The Pennsylvania State University

The Graduate School

UNIVERSAL LANDAU ZENER BEHAVIOR IN THE DYNAMICS OF

TOPOLOGICAL PHASE TRANSITIONS

A Dissertation in

Physics

by

Yang Ge

© 2021 Yang Ge

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

May 2021

The dissertation of Yang Ge was reviewed and approved by the following:

Marcos Rigol

Professor of Physics

Dissertation Advisor

Chair of Committee

Chao-Xing Liu

Associate Professor of Physics

Mikael C. Rechtsman

Downsbrough Early Career Development Professor of Physics

Paul Baum

Evan Pugh Professor of Mathematics

Nitin Samarth

Professor of Physics

George A. and Margaret M. Downsbrough Department Head

ii

Abstract

Topological (quantum) matter displays fascinating response properties and robust edgestates that put them at the center of continuous research. Generation, control andmanipulation of topological matter proves to be a colorful theoretical playground andpromises next-generation applied technologies such as quantum computing. The workpresented in this dissertation focuses on the generation of the simplest topologicalmatter—the Chern insulators by driving a topologically trivial system.

The topological index for a Chern insulator in translationally invariant systems isthe first Chern number. Unfortunately, in the thermodynamic limit, a no-go theoremstates that the Chern number of a system is invariant under unitary time evolutionsthat are smooth in quasimomentum space, which excludes almost all practical drives.However various experiments have successfully imprint topology into static systems usingtime-periodic drives. In numerics a real-space counterpart of the Chern number, theBott index, also changes in driven systems with open boundary conditions. The distinctbehavior of different topological indices, in systems with different boundary conditionsand the ultimate fate of these systems constitute the subject of my doctoral work.

My first paper shows that the prohibition of a unitary generation of topological phasesonly exists in the thermodynamic limit. I show that the Bott index and the Chernnumber are effectively equivalent. Hence there is no fundamental difference betweenthese topological indices. Then, I demonstrate how in finite-size translationally invariantsystems, a trivial Fermi sea under a slowly-turned-on drive can acquire a nontrivialtopology. This can happen provided that the gap-closing points in the thermodynamiclimit are absent in the discrete Brillouin zone. This provides a hint to resolving theconflict between the no-go theorem and the numerical and experimental studies. Studyingthese systems in momentum space also allows one to identify a simple Landau-Zenerbehavior when crossing the topological transition at different speeds.

In my second work, I show that Landau-Zener behavior is universal across boundaryconditions. In the excitation statistics, and in the critical field when a topological indexjumps, these driven systems are found to display three regimes with increasing speeds:the near-adiabatic regime, followed by the Landau-Zener regime, and then the rapiddrive regime where the topological indices do not change. In the thermodynamic limit,only the fast ramp and Landau-Zener regimes survive for nonvanishing ramp speeds. Atthe end it is shown that the dc Hall response can be used to detect topological phasetransitions independently of the behavior of the topological indices.

iii

Table of Contents

List of Figures vi

List of Symbols and Notations viii

Acknowledgments x

Chapter 1Introduction 11.1 Topological band theory of Chern insulators . . . . . . . . . . . . . . . . 3

1.1.1 The Chern number . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Winding on a Bloch sphere . . . . . . . . . . . . . . . . . . . . . . 71.1.3 The Bott index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.4 The Haldane model . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Floquet quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.1 Floquet theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.2 Perturbative expansion of the Floquet Hamiltonian . . . . . . . . 151.2.3 Notes on symmetry and topology . . . . . . . . . . . . . . . . . . 15

1.3 Dynamical preparation of Floquet Chern insulators, a preliminary discussion 161.3.1 Invariance of the Chern number . . . . . . . . . . . . . . . . . . . 161.3.2 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 171.3.3 Preparation protocol . . . . . . . . . . . . . . . . . . . . . . . . . 191.3.4 Dynamics in open boundary lattices . . . . . . . . . . . . . . . . . 191.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Chapter 2Topological phase transitions in finite-size periodically driven trans-

lationally invariant systems 242.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Equivalence between the Bott index and the Chern number . . . . . . . . 272.3 Model Hamiltonian and Floquet topological phases . . . . . . . . . . . . 322.4 Dynamics of the Bott index in finite systems . . . . . . . . . . . . . . . . 362.5 Landau-Zener dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

iv

Chapter 3Universal Landau-Zener regime in topological phase transitions 463.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Model, drive protocol, and geometries . . . . . . . . . . . . . . . . . . . . 483.3 Scaling of the critical field in the Floquet Hamiltonian . . . . . . . . . . 503.4 Scaling of the critical field in the dynamics . . . . . . . . . . . . . . . . . 52

3.4.1 Translationally invariant systems . . . . . . . . . . . . . . . . . . 533.4.2 Cylinder geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4.3 Patch geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.5 Hall Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.5.1 Floquet ground state . . . . . . . . . . . . . . . . . . . . . . . . . 683.5.2 Time-evolved state . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.6 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Chapter 4Summary and outlook 74

Appendix AComputations and some properties of the Bott index 77A.1 Bott index in cylinder geometry . . . . . . . . . . . . . . . . . . . . . . . 77A.2 Relation to the local Chern marker . . . . . . . . . . . . . . . . . . . . . 78A.3 Efficient calculation of the dynamics of topological indices . . . . . . . . 79

A.3.1 Translational invariant systems . . . . . . . . . . . . . . . . . . . 79A.3.2 Cylinder geometries . . . . . . . . . . . . . . . . . . . . . . . . . . 79A.3.3 Patch geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Appendix BBehavior of the exact solution to the Landau-Zener problem 82

Bibliography 84

v

List of Figures

1.1 Experimental realization of the Haldane model by Jotzu et al. . . . . . . 2

1.2 Haldane model, lattice, Brillouin zone and phase diagram . . . . . . . . . 10

1.3 Sketch of the Floquet Brillouin zone and quasienergy modes . . . . . . . 14

1.4 Bott index in a patch geometry lattice for systems in equilibrium and ainitially trivial system driven into a topological regime . . . . . . . . . . 20

1.5 Scaling of the dynamical critical field in patch geometry with system sizeand ramp speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1 Chern number phase diagram of the driven model with Berry flux snapshots 34

2.2 Critical field strength in finite translationally invariant lattices . . . . . . 35

2.3 Time evolution of the Bott index, occupation number and overlap intranslationally invariant systems . . . . . . . . . . . . . . . . . . . . . . . 38

2.4 Berry curvature in the static, Floquet, and time-evolved Fermi seas offinite translationally invariant systems . . . . . . . . . . . . . . . . . . . 40

2.5 Landau-Zener collapse of the final occupation of momentum states nearthe gap closing momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1 Honeycomb lattice with different boundary conditions . . . . . . . . . . . 49

3.2 Critical strength of the driving field in finite lattices of cylinder and patchgeometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Landau-Zener collapse of maximum excitation across the Brillouin zonein translationally invariant systems . . . . . . . . . . . . . . . . . . . . . 56

vi

3.4 Landau-Zener collapse of wavefunction overlap in translationally invariantystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5 The collapse of the excitation density in translationally invariant ystems 58

3.6 Landau-Zener collapse of the dynamical critical field in translationallyinvariant systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.7 Landau-Zener collapse of wavefunction overlap in cylinders . . . . . . . . 62

3.8 The collapse of the excitation density in cylinders . . . . . . . . . . . . . 63

3.9 Landau-Zener collapse of the dynamical critical field in cylinders . . . . . 65

3.10 Landau-Zener collapse of the dynamical critical field in patch geometries 66

3.11 Time-averaged Hall response of the ground state of the Floquet Hamilto-nian in translationally invariant systems . . . . . . . . . . . . . . . . . . 69

3.12 Time-averaged Hall response of the time-evolved state prepared underdifferent ramp times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A.1 Bott index of Floquet eigenstates at A = 1 in a patch geometry withdifferent contiguous fillings . . . . . . . . . . . . . . . . . . . . . . . . . . 81

B.1 Plots of the analytic solution to the Landau-Zener transition problem. . . 83

vii

List of Symbols and Notations

〈a|, |a〉 Dirac bracket notation of a wavefunction a.

i Imaginary unit.

e Natural exponential.

Z Integers.

N Natural numbers.

ln Complex natural logarithm, branch cut on negative real axis.

Tr Trace of a matrix or an operator.

Im Imaginary part of.

Re Real part of.

a Tensor/Matrix/Vector a.

~a Vector a.

Ck Differentiability class of order k.

δij Kronecker delta.

εijk Levi-Civita symbol.

e Electron charge. Set to 1 in most of the discussion.

h Planck’s constant.

~ Reduced Planck’s constant, h/2π. Set to 1 in most of the discussion.

Ψ Many-body Schrödinger wavefunction.

ψ Single particle Schrödinger wavefunction (orbital).

viii

ψk Single particle Bloch wavefunction at quasimomentum k.

uk The lattice periodic part of ψk.

Ch Chern number, p. 5.

Cb Bott index, p. 8.

PA Probability to remain adiabatic, i.e. the occupation number in the ground state.

PE Probability to become excited, i.e. 1− PA.

~σ,σi Pauli matrices, (σx, σy, σz) or (σ1, σ2, σ3).

σij Conductivity tensor. σxy is the Hall conductivity.

X The quantum operator of X.

XT Transpose of X.

X† Hermitian conjugate, a.k.a. Hermitian adjoint, that is conjugate transpose of X.

c.c. Complex conjugate of preceding terms.

H.c. Hermitian conjugate of preceding terms.

c, c† Fermionic annihilation and creation operators (of an electron).

n Number operator (of an electron).

〈i, j〉 Lattice nearest-neighbor pair i and j.

⟪i, j⟫ Lattice next-nearest-neighbor pair i and j.

J Nearest neighbor hopping amplitude. Set to 1 in most of the discussion.

d Nearest neighbor distance. Set to 1 in most of the discussion.

~ai Primitive lattice vectors.

~bi Reciprocal primitive vectors. ~ai ·~bj = 2πδij.

BZ Brillouin zone.

ix

Acknowledgments

There are many that I am thankful to in my pursue for a doctorate. Foremost I wouldlike to thank my advisor, Prof. Marcos Rigol, for his immense patience and support. Ideeply value his warm encouragements and wise advice. He sets a great example for mewith his passion, insight and integrity. Along with my classmates, I also benefited a lotfrom his graduate quantum mechanics course sequence.

I would like to express my gratitude to the committee. Prof. Chao-Xing Liu hastaught me a great deal with his keen lectures and genuine characters. His joint groupmeeting with Prof. Jain has been a fruitful opportunity for me to learn from peers andexperts. I would like to thank Prof. Mikael Rechtsman for discussions as well as theexcitement and devotion that he brings about with his presence. I am very grateful forProf. Paul Baum to be on my committee.

The Department of Physics has provided me the essential environment to nourish asan uninitiated physicist. I vividly remember the joy and wisdom radiating from the greatlectureship of Prof. John Collins, Prof. Radu Roiban, and Prof. Eugenio Bianchi. Theancient society of AMO journal club brings science, cheers and pizzas to the otherwisedull life on Fridays, along with the priceless sharp comments and discussions amongthe faculties. On the research life, I would like to thank Prof. Richard Robinett for hisexchanges and support.1 My gratitude also goes to Carol Deering, Melissa Diamanti,Julianne Mortimore and the many supporting staff here at the department.

My peers have been the most vibrant constituent of my graduate life. I havealways gained knowledge, perspectives and pleasure when discussing with fellow graduatestudents in our research group, Krishnanand Mallayya, Yicheng Zhang and Tyler LeBlond,as well as the postdocs, Lev Vidmar, Rubem Mondaini and Luca D’Alessio. A tributeto Luca is in order, as it was him who patiently coached me at the beginning with hisinsights and computer codes. I profoundly treasure my friendship and conversationswith Jiabin Yu, about life, science and reason, and the universe and everything. It hasbeen my pleasure to spend time and discuss with Tsung-Yao Wu, Songyang Pu, WilliamFaugno, Jianxiao Zhang2, Neel Malvania, Peng Du, Jiho Noh, Qing-Ze Wang, and manyothers. I have also been amazed and inspired by the condensed matter discussion group

1From now on titles will be arbitrarily omitted.2Hyphens in Chinese names are sometimes arbitrary.

x

orchestrated by Rui-Xing Zhang. I am grateful to Di Xiao, Leonardo Estêvão SchendesTavares, Kai-Jie Yang, Boyang Zheng, Jianyun Zhao, Tongzhou Zhao, Jiali Lu, BrettGreen, Zhong Lin, Yufei Sheng, Pavlo Bulanchuk and a great many others for cherishedmemories.

My undergraduate experience at the University of Minnesota is indispensable for mygraduate studies and research. I would like to express the gratitude to my undergraduateresearch advisor, Prof. Martin Greven for providing me with a wealth of opportunities. Mystudies in physics really began with the marvelous and penetrating introductory coursesdelivered by Prof. Paul Crowell. To wind further back to the beginnings but never be itthe endings, I would like to thank my classmate in high school and undergraduate times,Gong Chen, for his friendship and support, laughs and tears. The same gratitude extendsto my undergraduate friend Chenzi Zhang, who revealed to me the art of computing,and to my middle school friend Junhua Zhang, who explained to me the heat equationwhen I was a junior shouting officer.

My life at State College is colorfully enriched by my roommates, all fellow graduatestudents at a time. I would like to express my gratitude towards Sheng-Hsun Lee, for hisfriendliness, stewardship and master cookery. I also cherish my memories with MingkunCui and Emmy Titcombe, under the same roof of our lovely townhouse. During my lastsemester I was kindly taken in by Yao Duan and Lingjie Zhou. I am grateful for theirhospitality.

Towards the end I want to thank my parents for their everlasting support, comfort,patience and concern. My parents and my larger family have taught me the power anduniqueness of kinship. Their unconditional love and teachings never cease.

I would like to thank Prof. Tilman Esslinger for granting the right to reproduceFig. 1.1. The work presented in this dissertation was supported by the Office of NavalResearch under Grant No. N00014-14-1-0540, and the National Science Foundation underGrant No. PHY-2012145. The computations were done in the Roar supercomputer ofthe Institute for Computational and Data Sciences (ICDS) at Penn State. Any opinions,findings, and conclusions or recommendations expressed in this publication are those ofme and my collaborators, and do not necessarily reflect the views of the funding agenciesor computing centers.

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Chapter 1 |

Introduction

The discovery of quantum Hall effect [1] in 1980 set in motion the era of topological

condensed matter physics [2–4]. In Ref. [1], a pristine sample, which was effectively

a 2D electron gas, was seated under a magnetic field under a low temperature; then

a current was driven through and the induced voltage in the transverse direction was

measured. The measurement gave a quantized Hall conductivity σxy ≡ Ix/Vy = ne2/h,

n ∈ Z. The quantization is so precise that it was immediately proposed to be the new

standard for SI units [1] and finally got adopted in 2019 [5], winning its discoverer a

Nobel prize in between [6]. As such it is genuinely another triumph of quantum physics.

The quantization is associated to the momentum space topology of the 2D electron gas,

with the integer identified with the first Chern number [7–9]. In 1988 Haldane proposed a

model, now called the Haldane model, with intrinsic quantum Hall effect, i.e. a quantum

anomalous Hall effect [10], which became the first model in topological band theory.

The field thrived vigorously after the experimental realization of the quantum spin Hall

effect [11–13] followed by the quantum anomalous Hall effect [14–16] in the past two

decades. Recent developments in topological band theory have much accelerated the

search for new topological materials [17–20].

Meanwhile the Haldane model remained a theoretical toy, until Jotzu et al. in

Esslinger’s group realized it with ultracold atoms in a “shaking” honeycomb optical

1

Figure 1.1. Experimental realization of the Haldane model by Jotzu et al. in Ref. [21]. (a) Theultracold atoms are trapped by the two-dimensional optical lattice built-up from laser beamsbetween retro-reflecting mirrors. Two piezoelectric actuators shake the lattice to create aneffective Floquet Hamiltonian realizing the Haldane model. The acoustic-optical modulators(AOM) stabilizes the lattice. (b) Experimental result of the differential anomalous drift (seeEq. (1.4)) mapping out the phase diagram of the Haldane model realized (Adapted fromRef. [21]).

lattice in 2014 [21], see Fig. 1.1. Since then a plethora of topological models have

been realized with ultracold quantum gases in optical lattices [21–23], in photonic

systems [24–26], in optical waveguides [25, 27–34], as well as in electrical circuits and

mechanical networks [35–44]. The ability to assemble and synthesize topological systems

provides novel platforms and probes to manipulate and study the interplay between

topology and physics, all while offering new opportunities for applications [45–47].

An effective and powerful method to synthesize topological systems is by applying

periodic drives on trivial systems, e.g., by shining light [48–50], by shaking an optical

lattice [22,23,51], or by spacial modulation of a waveguide [27,28,30,31,33,34,52]. In

the steady state, these systems are described by Floquet Hamiltonians, and they fall

under the realm of Floquet topological systems.

