Many-Body Localization

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.

A system of interacting degrees of freedom in the presence of disorder hosts a variety 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 quantum 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 understanding, 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 classical 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 quantum 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 importance 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

ListofTables ...... xii

ListofFigures...... xiii

1 Introduction 1

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

1.2 RandomMatrixTheory ...... 10

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

1.3.1 ClassicalChaos ...... 15

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

1.3.3 Eigenstate Thermalization Hypothesis ...... 18

1.4 DisorderandInteractions...... 20

1.4.1 Variablerangehopping...... 21

1.4.2 Fermiglass ...... 23

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

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

1.6 Thesisoutline ...... 35

ix 2 The quantum many-body localization 37

2.1 Themodel...... 38

2.2 Single-siteobservable ...... 44

2.3 Transportofconservedquantities ...... 47

2.4 Energy-levelstatistics...... 51

2.5 Spatialcorrelations ...... 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,trajectoriesandtransport ...... 83

3.2.1 TheModel...... 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 Currentautocorrelations ...... 90

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

3.4 Furtherexplorationsandoutlook ...... 97

3.5 Finitesizeeﬀects ...... 101

3.6 Chaosampliﬁcationofround-oﬀerrors ...... 102

3.7 Summary ...... 104

4 Conclusion and Future outlook 106

4.1 QuestionofUniversality ...... 108

x 4.2 Symmetries ...... 109 4.3 Topologicalorder ...... 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 Chaoticandregulartrajectories ...... 16

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

1.8 BAAphasediagram ...... 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 thinﬁlm ...... 34

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

2.2 Decoupled precessing spins ...... 43

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

(n) 2.4 Probability distribution of miα ...... 47 2.5 Probability distribution of m(n) m(n+1) ...... 48 | iα − iα |

xiii 2.6 Dynamic part of initial spin-polarization ...... 50 2.7 Probability distribution of r(n) ...... 52 2.8 Ratioofadjacentenergygaps ...... 55

zz 2.9 Spin-spin correlation (Cnα)inenergyeigenstates ...... 56 2.10 Measure of anti-correlation at long distances ...... 59

zz 2.11 Probability distributions of ln Cnα(i, i + L/2) ...... 60

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

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

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

(n) 2.15 Probability distribution of Pα ...... 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) κ (t) κ (2t) ...... 96 ≡ L − L 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-sizeeffects ...... 101 3.11 Round-offeffects ...... 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 important 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 condensed 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 performed in George Feher’s group at the Bell Laboratories [2–4]. They were particularly 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 conceptualized 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 slightly impure due to the presence of Si29. (Figure from [5])

can be written as

† H = Einˆi + Vijcˆi cˆj + h.c. (1.1) i i,j X X

† where cˆi 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

z z the Jordan-Wigner transformation, it amounts to neglecting the Si Sj 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.

1 ψ(~k)= ei~k·~ri i (1.2) √ | i N i X

3 Figure 1.2: Electron spin resonance signal for diﬀerent phosporus concentration across the Anderson transition. (a) In the localized phase (d) In the delocalized phase. (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 qualitatively changes the transport from ballistic to diﬀusive but the states still remain delocalized. While if the perturbation theory is performed in the localized limit ˆ where the unperturbed states are eigenstates of H0 = i Einˆi while the hopping is a perturbation (technically referred to as the locatorP 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 | i represented as ψ = a i i| i i X

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

ia˙ j = Ejaj + Vjkak Xk6=j

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 a . If the → ∞ i 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

G (t)= i eiHtcˆ†e−iHtcˆ i ij h | j i| i

5 Thus, the problem becomes amenable to a perturbative treatment where the basis states are the eigenstates of Hˆ0 while the hopping terms are treated perturbatively. 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 from one site to the next and performs a quantum coherent random walk. Hopping back and forth between lattice sites 4 and 5 represents resonant tunneling. Figure from [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, G (t) 2 averaged | ij | over disorder realizations does capture the transition between the localized and diffusive phases while one needs to evaluate the probability distribution of Im Gij(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.

∆E e−L/ξ (1.3) localized ∼

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 π . Hence, the Thouless energy in the diffusive phase is inverse of the decay time ∼ L of the mode and obeys the following relation.

~ L2 ∆Ediffusive = = (1.4) tdiff D

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

dE δW = L−d (1.5) dn

dn where d is the dimension of the space and dE 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

∆E expressed as a universal function parametrized by a single parameter δW .

