Localization and slow thermalization in quantum many-body systems

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Thesis

Reference

Localization and slow thermalization in quantum many-body systems

HO, Wen Wei

Abstract

There has been a surge of interest in studying truly nonequilibrium quantum many-body phenomena, motivated both by theoretical advances such as in the understanding of many-body-localization (MBL), and by experimental advances in controlling synthetic quantum systems that serve as rich playgrounds to probe these phenomena. In this thesis, we show that looking at the physical structure of quantum systems, such as the locality of Hamiltonians and the entanglement of quantum states -- concepts with a quantum informational flavor to them -- is a useful lens with which to explore nonequilibrium many-body physics. We use these ideas to study localization -- the breakdown of ergodicity in certain classes of many-body systems, and also the timescales of thermalization in periodically driven many-body systems. We also explain the recent observations in a dipolar system of the discrete time crystal phase, an exemplary nonequilibrium phase of matter, and further consider the dynamics of entanglement in many-body systems.

HO, Wen Wei. Localization and slow thermalization in quantum many-body systems. Thèse de doctorat : Univ. Genève, 2017, no. Sc. 5083

DOI : 10.13097/archive-ouverte/unige:95354 URN : urn:nbn:ch:unige-953545

Available at:

http://archive-ouverte.unige.ch/unige:95354

Disclaimer: layout of this document may differ from the published version.

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UNIVERSIT É DE GEN ÈVE FACULT É DES SCIENCES

Section de Physique Professor Dmitry Abanin

D´epartment de Physique Th´eorique

Localization and Slow Thermalization in Quantum Many-body Systems

TH `ESE

présentée à la Faculté des Sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention Physique

par

Wen Wei Ho

de Singapour

Th`ese No. 5083

GEN `EVE

Atelier de reproduction ReproMail

Juin 2017

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Abstract

There has been a recent surge of interest in studying truly nonequilibrium quantum many-body phenomena. This has been motivated in part by theoretical advances, such as in the understanding of many-body localization (MBL), and in part by experimental advances in designing and controlling synthetic quantum systems that serve as rich playgrounds to probe these phenomena. However, our understanding of highly nonequilibrium quantum physics is still rather limited compared to that of equilibrium physics, and so we need new tools, conceptual insights and theoretical approaches to make headway in this largely unexplored frontier of condensed matter physics.

In this thesis, we show that looking at the physical structure of quantum systems, such as the locality of Hamiltonians and the entanglement structure of quantum states – concepts with a quantum informational flavor to them – is a useful lens with which to explore nonequilibrium many-body physics. We use these ideas to study localization – the breakdown of ergodicity in certain classes of many-body systems, and also the timescales of thermalization in periodically driven many- body systems.

Specifically, in Chapter 2, we study the stability of a potential nonergodic phase of matter arising naturally in disordered systems with a global, continuous non- Abelian symmetry, and which goes beyond an MBL prescription. We find that for arbitrarily weak perturbations, the phase is unstable in the thermodynamic limit, implying that the presence of the symmetry leads to thermalization. In Chapter 3 we study the timescales of heating in high-frequency periodically-driven many- body systems, and show that such systems generically exhibit the phenomenon of prethermalization. We further consider the experimentally relevant question of dynamically preparing such a system in its effective Hamiltonian’s ground state whilst minimizing heating and non-adiabatic effects, in Chapter 4. In Chapter 5, motivated by recent experimental observations of discrete time crystalline (DTC)

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order in long-ranged dipolar systems, we elucidate the mechanism by which the DTC order is stabilized, in periodically-driven, disordered systems with long-range interactions. Lastly, in Chapter 6 we explore the mechanism of entanglement dynamics in quantum many-body systems, and provide a toy model that explains salient features of the apparent universal linear in time growth of entanglement seen in ergodic systems.

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Abstrait

Il y a eu rcemment un regain d’interłt dans l’tude des phnomne quantiques n corps dans les systmes fortement hors quilibres. Ceci a t motiv en partie par des avances thoriques, telles que la comprhension de la localisation n corps (MBL), et en partie par des avances exprimentales dans le design et le contrle de systmes quantiques synthtiques qui sont des terrains de jeu riches (systmes modles riches) pour sonder ces phnomnes. Cependant, notre comprhension de la physique fortement hors quilibre est encore limite en comparaison avec la physique l’quilibre.

Ainsi, il nous faut des nouveaux outils, des ides conceptuelles et des approches thoriques pour faire une perce dans cette frontire largement inexplore de la physique de la matire condense.

Dans cette thse, nous montrons que l’observation de la structure physique des systmes quantiques - comme la localit des Hamiltoniens ou la structure de l’intrication des tats quantiques (concepts comprenant des notions d’information quantique) - offre une perspective utile pour l’exploration des systmes quantiques n corps.

Nous utilisons ces ides pour tudier la dynamique et la stabilit de la matire hors quilibre.

Spcifiquement, dans le chapitre 2, nous tudions la stabilit d’une potentielle phase non ergodique de la matire qui se manifeste naturellement dans les systmes dsordonns avec une symtrie non-ablienne globale et continue et qui dpasse une prescription MBL. Nous constatons que pour des perturbations arbitrairement faibles, la phase est instable dans la limite thermodynamique, ce qui implique que la prsence de la symtrie conduit la thermisation. Dans le chapitre 3, nous tudions galement l’chelle de temps de thermalisation d’un systme n corps priodiquement entrain sous champ haute frquence et montrons que de tels systmes prsentent gnreusement le phnomne de la prthermalization. Nous considrons en outre la question exprimentale de la prparation dynamique d’un tel systme dans l’tat fon- damental de l’hamiltonien effectif tout en minimisant les effets de chauffage et

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non adiabatique, au chapitre 4. Dans le chapitre 5, motiv par les rcentes observations exprimentales d’un ordre de cristal temporel discret (DTC) dans des systmes dipolaires longue porte, nous avons lucid le mcanisme par lequel l’ordre DTC est stabilis dans un systme dsordonn, interaction longue porte, priodiquement entrain. Enfin, dans le chapitre 6, nous explorons le mcanisme de la dynamique de l’enchevłtrement dans les systmes quantiques de nombreux corps et fournissons un modle de jouet qui explique les caractristiques saillantes de la croissance linaire linaire apparente de l’enchevłtrement vu dans les systmes ergodiques.