My doctoral work, among the work of many others [53–70], focuses on the preparation

2

of topological states from topological trivial ones in the lines of experiments in [21,71].

In the following I briefly review some preliminaries, first the topological band theory

with emphasis on the topological indices of Chern insulators and the Haldane model,

then the Floquet Hamiltonian. Next I review results from the work of D’Alessio and

Rigol [56], which generated some of the questions that I address in my work. At the end

of this chapter I provide an outline of the remaining chapters.

1.1 Topological band theory of Chern insulators

1.1.1 The Chern number

Translation symmetry is broken for electrons in a crystal structure setup by positive

ions in solids, and for ultracold quantum gases in optical lattices, namely, for ultracold

atoms that sit in a spatially modulated periodic potential made of light. There remains

a discrete translational symmetry by primitive lattice vectors ~ai that carries to the

Hamiltonian governing the dynamics of the system. Stationary states can then be charted

with unitary eigenvalues of the primitive lattice translation operators. These are the

Bloch states [72,73],

|ψkα〉 = eikx|ukα〉. (1.1)

They are labeled by the crystal momentum ~k1, together with other labels α indicating

different eigenstates at the same momentum. The Bloch states each consists of a plane

wave part eikx and a lattice periodic part |ukα〉. The latter feels the representation of the

Hamiltonian at k, termed the Bloch Hamiltonian Hk, defined through

H|ψkα〉 = eikxHk|ukα〉. (1.2)1I set ~ = 1 in most of this work. I also refer to crystal momentum, a.k.a. quasimomentum, simply

as momentum when its meaning is clear from the context.

3

The Bloch Hamiltonian has the periodicity of the unit cells. Its eigenenergies are denoted

by εkn, with n being the band index of energies.

The reciprocal space of crystal momentum repeats itself since a crystal momentum

k is equivalent to k + bi, where the reciprocal primitive vectors bi are defined through

bi · aj = 2πδij. Thus it is sufficient to identify a single unit cell in the reciprocal space,

called the Brillouin zone (BZ). Assuming the whole lattice is finite in size and taking

the Born-von Karman boundary condition, which is the periodic boundary condition at

the edges of the whole lattice, the crystal momentum are discrete points in the Brillouin

zone. We choose the normalization convention such that

〈ψkβ|ψkα〉 = δkk′δαβ. (1.3)

The introduction of crystal momentum allows us to write down semiclassical equations

of motion for particles moving in a lattice under a small external force F, [73–79],

ri = ∂εk

~∂ki+

d∑j,k=1

εijkkjΩk,

~ki = Fi, (1.4)

where d is the spatial dimension. The transverse term in the velocity

vH = k×Ω (1.5)

is the so-called anomalous velocity that contributes to the Hall current. The symbol Ω

here is the Berry curvature. To define it, we first note that there is a U(1) gauge freedom

in the momentum space, that each |uk〉 is free to take up a phase factor. With a smooth

gauge defined, |uk〉 at different momenta form a smooth connection A, called the Berry

4

connection, defined as

Ai = i〈uk|∂ki |uk〉. (1.6)

Electrons moving adiabatically through the Brillouin zone due to Eq. (1.4) under a small

external field will pickup an additional phase due to the Berry connection exp (∫

A · dk).

Its linear response, e.g., the velocity, will then pickup the contribution from the Berry

curvature Ω, defined as

Ωµ =d∑

ν,ρ=1εµνρ∂kνAρ, (1.7)

where d is the spatial dimension. It is invariant under a smooth gauge transformation.

In two dimensions (2D), for a band isolated in energy from other bands, the integral

of Berry curvature over the whole Brillouin zone is an integer multiple of 2π. That integer

factor is the first Chern number of the band,

Ch = 12π

∫BZ

d2k ∇×A. (1.8)

An alternative gauge-invariant formulation of the Chern number is given by the projection

operator, Pk = ∑i |ψki〉〈ψki|, with i running over filled bands,

Ch = 12π Im

∫BZ

d2kTr(Pk[∂kxPk, ∂ky Pk]). (1.9)

For 2D insulating states, the valence bands are completely filled while the conduction

bands are completely empty. Thus at low temperatures, a small electric field or other

external force cannot excite the Fermi sea, and the filled states just move around according

to Eq. (1.4). It gives rise to an integer Hall effect [8]. An important criterion for a

nonzero Berry curvature is the symmetry of the system. By applying spatial inversion to

both sides of Eq. (1.5), the velocity and force (k) both change sign, thus Ω(k) = Ω(−k).

Under time reversal, with a similar arguement one finds Ω(k) = −Ω(−k). Therefore

5

time reversal symmetry must be broken to allow for a nonzero Chern number.

The Chern number reflects a topological property of an isolated energy band. One

can adiabatically deform the band and the Chern number will remain the same as long

as the energy gap to other bands remains open and no band touching occurs. Such band

structures are said to be adiabatically connected, and this condition defines an equivalent

class of insulating states. Thus we can classify insulating states by their Chern number.

Those with a nonzero Chern numbers are called Chern insulators. States with different

Chern numbers are in different topological phases. The distinction of topological phases

is beyond the traditional Landau paradigm of symmetry breaking phase transitions.

Aside from a quantized Hall conductivity, a nontrivial Chern number also entails

edge modes at the interface between a Chern insulator and another material that has a

different Chern number or just the vacuum. This is the consequence of bulk-boundary

correspondence principle of topological invariants [80–83], which can be seen from the

following argument. Suppose one starts from the bulk of an insulating region with one

Chern number and moves towards the bulk of another insulating region with a different

Chern number. During this process the local Hamiltonian must have its gap closed,

which would happen around the interface between them. Hence the interface between

two topologically distinct regions must have these so-called topologically protected edge

states cross the gap of the bulk Hamiltonian on either side. Note that the common

vacuum is also an insulating state with zero Chern number. So topologically protected

edge states exists at the edge of a piece of Chern insulator.

Other than the Chern number, there is a family of stable topological indices that

applies to systems of different symmetry classes [84] or protected by crystalline symmetries

[85–87]. They are in general beyond the scope of this work. A common feature share by

them is that if one stays within the given symmetry class, insulating states with different

topological indices cannot be adiabatically connected, and protected gapless edge modes

6

live at the interface between them. The Chern number class do not require any symmetry

to be well-defined.

1.1.2 Winding on a Bloch sphere

A two-band system in 2D admits a convenient visualization of the Berry curvature and

the Chern number using the Bloch sphere [88]. An eigenstate of a two-band system in

general can be parameterized by angular directions of a spinor

|uk〉 =

cos( θk2 )

sin( θk2 )eiφk

, (1.10)

parameterized by θk ∈ [0, π] and φk ∈ [0, 2π). This parametrization naturally maps the

eigenstate to the surface of a unit sphere. In these variables, integral over the Berry

curvature becomes integral over the differential solid angle,

∫Ω(kx, ky)dkx ∧ dky =

∫ 12 sin θkdθk ∧ dφk. (1.11)

Hence in the Bloch sphere picture Chern number counts how many time the eigenstate

spinors over the Brillouin zone wraps around the sphere.

1.1.3 The Bott index

Since topology of an insulating state is stable against small deformations of the band

structure that does not close the gap, one would expect the Chern number to stay

unchanged when a small disorder is added to the system. The Chern number as

defined in momentum space in Eqs.(1.8), (1.9) is not convenient for systems that are not

translationally invariant. The alternative formulation is given by the Bott index [89–91],

defined as follows.

7

Consider a lattice on the surface of a torus of size Lx × Ly, where x, y are the two

orthogonal periodic directions. One can define the operators

U := exp(2πix/Lx), (1.12)

V := exp(2πiy/Ly). (1.13)

One can then use the projection operator of occupied states P to produce the reduced

coordinate-phase matrices U , V given by

P U P $

0 0

0 U

, (1.14)

P V P $

0 0

0 V

, (1.15)

in the basis of occupied (lower right) and unoccupied (upper left) single particle states.

The Bott index for this state is given by

Cb(P ) = 12π Im Tr ln(V U V †U †). (1.16)

If P represents a conducting state, the Bott index will be ill defined because the

corresponding V U V †U † matrix is singular [89]. As with the Chern number, a topologically

trivial state has zero Bott index, and a nonzero Bott index counts the number of

topologically protected edge modes at the boundaries of the system [91] (see also Fig. 1.4).

In the atomic limit, V and U commutes. Thus the Bott index takes a trivial value. A

nontrivial Bott index indicates that the occupied band cannot be completely expanded

in a localized Wannier basis. The presence of protected edge states really comes from

the nonlocalized states terminating at the boundaries of the system [82].

The Bott index is more flexible than the Chern number definition in that it requires

8

neither translational invariance nor completely filled energy bands. The requirement that

the lattice be sitting on a torus can be relaxed. Since the Bott index can be computed

in disordered systems, one can introduce a large repulsive disorder along a cut on the

torus to open the boundary in one direction, or both. In fact, one can just cut the lattice

directly and compute it on open boundary lattices. The exact coordinate where the

edges sit is of no consequence, and when a open boundary system is placed on a torus

there is some tolerance in Lx and Ly. The Bott index can also be generalized to other

topological indices, e.g., the Z2 index for the quantum spin Hall effect [90, 92, 93]. It has

been applied to a variety of geometries including quasicrystals, amorphous systems, and

fractals [26,94–97].

We note that there are other Chern number equivalents defined in real space that

can be used with different boundary conditions and disorder. It can further map the

topological phase locally. Examples are found in Kitaev’s work in Ref. [98], the work

by Bianco and Resta in Ref. [99] and the work by Loring in Ref. [100]. In App. A.2 we

discuss the relation between the Bott index and the local Chern marker.

1.1.4 The Haldane model

Now we introduce the Haldane model as a model for Chern insulators. It is also the

model on which my works focus on. The Haldane model consists of a honeycomb lattice

with real nearest-neighbor hoppings J , complex next-nearest-neighbor hoppings J2, and

a staggered sublattice potential ∆/2. The complex next-nearest-neighbor hopping comes

from a spatially alternating magnetic flux going though different parts of the lattice. As

seen in Fig. 1.2(a), magnetic flux ±Φ passes through region a and b in the lattice, so

that J2 contains a phase exp(iΦ) but the total flux enclosed by a unit cell remains zero.

9

A sublatticeB sublattice

-π 0 π

Φ

-6

-4

-2

0

2

4

6

∆/2

J 2

+-

+

++

+

+ --

--

-

b2

K

b1

K’

M

Γ-1 +1

(a) (b) (c)

0

0

Figure 1.2. (a) The Haldane model lattice. Different onsite energies on A and B sublatticesites constitute the onsite potential. Spinless electrons can hop with a real amplitude alongnearest-neighbor links drawn with solid lines, and a complex amplitude along next-nearest-neighbor links drawn with green dashed arrows in the center unit cell. The complex phase ofnext-nearest-neighbor hopping is determined by the alternating magnetic fluxes ±Φ throughtriangular regions in the figure. Hoppings along the arrows acquire positive phases. (b) TheBrillouin zone of honeycomb lattice in reciprocal space, with reciprocal primitive vectors ~b1 and~b2. High symmetry points M , K, K ′ and Γ are marked. (c) The Chern number phase diagramof the Haldane model, as a function of the ratio between the strength of the staggering potentialand the next-nearest-neighbor hopping amplitude ∆/2J2 and of the next-nearest-neighborphase Φ. Inspired by Ref. [10].

The Hamiltonian is given by,

H = −J∑〈i,j〉

(c†i cj + H.c.

)+∑⟪j,l⟫

(J2c†j cl + H.c.

)+ ∆

2∑i∈Aj∈B

(ni − nj). (1.17)

With only nearest-neighbor hopping and real next-nearest-neighbor hopping, the

Haldane model has time reversal (Θ), inversion (I), C3 rotation symmetry, and others [101].

Protected by Θ and I together, there are two Dirac cones located at K and K ′ [see

Fig. 1.2(b)] where the two bands touch. The staggered sublattice potential ∆ breaks

inversion symmetry. A nonzero imaginary part of J2 breaks time reversal symmetry. As

discussed in Sec. 1.1.1, it is the breaking of time reversal symmetry that allows for a

nonzero Chern number.

Figure 1.2(c) gives the Chern number phase diagram of the valence band of the

10

Haldane model. The phase boundary is traced out by the gapless condition of the

Hamiltonian. Note that one must keep |J2/J | < 1/3 to keep the system insulating at half

filling. The fact that a lattice without net flux can exhibit quantum Hall effect maybe

understood as the following. When a unit cell encloses exactly an integer multiple of h/e

flux, the total phase due to magnetic field an electron traveling around the edges of a

hexagon will pick up is integer multiple of 2π. This is equivalent to zero flux. However

part of the flux may be picked up by next-nearest-neighbor hoppings. Hence one can

just treat the Haldane model as the result of a single h/e flux per unit cell.

We have mentioned that there is so far no real material that can be described by the

Haldane model. However an effectively imaginary next-nearest-neighbor hopping can be

generated by a Floquet drive, introduced below.

1.2 Floquet quantum systems

Cyclic processes are central to engineering. The Floquet theory provides a powerful

means to analyze and design quantum systems [102] where the periodic drive may be

treated classically [103]. This section presents the Floquet theorem and the Floquet

Hamiltonian. At the end there is a short comment on symmetry and topology in Floquet

systems.

1.2.1 Floquet theory

A system under some periodic driving with period T is described by a Hamiltonian

periodic in time, i.e., H(t+ T ) = H(t). The Floquet theorem states that the evolution

operator U(t) of any periodic Hamiltonian H(t) can be decomposed into the action of a

time-independent Hamiltonian HF and a periodic evolution operator P [103–105]

U(t) = P (t− t0 − nT )e−inHF [t0]T P (t0), (1.18)

11

where 0 ≤ t0 < T , n = b(t− t0)/T c. Here HF [t0] is the Floquet Hamiltonian, explicitly

defined with

HF = iT

ln[U(t0 + T )U †(t0)]. (1.19)

P (t) is the micromotion operator. It is unitary with the same period T . From Eq. (1.18)

one can be that P (nT ) = 1 for n ∈ Z. HF depends on the initial time t0.

Obviously HF [t0] of different initial times t0 are unitarily equivalent. Thus all HF [t0]

have identical spectra and their eigenstates are adiabatically connected. In terms of

topological properties it suffices to consider HF := HF [0]. I will henceforth drop the

initial time label.

Floquet Hamiltonian functions as a discrete time translation operator of periodically

driven systems. As such, it greatly simplifies the analysis of stroboscopic time behavior.

Eigenvalues of the Floquet Hamiltonian are called quasienergies. The associated eigen-

states are quasienergy states or steady states. They only change by a phase factor at

stroboscopic times. By analogy with Bloch’s theorem2, we can extract their stroboscopic

phase [106],

|ψαn(t)〉 = exp(−iεαnt)|φαn(t)〉, (1.20)

where εαn ∈(− πT, πT

]is the nth quasienergy associated with quantum number(s) α, and

|φαn(t)〉 is a state periodic in time. The quasienergies are defined modulo the driving

frequency ω = 2π/T , and there is some freedom on choosing the branch cut in Eq. (1.18).

The time-periodic state can be further decomposed in to Fourier harmonics of ω,

|φαn(t)〉 =∑m

e−imωt|φ(m)αn 〉. (1.21)

Without integrating the Schrödinger equation to get U(T ), one may expand the

problem in Fourier harmonics and directly solve the eigenvalue problem in an extended2Historically, the Floquet theorem was formulated before Bloch’s theorem.

12

Hilbert space. This extend Hilbert space has the form H = H0 ⊗ Z, with the Fourier

modes labeled by their orders (Z), and different orders are orthorgonal. In the basis of

the Fourier harmonics, the eigenvalue problem reads,

∑l∈Z

(H(ml) −mωδml)|φ(l)αn〉 = εαn|φ(m)

αn 〉, (1.22)

where H(ml) is the Fourier component of H, as in

H(ml) ≡ H(m−l) = 1T

∫ t0+T

t0ei(m−l)ωtH(t)dt. (1.23)

The initial time t0 is explicitly restored to show the dependence. In matrix form

. . . ...

H(0) + 2ω H(1) H(2)

H(−1) H(0) + ω H(1) H(2)

H(−2) H(−1) H(0) H(1) H(2)

H(−2) H(−1) H(0) − ω H(1)

H(−2) H(−1) H(0) − 2ω H(1)

... . . .

...

|φ(1)αn〉

|φ(0)αn〉

|φ(−1)αn 〉...

. (1.24)

This form explicitly shows how the drive modifies the static Hamiltonian H(0). Similar

to the case of Bloch’s theorem, the eigenvalues ε in Eq. (1.22) are repeated in frequency

space by period ε+mω, with the eigenstate shifting upward m entries. As depicted in

Fig. 1.3, they form Floquet Brillouin zones of period ω. This artificial enlargement in

Hilbert space dimension goes away when one assembles the quasieigenstates of HF with

a branch cut convention [Eqs. (1.19-1.21)].

Let us point out two properties of the Floquet Hamiltonian and its quasieigenstates.

