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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
Stony Brook University
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)
Stony Brook University
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
2.1 Physics Rediscovery of the Mathematical Concepts 6
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
2.1 Physics Rediscovery of the Mathematical Concepts 7
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.
2.1 Physics Rediscovery of the Mathematical Concepts 8
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)
2.1 Physics Rediscovery of the Mathematical Concepts 9
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
2.2 Discovery of TI’s in Experiments 11
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,
2.2 Discovery of TI’s in Experiments 12
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).
2.2 Discovery of TI’s in Experiments 13
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
2.2 Discovery of TI’s in Experiments 14
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
2.2 Discovery of TI’s in Experiments 15
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-
2.2 Discovery of TI’s in Experiments 16
(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).
2.2 Discovery of TI’s in Experiments 17
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.
2.2 Discovery of TI’s in Experiments 18
Figure 4: Adopted from D. Hsieh et al. Nature 452, 970-974 (2008).
2.2 Discovery of TI’s in Experiments 19
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-
3.2 Quantum Hall Effects 24
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).
3.2 Quantum Hall Effects 25
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
3.2 Quantum Hall Effects 26
(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.
3.2 Quantum Hall Effects 27
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
3.2 Quantum Hall Effects 28
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.
3.2 Quantum Hall Effects 29
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
3.3 IQHE vs Topology 31
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
3.3 IQHE vs Topology 32
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)
3.3 IQHE vs Topology 33
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.
3.4 Monopole in Complex Momentum Space 35
?
?
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?
3.4 Monopole in Complex Momentum Space 36
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.
3.4 Monopole in Complex Momentum Space 37
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).
3.4 Monopole in Complex Momentum Space 38
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
3.4 Monopole in Complex Momentum Space 39
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)
3.4 Monopole in Complex Momentum Space 40
++
+ +
+
+++
+
+++
+ +
++
+++
+++
+++++
+++
++
+ +++++
+++
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.
3.4 Monopole in Complex Momentum Space 41
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
3.4 Monopole in Complex Momentum Space 42
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
3.4 Monopole in Complex Momentum Space 43
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
3.4 Monopole in Complex Momentum Space 44
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
3.4 Monopole in Complex Momentum Space 45
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.
3.4 Monopole in Complex Momentum Space 46
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)
3.4 Monopole in Complex Momentum Space 47
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
3.4 Monopole in Complex Momentum Space 48
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.
4.1 Higher Chern Number Model 51
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)
4.1 Higher Chern Number Model 52
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
4.1 Higher Chern Number Model 53
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
4.1 Higher Chern Number Model 54
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
4.2 Real Space Models 56
-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.
4.2 Real Space Models 57
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)
4.2 Real Space Models 58
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
4.3 Phase Transition with Fixed Mass Term 60
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).
4.3 Phase Transition with Fixed Mass Term 61
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.
4.3 Phase Transition with Fixed Mass Term 62
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
5.1 Band Structures of TI’s 65
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-
5.1 Band Structures of TI’s 66
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
5.1 Band Structures of TI’s 67
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.
5.1 Band Structures of TI’s 68
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.
5.1 Band Structures of TI’s 69
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-
5.2 Symmetry Breaking Instabilities 71
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
5.2 Symmetry Breaking Instabilities 72
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-
5.2 Symmetry Breaking Instabilities 73
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.
5.2 Symmetry Breaking Instabilities 74
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
5.2 Symmetry Breaking Instabilities 75
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
5.2 Symmetry Breaking Instabilities 76
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)
6.1 Supersymmetry in Quantum Mechanics 79
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)
6.1 Supersymmetry in Quantum Mechanics 80
⇒ 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
6.1 Supersymmetry in Quantum Mechanics 81
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
6.2 Supersymmetry in TI 83
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)
6.2 Supersymmetry in TI 84
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-
6.2 Supersymmetry in TI 85
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)
7.1 Bulk Signature of the Topological Phase Transition 88
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
7.1 Bulk Signature of the Topological Phase Transition 89
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
7.1 Bulk Signature of the Topological Phase Transition 90
/
← →
∼ +← ←
/
← →
/
← ←
+← ←
/
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.
7.1 Bulk Signature of the Topological Phase Transition 91
#!&
#
"$
%
#!&
#
%
#!&
#
%
#!&
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
7.1 Bulk Signature of the Topological Phase Transition 92
" #
!
! "
" #
!
" #
−
⋅
−
⋅
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
7.1 Bulk Signature of the Topological Phase Transition 93
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-
7.1 Bulk Signature of the Topological Phase Transition 94
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
7.1 Bulk Signature of the Topological Phase Transition 95
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
7.1 Bulk Signature of the Topological Phase Transition 96
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
7.1 Bulk Signature of the Topological Phase Transition 97
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.
8.1 High Resistance of the In-doped Pb1−xSnxTe 100
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
8.1 High Resistance of the In-doped Pb1−xSnxTe 101
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
8.1 High Resistance of the In-doped Pb1−xSnxTe 102
for T < 20K. The saturation pattern indicates the quantum behavior of the
conductance in the sample.
8.1 High Resistance of the In-doped Pb1−xSnxTe 103
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.
8.1 High Resistance of the In-doped Pb1−xSnxTe 104
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.
REFERENCES 112
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