G g = (1.6) e2/~

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) = β(g(L)) (1.7) d ln L

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

−αL g = g0e (1.9)

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

and β ln(g/g ). Assuming the beta function is continuous and doesn’t have → 0 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 random 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 matrices. 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

 ∗  H12 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

1 E = H + H (H H )2 +4 H 2 (1.11) ± 2 11 22 ± 11 − 22 | 12| p

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 = E E and d˜ is the number of free parameters to be adjusted. ∼ i+1 − i

1Riemann hypothesis: It is conjectured that ζ function:

∞ 1 1 ζ(z)= = = (1.10) nz 1 p−z n=1 X p∈{setY of primes} − 1 has zeros lying on the line z = 2 + iEi where Ei is real. Interestingly, the statistical ﬂuctuations of Eis behave like the eigenvalues of a random Hermitian matrix. This has been numerically veriﬁed for a large number of zeros of the function. Thus, it might be possible that this abstract mathematical problem is related the quantum chaotic behaviour of a physical system without time-reversal symmetry.

11 The distribution of level spacing for the Gaussian orthogonal ensemble is approximately given by

π s π s 2 P (s) exp (1.12) GOE ≈ 2 δ2 − 4 δ where δ = s and the angular brackets ... imply ensemble averaging. In the h i h i 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.

1 s P (s)= exp (1.13) Poisson δ −δ 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 collection 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 understanding 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 classical 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 I θ (i =1,...,d) where the action variable is conserved and the angle variable i − i oscillates at a frequency. Ii is the integral of motion. A particular set of conserved actions I deﬁnes a d dimensional torus in angle space (θ ). i − i

∂I i = 0 (1.14) ∂t

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 in an 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 excited 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 random 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 (X~ ) = exp(iP~ X~ ) (1.16) P~ ·

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 (X~ ) where the wavefunctions vanish outside the domain . The P~ D domain is deﬁned as

= (~x ,...,~x ): L xµ L; ~x ~x > 2a (1.17) D { 1 N − ≤ i ≤ | i − j| }

17 and ~ ~ Ψn(X)= Cn,P~ ΦP~ (X) (1.18) XP~ The momenta are also constrained by the total energy condition. In the case of the hard-sphere gas

N ~p2 E = i (1.19) n 2m i=1 X 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 = m) are also completely uncorrelated. Hence, a typical 6 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) = αnΨn (1.20) n X E¯ = α 2E (1.21) | n| n n X ∆2 = α 2(E E¯)2 (1.22) | n| n − n X

Eigenstate thermalization hypothesis [17, 18] states that the expectation value of local observables ˆ at long times equilibrates to the microcanonical average. O This equilibrium average can be well represented by just the expectation value in a typical eigenstate within the microcanonical energy span.

ˆ |En−E¯|≤∆ Ψn Ψn lim ψ(t) ˆ ψ(t) = h |O| i = typ Ψn ˆ Ψn typ (1.23) t→∞h |O| i P N∆ h |O| i

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

= α 2 + α∗ α e−i(En−Em)t (1.24) hOit | n| Onn m n Onm n X nX6=m

is the matrix element of the operator between eigenstates n and m. The Onm 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, also tend to zero exponentially in Onm 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 ~/τ i.e., diffusive relaxation in a finite system occurs at a much shorter time ≪ diff 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 challenge 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 distances 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 localization centers separated by distance R. The probability to produce excitations of order E E = ǫ in the heat bath goes as exp( ǫ/k T ). On the other hand ∼ 1 − 2 − B 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 2R ǫ σ(T ) exp( ) (1.25) ∼ − ξ − kBT

The typical separation between the states is given by

1 − d dn(EF ) Rtyp = (E E ) (1.26) dE 1 − 2

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

d ǫ (k T ) 1+d (1.27) optimal ∼ B

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

1 T 1+d σ (T )= σ (T ) exp 0 (1.28) variable 0 − T !

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 E E σ (T )= σ′ exp 1 − c (1.29) activation 0 − k T B

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. conductivity 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 important 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 localization, 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. 1 G(ω)= (1.30) ω Hˆ Σ(ˆ ω) − 0 −