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Acknowledgments

I would like to thank my advisor, Dima Abanin, for his support, advice and guidance over these past few years. Dima is an excellent physicist, and I am lucky to have been his student and seen his way of doing physics: he has taught me much in how to think and do research. Much of what I know about many-body physics is attributed to him. Furthermore his exceptional physical intuition is something that I can only hope to emulate. Working with him has thus always been both a pleasure and a learning experience. Also, Dima has been very supportive of what- ever half-baked ideas I might have, and concerned with my welfare throughout my PhD studies, and I am grateful to him for that.

I would also like to thank my other advisor, Guifr´e Vidal, for being very supportive and encouraging despite abandoning him at Perimeter Institute and moving to Geneva. He has also taught me how to think about many-body physics but from a more quantum informational viewpoint, and I find that such a perspective helps me to approach a problem in a very useful way that is complementary to standard ones.

I thank too Julian Sonner and Mark Rudner for agreeing to serve on my thesis committee, and for carefully reading my thesis and offering many constructive comments to help improve it.

I am also grateful to my family and friends, especially my parents, for their unwavering love, encouragement and confidence in me during my studies. Last but not least, I thank Irene Lo for all her support, and for being a good friend to me.

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Chapter 1 Introduction

1.1 Motivation and results

Historically, condensed matter physics has focused on equilibrium or near equilibrium many-body physics, such as that of the ground state and its low lying excitations. As such, while far from being completely understood, we have accu- mulated a wealth of understanding of such physics as well as a collection of tools to study it. Conversely, truly nonequilibrium many-body physics – such as that of highly excited eigenstates or driven systems – remains comparatively speaking a largely unexplored frontier of condensed matter physics.

From a conceptual standpoint, one might inquire into why it is worthwhile to explore this nonequilibrium frontier. Underpinning equilibrium physics is the concept of universality: that different quantum systems, even though described by different microscopic constituents, can have emergent macroscopic phenomena that do not depend on the ultra-violet (UV) physics and which thus allows for their physics to be systematically analyzed. Naively, one might argue that without this concept, the study of highly nonequilibrium physics would be extremely messy, rather system specific, and possibly featureless. However, as we will see, it turns out to be quite the contrary: it is actually possible to define quantum order even for highly excited or non-equilibrium quantum states, leading to novel new phases of matter with no static analogue. For example, we will discuss the very recent excit-

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1.1. MOTIVATION AND RESULTS

ing theoretical prediction and experimental observation of the Discrete Time Crys- talline (DTC) phase in this thesis. Furthermore, the study of quantum dynamics leads to many important, fundamental questions that challenge our foundational assumptions of basic physics, such as those of thermalization and ergodicity in statistical mechanics. The nonequilibrium frontier thus represents not just unexplored country, but rather also exciting new territory, which very much deserves our attention as condensed matter physicists.

Indeed, as mentioned in the abstract, there has been a recent surge of interest in studying such truly nonequilibrium many-body phenomena. This has been in part due to theoretical advances, such as in our understanding of the long conjec- tured but only recently almost conclusively confirmed phenomenon of many-body localization (MBL), in which the interplay of strong disorder and interactions in a quantum system leads to a many-body version of Anderson localization. A strik- ing implication of this phenomenon is that our assumption that generic interacting systems are always ergodic and hence thermalizing – a foundational tenet of statistical mechanics – is violated. Instead, MBL systems belong to a dynamical phase of quantum matter that is robustly nonergodic, thus allowing for quantum order to be defined in their highly excited eigenstates. On the experimental front, advances in designing and controlling synthetic quantum systems, such as with trapped ions and cold atoms, allow for rich playgrounds to explore such nonequilibrium phenomena in, thereby providing even more impetus and stimulus for their study.

Another fruitful direction that has emerged is the use of strong driving by light to control and manipulate quantum states of solid-state or synthetic quantum systems: driving has been shown to be able to change the topology of band structures, or enhance supercondutivity, or even produce new phases of matter that are not possible in a static system.

I hope to have made a strong case for the study of truly nonequilibrium quantum many-body physics. One of the main challenges that we face in the exploration of this frontier is that many of the concepts and tools of equilibrium physics are often times not the best in a truly nonequilibrium setting, and as such, we need

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new conceptual insights and theoretical approaches. That is precisely the point of this thesis: the overarching theme posited here is that looking at the physical structure of quantum systems, such as the locality of Hamiltonians and the entanglement structure of quantum states – concepts with a quantum-informational flavor to them – is a useful lens with which to explore nonequilibrium many-body physics. Armed with this viewpoint, we present a collection of results on some truly nonequilibrium quantum many-body phenomena, such as localization and the lack of ergodicity in certain classes of systems, and also the timescales of heating and thermalization in periodically-driven systems. We study too the stability of a novel nonequilibrium phase of matter, the discrete time crystal (DTC), and also the dynamics of entanglement in many-body systems.

The main results of this thesis are as follows:

Effect of continuous non-Abelian symmetry on many-body localization

This chapter is based on the work [120].

In chapter 2, we study the effect of a continuous non-Abelian symmetry on many-body localization. Our motivation here is as follows: can a fully-MBL description, in which the system has a complete set of quasi-local integrals of motion with area-law entanglement scaling of all eigenstates, survive under the presence of a non-Abelian symmetry? After all, very general representation theoretic arguments dictates that eigenstates necessarily come in multiplets with dimensionalities larger than one – how does the presence of these higher dimensional multiplets affect the stability of a putative MBL phase?

