Topological Materials - Appunti

Topological Quantum Materials

Tomarchio Luca

July 24, 2019

2

Contents

1 Berryology 7

1.1 Berry Phase [1][2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.1 Construction of the Geometric Phase . . . . . . . . . . . . . . . . . 8

1.1.2 Chern Number [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Parallel-Transport Gauge [3] . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Bloch Geometry [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.1 Connection, Curvature and Metric . . . . . . . . . . . . . . . . . . 12

1.4 Non-Abelian Geometry [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4.1 Exterior Product and Differentiation . . . . . . . . . . . . . . . . . 13

1.4.2 Non-Abelian Connection and Curvature . . . . . . . . . . . . . . . 14

1.4.3 Chern-Simons 2-form . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 The Quantum Hall Effect 17

2.1 The Classical Hall effect [1] . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Landau Levels [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 Landau Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.2 Symmetric Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 The Integer Quantum Hall effect [1] . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 Robustness and Disorder . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.2 The Role of Topology . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.3 Edge Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Particles on a Lattice [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.1 Particles on a Lattice in a Magnetic Field . . . . . . . . . . . . . . 29

2.5 The Fractional Quantum Hall Effect [1] . . . . . . . . . . . . . . . . . . . . 30

2.5.1 Laughlin States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5.2 Quasi-Holes and Quasi-Particles . . . . . . . . . . . . . . . . . . . . 31

3 Chern-Simons Theories 33

3.1 Integral QHE: Chern-Simons Term [1] . . . . . . . . . . . . . . . . . . . . . 34

3.1.1 2+1 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.2 Connection Between Partition Functions . . . . . . . . . . . . . . . 35

3.1.3 Quantisation of C-S Level . . . . . . . . . . . . . . . . . . . . . . . 36

3.1.4 Edge States and Anomaly Inflow . . . . . . . . . . . . . . . . . . . 37

3

4 CONTENTS

4 Topological Phases of Matter 39

4.0.1 Topological Ordering [5] [6] . . . . . . . . . . . . . . . . . . . . . . 40

4.1 Berry Phase Intrusion [25] . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.1 Degeneracies and Level Crossing . . . . . . . . . . . . . . . . . . . . 42

4.2 Many Body Intrusion [25] . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3 Hall Conductance and Chern Numbers [25] . . . . . . . . . . . . . . . . . . 46

4.4 Time-Reversal Symmetry [25] . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.5 Anomalous Hall Conductivity [3] . . . . . . . . . . . . . . . . . . . . . . . 50

4.5.1 AHC in Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.5.2 Haldanium [10] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.6 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.6.1 Symmetries of Graphene . . . . . . . . . . . . . . . . . . . . . . . . 54

4.7 Models for the Chern Insulator [25] . . . . . . . . . . . . . . . . . . . . . . 55

4.7.1 Chern Insulator in a Magnetic Field . . . . . . . . . . . . . . . . . . 57

4.8 Bulk-Edge Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Topological Insulators [25] 59

5.1 SOC and Quantum Spin Hall Insulators . . . . . . . . . . . . . . . . . . . 60

5.1.1 Inverted Quantum Wells . . . . . . . . . . . . . . . . . . . . . . . . 62

5.2 Z Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2

5.2.1 Charge Polarization and Thouless Pumping . . . . . . . . . . . . . 64

5.3 Band Crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.4 3D Topological Insulators [26] . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.4.1 Surface Dirac Cones . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6 Weyl and Dirac Semimetals 73

6.1 Dirac and Weyl Electrons [10][4] . . . . . . . . . . . . . . . . . . . . . . . . 74

6.1.1 One Dimensional Crystals . . . . . . . . . . . . . . . . . . . . . . . 75

6.1.2 Three Dimensional Crystals . . . . . . . . . . . . . . . . . . . . . . 76

6.2 The Nielsen-Ninomiya Theorem . . . . . . . . . . . . . . . . . . . . . . . . 77

6.2.1 The Berry Connection . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.3 Weyl Electrons in Condensed Matter [9] . . . . . . . . . . . . . . . . . . . 79