Hˆ0 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 | i feature which makes it particularly useful is that Im(Σ(ω)) tends to zero for ω µ. →

lim Im(Σαα′ (ω))=0 (1.31) ω→µ

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 perturbative 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 expansion 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 equal to zero. At high temperatures, the system becomes ergodic and the delocalized and has ﬁnite conductivity. λ is the strength of the interaction and δζ is the mean level 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 = nα ,...,nα N (1.32) | i | 0 2 i

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

1 Hˆ = ǫ cˆ† cˆ + V cˆ† cˆ† cˆ cˆ (1.33) α α α 2 αβγµ α β γ µ α X αβγµX

The matrix element Vαβγµ is restricted in energy and space. Due to the exponential localization the matrix elements are chosen to be ﬁnite only for states satifying

~r ~r . ξ | α − β| ~r ~r . ξ | β − γ | . .

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).

ǫ ǫ , ǫ ǫ . δ | α − γ| | β − µ| ξ ǫ ǫ , ǫ ǫ . δ | α − µ| | β − γ | ξ

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 function of energy. While in the localized phase (solid line), the delta function is smeared out due to the dissipation added by hand (ﬁnite η; at the end the limit η 0 is → taken). (b) The probability distribution of Γ in the loclaized (solid line) and the ergodic phase (dashed line). (Figure from [25])

Since, Γ varies from sample-to-sample, a naïve average over disorder realizations 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 localized 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 distribution 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.

finite for a metal lim lim P (Γ > 0) =  (1.35) η→0 L→∞  0 for an insulator   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 electrons 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 primarily 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 taking into account electron-elctron or electron-phonon interactions. Since, these are

30 usually not within an experimentalist’s control in real materials, a direct observation 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 potential 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 phenomena 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 superconducting qubits. Therefore, the dressed photons behave as eﬀective particles with

31 Figure 1.10: Atomic density profile of the BEC cloud. The condensate wavefunction is exponentially localized with the tails fitted to an exponential. The inset of figure (d) shows that in the absence of random potential (VR =0) the rms width of cloud grows linearly while when it is switched on (V =0) it stops growing after R 6 some time. Plot from [28]

32 Figure 1.11: Superconducting (SC) circuits as qubits: (a) A transimission line resonator with an array of SC qubits (b) The basis building block for the array - Superconducting quantum interference device (SQUID). It consists of 2 superconducting 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 characteristics 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 insulating phase 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 excited 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 measures (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 insulating phases. This will lead to throwing some light on the properties of the critical point separating these two phases. Chapter 3 will explore the possibility 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 system 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 temperature, 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 statistical mechanics: it is a quantum “glass transition” where equilibrium quantum statistical mechanics breaks down. In the localized phase the system fails to thermally 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]:

L ˆ ˆ H = [h Sˆz + JS~ S~ ] , (2.1) i i i · i+1 i=1 X

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 = h = 3.5 1.0 does occur in this c ∼ ± 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 Sˆz. 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 Sˆz.

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 infinite temperature we remove one parameter from the problem, and use all the eigenstates from the exact diagonalization (within the zero total Sˆz 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 temperature 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 localization 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/J at T = . The critical point is (h/J) 3.5. For h < h the system is ergodic ∞ c ≈ c while 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 GOE • Many-body eigenenergies have Pois- • • level statistics son level statistics System achieves local thermal equi- Doesn’t thermally equilibrate- quan- • • librium tum glass Finite D.C. transport of energy and D.C. transport is zero other• globally conserved quantities • Extensive entanglement in eigen- “Area-law” entanglement in eigenstates• 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 entanglement 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.

L ˆz H = hiSi (2.2) i=1 X

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 local hi magnetic ﬁeld hi with frequency Ωi = ~

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 = ρ m n . The eigenstates have mn mn| ih | different energies, so as time progresses theP off-diagonal density matrix elements m = n dephase from the particular phase relations of the initial condition, while 6 the diagonal elements ρ do not change. In the ergodic phase for L all the nn →∞ 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) = n Sˆz n (2.3) iα h | i | iα

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) m(n+1) ] for | iα − iα | 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.

−0.5

−1

−1.5 |]

(n+1) −2 α i

−2.5 − m 8.0 (n) α

i 5.0 −3 3.6

ln [|m −3.5 2.7 2.0 −4 1.0 0.6 −4.5 7 8 9 10 11 12 13 14 15 16 17 L

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 are indicated in the legend. In the ergodic phase (small h) where the eigenstates are thermal these differences vanish exponentially in L as L is increased, while they remain 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

ˆz this exponentially small amount, so the expectation value of Si 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.