We specifically consider the case of SU(2) symmetric disordered spin chains, and find that, owing to the continuous symmetry, such a fully-MBL description is not possible (this was also argued in [118]). Instead, the most ‘localized’ yet still symmetry respecting eigenstates all have logarithmic scaling of entanglement entropy (akin to a ‘quantum critical glass’), and which therefore realizes a different nonergodic phase of matter than that of MBL, characterized by an incomplete set

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of quasi-local integrals of motion. If found stable, such a system would represent an example of an exciting new dynamical phase of matter that is nonergodic, i.e. it violates the fundamental tenets of ergodicity that generic interacting systems are believed to possess in statistical mechanics.

We therefore construct a fixed-pont Hamiltonian associated with this phase, and study the stability of the phase to local symmetric perturbations. For arbitrarily weak perturbations, we find that the phase is unstable in the thermodynamic limit, implying that the presence of the SU(2) symmetry implies thermalization.

Our method can be used to probe the stability of other potentially nonergodic phases, such as with other non-Abelian symmetries, and can also be extended to understand the timescales of thermalization in these systems.

Prethermalization, effective Hamiltonians, and slow heating in periodically driven many-body systems

This chapter is based on the works [1, 2].

Motivated by theoretical proposals to use periodic driving to modify and control parameters in quantum systems, such as to create Floquet topological insulators (FTIs), we analyze the effect of interactions on the dynamics and stability of high-frequency periodically-driven (Floquet) systems in chapter 3. Specifically, we study the timescales of heating and thermalization in such systems, in order to, for example, understand the lifetime for which a Floquet system can host the interesting physics desired in Floquet engineering.

We employ a sequence of quasi-local unitaries to systematically rotate the original driven Hamiltonian and reduce the effect of the drive, thereby ending up with a mostly-static quasi-local Hamiltonian. We find that for generic periodically-driven many-body systems, there is an optimal order to which the sequence should be carried out to, and which depends on the locality of the Hamiltonian. At that optimal order, the resulting quasi-local effective Hamiltonian governs well strobo- scopic dynamics of the system for a time that is exponentially large in frequency.

The error between measurements of local observables evolved in time due to this

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quasi-local effective Hamiltonian and by the actual time evolution operator, can be bounded by Lieb-Robinson type arguments [87, 100]¹. Since the timescale of dynamics generated by the effective Hamiltonian can be parametrically smaller than the timescale that this Hamiltonian is itself valid for, this implies that the system can settle down to a thermal state given by the effective Hamiltonian for a long time before potentially eventually heating up to infinite temperature. Thus, high-frequency driven many-body systems generically exhibit the phenomenon of prethermalization. Proposals to use high-frequency driving to modify band structures of, for example, topologically trivial insulators into topologically non-trivial ones, are thefore viable for at least this long heating timescale.

Quasi-adiabatic dynamics and state preparation in periodically driven many- body systems

In chapter 4, we next turn to the dynamic preparation of a periodically driven many-body system in the ground state of its effective prethermal Hamiltonian found in the previous chapter. Since the ground state of the undriven Hamilto- nian (from which we presumably begin with) and the ground state of the effective prethermal Hamiltonian are potentially very different, how does one dynamically coax this highly nonequilibrium system to settle into the desired state? This issue is a rather relevant one from an experimental standpoint.

We thus imagine a protocol in which the ground state of the undriven Hamil- tonian is initially prepared, and the driving term slowly ramped up from zero to some finite value over some timescale. Such a driving protocol thus goes beyond the ‘Floquet’ picture since the drive is no long strictly periodic. However, we show, utilizing a sequence of quasi-local unitaries to rotate the Hamiltonian – similar to the ones used to derive effective prethermal Hamiltonians but now accounting also for the slow ramp – that there is a slowly changing effective Hamiltonian that

1These are bounds that govern the speed of information spreading in local quantum many-body systems, even those that are non-relativistic.

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emerges, provided that the ramp timescale is slower than the heating timescale (which is exponentially long in the driving frequency).

Thus, while there is no strict adiabatic limit possible, there still exists quasi- adiabatic dynamics in a slowly ramped periodically driven system: beginning from the ground state of the undriven model, the system ‘adiabatically’ tracks the ground state of an instantaneous effective Hamiltonian. We also consider a ramp through a quantum critical point (QCP) – necessarily the case if we wish to induce a non-trivial change between the ground states of the undriven Hamiltonian and the effective prethermal Hamiltonian – and obtain the optimal ramp speed which minimizes the effects of both heating and the non-adiabaticity passage through the QCP. We find that this optimal ramp speed is exponentially small in the driving frequency.

Critical time crystals in dipolar systems

The understanding that many-body localization can lead to stabilization and therefore quantum order in highly excited eigenstates naturally gives rise to the question of whether Floquet many-body systems, which might heat up to infinite temperature due to the drive, can similarly be stabilized by MBL. This would potentially lead to novel, new phases of matter with no static analogue. Indeed, such a phase of matter, called the Discrete Time Crystalline (DTC) phase was the- oretically predicted and subsequently, with astounding speed, experimentally observed in a number of experiments. Such a phase of matter is characterized by a spontaneously broken discrete² time translationally symmetry, with the existence of local observables that robustly respond at periods that are an integer multiple of the driving period.

However, one of the experimental observations of DTC order was in disordered dipolar systems [29], which have long-range interactions and which therefore pre-

2Because we are periodically driving the system, the continuous time translation symmetry is reduced to a discrete time translation symmetry, given by the groupZ.

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sumably does not exhibit MBL. If MBL is not the mechanism to stabilize the DTC order, then what is the origin of the robustness of the DTC signature? In chapter 5, we answer this question: we analyze the stability of DTC order in driven, disordered systems with long-range interactions. We find that the interplay of driving, disorder and long-range interactions gives rise to ‘critically-slow’ dynamics, leading to a decay rate that is asymptotically small in the perturbing parameter. We obtain this result via a resonance counting argument of pairwise spin flips that are seen to give rise to the dominant channel for which DTC order is lost by, and for- malize it with a quasi-local rotation incorporating the effect of disorder to extract the dominant decay physics.