6.3.1 Robustness of the Phase . . . . . . . . . . . . . . . . . . . . . . . . 81

6.3.2 Chiral Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.4 Fermi Arc Surface States [9] [10] . . . . . . . . . . . . . . . . . . . . . . . . 84

6.4.1 Boundary Modes for Dirac Fermions [10] . . . . . . . . . . . . . . . 86

6.5 Explicit Lattice Model and Phase Diagram [14] . . . . . . . . . . . . . . . 88

6.5.1 Surface States Phenomenology . . . . . . . . . . . . . . . . . . . . . 89

T P

6.5.2 Weyl Semimetals with Broken or Symmetry [9] . . . . . . . . . 90

6.6 Dirac Semimetals in 3D [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.6.1 Fermi Arcs in Dirac Semimetals . . . . . . . . . . . . . . . . . . . . 92

6.7 Materials Considerations [9] . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.8 Identification Through Band Structure [9] . . . . . . . . . . . . . . . . . . 95

6.9 Semiclassical Linear Transport [9] . . . . . . . . . . . . . . . . . . . . . . . 96

CONTENTS 5

6.10 Quantum Effects in Transport [9] . . . . . . . . . . . . . . . . . . . . . . . 98

6.10.1 Chiral Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.10.2 Charge Pumping Effect [16] . . . . . . . . . . . . . . . . . . . . . . 100

6.10.3 Surface States Transport . . . . . . . . . . . . . . . . . . . . . . . . 104

6.11 Conductivity for Real Weyl semimetal [20] . . . . . . . . . . . . . . . . . . 104

6.11.1 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.12 Nonlinear Responses [23] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.12.1 Shift Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.12.2 Injection Currents [24] . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.12.3 TaAs Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.12.4 Second Harmonic Generation . . . . . . . . . . . . . . . . . . . . . 114

7 Weyl-Kondo Semimetals [27] 115

7.1 3D Periodic Anderson Model . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8 Topological Superconductors [25] 119

8.1 Majorana Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8.2 Spinless p-wave Superconductors . . . . . . . . . . . . . . . . . . . . . . . . 122

8.2.1 2D Chiral p-wave Superconductor . . . . . . . . . . . . . . . . . . . 125

8.3 From Theory to Experimental Wires . . . . . . . . . . . . . . . . . . . . . 128

8.3.1 Perfect Andreev Reflection . . . . . . . . . . . . . . . . . . . . . . . 129

8.3.2 Flux-Induced Fermion Parity Switch . . . . . . . . . . . . . . . . . 130

8.4 Non-Abelian Quantum Statistics . . . . . . . . . . . . . . . . . . . . . . . 131

8.4.1 Quantum Computing Application . . . . . . . . . . . . . . . . . . . 134

6 CONTENTS

Chapter 1

Berryology

The Berry phase concept is the simplest demonstration of how geometry and topology

can emerge from quantum mechanics. It takes its name from a very influential paper by

Michael Berry, circulated in 1983 although published in 1984. Berryology is used here

as a synonymous for geometry in nonrelativistic quantum mechanics. Topological (i.e.

quantized) quantities are defined via geometrical quantities, but many of them are purely

geometrical. |ψ i |ψ i

The founding concept of geometry is distance. Let and be two quantum

1 2

states in the same Hilbert space: we adopt for their distance an appropriately modified

form of the Bures distance: 2 2

− |hψ |ψ i|

D = ln 1 2

12

The states are defined up to an arbitrary phase factor, and fixing it means a gauge

choice. The distance is clearly gauge-invariant. However, besides distance, an additional

geometrical concept is needed: the connection, which fixes the relative phases between

two states in the Hilbert space. The connection is arbitrary and cannot have any physical

meaning by itself. Nonetheless, after the 1984 paper by Berry, several physical observables

are expressed in terms of the connection and related quantities.