(n) In the two phases the probabability distribution of miα has distinctive behaviour. At inﬁnite temperature, in equilibrium we expect m(n) 0 which mani- iα ≈ 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 + 1 and 1 . The tendency of the eigenstates to have 2 − 2 the maximal z-component of spin at a single site suggests that the energy eigenstates are approximate product states i.e. n at strong but finite | i ∼| ↑↓ ···↑↓↓i 1 disorder. In the localized phase, we find the emergence of an effective spin- 2 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 and h = 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 conserved total Sˆz, 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:

ˆ ˆz M1 = Sj exp (i2πj/L) (2.4) j X

47 Figure 2.5: The probability distribution of difference of local magnetization between adjacent eigenstates for disorder strengths h = 0.6, 2.0, 3.0 and 6.0. The different systems sizes are marked in the legend. For low disorder in the ergodic phase, the difference distribution becomes more sharply peaked around zero. While in the localized phase the distribution develops a bump close to 1. At strong disorder 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

ǫMˆ † e 1 ρ = (1 + ǫMˆ †)/Z (2.5) 0 Z ≈ 1

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

Mˆ = Tr(ρ Mˆ ) (2.6) h 1it t 1 −iHt iHt = Tr(e ρ0e Mˆ 1) (2.7) ǫ = n Mˆ m 2ei(En−Em)t (2.8) Z |h | 1| i| n,m X

The initial spin polarization of this mode is then

ǫ Mˆ = n ρ Mˆ n = n Mˆ †Mˆ n . (2.9) h 1i0 h | 0 1| i Z h | 1 1| i n n X X 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

ǫ Mˆ = n Mˆ † n n Mˆ n . (2.10) h 1i∞ Z h | 1 | ih | 1| i n X

1 8 10 0.9 12 14 0.8 16

0.7

] 0.6 (n) α [f 0.5

0.4

0.3

0.2

0.1 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 h

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) the polarization 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

n Mˆ † n n Mˆ n f (n) =1 h | 1 | ih | 1| i . (2.11) α − n Mˆ †Mˆ n h | 1 1| i In the ergodic phase, the system does thermalize, so the initial polarization

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

50 (n) hand, there is no long-distance spin transport, so fα 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

(n) (n) (n−1) (n) spacings δα = Eα Eα , where Eα is the many-body eigenenergy of | − | (n) eigenstate n in sample α. Then we obtain the ratio of adjacent gaps as rα =

(n) (n+1) (n) (n+1) min δα , δα / max δα , δα , 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, 4.0 and 6.0. For h < hc the spectrum’s ﬁnite level-repulsion can be seen as a peak at (n) ﬁnite r in the distribution. While for large disorder h > hc, the distribution is peaked at zero.

52 single gap distribution. A single gap distribution (p(s) = [ δ(s δn ) ]) requires h − hδni i an appropriate definition of the mean gap δ . The mean gap has significant vari- h ni 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 exponential 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

min{δ(n),δ(n+1)} be derived by calculating δ r n n − max{δ( ),δ( +1)} D E

min δ(n), δ(n+1) P (r) = δ r { } P − max δ(n), δ(n+1) D { } E 1 ∞ ∞ min δ(n), δ(n+1) δn δn+1 = d δ d δ δ r { } exp exp s2 n n+1 − max δ(n), δ(n+1) − s − s Z0 Z0 { }

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).

1 ∞ δn δ δn δn+1 P (r) = d δ d δ δ r n+1 exp exp P s2 n n+1 − δ − s − s Z0 Z0 n 1 ∞ δn+1 δ δn δn+1 + d δ d δ δ r n exp exp s2 n+1 n − δ − s − s Z0 Z0 n+1 2 ∞ δn δ δn δn+1 = d δ d δ δ r n+1 exp exp s2 n n+1 − δ − s − s Z0 Z0 n

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.