13

k

-ω /2

ω /2ε

Edge stateCh = -1Ch = 1

k k

(a) HS

(b) (c) HF

Figure 1.3. A sketch of the effect of Floquet driving and the Floquet Brillouin zone. (a) Aperiodic drive of frequency ω is to be applied to this static system HF with two bands of Chernnumber ±1, and a topologically protected edge state in the gap. Generally states in resonanceare most strongly coupled. (b) The Floquet Brillouin zone repeats with a period of ω. Statescoupled by the drive are mixed and show level repulsion. Note that the choice of the branchcut changes location of the edge state. The choice used in this thesis is the dashed line. Theother is the dotted line. (c) The quasienergy spectrum of the Floquet Hamiltonian. Inspiredby Ref. [107].

First [103],

det U(T ) = exp(−i Tr HFT ) = exp[−i∫ T+τ

τTr H(t)dt

], for any τ. (1.25)

The second interesting property is about Floquet quasieigenstates. For an observable O

without time dependence, its expectation in a Floquet eigenstate is related to its time

averaged expectation [108],

〈φαn|O|φαn〉 = 1T

∫ T+τ

τdt〈ψαn(t)|O|ψαn(t)〉, for any τ. (1.26)

Therefore other than being solutions at stroboscopic times, the quasieigenstates have

14

information about the time average over a period.

1.2.2 Perturbative expansion of the Floquet Hamiltonian

There are many ways to get HF from a perturbation series, ∑l H(l)F [105,109,110]. We

adopt the high frequency expansion as in Ref. [105]. Then up to O(ω−1),

H(0)F = H(0).

H(1)F = 1

ω

∞∑l=1

1l[H(l), H(−l)]. (1.27)

This also agrees with the Brillouin-Wigner perturbation approach based on Eq. (1.22) [109].

1.2.3 Notes on symmetry and topology

Out of equilibrium, symmetries of H(t) do not necessarily transfer to U(t) or HF . This

lead to different topological classifications [111]. Even when they do, the topological

invariants for static systems do not fully characterize a driven system. This is already

hinted in Fig. 1.3(b), where the choice of branch cut can change the number of protected

edge states. More generally the Chern number only determines the difference in the

number of edge states above and below a quasienergy band. The same number of edge

states can exist about ε = 0 and ω/2, respectively called zero modes and π modes,

when all the bands in Fig. 1.3(b) have zero Chern number. This is the case of an

anomalous Floquet topological insulator [107]. More general topological characterization

and classification of Floquet systems can be found in Ref. [112–117]. The π mode has

received much research attention as it is related to the intriguing concept of discrete time

crystals [118–125]. Majorana π modes has also been propose as a candidate in quantum

computing [45].

15

1.3 Dynamical preparation of Floquet Chern insulators,

a preliminary discussion

In this section I introduce the Hamiltonian and driving protocol of interest to this work,

and review and discuss the results by D’Alessio and Rigol in Ref. [56], which set the

stage for my work discussed in later chapters.

1.3.1 Invariance of the Chern number

In Refs. [56,63], the Chern number was shown to be invariant in two-band systems under

unitary time evolutions, if the drive is C2 in momentum space and the initial state is C1

in momentum space. There is no restriction on the smoothness of the drive in time. This

can be seen as a consequence of the fixed point theorem in topology. Given two filled

bands as continuous mappings from the Brillouin zone to the space of states, for example

the Bloch sphere, bands with different winding numbers will be orthogonal to each other

since there will always be a k point where the states are orthogonal. In unitary time

evolutions, if a state |Ψ〉 can evolve into a different topology at some t∗, then it must be

the case that 〈Ψ(t∗+ dt)|Ψ(t∗− dt)〉 = 0. This is impossible for a state with finite energy.

Explicitly using the formulation of the Chern number in Eq.(1.9), and adopting the

gauge choice such that Pk is periodic in reciprocal space [126], one has

dChdt = −Re

∫BZ

d2k2π Tr

([Hk(t), Pk][∂kxPk, ∂ky Pk] + Pk

[∂kx

[Hk(t), Pk

], ∂ky Pk

]+Pk

[∂kxPk, ∂ky [Hk(t), Pk]

])= −Re

∫BZ

d2k2π Tr

([Hk(t), Pk][∂kxPk, ∂ky Pk]− ∂kxPk

[[Hk(t), Pk

], ∂ky Pk

]−∂ky Pk

[∂kxPk, [Hk(t), Pk]

])= −3 Re

∫BZ

d2k2π Tr

([Hk(t), Pk][∂kxPk, ∂ky Pk]

). (1.28)

16

Being idempotent, a projection operator has the following properties

∂xP = P (∂xP ) + (∂xP )P , (1.29)

P (∂xP )P = 0. (1.30)

With them, one can show that Tr([Hk(t), Pk][∂kxPk, ∂ky Pk]

)is zero. Therefore,

dChdt = 0. (1.31)

Thus, the Chern number is invariant in a multiband setting in general.

1.3.2 Model Hamiltonian

The contradiction between the experimental observation in the drift velocity measurements

prompted a study of the system in Ref. [21]. The experimental system is modeled by a

tight-binding model of spinless fermions on a honeycomb lattice with nearest-neighbor

hopping and a sublattice staggered potential at half filling. An in-plane circularly

polarized electric field, which is uniform in space, provides the time-periodic drive. In

units of ~ = 1, the Hamiltonian is

H(t) = −J∑〈i,j〉

[eie ~A(t)~dij c†i cj + H.c.

]+ ∆

2∑i∈Aj∈B

(ni − nj). (1.32)

The 2D vector potential ~A(t) = A (sinωt, cosωt) accounts for the electric field. It

introduces a phase when particles hop from site i to one of its nearest-neighbor sites j,

separated by a distance d = |~dij|. The second term in H(t), with site number operators

nj, describes the staggered potential (of strength ∆) between the A and B sublattices

in the honeycomb lattice. In a translationally invariant system, this Hamiltonian is

block diagonal in momentum space. Each 2 × 2 momentum block is described by a

17

pseudomagnetic field − ~Bk · ~σ acting on the sublattice spinor (ck,A, ck,B)T, where ~σ are

the Pauli matrices. From now on, we adopt the unit e = d = 1.

Same as in the Haldane model, when both A and ∆ are zero, the energy bands are

gapless at K and K ′ in the Brillouin zone, protected by the combination of Θ and I. In

the static case (A = 0), a nonzero ∆ introduces a Bz of equal magnitude at K and K ′

and opens a gap. Here and in the remaining text, we set ∆ = 0.15J to be close to the

experimental parameters in Ref. [21]. Both static bands have zero Chern numbers. The

Floquet spectrum is symmetric about zero due to Eq. (1.25).

Time reversal symmetry requires the existence of some time t0 such that H(t0 + t) =

H(t0 − t) for all t. This is broken by the circular polarization of the light. The result

is that mass terms of opposite sign added to K and K ′. It shrinks the gap opened by

inversion breaking mass at K ′. Since a change of topological phase needs closing of the

bulk gap, the transition comes when the time reversal breaking strength exceeds that

of inversion breaking. At that instance, the band gap closes, and the Berry curvature

diverges and flips sign.

The C3 symmetry, however, is preserved by HF . Despite the fact that H(t) does not

have C3 at any time, HF is invariant under the combination U(T/3)C3. Hence when any

of Θ or I is broken, a topological phase transition is most likely to change the Chern

number by 1 or 3. I defer further discussion of the topologcal phases of this model until

Sec. 2.3.

The driving frequency ω in Ref. [56] and in most of this thesis is selected to be ω = 7J

to match the experiment. This driving frequency is higher than the band width of both

of the bands together. Thus there is a well-defined connection from the Floquet system

to the static system. The Floquet valence band and conduction band are also clearly

ordered.

18

1.3.3 Preparation protocol

Using the Hamiltonian in Eq. (1.32), the driving field A is turned on linearly in time,

such that

A(t) = t/τ, (1.33)

following the protocol used by Jotzu et al. [21]. This way one can dynamically prepare a

topological state starting with the trivial band insulator at half filling at A = 0, and drive

it towards the topological phase at A = 1. The ramp up time τ affects how adiabatic the

preparation is. To analyze the dynamics, we monitor the topological indices together

with wavefunction overlap and excitation statistics.

1.3.4 Dynamics in open boundary lattices

One possibility to reconcile the experimental observations with the no-go theorem is to

break translational invariance. The Bott index enabled an investigation into the open

boundary systems. In Ref. [56], the Bott index was for the first time used on an open

patch of honeycomb lattice, displayed in Fig. 1.4(b-c). The Bott index computed at

different Fermi energies follows the behavior of the (cumulative) local density of state. It

is zero inside bulk bands and reports the topological character of the band inside the

band gap [Fig. 1.4(c)]. A Fermi sea with a nonzero Bott index supports edge modes in

the bulk gap.

Surprisingly, the dynamical results revealed that the system can start from a trivial

state and be prepared into a topological one [see Fig. 1.4(d)]. The first results to look

at are the stroboscopic overlap of the time-evolved state with the initial ground state,

defined as∣∣∣〈ΨS

0|Ψ(t)〉∣∣∣2, and the overlap with the final Floquet ground state, defined as∣∣∣〈ΨF

0 |Ψ(t)〉∣∣∣2. These overlaps obey the fixed-point theorem. They are never simultaneously

close to one. The time-evolved state starts off with perfect overlap with the initial ground

19

Figure 1.4. Results obtained by D’Alessio and Rigol (I). (a) The honeycomb lattice on anopen patch. Highlighted are A and B sublattice sites, as well as the center and edge sites usedfor cumulative density of state (CLDOS) calculations. (b-c) The Bott index as a function ofFermi (quasi)energy in (b) the trivial Ch = 0 regime and (c) the nontrivial Ch = 1 regime.The edge and bulk cumulative density of states clearly shows the bulk gap and whether anyedge modes is present. Inset of (c): Blown up view of A and B sublattice sites together withthe three nearest neighbor hoppings of an A site. (d) A topologically trivial system is drivenfrom the trivial regime, where A = 0, into the topological one, where A = 1, with A = τ/Tincreasing linearly in time. Plotted are the dynamics of the stroboscopic overlap with initialand final half filled Fermi sea and the stroboscopic Bott index. The inset shows the final overlapfor different turn-on time lengths τ of the driving field. Adapted from Ref. [56].

20

state, and ends up with greater overlap with the final Floquet ground state. This result

is then supported by the jump of the Bott index near the end of the ramp, indicating a

topological phase transition. Even though the behavior of the overlap is not monotonic

during the dynamics, the final overlap does generically increases with longer τ . One last

remark is that the stroboscopic overlap with an eigenstate of the final Hamiltonian is

guaranteed not to change when the ramp ends and the field is held constant for t >= τ .

Aside from the fact that a topological phase transition is possible in open boundary

systems, the scaling of the dynamical critical field at which the Bott index jumps, denoted

by E∗c , is also remarkable. As seen in Fig. 1.5, for longer ramp lengths the evolution

becomes more adiabatic and decreases E∗c , as one would expect. However, E∗c also

decreases with system size for a fixed ramp length. This seems to suggest that the

topological phase of open boundary systems changes in the thermodynamic limit. As I

will discuss in Chapter 3, this is an artifact due to the decreasing equilibrium critical

field with increasing lattice size. The difference between the dynamical and equilibrium

critical field in finite-size systems actually increases with lattice size, until the lattice size

is too large for a dynamical phase transition to occur.

Towards the end of Ref. [56] the current running inside equilibrium and dynamically

prepared systems were measured. Unfortunately, in Floquet systems the bulk quasienergy

states themselves carry current and it is difficult to disentangle contributions of the entire

many body state, even though the edge currents can be identified. Followup works on

the intrinsic current redistribution during the dynamics can be found in Ref. [64,127].

1.3.5 Summary

The work by D’Alessio and Rigol showed that despite the invariance of Chern number

during generic unitary dynamics, a topological phase transition can be detected in open

boundary lattices with the Bott index. That result was further supported by an analysis

21

0 2 4 6 8 10 12 14 16 18 20τ / T (x 100)

5

5.25

5.5

5.75

6

6.25

6.5

Ec*

aN

sites= 928

5 10 15 20 25 30 35 40 45 50N

sites(x 100)

5

5.5

6

6.5

Ec*

τ = 80 Τ

b

Figure 1.5. Results obtained by D’Alessio and Rigol (II). The scaling of the dynamicalcritical field E∗c upon Bott index jump in patch geometries. (a) E∗c versus linear ramp time τin a lattice with 928 sites. (b) E∗c versus system size for a fixed linear ramp length, τ = 80T .Adapted from Ref. [56].

of the overlap between time-evolved many body state and the half filled ground state of

the trivial initial system and the nontrivial final system.

Those results opened a number of questions. Why would the system behave differently

for different boundary conditions? What allows the no-go theorem to be violated? Is

there a single thermodynamic limit for systems with different boundary conditions? If

there is, how could the behavior of translationally invariant systems and open boundary

systems reconcile? These questions are answered in my work as discussed in the following

chapters.

1.4 Outline

The remaining text of my dissertation closely follows the two papers I wrote as part of

my graduate research. Chapter 2 contains my first work on the violation of the no-go

theorem in finite translationally invariant systems [65]. It highlights how finite-size effects

manifest as a commensurability condition on the lattice sizes in translationally invariant

systems. In the thermodynamic limit, the Bott index is translated into momentum space

and found to be equivalent to the Chern number. Hence the open-boundary patch lattices

22

and translationally invariant lattices should share the same thermodynamic limit. I also

give a pictorial explanation for why the no-go theorem holds in the thermodynamic limit

and yet fails in finite size lattices. Initial links between the dynamics of the topological

phase transition and the Landau-Zener theory are also reported.

In Chapter 3, I explore the Landau-Zener physics in the dynamics of topological

phase transitions across all boundary conditions [70]. The lattice types studied include

translationally invariant, cylindrical, and open-boundary patch geometries. A universal

Landau-Zener regime is demonstrated with the overlap and the dynamical critical field.

The Landau-Zener theory also leads to an expected Kibble-Zurek scaling in excitation

density in these systems. Two other regimes of ramp speed are explored: a fast regime

where the dynamics is almost system-size independent, and a slow adiabatic regime

that does not exist in the thermodynamic limit. I also study the Hall responses of

dynamically prepared systems. They show a nontrivial Hall conductance corresponding

to the topological character of the final Hamiltonian.

In Chapter 4, I summarize my results and discuss interesting problems that await

further research.

23

Chapter 2 |

Topological phase transitions infinite-size periodically driven trans-lationally invariant systems

This chapter presents the work reported in Ref. [65]. It first takes the Bott index to

momentum space and equated it with a Wilson loop based calculation of the Chern number

in the discrete Brillouin zone [128]. Then I examine the phase diagram of the model

used (Eq.(1.32)). Tracking the temporal dynamics of wavefunction overlaps, occupation

number, and the Bott index reveals the different behavior between commensurate and

incommensurate lattice sizes. As a result even in translationally invariant systems the

Bott index may change. It also leads to the identification of the Landau Zener dynamics

governing the topological phase transition. For that I provide a modified Bloch sphere

picture to explain why the Chern number is invariant in the thermodynamic limit but

may change in finite systems.

24

2.1 Introduction

Topological insulators have attracted much attention in the last decade [13,129]. While

they might appear as equilibrium phases in some materials, applying a time varying

potential provides a flexible way to induce topological phases in insulators that are

topologically trivial otherwise. In particular, a system driven periodically in time can

exhibit so-called Floquet topological phases [49,50,113]. For example, a high-frequency

periodic drive can modify the topological structure of energy bands giving rise to a rich

realm of exotic states [109,130,131]. A few experiments have been carried out to explore

topological phases in periodically driven systems [21, 27, 132]. Closely related to the

model studied here, in Ref. [21] a driven Fermi sea of ultracold atoms in a honeycomb

lattice acquired a nontrivial topology as predicted by the Haldane model [10].

The unitary time evolution of the topological properties of a Fermi sea, as it turns

out, is fundamentally different from just tuning the Floquet Hamiltonian across different

phases. In Ref. [56], a no-go theorem was proved for dynamics under a simple two-band

Hamiltonian in two dimensions. It states that the topological index, the Chern number,

of a Fermi sea that is in a pure state does not change during unitary dynamics under

rather general conditions. The system studied consisted of spinless fermions on an infinite

translationally invariant (periodic boundary conditions) honeycomb lattice. As a result,

the Hamiltonian is block diagonal in the crystal momentum space. At each crystal

momentum k, the Bloch Hamiltonian has the form Hk(t) = − ~Bk(t) · ~σ, where ~σ are the

Pauli matrices and ~Bk(t) is the pseudomagnetic field. If the initial state is pure and

both the pseudomagnetic field and the pseudospin are smooth in k space, then by the

no-go theorem the first Chern number is a constant of motion [56, 63], as seen in studies

of quantum quenches [53, 55, 133]. This implies that, in infinite systems, an adiabatic

annealing that changes the Chern number can never be achieved [62].