Our results thus provide an explanation for the mechanism of stability of the DTC order observed in the experiments. Furthermore the method we employed can be used to understand non-equilibrium phenomena in other disordered, driven long-range systems, for example in reduced dimensionalities or with different power law interactions.

Entanglement dynamics in quantum many-body systems

Lastly, in chapter 6, we study the dynamics of entanglement growth of quantum many-body systems beginning from a quench of random initial states that have zero entanglement.

We were motivated by numerical observations that the entanglement growth speed of ergodic and MBL systems are qualitatively different: in the former, entanglement across a cut is seen to grow universally linearly, S(t) ∼ t, while for MBL systems, it is seen to grow logarithmically,S(t)∼logt. An explanation of the logarithmic growth in MBL systems was provided via a dephasing picture [128], but an understanding of the linear growth in ergodic systems was not present.

We show that the spread of entanglement is intimately related to the growth of the physical size of the operators in space under time evolution. In time, an initially local operator with support localized in a region in space grows physically larger

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1.2. MANY-BODY LOCALIZATION

within its light cone dictated by Lieb-Robinson bounds. Furthermore, it becomes scrambled – it becomes not just physically larger, but is also a complicated sum of all possible basis operators within its light cone. Decreases in the measurements of certain basis operators in the system translates to an increase in entanglement, leading to entanglement growth.

We provide a toy model motivated by random matrix theory, in which local operators fully scramble within its light cone, that we believe captures the salient features of the universal linear in time growth of entanglement seen in ergodic systems. We also propose a method to experimentally measure entanglement entropy growth via a measurement of a Loschmidt echo in a replicated system setup.

The rest of the introduction provides some background material for the reader:

we explain the phenomena of many-body localization (MBL), how quantum entanglement is a useful tool to understand the nature of eigenstates of such a phase, relations to questions on ergodicity and thermalization and also the theory and methods employed to study periodically driven (Floquet) many-body systems.

1.2 Many-body localization

1.2.1 Anderson localization

In 1958, P. W. Anderson produced his seminal Nobel-prize winning paper that studied how transport and conductance in solids is affected by the presence of disorder or impurities [10]. He was motivated to explain certain experimental observations by Fletcher and Feher at Bell Laboratories of anomalously long relaxation times of electron spins in Si semiconductors doped with impurities [44, 45, 43].

What Anderson found was that in non-interacting systems of electrons moving in a disordered potential, single-particle eigenfunctions could be exponentially localized for strong enough disorder, contrary to the delocalized nature of eigenfunctions (i.e. Bloch states) at no disorder. This is the phenomenon dubbed ‘Ander-

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son localization’. Since exponentially localized electrons cannot transport heat or charge, disordered materials can thus undergo a so-called ‘Anderson transition’

from conducting to insulating at strong enough disorder strengths.

More concretely, Anderson considered a simplified model to describe Feher’s experiments, a variant of which is given by

H =X

hi,ji

t(c^†_ic_j +c^†_jc_i) +X

i

V_ic^†_ic_i. (1.1) In the above, electron-electron interactions that would be present in the experiments have been neglected, thus rendering the model a non-interacting one. Here c^†_i is the creation operator for an electron at sitei,tis the amplitude for short-range hopping, and the onsite potentialsV_i are independent and identically distributed random variables, for example drawn from a box distribution of widthW.

A heuristic analysis can be performed in the ‘locator limit’ (wheret W) by treating the hopping terms as perturbations to the onsite disorder. In this case, each unperturbed single-particle eigenstate is simply fully localized at site i, so that|ii=c^†_i|0i. Turning on the hopping terms, different single-particle eigenstates start to mix, but the degree of hybridization (and thus delocalization) depends on the ratio of the hopping amplitudetto the difference between energies on two sites

|E_i−E_j| ∼W, which could potentially be large if by chance the energy difference is small. In that case, such a process is said to be resonant. However, one can argue that due to the presence of disorder, and the fact that the hopping is weak t W, typical perturbations are off-resonant, and therefore unperturbed eigenstates a distancerapart mix with a strength that goes ast(t/W)^r ∼te^−r/ξ. Thus, this leads to Anderson’s result of localization: single-particle eigenstates have amplitudes that fall off exponentially,

|ψ_i(r)|² ∼e⁻

|r−ri|

ξ , (1.2)

which is to be contrasted to the case of a delocalized wavefunction

|ψi(r)|² ∼ 1

L^d, (1.3)

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whereLis the system size (in units of the lattice constant) anddthe dimensionality of the system. Of course, a more careful analysis requires properly taking care of the probability of dangerous resonance processes which would be detrimental to localization: Anderson’s treatment of this problem involved resumming the perturbative series to all orders to account for such delocalization processes.

Anderson’s important initial calculation sparked a (still on-going) investigation into the nature of localization-delocalization transitions due to the presence of disorder. Via the renormalization group, the theory of Anderson localization was in a milestone achievement formulated in terms of a single-parameter scaling theory [6]. In the theory, the Thouless conductance [132]

g = ET

δ = G

e²/h (1.4)

is the scaling variable, whereE_T is the Thouless energy identified by Edwards and Thouless,δis the mean level spacing, andGis a dimensionless conductance. With an upgrade of the scaling theory to an effective field-theoretic description of An- derson localization in terms of a non-linear σ model [144], more rigorous results arrived, and our current understanding of the phenomenon is as follows: in 1D and 2D, all electronic states³ are localized for arbitrarily weak disorder, implying a vanishing conductivityσ(T) = 0for any temperature T. In dimensions 3D and higher, there exists a so-calledmobility edge(introduced by Mott in [99]) in energy that separates localized and extended states, giving rise to an exponentially sup- pressed conductivityσ(T)∼e^−E^c^/T in the localized regime, whereE_cis the energy difference from the Fermi level to the mobility edge.

3Barring important cases of the inclusion of a topological term [121] in the non-linearσmodel, which is needed to explain the Quantum Hall effect in 2D. Such systems necessarily imply the existence of some extended bulk states, contrary to the Anderson prediction of all eigenstates being localized.

