Theory of the many-body localized phase and phase transition...Theory of the many-body localized phase and phase transition Ehud Altman – Weizmann Institute of Science Minerva foundation

Theory of the many-body localized phase and phase transition

Ehud Altman – Weizmann Institute of Science

Minerva foundation ISF

Collaborators: Ronen Vosk (WIS), David Huse (Princeton),

G. Refael (Caltech), Y. Bahri, A. Vishwanath (UCB), E. Demler (Harvard), V. Oganesyan (CUNY), D. Pekker (Pittsburg), Mark Fischer (WIS)

Conventional wisdom: Quantum mechanics is manifest only in the ground state

(and lowest energy excitations)

Fermi liquid:

Quantum Hall effect: Topological insulators:

Quantum critical points

Quantum dynamics at high energies: two generic paradigms

Thermalization

Classical hydro description of remaining slow modes (conserved quantities, and order parameters).

Quantum correlations in local d.o.f are rapidly lost as these get entangled with the rest of the system.

Many-body localization

Need a fully quantum description of the long time dynamics!

Local quantum information persists indefinitely.

?

elusive interface between quantum and classical worlds

The many-body localization transition =

Alternative perspective: structure of eigenstates at high energies

Thermalizing

Energy eigenstates are highly entangled:

SA ⇠ Ld

Many-body localized

?

Eigenstates have low entanglement

SA ⇠ Ld�1 (area law) (volume law)

Localization transition: fundamental change in entanglement pattern. More radical than in any other phase transition we know !

What is the conceptual framework to describe such dynamical quantum phases and phase transitions?

In equilibrium: renormalization group framework

Quant. Phases QCP

Out of equilibrium: cannot focus on low energies! Need a new RG philosophy

Outline

•  Thermalization in closed quantum systems Eigenstate thermalization hypothesis and its breaking

•  Effective description of the many-body localized phase RG, Quasi-local integrals of motion

•  The many-body localization phase transition

RG approach: transport, entanglement scaling and a surprise!

Eigenstate thermalization hypothesis (ETH)

L

A

In a high energy eigenstate:

⇢A =1

ZAe��HA

Extensive Von-Neuman entropy:

Anderson localization is an example where ETH fails:

L

SA / Ld�1

“Area law” entropy even in high energy eigenstates

The localized area-law states are stable to interaction!

Deutsch 91, Srednicki 94

SA ⌘ tr [⇢A ln ⇢A] / Ld

Many-Body Localization transition

Basko, Aleiner, Altshuler (2005); Gornyi, Mirlin, Polyakov (2005): Insulating phase stable below a critical T or E; metal above it.

Disorder tuned transition at    𝑇=∞   in a system with bounded spectrum Oganesyan and Huse (2007), Pal and Huse (2010)

delocalized thermalizing

Localized (κ =0, σ =0) non thermalizing

𝑇=∞

Disorder strength

T, E

Numerical studies: focus on random spin chains

Poisson r=0.39

GOE r=0.53

= interacting lattice fermions:

r

Exact diagonalization is limited to very small sizes (<18 spins). How can we do better? Disorder strength (h)

(Ratio of adjacent energy gaps: )

Bounded spectrum: allows to study disorder tuned transition in generic eigenstates (= infinite temperature)

Pal & Huse 2010 found indication of the transition in the many-body level statistics

Outline

•  Thermalization in closed quantum systems Eigenstate thermalization hypothesis and its breaking

•  Description of the many-body localized phase Computability, RG, Quasi-local integrals of motion

•  The many-body localization phase transition

RG approach: transport, entanglement scaling and a surprise!

Computability of the MBL state

•  Localized eigenstates admit efficient encoding with Matrix-product-states (thanks to area law entanglement).

•  Time evolution? Quench from a non entangled state:

A B

localized (?)

SA

t

Naïve expectation: Saturation of the entanglement entropy after quasi particles have reached a localization length ξ from the interface?

If true implies computability of dynamics! ⇠

Numerical result: Slow logarithmic growth of the entanglement entropy

A B

Znidaric et. al. 2008; Bardarson, Pollmann & Moore. 2012

Unbounded Growth of Entanglement in Models of Many-Body Localization

Jens H. Bardarson,1,2 Frank Pollmann,3 and Joel E. Moore1,2

1Department of Physics, University of California, Berkeley, California 94720, USA2Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

3Max Planck Institute for the Physics of Complex Systems, D-0118 Dresden, Germany(Received 5 March 2012; published 3 July 2012)

An important and incompletely answered question is whether a closed quantum system of many

interacting particles can be localized by disorder. The time evolution of simple (unentangled) initial states

is studied numerically for a system of interacting spinless fermions in one dimension described by the

random-field XXZHamiltonian. Interactions induce a dramatic change in the propagation of entanglement

and a smaller change in the propagation of particles. For even weak interactions, when the system is

thought to be in a many-body localized phase, entanglement shows neither localized nor diffusive

behavior but grows without limit in an infinite system: interactions act as a singular perturbation on

the localized state with no interactions. The significance for proposed atomic experiments is that local

measurements will show a large but nonthermal entropy in the many-body localized state. This entropy

develops slowly (approximately logarithmically) over a diverging time scale as in glassy systems.

