Many-body localization

Arijeet Pal

A Dissertation

Presented to the Faculty

of Princeton University

in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance

by the Department of

Physics

Adviser: Professor David A. Huse

September 2012

c© Copyright by Arijeet Pal, 2012.

All rights reserved.

Abstract

A system of interacting degrees of freedom in the presence of disorder hosts a vari-

ety of fascinating phenomena. Disorder itself has led to the striking pheonomena of

localization of classical waves and non-interacting quantum mechanical particles.

There are even phase transitons (like the glass transition) which are driven largely

due the effects of disorder. The work in this dissertation primarily addresses the

interplay of interactions and disorder for the fate of ergodicity in classical and quan-

tum systems. We specifically question the assumption of ergodicity in a generic,

isolated spin-system with interactions and disorder in the absence of coupling to an

external heat bath. Our results predict the existence of a novel phase transition at

finite temperature (even at ‘infinite’ temperature) in the quantum regime driven

by the strength of disorder. At relatively low disorder in the ergodic phase, an

isolated quantum system can serve as its own heat bath allowing any subsystem

to thermalize. While at strong disorder due to the localization of excitations, the

isolated system fails to serve as a heat bath. In the limit of infinite system size,

there is a quantum phase transition between the two phases with the critical point

showing infinite-randomness like scaling properties. Based on our conventional un-

derstanding, the low frequency dynamics of quantum systems at finite temperature

are often describable in terms of an effective classical model. With this motiva-

tion in mind, we also studied the dynamics of an interacting, disordered classical

spin-model. Our results exclude the possibility of many-body localization in clas-

sical systems. A classical many-body system at strong enough disorder becomes

chaotic under the dynamics of its own hamiltonian thus converging to thermal

iii

equilibrium at long times. Hence, many-body localization is a macroscopic quan-

tum phenomenon at extensive energies without a classical counterpart.

iv

Acknowledgements

A PhD dissertation is the culmination of five formative years of one’s life. It means

much more than just the hundred odd pages of words and figures which are part

of its final form. There are many stories told and lessons taught which can only

be read in between the lines. And countless people are a part of this experience

which can hardly be captured in this section. First and foremost, I would like to

thank my adviser, Professor Huse. Any words of appreciation will fall short of

his actual contribution to the work and my experience in graduate school. I am

sure his acumen as a physicist has been appreciated by many but as his student,

I can vouch that he is also a wonderful teacher of subtle and intricate ideas. The

extreme care with which he chooses his words is a rare quality to find. Starting

from the summer of 2006 as an undergraduate till now, he has introduced me to

the art of research and taken me through its various rigmaroles quite seamlessly.

Without his boundless patience and constant encouragement, my introduction to

a career in physics would be an entirely different experience. I also appreciate that

he introduced me to the beautiful problem of many-body localization at a very

early stage in graduate school. Exploring the cracks and corners of this problem

with him has indeed been a great learning and enriching experience.

I would also like to thank Professor Sondhi for fascilitating my first sojourn to

Princeton as a summer student. Over the years his advice on matters of impor-

tance have been invaluable. Without his support my move into Princeton, and now

as I leave the place, would be quite a different story. I hope to continue working

on problems we have identified and look forward to further scientific interactions.

The experience of working in Professor Hasan’s lab as a beginning graduate stu-

v

dent gave a really good perception of the nature of research in Condensed matter

physics. I would like to thank him for giving me the chance to work on topological

insulators for my experimental project when the field was in its infancy. I also

appreciate the advice and support he provided on taking the next step outside

graduate school. I would also like to thank Vadim for his support and useful sug-

gestions not just on physics but academic-life in general. I always eagerly looked

forward to his trips from New York and the interesting discussions they led to. I

hope to continue this discourse in the future.

At a personal level, graduate school has given me some great friends to cherish

for the years to come. The many hours spent in Jadwin would have seemed much

longer without discussions with Hans, Miro, BingKan, Anand, Anushya, Bo, Chris,

Charles, Sid, Hyungwon on physics and other random thoughts. Then there was

also the life outside Jadwin. Sharing the sentiments of winning and losing on the

soccer field with Pablo, John, Eduardo, Pegor and many others can hardly be

replicated outside the sports field. Finding the right tennis partners in Richard,

Bo and Hans helped me fulfil my childhood desire to play the sport. I also spent

4 memorable years in 3V Magie with Darren and Ketra who were my partners in

crime on many occassions from Bollywood choreography at Dbar to cooking meals

for friends. It was always comforting to know that on occassions when I needed a

‘break ’ from physics, I was only a walk away from interesting conversations over

lunch or coffee with Rohit D, Rohit L, Anna, Radha, Vinay, Franziska, Rotem and

Udi. Last but definitely not the least, my impressions on Princeton will be far

from complete without Sare’s companionship and her always being there when I

needed.

vi

Then there are the people outside Princeton who had as much influence. On

this occassion, I would also like to thank my mentors in Boys’ High School, Dr.

Aditi Mukhopadhyaya and St.Stephen’s College, Dr. Bikram Phookun. They

channeled my youthful exuberance and gave form to my random ideas. The

late-night, light-hearted conversations with Ranit, one of my closest friends from

yesteryears, gave a lot more perspective on ‘life’ than we had expected! I literally

cannot describe in words the contribution of my parents, Sripati and Paulina and,

brother and sister-in-law, Shubhojit and Manpreet. Without their efforts, reach-

ing this stage of my life would not just be impossible but inconceivable. Had it

not been for the train journey from Guwahati to Allahabad, I would very well be

telling a different story.

vii

To my parents.

viii

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1 Introduction 1

1.1 Scaling theory for Anderson transition . . . . . . . . . . . . . . . . 7

1.2 Random Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Ergodicity (Thermalization) . . . . . . . . . . . . . . . . . . . . . . 13

1.3.1 Classical Chaos . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.2 Berry’s conjecture (Quantum Chaology) . . . . . . . . . . . 16

1.3.3 Eigenstate Thermalization Hypothesis . . . . . . . . . . . . 18

1.4 Disorder and Interactions . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.1 Variable range hopping . . . . . . . . . . . . . . . . . . . . . 21

1.4.2 Fermi glass . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.4.3 Many-Body Localization: Basko, Aleiner, Altshuler (BAA) . 25

1.5 Possible signatures in experiments . . . . . . . . . . . . . . . . . . . 30

1.6 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

ix

2 The quantum many-body localization 37

2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2 Single-site observable . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3 Transport of conserved quantities . . . . . . . . . . . . . . . . . . . 47

2.4 Energy-level statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.5 Spatial correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.6 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.7 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3 Energy transport in disordered classical spin chains 78

3.1 Classical many-body localization? . . . . . . . . . . . . . . . . . . . 78

3.2 Model, trajectories and transport . . . . . . . . . . . . . . . . . . . 83

3.2.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.2.2 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.2.3 Finite-size and finite-time effects . . . . . . . . . . . . . . . 89

3.3 Results: Macroscopic diffusion . . . . . . . . . . . . . . . . . . . . . 90

3.3.1 Current autocorrelations . . . . . . . . . . . . . . . . . . . . 90

3.3.2 DC conductivity: extrapolations and fits . . . . . . . . . . . 92

3.4 Further explorations and outlook . . . . . . . . . . . . . . . . . . . 97

3.5 Finite size effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.6 Chaos amplification of round-off errors . . . . . . . . . . . . . . . . 102

3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4 Conclusion and Future outlook 106

4.1 Question of Universality . . . . . . . . . . . . . . . . . . . . . . . . 108

x

4.2 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.3 Topological order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.4 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Bibliography 111

xi

List of Tables

2.1 Properties of the ergodic and localized phases . . . . . . . . . . . . 41

3.1 Estimates of the D.C. conductivity κ . . . . . . . . . . . . . . . . . 98

xii

List of Figures

1.1 Disorder in phosphorus doped silicon . . . . . . . . . . . . . . . . . 3

1.2 ESR measurements of p-doped silicon . . . . . . . . . . . . . . . . . 4

1.3 A typical diagrammatic term in the locator expansion . . . . . . . . 6

1.4 Scaling function for single-particle localization . . . . . . . . . . . . 10

1.5 Non-equilibrium initial conditions . . . . . . . . . . . . . . . . . . . 13

1.6 Chaotic and regular trajectories . . . . . . . . . . . . . . . . . . . . 16

1.7 Variable-range hopping between localized energy-levels . . . . . . . 21

1.8 BAA phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.9 Probability distribution of the relaxation rate (Γ) . . . . . . . . . . 28

1.10 Density profile of Anderson-localized condensate . . . . . . . . . . . 32

1.11 Schematic diagram of coupled SC qubits in microwave resonator . . 33

1.12 I-V characteristic of InOx thin film . . . . . . . . . . . . . . . . . . 34

2.1 Phase diagram of many-body localization transition . . . . . . . . . 41

2.2 Decoupled precessing spins . . . . . . . . . . . . . . . . . . . . . . . 43

2.3 Difference of m(n)iα between adjacent eigenstates vs L . . . . . . . . . 45

2.4 Probability distribution of m(n)iα . . . . . . . . . . . . . . . . . . . . 47

2.5 Probability distribution of |m(n)iα −m

(n+1)iα | . . . . . . . . . . . . . . 48

xiii

2.6 Dynamic part of initial spin-polarization . . . . . . . . . . . . . . . 50

2.7 Probability distribution of r(n) . . . . . . . . . . . . . . . . . . . . . 52

2.8 Ratio of adjacent energy gaps . . . . . . . . . . . . . . . . . . . . . 55

2.9 Spin-spin correlation (Czznα) in energy eigenstates . . . . . . . . . . . 56

2.10 Measure of anti-correlation at long distances . . . . . . . . . . . . . 59

2.11 Probability distributions of lnCzznα(i, i+ L/2) . . . . . . . . . . . . . 60

2.12 Scaled width of the distribution of lnCzznα(i, i+ L/2) . . . . . . . . . 62

2.13 Level spacing and ET in the ergodic and localized phases . . . . . . 64

2.14 Contribution to the dynamic fraction from adjacent eigenstates (P (n)α ) 65

2.15 Probability distribution of P (n)α . . . . . . . . . . . . . . . . . . . . 67

2.16 System + bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.17 Entanglement entropy of energy eigenstates vs L/2 . . . . . . . . . 75

2.18 Entanglement spectrum of energy eigenstates . . . . . . . . . . . . . 76

3.1 Diffusion constant vs relative spin-spin interaction strength . . . . . 82

3.2 Short time behavior of current autocorrelation C(t) . . . . . . . . . 92

3.3 Current autocorrelations on medium time scales . . . . . . . . . . . 93

3.4 Long-time tail of the current autocorrelation function . . . . . . . . 94

3.5 Estimation of the exponent of long-time tails . . . . . . . . . . . . . 95

3.6 Long time tails in terms of η(t) ≡ κL(t)− κL(2t) . . . . . . . . . . . 96

3.7 Variation of the D.C. conductivity κ(t) vs t . . . . . . . . . . . . . . 97

3.8 Rescaled κ(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.9 Long time limit of κ(t) . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.10 Finite-size effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.11 Round-off effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

xiv

Chapter 1

Introduction

Understanding the role of disorder in natural phenomena has been one of the most

puzzling questions in physical sciences which has received its requisite attention

only in the past few decades. Given the ubiquitous nature of disorder, it is im-

portant to understand if disorder fundamentally changes our predictions which

are usually based on idealized and clean theoretical models. Although disorder is

present on all scales, it is interesting to note, its significance for current empirical

observations is probably most pronounced in condensed matter physics. In con-

densed matter physics itself, this paradigm was brought to the forefront by the

seminal work of P. W. Anderson (1958) [1], where he was able to show that the

quantum mechanical wavefunction of a non-interacting particle is exponentially

localized at all energies for sufficiently strong but finite disorder. As the localized

states do not carry currents over macroscopic length scales hence, this had dra-

matic consequences for the transport properties of a material. Thus, a complete

1

description of transport in solid-state systems requires taking into consideration

effects due to disorder on an equal footing. At the time of Anderson’s paper, this

result was a paradigm shift from the conventional way of thinking. It took some

time before the community grew to realize the significance of this work. Neville

Mott and David Thouless were probably some of the first people to understand

the impact of this work and found its connection to physical realizations of metal-

insulator transitions.

Anderson’s theoretical work at that time was motivated by experiments per-

formed in George Feher’s group at the Bell Laboratories [2–4]. They were particu-

larly interested in the phenomenon of spin relaxation in phosphorus doped silicon

using electron spin resonance techniques. The electronic wavefunction localized

on a phosphorus atom in doped silicon has a Bohr radius of ∼ 20 Å. The electron

in this state felt the random environment of Si29 defects in Si28. The relaxation

time of the spins on these donor atoms was of the order of minutes as opposed

to milliseconds which was predicted by theoretical calculations based on Fermi

Golden Rule taking into account phonons and spin-spin interactions.

At low dopant concentration, the electron spin resonance signal is inhomo-

geneously broadened while as the dopant concentration increased the signal is

homogeneously broadened signifying the localization transition.

Anderson was considering energy transport in this spin system and conceptu-

alized it as a collection of interacting spins in a disordered environment. This is

in general an interacting (nonlinear) problem. In order to capture the essential

physics he made the “ linear ” approximation which made the problem tractable

compared to the fully interacting case. The hamiltonian for this simplified model

2

Figure 1.1: The electron on the donor phosphorus impurities are bound in a hy-drogenic wavefunction with a large Bohr radius. The Si environment is slightlyimpure due to the presence of Si29. (Figure from [5])

can be written as

H =∑

i

Eini +∑

i,j

Vij c†i cj + h.c. (1.1)

where c†i is the creation operator on a localized state at site i. It is important

to note that when this hamiltonian is expressed in terms of spin operators through

the Jordan-Wigner transformation, it amounts to neglecting the Szi Szj term in

the hamiltonian. In terms of annhilation-creation operators this implies that the

model is linear and there are no interactions between the occupied states on the

sites. In a clean system with uniform nearest neighbor hopping, this hamiltonian

is diagonalized by Bloch states.

ψ(~k) =1√N

∑

i

ei~k·~ri|i〉 (1.2)

3

Figure 1.2: Electron spin resonance signal for different phosporus concentrationacross the Anderson transition. (a) In the localized phase (d) In the delocalizedphase. (Plot from [3])

4

A perturbation theory setup in three dimensions or higher using the bloch states

as the unperturbed part while treating the disorder as a perturbation though qual-

itatively changes the transport from ballistic to diffusive but the states still remain

delocalized. While if the perturbation theory is performed in the localized limit

where the unperturbed states are eigenstates of H0 =∑

iEini while the hopping

is a perturbation (technically referred to as the locator expansion), the localized

states remain stable in the presence of the hopping terms at strong disorder. In

one or two dimensions all states are localized for arbitrarily small disorder (with

time-reversal symmetry and without spin-orbit coupling).

Let |i〉 be a localized state at site i. A general single-particle state can be

represented as

ψ =∑

i

ai|i〉

The dynamics of the amplitudes are governed by the Schrodinger time-

evolution.

iaj = Ejaj +∑

k 6=j

Vjkak

If we initialize our system with the particle localized at site i, the question of

localization is related to the t → ∞ limit of the probability amplitude ai. If the

system is localized at that energy the probability at site i remains finite while in

the delocalized state it diffuses and decays to zero. This can be formulated in

terms of the Green’s function

Gij(t) = 〈i|eiHtc†je−iHtci|i〉

5

Thus, the problem becomes amenable to a perturbative treatment where the

basis states are the eigenstates of H0 while the hopping terms are treated per-

turbatively. This gives rise to many nuances. Firstly, the unperturbed energies

are randomly distributed. So to maintain conservation of energy while hopping

becomes a probabilistic statement.

Figure 1.3: A typical path in the perturbation theory where the particle hops fromone site to the next and performs a quantum coherent random walk. Hoppingback and forth between lattice sites 4 and 5 represents resonant tunneling. Figurefrom [1]

Secondly, the perturbation theory may have resonances which are due to almost

degenerate states with relatively large tunneling between them. These resonances

6

could affect the convergence of the perturbation theory. This can be addressed by

renormalizing the bare energy levels of the resonating sites self consistently which

mitigates the divergence. Thus, by taking into account terms at all orders of the

perturbation theory, for weak enough hopping and infinite system size, the initial

state has an infinite life time with probability one.