25

Much work has been done to explain the change in topology as observed in experiments.

One approach is to embed the out-of-equilibrium system in a dissipative setting. This

can be achieved by introducing a thermal bath [54, 57] or dephasing noise [61]. Non-

unitary evolutions destroy coherence within the quantum state. Consequently, the Hall

conductance and a generalized version of the Chern number can change [61,134]. Similar

results can be obtained in the context of the diagonal ensemble [135, 136]. Another

important point is that equivalent formulations of a topological index in equilibrium may

not be equivalent any more out of equilibrium, and the response function of the system

can be contained in a non-conserved formulation [53,64,136–138].

Intriguingly, boundary conditions also appear to determine whether a topological

index can change under a periodic drive, as demonstrated in the study of systems with

open boundary conditions [56]. In those systems, which lack translational symmetry, the

Bott index is a topological index that can be used in place of the Chern number. The

Bott index is defined in real space and does not require transitional invariance [89–91].

In Ref. [56], when evolving a finite Fermi sea under a periodic drive that was turned

on slowly, it was found that the Bott index can change from a value determined by the

initial Hamiltonian to that of the equilibrium Floquet bands. Thus, for open boundary

conditions (the case in experiments), the Bott index is not a conserved quantity.

The contrast between the no-go theorem for infinite translationally invariant systems

and the fact that the Bott index can change under unitary dynamics in finite lattices

with open boundary conditions motivates us to further explore the relation between

the Chern number and the Bott index and to study the time evolution of the Bott

index in finite lattices with periodic boundary conditions. First, we reformulate the Bott

index in momentum space and prove that it is equivalent to the Chern number in the

thermodynamic limit. When written in momentum space, the Bott index is nothing but

the integer formulation of the Chern number in finite lattices as derived in Ref. [128] from

26

lattice gauge theory. In addition to being a gauge-independent integer by definition, this

topological index has the advantage that, with increasing system size, it converges to the

thermodynamic result much more rapidly than the usually used discretized integration

of the traditional Chern number. Our second goal is to understand the dynamics of the

Bott index in finite translationally invariant systems under the same model Hamiltonian

as in Ref. [56]. We show that the Bott index can change in incommensurate lattices,

as it does in systems with open boundary conditions. There is a finite time scale for

the turn on of the periodic drive that enables this topological transition to occur. This

time scale diverges with increasing system size, as expected from the no-go theorem for

infinite systems.

The presentation is organized as follows. In Sec. 2.2, we reformulate the Bott index in

momentum space and prove its equivalence with the Chern number in the thermodynamic

limit. In Sec. 2.3, we introduce the model Hamiltonian, and use the Bott index and a

finite-size scaling analysis in translationally invariant lattices to determine the phase

diagram of the Floquet Hamiltonian in the thermodynamic limit. The dynamical behavior

of the Bott index in finite translationally invariant systems is studied in Sec. 2.4. A

summary of our results is presented in Sec. 2.6.

2.2 Equivalence between the Bott index and the Chern

number

The Chern number of an energy band is defined as the integral of the Berry curvature

over the Brillouin zone (BZ) [8,129]. By definition, the Chern number is a topological

index for translationally invariant two-dimensional (2D) systems. For 2D systems that

lack translational invariance, one can use the Bott index introduced by Loring and

Hastings [89] as the analog of the Chern number [Eq. (1.16)]. This was done in Ref. [56]

27

to study topological properties of Fermi seas in patch geometries and their unitary time

evolution under a periodic drive. Below we formulate the Bott index on a translationally

invariant system.

Consider a 2D lattice spanned by primitive (right-handed) vectors ~aµ, µ = 1 and 2,

with Lµ lattice sites along each ~aµ such that it has L1 × L2 lattice sites. Let lµ ∈ [0, Lµ)

be the spatial coordinate along ~aµ such that a lattice site (l1, l2) is at l1~a1 + l2~a2. Since

the Bott index is formula is coordinate system invariant, we can just redefine the U and

V operators in Eq. (1.12) to be

U := e2πil1/L1 , (2.1)

V := e2πil2/L2 . (2.2)

In finite translationally invariant systems, one can rewrite the Bott index in crystal

momentum space. Here we use the coordinate system (k1, k2), with kµ ∈ [0, 2π/aµ)

being the component along primitive reciprocal vectors ~bµ, for which ~aµ ·~bν = 2πδµν .

Our set of k points of interest is within the parallelogram bounded by ~b1 and ~b2, which

is equivalent to the first Brillouin zone. The infinitesimal momentum space distances

between neighboring momentum space points are

δk1 =[ 2πL1a1

, 0], δk2 =

[0, 2πL2a2

]. (2.3)

Let

q0 := k, q1 := k− δk1, q2 := k− δk2, q3 := k− δk1 − δk2. (2.4)

The operators U and V are infinitesimal translation operators in momentum space, i.e.,

〈ψn (k) |U |ψm (k′)〉 = 〈un (k)|um (k′)〉 δq1,k′ , (2.5)

〈ψn (k) |V |ψm (k′)〉 = 〈un (k)|um (k′)〉 δq2,k′ , (2.6)

28

where we adopted the following normalization for Bloch states |ψn (k)〉,

〈ψn (k)|ψn′ (k′)〉 = δnn′δkk′ . (2.7)

In the Bloch-state basis, the matrix elements of V U V †U † can be written as

〈ψn (k) |V U V †U † |ψn′ (k′)〉 = δkk′∑jlm

U02njU23

jl U31lmU10

mn′ , (2.8)

where

Uαβnj := 〈un (qα)|uj (qβ)〉 , (2.9)

with α, β = 0, 1, 2, and 3, and the indices j, l,m run over filled bands for a given k.

Thus, V U V †U † is block diagonal in momentum space.

For a single band, we then have that

Cb(n) = 12π

∑k∈BZ

Im ln(U02nnU23

nnU31nnU10

nn) (2.10)

This expression was derived for the Chern number in finite translationally invariant sys-

tems in Ref. [128] using Wilson loops. As discussed there, the result of Eq. (2.10) in finite

systems converges much faster to the value of the Chern number in the thermodynamic

limit than the discretized version of Eq. (1.8). In addition, Cb(n) is gauge independent.

The physical picture of this formula is an Aharonov–Bohm interference experiment in

momentum space. The wavefunction in transported in small loops in k space to reflect

the Berry flux that permeates the Brillouin zone.

The Bott index was shown to give the Hall conductance in Ref. [90]. Hence, it

is equivalent to the Chern number. Below, we give an elementary proof that, in the

thermodynamic limit, the Bott index is identical to the Chern number. The only

requirement for this proof is that the occupied single-particle Bloch states be locally C2

29

in momentum space.

First, we expand |un (qα)〉, with α = 1, 2, and 3, about k. It gives

|un (q1)〉 = |un〉 − δk1∂|un〉∂k1

+ (δk1)2

2∂2|un〉∂k2

1+O

(δk3

1

), (2.11)

where, on the right-hand side, we omitted the momentum argument as all kets and

their derivatives are evaluated at k (we follow this convention in the expressions below).

Similarly, one can expand |un (q2)〉 and |un (q3)〉. Plugging those expansions into Eq. (2.8),

one finds that the offdiagonal matrix elements scale as

〈ψn (k) |V U V †U † |ψn′ (k)〉 ∼ O[(δk)2

]for n 6= n′, (2.12)

where we assumed that δk ∼ δk1 ∼ δk2. For the diagonal entries, on the other hand,

〈ψn (k) |V U V †U † |ψn (k)〉 = 1 + δk1δk2

(∂〈n|∂k2

∂|n〉∂k1− c.c.

)

+δk1δk2∑m

[〈n|∂|m〉

∂k1

∂〈m|∂k2|n〉 − c.c.

]

+(δk1)2

∑m

∣∣∣∣∣〈n|∂|m〉∂k1

∣∣∣∣∣2

−∣∣∣∣∣∂|n〉∂k1

∣∣∣∣∣2+ (1→ 2)

+O(δk3). (2.13)

Given these results for the diagonal and off-diagonal matrix elements of V U V †U †,

one can evaluate Eq. (1.16) using the fact that, for a general matrix A decomposed as

A = 1 + AD + AO, (2.14)

30

where AD is the diagonal part of A− 1 and AO is the off-diagonal part, one can write

Tr lnA = TrAD + Tr[O(A2D, A

2O, ADAO)]. (2.15)

Since (V U V †U †)D and (V U V †U †)O are order δk2 or higher, it follows from Eq. (2.15)

that

Tr ln V U V †U † = Tr(V U V †U †)D + Tr[O(δk4

)]. (2.16)

Taking the imaginary part gives

Im Tr ln V U V †U † = 1i∑n

∑k1k2

δk1δk2

(∂〈n|∂k2

∂|n〉∂k1− c.c.

)+O (δk) . (2.17)

In the limit L1, L2 →∞, ∑k1k2 δk1δk2 →∫

BZ d2k, and all higher-order terms vanish. The

trace becomes the integral of Berry curvature. Therefore, in the thermodynamic limit,

for a Fermi sea occupying all n 6 N bands

Cb(Pn6N) =∑n6N

Cb(n) =∑n6N

Ch(n). (2.18)

Hence, for a Fermi sea that is locally C2 in k space, the Chern number and the Bott

index are identical in the thermodynamic limit. Each term of the sum in Eq. (2.10) is

simply the local Berry curvature times the area element δk2, in other words the Berry

connection around the boundary of that area element, as shown in Ref. [139]. If the

Fermi level is in the middle of a band, which corresponds to a conducting state, some k

points in the Brillouin zone will have underfilled neighbors q1 and/or q2. That k block

is then singular and so is V U V †U †. Thus, in this case the Bott index is ill defined. For

Fermi seas with a well defined Bott index or Chern number, these two topological indices

are well-defined and equivalent during unitary time evolutions under Hamiltonians that

are C2 in k space.

31

2.3 Model Hamiltonian and Floquet topological phases

Having established the equivalence between the Bott index and the Chern number, in

what follows we study the dynamics of the Bott index in systems with periodic boundary

conditions. Our goal is to understand how it compares to the dynamics of the same

topological index in systems with open boundary conditions [56].

Recalled that for the model we use in Eq. 1.32, when both A and ∆ are zero, the

energy bands are gapless at K and K ′ in the Brillouin zone. Let the lattice constant

be a, then the coordinates of K and K ′ are (2π3a ,

4π3a ) and (4π

3a ,2π3a ), respectively. Those

band-touching points are protected by the combination of inversion symmetry and time-

reversal symmetry. In the static case (A = 0), a nonzero ∆ introduces a Bz of equal

magnitude at K and K ′, still related by time-reversal symmetry, and opens a gap. In this

work, we set ∆ = 0.15J in order to be close to the experimental parameters in Ref. [21].

Both static bands have zero Chern numbers; that is, they are topologically trivial.

The periodically driven system we use can be effectively described with the Floquet

Hamiltonian. Under high driving frequencies ω = 2π/T , HF can be extracted from a high-

frequency expansion [105,109]. To O(ω−1), the rotating electric field renormalizes the

nearest-neighbor hopping amplitude and induces next-nearest-neighbor (⟪j, l⟫) hoppings.The Floquet Hamiltonian reads

HF (t) = −JJ0(A)∑〈j,l〉

(c†j cl + H.c.

)+ J2

ω

∑⟪j,l⟫

(iKjlc

†j cl + H.c.

)

+∆2∑j∈Al∈B

(nj − nl) +O(ω−2), (2.19)

where Jn are the Bessel functions of the first kind, and

Kjl = sjl∞∑n=1

2nJ 2n (A) sin 2nπ

3 . (2.20)

32

The sign sjl is + (−) if the two-step hopping, going around the hexagon corners, has the

same (opposite) chirality as the polarization of the electric field.

In momentum space, next-nearest neighbor hoppings contribute to Bz of the Hamilto-

nian at K and K ′. For 0 < A < 1.69, its sign is the opposite to (same as) Bz generated

by ∆ at K ′ (K). As a result, the Floquet band gap closes at K ′ upon increasing the

magnitude of the vector potential. This results in a topological phase transition in

which the Chern number changes from 0 to 1 [see dashed line in Fig. 2.1(a)]. A second

topological phase transition in which the Chern number changes from 1 to 0, and the

gap closes once again at K ′ occurs upon further increasing the magnitude of the vector

potential.

Higher order terms in ω−1 introduce further neighbor hoppings that can, in turn,

generate new topological phase transitions if ω is not too large. In Fig. 2.1(a), we

show the Chern number phase diagram (solid lines) obtained using a numerically exact

calculation of U(T, 0) [56]. One can see that, as a result of terms O(ω−2) and higher, two

additional phase transitions appear in the regime studied. Interestingly, the corrections

to the critical values obtained for the transitions between phases with Chern numbers of

0 and 1 are small. A detailed discussion of the phase diagram of the system studied here,

for ∆ = 0, can be found in Ref. [109]. In particular the C3 symmetry of the Floquet

Hamiltonian leads to the Berry flux pattern near phase boundaries in Fig. 2.1(iii) and

Fig. 2.1(v). In general the Berry flux depends on the choice of initial time of HF [t0] in

Eq. (1.19). Thus the Berry flux near K ′ in Fig. 2.1(iv) rotates about K ′ as t0 varies.

We note that the Chern number of a Floquet system does not directly give the number

of topologically protected edge states, as is the case for anomalous Floquet topological

states [107].

In finite systems, the topological index of a Floquet band can be calculated either

using a discretized version of the integration in Eq. (1.8) for the Chern number, or using

33

0 0.5 1 1.5 2 2.5 3

A

7

8

9

10

11

12

13

14

Ω/J

1 -2 100

(i)

(ii)

(iii)

(iv)

(v)

b₁

b₂

Figure 2.1. Chern number phase diagram in the driving frequency ω and A plane for HF

obtained from a high-frequency expansion to O(ω−1) (dashed lines) and from numerically exactcalculations (solid lines). The Chern number is computed using the Bott-index formula inEq. (1.16) in finite commensurate systems that are sufficiently large so that the result does notchange (within machine precision) with increasing system size. Insets: Berry flux at drivingω = 7J and field strengths (i) A = 0, (ii) A = 1, (iii) A = 2.42, (iv) A = 2.45, and (v) A = 2.52.The color scales are normalized differently for each field strength.

the Bott index in Eq. (1.16). As mentioned before, the Bott index calculation converges

much more rapidly to the thermodynamic limit result with increasing system size [128].

How rapidly the critical value A∗ (obtained using the Bott index) for the topological

transition converges depends on whether the momentum at which the gap closes in

the thermodynamic limit is present in the discrete Brillouin zone of the finite system.

In Fig. 2.2, we plot results for the critical value A∗L obtained for the first topological

transition in systems with L1 = L2 ≡ L as a function of L (for ω/J = 7). Only lattices in

which L = 3ι (ι ∈ Z) contain the K and K ′ points (are commensurate), where the Berry

34

0 6 12 18 24 30

L

0.5

1

AL*

3n3n+13n+2

10L

0.01

0.1

∆A

L*

(a) (b)

Figure 2.2. (a) Critical value of the magnitude of the electric field A∗L for the first topologicaltransition when ω = 7J [black dot in Fig. 2.1] in systems with L1 ×L2 lattice sites plotted as afunction of L1 = L2 ≡ L. (b) Scaling of the critical value for incommensurate systems. We plot∆(A∗L) := (A∗L −A∗∞) vs L for lattices with L = 3ι+ 1 and L = 3ι+ 2 (ι ∈ Z), as well as fits to∆(A∗L) = γ L−2x for L ≥ 10. The fits yield x ≈ 1.07 and 0.98, respectively.

curvature is concentrated near the transition. They can be seen to produce a critical

value that is system size independent starting from L = 3. On the other hand, lattices

with L = 3ι + 1 and L = 3ι + 2 exhibit a power-law approach of the critical value to

the thermodynamic limit result [see Fig. 2.2(b)]. Whether K ′ is included in the discrete

Brillouin zone of the finite system plays a fundamental role in the unitary dynamics of

the Bott index studied in what follows.

35

2.4 Dynamics of the Bott index in finite systems

As mentioned before, the Chern number (Bott index) is a constant of motion in trans-

lationally invariant systems in the thermodynamic limit. However, there is nothing

preventing the Bott index from changing during unitary time evolutions in finite sys-

tems, even if those systems are translationally invariant. One can consider two extreme

cases of dynamics: (i) In a sudden quench of a Fermi sea, the finite system size limits

the resolution of reciprocal space. As a result, a quenched Fermi sea can develop an

increasingly complicated Berry curvature with time, such that the Bott index calculated

in finite systems may be strongly dependent on time (for times larger than the linear

system size divided by the maximal group velocity) and system size [55, 63,64]. (ii) In a

system driven at a high frequency with a slowly increasing driving term, the Bott index

can change if the system is able to evolve adiabatically in a Floquet picture [113,140]. A

finite time scale for adiabatic evolution can exist only if the k point at which the gap

closes in the Floquet Hamiltonian (in the thermodynamic limit) is absent in the finite

system.