DOI: 10.1103/PhysRevLett.109.017202 PACS numbers: 75.10.Pq, 03.65.Ud, 71.30.+h

One of the most remarkable predictions of quantummechanics is that an arbitrarily weak random potential issufficient to localize all energy eigenstates of a singleparticle moving in one dimension [1,2]. In experimentson electronic systems, observation of localization is lim-ited to low temperatures because the interaction of anelectron with its environment results in a loss of quantumcoherence and a crossover to classical transport. Recentwork has proposed that, if there are electron-electroninteractions but the electronic system is isolated fromother degrees of freedom (such as phonons), there canbe a ‘‘many-body localization transition’’ even in a one-dimensional system for which all the single-particle statesare localized [3–8].

Two important developments may enable progress onmany-body localization beyond past efforts using analyti-cal perturbation theory. The first is that numerical methodslike matrix-product-state based methods and large scaleexact diagonalizations enable studies of some, not all,important quantities in large systems. The second is thatprogress in creating atomic systems where interactionsbetween particles are strong but the overall many-atomsystem is highly phase coherent [9] suggests that thismany-body localization transition may be observable inexperiments [10,11]. Note that many-body localization isconnected to the problem of thermalization in closed quan-tum systems as a localized system does not thermalize.

The goal of the present Letter is to show that the many-body localized phase differs qualitatively, even for weakinteractions, from the conventional, noninteracting local-ized phase. The evolution of two quantities studied, theentanglement entropy and particle number fluctuations,show logarithmically slow evolution more characteristicof a glassy phase; however, the long-term behavior of these

quantities is quite different. The growth of the entangle-ment entropy has previously been observed [12,13] toshow roughly logarithmic evolution for smaller systemsand stronger interactions. We seek, here, to study thisbehavior systematically over a wide range of time scales(up to t ! 109J"1

? ), showing that the logarithmic growthbegins for arbitrarily weak interactions. We show that theentanglement growth does not saturate in the thermody-namic limit, and obtain additional quantities that distin-guish among possible mechanisms. Further discussion ofour conclusions appears after the model, methods, andnumerical results are presented.Model system.—One-dimensional (1D) s ¼ 1

2 spin chainsare a natural place to look for many-body localization [4] asthey are equivalent to 1D spinless lattice fermions. To start,consider the XX model with random z directed magneticfields so that the total magnetization Sz is conserved:

H0 ¼ J?X

i

ðSxi Sxiþ1 þ Syi Syiþ1Þ þ

X

i

hiSzi : (1)

Here, the fields hi are drawn independently from the interval[" !, !]. The eigenstates are equivalent via the Jordan-Wigner transformation to Slater determinants of free fermi-ons with nearest-neighbor hopping and random on-sitepotentials; particle number in the fermionic representationis related to Sz in the spin representation, so the z directedmagnetic field is essentially a random chemical potential.Now every single-fermion state is localized by any !> 0,and the dynamics of this spin Hamiltonian are localizedas well: a local disturbance at time t ¼ 0 propagates onlyto some finite distance (the localization length) as t ! 1. Asan example, consider the evolution of a randomly chosen Sz

basis state. The coupling J? allows ‘‘particles’’ (up spins) tomove, and entanglement entropy to develop, between two

PRL 109, 017202 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending6 JULY 2012

0031-9007=12=109(1)=017202(5) 017202-1 ! 2012 American Physical Society

subregions A and B. But the total amount of entanglemententropy generated remains finite as t ! 1 (Fig. 1), and thefluctuations of particle number eventually saturate as well(see below). The entanglement entropy for the pure stateof the whole system is defined as the von Neumann entropyS ¼ "tr!A log!A ¼ "tr!B log!B of the reduced densitymatrix of either subsystem. We always form the two biparti-tions by dividing the system at the center bond.