The distinction between the localized and delocalized states at this single-

particle level may require taking into consideration the probability distributions

rather than the averages of the chosen observable (For example, |Gij(t)|2 averaged

over disorder realizations does capture the transition between the localized and dif-

fusive phases while one needs to evaluate the probability distribution of ImGij(ω)

to probe the transition). Averaging over the myriad realizations of disorder or

over the various lattice sites washes away the effects of localization and results in

a finite decay time. This is reasonable from a physical point of view as any real

system has one particular realization of disorder.

1.1 Scaling theory for Anderson transition

Since Anderson’s perturbative approach to localization there have been many other

ways of addressing the problem which have uncovered the richness of the problem.

A scaling theory of the single-particle localization transition crucially depends on

the idea of Thouless energy. It is a measure of the shift of the eigenenergies (∆E)

for a finite-size system due to changing the boundary conditions from periodic

to anti-periodic. Intuitively speaking, a change in boundary conditions does not

appreciably affect the energy of an eigenstate exponentially localized in the bulk.

7

Hence, the shift in energy is only exponentially small in system size L.

∆Elocalized ∼ e−L/ξ (1.3)

where ξ is the localization length. On the other hand in the delocalized part

of the spectrum, a change in the boundary condition completely changes the state

and the shift in energy is comparable to the inverse of the diffusion time across

the finite-size sample. In a clean system the change in the boundary conditions on

the Hamiltonian can also be conceived of as a density modulation of wavenumber

∼ πL. Hence, the Thouless energy in the diffusive phase is inverse of the decay time

of the mode and obeys the following relation.

∆Ediffusive =~

tdiff=L2

D(1.4)

where D is the diffusion constant. Thus, the ratio of the energy shift to the

energy spacing (δW ) is a useful measure of localization proposed by Edwards and

Thouless in 1972 [6, 7]. And this was an important ingredient for proposing the

scaling theory of the transition later in the decade by the Gang of Four [8]. The

average level-spacing for single-particle states in a finite system scales as a power-

law in the middle of the band and is given by

δW =

(

dE

dnL−d

)

(1.5)

where d is the dimension of the space and dndE

is the density of states per unit

volume. In order to develop a scaling theory, the eigenstates of a system of linear

dimension aL has to be expressible as an admixture of states of ad sub-systems

8

of linear dimension L. The energy levels within the various subsystems are mixed

and broadened due to tunneling matrix elements at the boundary between adjacent

subsystems. The crucial insight from Thouless’ work was that the physical quantity

which behaves universally (in the RG sense) is the conductance G defined in units

of e2/~ and not the conductivity (σ). Also, the dimensionless conductance can be

expressed as a universal function parametrized by a single parameter ∆EδW

.

g =G

e2/~(1.6)

As we combine ad blocks of linear dimension L to form a larger block of size

aL, the dimensionless conductance can be expressed as a one parameter scaling

function which satisfies the following renormalization group equation

d ln g(L)

d lnL= β(g(L)) (1.7)

For large conductance (weak disorder) the system must obey Ohm’s law for

weak scattering providing the system with finite conductivity. Hence,

G(L) = σLd−2 (1.8)

Therefore, for g → ∞, β → d − 2. In the other limit of small conductance

(strong disorder), the leading order behaviour at long distances is

g = g0e−αL (1.9)

9

Figure 1.4: For d > 2 there is a critical value of conductance gc above which underthe RG flow the conductance flows to infinity implying the system behaves as ametal. While for d = 1, 2 for any value of initial conductance g0 the conductanceat scale L renormalizes to 0 and the system is localized. Figure from [8]

and β → ln(g/g0). Assuming the beta function is continuous and doesn’t have

singularities, the behaviour of the system can be represented as in Fig. 1.4.

1.2 Random Matrix Theory

Freeman Dyson and Eugene Wigner had studied the spectral properties of ran-

dom matrices in the 50s and 60s in an attempt to describe spectral properties of

complex nuclei [9–13]. It was found that when the elements of matrices satisying

certain global symmetries (e.g. orthogonality, unitarity, symplectic) are chosen

from a Gaussian distribution, the eigenvalue distribution of the matrix has uni-

10

versal characteristics1. Specifically, behaviour of the level spacing (δ) distribution

close to zero is only dependent on the global symmetry of the ensemble of ma-

trices. The disappearance of the weight of the probability distribution at zero is

a signature of spectral rigidity. An intuitive understanding of this effect can be

developed in terms of eigenvalues of 2× 2 matrices.

H11 H12

H∗12 H22

where Hij is the matrix element between two adjacent states in energy. The

off-diagonal part is due to a perturbation coupling the states. The eigenvalues of

the matrix are

E± =1

2

(

H11 +H22 ±√

(H11 −H22)2 + 4|H12|2)

(1.11)

For an orthogonal matrix, H12 is purely real. Therefore, only 2 parameters

need to be tuned for the perturbed energies to be degenerate i.e., H11 = H22 and

H12 = 0. While for a unitary matrix, the number of such real parameters to be

tuned is 3. Thus, close to the s = 0, the level spacing distribution behaves as

∼ sd−1 where s = Ei+1−Ei and d is the number of free parameters to be adjusted.

1Riemann hypothesis: It is conjectured that ζ function:

ζ(z) =

∞∑

n=1

=1

nz=

∏

p∈set of primes

1

1− p−z(1.10)

has zeros lying on the line z = 1

2+ iEi where Ei is real. Interestingly, the statistical fluctuations

of Eis behave like the eigenvalues of a random Hermitian matrix. This has been numericallyverified for a large number of zeros of the function. Thus, it might be possible that this abstractmathematical problem is related the quantum chaotic behaviour of a physical system withouttime-reversal symmetry.

11

The distribution of level spacing for the Gaussian orthogonal ensemble is ap-

proximately given by

PGOE(s) ≈π

2

s

δ2exp

(

−π4

(s

δ

)2)

(1.12)

where δ = 〈s〉 and the angular brackets 〈. . . 〉 imply ensemble averaging. In the

case, where the matrix is sparse which is what physical Hamiltonians correspond

to, the level spacing distribution captures the effects of localization. In the regime

of extendend states, the spectrum of the Hamiltonian experiences level repulsion

and shares the same universal properties as the GOE ensemble. The delocalized

eigenstates have finite matrix elements due to the disorder which provides it the

spectral rigidity. While at strong disorder when the states are localized, the off-

diagonal terms are exponentially suppressed in L as two adjacent states in energy

are typically localized far apart in space. This amounts to energy eigenvalues

being completely random without any correlations between them. Hence, the

level spacing has a Poisson distribution in the localized phase.

PPoisson(s) =1

δexp

(

−sδ

)

(1.13)

This argument is rather general and doesn’t assume if the Hamiltonian is that

of a single-particle or many-particles. As long as the eigenstates are localized

in real-space, the off-diagonal matrix elements for the disorder potential will be

exponentially suppressed in the localized phase. In the many-body case the matrix

elements Hij are evaluated between states in Fock space which would be eigenstates

of the clean Hamiltonian.

12

1.3 Ergodicity (Thermalization)

Figure 1.5: Particles in a box start from an initial non-equilibrium distribution

A classic textbook example used to motivate the idea of ergodicity is a collec-

tion of atoms in a box. The atoms begin from an arbitrary initial state where they

are manifestly out of equilibrium (for example, either localized in a part of the box

or all atoms moving in one direction). How does this gas of atoms reach a steady

state describe by thermodynamic quantities like pressure and temperature whose

statistical fluctuations are governed by equilibrium statistical mechanics? Given

the generality of this phenomena it is quite striking how nascent our understand-

ing is of this phenomena not just in the quantum but, arguably, to some extent

in the classical realm as well. Though in the first instance the quantum and clas-

sical world seem disparate. But if we believe that the description of phenomenon

is always quantum at the microscopic level and the classical description is valid

only in a coarse-grained sense at the macroscopic scale, a complete description of

thermalization must have elements of the classical as well as quantum.

13

The relevance of localization for many quantum interacting degrees of freedom

to thermalize in the absence of coupling to a heat bath was though not directly

addressed but was indeed recognized in Anderson’s 1958 paper. Ever since, the

connection between localization and ergodicity in a many-body quantum system

is relatively unexplored. How does an isolated system reach a state of thermal

equilibrium from a generic initial condition? Under what conditions can a system

serve as its own heat-bath? This is a question of fundamental importance not

just for quantum mechanical but also classical systems. The equations of motion

governing the dynamics of observables are time reversal invariant and yet at long

times in a statistical sense an arrow of time emerges. Hence, at long times effective

equations of motion become Markovian. What permits the existence of such a

solution to the dynamical equations remains a question of broad interest relevant

to many fields in physics.

There are some differences between classical and quantum systems in their

theoretical treatment which a priori is not clear if they are relevant. Nonetheless,

let me highlight them for the sake of completeness. For a classical system Hamil-

tonian dynamics is completely deterministic. Therefore, if we measure a specific

observable at a fixed time for a fixed initial condition, there will be no fluctuations

in this quantity. Hence, in order to have a reasonable definition which results in a

distribution for the observed quantity, one either needs to average over an ensemble

of initial conditions or perform an average in time. While for a quantum system,

even if we start from the same initial state, quantum dynamics inherently allows

for fluctuations. Thus, several measurements of a specific observable generates a

distribution and if the final state is indeed thermalized, this distribution function

should coincide with the Gibb’s measure. Also, for classical systems phase space is

14

continuous even for a finite system. The equivalent concept in a quantum system

is the Hilbert space spanned by its basis states. Even for a many paricle system,

this space (Fock space) is discrete for any finite system and the notion of distance

(geometry) in this space is very different from the classical phase space.

1.3.1 Classical Chaos

Classical degrees of freedom in the presence of strong enough non-linearities is

expected to exhibit chaos where at long times the trajectory of a classical system

uniformly visits all points of the phase space on the constant energy surface. The

Kolmogorov-Arnol’d-Moser (KAM) theorem addresses this issue to some extent.

A classical integrable system can be represented in terms action-angle co-ordinates

Ii − θi (i = 1, . . . , d) where the action variable is conserved and the angle variable

oscillates at a frequency. Ii is the integral of motion. A particular set of conserved

actions Ii defines a d−dimensional torus in angle space (θi).

∂Ii∂t

= 0 (1.14)

θi = 2πωit (1.15)

A simple example of a classically integrable system is a freely propagating

particle in a rectangular box. In this case the integrals of motion are the two

orthogonal components of linear momentum. According to the KAM theorem,

for small enough perturbations from the integrable system, the dynamics of the

system still preserves most/some (depending on the nature of the integrability

breaking term) of the invariant tori. Hence, the system is not fully chaotic even

though some of the action variables do cease to be conserved i.e., corresponding

15

trajectories become stochastic at long times. The system is fully ergodic when the

total energy is the only integral of motion.

Figure 1.6: (a) Chaotic trajectory of a particle in a stadium (b) Regular orbit inan integrable system (Figure from [14])

1.3.2 Berry’s conjecture (Quantum Chaology)

Due to the linearity of the Schrödinger equation, the definition of chaos for a

quantum mechanical system is subtle. It is not analogous to classical chaos which

implies exponential sensitivity to initial conditions. Some of the subtleties also

arise from the

~ → 0 limit of quantum mechanics. This limit in a sense is singular. As opposed to

the case of special relativity where the classical newtonian regime can be reached

perturbatively in orders of (v/c)2, there is no such correspondence where classical

mechanics can be developed from quantum mechanics perturbatively in ~. There-

fore, Michael Berry defines quantum chaology as “the study of semiclassical, but

non-classical, behaviour characteristic of systems whose classical motion exhibits

16

chaos". Hence, the question of quantum chaos is well-posed for the highly ex-

cited states of a hamiltonian which in its classical limit behaves chaotically. One

of the nonclassical measures of quantum chaos is in terms of the statistics of the

spectrum of the Hamiltonian for a bounded system. For a hamiltonian exhibiting

quantum chaos the spectrum exhibits level repulsion while an integrable system

has Poissonian statistics. This distinction is exactly like the difference between

the extended and localized phases of a single particle Anderson model.

The other measure concerns the properties of the wavefunctions of the highly

excited states where the system behaves semiclassically (~ → 0) [15]. For a classi-

cally chaotic system, Berry conjectured that the energy eigenstates when expressed

as a linear combination of the basis states, the amplitudes behave as gaussian ran-

dom functions of the quantum number corrsponding to the basis states [16]. For a

concrete example, let us consider the case of a gas of hard spheres of radius a in a

box of linear dimension 2L. The phase space of the classical system is known to be

fully chaotic. In this case, the natural basis states are the momentum eigenstates

Φ~P (~X) = exp(i ~P · ~X) (1.16)

where ~P = (~p1, . . . , ~pN) and ~X = (~x1, . . . , ~xN) are the momenta and positions

of the N hard spheres. The energy eigenstates Ψn can be expressed as a linear

combination of Φ~P (~X) where the wavefunctions vanish outside the domain D. The

domain is defined as

D = (~x1, . . . , ~xN) : −L ≤ xµi ≤ L; |~xi − ~xj | > 2a (1.17)

17

and

Ψn( ~X) =∑

~P

Cn, ~PΦ~P (~X) (1.18)

The momenta are also constrained by the total energy condition. In the case

of the hard-sphere gas

En =

N∑

i=1

~p2i2m

(1.19)

In the limit of large N and L with the density held fixed, Berry’s conjecture is

equivalent to assuming that Cn, ~P is an uncorrelated gaussian random variable in

~P only to be limited by the energy of the eigenstate. Also, Cn, ~P and Cm,~P for two

different eigenstates (n 6= m) are also completely uncorrelated. Hence, a typical

eigenstate at the chosen energy satisfies the statistical properties of a Gaussian

ensemble. This property of the Berry’s conjecture forms the basis of Eigenstate

Thermalization hypothesis (ETH).

1.3.3 Eigenstate Thermalization Hypothesis

Assuming that highly excited energy eigenstates satisfy Berry’s conjecture, what

can one say about approach to thermal equilibrium of an isolated quantum system?

Let us start the system in some pure quantum state (ψ(0)) with a well defined

average energy (E) with small fluctuations (∆ ≪ E; this implies that the energy

eigenstates contributing to the initial state are within an energy window ∼ ∆ -

energy window in a microcanonical ensemble). For an out-of-equilibrium initial

18

condition the co-efficients αn have a very detailed and specific arrangement.

ψ(0) =∑

n

αnΨn (1.20)

E =∑

n

|αn|2En (1.21)

∆2 =∑

n

|αn|2(En − E)2 (1.22)

Eigenstate thermalization hypothesis [17, 18] states that the expectation value

of local observables O at long times equilibrates to the microcanonical average.

This equilibrium average can be well represented by just the expectation value in

a typical eigenstate within the microcanonical energy span.

limt→∞

〈ψ(t)|O|ψ(t)〉 =∑

|En−E|≤∆〈Ψn|O|Ψn〉N∆

= typ〈Ψn|O|Ψn〉typ (1.23)

where N∆ is the number of states in the energy window. For any finite t the

expectation value is

〈O〉t =∑

n

|αn|2Onn +∑

n 6=m

α∗mαne

−i(En−Em)tOnm (1.24)

Onm is the matrix element of the operator between eigenstates n and m. The

off-diagonal terms gives rise to dephasing on time evolution. For an ergodic system,

decoherence occurs possibly for two reasons. For generic initial conditions, the

complex phases are randomized over time ∼ ~/∆. But for the decoherence of finely

tuned non-equilibrium initial conditions, Onm also tend to zero exponentially in

system size. The decay of the off-diagonal matrix element can be argued based

on Berry’s conjecture but this behaviour has not been rigorously shown. Once the

19

second term has decayed to zero, the diagonal term survies which still depends on

the intial conditions αn.

It is important to remember some of the limitations of ETH. Intuitively, it must

depend on the time scales of dynamics. In the case of a few degrees of freedom

at equilibrium, the relevant time scale is the time needed to diffusively relax an

excitation in a finite system (τdiff ) while the mean level spacing (δ) governs the

time scale of the fast dynamics [19–22]. Hence, the limit in which ETH is valid is

δ ≪ ~/τdiff i.e., diffusive relaxation in a finite system occurs at a much shorter time

scale compared to δ. This is exactly the condition which is violated for localized

states and results in the breakdown of ETH. The semiclassical limit also implicitly

assumes that the states under study are highly excited (The mean level spacing is

small between high energy states). Hence, one should expect ETH to break down

at low energies for finite systems. For instance, it is evident that the ground state

will not satisfy ETH because there is a lot of structure in the wavefunction which

gives the state its special status (Also entanglement entropy (to be discussed later)

doesn’t satisfy a volume law). It remains to be explored if the breakdown of ETH

indeed means non-thermalization or is there another mechanism which can still

result in thermalization at lower energies. This suggests that for energies close to

the ground state the excitations can only thermalize by coupling to an external

heat bath.