A note is in order about the computation of the Chern number in finite systems out

of equilibrium. When the magnitude of the driving term is increased slowly (in a system

driven at high frequency), states away from the band gap of the Floquet Hamiltonian

mainly evolve adiabatically. If the new band that is generated (in the Floquet picture)

with increasing the strength of the driving term changes topology, then the Berry

curvature of the original Fermi sea will accumulate about the gap-closing point(s) and

vary rapidly about it (them), in order to observe the no-go theorem. The computation of

the Chern number using Eq. (1.8) in finite systems then becomes numerically unstable.

The Bott-index formula in Eq. (1.16) should be the one used to study those systems out

of equilibrium.

36

In our numerical calculations, we turn on the vector potential smoothly [its magnitude

is increased linearly from zero, A(t) ∝ t] to drive the ground state of the static Hamiltonian

into the ground state of the Floquet one with Ch = 1, shown in Fig. 2.1. We consider

only driving frequencies greater than the band width, so that the Floquet bands are

ordered unambiguously. We work in a regime in which A(t) ω. In this regime, one

can think of the evolution of the time-dependent state as being dictated by a slowly

changing Floquet Hamiltonian, so that traditional concepts such as adiabaticity can be

applied [140].

The Bott indexes of two time-evolving Fermi seas in which the magnitude of the

electric field is slowly ramped from A = 0 to 1 are plotted in Figs. 2.3(a) and 2.3(b)

as a function of A(t) (bottom labels) at stroboscopic times t (top labels) for ω = 7J .

Figure 2.3(a) shows results for a 150×150 (commensurate) lattice, while Fig. 2.3(b) shows

results for a 151× 151 (incommensurate) lattice. The Bott index of the commensurate

lattice [Fig. 2.3(a)] observes the no-go theorem, namely, it is conserved during the

dynamics. On the other hand, the Bott index of the incommensurate lattice [Fig. 2.3(b)]

changes from 0 to 1 when the magnitude of the electric field exceeds the critical value A∗

in Fig. 2.1. Namely, the initial topologically trivial Fermi sea evolves into a topologically

nontrivial state under unitary dynamics. A first insight into the origin of the different

behavior of the Bott index in those two lattices can be gained by studying the overlap

between the time-evolving wave function and the instantaneous Floquet ground state,

also shown in Fig. 2.3. For the commensurate lattice [Fig. 2.3(a)], that overlap is

essentially 1 (near adiabatic evolution) up to about A∗, but then, when A(t) becomes

larger than A∗, the overlap vanishes, and the time-evolving state becomes orthogonal to

the instantaneous (topologically nontrivial) Floquet ground state. On the other hand, for

the incommensurate lattice [Fig. 2.3(b)], the overlap remains close to 1 (near adiabatic

evolution) at all times. The smallest overlaps occur about A∗, but they are still higher

37

0 0.2 0.4 0.6 0.8 1

A(t)

0

0.2

0.4

0.6

0.8

1

|⟨Ψ

(t)|

ΨF GS(t

)⟩|2

0 200 400 600 800 1000

t /T

0

0.2

0.4

0.6

0.8

1

Cb(t

)

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1C

b(t

)0 0.5 1

A(t)

0

0.2

0.4

0.6

0.8

1

|⟨Ψ

(t)|

ΨF GS(t

)⟩|2

Cb(t)

|⟨Ψ(t)|ΨFGS(t)⟩|

2

PA

at K’

min PA

without K’

average PA

(a)

(b)

Figure 2.3. Bott index of the time-evolving wave function |Ψ(t)〉, and overlap between |Ψ(t)〉and the ground state of the instantaneous Floquet Hamiltonian |ΨF

GS(t)〉 as a function of A(t)(bottom labels) at stroboscopic times t (top labels). We also show how the lower band of theinstantaneous Floquet Hamiltonian is occupied in the time-evolving state. Specifically, weplot the occupation at the K ′ point, the lowest occupation of any k state in the lower bandexcluding the K ′ point, as well as the average occupation of the k states in the lower band. (a)Results for a 150× 150 (commensurate) lattice. (b) Results for a 151× 151 (incommensurate)lattice. Note that the latter does not contain the K ′ point, so no result is reported for itsoccupation. In both lattices, the magnitude of the electric field is ramped up linearly from 0 toA = 1 in 1000 driving periods, for ω = 7J [corresponding to the black dot in Fig. 2.1].

38

than 0.8 and can be made arbitrarily close to 1 by decreasing the ramp speed.

For the commensurate lattice in Fig. 2.3(a), we also plot the occupation of the K ′

point of the lower band of the instantaneous Floquet Hamiltonian in the time-evolving

state, obtained by computing∣∣∣〈u(K ′, t)|uFGS(K ′, t)〉

∣∣∣2, where |u(K ′, t)〉 is the time-evolving

wave function at time t at K ′ and |uFGS(K ′, t)〉 is the ground-state wave function of the

instantaneous Floquet Hamiltonian at time t at K ′. The occupation of the K ′ point can

be seen to vanish when the magnitude of the vector potential exceeds A∗. This is the

reason behind the vanishing of the overlap between the time-evolving wave function and

the instantaneous Floquet ground state and, ultimately, behind the conservation of the

Bott index. The next lowest occupied k state of the Floquet ground-state band, also

shown in Fig. 2.3(a), is very close to 1. Namely, all but the K ′ point evolve (nearly)

adiabatically during the dynamics. As a result, the arithmetic mean of the occupation of

k states of the Floquet ground-state band in the time-evolving state is very close to 1

[see Fig. 2.3(a)].

For the incommensurate lattice in Fig. 2.3(b), for which there is no vanishing gap in

the Floquet Hamiltonian, the minimally occupied k state of the Floquet ground-state

band during the dynamics is very close to 1 at all times, with the largest departure

from 1 occurring when the magnitude of the vector potential is about A∗ (similarly to

what is seen for the wave-function overlaps). As for the wave-function overlaps, the final

occupation can be arbitrarily close to 1 if the ramp speed is decreased (the gap provides

a well-defined time scale for adiabaticity). In the incommensurate lattice [Fig. 2.3(b)],

the average occupation of k states of the Floquet ground-state band in the time-evolving

state can also be seen to be very close to 1. We should stress that the magnitude of the

vector potential at which the Bott index jumps in Fig. 2.3(b) and the overlap vanishes in

Fig. 2.3(a) can change if one changes the ramping speeds. However, it converges to A∗

when A ω.

39

0

1/3

2/3

1

0 1/3 2/3 10

1/3

2/3

1

0 1/3 2/3 1

1/6

1/3

1/2

1/2 2/3 5/61/6

1/3

1/2

1/2 2/3 5/6

k2a/2π

k1a/2π

−4000

−2000

0

2000

4000

k2a/2π

k1a/2π

−400

−200

0

200

400k2a/2π

k1a/2π

-48000-24000

...

0

200

400

k2a/2π

k1a/2π

−400

−200

0

200

400

(a) (b)

(c) (d)

Figure 2.4. Berry curvature in the static, Floquet, and time-evolved Fermi seas of finitesystems. The discrete k space is displayed in coordinates of the primitive reciprocal latticevectors with unit lattice constants. K and K ′ are located at (k1, k2) = (2π/3, 4π/3) and(4π/3, 2π/3), respectively. (a) Berry curvature of the static, topologically trivial, ground statein a lattice with L = 900. The signs are positive around K and negative around K ′. (b) Berrycurvature of a Floquet, topologically nontrivial, ground state (A = 1 and ω = 7J) in a latticewith L = 900. All signs around K and K ′ are positive. (c) Berry curvature about K ′ in atime-evolved commensurate system with L = 150. Note the accumulation of negative Berrycurvature at K ′. The four data points in the center forming a 2×2 square are negative, oppositeto all the surrounding points. (d) Berry curvature about K ′ in a time-evolved incommensuratesystem with L = 151. All signs around K ′ are positive. In (c) and (d), we show results from thetime evolution of an initial topologically trivial Fermi sea after the magnitude of the electric fieldis ramped up linearly from A = 0 to 1 (with ω = 7J) in 8000 driving periods. Here the ramp is8 times slower than that in Fig. 2.3 to make the Berry curvature about K ′ indistinguishable(in the scale of these plots) between (b) and (d).

40

Next, we study the Berry curvature of the static and Floquet ground states, as well

as of the time-evolved Fermi seas in finite systems. We compute them from each term

in Eq. (2.10), dividing by the area element δk2. The Berry curvature of the static,

topologically trivial ground state is shown in Fig. 2.4(a). In this case, the Berry curvature

is mostly zero everywhere in the band and then large and positive (negative) about the

K (K ′) point. This results in a vanishing Chern number. Figure 2.4(b) shows the Berry

curvature of a topologically nontrivial Floquet ground state (corresponding to A = 1 and

ω = 7J). In this case, the Berry curvature is once again mostly zero everywhere in the

band, but then it is large and positive about the K and K ′ points (larger about K ′).

This results in Ch = 1.

As previously discussed, when one increases the magnitude of the electric field in our

driven systems from zero, the first topological transition in the Floquet Hamiltonian

occurs via a band-gap closing at K ′, as a result of which the Chern number changes from

0 to 1. As hinted by our results in Fig. 2.3, something fundamentally different happens

to the Berry curvature about that K ′ point in commensurate and incommensurate

lattices when one evolves unitarily the topologically trivial Fermi sea of the static

Hamiltonian by slowly ramping up the magnitude of the electric field. This is shown

in Figs. 2.4(c) and 2.4(d), respectively. In the time-evolved state of the commensurate

lattice [Fig. 2.4(c)], the Berry curvature close to (but not at) K ′ is very similar to that in

the Floquet Hamiltonian [Fig. 2.4(b)]. However, at K ′ the Berry curvature in the former

is very large and negative, in contrast to the positive Berry curvature in the Floquet

Hamiltonian. This is how the Bott index remains zero during the dynamics. On the

other hand, in the time-evolved Fermi sea of the incommensurate lattice [Fig. 2.4(d)],

the Berry curvature about K ′ is indistinguishable from that in the Floquet Hamiltonian

[Fig. 2.4(b)].

41

2.5 Landau-Zener dynamics

As mentioned before, our study focuses on a regime in which the evolution of the time-

evolving state is effectively dictated by a slowly changing Floquet Hamiltonian. Close to

K ′, the evolution is essentially a Landau-Zener problem [61,62,140–142], in which the

ground state evolves under a level-crossing Hamiltonian

Hk(t) ∼ βktσz + Vkσx. (2.21)

The Hamiltonian (2.21) is written in the basis of the level-crossing eigenstates, which

are the eigenstates of σz. βk gives the rate at which the level-crossing point is passed,

and Vk is the perturbing off-diagonal term, which is zero at K ′ at the time at which the

magnitude of the vector potential is A∗. (See the inset of Fig. 2.5 for an illustration.)

The probability PA to remain adiabatic, i.e., in the ground state of the final Floquet

Hamiltonian, is given by the Landau-Zener formula

PA(k) = 1− exp(−π|Vk|2

βk

). (2.22)

The Landau-Zener parameters are extracted from HF (A) calculated shortly before and

shortly after A becomes equal to A∗, A±, which are ∆t away from each other (we used

A± = A∗ ± 0.0015 for the results shown). The Floquet Hamiltonians for those two

values of A allow us to extract both βk from [HF,k(A+) − HF,k(A−)]/∆t and Vk from

[HF,k(A+) + HF,k(A−)]/2. Results obtained for 1 − PA, for several values of k, lattice

sizes, and ramping rates are plotted in Fig. 2.5 vs π|Vk|2/βk. They exhibit an excellent

collapse to the Landau-Zener prediction.

The different behaviors between commensurate and incommensurate and, ultimately,

finite- and infinite-size lattices have a geometric interpretation. An infinite-size lattice

42

0 5 10 15 20

πV2/β

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

1-P

A

Landau Zener

ω =7J

ω =14J

PA

A1-P

t

E

Figure 2.5. Collapse of the final occupation of momentum states near K ′ to the Landau-Zenerprediction (straight line). β and V are extracted from HF [A(t)] before and after the gap closing.In all simulations, the magnitude of the electric field is turned on linearly from A = 0 to 1. Weshow results for several ramping rates (six), lattice sizes (ten, in which L ranges from 600 to16384), and values of k (about ten for each lattice size and ramping rate). Inset: a diagramshowing the two energy levels in a Landau-Zener process, as well as their occupation numbers,PA and 1 − PA. The dotted lines are the asymptotic with slopes of ±βk. At t = 0 the levelspacing is 2Vk.

has a continuous Brillouin zone. The eigenstates of a filled band can be mapped

onto a closed surface. The topological index of a given band reflects the topological

charges enclosed by it [74, 129]. In a time-evolving traceless two-band model, one can

map each state spinor[cos θk(t)

2 , sin θk(t)2 eiφk(t)

]Tonto a point in three dimensions, with

polar coordinates [|Ek(t)|, θk(t), φk(t)], where Ek(t) is the energy of the state in the

43

instantaneous (Floquet in our case) Hamiltonian. The topological charge, corresponding

to a band-touching point, sits at the origin. Under a slow evolution, the surface “follows”

the time-dependent Hamiltonian Hk(t); that is, it shifts and deforms. But whenever a

patch of the surface moves close to the origin, the dynamics of that patch freezes due

to vanishing Ek. Consequently, topological charges always stay on the same side of the

surface corresponding to the original energy band. Hence, the Chern number of such a

filled band cannot change under unitary time evolutions, dCh(n, t)/dt = 0. Intuitively,

this geometric picture should generalize to higher dimensions for systems with more than

two bands.

For a finite system, one has a discrete set of points rather than a closed surface.

Therefore, the topological charge can easily “leak” through during the dynamics. This is

allowed to happen unless the critical k point(s) at which Ek(t) vanishes is present to

prevent it, such as K ′ in our commensurate lattices. In general, the critical k points

are model dependent; for example, they can depend on the presence of disorder and

boundary conditions. For the transition to the Ch = −2 phase in Fig. 2.1, the location of

the three band touching points depends on both the strength of the staggered sublattice

potential and the driving frequency. In such situations, one usually will not find that the

Chern number is conserved during dynamics in finite lattices.

2.6 Summary

Using elementary methods, we proved that the Bott index (generally used to study systems

that lack translational symmetry) is equivalent to the Chern number in translationally

invariant systems in the thermodynamic limit. As a byproduct of our proof, we showed

that, when written in momentum space, the Bott index is nothing but the Chern number

introduced in Ref. [128] for finite translationally invariant systems.

We used the Bott index in finite honeycomb lattices with periodic boundary conditions

44

to determine the topological phase diagram of the Floquet Hamiltonian of driven spinless

fermions with nearest-neighbor hoppings and a staggered potential. We then studied

the dynamics of initial topologically trivial Fermi seas when the driving term is slowly

ramped through a topological phase transition. We showed that while the Bott index

is conserved in commensurate lattices, those that contain the k point at which the gap

closes in the Floquet Hamiltonian, it is not conserved in incommensurate lattices. The

latter behavior was the one observed in systems with open boundary conditions [56]. We

argued that, in incommensurate lattices, adiabatic dynamics allows the Bott index to

change at the critical value computed for the Floquet Hamiltonian. Hence, regardless of

the no-go theorem for the thermodynamic limit [56], topological phase transitions can

occur in finite translationally invariant systems provided there is a well-defined adiabatic

limit for the effective Floquet dynamics.

45

Chapter 3 |

Universal Landau-Zener regime intopological phase transitions

This chapter presents the work reported in Ref. [70]. It looks at systems across boundary

conditions and answers the questions raised by D’Alessio and Rigol in Ref. [56]. The

dynamical results show that, in the thermodynamic limit, the Chern number is conserved

in all lattice types. For that, I will reformulate the Bott index to work in cylinder

geometries, where one direction is open and the other is periodic. The wavefunction

overlap and reduced deviation of dynamical critical field show a dominating Landau

Zener regime that is universal. This leads to the Kibble-Zurek scaling of the excitation

density. Two other regimes are identified. Fast ramps sit inside a regime in which the

Bott index does not change and the dynamics behave in a system size independent way.

Very slow ramps sit inside a near-adiabatic regime that only exist at zero ramp speed in

the thermodynamic limit. Finally I show that the Hall response of time-evolved states

conform to the topology of the final Hamiltonian, instead of that of the states themselves.

46

3.1 Introduction

Dynamically generating topological phases of matter is a powerful way to explore a variety

of topological states and theoretical models that cannot or are very challenging to explore

in equilibrium [21–23, 30, 48, 143]. Periodic driving, e.g., shining circularly polarized

light on a material, is one of the most promising protocols that has been proposed to

dynamically generate (Floquet) topological states [25,45,49,50,102,130,144]. A common

feature of most of those protocols is that the Floquet topological states of interest are

not adiabatically connected to the initial (trivial) equilibrium states of the systems that

are periodically driven. This has generated much interest in what happens in real time

when systems are driven across topological phase transitions [53, 133,145–154].