2 ∞ (1 + r)δn PP (r)= 2 d δn δn exp (2.12) s 0 − s Z h i 2 On evaluating the above integral the distribution of r is given by Pp = (1+r)2 and [r] 2 ln 2 1 = 0.39. Note that our model is integrable at h = 0, so will p → − ∼ 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 0.54 8 10 0.52 12 14 0.5 16

0.48 ]

(n) α 0.46 [r

0.44

0.42

0.4

0.38 0.5 2.5 4.5 6.5 8.5 10.5 12.5 h

Figure 2.8: The ratio of adjacent energy gaps (deﬁned in the text). The sample size L is indicated in the legend. In the ergodic phase, the system has GOE level statistics, 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 | i

Czz (i, j)= n SˆzSˆz n n Sˆz n n Sˆz n . (2.13) nα h | i j | iα −h | i | iαh | j | iα

−3 h=0.6 −4

−5

−6

−7 h=3.6 (i,i+d)|]

α −8 zz n C −9 [ln | −10 8 10 −11 12 h=6.0 14 −12 16

−13 0 1 2 3 4 5 6 7 8 9 d

Figure 2.9: The spin-spin correlations in the many-body eigenstates as a function of the distance d. The sample size L is indicated in the legend. The correlations decay exponentially with d in the localized phase (h =6.0), while they are independent of 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 − | i 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.

zz Cnα(i, j)=0 . (2.14) j X In Fig. 2.9 we show the mean value [ln Czz (i, i + d) ] as a function of the | nα | 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 Sˆz that we look at,

n Sˆz n = 0 in the thermal eigenstates of the ergodic phase. The thermal corre- h | i | i ∼ 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-

zz tion on the same site is: Cnα(i, i)=1/4. Therefore,

1 Czz (i, j)= . (2.15) nα −4 Xj6=i The conservation of total Sˆz does result in anti-correlations so that Czz (i, j) nα ≈ 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 Sˆz n remains nonzero for L . If spin i is flipped, within a single eigenstate h | i | i →∞ that quantum of spin remains localized near site i, with its amplitude for being at site j falling off exponentially with the distance: Czz (i, j) exp ( i j /ξ), nα ∼ −| − | 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 localized case positive and negative correlations are equally likely in which case the fraction tends 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.

0.45 0.6 0.4 1.8 2.7 0.35 3.6 0.3 5.0 6.0 0.25 )

φ 8.0

P( 0.2

0.15

0.1

0.05

0 −30 −25 −20 −15 −10 −5 φ zz =ln|Cnα(i,i+L/2)| Figure 2.11: The probability distributions of the natural logarithm of the long distance spin-spin correlation in the many-body eigenstates for sample size L = 16 and 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 distributions 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 Czz (i, i +(L/2)) , (2.16) | nα |

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. Byp this measure, 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.

0.35 8 10 12 14 16 0.3 L σ 0.25

0.2

0.15 0.5 2.5 4.5 6.5 8.5 10.512.5 h

Figure 2.12: The scaled width σ of the probability distribution of the logarithm of the long-distance spin correlations (see text). The legend indicates the sample lengths L. In the ergodic phase at small h and in the localized phase at large h, this width decreases with increasing L, while near the transition it increases. To produce the one-standard-deviation error bars shown, we have calculated the σ (see text) for each sample by averaging only over sites and eigenstates within each sample, and then used the sample-to-sample variations of σ to estimate the statistical errors. We have also (data not shown) calculated σ by instead averaging φ and φ2 over all samples; this produces scaling behavior for σ that is qualitatively the same as shown here, but with σ somewhat larger in the localized phase and near 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 Sˆz. Their respective transport times (and hence their corresponding Thouless energy) in the ergodic phase may have different scaling properties close to the critical point. By studying the relaxation of the spin modulation, Mˆ1, 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 system size 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, Mˆ 1. If the scaling at the many-body localization 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 Mˆ should be from its matrix elements between adjacent energy levels, and this h 1i

64 0.25 8 10 12 0.2 14 16

0.15 ] (n) α [P 0.1

0.05

0 0.5 2.5 4.5 6.5 8.5 h

Figure 2.14: Contribution to the dynamic part of Mˆ from matrix elements h 1i between adjacent energy states (see text). In the ergodic and localized phase the contribution is decreasing to zero with increasing sample size. The sample size L is indicated in the legend. The maximum contribution from adjacent states is close 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 Mˆ is given by | i h 1i

(∆M )(n) = n Mˆ †Mˆ n n Mˆ n 2 . (2.17) 1 α h | 1 1| i−|h | 1| i|

(n) In the ergodic phase, (∆M1)α has signiﬁcant contributions from matrix elements 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) as the contribution to the dynamic part of Mˆ from the matrix elements iα h 1i between state n and states n i: ±

Q(n) = n i Mˆ n 2 + n Mˆ n + i 2 (2.18) iα |h − | 1| i| |h | 1| i| in sample α. Note that

(n) (n) Qiα = (∆M1)α . (2.19) Xi6=0

(n) (n) (n) We deﬁne Pα = Q1α /(∆M1)α as the fraction of the longest-wavelength “diffusive” 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 localization 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 E δ scaling, and thus dynamic exponent z . In T ∼ → ∞ 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.

