the interband absorption edge. As shown in FIG. 25(b),

the absorption edge systematically tilts towards higher photon energy from

2.4 to 4.2 GPa, suggesting that the band gap monotonically increases. Fi-


nally, a structural phase transition occurs at 4.5 GPa (at 70 K), causing a

dramatic overall decrease of absorbance. Pressure-induced band gap closing

and reopening were also observed in PbTe at low temperature.

The above discussion illustrates the complexity of analyzing the inter-

band absorption edge to identify gap closure and band inversion at a TPT.

Considering the apparent monotonic increase of spectral weight in Region III

[see FIG. 25(a)] as a function of pressure, one might conclude that a TPT

had not occurred. But the key signature of gap closure is obscured by overlap

with the intraband absorption, as well as by thermal broadening. This can

be circumvented by cooling the material, revealing both the redshift and then

blueshift of the absorption edge as the band gap closes and then reopens. The

situation for BiTeI is even more complicated due to the Rashba splitting and

the additional optical transitions among the split subbands. Cooling does

not alleviate this complication. Thus, inferring how the band gap changes in

that and similar materials from measurements to sense the absorption edge

is not practical.

In the rest of this Letter, we present two more signatures of pressure-

induced TPTs in PbSe, namely a steeper absorption edge in the topological

phase and a maximum in the pressure dependence of the Fermi level.

The absorption edge becomes steeper after the TPT, distinguishing the

topological phase from the trivial phase. Such behavior is clearly observed

in FIG. 25(b) and confirmed by the calculated optical conductivity shown

in FIG. 25(c). The results emphasize the hybridization nature of the band

gap in a TI (or TCI) as illustrated in FIG. 25(d), qualitatively different from


that in a trivial insulator. As demonstrated by the evolution of the band

structure across a TPT close to the direct band gap, shown in FIG. 23(a–b),

before the TPT, pressure suppresses the band gap and transforms the band

dispersion from a near-parabolic shape to almost linear. After the TPT, the

band dispersion briefly recovers the near-parabolic shape and then flattens.

[At even higher pressure, it develops a Mexican-hat feature as I mentioned in

Chapter 5 similar to that illustrated in FIG. 25(d).] The flat band makes the

joint density of states just above the band gap much greater than that of an

ordinary insulator with the same band gap size, yielding a steeper absorption

edge. It also gives rise to Van Hove singularities that differ from the typical

ones [53], shown as peaks in the optical conductivity for a/a0 = 1.01 and

1.00 in FIG. 25(c).

Such a peak feature was previously observed (although unexplained) in

the Bi2Te2Se TI material with low free carrier density [35, 36], but is ab-

sent in our infrared absorbance data shown in FIG. 25(a–b), possibly for two

reasons. First, the peak only appears deep in the topological phase, which

requires a high pressure that in reality causes a structural phase transition.

Second, the Burstein-Moss effect (see the next paragraph) in our sample pre-

cludes optical transitions connecting the states near the top valence band

and the bottom conduction band and thus the observation of Van Hove sin-

gularities. The dashed line in FIG. 25(c) demonstrates that holes with a

density of 1018 cm−3 completely smears the sharp peak.

Lastly, our calculation shows a maximum in the pressure-dependence of

the Fermi level |EF | at the gap-closing pressure [FIG. 23(c)], which can be


measured from Shubnikov-de Haas oscillations to support the TPT. This

maximum in |EF | happens because the density of states near the top valence

band diminishes as the band dispersion becomes linear, pushing the Fermi

level away from the top valence band to conserve the phase space for the

holes. This effect also manifests in the infrared spectra, although EF cannot

be easily determined from them. When free carriers are present, the band

gap associated with the absorption edge is not the true band gap in the

electronic band structure. As illustrated in FIG. 23(a), the holes shift the

Fermi level to below the top valence band, making the energy threshold for

direct interband transitions approximately Eg+2|EF |, known as the Burstein-

Moss effect. The absorption edge characterizes Eg+2|EF | instead of Eg. The

combined pressure effects on Eg and EF retain a minimum in Eg+2|EF |, but

the corresponding pressure could be different from the critical pressure for

band gap closing, shown in FIG. 23(c). Moreover, the absorption edge never

redshifts to zero photon energy even when Eg = 0. The Burstein-Moss effect

adds further difficulty to the identification of TPTs using the absorption edge.