1.1 Berry Phase [1][2]

Suppose to have a general Hamiltonian which depends upon two different kinds of variables

a i a

H(x ; λ ). The x are the degrees of freedom of the system, meanings the things that

evolve dynamically, the ones that we want to solve in any problem. In contrast, the other

i

variables λ are the parameters of the Hamiltonian, which are fixed values determined by

some external apparatus. We don’t usually exhibit the dependence of H from them. We

|ψi

now place the system in an energy eigenstate (the ground state for simplicity) and we

assume it to be unique. By varying very slowly the parameters λ, the Hamiltonian changes,

|ψ(λ(t))i.

and so does the state However, a theorem in quantum mechanics, known as

adiabatic theorem, states that, under the considerations done above, the system will

7

8 CHAPTER 1. BERRYOLOGY

cling to that energy eigenstate. This theorem doesn’t take in mind the possibility of level

crossing with varying parameters, indeed these events are rare in quantum mechanics.

The idea of the Berry phase arises if we reach again the initial parameters configura-

tion, doing a loop in the Hamiltonian space. Thus the final state must coincide with the

initial one, with the only exception of an uncertain phase

iγ

|ψi → |ψi

e

We often think of the phase as unphysical, but this is not the case due to the fact

that this can be a phase difference. For example, we could have started with two

states and taken only one of them on this journey while leaving the other unchanged. A

subsequent interference between them would exploit the acquired phase. This element will

−iEt/}

be composed of two contributions: a common dynamical phase e , and a geometrical

one, which is the mentioned Berry phase.

1.1.1 Construction of the Geometric Phase

The concept of ‘geometric phase’ made its first appearance in 1956 in a paper written by

the young Indian physicist S. Pancharatnam, which at the time went largely unnoticed

in the Western world. The ideas behind it can be applied with basically no change

to the phases of the quantum states of any physical system. The essential feature of

Pancharatnam’s work is that of considering discrete phase changes, at variance with the

more recent work, where continuous phase changes are usually addressed.

|ψ(λ(t))i

The state vectors are all supposed to reside in the same Hilbert space, mean-

ing that the wave functions are supposed to obey λ-independent boundary conditions. At

t = 0 we are sure to be in a non-degenerate ground state, and the degeneracy property is

maintained at every time. The most natural way of defining a phase difference is

hψ(λ )|ψ(λ )i

1 2

−i∆ϕ

e =

12 |hψ(λ )|ψ(λ )i|

1 2

However, in this way ∆ϕ will be unphysical, in fact we can always change in an arbi-

12

trary way the phase (gauge) of one of the states. This same problem remains between the

initial and final state of a discrete path in the Hamiltonian space, and only by closing it

into a loop we can make the phase difference to be gauge-invariant, i.e., physical.

Ex: Thinking back at the example of two identical states, the one that evolves with

the parameters will get a phase every time it is subjected to a change in λ, however, it

can be randomized by choosing a different global gauge (it is related to the Hamiltonian,

and now the first is different, hence can change the gauge without affecting the immobile

state). Thus, its interaction with the initial state won’t see any phase interference until

the initial λ configuration is reached again, where a global gauge change would afflict both

the states.

1.1. BERRY PHASE [1][?] 9

Since the phase acquired at every step (with the exception of a gauge change) is charac-

terised by the present and antecedent λ configuration, it seems obvious that the phase at

the end of the loop, γ = ∆ϕ + ∆ϕ + ∆ϕ + ∆ϕ , will depend on the chosen path.

12 23 34 41

Switching to a continuous formalism, if we assume that the gauge is so chosen that

the phase varies in a differentiable way along the path, then we get to leading order in