1.8 10 1.6 16

1.4

1.2

1 P(f)

0.8

0.6

0.4

0.2 0 0.2 0.4 0.6 0.8 1 f

Figure 2.15: Probability distribution of the dynamic fraction of Mˆ 1 for L = 10 and 16. Close to the transition for h =3.0, the distribution becomesh i broader and more bimodal with increasing L.

(n) The dynamic fraction [fα ] (Fig. 2.6) tends to 1 in the ergodic phase and

(n) decreases to 0 in the localized phase. The probability distribution of fα (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 entanglement. 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 . On dividing the Hilbert space into two parts S and H S¯ such that, = ¯, the state is expressible as a a linear combination of H HS ⊗ HS basis states of S and S¯

N M ψ = x i α ¯ (2.20) | i iα| iS ⊗| iS i=1 α=1 X X

where i and α ¯ are the basis states of and ¯ while N and M are their | iS | iS HS HS 1 dimensions respectively. The simplest example is of two localized spin- 2 degrees of freedom in a singlet state.

¯ ¯ ψ = | ↑iS ⊗| ↓iS −|↓iS ⊗| ↑iS | i √2

68 Figure 2.16: S + S¯ form an isolated system. S is the subsystem while S¯ serves as the environment to the subsystem. l . L

A pure state is said to be quantum mechanically entangled in space if it is H not expressible as a direct product of state vectors in and ¯. HS HS

ψ = φ ξ ¯ | iH 6 | iS ⊗| iS

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 bipartite 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 | ih | | i 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) ¯ the density matrix is | ↑iS ⊗| ↑iS

prod 1 0 ρS =   0 0     while for the singlet state the reduced density matrix is

1 entangled 2 0 ρS =  1  0 2     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 entanglement is by studying the spectrum of ρ - λ . For a general product state, the S { i} spectrum is always λ = 1 while all the other eigenvalues are 0. In the the case

1 of the singlet state, the eigenvalues are λ1 = λ2 = 2 . This is the maximally entangled 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 implies 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 = λ ln λ (2.22) − i i i=1 X Thus, a product state has zero entanglement entropy while a maximally en-

max tanglement state maximizes entropy. (Sentanglement = ln N; 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

1 1 N Sq = ln Tr (ˆρq )= ln λq (2.23) Rényi 1 q S S 1 q i i=1 ! − − X 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

Hˆ − S e kBT ρ = ρ = (2.24) S Gibbs Z

71 Assuming the entire system starts from a normalized pure state ψ = | i NM c n with a well-deﬁned mean energy E¯ with small ﬂuctuations around it. n=1 n| i TheP corresponding density matrix is given by

ρ = c∗ c n m 0 n m| ih | n,m X

n and m are the energy eigenstates of the entire system. At a later time, | i | i the reduced density matrix for S is given by

∗ ρ (t)= c c exp( i(E E )t) Tr ¯ ( n m ) (2.25) S n m − n − m S | ih | n,m X

For n = m, the relative phases are randomized with time and hence, average 6 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 Tr ¯ ( n n ) is typically independent of n. S | ih |

ρ ( ) Tr ¯ ( n n ) S ∞ ≈ S | ih |

What should be the entanglement properties of the eigenstate for the ﬁnal equilibrium 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 in S while of Greek letters would imply a state in S¯.

72 ρ = x x∗ i j α β (2.26) iα jβ| ih |⊗| ih | i,j X Xα,β M ρ = Tr ¯ ρ = β ρ β (2.27) ⇒ S S h | | i Xβ=1 N M = x x∗ i j (2.28) iα jα| ih | i,j=1 α=1 X X N = U i j (2.29) ij| ih | i,j=1 X

where U = XX† where U is a N N matrix while X is a N 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 XX (2.30) { } ∝ − ¯ ¯ The properties of such a random matrix has been studied extensively in statistics and goes by the name Wishart matrix. The spectrum of the Wishart random

N matrix with the normalization constraint i=1 λi = 1 had been studied from the point of view of quantum entanglement inP a random pure state as far back as 1978 [63]. The probability distribution of the eigenvalues is

N N M−N 2 PWishart( λ ) δ λ 1 λ (λ λ ) (2.31) { i} ∝ i − i j − k i=1 ! i=1 X Y Yj

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.