To summarize, we have established bulk signatures of pressure-induced

band inversion and thus topological phase transitions in Pb1−xSnxSe (x =

0.00, 0.15, and 0.23). Infrared reflectance shows a maximum in the bulk

free carrier spectral weight near the gap-closing pressure. The interband ab-

sorption edge tracks the change of the band gap across the topological phase

transition, however the free carriers complicate the picture due to the overlap

with the intraband transition and the shift of the Fermi level. The absorp-

tion edge becomes steeper in the topological phase due to the hybridization


nature of the band gap in topological insulators. A maximum in the pressure

dependence of the Fermi level is also expected. These robust bulk features

complement the surface-sensitive techniques and serve as a starting point to

investigate topological phase transitions in more complicated systems.

98

8 Chapter 8

8.1 High Resistance of the In-doped Pb1−xSnxTe

For applications in spintronics, it is important to have the resistivity domi-

nated by the topologically protected surface states. Substantial efforts have

been made on the TI material Bi2Se3 and its alloys to reduce the bulk carrier

density; however, while it has been possible to detect the signature of surface

states in the magnetic-field dependence of the resistivity at low temperature

[10–12], attempts to compensate intrinsic defects have not been able to raise

the bulk resistivity above 15 Ω·cm. Theoretical analysis suggests that even

with perfect compensation of donor and acceptor defects, the resulting ran-

dom Coulomb potential still highly limits the achievable bulk resistivity [54].

However, The solid solution Pb1−xSnxTe doped with small amount of In-

dium provides a fresh opportunity for exploration. The parent compound

goes through the topological phase transition with the changes of the com-

ponents: starting from x = 0 as a trivial insulator, then going through the

topological phase transition around xC ≈ 0.35 and staying in topological

insulating phase till x = 1. we observed a nonmonotonic variation in the

normal-state resistivity with indium concentration, with a maximum at 6%

indium doping. A systematic study has been performed, growing and charac-

terizing single crystals with six Pb/Sn ratios (x = 0.2, 0.25, 0.3, 0.35, 0.4, 0.5)

and a variety of In concentrations(y = 0− 0.2).

8.1 High Resistance of the In-doped Pb1−xSnxTe 99

Figure 26: (Color online) Temperature dependence of resistivity in(Pb1−xSnx)1−yInyTe for (a) x = 0.5, (b) x = 0.4, (c) x = 0.35, (d) x = 0.3,(e) x = 0.25, and (f) x = 0.2; the values of y are labeled separately in each panel.For each value of x, indium doping turns the metallic parent compound into aninsulator, with low-temperature resistivity increasing by several orders of magni-tude. The saturation of resistivity at temperatures below 30K suggests that thesurface conduction becomes dominant.


The measured resistivities, ρ (T ), for all samples, characterized by Sn

concentration x and In concentration y, are summarized in FIG. 26. For

each value of x, one can see that the resistivity of the parent compound

(y = 0, black open triangles) reveals weakly metallic behavior; furthermore,

the magnitudes of ρ in the In-free samples depend only modestly on x. With

a minimum of ≈ 2% indium doping, the low temperature resistivity grows

by several orders of magnitude, and the temperature dependence above ≈ 30

K exhibits the thermal activation of a semiconductor. The saturation of the

resistivity for T<30 K is consistent with a crossover to surface-dominated

conduction. The maximum resistivities, surpassing 106Ω·cm, are observed

for x = 0.25 − 0.3. Even for x = 0.35, doping with 6% In results in a rise

in resistivity of 6 orders of magnitude at 5 K; higher In concentrations tend

to result in a gradual decrease in ρ. With increasing y, one eventually hits

the solubility limit of In. Exceeding that point results in an InTe impurity

phase, which is superconducting below 4 K and appears to explain the low-

temperature drop in resistivity for x = 0.4 and y = 0.16 illustrated in FIG. 26

(b). Past studies [55, 56] of various transport properties in Pb1−xSnxTe and

the impact of In doping provide a basis for understanding the present results.

In the topological phase, the system is intrinsic hole-doped which means the

Fermi level is cut into the valence band. For In concentrations of <0.06,

the purities give rise to huge amount of localized impurity states which mix

with the valence band. As a result, the impurity levels stabilize the Fermi

level and deplete the mobile states of the system, which in turn reduce the

conductivity by orders. In a small range of Sn concentration centered about


x = 0.25, the chemical potential should be pinned within the band gap.

Hence, the very large bulk resistivities observed for x = 0.25 and 0.3 are

consistent with truly insulating bulk character.