1.4 Disorder and Interactions

Understanding the effects of disorder combined with interactions is a major chal-

lenge in condensed matter physics. Part of the difficulty lies in the fact that there

20

are few theoretical tools which allow the treatment of disorder and interactions

on an equal footing. The robustness of the Anderson insulator to interactions

has perplexed physicists from the early days of localization. On those lines, Mott

had posed the question - What is the result of coupling a single-particle localized

insulator to an external heat bath?

1.4.1 Variable range hopping

Figure 1.7

In the limit of strong enough disorder where all the single particle states near

the fermi-level are localized, two adjacent states in energy are localized far apart

in space. While a heat bath by definition has delocalized excitations for excitation

energies arbitrarily close to zero. In essence the intuitive picture suggests that

21

the localized states can exchange energy with the heat bath to hop over long dis-

tances from one localized state to another state close in energy thus resulting in

conduction as shown in Fig. 1.7. Assuming that we are dealing with fermions,

therefore at low temperatures there is a well-defined fermi level with long-lived

excitations restricted only close to EF . Lets consider the transport due to the

tunneling between two states with energies E1 > EF and E2 < EF and their local-

ization centers separated by distance R. The probability to produce excitations of

order ∼ E1 − E2 = ǫ in the heat bath goes as exp(−ǫ/kBT ). On the other hand

the tunneling matrix element decays as ∼ exp(−2R/ξ) where ξ is the localization

length of the states. Hence, at leading order the conductivity at low temperatures

behaves as

σ(T ) ∼ exp(−2R

ξ− ǫ

kBT) (1.25)

The typical separation between the states is given by

Rtyp =

(

dn(EF )

dE(E1 −E2)

)− 1d

(1.26)

where d is the dimension of the space. Hence, the two terms in the exponential

of Eq. 1.25 have competing dependence on ǫ. Mott argued that the conductivity

will be dominated by states where the tunneling and activation are optimal. Thus,

maximizing over ǫ one gets the result

ǫoptimal ∼ (kBT )d

1+d (1.27)

22

This gives the conductivity of the system at low temperatures to be

σvariable(T ) = σ0(T ) exp

(

−(

T0T

)1

1+d

)

(1.28)

σ0 and T0 depends on the details of the model. σ0 has weak dependence (power-

law) on T . If one had expected just naively that the transport in the presence of

a bath would be due to activation across the mobility edge, conductivity would be

given as

σactivation(T ) = σ′0 exp

(

−E1 − EckBT

)

(1.29)

Ec is the mobility edge of the sample. Mott’s variable range hopping argument

predicts a different exponent for the power in the exponential from the transport

just due to activation across the mobility edge. The difference is more conspicuous

in higher dimensions. The variation from variable range hopping conductivity at

low T to activated transport at high T has been experimentally observed in doped

semiconductors.

1.4.2 Fermi glass

An Anderson insulator without any coupling to a heat bath has zero D.C. conduc-

tivity at zero temperature (At finite temperature if the entire spectrum is localized

D.C. conductivity is still zero). At the same time Mott’s result of finite hopping

conductivity in the presence of the bath beckons the question if electron-electron

interactions can play a similar role in the absence of an external heat bath. Can

the electrons serve as their own heat bath? This was recognized to be an impor-

tant issue in order to completely understand transport phenomenon of electrons in

23

semiconductors. An early work [23, 24] attempted to address this problem using

a perturbative analysis much on the lines of Landau’s Fermi liquid theory. Albeit,

the breakdown of translational invariance due to disorder introduces complications.

Let us consider the case in which the fermi level is below the mobility edge for the

single-particle problem. Hence, all the low energy-excitations are exponentially

localized. Much like Anderson’s locator expansion predicting single-particle local-

ization, the important quantity to probe localization in the presence interaction

is the behaviour of the imaginary part of self energy (Im(Σ(ω))) of the Green’s

function.

G(ω) =1

ω − H0 − Σ(ω)(1.30)

H0 is the non-interacting disordered hamiltonian. One of the crucial ingredients

to setup a perturbative calculation is the basis in which it is performed. It was

realized that working in the basis of single-particle localized states (from now

on denoted by |α〉) helps to keep the perturbative expansion relatively clean. The

feature which makes it particularly useful is that Im(Σ(ω)) tends to zero for ω → µ.

limω→µ

Im(Σαα′(ω)) = 0 (1.31)

In this sense, local excitations have infinite lifetime close to the fermi-level.

For the purposes of decay of excitations due to inelastic scattering, Re(Σ(ω)) acts

only to renormalize the disorder potential. For strong enough disorder this has

negligible effect on the dynamics. In the α-basis, if Im(Σ(ω)) is a continuous

function of ω, it is a signature of decay of the single-particle excitations. On the

other hand if it is finite only at a discrete set of points in ω (pure-point spectrum:

24

dense set of points with measure zero), it implies that the excitations do not decay

via single or many-particle processes. The behaviour of the self energy has clear

features which can be understood at 1st order in perturbation theory.

In the single-particle channel at frequency ω, if the interactions are short ranged

two states α and α′ can have significant tunneling only if they are localized within

a finite distance off each other. But this imposes energy restrictions as two states

nearby in space are far apart in energy. In this channel both conditions are sat-

isfied only for a finite number of states and the probability of such an occurence

tends to zero. Hence, they only contribute as poles to the imaginary part of self

energy. A similar argument for the many-particle channel taking into account the

available phase space volume for scattering also contributes only isolated poles to

the Im(Σ(ω)). At the 1st order in perturbation theory the low ω part of spectral

support is discrete and the state are bound. This hints that Anderson insulator

with zero D.C conductivity is stable in the presence of short-range interactions.

This treatment of the problem has a few limitations. Firstly, being perturba-

tive in nature it cannot discount non-perturbative affects at strong interactions.

Secondly, a priori it is not clear if higher-order terms in the perturbative expan-

sion converge to the same conclusion. The work of Basko, Aleiner and Altshuler

[25] addresses at least one of these issues.

1.4.3 Many-Body Localization: Basko, Aleiner, Altshuler

(BAA)

Following the work of Fleishman, Licciardello and Anderson, there were many

efforts to resolve the question of localization in the presence of interaction but

25

only an inconclusive picture emerged. But a compelling evidence in favour of

many-body localization was reported by BAA based on a rigourous perturbative

treatment where they summed up Feynman diagrams up to all orders. They made

a striking claim that localization persists upto a finite temperature (or energies of

O(N) as temperature is ill-defined in the many-body insulator). There is a phase

transition from the insulating to the conducting phase.

Figure 1.8: Below a critical temperature Tc the D.C. condustivity is strictly equalto zero. At high temperatures, the system becomes ergodic and the delocalizedand has finite conductivity. λ is the strength of the interaction and δζ is the meanlevel spacing with the localization volume ζd. (Figure from [25])

The treatment of the problem shared many features with Anderson’s locator

expansion. In this case one works in the limit where all single-particle eigenstates

are localized. This is true in d = 1, 2 for arbitrarily weak disorder while for d > 2

above a critical disorder strength. There is no single-particle mobility edge as the

26

spectrum is bounded (in the tight-binding limit). The pertubation theory is in the

basis of occupied single-particle states.

|Ψα〉 = |nα0 , . . . , nα2N〉 (1.32)

nαi(= 0, 1 for spinless fermions) is the occupation number of the eigenstate

with energy Eαiand localized around site ~rαi

with localization length ξ. Ψα is a

state in Fock space corresponding to the occupation numbers. The Hamiltonian

in this basis is expressed as

H =∑

α

ǫαc†αcα +

1

2

∑

αβγµ

Vαβγµc†αc

†β cγ cµ (1.33)

The matrix element Vαβγµ is restricted in energy and space. Due to the ex-

ponential localization the matrix elements are chosen to be finite only for states

satifying

|~rα − ~rβ| . ξ

|~rβ − ~rγ | . ξ

...

Also, the matrix elements are neglected for states separated in energy by more

than the typical single-particle level spacing within the localization volume (δξ).

|ǫα − ǫγ|, |ǫβ − ǫµ| . δξ

|ǫα − ǫµ|, |ǫβ − ǫγ | . δξ

27

In this terminology the interaction term generates hops in the Fock space of

many-body states. It plays the same role as tunneling played in the single-particle

Anderson problem. The Anderson problem is studied in fixed dimension d in the

limit L→ ∞ while the way this problem is conceived it is the study of localization

in the very high-dimensional Fock space (d → ∞). BAA studied the statistics of

the imaginary part of the single-particle self-energy which governs the quasiparticle

relaxation for a finite size system. The limit L → ∞ is taken at the end of the

calculation to be discussed later.

Γα(ω) = Im(Σα(ω)) (1.34)

Figure 1.9: (a) In the delocalized phase (dashed line) Γ(ǫ) is a continuous functionof energy. While in the localized phase (solid line), the delta function is smearedout due to the dissipation added by hand (finite η; at the end the limit η → 0 istaken). (b) The probability distribution of Γ in the loclaized (solid line) and theergodic phase (dashed line). (Figure from [25])

Since, Γ varies from sample-to-sample, a naïve average over disorder realiza-

tions cannot distinguish between the two phases. For a single realization of disor-

28

der, in the delocalized phase Γ is expected to be a smooth function of ǫ in the limit

L → ∞ as the excitation decays into the continuum. This results in a gaussian

probability distribution for Γ peaked around the mean value. While in the local-

ized phase the spectrum is expected to be a discrete point spectrum. Hence, the

probability distribution is a delta function at zero. This kind of a singular distribu-

tion is difficult to analyse in a theoretical calculation. Thus, a method originally

employed by Anderson for the single-particle problem serves to be useful. The

self-energy is analytically continued to small imaginary values of ω (Im(ω) = η).

We’ll take the η → 0 at the end of the calculation. Physically, it is as if the sys-

tem is coupled to an external bath. This procedure leaves the delocalized phase

unaffected. But in the localized case it has the effect of broadening the δ-function

peaks in Γ into Lorentzians thus giving the states a finite lifetime. In this case,

the distribution function develops a tail and the peak shifts to η from zero as

shown in Fig. 1.9 (b). It is important to note that before taking the limit η → 0

one has to send the system volume to ∞, first. This limiting procedure has to

be treated carefully. η shouldn’t tend to zero faster than the mean level spacing:

η > exp(−Ld). In this case the order of limits are not interchangeable as for any

finite closed system the spectrum is always a sequence of delta functions. The

spectral weight for a single eigenstate (even for an infinite system) in the localized

phase is finite only at a discrete set of points but, it is a different set of points for

different eigenstates. For an arbitrary initial state which is a linear combination

of many eigenstates, this procedure must still produce a pure-point spectrum for

29

the system to be localized.

limη→0

limL→∞

P (Γ > 0) =

finite for a metal

0 for an insulator(1.35)

As shown in Fig. 1.9 (b), the probability of Γ > η behaves as ∼ η in the

many-body localized phase. Hence, the probability for any finite decay rate in the

insulating phase tends to zero as the coupling to the bath is switched off. While

in the delocalized phase, the decay rate stays finite in this limit.

1.5 Possible signatures in experiments

Manifestations of single-particle localization have been measured in early transport

experiments in doped semiconductors. As a matter of fact some of the theoretical

work was born out of attempts to understand impurity band conduction of elec-

trons and holes in doped semiconductors. This was verified by careful transport

experiments at low temperatures. Since, quantum coherence of the wavefunction

over large distances plays a crucial role in localization, its effects are only mea-

sureable at low temperatures. A direct measurement of exponential localization of

the single-particle wavefunction eluded experiments until recently. Since, it is pri-

marily a wave phenomena the first direct observations were using light or classical

photons [26, 27]. Because of the non-iteracting nature of photons, this is truly an

observation of Anderson localization.

In material systems, any description of localization is incomplete without tak-

ing into account electron-elctron or electron-phonon interactions. Since, these are

30

usually not within an experimentalist’s control in real materials, a direct observa-

tion of the localized wavefuntion remained illusive. With the advent cold atomic

systems, where the strength of the interactions is a finely tunable experimental

knob via a Feshbach resonance, Anderson localization was directly imaged in a

system of bosonic atoms [28, 29]. These experiments were performed in the limit

of negligible interactions due to the low density of the cloud. The disorder poten-

tial is realized by an optical speckle pattern whose strength can be controlled by

the intesity of the laser beam.

These experiments are particularly promising to study phenomena pertaining

to many-body localization not just due to the tunability of parameters but also,

the lack of an external heat bath makes the system extremely isolated to a very

good approximation. In real materials even though phonons are often neglected in

a calculation the assumption of thermal equilibrium pre-supposes the existence of

a heat bath at low temperatures. Due to the lack of a physically dynamic lattice

in cold atoms or other degrees of freedom which can serve as a bath in an obvious

way, this assumption may not be a bad approximation for realistic experiments.

Thus, the possibility of observing a signature of the many-body localized insulator

may not be a far-fetched one.

There are other experimental setups which are being developed to emulate phe-

nomena in materials. Most of them are being conceptualized as possible platforms

for quantum information processing. One such system is that of superconducting

circuits in transmission line resonators [30]. In this system, the photons inside

the cavity are prepared to interact with each other by coupling via the supercon-

ducting qubits. Therefore, the dressed photons behave as effective particles with

31

Figure 1.10: Atomic density profile of the BEC cloud. The condensate wavefunc-tion is exponentially localized with the tails fitted to an exponential. The inset offigure (d) shows that in the absence of random potential (VR = 0) the rms widthof cloud grows linearly while when it is switched on (VR 6= 0) it stops growing aftersome time. Plot from [28]

32

Figure 1.11: Superconducting (SC) circuits as qubits: (a) A transimission lineresonator with an array of SC qubits (b) The basis building block for the array -Superconducting quantum interference device (SQUID). It consists of 2 supercon-ducting islands connected by a tunnel junction. (Figure from [30])

33

on-site interaction and hopping on a lattice. This can be used to study quantum

many-body physics of photons [31, 32].

The other physical system where the effects of many-body localization may

be relevant for experiments is the problem of disordered superconductivity in two

dimensions. Experiments performed on InOx and TiN thin films have given some

intriguing results. These thin films undergo a superconducting transition at low

temperatures. At low temperatures in the presence of a magnetic field I-V charac-

teristics shows highly non-linear behaviour on applying a D.C. bias voltage [33–36]

on the insulating side.

Figure 1.12: InOx thin film showing a jump in I-V characteristic in the insulatingphase for T = 0.01K. (Figure from [34])

This was explained by invoking the idea of electron overheating. On applying

a voltage, the electrons are excited to a higher temperature than the phonon bath

(Tel > Tph) as the phonon and electron degrees of freedom are decoupled from

each other [36, 37]. Thus, the resistance of the sample is well-behaved in terms

34

of Tel (assuming the electrons are thermalized) and the apparent non-linearity is

due to the overheating of the electrons compared to phonons (Tph). The jump in

the I-V characteristic is reflecting the bistability of the electronic system where

on applying a strong enough voltage the electronic system goes to the metastable

state with the higher Tel. This phenomena hints that under suitable conditions

the electronic degrees of freedom can be decoupled from the phononic heat bath.

Thus, making the realization of a many-body localized insulator more feasible.

1.6 Thesis outline

In Chapter 2, I will be discussing the numerical treatment of localization of the ex-

cited states in the presence of interactions and disorder. We specifically search for

signatures of localization at infinite temperature. In our case the relative strength

of disorder is the only tunable parameter. I will explain in detail the various mea-

sures (motivated by ideas mentioned in the introduction) we used to probe the

physics of many-body localization. Continuing from there I will highlight some

of our results showing the existence and distictions between the ergodic and in-

sulating phases. This will lead to throwing some light on the properties of the

critical point separating these two phases. Chapter 3 will explore the possibil-

ity of realizing the phenomena of many-body localization in an effective classical

model with disorder. I will discuss the numerical method employed to study the

dynamics of the model and results on energy transport. Phase transitions at finite

temperature are mostly described by effective classical model. Since, the many-

localization transition is also at nonzero temperature in this work we explore if

a classical model can capture the transition. The concluding chapter will discuss

35

the overall picture of this interesting transition that our work has realized. Also,

discuss the prospects for future work on this problem and other open questions

related to it.