The focus of this chapter is the role of the boundary conditions in the dynamics

of topological indices. For systems with open boundary conditions, or in general for

systems that are not translationally invariant, the Bott index is the topological index

that is commonly used to characterize topological phases in equilibrium [89–91]. When

computed out of equilibrium in systems with open boundary conditions driven across

a topological phase transition, the Bott index can change [56, 155]. (The local Chern

marker [99] was found to exhibit similar behavior in Ref. [62].) One can also compute the

Bott index in finite translationally invariant systems [65], in which it is equivalent to the

discretized Chern number usually used in finite-system calculations [128]. In Ref. [65], we

showed that the discretized Chern number can change in finite translationally invariant

systems that are incommensurate, namely, in translationally invariant systems in which

the gap-closing point(s) is(are) missing in their Brillouin zone. On the other hand, the

discretized Chern number does not change in commensurate systems, namely, those that

contain the gap-closing point(s) in their Brillouin zone.

One of the goals of this chapter is to provide a unified picture of the real-time

47

dynamics of topological indices in finite periodically driven systems with periodic and

open boundary conditions. We also discuss the scaling with system size of the critical

field of topological transitions in the Floquet Hamiltonian in lattices with periodic and

open boundary conditions. Another goal is to show that the dc Hall response can be used

to detect topological phase transitions independently of the behavior of the discretized

Chern number or the Bott index.

The presentation is organized as follows. In Sec. 3.2, we discuss the model, the drive

protocol used to generate Floquet topological phases, and the geometries considered.

The finite-size scaling of the critical field of the topological transition of interest in

the Floquet Hamiltonian is discussed in Sec. 3.3. In that section, we review results

obtained in Ref. [65] for periodic boundary conditions and report results for patch and

cylinder geometries. In Sec. 3.4 we study the scaling of the critical field of the topological

transition during the unitary dynamics generated by ramping up the driving field. We

consider incommensurate translationally invariant systems as well as patch and cylinder

geometries. Section 3.5 is devoted to the study of the dc Hall response in the topologically

trivial and nontrivial ground states of the Floquet Hamiltonian in translationally invariant

systems, as well as in the nonequilibrium state generated after ramping up the driving

field. The chapter concludes with a summary and discussion of the results in Sec. 3.6.

3.2 Model, drive protocol, and geometries

We study spinless fermions on a honeycomb lattice with nearest neighbor hopping and a

staggered potential between the A and B sublattice sites, as displayed in Fig. 3.1. The

model Hamiltonian is the same as Eq. (1.32). Depending on the frequency and strength

of the drive, the resulting Floquet Hamiltonian HF displays various topological phases at

half filling [52,65,109]. Here we focus on the first transition in the high frequency regime,

for ω = 7J (this value of ω is greater than the band width). In this regime, to the lowest

48

A sublatticeB sublatticePatchCylinder

dd

d

x

y

Figure 3.1. Honeycomb lattice with nearest neighbor lattice spacing d, and highlightedsublattices A and B. The dotted line indicates the boundaries of the patch geometry (openboundaries in both directions), while the dashed line indicated the boundary of the cylindergeometry (translationally invariant in the horizontal direction and zigzag open boundaries inthe vertical one). In our calculations, all cylinders have the same number of unit cells alongboth lattice directions, while lattices in the patch geometry have roughly equal number of unitcells in the x and y directions.

order in the drive period [65,109], the Floquet Hamiltonian is the Haldane model [10].

The topological transition in the thermodynamic limit occurs at edA∗ ≈ 0.498. In what

follows, we set e = d = 1.

To dynamically prepare a topological state starting from the trivial band insulator

at A = 0, we turn on the electric field using a linear ramp with ramp time τ . Different

ramping speeds, system sizes, and in the translationally invariant case the commensura-

bility of the lattice, result in different behaviors of the topological indices as A(t) crosses

the critical field.

We study finite lattices with three different boundary conditions (see Fig. 3.1):

(i) Periodic (tori) so that the systems are translationally invariant, (ii) periodic in one

direction and open in the other one (cylinders), and (iii) open in both directions (patches).

To each of those lattice geometries we associate a specific topological index that allows one

to locate the topological transition in the Floquet Hamiltonian. For periodic boundary

49

conditions we use the discretized Chern number (in short, the Chern number) [65,128],

while for the patch geometry we use the Bott index [89,156].

For the cylinder geometry, Fourier expanding in momentum space along the transla-

tional invariant direction x, the Bott index can be rewritten using the projection operator

Pkx onto the periodic part of Bloch states within filled bands at each kx

Cb(P ) = 12π Im

∑kx

Tr ln(Pkxeiδy yPkxPkx−e−iδy yPkx−), (3.1)

where δy = 2π/Ly, kx− = kx−2π/Lx. The logarithm and the trace should be evaluated in

the subspace spanned by filled bands at each kx. This formula is a discretized integral that

calculates the Wannier center flow winding along x [83]. A technical complication arises

because gapless edge states make Fermi seas ill-defined in topologically nontrivial phases.

It is then necessary to include in the calculation the relevant (nearly) degenerate states at

the k points at which the edge modes cross, and then discard the kernel of the operator

product due to the extra state before taking the logarithm. Our dynamical calculations

are not affected by this complication because the initial state is a topologically trivial

insulating state. The above formulations of the Chern number and the Bott index are

equivalent in the thermodynamic limit.

3.3 Scaling of the critical field in the Floquet Hamilto-

nian

In finite lattices, the critical fields A∗L at which topological transitions occur in the

Floquet Hamiltonian (the fields at which the topological indices change value) depend

on the linear system size L. For translationally invariant systems, the size dependence of

the critical field for the topological transition of interest here was studied in detail in the

previous chapter [65]. There we showed that the commensurability of the lattice plays an

50

essential role. In commensurate lattices, with linear dimensions L = 3n (n ∈ Z) which

contain the K and K ′ points [(2π3 ,

4π3 ) and (4π

3 ,2π3 ), respectively] at which gaps may close

at the topological transitions, the critical field is system-size independent starting from

L = 3. On the other hand, in lattices with L = 3n+ 1 and L = 3n+ 2 the critical field

approaches the thermodynamic limit result as ∼ 1/L2. This occurs because in finite

incommensurate lattices the Chern number cannot resolve the singular behavior of Berry

curvature at the phase transition until it spreads out in momentum space as the systems

go deeper into the topological regime.

In this section, we use the Bott index (see Sec. 3.2) to determine A∗L for the Floquet

Hamiltonian in cylinder and patch geometries, and to study the system-size dependence

of A∗L. Figure 3.2 shows the results obtained for A∗L vs L. For the cylinder geometry L is

the (equal) number of unit cells in each lattice direction, while for the patch geometry L

is defined as the square root of the total number of unit cells and we keep roughly an

equal number of unit cells in the x and y directions. While A∗L depends smoothly on L

for the patch geometry, we find strong commensurability effects in the cylinder geometry.

In the latter, a smooth dependence on L is seen only if one looks independently at the

results for L = 3n, 3n+ 1, and 3n+ 2 (finite-size effects being strongest for L = 3n and

weakest for L = 3n+ 2). Still, for both geometries one can see that A∗L approaches the

thermodynamic limit result A∗ ≈ 0.498 with increasing system size.

In the inset in Fig. 3.2 we plot ∆A∗L = A∗L − A∗ vs L. These plots make apparent

that A∗L approaches the thermodynamic limit result as a power law in L. Power-law

fits of ∆A∗L for the largest systems sizes (depicted as dashed lines in the main panel

and inset in Fig. 3.2) reveal that ∆A∗L ∼ 1/L. (The local Chern marker exhibits similar

scalings in systems with open boundary conditions [66].) Our scalings in Fig. 3.2 are to

be contrasted to ∆A∗L ∼ 1/L2 found for incommensurate lattices with periodic boundary

conditions in Fig. 2.2. Since cylinders with L = 3n + 2 exhibit the smallest finite-size

51

effects, those will be the ones on which we focus in the reminder of this chapter.

3.4 Scaling of the critical field in the dynamics

In the thermodynamic limit, and in finite translationally invariant lattices that are

commensurate, both the Chern number and the Bott index of fully filled bands are

invariant upon unitarily driving the system [56, 64, 65, 155]. This comes from the

10 100

10-3

10-2

10-1

∆A

L*

0 10 20 30 40 50 60 70 80 90 100 110L

0.5

0.6

0.7

0.8

AL* Patch

Cylinder (3n)

Cylinder (3n+1)

Cylinder (3n+2)

Figure 3.2. Critical strength of the drive’s field A∗L plotted as a function of L for cylinderand patch geometries. For the cylinders L is the number of unit cells in each lattice direction,while for the patch geometry L is the square root of the total number of unit cells (we keepLx ' Ly). The inset shows the offset ∆A∗L = A∗L − A∗ from the critical field strength A∗ inthe thermodynamic limit. We fit power laws ∆A∗L ∝ L−2α to the largest system sizes depicted,obtaining α ≈ 0.483 for the patch geometry, 0.519 for cylinders with L = 3n, 0.515 for cylinderswith L = 3n+ 1, and 0.491 for cylinders with L = 3n+ 2. The dashed lines in the main paneland inset depict the results of our fits. Note that, in cylinders, finite-size effects are smallest forL = 3n+ 2. These are the cylinders on which we focus in the remainder of this chapter.

52

unavoidable diabatic bulk excitation at the phase transition, which occurs because of the

closing of a gap. Such a gap closing is inevitable in the transition studied here since all

symmetries are already broken. Hence, no detour via symmetry breaking exists [111],

unlike, for example, the case of the SSH model [157,158].

In this section we study the dynamical regimes that arise in translationally invariant

lattices that are incommensurate, as well as in patch and cylinder geometries, depending

on the system size and the speed at which one ramps up the strength of the drive’s field.

3.4.1 Translationally invariant systems

In Ref. [65] we showed that, in translationally invariant two-band systems [as described

by Eq. (1.32)], unitarily driving across a topological transition generates Landau-Zener

transitions [61, 62, 140]. In the topological transition explored there (the same of interest

here), the gap closes at the K ′ Dirac point so the momentum states considered were

near that point. In Ref. [65], we also showed that the Chern number can change in

incommensurate systems for slow enough ramps. Now we connect those results showing

that in incommensurate translationally invariant systems the finite-size scaling of the

excitations and of the critical field when crossing the topological transition are described

by the Landau-Zener formulae.

The Landau-Zener theory describes coupled two-state systems in which the diagonal

(uncoupled) energies change linearly in time [141,159,160],

H(t) =

βt V

V −βt

, (3.2)

with the system initially in the ground state at t = −∞. A finite coupling V allows

transitions to occur, and at t =∞ the probability of finding the system in the excited

53

state is

PE = exp(−πV 2/β). (3.3)

For our topological transition, the Landau-Zener formula gives the final occupation in the

higher Floquet band. For incommensurate translationally invariant systems, V ∼ 1/L and

β ∼ 1/τ , hence V 2/β ∼ τ/L2. Henceforth, we define τ/(L2T ) to be the dimensionless

Landau-Zener parameter (in short, the Landau-Zener parameter). Furthermore, we

evaluate Eq. (3.3) via determining the exact relationships between V and L, as well as β

and τ , using that: (i) at the gap-closing point at A∗, the coupling strength is given by

the system size and the Fermi velocity vF with

V = ηvF/L, (3.4)

where η depends on the k coordinate of the gap-closing momentum For our transition the

gap closes at the K ′ point, hence η = 2/9; (ii) β is the rate of increase of the time-reversal

breaking mass term generated by the Floquet drive near criticality [10,91].

While translationally invariant commensurate systems are fully excited at the gap-

closing momentum after the topological transition is crossed (V = 0), this momentum

point is absent in incommensurate systems. Figure 3.3 shows the highest occupation of

any k point in the conduction Floquet band (dubbed the maximum excitation) across

the Brillouin zone, in incommensurate lattices with L = 3n+ 1 and 3n+ 2, plotted as

a function of the Landau-Zener parameter. The dashed line shows the Landau-Zener

prediction [Eq. (3.3)]. Three regimes can be identified: (i) For fast ramps, the maximum

excitation (very close to 1) depends weakly on τ [see inset (a) in Fig. 3.3]. The maximum

excitation in this regime is essentially determined by the overlap between the initial

and final valence-band states, which depends on the system size. (ii) For intermediate

ramp speeds, one can see a collapse of the maximum excitation for different lattice

54

sizes when plotted against the Landau-Zener parameter. This is a regime in which

Landau-Zener dynamics dominates and the maximal excitation occurs in a k point close

to the gap-closing momentum. (iii) For slow ramps, the system follows near adiabatically

the drive. The maximum excitation no longer exhibits collapse when plotted against

the Landau-Zener parameter. In this regime, diabatic processes other than those about

the gap-closing momentum, or those about the critical field strength, may dominate the

Fermi sea excitation. Also, the maximum excitation is solely determined by the ramping

speed PE ∼ τ−2 as expected [108], see the inset (b) in Fig. 3.3.

Figure 3.3 makes apparent that, with increasing system size, the maximal excitation is

described by the Landau-Zener prediction for a wider range of Landau-Zener parameters.

The collapse extends to smaller [inset (a) in Fig. 3.3] and larger (main panel in Fig. 3.3)

values of τ/(L2T ). We have found that the three regimes discussed for PE in Fig. 3.3 are

also apparent in other quantities. For example, the Landau-Zener and near-adiabatic

regimes can be seen in Fig. 3.4 for the overlap between the entire time-evolved state |Ψτ 〉

and the lower Floquet band of the final Hamiltonian |ΨF 〉.

Another interesting aspect of crossing a topological phase transition in the thermody-

namic limit is the fact that the excitation density

ρE := 1L2

∑k

PE,k (3.5)

across the Brillouin zone should be governed by a Kibble–Zurek scaling ρE ∼ τα [161–165].

Integrating Eq. (3.3) over the Dirac cone, one finds that in the Landau-Zener regime

α = −1 [62,67,166]. In Fig. 3.5 we plot ρE vs τ/T for the same lattice sizes as in Fig. 3.4.

Those results show the existence of an expanding-with-system-size Landau-Zener regime

with Kibble–Zurek scaling of ρE (note that an increasing number of points collapse

onto the 1/τ line) that is preceded by a nearly size-independent fast-ramp regime and

succeeded by a near-adiabatic one. The analysis in the inset in Fig. 3.5, whose goal is

55

10-4

10-3

10-2

10-1

100

101

τ /L2T

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

max

PE

L = 64L = 128L = 256L = 512Landau-Zener

10-5

10-4

10-3

10-2

τ /L2T

10-3

10-2

10-1

1

1-m

ax P

E10

010

110

210

310

410

5

τ /T10

-10

10-8

10-6

10-4

10-2

100

(b)

(a)

Figure 3.3. Highest occupation of any k point in the conduction Floquet band across theBrillouin zone, in incommensurate lattices with L = 3n+ 1 and 3n+ 2, plotted as a functionof the Landau-Zener parameter. We show results for four lattice sizes. The dashed line is theLandau-Zener prediction [Eq. (3.3)]. Inset (a) shows the departure of the maximum occupationfrom 1 in the limit of fast ramps. The dashed line is the Landau-Zener prediction, while thevertical dotted lines mark the τ = 50T boundary between the fast-ramp and the Landau-Zenerregimes identified in Fig. 3.5. Inset (b) shows the maximum excitation plotted as a function ofthe ramp time τ , highlighting the near-adiabatic regime in which PE ∼ τ−2 (the dash-dottedline depicts τ−2 behavior).

to determine the power of a potential power-law decay for each pair of contiguous data

points shown in the main panel, allows us to identify the onset of the Kibble–Zurek 1/τ

regime (τ ≈ 50T , see vertical dotted lines). Note that τ ≈ 50T also marks the onset of

the Landau-Zener regime in Fig. 3.3(a). In Fig. 3.5 and its inset one can see that in the

56

10-4

10-3

10-2

10-1

100

τ /L2T

10-5

10-4

10-3

10-2

10-1

100

1-|

⟨Ψτ|Ψ

F⟩|

2L = 64L = 128L = 256L = 512

100

101

102

103

104

τ /T10

-6

10-5

10-4

10-3

10-2

10-1

100

Figure 3.4. 1 − |〈Ψτ |ΨF 〉|2, where |Ψτ 〉 is the time-evolved state and |ΨF 〉 is the groundstate of the Floquet Hamiltonian, both at the end of the ramp (A = 1), plotted as a function ofthe Landau-Zener parameter (main panel) and of the ramp time (inset). The dash-dotted linein the inset depicts τ−2 behavior. We show results for the same four lattice sizes as in Fig. 3.3.

near-adiabatic regime, which for the largest values of τ considered here is visible only in

the three smallest system sizes shown, the excitation density is much smaller than in the

Landau-Zener regime and scales as ρE ∼ τ−2.