The type of evolution considered here can be viewed as a‘‘global quench’’ in the language of Calabrese and Cardy[14] as the initial state is the ground state of an artificialHamiltonian with local fields. Evolution from an initialproduct state with zero entanglement can be studied effi-ciently via time-dependent matrix product state methodsuntil a time where the entanglement becomes too large fora fixed matrix dimension. Since entanglement cannotincrease purely by local operations within each subsystem,its growth results only from propagation across the

subsystem boundary, even though there is no conservedcurrent of entanglement.The first question we seek to answer is whether there is

any qualitatively different behavior of physical quantitieswhen a small interaction

Hint ¼ JzX

i

Szi Sziþ1 (2)

is added. With Heisenberg couplings between the spins(Jz ¼ J?), the model is believed to have a dynamical tran-sition as a function of the dimensionless disorder strength"=Jz [4,5,7]. This transition is present in generic eigenstatesof the system and hence exists at infinite temperature atsome nonzero ". The spin conductivity, or equivalentlyparticle conductivity after the Jordan-Wigner transforma-tion, is zero in the many-body localized phase and nonzerofor small enough"=Jz. However, with exact diagonalizationthe system size is so limited that it has not been possible toestimate the location in the thermodynamic limit of thetransition of eigenstates or conductivities.We find that entanglement growth shows a qualitative

change inbehavior at infinitesimalJz. Instead of the expectedbehavior that a small interaction strength leads to a smalldelay in saturation and a small increase infinal entanglement,we find that the increase of entanglement continues to timesorders of magnitude larger than the initial localization timein the Jz ¼ 0 case (Fig. 1). This slowgrowth of entanglementis consistent with prior observations for shorter times andlarger interactions Jz ¼ 0:5J? and Jz ¼ J? [12,13],although the saturation behavior was unclear. Note that ob-serving a sudden effect of turning on interactions requireslarge systems, as a small change in the Hamiltonian appliedto the same initial state will take a long time to affect thebehavior significantly. We next explain briefly the methodsenabling large systems to be studied.Numerical methodology.—To simulate the quench, we

use the time evolving block decimation (TEBD) [15,16]method which provides an efficient method to perform atime evolution of quantum states, jc ðtÞi ¼ UðtÞjc ð0Þi, inone-dimensional systems. The TEBD algorithm can be seenas a descendant of the density matrix renormalization group[17] method and is based on a matrix product state (MPS)representation [18,19] of the wave functions. We use asecond-order Trotter decomposition of the short time propa-gator Uð!tÞ ¼ expð"i!tHÞ into a product of term whichacts only on two nearest-neighbor sites (two-site gates).Aftereach application, the dimension of the MPS increases. Toavoid an uncontrolled growth of the matrix dimensions,the MPS is truncated by keeping only the states which havethe largest weight in a Schmidt decomposition.In order to control the error, we check that the neglected

weight after each step is small (< 10"6). Algorithms ofthis type are efficient because they exploit the fact that theground-state wave functions are only slightly entangledwhich allows for an efficient truncation. Generally theentanglement grows linearly as a function of time which

FIG. 1 (color online). (a) Entanglement growth after a quenchstarting from a site factorized Sz eigenstate for different inter-action strengths Jz (we consider a bipartition into two half chainsof equal size). All data are for " ¼ 5 and L ¼ 10, except forJz ¼ 0:1 where L ¼ 20 is shown for comparison. The insetshows the same data but with a rescaled time axis and subtractedJz ¼ 0 values. (b) Saturation values of the entanglement entropyas a function of L for different interaction strengths Jz. The insetshows the approach to saturation.

PRL 109, 017202 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending6 JULY 2012

017202-2

Bardarson et. al. 2012

subregions A and B. But the total amount of entanglemententropy generated remains finite as t ! 1 (Fig. 1), and thefluctuations of particle number eventually saturate as well(see below). The entanglement entropy for the pure stateof the whole system is defined as the von Neumann entropyS ¼ "tr!A log!A ¼ "tr!B log!B of the reduced densitymatrix of either subsystem. We always form the two biparti-tions by dividing the system at the center bond.

The type of evolution considered here can be viewed as a‘‘global quench’’ in the language of Calabrese and Cardy[14] as the initial state is the ground state of an artificialHamiltonian with local fields. Evolution from an initialproduct state with zero entanglement can be studied effi-ciently via time-dependent matrix product state methodsuntil a time where the entanglement becomes too large fora fixed matrix dimension. Since entanglement cannotincrease purely by local operations within each subsystem,its growth results only from propagation across the

subsystem boundary, even though there is no conservedcurrent of entanglement.The first question we seek to answer is whether there is

any qualitatively different behavior of physical quantitieswhen a small interaction

Hint ¼ JzX

i

Szi Sziþ1 (2)

is added. With Heisenberg couplings between the spins(Jz ¼ J?), the model is believed to have a dynamical tran-sition as a function of the dimensionless disorder strength"=Jz [4,5,7]. This transition is present in generic eigenstatesof the system and hence exists at infinite temperature atsome nonzero ". The spin conductivity, or equivalentlyparticle conductivity after the Jordan-Wigner transforma-tion, is zero in the many-body localized phase and nonzerofor small enough"=Jz. However, with exact diagonalizationthe system size is so limited that it has not been possible toestimate the location in the thermodynamic limit of thetransition of eigenstates or conductivities.We find that entanglement growth shows a qualitative