36

Chapter 2

The quantum many-body

localization

As originally proposed in Anderson’s seminal paper [1], an isolated quantum sys-

tem of many locally interacting degrees of freedom with quenched disorder may be

localized, and thus generically fail to approach local thermal equilibrium, even in

the limits of long time and large systems, and for energy densities well above the

system’s ground state. In the same paper, Anderson also treated the localization

of a single particle-like quantum degree of freedom, and it is this single-particle

localization, without interactions, that has received most of the attention in the

half-century since then. Much more recently, Basko, et al. [25] have presented a

very thorough study of many-body localization with interactions at nonzero tem-

perature, and the topic is now receiving more attention; see e.g. [38–48].

37

Many-body localization at nonzero temperature is a quantum phase transition

that is of very fundamental interest to both many-body quantum physics and sta-

tistical mechanics: it is a quantum “glass transition” where equilibrium quantum

statistical mechanics breaks down. In the localized phase the system fails to ther-

mally equilibrate. These fundamental questions about the dynamics of isolated

quantum many-body systems are now relevant to experiments, since such systems

can be produced and studied with strongly-interacting ultracold atoms [49]. And

they may become relevant for certain systems designed for quantum information

processing [50, 51]. Also, many-body localization may be underlying some highly

nonlinear low-temperature current-voltage characteristics measured in certain thin

films [37].

2.1 The model

Many-body localization appears to occur for a wide variety of particle, spin or

q-bit models. Anderson’s original proposal was for a spin system [1]; the specific

simple model we study here is also a spin model, namely the Heisenberg spin-1/2

chain with random fields along the z-direction [40]:

H =L∑

i=1

[hiSzi + J ~Si · ~Si+1] , (2.1)

where the static random fields hi are independent random variables at each

site i, each with a probability distribution that is uniform in [−h, h]. Except

when stated otherwise, we take J = 1. The chains are of length L with periodic

boundary conditions. This is one of the simpler models that shows a many-body

38

localization transition. Since we will be studying the system’s behavior by exact

diagonalization, working with this one-dimensional model that has only two states

per site allows us to probe longer length scales than would be possible for models on

higher-dimensional lattices or with more states per site. We present evidence that

at infinite temperature, β = 1/T = 0, and in the thermodynamic limit, L → ∞,

the many-body localization transition at h = hc ∼= 3.5 ± 1.0 does occur in this

model. The usual arguments that forbid phase transitions at nonzero temperature

in one dimension do not apply here, since they rely on equilibrium statistical

mechanics, which is exactly what is failing at the localization transition. We also

present indications that this phase transition might be in an infinite-randomness

universality class with an infinite dynamical critical exponent z → ∞.

Our model has two global conservation laws: total energy, which is conserved

for any isolated quantum system with a time-independent Hamiltonian; and total

Sz. The latter conservation law is not essential for localization, and its presence

may affect the universality class of the phase transition. For convenience, we

restrict our attention to states with zero total Sz.

For simplicity, we consider infinite temperature, where all states are equally

probable (and where the sign of the interaction J does not matter). The many-

body localization transition also occurs at finite temperature; by working at in-

finite temperature we remove one parameter from the problem, and use all the

eigenstates from the exact diagonalization (within the zero total Sz sector) of each

realization of our Hamiltonian. We see no reason to expect that the nature of the

localization transition differs between infinite and finite nonzero temperature (with

an extensive amount of energy in the system),although it is certainly different at

strictly zero temperature [52]. It is important to emphasize that temperature is

39

not a well-defined macroscopic observable in the many-body localized phase. In

cases, where the isolated system doesn’t thermalize to a mixed state with a Gibb’s

distribution at finite temperature one should consider the parameter being tuned

as the energy density. The the "temperature” T can be defined as the tempera-

ture that would give that energy density at thermal equilibrium. Note that this

is a quantum phase transition that occurs at nonzero (even infinite) temperature.

Like the more familiar ground-state quantum phase transitions, this transition is

a sharp change in the properties of the many-body eigenstates of the Hamiltonian,

as we discuss below. But unlike ground-state phase transitions, the many-body lo-

calization transition at nonzero temperature appears to be only a dynamical phase

transition that is invisible in the equilibrium thermodynamics [39].

The model we chose to study has a finite band-width. An infinite temperature

limit of such a system is studied by considering states at high energy densities i.e.

eigenstates in the middle of the band. We weigh the observables evaluated from

these states with equal probability in order to study their thermal expectation

values. A practical benifit of working in this limit is the utilization of all the data

we acquire from the full diagonalization of the Hamiltonian which is the most

computer time-consuming part of the calculation.

There are many distinctions between the localized phase at large random field

h > hc and the delocalized phase at h < hc. We call the latter the “ergodic” phase,

although precisely how ergodic it is remains to be fully determined [53]. The

distinctions between the two phases all are due to differences in the properties of the

many-body eigenstates of the Hamiltonian, which of course enter in determining

the dynamics of the isolated system.

40

Figure 2.1: The phase diagram as a function of relative interaction strength h/Jat T = ∞. The critical point is (h/J)c ≈ 3.5. For h < hc the system is ergodicwhile for h > hc, it is many-body localized.

Ergodic phase Many-body localized phase• An infinite system is a heat bath • An infinite system is not a heat bath• Many-body eigenenergies obey GOElevel statistics

• Many-body eigenenergies have Pois-son level statistics

• System achieves local thermal equi-librium

• Doesn’t thermally equilibrate- quan-tum glass

• Finite D.C. transport of energy andother globally conserved quantities

• D.C. transport is zero

• Extensive entanglement in eigen-states

• “Area-law” entanglement in eigen-states

• Eigenstate thermalization is true • Eigenstate thermalization is false

Table 2.1: Properties of the ergodic and localized phases

41

In the ergodic phase (h < hc), the many-body eigenstates are thermal [17, 18,

54, 55], so the isolated quantum system can relax to thermal equilibrium under

the dynamics due to its Hamiltonian. In the thermodynamic limit (L → ∞),

the system thus successfully serves as its own heat bath in the ergodic phase. In a

thermal eigenstate, the reduced density operator of a finite subsystem converges to

the equilibrium thermal distribution for L → ∞. Thus the entanglement entropy

between a finite subsystem and the remainder of the system is, for L → ∞, the

thermal equilibrium entropy of the subsystem. At nonzero temperature, this en-

tanglement entropy is extensive, proportional to the number of degrees of freedom

in the subsystem.

In the many-body localized phase (h > hc), on the other hand, the many-

body eigenstates are not thermal [25]: the “Eigenstate Thermalization Hypothesis”

[17, 18, 54, 55] is false in the localized phase. Thus in the localized phase, the

isolated quantum system does not relax to thermal equilibrium under the dynamics

of its Hamiltonian. The infinite system fails to be a heat bath that can equilibrate

itself. It is a “glass” whose local configurations at all times are set by the initial

conditions. Here the eigenstates do not have extensive entanglement, making them

accessible to DMRG-like numerical techniques [40, 56]. A limit of the localized

phase that is simple is J = 0 with h > 0.

H =L∑

i=1

hiSzi (2.2)

Here the spins do not interact, all that happens dynamically is local Larmor

precession of the spins about their local random fields. No transport of energy or

42

Figure 2.2: In the limit J = 0 randomly oriented spins precess around their localmagnetic field hi with frequency Ωi =

hi~

spin happens, and the many-body eigenstates are simply product states with each

spin either “up” or “down”.

Any initial condition can be written as a density matrix in terms of the many-

body eigenstates of the Hamiltonian as ρ =∑

mn ρmn|m〉〈n|. The eigenstates have

different energies, so as time progresses the off-diagonal density matrix elements

m 6= n dephase from the particular phase relations of the initial condition, while

the diagonal elements ρnn do not change. In the ergodic phase for L→ ∞ all the

eigenstates are thermal so this dephasing brings any finite subsystem to thermal

equilibrium. But in the localized phase the eigenstates are all locally different

and athermal, so local information about the initial condition is also stored in the

diagonal density matrix elements, and it is the permanence of this information

that in general prevents the isolated quantum system from relaxing to thermal

equilibrium in the localized phase.

43

Our goals in this work are (i) to present results in the ergodic and localized

phases that are consistent with the expectations discussed above, and (ii), more

importantly, to examine some of the properties of the many-body eigenstates of

our finite-size systems in the vicinity of the localization transition to try to learn

about the nature of this phase transition. Although the many-body localization

transition has been discussed by a few authors, there does not appear to be any

proposals for the nature (the universality class) of this phase transition or for

its finite-size scaling properties, other than some very recent initial ideas in Ref.

[45]. It is our purpose here to investigate these questions, extending the previous

work of Oganesyan and Huse [39], who looked at the many-body energy-level

statistics of a related one-dimensional model. Since the many-body eigenstates

have extensive entanglement on the ergodic side of the transition, it may be that

exact diagonalization (or methods of similar computational “cost” [45]) is the only

numerical method that will be able to access the properties of the eigenstates on

both sides of the transition.

2.2 Single-site observable

As a first simple measure to probe how thermal the many-body eigenstates appear

to be, we have looked at the local expectation value of the z component of the

spin.

m(n)iα = 〈n|Szi |n〉α (2.3)

44

at site i in sample α in eigenstate n. If the system of spins does thermalize, the

equilibrium properties of a single spin coupled to the rest of the spins has thermal

behaviour. For each site in each sample we compare this for eigenstates that are

adjacent in energy, showing the mean value of the difference: [|m(n)iα −m

(n+1)iα |] for

various L and h in Fig. 2.3, where the eigenstates are labeled with n in order of

their energy. The square brackets denote an average over states, samples and sites.

The number of samples used in the data shown in this work ranges from 104 for

L = 8, to 50 for L = 16 and some values of h.

7 8 9 10 11 12 13 14 15 16 17−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

L

ln [

|miα (

n) −

miα (

n+

1) |]

8.0

5.0

3.6

2.7

2.0

1.0

0.6

Figure 2.3: The natural logarithm of the mean difference between the local mag-netizations in adjacent eigenstates (see text). The values of the random field h areindicated in the legend. In the ergodic phase (small h) where the eigenstates arethermal these differences vanish exponentially in L as L is increased, while theyremain large in the localized phase (large h).

45

In our figures we show one-standard-deviation error bars. The error bars are

evaluated after a sample-specific average is taken over the different eigenstates and

sites for a particular realization of disorder. Here and in all the data in this work

we restrict our attention to the many-body eigenstates that are in the middle one-

third of the energy-ordered list of states for their sample. Thus we look only at

high energy states and avoid states that represent low temperature. In this energy

range, the difference in energy density between adjacent states n and (n+ 1) is of

order√L2−L and thus exponentially small in L as L is increased. If the eigenstates

are thermal then adjacent eigenstates represent temperatures that differ only by

this exponentially small amount, so the expectation value of Szi should be the same

in these two states for L → ∞. From Fig. 2.3, one can see that the differences

do indeed appear to be decreasing exponentially with increasing L in the ergodic

phase at small h, as expected. In the localized phase at large h, on the other

hand, the differences between adjacent eigenstates remain large as L is increased,

confirming that these many-body eigenstates are not thermal.

In the two phases the probabability distribution of m(n)iα has distinctive be-

haviour. At infinite temperature, in equilibrium we expect m(n)iα ≈ 0 which mani-

fests itself as a peak in the probability distribution around zero as shown in Fig.

2.4. With increasing strength of disorder, the probability distribution becomes

bimodal and peaked around +12

and −12. The tendency of the eigenstates to have

the maximal z-component of spin at a single site suggests that the energy eigen-

states are approximate product states i.e. |n〉 ∼ | ↑↓ · · · ↑↓↓〉 at strong but finite

disorder. In the localized phase, we find the emergence of an effective spin-12

degree

of freedom (two-level system) which is the dressed form of the original spin due to

finite interaction strengths.

46

Figure 2.4: The probability distribution of local magnetization for L = 14 andh = 0.6, 3.0, 4.0 and 10.0.

2.3 Transport of conserved quantities

Thermalization requires the transport of energy. In the present model with con-

served total Sz, it also requires the transport of spin. To study spin transport on

the scale of the sample size L, we consider the relaxation of an initially inhomo-

geneous spin density:

M1 =∑

j

Szj exp (i2πj/L) (2.4)

47

Figure 2.5: The probability distribution of difference of local magnetization be-tween adjacent eigenstates for disorder strengths h = 0.6, 2.0, 3.0 and 6.0. Thedifferent systems sizes are marked in the legend. For low disorder in the ergodicphase, the difference distribution becomes more sharply peaked around zero. Whilein the localized phase the distribution develops a bump close to 1. At strong dis-order the finite size effects are negligible.

48

is the longest wavelength Fourier mode of the spin density. Consider an initial

condition that is at infinite temperature, but with a small modulation of the spin

density in this mode, so the initial density matrix is

ρ0 =eǫM

†1

Z≈ (1 + ǫM †

1)/Z (2.5)

where ǫ is infinitesimal, and Z is the partition function. The time-evolution of

the initial magnetization is give by

〈M1〉t = Tr(ρtM1) (2.6)

= Tr(e−iHtρ0eiHtM1) (2.7)

=ǫ

Z

∑

n,m

|〈n|M1|m〉|2ei(En−Em)t (2.8)

The initial spin polarization of this mode is then

〈M1〉0 =∑

n

〈n|ρ0M1|n〉 =ǫ

Z

∑

n

〈n|M †1M1|n〉 . (2.9)

If we consider a time average over long times, then the long-time averaged

density matrix ρ∞ is diagonal in the basis of the eigenstates of the Hamiltonian,

since a generic finite-size system has no degeneracies and the off-diagonal matrix

elements of ρ each time-average to zero. As a result, the long-time average of the

spin polarization in this mode is

〈M1〉∞ =ǫ

Z

∑

n

〈n|M †1 |n〉〈n|M1|n〉 . (2.10)

49

0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.50.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1[f

α (n

) ]

h

8

10

12

14

16

Figure 2.6: The fraction of the initial spin polarization that is dynamic (see text).The sample size L is indicated in the legend. In the ergodic phase (small h) thepolarization decays substantially under the dynamics, while in the localized phase(large h) the decay is small, and this distinction gets sharper as L increases.

Thus for each many-body eigenstate in each sample we can quantify how much

it contributes to the initial and to the long-time averaged polarization. We then

define the fraction of the contribution to the initial polarization that is dynamic

and thus decays away (on average) at long time, as

f (n)α = 1− 〈n|M †

1 |n〉〈n|M1|n〉〈n|M †

1M1|n〉. (2.11)

In the ergodic phase, the system does thermalize, so the initial polarization

does relax away and f(n)α → 1 for L → ∞. In the localized phase, on the other

50

hand, there is no long-distance spin transport, so f (n)α → 0 for L → ∞. In Fig.

2.6 we show the mean values of f for each L vs. h. They show the expected

behavior in the two phases (trending with increasing L towards either 1 or 0), and

the phase transition is indicated by the crossover between large and small f that

occurs more and more abruptly as L is increased.

2.4 Energy-level statistics

A spectral distinction between the many-body localized and the ergodic phases is

based on the statistics of energy eigenvalues. The spectral rigidity in the ergodic

phase is reflected in the level repulsion. This repulsion between eigenvalues can be

understood heuristically from the point of view of second-order perturbation theory

where any local perturbation from the Hamiltonian in the ergodic phase leads to

the increase in gap between adjacent eigenvalues for a finite size system. While

in the many-body localized phase a local perturbation has exponentially small

overlap between two adjacent many-body eigenstates thus producing negligible

repulsion between the levels. This picture has been theoretically and numerically

substantiated for single-particle localization [57].

A qualitatively similar finite-size scaling plot to Fig. 2.6 also indicating the

phase transition is obtained by examining the many-body eigenenergy spacings

as was done in Ref. [39], and is shown as Fig. 2.8. We consider the level

spacings δ(n)α = |E(n)

α − E(n−1)α |, where E

(n)α is the many-body eigenenergy of

eigenstate n in sample α. Then we obtain the ratio of adjacent gaps as r(n)α =

minδ(n)α , δ(n+1)α /maxδ(n)α , δ

(n+1)α , and average this ratio over states and samples

at each h and L. A choice of a two-gap quantity was made as opposed to the

51

Figure 2.7: The probability distribution of r(n) for L = 16 and h = 1.0, 3.6, 4.0and 6.0. For h < hc the spectrum’s finite level-repulsion can be seen as a peak atfinite r(n) in the distribution. While for large disorder h > hc, the distribution ispeaked at zero.