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1.2.2 Basko, Aleiner and Altshuler (BAA)

A very natural question that arises from the previous discussion is: does the phenomenon of Anderson localization survive in the presence of weak, short-range interactions? After all, the original experimental setup of Fejer’s involved interacting spins in a disordered potential, but the Anderson model treats the system in a non-interacting way. To what extent does a notion of localization extend also in the many-body case⁴?

Such a question, while almost instantaneously considered since the inception of the non-interacting Anderson localization, remained an unresolved issue till the work of Basko, Aleiner and Altshuler (BAA) in 2006 [14], although there was some notable progress made by Fleishman and Anderson in [48].

BAA’s treatment of the many-body problem involved considering the effect of many-body interactions to localized single particle eigenstates rigorously in all orders of perturbation theory (a very difficult calculation indeed!). They predicted a many-body mobility edge separating localized and extended eigenstates in the many-body system, with implications that the conductivity is strictly zero σ(T) = 0for temperatures below some finite critical temperatureT_c(or more precisely, energy density, since temperature is an ill-defined concept without a bath).

There is thus a many-body metal-insulator transition separating the localized (non- conducting) and delocalized (conducting) regimes.

We do not present BAA’s formidable calculation here, but only very roughly sketch an outline. BAA considered a Hamiltonian of the form

H =X

α

ξ_ac^†_αc_α+X

αβγδ

V_αβγδc^†_αc^†_βc_γc_δ, (1.5) where c^†_α creates a localized single particle eigenstate with energy ξ_α, localization centerr_α and localization length ζ. The unperturbed many-body eigenstates are the Fock states |n_α₁, n_α₂, n_α₃,· · · i, n_α_i = 0,1 of the single particle Hamiltonian.

V_αβγδ is a short range electron-electron interaction which is treated as a perturba-

4We consider systems which are isolated, that is, not coupled to an external bath.

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tion, with matrix elements∼λδ_ζ when

|r_α−r_β|.ζ,

|ξ_α−ξ_δ|,|ξ_β−ξ_γ|.δ_ζ, or

|ξ_α−ξ_γ|,|ξ_β −ξ_δ|.δ_ζ, (1.6) whereδ_ζ = (νζ^d)⁻¹ ∼O(1)is the so-called local spectral gap within a region of size of the localization lengthζ.

In perturbation theory, repeated application of the perturbation V leads to a process where a particle-excitationα decays into some hole excitations and some particle excitations, and so on. Once again, similarly to the Anderson case, if the matrix element for such a process is comparable to the energy difference between the connected states then hybridization (and therefore delocalization) occurs in the Fock space. BAA’s difficult calculation involved using the self-consistent Born ap- proximation to obtain the imaginary part of the single-particle self energy ImΣto all orders in perturbation theory, and they found that the conductivity was strictly zero, σ(T) = 0, even for finite energy densities T_c > 0. This thus provided very strong evidence for the phenomenon of many-body localization.

1.2.3 Canonical model

Impressive as their perturbative calculation is, the model (1.5) that BAA utilized was not the most amenable to numerical simulations. The use of numerical methods is necessary, for example to go beyond the peturbative limit and investigate the transition, which is where perturbation theory breaks down. To that end, Pal and Huse [107] therefore instead considered a 1D interacting spin-1/2model of a Heisenberg model with random on-site fields,

H =X

i

J(S~_i·S~_i+1+h_iS_i^z), (1.7) where S~_i are spin-1/2 Pauli matrices and h_i are independent, identical random variables drawn from a box distribution[−h, h]. Note that such a model is map- pable via a Jordan-Wigner transformation to a fermionic Anderson model with

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nearest-neighbor Hubbard interactions. In the literature on MBL, the spin model has become the canonical, or ‘Ising model’ of studies of MBL, and is the model of choice for both theoretical and numerical work [107, 152, 13, 129].

Let us first heuristically describe some salient features of two limits of the model: (h/J) → ∞ and (h/J) → 0⁺. In the former limit, we can essentially ig- nore the Heisenberg part of the Hamiltonian, leaving us with H = P

ih_iS_i^z. So, the many-body eigenstates of (1.7) are simply product states in theS^z-basis, of the form|si = | ↑↑↓ · · · i, wheres represents a spin configuration. Upon turning on smallJ, there will be some hybridization of these spin configuration states. How- ever the calculation of BAA [14] suggests that ifJ h, due to the disorder,J will off-resonantly coupled different spin configurations and so that new many-body eigenstates are all⁵ still ‘close’ to the spin configurations |si. Conversely, in the latter limit, hybridization is so strong that many-body eigenstates are essentially delocalized over the space of all spin configuration states |si, so in between, the model must exhibit a many-body localization transition at some finite value of the disorder strength h_c. Many studies have pinpointed the MBL transition to be at h_c ' 3.5[107, 92, 129]. Note that the point (h/J) = 0 is singular as the Heisen- berg model without random fields is solvable via the Bethe ansatz, which renders the model integrable and non-generic. However, any weak but finite disorder is believed to break that integrability and leads to ergodicity.

What is the nature of the many-body localization phase transition? It is a quantum phase transition (QPTs) even at finite energy density, in which the nature of each eigenstate changes singularly⁶. For example, an intuitive way to diagnose the two sides of the phase diagram would be to calculate the degree of delocalization of energy eigenstates over theS^z-eigenbasis, with such states given by spin configurations|si. A measure could be the inverse participation ratio (IPR) of momentq

5By all, we therefore mean atinfinitetemperature.

6This quantum phase transition is not a thermodynamic one as equilibrium thermodynamic observables (i.e. measuring observables with respect to ensemble of eigenstates in some energy window) do not exhibit singularities as one crosseshc.