N S = ln N (2.32) random − 2M

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 = λ ln λ − i i i X At low disorder when the system is thermal, entanglement entropy tends to its thermal value L ln(2) which is extensive in subsystem size. The entropy ∼ 2 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. The legend 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 spectrum 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 versus i (It is deﬁned such λi > λi+1) for L = 14. The legend indicates the disorder strengths. For stronger disorder, λi 0 within machine precision for larger i and hence 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 E δ scaling at criticality, in apparent agreement with our earlier conclusion of T ∼ 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 universality 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 interacting 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 models 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 disordered classical many-body systems are capable of localization at nonzero temperature. 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 2J 2 + ∆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 J γ, 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 that D(J

— such ﬁts prove inconclusive as they produce estimates of Jc considerably smaller

81 log10D log J -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 10 -1

-2

-3

-4

-5

-6

Figure 3.1: Disorder-averaged energy diﬀusion constant D as a function of the spin-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

∂S j = H S , (3.1) ∂t j × j where Hj = ∂H/∂Sj is the total instantaneous ﬁeld acting on spin Sj. The Hamiltonians we consider are all of the form

H = (h S + JS S ), (3.2) j · j j · j+1 j X

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 = hj ˆnj, where the hj are independent

2 Gaussian random numbers with mean zero and variance ∆ , while the ˆnj 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

2J 2 + ∆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

J e (t)= h S (t)+ (S (t) S (t)+ S (t) S (t)) . (3.5) i i · i 2 i−1 · i i+1 · i

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 j (t)= J[S (t + 1) S (t)] [S (t) S (t)] , (3.6) i i − i · i+1 − i−1

while for i odd,

j (t)= J[S (t + 1) S (t)] [S (t + 1) S (t + 1)] . (3.7) i i − i · i+1 − i−1

−1 We are working at infinite temperature, or alternatively at β = (kBT ) = 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 = C(t) , (3.9) t X where

C(t)= [ j (0)j (t) ] (3.10) h 0 i i i X 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 h i

87 sample and time average within a given run. For our model (3.2) the average energy per site obeys

d[ e ] J 2 + ∆2 h i = (3.11) dβ 3

at β =0, and the energy diﬀusivity D is then obtained from the relation

κ d[ e ] = h i . (3.12) D dβ

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

t L 1 2 κ = lim [ ji(τ) ] . (3.13) L,t→∞ Lt h{ } i τ=1 i=1 X X 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

1 t L κ (t)= j (τ) 2 . (3.14) I Lt{ i } τ=1 i=1 X X If these estimates are then averaged over samples and over initial conditions for a given L and t, this results in the estimate κ (t) = [ κ (t) ] . These estimates L h I i

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

2 t > tL = CDL /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 κ (t) J 2/t in this L ∼ small-J regime. To explain this better we can rewrite the definition of κ as

89 t/2 1 t t 2 κ (t)= C(τ τ ′)= κ∗ (t ), (3.16) L Lt − Lt L av τ=1 ′ t =1 X τX=1 Xav where we have assumed an even t (there is an additional term otherwise) and κ∗ (τ) τ C(τ ′). Localization, i.e. zero DC conductivity, implies a rapidly L ≡ −τ vanishingPκ∗ as well as κ at long times. The latter however acquires a tail, κ (t) L ∼ ∗ 1/t whose amplitude is set by the short-time values of κL.

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 expect 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 J 2 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 J 3 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 amplitude: 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) κ (t) κ (2t), at a sequence of points t =2n, ≡ L − L n 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 CJ2 0.05 0.04 0.03 0.02 0.01 0.00 t 1 2 3 4 5 Figure 3.2: Short time behavior of C(t) for J = 0.32 (red, noticeably diﬀerent trace) and J =0.08, 0.12, 0.16 (these are almost identical data in this plot). Note rescaling of the vertical axis by J 2.