52

single gap distribution. A single gap distribution (p(s) = [〈δ(s − δn〈δn〉

)〉]) requires

an appropriate definition of the mean gap 〈δn〉. The mean gap has significant vari-

ations over the range of the spectrum. It is indeed exponentially small in system

size in the middle of the band but towards the edge of the band the gap decreases

as a power law in L. Such variations significantly affect the numerical estimates

of the single-gap distribution. Hence, looking at a two-gap quantity alleviates this

concern. Also, a particular realization of the random potential for small system

sizes is comprised of just a few numbers but the energy eigenvalues are exponen-

tial large in number leading to significant correlation between them even in the

localized phase where they are expected to be uncorrelated. It is assumed that

such effects decay as L→ ∞.

In the ergodic phase, the energy spectrum has GOE (Gaussian orthogonal

ensemble) level statistics and the average value of r converges to [r]GOE ∼= 0.53 for

L→ ∞. This can be verified by looking at an ensemble of large random matrices

and numerically estimating [r]GOE. While in the localized phase the level statistics

are uncorrelated and Poissonian. The distribution of r in the localized phase can

be derived by calculating⟨

δ(

r − minδ(n) ,δ(n+1)

maxδ(n) ,δ(n+1)

)⟩

PP (r) =⟨

δ

(

r − minδ(n), δ(n+1)maxδ(n), δ(n+1)

)

⟩

=1

s2

∫ ∞

0

d δn

∫ ∞

0

d δn+1δ

(

r − minδ(n), δ(n+1)maxδ(n), δ(n+1)

)

exp

(

−δn

s

)

exp

(

−δn+1

s

)

53

s is the mean level spacing. On splitting the integral into two parts, first part

has δ(n) > δ(n+1) while for the second part δ(n) < δ(n+1).

PP (r) =1

s2

∫ ∞

0

d δn

∫ δn

0

d δn+1δ

(

r − δn+1

δn

)

exp

(

−δn

s

)

exp

(

−δn+1

s

)

+1

s2

∫ ∞

0

d δn+1

∫ δn+1

0

d δnδ

(

r − δnδn+1

)

exp

(

−δn

s

)

exp

(

−δn+1

s

)

=2

s2

∫ ∞

0

d δn

∫ δn

0

d δn+1δ

(

r − δn+1

δn

)

exp

(

−δn

s

)

exp

(

−δn+1

s

)

Since r ≤ 1 by definition and the peak of the δ-funtion is at rδn therefore, the

integral over δn+1 picks up this value.

PP (r) =2

s2

∫ ∞

0

d δn

[

δn exp

(

−(1 + r)δn

s

)

]

(2.12)

On evaluating the above integral the distribution of r is given by Pp = 2(1+r)2

and

[r]p → 2 ln 2 − 1 ∼= 0.39. Note that our model is integrable at h = 0, so will

not show GOE level statistics in that limit, and this effect is showing up for our

smallest L and lowest h in Fig. 2.8.

The crossings of the curves for different values of L in Figs. 2.6 and 2.8 give

estimates of the location hc of the phase transition. Both plots show these estimates

“drifting” towards larger h as L is increased, with the crossings at the largest L

being slightly above h = 3. In both cases this “drifting” is also towards the localized

phase, suggesting the behavior at the phase transition is, by these measures, more

like the localized phase than it is like the ergodic phase. This drift towards the

localized phase could also be due to the choice of the observable. Due to the

lack of a finite-size scaling theory for the transition, our choice of r could have

54

0.5 2.5 4.5 6.5 8.5 10.5 12.50.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

[rα (

n) ]

h

8

10

12

14

16

Figure 2.8: The ratio of adjacent energy gaps (defined in the text). The samplesize L is indicated in the legend. In the ergodic phase, the system has GOE levelstatistics, while in the localized phase the level statistics are Poisson.

contributions from irrelevant operators in the RG sense whose finite size effects

have a slower decay with increasing system size. Thus, resulting in the drift of

the critical point. This issue can be be possibly addressed either by changing the

observable or the hamiltonian which may reverse the direction of the drift and/or

reduce the size of the finite-size effect from the irrelevant operator.

55

2.5 Spatial correlations

To further explore the finite-size scaling properties of the many-body localization

transition in our model, we next look at spin correlations on length scales of order

the length L of our samples. One of the simplest correlation functions within a

many-body eigenstate |n〉 of the Hamiltonian of sample α is

Czznα(i, j) = 〈n|Szi Szj |n〉α − 〈n|Szi |n〉α〈n|Szj |n〉α . (2.13)

0 1 2 3 4 5 6 7 8 9−13

−12

−11

−10

−9

−8

−7

−6

−5

−4

−3

[ln

|C

nα

zz(i,i+

d)|

]

d

810121416

h=0.6

h=3.6

h=6.0

Figure 2.9: The spin-spin correlations in the many-body eigenstates as a function ofthe distance d. The sample size L is indicated in the legend. The correlations decayexponentially with d in the localized phase (h = 6.0), while they are independentof d at large d in the ergodic phase (h = 0.6). Intermediate behavior at h = 3.6,which is near the localization transition, is also shown.

56

Due to periodic boundary conditions, the correlations are only shown up to

d = L/2. For d > L/2 the correlation is identically equal to the correlation at

distance L − d for each eigenstate |n〉. This particular correlation function has a

sum-rule due to the global Sz conservation which proves useful at evaluating the

thermal expectation value of the correlation at infinite temperature. For fixed i

(j), the correlation function summed over j (i) results in zero for every eigenstate

and realization of disorder.

∑

j

Czznα(i, j) = 0 . (2.14)

In Fig. 2.9 we show the mean value [ln |Czznα(i, i+ d)|] as a function of the

distance d between the two spins for representative values of h in the two phases

and near the phase transition. Data are presented for various L. This correlation

function behaves very differently in the two phases:

In the ergodic phase, for large L this correlation function should approach its

thermal equilibrium value. For the states with zero total Sz that we look at,

〈n|Szi |n〉 ∼= 0 in the thermal eigenstates of the ergodic phase. The thermal corre-

lation at infinite temperature at large distances for finite system sizes is entirely

constrained by the sum-rule as the Boltzmann weight (e−βH) tends to 1. Correla-

tion on the same site is: Czznα(i, i) = 1/4. Therefore,

∑

j 6=i

Czznα(i, j) = −1

4. (2.15)

The conservation of total Sz does result in anti-correlations so that Czznα(i, j) ≈

−1/(4(L − 1)) for well-separated spins. These distant spins at sites i and j are

57

entangled and correlated: if spin i is flipped, that quantum of spin is delocalized

and may instead be at any of the other sites, including the most distant one.

These long-range correlations are apparent in Fig. 2.9 for h = 0.6, which is in the

ergodic phase. Note that at large distance the correlations in the ergodic phase

become essentially independent of d = |i− j| at large L and d, confirming that the

spin flips are indeed delocalized. Although we only plot the absolute value of the

correlations, in fact these correlations are almost all negative, as expected, in this

large L ergodic regime.

In the localized phase, on the other hand, the eigenstates are not thermal and

〈n|Szi |n〉 remains nonzero for L→ ∞. If spin i is flipped, within a single eigenstate

that quantum of spin remains localized near site i, with its amplitude for being

at site j falling off exponentially with the distance: Czznα(i, j) ∼ exp (−|i− j|/ξ),

with ξ the localization length. In the localized phase the typical correlation and

entanglement between two spins i and j thus fall off exponentially with the distance

|i− j| (except for |i− j| near L/2, due to the periodic boundary conditions). This

behavior is apparent in Fig. 2.9 for h = 6.0, which is in the localized phase and

has a localization length that is less than one lattice spacing. We note that in

the localized phase, as well as near the phase transition, the long distance spin

correlations Czz are of apparently random sign.

The data of Figs. 2.3-2.10 show the existence of and some of the differences

between the ergodic and localized phases. We have also looked at entanglement

spectra of the eigenstates, which also support the robust existence of these two

phases. In addition to confirming the existence of these two distinct phases, we

would like to locate and characterize the many-body localization phase transition

between them. However, in the absence of a theory of this transition, the nature

58

Figure 2.10: The excess fraction of states with anti-correlations at distance |i−j| =L/2. In the ergodic side the correlations are mostly negative while in the localizedcase positive and negative correlations are equally likely in which case the fractiontends to zero for larger system sizes.

of the finite-size scaling is uncertain, which makes it difficult to draw any strong

conclusions from these data with their modest range of L. In studies of ground-

state quantum critical points with quenched randomness, very broadly speaking,

one first step is to classify the transitions by whether they are governed (in a

renormalization group treatment) by fixed points with finite or infinite randomness

[58–61]. In this real-space RG technique, as the local high-energy terms of the

Hamiltonian above the cutoff are traced out, infinite randomness at the critical

59

point results in broad distribution functions for the coupling constants and of long

distance correlations.

−30 −25 −20 −15 −10 −50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

φ=ln|Cnαzz (i,i+L/2)|

P(φ

)

0.6

1.8

2.7

3.6

5.0

6.0

8.0

Figure 2.11: The probability distributions of the natural logarithm of the longdistance spin-spin correlation in the many-body eigenstates for sample size L = 16and the values of the random field h is indicated in the legend.

To explore this question for our system, we next look at the probability distri-

butions of the long distance spin correlations. For quantum-critical ground states

governed by infinite-randomness fixed points, these probability distributions are

found to be very broad [58–60]. In particular, we look at

φ = ln |Czznα(i, i+ (L/2))| , (2.16)

60

whose probability distributions for L = 16 are displayed in Fig. 2.11 for various

values of h. Note the distributions are narrow, as expected, in the ergodic phase

and consistent with log-normal, as expected, in the localized phase. In between,

in the vicinity of the apparent phase transition, the distributions are quite broad

and asymmetric.

To construct a dimensionless measure of how these distributions change shape

as L is increased, we divide φ by its mean, defining η = φ/[φ]. Then we quantify the

width of the probability distribution of η by the standard deviation σ =√

[η2]− 1.

This quantity is shown in Fig. 2.12 vs. h for the various values of L. By this mea-

sure, in both the ergodic and localized phases the distributions become narrower

as L is increased, as can be seen in Fig. 2.12. This happens in the localized phase

because although the mean of −φ grows linearly in L, the standard deviation is

expected to grow only ∼√L. Over the small range of L that we can explore, σ is

found to decrease more slowly than the expected L−1/2 in the localized phase, but

it does indeed decrease.

This scaled width σL(h) of the probability distribution of φ as a function of

the random field h for each sample size L shows a maximum between the ergodic

and localized phases. In the vicinity of the phase transition, σ actually increases

as L is increased, suggesting that its critical value is nonzero, like for quantum-

critical ground states that are governed by an infinite randomness fixed point.

This suggests the possibility that this one-dimensional many-body localization

transition might also be in an infinite-randomness universality class. The peak in

this plot is close to h = 4, and is thus suggesting a slightly higher estimate of hc

than the crossings in Figs. 2.6 and 2.8.

61

0.5 2.5 4.5 6.5 8.5 10.512.50.15

0.2

0.25

0.3

0.35

h

σL

810121416

Figure 2.12: The scaled width σ of the probability distribution of the logarithmof the long-distance spin correlations (see text). The legend indicates the samplelengths L. In the ergodic phase at small h and in the localized phase at largeh, this width decreases with increasing L, while near the transition it increases.To produce the one-standard-deviation error bars shown, we have calculated theσ (see text) for each sample by averaging only over sites and eigenstates withineach sample, and then used the sample-to-sample variations of σ to estimate thestatistical errors. We have also (data not shown) calculated σ by instead averagingφ and φ2 over all samples; this produces scaling behavior for σ that is qualitativelythe same as shown here, but with σ somewhat larger in the localized phase andnear the phase transition.

62

2.6 Dynamics

In the study of the spectral and localization properties of noninteracting particles

in finite samples (such as quantum dots), there are two very important energy

scales: the level spacing δ and the Thouless energy ET . The Thouless energy is

~ times the rate of diffusive relaxation on the scale of the sample. The diffusive

(nonlocalized or ergodic) phase is where ET is larger than δ, and for d-dimensional

samples with d ≥ 3, the localization transition occurs when these two energy

scales are comparable. Since the single-particle level spacing in a d-dimensional

system of linear size L behaves as δ ∼ L−d and this sets the relaxation time at

the localization transition, the dynamic critical exponent for the single-particle

localization transition is z = d.

A possibility that we will now investigate is that the many-body localization

transition also occurs when the Thouless energy is of order the many-body level

spacing. Since the many-body level spacing behaves as log δ ∼ −Ld, this corre-

sponds to an infinite dynamic critical exponent z → ∞. Note also that even for

our model with d = 1 this is a stronger divergence of the critical time scales than

occurs at the known infinite-randomness ground-state quantum critical points,

where log δ ∼ −Lψ with ψ ≤ 1/2.

It is important to note that the model (1) we study has two globally conserved

quantities; total energy and total Sz. Their respective transport times (and hence

their corresponding Thouless energy) in the ergodic phase may have different scal-

ing properties close to the critical point. By studying the relaxation of the spin

modulation, M1, we are specifically probing the spin transport time which may

diverge differently from the energy transport time close to the critical point. Such

63

Figure 2.13: Energy levels in the middle of the band for h = 1.0 and 8.0 of systemsize L = 16. In the localized phase, ET ≈ δ while in the ergodic phase it is many> δ

a possibility has been discussed in the context of zero-temperature metal-insulator

transitions [62] and may play a role in deciding the universality class of the many-

body localization transition.

Naively, the Thouless energy is set by the relaxation rate of the longest-

wavelength spin density modulation, M1. If the scaling at the many-body lo-

calization transition is such that the Thouless energy is of order the many-body

level spacing, then at the transition a nonzero fraction of the dynamic part of

〈M1〉 should be from its matrix elements between adjacent energy levels, and this

64

0.5 2.5 4.5 6.5 8.50

0.05

0.1

0.15

0.2

0.25

[Pα (

n) ]

h

8

10

12

14

16

Figure 2.14: Contribution to the dynamic part of 〈M1〉 from matrix elementsbetween adjacent energy states (see text). In the ergodic and localized phase thecontribution is decreasing to zero with increasing sample size. The sample sizeL is indicated in the legend. The maximum contribution from adjacent states isclose to the critical point.

fraction should remain large as L is increased. In each sample α, the contribution

of a given eigenstate |n〉 to the dynamic part of 〈M1〉 is given by

(∆M1)(n)α = 〈n|M †

1M1|n〉 − |〈n|M1|n〉|2 . (2.17)

In the ergodic phase, (∆M1)(n)α has significant contributions from matrix ele-

ments with many other eigenstates, and the Thouless energy is a measure of the

energy range over which these contributions occur. To quantify this, we define

65

Q(n)iα as the contribution to the dynamic part of 〈M1〉 from the matrix elements

between state n and states n± i:

Q(n)iα = |〈n− i|M1|n〉|2 + |〈n|M1|n+ i〉|2 (2.18)

in sample α. Note that∑

i 6=0

Q(n)iα = (∆M1)

(n)α . (2.19)

We define P (n)α = Q

(n)1α /(∆M1)

(n)α as the fraction of the longest-wavelength “dif-

fusive” dynamics that is due to interference between adjacent (i = 1) many-body

energy levels. Fig. 2.14 shows this quantity averaged over disorder realizations

and states.

If at the localization transition the Thouless energy ET is proportional to the

many-body level spacing δ, then [P ] should remain nonzero in the limit L → ∞.

We do indeed find a strong peak in this fraction near the many-body localiza-

tion transition, and that the fraction is large and not decreasing much as L is

increased. Note that the level spacing decreases by almost a factor of 4 for every

increase of L by two additional spins, so near the transition the Thouless energy

is apparently decreasing by almost the same factor as L is increased. This seems

at least consistent with ET ∼ δ scaling, and thus dynamic exponent z → ∞. In

the localized phase, the dynamics is due to spin-moves that are short-range in real

space (probably of order the localization length). These spin-hops involve pairs of

many-body eigenstates that become far apart (large i) for large L; this is why [P ]

drops with increasing L in the localized phase. Note that the peak in [P ] occurs

a little below h = 3. If one ignores L = 8, the location of this peak is apparently

66

drifting to larger h with increasing L, consistent with our other rough estimates

of hc.