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[92], defined for an eigenstate|niwith energyE_nas S_IPR^(q)(|ni) = 1

1−q lnX

s

p^q_s, (1.8)

spslnps. Forh ≤ hc, eigenstates are found [92] to be delocalized with scaling of the IPR

S_IPR^(q) =a_qln(dimH), (1.9) wherea_q∼1for allq, dimHis the dimension of the Hilbert spaceHwhich is equal to2^Lfor a spin chain of lengthL, while forh > h_c, eigenstates are found to have a scaling behavior either with a behavior similar to above but witha_q 1ora_q = 0 and a slow logarithmic divergence

S_IPR^(q) =lqln(lndimH), (1.10) indicating a non-trivial multifractal behavior.

Of course, there are many other diagnostics that can be used, one popular one of which is to look at the spectral statisticss_n = (E_n+1−E_n)of adjacent energy lev- els in a many-body Hamiltonain [107]. In the infinite-randomness limit h_c → ∞, the Hamiltonian reduces toH =P

ih_iS_i^z, and a single spin flip from a given configuration costs energy∼ hwhich is much larger than the many-body level spacing

∼2^−L. Therefore nearby eigenstates in energy involve a large number of spin flips

∼ O(L), and so these eigenstates are spatially very dissimilar and therefore do not interact much. The energy spacingss_nare therefore expected to follow a Pois- sonian distribution which indicates a lack of level repulsion (i.e. the distribution P(s)of s_n has lims→0P(s) > 0). Conversely, in the h_c → 0⁺ limit, highly excited eigenstates essentially look like random vectors described by random matrix theory, implying that level statistics should follow that of one of the random matrix ensembles, with lim_s→0P(s) = s^β for some β > 0indicating level repulsion. For the case of the canonical model (1.7) which has an antiunitary symmetry implying time reversal symmetry, the ensemble is given by the Gaussian Orthogonal Ensem- ble (GOE) with β = 1. The mean value of the level spacing distributions of these

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two limiting distributions serves as a single number that can be used to distinguish between the localized and delocalized regimes.

Figure 1.1: Numerically obtained phase diagram of (1.7), as a function of energy densityand disorder strengthh. The transition into full MBL occurs ath_c ∼ 3.5. There is a many-body mobility edge for smaller values of h. Figure taken from [92].

Using such measures and more, state of the art exact diagonalization of up to L= 22spins has mapped out the phase diagram of (1.7) as a function of the energy densityand disorder strengthh, seen in Fig. 1.1.

1.2.4 Local integrals of motion, and phase of matter

The Hamiltonian (1.7) is very useful for numerical simulations, but the observation thatalleigenstates look ‘similar’ to physical spin configurations|siin the fully MBL (FMBL) regime suggests the following phenomenological picture: There exists anemergent, extensive number of local integrals of motion (LIOMs) (more precisely, quasi-local) which all commute amongst themselves and with the Hamilto-

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nianH, and which are just dressed versions of physical spins [127, 65].

In other words, an LIOM τ (also called an l-bit, l for local) can be expanded in terms of the physical degrees of freedomσ (also called p-bits, p for physical) as

τ_i^z =X

j

K_ij^ασ_j^α+K_ijk^αβσ_j^ασ_k^β +· · · (1.11) where α = 0, x, y, z,, and K_i^α₁¹_i₂^α_i²₃^α_···i³^···α_m ^m have magnitudes that decay with the maximum separation between any two spins in the set{i₁, i₂, i₃,· · ·i_m}. Thus, the support of the operatorτ_i^z is mostly localized on siteiand decays exponentially away from it.

Written in terms of the LIOMs, the Hamiltonian then takes the form H =X

i

h_iτ_i^z+X

ij

J_ijτ_i^zτ_j^z+X

ijk

J_ijkτ_i^zτ_j^zτ_k^z+· · · (1.12) whereJ_i₁_i₂···im have magnitudes which similary toK_i₁_i₂···im decay exponentially in the maximum separation between any two spins in the set{i1, i2, i3,· · ·im}. There- fore, this Hamiltonian is diagonal and quasi-local, and many-body eigenstates are simply product states in theτ^z basis, which are weakly rotated from a parent spin configuration basis|si. Note that in the non-interacting Anderson limit, onlyh_i’s are non-zero, while interaction termsJ’s are all zero.

We see that the dynamics of this Hamiltonian is straightforward (but non- trivial): each τ_i^α operator precesses about the z-axis set by the effective field ex- perienced at site i, which is dependent on the interactions J and the state of the other LIOMs. This leads to interesting predictions about dynamics such as the logarithmic in time growth of entanglement as we will see below [13]. Moreoever, sinceτ_i^zs all commute with the Hamiltonian, MBL systems retain memory of their local initial conditions for an infinitely long time, raising intriguing possibilities of them being used for novel platforms for quantum memory [130].

This phenomenological picture of LIOMs or l-bits was argued for in [127, 65], and has proven to be extremely useful in analytical calculations of various quantities in fully MBL systems. Furthermore, a proof that MBL systems indeed take the

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form (1.12) was mathematically given⁷in [68], lending even more credence to this picture.

One should also note that the presence of these LIOMs arerobust. That is to say, if one perturbs the original fully MBL Hamiltonian (1.12) by a local perturbation V, so that

H →H⁰ =H+V, (1.13)

then if the perturbation is weak enough, H⁰ will still be described by the phenomenology of (1.12), but with modifiedK, h_i, andJ in the expansion of the LI- OMs (1.11) and the Hamiltonian. Thus, the MBL prescription describes aphase of mattercharacterized by robust emergent integrability, leading to a lack of transport and therefore infinitely persistent memory of local initial conditions.

At this stage, let us restate the phenomenology of FMBL in slightly more con- crete terms. Let us be given a Hamiltonian written in terms of physical spin vari- ablesH(σ)such as (1.7). Then the FMBL regime is characterized by the existence of a (not necessarily unique) quasi-local unitaryU(σ), such that there is an extensive number of emergent quasi-local operators

τ_i^α(σ)≡U(σ)^†σ_i^αU(σ), (1.14) and the Hamiltonian can be re-expressed in terms of theτ variables⁸

H(σ) = H(σ(τ)) =X

i

h_iτ_i^z+X

ij

J_ijτ_i^zτ_j^z +· · · , (1.15)

7Well, almost, barring one very physical assumption about the amount of level attraction in quantum many-body systems.