The existence of a Landau-Zener regime that, with increasing system size, extends to

arbitrarily long ramp times allows us to describe analytically the scaling of the dynamical

critical field strength A∗L,τ at which the Chern number changes during the dynamics in

finite systems. For that, we use that the time dependence of the probability to remain

in the ground state (the “adiabatic probability”) for the Landau-Zener problem can be

57

100

101

102

103

104

τ /T

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

ρE

L = 64

L = 128

L = 256

L = 512

L = 1024

100

101

102

103

104

τ /T-3

-2

-1

0

α

Figure 3.5. Excitation density ρE in the time-evolved state plotted as a function of the ramptime. As in Fig. 3.4, we show results for the same four lattice sizes as in Fig. 3.3. In addition,for the longest ramp times, we show results for L = 1024. A fit of the L = 1024 results to1/τ is shown as a dashed line. (Inset) Discrete derivative α = ∆ ln ρE/∆ ln(τ/T ) for the datashown in the main panel (for clarity, we do not show the results for L = 64). Note the plateauat α = −1, whose onset (the onset of the Kibble–Zurek regime) is marked by a vertical dottedline (at τ = 50T . also shown in the main panel). Another plateau (α = −2, indicated by thearrow) shows the near-adiabatic regime. The latter moves to longer ramp times as the systemsize increases (the near-adiabatic regime is not visible for the two largest system sizes).

expressed in terms of parabolic cylindrical functions Da(z) [141],

PA

(√βt,

V 2

β

)=√V 2

2β e−πV 28β

[D− iV 2

2β −1

(ei 5π

4

√2βt

)]2. (3.6)

The dynamics in the Landau-Zener regime is then determined by both the Landau-

Zener parameter and the dimensionless time√βt, as one could have concluded from a

58

dimensional analysis.

For our ramps, the time at which A(t) = A∗L is the one that corresponds to t = 0 in

the Landau-Zener problem. We can then rewrite the dimensionless time at which the

Chern number changes (√βt∗) in terms of the deviation between the dynamical (A∗L,τ )

and Floquet (A∗L) critical field strengths (∆A∗L,τ = A∗L,τ − A∗L), namely,

√βt∗ ∼

√τ∆A∗L,τ . (3.7)

In Fig. 3.6, we plot ∆A∗L,τ (τ/T )1/2 as a function of the Landau-Zener parameter. The

results exhibit a clear data collapse for over a decade in the range of Landau-Zener

parameters and, for the largest values of the Landau-Zener parameter shown, the collapse

improves with increasing system size. We note that, on the other hand, for the smallest

values of the Landau-Zener parameter shown the data is cut at much larger values

of τ/(L2T ) than in Figs. 3.3 and 3.4. This is because the Chern number either does

not change or exhibits oscillatory behavior. In the presence of oscillations, we report

∆A∗L,τ (τ/T )1/2 at the time the Chern number changes for the first time only if the state at

the end of the ramp has a nonvanishing Chern number and remains so at long times under

the evolution driven by the final Hamiltonian. Figure 3.6 shows that, with increasing

system size, longer ramps are needed for the Chern number to change.

The data collapse in Fig. 3.6 indicates that PA(√

βt∗, V 2/β)is constant at the time at

which the dynamical critical field is crossed. Theoretically, we expect PA(√

βt∗, V 2/β)

=

1/2 at the dynamical critical field. This can be visualized in the Bloch sphere picture

for two-band systems [65,81]. In a trivial initial state near the gap crossing momentum

K ′, the states in are within one hemisphere, leaving the other unoccupied. A threshold

PA(√

βt∗, V 2/β)

= 1/2 marks the entrance of the time-evolved state to the other

hemisphere. Hence, we can compute A∗L,τ analytically by setting PA(√

βt∗, V 2/β)

= 1/2

in Eq. (3.6). We find that the range of Landau-Zener parameters for which PE in Eq. (3.3)

59

0.01 0.1

τ /L2T

0.01

0.1

1∆

AL

,τ*

(τ /

T)1

/2Landau ZenerL = 64L = 128L = 256L = 512L = 128, HF[A(t)]

0.1 1

τ /L2T

-0.01

0

0.01

∆A

L,τ*

(τ /

T)1

/2

Figure 3.6. Reduced deviation of the dynamical critical field ∆A∗L,τ (τ/T )1/2 plotted as afunction of the Landau-Zener parameter for the same lattice sizes as in the previous figures.The dashed line shows the analytical Landau-Zener prediction using Eq. (3.6). For too smallvalues of the Landau-Zener parameter no dynamical transition occurs (this is why the data iscut at much larger values of the Landau-Zener parameter than in the previous figures). (Inset)The same data plotted using a linear scale for the reduced deviation, whose absolute valueconverges to zero for all system sizes as τ →∞.

is smaller than 1/2 (so that PA > 1/2) is essentially the same for which we report values

of ∆A∗L,τ (τ/T )1/2 in Fig. 3.6. We focus on that range of Landau-Zener parameters when

solving for A∗L,τ . Also, the fact that PA(√

βt, V 2/β)vs√βt is oscillatory for the smallest

values of τ/(L2T ) shown in Fig. 3.6 allows us to understand the oscillatory behavior

observed in the Chern number in that regime. In the presence of oscillations that cross

1/2, τ ∗ is taken to be the first time at which PA(√

βt, V2

β

)crosses 1/2. The detailed

60

behavior of Eq. (3.6) is discussed in App. B.

Figure 3.6 shows that the theoretical curve extracted this way agrees well with the

numerical results when the latter exhibit data collapse. When the data departs from the

analytic prediction for large values of the Landau-Zener parameter, the dynamical critical

field can be lower than its equilibrium counterpart. This is shown in the inset of Fig. 3.6

and is the result of micromotion, namely, of the fact that the system can closely follow the

oscillatory electric field within each driving period [103–105]. To verify that micromotion

is indeed the cause, we carried out the evolution of a system with L = 128 using the

Floquet Hamiltonian corresponding to the instantaneous field strength HF [A(t)]. The

results are also reported in Fig. 3.6 with solid symbols (denoted as L = 128, HF [A(t)]).

In that case, ∆A∗L,τ remains positive as it approaches zero with increasing Landau-Zener

parameter.

3.4.2 Cylinder geometry

Here we carry out the same analysis for cylinders. As mentioned in Sec. 3.3, in order to

minimize finite-size effects, we focus on cylinders with L = 3n+ 2.

Quantifying how much, e.g., in terms of the amount of excitations, cylinders are

affected by crossing a topological transition is complicated by the presence of edge states.

Out of equilibrium, the far separated topological edges are effectively two disconnected

1D systems with possibly different chemical potentials. Hence, simply taking the ground

state of the final Floquet Hamiltonian to be the reference state for adiabatic dynamics

may not yield meaningful results [62,67,167]. As discussed by Privitera and Santoro [62],

in the thermodynamic limit edge states of the instantaneous Floquet Hamiltonian (with

vanishing coupling to any other state, including those at the opposite edge) can move

around in the bulk gap during the dynamics. As a result, when the crossing point of the

dispersion of opposite edge states shifts, occupied states end up having a higher energy

61

0.001 0.01 0.1 1

τ /L2T

0

0.2

0.4

0.6

0.8

1

1-|

⟨Ψτ|Ψ

F⟩|

2

L = 59L = 89L = 128L = 158L = 200L = 512, TI

10-3

10-2 10

-12

10-8

10-4

100

|⟨Ψ

τ|Ψ

F⟩|

2

Figure 3.7. 1− |〈Ψτ |ΨF 〉|2, where |Ψτ 〉 is the time-evolved state and |ΨF 〉 is the appropriatereference state of the Floquet Hamiltonian (see explanation in the text), both at the endof the ramp (A = 1), plotted as a function of the Landau-Zener parameter in the cylindergeometry. For comparison, the dash-dotted line (L = 512, TI) shows the results obtainedfor the translationally invariant system with L = 512 reported in Fig. 3.4. (Inset) Overlap|〈Ψτ |ΨF 〉|2 vs the Landau-Zener parameter in the fast-ramp regime. The vertical dotted linesin the inset mark τ = 50T , the ramp time at which the onset of the Landau-Zener regime wasidentified in Fig. 3.4 for incommensurate translationally invariant systems.

than unoccupied ones even though no true excitation has occurred. Those occupied states

need to be identified and included in the reference state used to quantify excitations.

Failing to do this can result, e.g., in a vanishing overlap between the reference state and

a near-adiabatic time-evolved one.

For the Floquet Hamiltonian in cylinders, our reference state for adiabatic dynamics

|ΨF 〉 contains the lower bulk band and all edge states on both sides of the cylinder.

Consequently, we do not resolve the contributions of the bulk and the edges independently.

62

1 10 100 1000

τ /T

10-5

10-4

10-3

10-2

ρE

L = 59

L = 89

L = 128

L = 158

L = 200

L = 512, TI

Figure 3.8. Excitation density ρE in the time-evolved state plotted as a function of the ramptime. For comparison, the dash-dotted line (L = 512, TI) shows the results obtained for thetranslationally invariant system with L = 512 reported in Fig. 3.5. The vertical dotted linemarks τ = 50T , the ramp time at which the onset of the Landau-Zener regime was identified inFig. 3.4 for incommensurate translationally invariant systems.

For the zigzag edges considered here, the edge modes extend between the two Dirac

points, 2π/3 ≤ ka ≤ 4π/3 where a =√

3d is the unit cell spacing [81]. Hence, we

take the lowest L + 1 states for each k within this range and L states elsewhere to

form the reference basis. We then compute the occupation matrices 〈ψiτ (k)|ψjF (k)〉, with

i = 1, . . . , L and j = 1, . . . , L(+1) outside (inside) the edge state momentum range. The

L singular values of this matrix are the square roots of occupation numbers. They are

used to compute the overlap (the product of the square of the singular values) and the

excitation density (the mean of the square of the singular values). One can verify the

63

correctness of the reference state looking at the spectrum of these singular values. It

should be continuous in k and change sharply when the reference state filling number is

one too many.

The overlap obtained this way is shown in Fig. 3.7. It exhibits collapse when plotted

against the Landau-Zener parameter, for τ & 50T (see inset), even though the falloff

is not as fast as in the incommensurate translationally invariant geometry (see the

dash-dotted line for the results for L = 512). The corresponding excitation density in

Fig. 3.8 obeys the Kibble-Zurek prediction. Notice the collapse with increasing system

size to the results obtained for the incommensurate translationally invariant geometry

with L = 512 (dash-dotted line).

Given the results reported in Fig. 3.7, it is natural to expect that the critical field for

the dynamical topological phase transition in the cylinder geometry exhibits a Landau-

Zener regime that extends to larger values of τ as the system size increases. Indeed,

the behavior of the reduced dynamical critical field√τ∆A∗L,τ in Fig. 3.9 displays the

advanced scaling collapse and convergence with increasing cylinder size. Interestingly,

and in contrast to the results for incommensurate translationally invariant systems, for

the first decade of Landau-Zener parameters shown√τ∆A∗L,τ is not monotonic. However

we should stress that, as shown in the inset in Fig. 3.7, the dynamical critical field

itself is mostly monotonic, up to small oscillations similar to the ones observed in the

translationally invariant case. For very slow ramps micromotion again sometimes results

in negative values of ∆A∗L,τ .

3.4.3 Patch geometry

The patch geometry has the largest finite-size effects, as seen in the finite-size scaling

analysis of the critical field for the topological transition in the Floquet Hamiltonian (see

Fig. 3.2). Also, in the topological phase, the distinction between edge and bulk states

64

0.01 0.1 1

τ /L2T

0

0.5

1

1.5

2

∆A

L,τ* (τ

/T

)1/2

L = 59L = 89L = 128L = 158L = 200L = 272

100 1000 10000

τ /T

0

0.1

0.2

∆A

L,τ*

Figure 3.9. Reduced deviation of the dynamical critical field ∆A∗L,τ (τ/T )1/2 plotted as afunction of the Landau-Zener parameter in the cylinder geometry. For too small values ofthe Landau-Zener parameter no dynamical transition occurs (this is why the data is cut atlarger values of the Landau-Zener parameter than in Fig. 3.7). For the slowest ramps, ∆A∗L,τsometimes becomes negative (∼−10−5). (Inset) The unscaled ∆A∗L,τ vs τ/T , which decreases(mostly) monotonically. Using the approach discussed in App. A.3.2, we are able to computeA∗L,τ for larger lattices and longer ramp times than for the overlaps and excitation densitiesreported in Fig. 3.7.

is ambiguous. In fact, there are (sometimes discontinuous) ranges of fillings within the

bulk gap for which the Bott index is nonzero [56]. These issues make it difficult to define

a reference state |ΨF 〉, and to carry out analyses such as the ones reported in Fig. 3.7 for

cylinders. Hence, for patches we focus on the scaling of the dynamical critical field.

In Fig. 3.10 we show the reduced deviation of the dynamical critical field plotted as a

function of the Landau-Zener parameter. The results are qualitatively similar to the ones

in Fig. 3.9 for cylinders. As in the cylinder geometry, we find that√τ∆A∗L,τ exhibits

nonmonotonic behavior, while the dynamical critical field itself is mostly monotonic (see

65

0.01 0.1 1

τ /L2T

0.5

1

1.5

2

∆A

L,τ* (τ

/T

)1/2

L ≈ 30L ≈ 40L ≈ 48L ≈ 60L ≈ 68L ≈ 76

100 1000 10000

τ /T

0

0.1

0.2

∆A

L,τ*

Figure 3.10. Reduced deviation of the dynamical critical field ∆A∗L,τ (τ/T )1/2 plotted as afunction of the Landau-Zener parameter in the patch geometry. For too small values of theLandau-Zener parameter no dynamical transition occurs. (Inset) The unscaled ∆A∗L,τ vs τ/T ,which decreases (mostly) monotonically. For L ≈ 68 and 76 we only show results for three rampspeeds. They help gauging convergence with increasing the size of the patch. Qualitatively, thebehavior of the dynamical critical field is similar to that in the cylinder geometry reported inFig. 3.9. For the longest ramps and the largest system sizes, the Bott index is computed withthe method in App. A.3.3.

inset in Fig. 3.10). Also, the data is consistent with a collapse that improves and extends

to larger values of the Landau-Zener parameter with increasing patch size. We note that,

in Fig. 3.10, only patches with L ≈ 30, 40, 48, and 60, were systematically studied for

many ramp times τ . Due to their high computational cost, we report results for L ≈ 68

and 76 only for three ramp times. They are consistent with the trend seen for smaller

patches.

66

3.5 Hall Responses

Our results in Sec. 3.4 show that, when ramping up the strength of the field of the drive

in increasingly large systems (no matter the boundary conditions), topological indices

such as the Chern number and the Bott index fail to indicate the crossing of a topological

transition. Here we explore what happen with the more experimentally relevant Hall

responses. We focus on translationally invariant systems as, based on our results in

Sec. 3.4, we expect cylinder and patch geometries to behave similarly for large systems

sizes.

For a static insulating state, the Hall conductivity given by linear response depends

exclusively on the Berry curvature of the occupied band [8, 74]. Out of equilibrium, the

Hall conductivity depends both on the time-evolving state and the Hamiltonian [64,67,168].

In this section we report results for the Hall response of the ground state of the Floquet

Hamiltonian, as well as of the out-of-equilibrium state generated by ramping up the

strength of the driving field, in translationally invariant systems.

We probe those states via turning on a weak constant electric field Ex and measuring

the transverse displacement. The displacement per unit cell along the primitive lattice

vectors a1 and a2 are computed evaluating [169],

Yi(t) = a

2πL∑

kIm ln〈uk(t)|uk+dki(t)〉, i = 1, 2, (3.8)

where uk is the periodic part of the Bloch state at momentum k, and dk1,2 is the

displacement between neighboring k points along the two reciprocal vectors. From

the stroboscopic transverse displacement Y = (Y1 − Y2)/2 over time, we obtain the

time-averaged Hall conductivity computing

σxy(t) = 2πS

Y (t)Ext

(3.9)

67

in units of e2/h, where S is the area of a unit cell. Each term in the sum in Eq. (3.8)

needs to be monitored over time to keep them on a continuous Riemann sheet. We favor

the numerical evaluation of the displacement over the current because within each cycle

the current produced by the Floquet drive and nonequilibrium state overwhelms the

response to the applied field.

3.5.1 Floquet ground state

In Fig. 3.11, we plot the time-averaged Hall conductivity during the first 50 periods of

the drive after turning on a weak constant electric field Ex = 10−5. The initial state

is taken to be the ground state of the Floquet Hamiltonian for driving field strengths

between 0 and 1 (see legend at the top of Fig. 3.11). We report results for commensurate

lattices L = 63 [Fig. 3.11(a)] and L = 126 [Fig. 3.11(b)], as well as for incommensurate

ones L = 64 [Fig. 3.11(c)] and L = 127 [Fig. 3.11(d)].