0 0.2 0.4 0.6 0.8 10.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

P(f

)

f

1016

Figure 2.15: Probability distribution of the dynamic fraction of 〈M1〉 for L = 10and 16. Close to the transition for h = 3.0, the distribution becomes broader andmore bimodal with increasing L.

The dynamic fraction [f(n)α ] (Fig. 2.6) tends to 1 in the ergodic phase and

decreases to 0 in the localized phase. The probability distribution of f (n)α (P (f)) is

strongly peaked around 1 and 0 in these respective phases. At the phase transition,

this distribution could either be peaked at the critical fc, broadly distributed, or

even bimodal with peaks near both zero and one. In Fig. 2.15, we show P (f) for a

disorder strength h = 3.0 close to the estimated transition, for system sizes 10 and

16. This distribution P (f) becomes broader and more bimodal with increasing L.

67

This feature of the distribution is consistent with the indication from Fig. 6 that

the critical point may be governed by a strong disorder fixed point.

2.7 Entanglement

One of the intriguing outcomes of quantum mechanics is the notion of entangle-

ment. A collection of degrees of freedom (e.g., a system of many-particles or just

two disparate degrees of freedom of a single particle but with different quantum

numbers (spin and orbital angular momentum)) can exist in a state which has

correlations not describable by correlations between effective classical degrees of

freedom. It is one of the fundamental features which renders a system its quantum

nature (and gives rise to uncertainty). It is more easily described mathematically

rather than through an intuitive picture. Consider a pure state ψ of a system

defined on a Hilbert space H. On dividing the Hilbert space into two parts S and

S such that, H = HS ⊗ HS, the state is expressible as a a linear combination of

basis states of S and S

|ψ〉 =N∑

i=1

M∑

α=1

xiα|i〉S ⊗ |α〉S (2.20)

where |i〉S and |α〉S are the basis states of HS and HS while N and M are their

dimensions respectively. The simplest example is of two localized spin-12

degrees

of freedom in a singlet state.

|ψ〉 = | ↑〉S ⊗ | ↓〉S − | ↓〉S ⊗ | ↑〉S√2

68

Figure 2.16: S + S form an isolated system. S is the subsystem while S serves asthe environment to the subsystem. l . L

A pure state is said to be quantum mechanically entangled in space H if it is

not expressible as a direct product of state vectors in HS and HS.

|ψ〉H 6= |φ〉S ⊗ |ξ〉S

If this is true, the degrees in these two parts of the space are entangled. This

idea of defining entanglement between exactly two parts is specifically called bi-

partite entanglement. There is still no unambiguous way to define multipartite

entanglement and is a question of ongoing research.

A more precise mathematical definition of entanglement can be given in terms

of a density matrix. The density matrix ρ of a pure state is |ψ〉〈ψ|. If |ψ〉 is

normalized ρ satisfies the constraint

TrH [ρ] = 1 (2.21)

The reduced density matrix is evaluated from ρ by taking a partial trace over

one part of the Hilbert space

69

ρS = TrS [ρ]

ρS = TrS [ρ]

For the simple case of two spins, tracing out spin B for a product state (spin-

triplet with maximal spin) | ↑〉S ⊗ | ↑〉S the density matrix is

ρprodS =

1 0

0 0

while for the singlet state the reduced density matrix is

ρentangledS =

12

0

0 12

For these simple cases, the matrix only has diagonal entries. In general, there

will be off-diagonal matrix elements. One particular of way of probing entangle-

ment is by studying the spectrum of ρS - λi. For a general product state, the

spectrum is always λ = 1 while all the other eigenvalues are 0. In the the case

of the singlet state, the eigenvalues are λ1 = λ2 = 12. This is the maximally en-

tangled state. The contraint in Eq. 2.21 gives an additional meaning to the λs.

The normalization and positivity of λ allows us to treat the distribution of λ as

a probability distribution. A distribution dominated by one value of λ near 1 im-

plies a less entangled state while a more uniform distribution means closer to being

maximal entanglement. What are the physically measurable consequences of this

distribution remains to be explored (In analogy with energy-level statistics where

70

they are tied to conductance fluctuations in mesoscopic systems). Drawing from

our knowledge of statistical physics, this probability distribution can be assigned

an entropy (a.k.a. von Neumann entropy), called the entanglement entropy.

Sentanglement = −N∑

i=1

λi lnλi (2.22)

Thus, a product state has zero entanglement entropy while a maximally en-

tanglement state maximizes entropy. (Smaxentanglement = lnN ; Note this has the right

scaling for entropy to be extensive. We should expect the thermal state to be

similar to a maximally entangled state) There are other measures which have also

been studied like the Rényi entropy. This is analogous to the moments of the

distribution of λ

SqRényi =1

1− qln TrS (ρ

qS) =

1

1− qln

(

N∑

i=1

λqi

)

(2.23)

Entanglement is now beginning to be understood as the fundamental source of

local decoherence in a system. Consider a spin which is coupled to a collection of

other spins just like our spin chain. Even though this entire system is isolated from

external noise, under the dynamics of its own hamiltonian any particular spin or

cluster of spins generically loses the coherence of the intial state by entangling with

the rest of the spins with time. Eventually, if the system is ergodic any subsystem

must reach the equilibrium state described by the Gibb’s distribution

ρS = ρGibbs =e−

HS

kBT

Z(2.24)

71

Assuming the entire system starts from a normalized pure state |ψ〉 =∑NM

n=1 cn|n〉 with a well-defined mean energy E with small fluctuations around it.

The corresponding density matrix is given by

ρ0 =∑

n,m

c∗ncm|n〉〈m|

|n〉 and |m〉 are the energy eigenstates of the entire system. At a later time,

the reduced density matrix for S is given by

ρS(t) =∑

n,m

c∗ncm exp(−i(En − Em)t) TrS (|n〉〈m|) (2.25)

For n 6= m, the relative phases are randomized with time and hence, average

to zero. For the purposes of thermalization, the behaviour of the diagonal term

(n = m) becomes important. One of the consequences of Eigenstate thermalization

hypothesis is that TrS (|n〉〈n|) is typically independent of n.

ρS(∞) ≈ TrS (|n〉〈n|)

What should be the entanglement properties of the eigenstate for the final equi-

librium state to be thermal? For an arbitrary state given by Eq. 2.20 the density

matrix is given by 1

1I will drop the subscript S and S from now on. Use of Roman letters would mean a state inS while of Greek letters would imply a state in S.

72

ρ =∑

i,j

∑

α,β

xiαx∗jβ|i〉〈j| ⊗ |α〉〈β| (2.26)

⇒ ρS = TrS ρ =M∑

β=1

〈β|ρ|β〉 (2.27)

=

N∑

i,j=1

M∑

α=1

xiαx∗jα|i〉〈j| (2.28)

=

N∑

i,j=1

Uij|i〉〈j| (2.29)

where U = X¯X¯† where U is aN×N matrix whileX is aN×M matrix. Inspired

by the Berry conjecture where the amplitude of a quantum chaotic wavefunction

is a gaussian random function and the results from random matrix theory, let us

consider the case where the elements of the matrix X are drawn from a gaussian

distribution and X is a complex matrix.

P (Xij) ∝ exp(

−Tr(

X¯X¯†))

(2.30)

The properties of such a random matrix has been studied extensively in statis-

tics and goes by the name Wishart matrix. The spectrum of the Wishart random

matrix with the normalization constraint∑N

i=1 λi = 1 had been studied from the

point of view of quantum entanglement in a random pure state as far back as 1978

[63]. The probability distribution of the eigenvalues is

PWishart(λi) ∝ δ

(

N∑

i=1

λi − 1

)

N∏

i=1

λM−Ni

∏

j<k

(λj − λk)2 (2.31)

73

The average entropy of a subsystem was calculated by several authors [63–65]

in the limit N ≪M and they found the state to be very close to being maximally

entangled.

Srandom = lnN − N

2M(2.32)

Thus, a random pure state has an extensive amount of entropy in a subsystem.

The treatment in terms of a random pure state assumes the state being sampled

uniformly from the space of states without any reference to the Hamiltonian. This

misses out that in a microcanonical ensemble the states are limited to an energy

shell. There has been some work done mostly from a mathematical physics point of

view where the microcanonical condition was imposed [66, 67] along with studying

the dynamics [68, 69]. But these also fall short of studying realistic quantum many-

body hamiltonians and in the presence of disorder. They are successful at proving

the approach to thermal equilibrium under some very general assumptions.

In Fig. 2.17 the entanglement entropy of the energy eigenstates averaged over

the mid one-third states and disorder realizations are shown. The eigenstates were

evaluated with open boundary conditions and the entanglement spectrum is for

one half of the system traced out.

Sentanglement = −∑

i

λi lnλi

At low disorder when the system is thermal, entanglement entropy tends to

its thermal value ∼ L2ln(2) which is extensive in subsystem size. The entropy

per site is ln 2 consistent with the infinite temperature result. While deep in the

74

Figure 2.17: Entanglement entropy as a function of system size: L = 8 to 14. Thelegend indicates the disorder strengths. The dashed line has slope ln(2).

localized phase, the entanglement entropy is independent of system size for this

one-dimensional system.

Besides the entanglement entropy, even the distribution of entanglement spec-

trum has distintive features in the 2 phases. In the localized phase, it is dominated

by a few values close to 1 showing that the eigenstates have very low entanglement.

While the delocalized phase the spectrum is distributed much more evenly.

75

Figure 2.18: Average of the logarithm of the entanglement spectrum plotted versusi (It is defined such λi > λi+1) for L = 14. The legend indicates the disorderstrengths. For stronger disorder, λi ≈ 0 within machine precision for larger i andhence are left out of the plot.

2.8 Summary

This study of the exact many-body eigenstates of our model 2.1 has demonstrated

some of the properties of the ergodic and localized phases. We also find a rough

estimate of the localization transition using various different diagnostics. Based

on earlier work by Oganesyan and Huse [39], the many-body energies go from

having GOE to Poisson level statistics with increasing disorder. The scaling of the

probability distributions of the long-distance spin correlations suggests that the

transition might be governed by an infinite-randomness fixed point with dynamic

76

critical exponent z → ∞. We also study the relaxation of spin modulation under

the dynamics of the Hamiltonian. In this case our results are consistent with

ET ∼ δ scaling at criticality, in apparent agreement with our earlier conclusion of

z → ∞ at the transition. These results suggest that efforts to develop a theory of

this interesting phase transition should consider the possibility of a strong disorder

renormalization group approach [70]. Of course, the model we have studied is only

one-dimensional, and the behavior of this transition in higher dimensions might

be different in important ways.

77

Chapter 3

Energy transport in disordered

classical spin chains

3.1 Classical many-body localization?

Setting aside the question of the existence of a many-body localization transition

(i.e., assuming it does exist), one might wonder about its nature, e. g., the univer-

sality class. On the one hand, the theoretical analysis of Basko, et al. [25] relies

entirely on quantum many-body perturbation theory. Rather generally, however,

one expects macroscopic equilibrium and low-frequency dynamic properties of in-

teracting quantum systems at nonzero temperature to be describable in terms of

effective classical models. This expectation is certainly borne out in a variety of

symmetry-breaking phase transitions with a diverging correlation length, such as,

78

e. g., a finite temperature Néel ordering of spin-1/2 moments. One can begin to

understand the microscopic mechanism behind such a many-body “correspondence

principle" as a consequence of an effective coarse-graining, whereby the relevant

degrees of freedom are correlated spins moving together in patches that grow in

size as the phase transition is approached and therefore become “heavy” and pro-

gressively more classical. Further extension of these ideas to general, non-critical,

dynamical response is more involved: roughly speaking, it requires that the typical

many-body level spacing in each patch be much smaller than the typical matrix

element of interactions with other patches. If this is true (as it is in most mod-

els at finite temperature, though not necessarily in the insulating phase analysed

by Basko and collaborators) one replaces microscopic quantum degrees of freedom

with macroscopic classical ones, which typically obey “hydrodynamic” equations of

motion at low frequencies [71]. Since it is expected that the many-body localization

transition is accompanied by a diverging correlation length (akin to the Anderson

transition) one might expect some sort of classical description to emerge en-route

from the localized phase to the diffusive phase. It was this thinking that initially

motivated us to consider the possibility of classical many-body localization.

The process by which collective classical (hydro-) dynamics emerges from a

microscopic quantum description is subtle and may or may not be relevant to the

many-body localization discussed above. A somewhat less subtle, but apparently

largely unexplored related question, is whether nonlinear, interacting and disor-

dered classical many-body systems are capable of localization at nonzero tem-

perature. To be precise, a many-body classical dynamical system with a local

Hamiltonian (including static randomness) should show hydrodynamic behavior,

e.g. energy diffusion, provided the local degrees of freedom are nonlinear and inter-

79

acting, and the disorder is not too strong. In this regime, the isolated system can

function as its own heat bath and relax to thermal equilibrium. Diffusive energy

transport must stop if the interactions between the local degrees of freedom are

turned off. How is this limit approached? Can there be a classical many-body

localization transition where the energy diffusivity vanishes while the interactions

remain nonzero? These are the basic questions we set out to investigate in this

work.

Our preliminary conclusion is that classical many-body systems with quenched

randomness and nonzero nonlinear interactions do generically equilibrate, so there

is no generic classical many-body localized phase. Our picture of why this is true is

that generically a nonzero fraction of the nonlinearly interacting classical degrees

of freedom are chaotic and thus generate a broad-band continuous spectrum of

noise. This allows them to couple to and exchange energy with any other nearby

degrees of freedom, thus functioning as a local heat bath. Random classical many-

body systems generically have a nonzero density of such locally-chaotic “clusters”,

and thus the transport of energy between them is over a finite distance and can

not be strictly zero, resulting in a nonzero (although perhaps exponentially small)

thermal conductivity. Quantum systems, on the other hand, can not have a finite

cluster with a truly continuous density of states: the spectrum of a finite cluster

is always discrete. Thus the mechanism that we propose forbids a generic classical

many-body localized phase, yet it does not appear to apply to the quantum case.

The proposed existence of the many-body insulator in quantum problems is then

a remarkable manifestation of quantum physics in the macroscopic dynamics of

highly-excited matter. In this work we shall primarily focus on macroscopic low

frequency behavior, postponing detailed analysis of local structure of noise and

80

its relation to transport. Our conclussions are broadly consistent with findings

of Dhar and Lebowitz [72] although given the rather major differences in models,

methods and, most importantly, the extent to which the strongly localized regime

is probed we refrain from making direct comparisons.

We study energy transport in a simple model of local many-body Hamiltonian

dynamics that has both strong static disorder and interactions: a classical Heisen-

berg spin chains with quenched random fields. For simplicity, we consider the

limit of infinite temperature, defined by averaging over all initial conditions with

equal weights. Our systems conserve the total energy and should exhibit energy

diffusion; they have no other conservation laws. The energy diffusion coefficient,

D, can be deduced from the autocorrelations of the energy current (as explained

below) and is shown in Fig. 3.1 as a function of the strength of the spin-spin

interactions, J . The mean-square random field is ∆2, and as we vary J we keep

2J2 + ∆2 = 1, as explained below. The limit J → 0 is where the interactions

vanish, so there is (trivially) no energy transport.

As the interaction J is decreased, the thermal diffusivity D decreases very

strongly; we have been able to follow this decrease in D for about 5 orders of

magnitude before the systems’ dynamics become too slow for our numerical studies.

For most of this range, we can roughly fit D(J) with a power-law, D ∼ Jγ, with

a rather large exponent, γ ∼= 8, as illustrated in Fig. 3.1. This large exponent

suggests that the asymptotic behavior at small interaction J may be some sort of

exponential, rather than power-law, behavior, consistent with the possibility that

the transport is actually essentially nonperturbative in J . In principle, it is also

possible to fit these data to a form with a nonzero critical Jc, so thatD(J < Jc) = 0

— such fits prove inconclusive as they produce estimates of Jc considerably smaller

81

-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2log10J

-6

-5

-4

-3

-2

-1

log10D

Figure 3.1: Disorder-averaged energy diffusion constant D as a function of thespin-spin interaction J . The line has slope 8 on this log-log plot.

than the values of J where we can measure a nonzero D. Since we are not aware

of any solid theory for the behavior of D(J), these attempts at fitting the data

are at best suggestive. The large range of variation of the macroscopic diffusion

constant D across a rather modest range of J is the most clearly remarkable and

robust finding that we wish to present in this work.