8To express σ in terms of τ, as in σ(τ) = · · ·, is a straightforward albeit convoluted af- fair: from (1.14), one can define the map between the space of all tensor products of operators of τ and all tensor products of operators of σ, i.e. (τ_i^α₁¹τ_i^α₂²· · ·) = U(σ)^†(σ_i^α₁¹σ^α_i₂²· · ·)U(σ).

Furthermore we can parameterize U(σ) in terms of its Hermitian generators σ and their real coefficients c, via U(σ) = U(c, σ) = exp(−iPc^α_i₁¹_i^α2₂_···^···(σ_i^α₁¹σ^α_i₂²· · ·)). Then we have (τ_i^α₁¹τ_i^α₂²· · ·) = P

(β;j)R(α₁α₂···;i₁i₂···),(β₁β₂···;j₁j₂···)(σ^β1_j₁σ_j^β₂²· · ·)whereR is the rotation matrix associated with the adjoint action ofU(σ). Multiplying both sides byR^T then gives(σ_i^α₁¹σ^α_i₂²· · ·) = U(c, τ)(τ_i^α₁¹τ_i^α₂²· · ·)U(c, τ)^†, which impliesσ^α_i =U(c, τ)τ_i^αU(c, τ)^†andU(σ(τ)) =U(c, τ).

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1.3. QUANTUM THERMALIZATION AND ERGODICITY

so thatτ_i^zs are the quasi-local integrals of motion

[H, τ_i^z] = 0 (1.16)

for all i. Quasi-locality means that the expansionτ_i^α(σ)of (1.14) in theσ operator space is localized at sitei. Eigenstates ofH(σ)are then given by

|ni=U(σ)^†|si, (1.17)

which we see are just small rotations of the physical spin configuration |si if the disorder is strong.

The requirement of quasi-locality ofU(σ)is important: if we relax such a condition, then we can always find many integrals of motion, such as the 2^L projec- tors onto eigenstates |nihn|. However, that is obviously not a very meaningful statement as this will be true for any quantum mechanical system. The property of quasi-locality therefore imposes a stringent condition to satisfy and leads to a sharp distinction between FMBL and not-FMBL phenomenology.

1.3 Quantum thermalization and ergodicity

Thus far, we have described the MBL phase as one that is characterized by a lack of transport and infinitely persistent memory of local initial conditions. However, such characteristics are rather operational, and we would like a more fundamental and more abstract understanding of the crux of the phenomenon of many-body localization. It turns out that MBL should be thought of as a noveldynamicalphase of matter, with the crucial feature that it is nonergodic.

In the fields of quantum dynamics and quantum statistical mechanics (see [104]

for a review), a very general and important, yet not very well understood question that can be asked is the following: what is the fate of generic, interacting, isolated quantum many-body systems? More precisely, if one prepares an out of equilibrium state (by quenching the system for example) and let it evolve under its own dynamics, does the systemequilibrate after some long time? In other words, does it

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reach a steady state? Even more stringently, does itthermalize, i.e. can expectation values of microscopic variables be described, in this limit, by some small number of macroscopic variables relating to global conserved quantities in the system such as energy or particle number?

Of course first we have to clarify what it means for equilibration and thermalization to take place in the context of isolated systems, i.e. without coupling to a bath. After all, quantum dynamics is unitary, so information preserving; the full density matrix of the system just picks up phase factors when written in the energy eigenbasis|ni:

ρ(t) = X

nm

c_nm|nihm|e^−i(Eⁿ^−E^m^)t, (1.18) and clearly does not have a steady state limitlim_t→∞ρ(t)6=ρ_eq.

However, the key point is that while indeed all the quantum information is preserved within ρ(t), the information gets scrambled between different degrees of freedom over time, some of which could be macroscopically physically large (i.e. global variables with support on the order of the size of the system). If we are only allowed to perform local measurements on only a few degrees of freedom, as is the case of the physical world, then we will be unable to recover all the quantum information. Then, measurements of local operators can indeed reach a steady limit, which is equilibriation. Thus the correct way to think about this dynamics is to consider partitioning the isolated system in to two parts, Aand B, where Ais a small subsystem compared toB, such that the ratio of the number of degrees of freedom between the two is vanishingly small in the thermodynamic limit. Then we only consider measurements restricted toA, which amounts to tracing out the contributions ofBto give the reduced density matrix onA

ρ_A(t) = Tr_B(ρ(t)) = Tr_B(U(t)ρ(0)U^†(t)), (1.19) which could have a steady state limit limt→∞limL→∞ρ_A(t) = ρ_A,eq. If in addition this limit is given by the equilibrium Gibbs ensemble

ρ_A,eq = 1

ZTrB e^−βH

, (1.20)

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where the inverse temperatureβ is obtained implicitly from the energy density of the initial state (which is conserved) via the following equation

Tr(Hρ(0)) = 1

ZTr He^−βH

, (1.21)

then we say the system is thermalizing and hence has ergodic dynamics.

The link of quantum dynamics to quantum statistical mechanical mechanics emerges when we realize that if (1.20) is true, then essentially the subsystemB is serving as a bath forAallowing it to thermalize, even though the combined system A∪B is isolated. The original question that we posed at the beginning can then also be understood as:Does an interacting quantum many-body system serve as its own heat bath? For a long time, it was assumed that generic interacting systems are always ergodic and thermal – that a statistical mechanical description holds given long enough times. However, the discovery of MBL reveals that this assumption is not correct. The MBL phase is therefore to be considered as a new dynamical phase of matter, characterized a lack of ergodicity, a failure to thermalize, and an inability for quantum information to be scrambled globally in the system.

1.3.1 Eigenstate thermalization hypothesis (ETH)

What implications are there for a quantum mechanical many-body system that does thermalize? It turns out that one way of reconciling quantum statistical mechanics and the dynamics of quantum thermalization involves imposing strong and surprising constraints on the nature of eigenstates and observables in such systems, captured by the eigenstate thermalization hypothesis (ETH).