The results in Fig. 3.11 show that, deep in the trivial (A → 0) and topological

(A → 1) phases the time-averaged Hall conductivity rapidly approaches its quantized

0 and 1 values, respectively. As the ground state of the system is taken closer to the

transition point (A∗ ≈ 0.498 in the thermodynamic limit) it takes much longer for the

time-averaged Hall conductivity to become 0 (if A < A∗) or 1 if (if A > A∗). The presence

of the gap-closing momentum in finite commensurate systems [Figs. 3.11(a) and 3.11(b)]

produces a “jump” between the dynamical responses for A smaller and greater than A∗,

which is absent in the incommensurate systems [Figs. 3.11(c) and 3.11(d)]. The jump

in the former is the result of the large Berry curvature concentrated at the gap-closing

momentum about the topological phase transition.

We expect that, with increasing system size, the Hall responses of commensurate and

incommensurate systems become similar to each other, with no jump (as a result of the

smooth interpolation of the Berry curvature) and a wide range of values of σxy about the

68

0

1

σ x

y

0 10 20 30 40t /T

0

1

σ x

y

0 10 20 30 40 50t /T

0 0.2 0.4 0.6 0.8 1A

(a) L = 63 (c) L = 64

(b) L = 126 (d) L = 127

Figure 3.11. Time-averaged Hall response σxy [see Eq. (3.9)] of the ground state of theFloquet Hamiltonian in translationally invariant systems, in units of e2/h. We show resultsfor commensurate lattices, (a) L = 63 and (b) L = 126, as well as for incommensurate ones,(c) L = 64 and (d) L = 127. The strength of the Floquet driving field is taken between 0 and1 and is color coded in the lines according to the legend at the top. The ticks in the colorbar mark the field strengths A for which results are reported (notice the finer 0.001 resolutiongrid about the critical field for the topological transition). The dashed lines in (b) show theresponse for A = 0.497 and 0.498 in a lattice with L = 300. Those are the fields closest to thecritical one for which results are reported in all panels.

69

topological transition at the longest times shown (because of the increasing resolution of

momenta about the gap-closing point). Our results in Fig. 3.11 are consistent with those

expectations. Notice that in commensurate systems the magnitude of the jump decreases

with increasing system size. This is apparent when comparing the results for A = 0.497

and 0.498 in the lattice with L = 63 [Fig. 3.11(a)] and in the lattices with L = 126 and

300 [Fig. 3.11(b)]. In incommensurate systems, on the other hand, the response near the

phase boundary fans out with increasing system size. In Fig. 3.11, the average response

has mostly acquired a quantized value after fifty driving periods, except for fields close

to the critical one (0.45 ≤ A ≤ 0.55).

3.5.2 Time-evolved state

Before turning on Ex to study the response of the time-evolving states obtained at the

end of the ramp, we let those states equilibrate by time evolving them for long times with

the final Hamiltonian (with A = 1). The equilibration simulations are ran for between 103

and 104 periods under the final Hamiltonian. They allow the currents that accumulate

during the ramp relax. Such currents can be generated, e.g., by the higher strength of

the driving field during the second half of the driving cycles. After equilibration, we

measure the displacement after turning on a weak constant electric field Ex = 10−5 and

also without the field. We define the net displacement as the difference between the two.

The Hall response extracted using the net displacement is shown in Fig. 3.12. Re-

gardless of whether the lattices considered are commensurate or not, and of whether the

Chern number changes or not during the ramp (it can only change in incommensurate

lattices), the time-evolved states exhibit nontrivial Hall responses that converge to the

equilibrium value with increasing ramp time τ and system size L. This demonstrates

that when crossing a Floquet topological transition under driven unitary dynamics it is

possible to create states with nearly quantized Hall response, despite the fact that those

70

0.6

0.8

1

σ x

y

0 10 20 30 40t /T

0.6

0.8

1

σ x

y

0 10 20 30 40 50t /T

τ = 50τ = 100τ = 500τ = 1000

(a) L = 63 (c) L = 64

(b) L = 126 (d) L = 127

Figure 3.12. Time-averaged Hall response σxy [see Eq. (3.9)] of the time-evolved stateprepared under different ramp times τ (the strength of the field is always increased from A = 0to 1) in translationally invariant systems, in units of e2/h. After the ramp, we evolve the systemsfor long times (at fixed A = 1) to probe the state after equilibration (see text). We show resultsfor commensurate lattices, (a) L = 63 and (b) L = 126, as well as for incommensurate ones, (c)L = 64 and (d) L = 127, as in Fig. 3.11. Independently of whether the Chern number changesor not, the Hall conductivity converges to the equilibrium quantized value with increasing ramptime and system size.

71

states may have a trivial Chern number. The oscillatory behavior observed in Fig. 3.12

is a consequence of the coherent superposition of states in the lower and upper Floquet

band [67, 168]. The magnitude of the oscillations can be seen to decrease with increasing

ramp time and system size. Beyond ramp times τ ' 500, the change in Hall response is

not discernible in Fig. 3.12.

3.6 Summary and discussion

We have identified and explored the three dynamical regimes (fast-ramp, Landau-Zener,

and near-adiabatic regimes) that occur in various observables when ramping up the

strength of a driving field across a topological phase transition in finite incommensurate

periodic lattices as well as in cylinder and patch geometries.

For observables such as the maximal excitation, the overlap between the time-evolved

state and the appropriate lowest energy Floquet reference state, and the average excitation

energy; whenever they can be meaningfully defined, we find that the fast-ramp regime

is nearly system-size independent. This regime is followed (for longer ramp times) by

a Landau-Zener one whose extent in ramp times increases with increasing system size.

In the Landau-Zener regime we showed that the excitation density is governed by a

Kibble–Zurek scaling ρE ∼ τ−1. On the other hand, for very slow ramps, we showed that

there is a near-adiabatic regime in which the magnitude of the observables is governed

by the square of the ramp speed.

In the fast-ramp regime, and in the initial faster ramp part of the Landau-Zener regime

identified via the previously mentioned observables, we find that with increasing system

size the topological indices do not change (or oscillate) when crossing the topological

transition. Consistent with the fact that the topological indices are invariant in the

thermodynamic limit, we have shown that as the size of the lattices increase longer

ramp times are needed for the topological indices to change. When they do change, we

72

observe a robust regime in which the dynamical critical field scales with the Landau-Zener

parameter. That regime extends to larger values of the Landau-Zener parameter with

increasing system size.

Finally, we showed that the dc Hall response allows one to identify the dynamical

crossing of a Floquet topological phase transition independently of the behavior of the

topological indices. Specifically, we showed that the states created under driven unitary

dynamics after crossing a Floquet topological transition have nearly quantized Hall

response when the strength of the driving field is increased using slow enough ramps. In

contrast to the Chern number and the Bott index, the ramp times needed to observe a

nearly quantized Hall response are O(1), namely, they do not diverge with increasing

system size.

73

Chapter 4 |

Summary and outlook

Floquet engineering of topological phases greatly expands the scope of systems and

operations that can be applied in the future application of topological systems. The

dynamics of topological phase transition in experimental realizations of topological models

called into question how a change of topology is achieved in such processes. My doctoral

work presented in this dissertation addressed those questions in the context of unitary

time evolution.

In Chapter 1, I reviewed some necessary background on the Chern number, the

Bott index, and the Chern insulator model known as the Haldane model. The basics of

Floquet systems was also presented there. Then I reviewed the work by D’Alessio and

Rigol [56] on the unitary driving of a topologically trivial system towards a topological

phase. The naive idea of adiabatic following the ground state of the Floquet Hamiltonian

was challenged by a no-go theorem showing dynamical invariance of the Chern number.

But when the Bott index was applied to an open boundary system, the topology of the

driven state was observed to change in time.

In Chapter 2, I presented the results reported in Ref. [65], which unifies the behavior

of the Chern number and the Bott index in the thermodynamic limit. For finite-size

lattices even translationally invariant systems may admit a change in their topological

index if the lattice size is incommensurate, i.e., if the gap closing momentum is missing.

74

The gap closing mechanism also allow us to observe the Landau Zener dynamics that

dominates the unitary topological phase transition in finite translationally invariant

systems.

In Chapter 3, reporting the results from Ref. [70], I provided a unified view of dy-

namical topological phase transitions across different boundary conditions, including the

cylinder and patch geometries. The indicators monitored were wavefunction overlaps,

excitation density, and the Chern number or the Bott index. In dynamics under differ-

ent turn-on speeds, a universal Landau Zener regime was established. It governs the

dynamical critical field for the topological phase transition. A faster turn-on, including a

quench, falls into another regime where the dynamics is system size independent and the

topological indexes do not change. These are the only two regimes that are relevant in

the thermodynamics limit. The Hall response of dynamically prepared state shows the

quantized Hall conductance associated with the final Hamiltonian, independent from the

behavior of the Bott index or the Chern number.

In short, the topological phase transition in Chern insulators is distinguished by a

guaranteed zero overlap between states of different phases. This entails that for generic

driving protocols, the dynamics of the system will be governed by a Landau Zener

transition. This is universal across lattice types and boundary conditions.

Some questions remain to be answered. A technical question is what causes the

nonmonotonicity in the reduced deviation of dynamical critical field in systems with

open boundaries (Fig. 3.9 and 3.10). Furthermore, if the Hall response of the system

does not depend on the topological indices, what character of the state does the Bott

index quantify? Local versions of the topological indices may allow one to better

identify observables correlated with them [66,69]. It would be interesting to construct a

measurable quantity to see if one can probe the topological memory due to the no-go

theorem. Wavefunction overlap is not practical in this regard. Some effort has been made

75

to build such an observable, e.g., in Ref. [137]. Others have used the linking invariant

or Euler class to characterize the whole quench process [148,154,170,171]. Finally, one

can apply the analysis to other topological classes and topological order. This has much

overlap with advances in the understanding of dynamical quantum phase transition in

the past decade [145].

76

Appendix A|

Computations and some proper-ties of the Bott index

A.1 Bott index in cylinder geometry

Here I discuss the Bott index expression in Eq. (3.1). Note that the original Bott

index formula, albeit more costly, can be used in cylinders. The Bott index formula in

Eq. (1.16),

Cb(P ) = 12π Im Tr ln(V U V †U †), (A.1)

where U , V are the operators e2πix/Lx , e2πiy/Ly respectively projected to the occupied

states. In a cylinder geometry translationally invariant in x, the U operator becomes

U =∑kx,α

∣∣∣∣kx + 2πLx, α⟩⟨kx, α

∣∣∣∣ =∑kx

∣∣∣∣kx + 2πLx

⟩⟨kx

∣∣∣∣⊗ Iα, (A.2)

where α are other degrees of freedom, such cell index in y direction and sublattice sites,

and Iα is the identity operator in these degrees of freedom. Eigenstates of the Hamiltonian

77

are Bloch states along x. Hence the projection operator have the form

P =∑kxn

|ψkxn〉〈ψkxn| =∑kxn

eikxx|ukxn〉〈ukxn|e−ikxx =∑kx

eikxxPkxe−ikxx, (A.3)

where |ψkxn〉 is the Bloch eigenstates of band n, |ukxn〉 is the lattice periodic part of

the Bloch state, and Pkx := ∑n |ukxn〉〈ukxn| is the projection into occupied bands at kx.

Hence

U = P U P =∑kx

Pkx+ 2πLxPkx . (A.4)

Plug this into Eq. (A.1) and one obtains Eq. (3.1).

A.2 Relation to the local Chern marker

Another topological index of choice for systems with open boundary conditions is the

local Chern marker [99]. It can be used to probe the Chern number locally in the real

space. It is defined as

c(r) = 4πS

Im∑α

〈rα|P xP yP |rα〉, (A.5)

where S the unit cell area, and the sum takes place over the orbitals α in the unit cell at

r. When averaged over an entire patch, the local Chern marker yields zero, reflecting the

fact that an open boundary system is topologically trivial as a whole [99]. This comes

from the large negative values that the local Chern marker takes at the edges. The

physical picture is that the cyclotronic bulk and the skipping orbits at the edge have

opposite chiralities. The Bott index, on the other hand, eliminates the edge contribution

by gluing opposite edges together and allowing their chiralities to cancel. Therefore, the

Bott index can be seen as the bulk average of the local Chern marker.

78

A.3 Efficient calculation of the dynamics of topological

indices

There are a few ways to simplify the computation of topological indices in detecting a

phase transition, as discussed below. This part complements the discussion in Sec. 3.4.

A.3.1 Translational invariant systems

To detect a topological phase change for long enough ramps in translationally invariant

systems, it is sufficient to track the time evolution of states near the gap-closing momenta.

It is not necessary to carry out the time evolution of the entire Brillouin zone. The same

turns out to be true for the other two geometries considered in this work.

A.3.2 Cylinder geometries

In cylinders, we also find that only states near the gap-closing momenta need to be time

evolved for long enough ramps. One can further notice that, in the initial Hamiltonian,

there are already edge states at the top of the valence band. They also end up with the

highest overlap with the topological edge states. For very long ramps only the top two

states of the initial valence band at each momentum are needed to capture the jump in

Berry curvature. Compared to the full band calculations, the error introduced in A∗L,τ

and ∆A∗L,τ (τ/T )1/2 in slow ramps is smaller than the size of the symbols in Fig. 3.9.

A.3.3 Patch geometries

Similar to the case of the cylinder geometry, the calculations for very long ramps also

benefit from the presence of initial edge states. Here the Bott index of a valence band

can be similarly computed starting from the top of the valence band as follows. Denote

79

a generic projection matrix by P (l, h) = ∑hi=l |i〉〈i|, where |i〉 is the ith eigenstate. In

a typical Bott index calculation the indices [l, h] enclose the valence band and possibly

topological edge states. Instead, one can let l start somewhere in the bulk and still obtain

a well-behaved Bott index across in the phase diagram. The Bott index computed with

different contiguous fillings of Floquet eigenstates at A = 1 is displayed in Fig. A.1. The

continuous band shows that computing from the Bott index is reliable as long as l is

within the bulk band and h is within the bulk gap. Thanks to the initial edge states, we

can also track the dynamics of the Bott index in this way. The resulting error in A∗L,τ

and ∆A∗L,τ (τ/T )1/2 is again indiscernible in our results in Fig. 3.10.

80

0

100

200

300

400

500

600

700

800

900

100 200 300 400 500 600 700 800 900

h

l

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Figure A.1. Bott index of Floquet eigenstates at A = 1 in a patch geometry with differentcontinuous fillings between state index l and h (order insensitive). The graph is triviallysymmetric about l = h. The system size is L ≈ 22. When h is located within the bulk bandgap, l does not have to start from 1 to produce a reliable Bott index. The irregularities nearl = h is simply due to too small a range of states.

81

Appendix B|

Behavior of the exact solution tothe Landau-Zener problem

Sec. 3.4 discussed that the solution to a Landau-Zener problem, H(t) = βtσz + V σx, is

given by the parabolic cylindrical functions. The occupation number in the ground state

is

PA

(√βt,

V 2

β

)=√V 2

2β e−πV 28β

[D− iV 2

2β −1

(ei 5π

4

√2βt

)]2, (B.1)

and

PA(+∞, V 2/β) = 1− exp(−πV 2/β). (B.2)

The behavior of PA is shown in Fig. B.1. For small Landau-Zener parameters, it

oscillates and cross 0.5 multiple times. Consequently the Bott index oscillates in time

for V 2/β parameters less than around 0.3. When extracting the critical field in Sec. 3.4,

I use two necessary criteria for the data points: (a) at t = +∞, PA > 0.5 and (b) at

t = τ , Cb = 1. The A∗L,τ is then calculated using the smallest t when PA reaches 0.5,

corresponding to the lowest monotonic part of the green contour in Fig. B.1(b-c). On

the PA = 0.5 contour,√βt ∼ exp(−V 2/β) for large V 2/β. On contours where PA > 0.5,

√βt is not monotonic in V 2/β even for large V 2/β’s.

82

(a) The function PA(√βt) for different Landau Zener parameters

V 2/β.

(b) PA in the reduced time-Landau Zener pa-rameter plane. The contour separations are0.1 uniformly, with PA = 0.5 highlighted ingreen.

(c) PA in the reduced time-Landau Zener parameterplane, with log-log scale. The contour at PA =0.5 is highlighted in green.

Figure B.1. The probability to remain adiabatic in the Landau-Zener transition problemusing Eq. (B.1). (a) The probability as a function of reduced time

√βt for different Landau-

Zener parameters. (b-c) The same solution in the reduced time-Landau Zener plane, in (b)normal scale, and (c) log-log scale.

83

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VitaYang Ge

Yang Ge was born in a hospital and grew up in Fuzhou, Fujian, China. His teenageyears were spent in Shi Dai Middle School and Fuzhou Middle School I. Into the adulthoodhe first attended the Hong Kong Polytechnic University. Later he transferred to Universityof Minnesota and graduated with a Bachelor of Science in Physics in 2014. During hisyears in Minnesota he worked in Prof. Martin Greven’s group working with mercury-based cuprate high temperature superconductors. After B.S. he enrolled in the doctoralprogram in physics at the Pennsylvania State University and worked under the guidanceof Prof. Marcos Rigol. He was awarded David C. Duncan Graduate Fellowship in Physicsin 2014 and Downsbrough Graduate Fellowship in Physics in 2016. During doctoralcareer he had 16 semesters of teaching experience, and was awarded Teaching AssistantAward in 2019.