Our model and general methods employed will be presented and discussed in

the next Section. Much of what we present is based on the analysis of energy

current fluctuations in isolated rings. For various reasons we have found it benefi-

cial to focus on these rather than fluctuations of the energy density or on current

carrying states in open systems (we have spot-checked for quantitative agreement

among these three methods). In section 3.3 we present our results for macroscopic

82

transport starting from short time behavior that is relatively easy to understand

and working up to long time, DC behavior that is both difficult to compute and

as of yet poorly understood. One particularly interesting observation we make

here is that of a subdiffusive behavior over a substantial time range at weaker

interactions, apparently distinct from the much discussed mode-coupling behavior

well representative of linear diffusion in the presence of disorder. We discuss some

afterthoughts and open problems in the Summary, with some important additional

details in Sections 3.6, 3.5 (such as quantitative explorations of finite-size effects,

roundoff, many-body chaos and self-averaging).

3.2 Model, trajectories and transport

The classical motion of N interacting particles is usually defined by a system of

coupled differential equations of motion. The “particles" we study here are classical

Heisenberg spins – three-component unit-length vectors, Si, placed at each site i

of a one-dimensional lattice. With a standard angular momentum Poisson bracket

and a Hamiltonian, H , the equations of motion are

∂Sj∂t

= Hj × Sj, (3.1)

where Hj = ∂H/∂Sj is the total instantaneous field acting on spin Sj. The

Hamiltonians we consider are all of the form

H =∑

j

(hj · Sj + JSj · Sj+1), (3.2)

83

with uniform pairwise interaction J between nearest-neighbor spins, and

quenched random magnetic fields, hj . For almost all of the results in this work,

we choose the random fields to be hj = hjnj , where the hj are independent

Gaussian random numbers with mean zero and variance ∆2, while the nj are

independent randomly-oriented unit vectors, uniformly distributed in orientation.

Because of the random fields, total spin is not conserved and we can focus on

energy diffusion as the only measure of transport in this system. For J = 0 and

∆ > 0 any initial distribution of energy is localized, as the spins simply precess

indefinitely about their local random fields, so the diffusivity is D = 0. In the

opposite limit, where ∆ = 0 and J > 0, there is diffusive transport with D ∼ J

(with nonlinear corrections due to the coupling between energy and spin diffusion

[71, 73]). We are interested in the behavior of D as one moves between these two

limits, especially as one approaches J = 0 with ∆ > 0.

Given initial spin orientations, it is in principle straightforward to integrate

the equations of motion numerically, thus producing an approximate many-body

trajectory. Correlation functions can then be computed and averaged over a such

trajectories and over realizations of the quenched random fields. The transport

coefficients can thereby be estimated via the fluctuation-dissipation relations.

3.2.1 The Model

Before we embark on this program, however, we start by making a change to the

model’s dynamics (3.1), but not to its Hamiltonian (3.2), in order to facilitate

the numerical investigation of the long-time regime of interest to us, where the

diffusion is very slow. In order to get to long times with as little computer time

84

as possible, we want our basic time step to be as long as is possible. What we are

interested in is not necessarily the precise behavior of any specific model, but the

behavior of the energy transport in a convenient model of the type (3.2). Since

we are studying energy transport, it is absolutely essential that the numerical

procedure we use does conserve total energy (to numerical precision) and that the

interactions and constraints remain local. Thus we modify the model’s dynamics

to allow a large time step while still strictly conserving total energy.

We change the equations of motion (3.1) of our model so that the even- and

odd-numbered spins take turns precessing, instead of precessing simultaneously.

We will usually have periodic boundary conditions, so we thus restrict ourselves

to even length (thus bipartite) chains. We use our basic numerical time step as

the unit of time (and the lattice spacing as the unit of length). During one time

step, first the odd-numbered spins are held stationary, while the even-numbered

spins precess about their instantaneous local fields,

Hr(t) = hr + JSr−1(t) + JSr+1(t) , (3.3)

by the amount they should in one unit of time according to (3.1). Note that

since the odd spins are stationary, these local fields on the even sites are not

changing while the even spins precess, so that this precession can be simply and

exactly calculated, and the total energy is not changed by this precession. Then

the even spins are stopped and held stationary in their new orientations while the

odd spins “take their turn” precessing, to complete a full time step. Although this

change in the model’s dynamics from a continuous-time evolution to a discrete-

time map is substantial, we do not expect it to affect the qualitative long-time,

85

low-frequency behavior of the model that is our focus in this work. In particular,

we clearly observe correct diffusive decay of local correlations for weak disorder

and essentially indefinite precession of spins at very strong disorder.

We have decided to use parameters so that the mean-square angle of precession

of a spin during one time step is one radian (at infinite temperature), which seems

about as large as one can make the time step and still be roughly approximating

continuous spin precession. This choice dictates that the parameters satisfy

2J2 +∆2 = 1 . (3.4)

We will generally describe a degree of interaction by quoting the J ; the strength

∆ of the random field varies with J as dictated by (3.4).

3.2.2 Observables

The basic observable of interest, the instantaneous energy ei(t) at site i is

ei(t) = hi · Si(t) +J

2(Si−1(t) · Si(t) + Si+1(t) · Si(t)) . (3.5)

Note that with this definition, the interaction energy corresponding to a given

bond is split equally between the two adjacent sites. When updating the spin at

site i, only the energies of the three adjacent sites, ei and ei±1, change, due to

the change in the interaction energies involving spin i. This rather simple pattern

of rearrangement of energy allows for an unambiguous definition of the energy

current at site i during the time step from time t to t + 1. If site i is even, so it

precesses first, then the current is

86

ji(t) = J [Si(t + 1)− Si(t)] · [Si+1(t)− Si−1(t)] , (3.6)

while for i odd,

ji(t) = J [Si(t + 1)− Si(t)] · [Si+1(t + 1)− Si−1(t+ 1)] . (3.7)

We are working at infinite temperature, or alternatively at β = (kBT )−1 = 0.

The conventionally-defined thermal conductivity vanishes for β → 0 [71]. Instead,

here we define the DC thermal conductivity κ so that the average energy current

obeys

j = κ∇β (3.8)

in linear response to a spatially- and temporally-uniform small gradient in

β = 1/(kBT ). The Kubo relation then relates this thermal conductivity at β = 0

to the correlation function of the energy current via

κ =∑

t

C(t) , (3.9)

where

C(t) =∑

i

[〈j0(0)ji(t)〉] (3.10)

is the autocorrelation function of the total current, where the square brackets,

[. . .], denote a full average over instances of the quenched randomness (“samples”)

and the angular brackets, 〈. . .〉, denote an average over initial conditions in a given

87

sample and time average within a given run. For our model (3.2) the average energy

per site obeys

d[〈e〉]dβ

=J2 +∆2

3(3.11)

at β = 0, and the energy diffusivity D is then obtained from the relation

κ

D=d[〈e〉]dβ

. (3.12)

In a numerical study, if a quantity (such as κ) is non-negative definite, then it

is helpful to measure it if possible as the square of a real measurable quantity. We

use this approach here, noting that

κ = limL,t→∞

1

Lt[〈

t∑

τ=1

L∑

i=1

ji(τ)2〉] . (3.13)

For a particular instance of the random fields in a chain of even length L with

periodic boundary conditions and a particular initial condition I run for time t,

we thus define the resulting estimate of κ as

κI(t) =1

Lt

t∑

τ=1

L∑

i=1

ji(τ)2 . (3.14)

If these estimates are then averaged over samples and over initial conditions

for a given L and t, this results in the estimate κL(t) = [〈κI(t)〉] . These estimates

κL(t) must then converge to the correct DC thermal conductivity κ in the limits

L, t→ ∞.

88

3.2.3 Finite-size and finite-time effects

In a sample of length L, we expect finite-size effects to become substantial on time

scales

t > tL = CDL2/Deff , (3.15)

where Deff is the effective diffusion constant at those time and length scales, and

we find CD ∼= 10 (remarkably Eq. 3.15 remains valid more or less with the same

value of CD across the entire range of parameters – see Section 3.5). With periodic

boundary conditions (which is the case in our simulations) this means that κL(t)

saturates for t > tL to a value different from (and usually above) its true DC value

in the infinite L limit, while with open boundary conditions (no energy transport

past the ends of the chain) the infinite-time limit of κL(t) is instead identically

zero for any finite L. We simply avoid this purely hydrodynamic finite-size effect

by using chains of large enough length L, which is relatively easy, especially in the

strongly-disordered regime of interest, where Deff is quite small. Thus, when the

subscript L is dropped, this means the results being discussed are at large enough

L so that they are representative of the L→ ∞ limit.

For the smallest values of J that we have studied, the system is essentially

a thermal insulator, and the Deff is so small that finite-size effects are just not

visible at accessible times even for small values of L, such as L = 10. Instead,

given the way we are estimating κ, a finite-time effect, due to the sharp “cutoffs”

in time at time zero and t in (3.14), dominates the estimates κL(t) ∼ J2/t in this

small-J regime. To explain this better we can rewrite the definition of κ as

89

κL(t) =1

Lt

t∑

τ=1

t∑

τ ′=1

C(τ − τ ′) =2

Lt

t/2∑

tav=1

κ∗L(tav), (3.16)

where we have assumed an even t (there is an additional term otherwise) and

κ∗L(τ) ≡∑τ

−τ C(τ′). Localization, i.e. zero DC conductivity, implies a rapidly

vanishing κ∗ as well as κ at long times. The latter however acquires a tail, κL(t) ∼

1/t whose amplitude is set by the short-time values of κ∗L.

For the intermediate values of J that are of the most interest to us in this

work, there is also another, stronger finite-time effect due to an apparently power-

law “long-time tail” in the current autocorrelation function, C(τ), as we discuss in

detail below. Importantly, at long times this intrinsic finite-time effect dominates

the extrinsic, cutoff-induced, 1/t effect discussed above, so κL(t) remains a useful

quantity to study in this regime.

3.3 Results: Macroscopic diffusion

3.3.1 Current autocorrelations

Since the total current is not dynamically stationary, its autocorrelation function,

C(t), should decay in time. In a strongly disordered dynamical system we ex-

pect the DC conductivity, which is the sum over all times of this autocorrelation

function, to be very small due to strong cancelations between different time do-

mains (i.e., C(t) changes sign with varying t). The basic challenge of computing

the DC thermal conductivity κ boils down to computing (and understanding) this

cancelation.

90

The autocorrelation function C(t) has three notable regimes as we vary J and

t. First, C(t) is positive and of order J2 at times less than or of order one, as

illustrated in Fig. 3.2. It quickly becomes negative at larger times. For small

J it is negative and of order J3 in magnitude for times of order 1/J (see Fig.

3.3). For very small J , this negative portion of C(t) almost completely cancels the

short-time positive portion, resulting in an extremely small κ∗ (see inset in Fig.

3.3). This cancelation is a hallmark of strong localization and can be observed,

e.g. in an Anderson insulator where it is nearly complete. While the very short

time behavior at small J is easily reproduced analytically by ignoring dynamical

spin-spin correlations, the behavior out to times of order 1/J is representative of

correlated motion of few spins (likely pairs). Although likely non-integrable, this

motion is nevertheless mostly quasiperiodic — we recorded indications of this in

local spin-spin correlation functions (not shown here).

Finally, there is apparently a power-law long-time tail with a negative ampli-

tude: C(t) ∼ −t−1−x, with an exponent that we find is approximately x ∼= 0.25

over an intermediate range of 0.2 . J . 0.4 (and more generally, perhaps).

To observe this with the least amount of effort it is best to average C(t) at long

times over a neighborhood of t (see Figs. 3.4, 3.5) or to measure κL(t) and compute

its “exponential derivative”, η(t) ≡ κL(t)−κL(2t), at a sequence of points tn = 2n,

n = 1, 2, 3 . . . (see Fig. 3.6). The apparent value x ∼= 0.25 of this exponent is

something that we do not understand yet theoretically. However, we find that it

does provide a good fit to the data over a wide dynamic range, providing some

support for our use of it to extrapolate to infinite time and thus estimate the DC

thermal conductivity, as discussed below.

91

1 2 3 4 5t0.00

0.01

0.02

0.03

0.04

0.05

CJ2

Figure 3.2: Short time behavior of C(t) for J = 0.32 (red, noticeably differenttrace) and J = 0.08, 0.12, 0.16 (these are almost identical data in this plot). Noterescaling of the vertical axis by J2.

3.3.2 DC conductivity: extrapolations and fits

Our extrapolations of the DC conductivity will be based entirely on the long time

behavior of κL(t) evaluated at a set of times tn = 2n with integer n and for large

enough L to eliminate finite-size effects (so we drop the subscript L). We start

by describing the procedure used to arrive at the numerical estimates of the DC

conductivity, then turn to the subject of uncertainties.

A typical instantaneous value of the energy current is set by the strength of the

exchange, J . As a consequence κ(t) ∼ J2 for small J at short and intermediate

times (t of order 1/J or less). Given the time-dependence at intermediate and long

times, as discussed above, we adopt the variable s = (1 + Jt)−0.25 as a convenient

92

2 3 4 5 6 7 8 t J

-0.015

-0.010

-0.005

0.000

0.005CJ3

2 4 6 8t J0.00

0.02

0.04

0.06

0.08Κ*J2

Figure 3.3: Current autocorrelations on medium time scales ∼ 1/J for J = 0.32,0.16, 0.12, 0.08, from top (red) to bottom (green) trace at tJ = 2). Note therescaling of both the vertical and time axes. The inset shows near cancellationbetween short and medium times.

“scaling” of time for displaying our results. These rescalings “collapse” the observed

values of κ(t) for short to intermediate times across the entire range of J studied,

as shown in Fig. 3.7.

The extrapolated values of the DC conductivity decrease strongly as J is re-

duced. Extrapolation of κ(t) to s = 0 and thus DC is fairly unambiguous for

J ≥ 0.32, as can be seen in Fig. 3.7. To display the long-time results at smaller J ,

in Figs. 3.8,3.9 we instead show κ/J10. Here one can see that as we go to smaller

J the extrapolation to the DC limit (s = 0) becomes more and more of “a reach”

as J is reduced.

93

2 3 4 5log10 t

-9

-8

-7

-6

-5

-4log10 H-CL

Figure 3.4: Long-time tail in the current autocorrelation function for J = 0.20,0.24, 0.28, 0.32, 0.40 shown bottom to top in yellow, light green, dark green, lightblue and dark blue, respectively.

The outcomes of these extrapolations and rough estimates of the uncertainties

are summarized in Table 3.1.

There are several sources of uncertainty in the estimates of the DC thermal

conductivity κ reported in Table 3.1. These can be separated into those originating

with the measured values of κL(t) and those due to the extrapolation to DC.

The statistical uncertainties in the measured values of κL(t) were estimated

(and shown in the figures) from sample-to-sample fluctuations which we find follow

gaussian statistics to a good approximation for these long (large L) samples. We

did look for a possible systematic source of error originating with roundoff and its

amplification by chaos (see Section 3.6) and found it not to be relevant for the

values of J and t studied.

94

1 2 3 4 5log10 t

-3.6

-3.4

-3.2

-3.0

-2.8

-2.6

log10 H-C t54L

Figure 3.5: To estimate the exponent we multiply the data by t5/4 (and alsodisplay lines with slope ±0.05). Although these data do not exclude an exponentthat varies with J , we interpret these results as supportive of a single exponentx ≈ 0.25 at asymptotically long times but with a more pronounced short-timetransient at smaller J .

The uncertainties in our estimates of the DC κ from the extrapolation proce-

dure begin with the assumed value of the long-time powerlaw, x ∼= 0.25. Clearly,

using a different exponent will change the extrapolated DC values of κ somewhat.