Consider a pure quantum state of a thermodynamically large many-body system, which is in a superposition of energy eigenstates|nithat have energiesEn,

|ψ(0)i=X

n

c_n|ni, (1.22)

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so that it has a mean energyE¯ and small energy uncertainty∆, E¯ =X

n

|c_n|²E_n

∆ = q

hψ(0)|H²|ψ(0)i −E¯² E.¯ (1.23) The measurement of a local operatorOin time is

hψ(t)|O|ψ(t)i=X

n

|c_n|²Oⁿⁿ+X

n6=m

c^∗_nc_me^−i(E^m^−Eⁿ^)tO^nm. (1.24) In the infinite time limit, the off-diagonal terms dephase, interfere, and cancel, and so the measurement should reach a steady value given by the time-average

hOi= lim

T→∞

1 T

Z T 0

dthψ(t)|O|ψ(t)i=X

n

|cn|²Oⁿⁿ. (1.25) On the other hand, if the system is thermalizing, then the expectation value ofO can be described well by quantum statistical mechanics, as discussed previously.

In this case,hOican be equally well calculated from hOi= 1

ZTr Oe^−βH

Canonical ensemble, (1.26) with inverse temperature fixed byE¯, or

hOi= 1

NTr O

N

X

n=1

|nihn|

!

Micro-canonical ensemble, (1.27) where the sum is over theN states in the small energy window[ ¯E−∆,E¯+ ∆].

If we concentrate on the latter, we get hOi=X

n

|c_n|²Oⁿⁿ =

N

X

n=1

1

NOⁿⁿ. (1.28)

How can this expression be equal for generic initial states that have different values coefficientsc_nbut with similarE¯ and∆?

One way this can be made equal is if the matrix elements Onn form a slowly varying function of energy on the order of ∆, Oⁿⁿ ∼ O( ¯E), and so equality can be achieved since P

n|c_n|² = 1 = PN n=1

1

N. Coupled with predictions on how far

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the instantaneous value of the observable should deviate from the thermal or equilibrium value in the long time limit leads to Srednicki’s eigenstate thermalization hypothesis (ETH) for the matrix elements of local operators [35, 131]:

hn|O|mi ≡ O^nm=O( ¯E)δ_nm+e^{−S( ¯}^E)/2fO( ¯E, ω)R_nm, (1.29) where in this situationE¯ ≡ (E_n+E_m)/2, ω ≡ E_n−E_m, and S(E)is the thermodynamic entropy at energyE. O( ¯E)is a smooth slowly varying function that only depends on the average energy, while fO( ¯E, ω)is a smooth, function that varies slowly with E¯ but can vary quickly with ω. Rnm is a random real or complex number with zero mean and unit variance, which should be understood in the statistical context of all pairs of eigenstates, since a given realization of a Hamiltonian does not yield probabilistic eigenstates.

Figure 1.2: Figure from [49]. Trace norm distance between the two density matrices of the LHS and RHS of (1.31), for various system sizesL. (Left): Trace norm distance for all eigenstates in some inverse temperature windowβfor a subsystem of sizeL_A = 5. (Right): Scaling of mean and standard deviation of trace norm distance withL, for various subsystem sizesL_A. For fixedL_A, one sees a decrease in the distance asL(or equivalently, the size of the bathB) is increased: this provides evidence that the equality (1.31) holds in the thermodynamic limit.

Another formulation of the ETH which constrains the nature of eigenstates is to take the limiting case when |ψ(0)i is indeed just an eigenstate |ni. Then,

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hψ(t)|O|ψ(t)i=hn|O|niis a static quantity. But if one assumes that in a thermalizing system all initial states eventually reach thermal equilibrium, which includes the special case of a single eigenstate⁹, then it must mean that the expectation value of a local operatorOis given by

hn|O|ni= 1

ZTr Oe^−βH

, (1.30)

withβimplictly defined fromhn|H|ni= Tr He^−βH

/Z. However,Ocould be any generic local operator. In particular, if (1.30) is true for all basis operators in some small subregion A, then the equality must hold at the level of reduced density matrices of this small subregion in the thermodynamic limit:

ρ_A(|ni) = Tr_B|nihn|= 1

ZTr_B e^−βH

, (1.31)

which is a very strong statement about the character of individual eigenstates of a thermalizing system.

It important to note that the ETH is a hypothesis, with no rigorous analytical understanding of its validity or applicability. Numerically though, it has been tested extensively and seen to hold in systems where thermalization holds. For example, Fig. 1.2 describes numerical work in probing (1.31): indeed, they see that equality of the equation, measured via the trace norm distance of two density matrices, is achieved in the thermodynamic limit.

Of course, the ETH does not hold true in systems which do not thermalize.

Connecting to the previous discussion on MBL and ergodicity, we see that yet another characterization of the MBL phase is the breakdown of the eigenstate thermalization hypothesis.

9The assumption of all eigenstates being thermal leads to the notion ofstrongETH[76]; if almost all eigenstates are thermal then we haveweakETH.

Localization and slow thermalization in quantum many-body systems

Thesis

Reference

Localization and slow thermalization in quantum many-body systems

Localization and Slow Thermalization in Quantum Many-body Systems

TH `ESE

présentée à la Faculté des Sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention Physique

par

Wen Wei Ho

de Singapour

Th`ese No. 5083

GEN `EVE

Atelier de reproduction ReproMail

Juin 2017

Abstract

Abstrait

Acknowledgments

Contents

Chapter 1

Introduction

1.1 Motivation and results

1.2 Many-body localization

1.2.1 Anderson localization

1.2.2 Basko, Aleiner and Altshuler (BAA)

1.2.3 Canonical model

1.2.4 Local integrals of motion, and phase of matter

1.3 Quantum thermalization and ergodicity

1.3.1 Eigenstate thermalization hypothesis (ETH)