3.3.2 DC conductivity: extrapolations and ﬁts

Our extrapolations of the DC conductivity will be based entirely on the long time

n behavior of κL(t) evaluated at a set of times tn =2 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) J 2 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 CJ3 0.005

0.000 2 3 4 5 6 7 8 t J

Κ*J2 -0.005 0.08 0.06

-0.010 0.04 0.02 0.00 t J -0.015 2 4 6 8 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 the rescaling of both the vertical and time axes. The inset shows near cancellation between 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 reduced. 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/J 10. 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 log10 H-CL -4

-5

-6

-7

-8

-9

log t 2 3 4 5 10

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, light blue 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 54 log10 H-C t L -2.6

-2.8

-3.0

-3.2

-3.4

-3.6 log t 1 2 3 4 5 10

Figure 3.5: To estimate the exponent we multiply the data by t5/4 (and also display lines with slope 0.05). Although these data do not exclude an exponent ± that varies with J, we interpret these results as supportive of a single exponent x 0.25 at asymptotically long times but with a more pronounced short-time transient≈ at smaller J.

The uncertainties in our estimates of the DC κ from the extrapolation procedure 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 log10 Η

-3.0

-3.5

-4.0

log10 t 3 4 5 6 7 -5.0

-5.5

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− autocorrelation 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

4 n polynomial 0 ans to better capture the curvature apparent in the data – these

fits are shownP 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 ΚJ 2

0.15

0.10

0.05

0.00 0.0 0.2 0.4 0.6 0.8 1.0 s 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 guides to 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 extrapolated 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 ΚJ 10 5000

4000

3000

2000

1000

0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 s Figure 3.8: Refer to the caption of Fig. 3.9

J κ δ κ L log2 T samples 0.64 0.18J 2 0.01J 2 5000 20 1000 0.56 0.09J 2 0.01J 2 2000 20 1000 0.48 0.045J 2 0.005J 2 1000 20 4000 0.40 0.020J 2 0.003J 2 1000 20 1000 0.36 0.014J 2 0.003J 2 1000 21 2200 0.32 50J 10 8J 10 1000 21 16000 0.28 70J 10 10J 10 1000 24 912 0.24 95J 10 20J 10 1000 25 558 0.20 130J 10 30J 10 1000 26 1179 0.18 175J 10 50J 10 500 27 2000 0.16 250J 10 100J 10 500 27 1550 0.14 400J 10 200J 10 500 27 520 0.12 600J 10 400J 10 500 27 1116

Table 3.1: Extrapolated estimates of the DC conductivity κ, estimated uncertainties, length L of samples, and the number of time steps T of the runs.

98 ΚJ 10 1000

800

600

400

200

0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 s Figure 3.9: Same data as in Fig. 3.7, but now scaled and displayed in a way that allows one to see the extrapolations to s = 0 (t ) for small J. Note the diﬀerent rescaling schemes used in preceeding and→ curre ∞nt plots to focus on “collapse” of short time data (previous plot) vs. long time extrapolations (present plot). 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 and 3.9 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

ΚJ 2 0.08 ΚJ 10 60 50 40 0.06 30 20 10 0 0.0 0.1 0.2 0.3 0.4 s 0.04

0.02

0.00 0.0 0.2 0.4 0.6 0.8 1.0 s Figure 3.10: Finite size effects for J = 0.16, 0.32, 0.40: 100 vs. 20 spins for J =0.40, 10 and 20 vs. 100 spins for J =0.32, 20 vs. 100 spins for J =0.16. Red color 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 ∗ 2 2π/t corresponding to ∼= CDD(2π/L) , with D ∼= 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 trajectories 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 angle 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

−15 intrinsic noise (which at double precision corresponds to Pintr . 10 ). 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 ΚJ 2 0.07 ΚJ 10 0.06 6000 5000 0.05 4000 3000 2000 0.04 1000 0 0.00 0.02 0.04 0.06 0.08 0.10s 0.03

0.02

0.01

0.00 0.0 0.2 0.4 0.6 0.8 1.0 s Figure 3.11: Roundoﬀ eﬀects at J = 0.14: low frequency, long-time conductivity is larger for larger values of P but is essentially indistinguishable between P = 0 and 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 insight 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 dynamics 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 localization 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 systems 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 dimensions. 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 presented 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 “conventional” 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 localized 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 understanding 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 h )−zν (4.1) ∼ − c

4.1 Question of Universality

The notion of universality is extremely crucial for a full theory of a phase transition. 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 challenging, 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 transition? 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 interesting dynamical eﬀects [77, 78]. Hamiltonians with topological order are also predicted 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 combined 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.

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