This uncertainty increases with decreasing J as the ratio of the κL(t) at the last

time point to the extrapolated value increases. At our smallest J values, the

curvature in our κ vs. s plots due to the crossover to the earlier-time insulating-

like ∼ 1/t ∼ s4 dependence becomes more apparent and further complicates the

extrapolation. Although we have experimented some with different schemes for

extrapolating to DC, including different choices of exponent x, in the end the fol-

95

3 4 5 6 7log10 t

-5.5

-5.0

-4.0

-3.5

-3.0

log10 Η

Figure 3.6: Long time tails as seen from η(t) for J = 0.20, 0.24, 0.28, 0.32, 0.36,0.40, 0.48 (bottom to top). Black line is a guide to the eye with slope −1/4.Note that the short-time transients are stronger here, as compared to the auto-correlation data in Fig. 3.4.

lowing procedure appeared to capture the overall scale of the diffusion constant,

and with a generous estimate of the uncertainty: i) we start by removing early

data with s & 0.5 to focus strictly on the long-time behavior; ii) this long time

dependence is further truncated by removing 5 latest points and then fitted to a

polynomial∑4

0 ansn to better capture the curvature apparent in the data – these

fits are shown in Fig. 3.9 and a0 are the DC values reported in Table 3.1; iii) the

uncertainty is estimated as the greater of statistical error in the last point (which

is negligible for most of our data) and the difference between a0 and a simple linear

extrapolation performed on latest five data points not included in (ii).

96

0.0 0.2 0.4 0.6 0.8 1.0s0.00

0.05

0.10

0.15

ΚJ2

Figure 3.7: Variation of κ(t) for J = 0.64, 0.56, 0.48, 0.40, 0.36, 0.32, 0.28, 0.24,0.20, 0.18, 0.16, 0.14, 0.12 plotted vs. s = (1 + Jt)−0.25. Lines are merely guidesto the eye, and statistical errors are too small to be seen on most of these points.This figure is used for obtaining J ≥ 0.36 entries in Table 3.1.

Overall, we deem the values presented in Table 3.1 as “safe” since all extrapo-

lated κ’s differ by at most a factor 2 from κ’s actually measured, in other words

our extrapolations are reasonably conservative (with the exception of two smallest

J ’s where the extrapolation yields stronger reductions).

3.4 Further explorations and outlook

In summary, we considered a rather generic model of classical Hamiltonian many-

body dynamics with quenched disorder, and explored the systematic variation

in the thermal diffusivity between conducting and insulating states. We found

97

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14s0

1000

2000

3000

4000

5000ΚJ10

Figure 3.8: Refer to the caption of Fig. 3.9

J κ δ κ L log2 T samples0.64 0.18J2 0.01J2 5000 20 10000.56 0.09J2 0.01J2 2000 20 10000.48 0.045J2 0.005J2 1000 20 40000.40 0.020J2 0.003J2 1000 20 10000.36 0.014J2 0.003J2 1000 21 22000.32 50J10 8J10 1000 21 160000.28 70J10 10J10 1000 24 9120.24 95J10 20J10 1000 25 5580.20 130J10 30J10 1000 26 11790.18 175J10 50J10 500 27 20000.16 250J10 100J10 500 27 15500.14 400J10 200J10 500 27 5200.12 600J10 400J10 500 27 1116

Table 3.1: Extrapolated estimates of the DC conductivity κ, estimated uncertain-ties, length L of samples, and the number of time steps T of the runs.

98

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14s0

200

400

600

800

1000ΚJ10

Figure 3.9: Same data as in Fig. 3.7, but now scaled and displayed in a waythat allows one to see the extrapolations to s = 0 (t → ∞) for small J . Notethe different rescaling schemes used in preceeding and current plots to focus on“collapse” of short time data (previous plot) vs. long time extrapolations (presentplot). As before black lines are drawn through the data for guiding the eyes.Colored lines are results of polynomial fits, as explained in the text. Figs. 3.8 and3.9 are used for obtaining entries in Table 3.1 for J ≤ 0.32.

a rapid variation of the diffusion constant and presented quantitative estimates

of the latter across more than 5 orders of magnitude of change. The origin of

this behavior may be traced to spatial localization of classical few-body chaos.

Qualitatively, such a scenario is rather plausible at very low J , where most spins

are spectrally decoupled due to disorder and essentially just undergo independent

Larmor precessions. As long as J is nonzero, however, there will always be a

fraction of spins in resonance with some of their immediate neighbors. These

clusters are then deterministically chaotic and thus generate broad-band noise,

99

which allows them to exchange energy with all other nearby spins. Importantly,

in the entire parameter range studied this heterogeneous regime eventually gives

way at long time to a more homogeneous conducting state in the DC limit. Thus,

we suspect that internally generated but localized noise always causes nonzero DC

thermal transport even in the strongly disordered regime, as long as the spin-spin

interaction J is nonzero.

Additionally, we also discovered and characterized an apparent, novel finite-

time (frequency) correction to diffusion, with the diffusivity varying as D(ω) ≈

D(0) + a|ω|x with x ∼= 0.25. Previous theoretical work on corrections to diffusion

due to quenched disorder [74] have instead found a correction with exponent x =

1/2, which is quite inconsistent with our numerical results. This powerlaw behavior

is apparently not due to the localization of chaos discussed above, as it persists well

into the strongly conducting regime (larger J) and also exists in models without a

strong disorder limit at all (e.g. with random fields of equal magnitude but random

direction; data not shown). So far we have not found a theoretical understanding

of these interesting corrections to simple thermal diffusion.

100

3.5 Finite size effects

0.0 0.2 0.4 0.6 0.8 1.0s0.00

0.02

0.04

0.06

0.08ΚJ2

0.0 0.1 0.2 0.3 0.4s0

102030405060

ΚJ10

Figure 3.10: Finite size effects for J = 0.16, 0.32, 0.40: 100 vs. 20 spins forJ = 0.40, 10 and 20 vs. 100 spins for J = 0.32, 20 vs. 100 spins for J = 0.16. Redcolor is used to indicate the data influenced by finite size effects according to Eq.3.15 with CD = 10. Inset: J = 0.16 data replotted.

We have checked that all of the extrapolations above are free from finite size

effects (by comparing against simulations on smaller, and in some cases, larger

samples). Nevertheless, it is interesting to consider the expected hydrodynamic

size effects somewhat quantitatively, via Eq. 3.15. To illustrate this we display in

Fig. 3.10 some results on shorter systems for J = 0.16, 0.32, 0.40 that do show size

effects: due to the periodic boundary conditions, the conductivity in smaller rings

saturates in the DC limit at a value corresponding to the AC value at a “frequency”

101

2π/t∗ corresponding to ∼= CDD(2π/L)2, withD ∼= 3κ. Our results are qualitatively

consistent this with CD ∼= 10 or slightly larger. It perhaps remarkable that despite

orders of magnitude of variation in the diffusion constant in going from J = 0.4

to J = 0.16 the crossover from bulk to finite system behavior is characterized by

roughly the same constant CD ∼= 10.

3.6 Chaos amplification of round-off errors

No numerical study of a nonlinear classical dynamical system is complete without

some understanding of the interplay of discretization and round-off errors and

chaos. We are studying a Hamiltonian system that conserves total energy, so the

chaos is only within manifolds of constant total energy in configuration space.

Thus although round-off errors introduce tiny violations of energy conservation,

these changes in the total energy are not subsequently amplified by the system’s

chaos; we have numerically checked that this is indeed the case. As a result of

this precise energy conservation the energy transport computation remains well-

defined. The simulation is far less stable within an equal-energy manifold, where

nearby trajectories diverge exponentially due to chaos. In particular, this means

that the component of any round-off error that is parallel to the equal-energy

manifold is exponentially amplified by the chaos. At large J this happens rather

quickly, while for small J the chaos is weaker and longer individual trajectories can

be retraced back to their respective initial conditions. However, at small J very

long runs are necessary to extrapolate to the DC thermal conductivity: in the end

all of our extrapolations are done in the regime where all individual many-body

trajectories are strongly perturbed by chaos-amplified round-off errors.

102

Ultimately, however, we are only concerned with the stability of the current

autocorrelations that enter in the Kubo formula for κ. Although the precise tra-

jectories may diverge due to chaos-amplified round-off errors, this need not have

a strong effect on C(t). To study this issue quantitatively we simulated roundoff

noise of different strength in our computations. Specifically, we add extra random

noise to the computation without altering the total energy by multiplying the an-

gle each spin precesses in each time step by a factor of 1 + ηi(t), where the ηi(t)

are independent random numbers uniformly distributed between P and −P (P =

noise strength).

In 400 rings of 500 spins coupled with J = 0.14 we simulated the same initial

condition with different ηi(t), and with different values of simulated noise P =

100, 10−1, 10−2, 0 – these results are presented in Fig. 3.11 below.

As expected, the long-time insulating behavior is weakened by the presence of

noise. Quantitatively, however, we observe little or no difference between results

obtained in the presence of simulated noise with P = 10−2 vs. ones obtained for

intrinsic noise (which at double precision corresponds to Pintr . 10−15). Clearly,

this statement heavily depends on the duration of the simulation, value of J , etc.

Judging from Fig. 3.11 roundoff errors are not a serious source of uncertainty

in our results in main text. Interestingly, it is also possible for strong noise to

suppress κ, as indeed happens at shorter times, which can be traced here to a sort

of “dephasing” of sharp response of quasiperiodic localized states.

103

0.0 0.2 0.4 0.6 0.8 1.0s0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07ΚJ2

0.00 0.02 0.04 0.06 0.08 0.10s0

100020003000400050006000ΚJ10

Figure 3.11: Roundoff effects at J = 0.14: low frequency, long-time conductivityis larger for larger values of P but is essentially indistinguishable between P = 0and P = 0.01. Inset: data with P = 0.1, 0.01, 0.

3.7 Summary

Our work on the dynamics of interacting, disordered systems has given some in-

sight into the physics of the many-body localization transition. But, it has also

raised some challenging theoretical questions. This preliminary work seems to

suggest that though the many-body localization transition is a finite-temperature

transition, there is no effective classsical description of it. Hence, quantum dynam-

ics even at finite temperature can be distinct from classical dynamics when the

system is isolated. It is imperative to establish universality for this class of phase

transitions. Unfortunately, there doesn’t appear to be any obvious theoretical tool

to address this question. We believe that the transition is a strong-disorder fixed

104

point where the disorder is relevant at long length scales with a dynamic critical

exponent z → ∞. Thus, one possible direction to proceed would be to perform

a real-space strong-disorder renormalization group but at finite temperature. A

controlled understanding of this method only exists for quantum ground states.

Other numerical methods may also prove to be useful in understanding the physics

of this phenomena. Specifically, in the localized phase where the entanglement of a

subsystem with the rest of the system is low, methods like time-dependent density-

matrix renormalization group can study larger system sizes.

Given the ubiquitous nature of disorder in real systems, many-body localiza-

tion may be a phenomena not very unaccesible to certain experimental systems.

Specifically, experiments in cold atoms where the system is isolated to a very good

approximation. Dynamic measurements of transport and relaxation in such sys-

tems look the most promising to explore many-body localization. Even in solid

state settings, where the coupling to a heat-bath is weak, it is conceivable that

many-body localization can strongly alter the conduction properties in low di-

mensions. For instance, many-body localization may be underlying some highly

nonlinear low-temperature current-voltage characteristics measured in certain thin

films [37].

105

Chapter 4

Conclusion and Future outlook

There are many unsettled aspects of localization involving interacting degrees of

freedom in the presence of disorder which remain to be explored. Our work pre-

sented in this thesis has probably only scratched the surface of this edifice. The

salient features of our work can be highlighted in the following points:

1. Some of the interesting aspects of the many-body localization transition are

accessible to computational techniques currently prevalent. Contrary to “con-

ventional ” equilibrium phase transitions which do not exist in d = 1, many-

body localization transition is expected to exist in all dimensions. This makes

it quite amenable for further numerical work, for example using DMRG-like

techniques. Leaving aside the aspects of the transition, the many-body lo-

calized phase in itself can host interesting phenomena arising due to the

106

interplay of disorder and interactions. Since the localized eigenstates are

relatively less entangled, the localized phase is more numerically tractable.

2. Quantum dynamics of an isolated system at high (extensive) energies can be

different from effective classical Langevin dynamics. The conventional un-

derstanding suggests that the quantum nature of a system becomes irrelevant

for highly excited states. It is indeed true that on coupling to an external

heat bath a classical description is sufficient to capture the low-frequency dy-

namics at such an energy scale. But, dynamics of an isolated system which

is many-body localized doesn’t fit into this paradigm. An isolated system

can maintain its quantum coherence for relatively long times (large T2 time)

under its own dynamics. The lack of many-body localization in a classical

system with disorder strongly suggests that although the quantum transition

is at nonzeor temperature there is no classical description of it.

3. The work by Basko, Aleiner and Altshuler [25] had put forward a picture

of many-body localization which is an analogue of single-particle Anderson

localization but in the many-particle Fock space. We find based on our

numerics that there is a sense in which the highly excited states are local-

ized in real space as well i.e., a local operator creating an excitation in an

eigenstate, with an extensive amount of energy, has exponentially decaying

support in real space. This has implications for the growth with time of the

entanglement of a subsystem in real space (in which the Hamiltonian is local)

with the remainder of the system in the many-body localized insulator. In

a single-particle Anderson insulator the entanglement remains finite at long

times after starting in a product state. While in the many-body localized

107

insulator due to the finite interactions the entanglement grows without limit

as the logarithm of time [56].

4. We have shown that the critical point between the localized and the ergodic

phases may belong to the infinite randomness class. On coarse-graining

in a renormalization group sense, this would imply the system appears in-

creasingly disordered at larger length scales. Also, there is evidence that

the dynamical critical exponent z of this transition is ∞. Recent analyti-

cal studies suggest that this is indeed the case at the transition [70]. Thus

the relaxation time diverges exponentially on approaching the critical point

(assuming that the correlation length critical exponent ν is bounded from

below: ν > 2/d > 0 [75]).

τ ∼ (h− hc)−zν (4.1)

4.1 Question of Universality

The notion of universality is extremely crucial for a full theory of a phase transi-

tion. In our work and others’ the existence of the two phases which have distinct

dynamical behaviour has been established. Strictly speaking, a renormalization

group treatment to describe the critical behaviour seems essential for establishing

the universal properties of the transition. But such a method is relatively challeng-

ing, though there have been some efforts in this direction [70]. One needs to coarse

grain to “flow ” to a state which is an excited state with an extensive amount of

energy. Real-space renormalization group technique appears to be amenable for

such a treatment but it may only be a controlled calculation in the localized phase.

Also, performing an RG for the long-time hamiltonian dynamics is fraught with

108

pitfalls (such as vanishing energy denominators in the perturbative treatment) and

is a challenging open question.

4.2 Symmetries

An intriguing question pertaining to the many-body localization transition is the

effect of various kinds of global symmetries. Symmetries play an important role in

distinguishing the different universality classes of finite temperature equilibrium

transitions. Do they play an analogous role in the many-body localization transi-

tion? The symmetries correspond to different globally conserved quantities, and

their transport can tend to zero as one approaches the critical point from the

ergodic phase in distinct ways. Also, if the system is susceptible to a symmetry-

breaking transition at equilibrium could it affect this dynamical transition? It is

not possible to have many-body localization in the presence of a spontaneously

broken continuous symmetry due to non-localizability of the resulting Goldstone

mode, except maybe for cases where the Goldstone mode is gapped because of

the Anderson-Higgs mechanism. When the symmetry is discrete, it may very well

be that the many-body localization transition is not affected by such a discrete

spontaneous symmetry breaking.

4.3 Topological order

There is an interesting class of ground state quantum phase transitions on tun-

ing a parameter of the hamiltonian where the transition is not accompanied by

the breaking of any symmetry. The transition manifests itself through a change

109

in the topological properties of the system [76]. In the presence of disorder the

topological order in the ground state is expected to be robust at least for weak

disorder, but the existence of a many-body localization transition could bear inter-

esting dynamical effects [77, 78]. Hamiltonians with topological order are also pre-

dicted to be particularly useful for performing fault-tolerant quantum computation

[79]. Hence, many-body localization combined with topological order in Kitaev-

like models could putatively change the bounds for quantum error-correction by a

significant amount.

4.4 Decoherence

Decoherence is a major issue affecting almost all experimental realizations under

study for the purposes of quantum computation. Interactions result in entangling

the qubit with the environment and other qubits, giving rise to decoherence. Thus,

having strong enough interactions allowing for sufficient control of the qubits com-

bined with long coherence times for performing many quantum operations with

high fidelity becomes a challenging problem. In this regard spin-echo techniques

have been able to undo some of the decoherence due to interactions. A many-body

localized state can possibly be used as an effective quantum memory because of the

slow growth of entanglement, perhaps rendering spin-echo methods more effective.

110

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