∆λ −i∆ϕ ' hψ(λ)|∇ ·

ψ(λ)i ∆λ

λ

with λ a vector of general dimension. In the limiting case of a set of points which be-

comes dense on the continuous path, the total phase difference γ will converge to a circuit

integral: M I

I

I

X A(λ) ·

hψ(λ)|∇ · −

→ dλ

ψ(λ)i dλ =

dϕ = i

γ = ∆ϕ λ

s,s+1 C

C

C

s=1 A,

where the quantity known as Berry connection, has been introduced. This is defined

up to a gauge transformation, in fact it will depend upon the gauge used to make the

phase change differentiable. We could have also used a different general form

0 0

iω(λ)

|ψ |ψ(λ)i → A A(λ) ∇

(λ)i = e (λ) = + ω

λ

Following the analogy with electromagnetism, we might expect to find the physical infor-

mation in the curvature of the connection

∂A ∂A ∂ψ(λ) ∂ψ ∂ψ(λ) ∂ψ

i j

F − −= h | i−h | i

(λ) = =

ij j i i j j i

∂λ ∂λ ∂λ ∂λ ∂λ ∂λ

such that Z ij

− F

γ = dS

ij

S

where S is a two dimensional surface in the parameter space bounded by the path C. The

Berry connection and curvature are also spelled out as gauge potential and gauge field.

One being gauge-dependent and the other gauge-invariant.

The Berry phase will be identified with an observable quantity which cannot be ex-

pressed in terms of any Hermitian operator. This strangeness is due to the fact that

an isolated system cannot show a parametric dependence, hence a Berry phase. Its ap-

pearance is the consequence of an underlying interaction of the system with an external

universe, and its nature is just a trade-off to exploit some observable effects not included

in the initial treatment of through the incomplete Hamiltonian.

10 CHAPTER 1. BERRYOLOGY

1.1.2 Chern Number [3]

If we now take a three dimensional parameter space, the S surface, which is enclosed in

2

the path to compute the Berry phase, can be used to identify a sphere S which contains

it as its equator (supposed to be a circle). If we develop the flux integral of the curvature

over this surface, and divide by 2π, we would get an integer value called Chern number

of the first class Z

1 ij

F

3 dS

C =

Z ij

1 2π 2

S

The proof of such a statement is obtained through the Gauss-Bonnet-Chern theorem.

This proves the Chern number to be a robust topological invariant of the wave function,

and is at the origin of observable effects.

To give some insights about the translation of the system symmetries into the Chern

number computation, we introduce the density of states (projector) of the ground state

P (λ) and its complement Q(λ), i.e.

|ψ −

P (λ) = (λ)ihψ (λ)|, Q(λ) = P (λ)

I

0 0

When λ is even under time reversal (like e.g. a nuclear coordinate), then time-reversal

invariance implies that both H(λ) and P (λ) are real for any λ, and warrants that the

curvature is everywhere vanishing, in fact it can be computed as

F −2=Tr{∇

(λ) = P (λ)Q(λ)∇ P (λ)}

i j

ij λ λ

The Berry phase γ can be nonzero (modulo 2π) only if the curve C loops around a

singularity or, more generally, it does not lie in a simply connected domain; the only

allowed values are γ = 0mod2π or γ = πmod2π.

When instead λ is odd under time reversal (like momentum) then this symmetry

∗

requires P (λ) = P (−λ), therefore the curvature is odd. The Berry phase along an

inversion-symmetric path vanishes and the Chern number vanishes as well. Thus, a non

vanishing Chern number can only occur in absence of time-reversal symmetry.

In the case of reflection symmetry, instead, the condition is for the curvature to be even,

thus making possible the presence of a non-vanishing Chern number under this symmetry.

If both the symmetries are present, then the curvature is everywhere vanishing, and the

Berry phase can be only 0 or π, with the latter case requiring a domain which is not

simply connected.

1.2 Parallel-Transport Gauge [3] |∇

The main ingredients in Berryology are the λ−derivatives of the ground state ψ (λ)i.

λ 0

To lowest order we have

1.3. BLOCH GEOMETRY [?] 11

!

hψ |∆H|ψ i

X n 0

−iϕ(λ)

|ψ |ψ |ψ

(λ + ∆λ)i = e (λ)i + (λ)i

0 0 n −

E (λ) E (λ)

0 n

n>0

−

where ∆H = H(λ + ∆λ) H(λ). The overall phase factor is generally omitted: this

corresponds to a specific gauge, called the parallel-transport gauge. The name indicates

that the first order change in the ground state is chosen to be orthogonal to the state

itself. Thus, within this gauge, th