If only the surface states contribute the conductance, one may expect

that the resistance should not depend on the thickness of the sample. Thus

we concentrate on testing the character of the x = 0.35 and y = 0.02 sample,

where we anticipate topological surface states. To test the contribution of the

surface states to the sample conductivity, we have measured the resistance

R(T ) as a function of sample thickness. The measurements involved sanding

the bottom surface of the crystal with the top contacts remaining nominally

constant. In FIG.( 27(a)) we plot the ratio r(T ) = R (T )/R (300K) for several

thicknesses. Assuming parallel conductance channels for the surface and the

bulk, with the bulk conductance being thermally activated, we fit r(T ) with

r (T )−1 = r−1s + r−1

b e−∆/kBT , (99)

where subscripts s and b label the surface and bulk contributions, respec-

tively. The fitted results for rs and rb are plotted in FIG.( 27(c) and (d));

for the gap, we obtain ∆ = 14.6 ± 0.3 meV. The parameter rs , essentially

the ratio of the bulk conductance at 300K to the surface conductance, lin-

early extrapolates to zero in the limit of zero thickness. Alternatively, we

can calculate the fraction of the conductivity in the surface channel, which

is plotted in FIG.( 27(b)). Despite the fact that the sample thicknesses are

quite large, we find that the surface states provide > 90% of the conduction


for T < 20K. The saturation pattern indicates the quantum behavior of the

conductance in the sample.


Figure 27: (a) Resistance normalized to its room temperature value for severalthicknesses of (Pb0.65Sn0.35)0.98In0.02Te. Lines are fits as described in the text.Results for fitting parameters rs and rb are shown in panels (c) and (d), respec-tively. (b) Fraction of conductivity due to surface states calculated from the fitparameters.


We conclude that small amount doping of indium of topological Pb1−xSnxTe

helps to increase the resistance of the sample into the insulating regime,

which does not change significantly the quantum conductance behavior of

the parent TCI compound. This allows one to exploit the unusual properties

of the surface states in transport measurements without the need to apply

a bias voltage to the surface. Another interesting phenomena coming from

indium doped Pb1−xSnxTe is that with extra more impurity, the system may

be driven into superconducting phase, which may support our theoretical

results of the instability of TI’s (or TCI’s) in Chapter 5 [53].

105

9 Conclusions

The topological insulator is one of the most important discoveries in con-

densed matter physics in recent years. Its exotic properties, such as the

topological protected edge states and spin-momentum locking current, have

promising applications on the devices of spintronics and quantum informa-

tion. The topological insulating phase can be understood from the topolog-

ical property of the Hamiltonian, says mapping the k-space model to the

d-space. Its unique property can be described by the fiber bundle of the

(U(1) in our model) gauge theory. As I present in the thesis, one can also

study it from another point of view–the complex analytic continuation of the

electronic band structure. In the physical k-space, the effective monopoles

reside right at the branch points that are the double degenerate band points

belonging to both the upper and lower bands (in two-band system). The to-

tal magnetic charges below the real k-axis correspond to the Chern number

which is the topological invariant of the system. Then, the only way to go

through the topological phase transition is to swap the branch points with

different cumulative half quantum magnetic charge, which actually become

the Dirac point touching the real k-axis, and then the system is metallic. Fol-

lowing this logic, one can construct higher Chern number model with more

magnetic charge associated to the swapping branch points, which presents

even more rich phases and can be described intuitively by dx + idy winding.

Furthermore, the topological insulator also can show the instability due to

the Mexican hat band structure. Another interesting facet relates the topo-

106

logical insulator model to the Witten’s supersymmetry quantum mechanics

model. To approach the real applications, some experiments are crucial,

such as detecting topological phase by bulk signal and creating real bulk in-

sulating state. Two relevant experimental results are included, supported by

theoretical calculations.

107

Appendices

A Landauer-Büttiker Formalism

A.1 Quantum Hall Effect

The Landauer-Büttiker equation reads

Ii = e2

h

∑j

(TjiVi − TijVj). (18)

For QH, one has

T (QH)i+1,i = 1, for i = 1, ...N, (100)

and the rest of elements in transmission matrix are vanishing. Then we have,

I1

I2

I3

I4

I5

I6

=

G16 (V1 − V6)

G21 (V2 − V1)

G32 (V3 − V2)

G43 (V4 − V3)

G54 (V5 − V4)

G65 (V6 − V5)

= G

1 0 0 0 0 −1

−1 1 0 0 0 0

0 −1 1 0 0 0

0 0 −1 1 0 0

0 0 0 −1 1 0

0 0 0 0 −1 1

V1

V2

V3

V4

V5

V6

,

(101)

where we have used the fact that all Gij = e2

hTij are the same due to the

dissipationless transport of the QH effect and G = e2/h. The transmission

A.1 Quantum Hall Effect 108

matrix is singular because of the constraint–∑iIi = 0. Then it is free to drop

the variable I6 that can be determined after we solve the other currents in

question. Further, we assume V6 = 0 as the reference voltage. Then we can

solve the linear equation,

V1

V2

V3

V4

V5

= G−1

1 0 0 0 0

1 1 0 0 0

1 1 1 0 0

1 1 1 1 0

1 1 1 1 1

I1

I2

I3

I4

I5

. (102)

If one consider the current leads on electrodes 1 and 4, and voltage leads

on electrodes 2, 3, 5 and 6, then one may require that I1 = −I4 = I14 and

other currents are vanishing.

V1 = G−1I14,

V4 = G−1I14 −G−1I14 = 0.(103)

The two-terminal resistance is

⇒ R14,14 = G−1 = h

e2 . (104)

In the same way, we obtain R14,23 = 0.

A.2 Quantum Spin Hall Effect 109

A.2 Quantum Spin Hall Effect

There are two chiral conducting channels in QSHE which are propagating in

opposite directions along the sample edges, and transmission probability to

each direction is 1 due to no backscattering between them.

T (QSH)i+1,i = T (QSH)i,i+1 = 1, for i = 1, ...N, (105)

and again, the rest of elements in transmission matrix are vanishing. Then

Landauer-Büttiker becomes

I1

I2

I3

I4

I5

I6

=

G16 (V1 − V6) +G12 (V1 − V2)

G21 (V2 − V1) +G23 (V2 − V3)

G32 (V3 − V2) +G34 (V3 − V4)

G43 (V4 − V3) +G45 (V4 − V5)

G54 (V5 − V4) +G56 (V5 − V6)

G65 (V6 − V5) +G61 (V6 − V1)

= G

2 −1 0 0 0 −1

−1 2 −1 0 0 0

0 −1 2 −1 0 0

0 0 −1 2 −1 0

0 0 0 −1 2 −1

−1 0 0 0 −1 2

V1

V2

V3

V4

V5

V6

,

(106)

which has the identical redundancy as in QH. We also drop I6, choose V6 = 0

as the voltage reference, and still G = e2/h.

110

Solving the linear equation, we get

V1

V2

V2

V4

V5

= G−1

6

5 4 3 2 1

4 8 6 4 2

3 6 9 6 3

2 4 6 8 4

1 2 3 4 5

I1

I2

I3

I4

I5

. (107)

Then with the same connection in QH, we obtain the two-terminal resis-

tance, V1 = G−1

6 (5I14 − 2I14) ,

V4 = G−1

6 (2I14 − 8I14) .⇒ R14,14 = 3

2G−1 = 3

2h

e2 . (108)

In the same way, the four-terminal resistance is,

V2 = G−1

6 (4I14 − 4I14) ,

V3 = G−1

6 (3I14 − 6I14) .⇒ R23,14 = 1

2G−1 = h

2e2 . (109)

B Invariance of Chern Number Formalism

Consider the following coordinate transformation,

(kx, ky)→ (kα, kβ) , (110)

111

where kα and kβ does not need to be perpendicular to each other, but to be

linear independent. The area element transforms as,

dkxdky = ∂ (kx, ky)∂ (kα, kβ)dkαdkβ, (111)

where ∂ (kx, ky)/∂ (kα, kβ) is the Jacobian.

Then we check the integrand,

∂d

∂kx× ∂d

∂ky= ∂d

∂kα

∂kα∂kx

+ ∂d

∂kβ

∂kβ∂kx

× ∂d

∂kα

∂kα∂kx

+ ∂d

∂kβ

∂kβ∂ky

=

∂d

∂kα× ∂d

∂kβ

∂kα∂kx

∂kβ∂ky

+ ∂d

∂kβ× ∂d

∂kα

∂kβ∂kx

∂kα∂kx

=

∂d

∂kα× ∂d

∂kβ

(∂kα∂kx

∂kβ∂ky− ∂kβ∂kx

∂kα∂kx

)

=

∂d

∂kα× ∂d

∂kβ

∂ (kα, kβ)∂ (kx, ky)

.

(112)

Combing Eq. ( 111) and ( 112), we conclude that the Chern formula

( 63) is invariant for any crystal lattices. Furthermore, our real space higher

Chern number model Eq. ( 71) is also available, though the position vectors~Rnm’s are written along the new primitive vectors.

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