change inbehavior at infinitesimalJz. Instead of the expectedbehavior that a small interaction strength leads to a smalldelay in saturation and a small increase infinal entanglement,we find that the increase of entanglement continues to timesorders of magnitude larger than the initial localization timein the Jz ¼ 0 case (Fig. 1). This slowgrowth of entanglementis consistent with prior observations for shorter times andlarger interactions Jz ¼ 0:5J? and Jz ¼ J? [12,13],although the saturation behavior was unclear. Note that ob-serving a sudden effect of turning on interactions requireslarge systems, as a small change in the Hamiltonian appliedto the same initial state will take a long time to affect thebehavior significantly. We next explain briefly the methodsenabling large systems to be studied.Numerical methodology.—To simulate the quench, we

use the time evolving block decimation (TEBD) [15,16]method which provides an efficient method to perform atime evolution of quantum states, jc ðtÞi ¼ UðtÞjc ð0Þi, inone-dimensional systems. The TEBD algorithm can be seenas a descendant of the density matrix renormalization group[17] method and is based on a matrix product state (MPS)representation [18,19] of the wave functions. We use asecond-order Trotter decomposition of the short time propa-gator Uð!tÞ ¼ expð"i!tHÞ into a product of term whichacts only on two nearest-neighbor sites (two-site gates).Aftereach application, the dimension of the MPS increases. Toavoid an uncontrolled growth of the matrix dimensions,the MPS is truncated by keeping only the states which havethe largest weight in a Schmidt decomposition.In order to control the error, we check that the neglected

weight after each step is small (< 10"6). Algorithms ofthis type are efficient because they exploit the fact that theground-state wave functions are only slightly entangledwhich allows for an efficient truncation. Generally theentanglement grows linearly as a function of time which

FIG. 1 (color online). (a) Entanglement growth after a quenchstarting from a site factorized Sz eigenstate for different inter-action strengths Jz (we consider a bipartition into two half chainsof equal size). All data are for " ¼ 5 and L ¼ 10, except forJz ¼ 0:1 where L ¼ 20 is shown for comparison. The insetshows the same data but with a rescaled time axis and subtractedJz ¼ 0 values. (b) Saturation values of the entanglement entropyas a function of L for different interaction strengths Jz. The insetshows the approach to saturation.

PRL 109, 017202 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending6 JULY 2012

017202-2

SA(t) ⇠ log t

•  Unbounded slow-growth of SA in an infinite system.

•  Saturates to a sub-thermal volume law in a finite sub system

RG Solution of time evolution

Dynamical quantum phase transitions in random spin chains

Ronen Vosk and Ehud AltmanDepartment of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel

Quantum spin chains and related systems undergo interesting phase transitions in their groundstates. The transition of the transverse-field Ising model from a paramagnet to a magneticallyordered state is a paradigmatic example of a quantum critical point. On the other hand, quantumtime evolution of the same systems involves all energies and it is therefore thought to be muchharder, if at all possible, to have sharp transitions in the dynamics. In this paper we show thatthe non-equilibrium dynamics of random spin chains do exhibit phase transitions characterized byuniversal singularities. The sharpness of the transitions and integrity of the phases owes to many-body localization, which prevents thermalization in these systems. Using a renormalization groupapproach, we solve the time evolution of random Ising spin chains with generic interactions startingfrom initial states of arbitrary energy. As a function of the Hamiltonian parameters, the system istuned through a dynamical transition, similar to the ground state critical point, at which the localspin correlations establish true long range temporal order. In the state with dominant transversefield, a spin that starts in an up state loses its orientation with time, while in the ”ordered” stateit never does. As in ground state quantum phase transitions, the dynamical transition has uniquesignatures in the entanglement properties of the system. When the system is initialized in a productstate the entanglement entropy grows as log(t) in the two ”phases”, while at the critical point itgrows as log

↵(t), with ↵ a universal number. This universal entanglement growth requires generic(”integrability breaking”) interactions to be added to the pure transverse field Ising model.

Closed systems evolving with Hamiltonian dynamics,are commonly thought to settle to a thermal equilibriumconsistent with the energy density in the initial state.Any sharp transition associated with the long time be-havior of observables must in this case correspond to clas-sical thermal phase transitions in the established thermalensemble. Accordingly in one dimension where thermaltransitions do not occur, dynamical transitions are notexpected either.

But systems with strong disorder may behave di↵er-ently. Anderson conjectured already in his original paperon localization, that closed systems of interacting parti-cles or spins with su�ciently strong disorder would fail tocome to equilibrium[1]. Recently, Basko et. al. [2] gavenew arguments to revive this idea of many-body localiza-tion, which has since received further support from the-ory and numerics[3–7]. An important point for our dis-cussion is that localized eigenstates, even at macroscopicenergies are akin to quantum ground states in their en-tanglement properties[7, 8]. In particular, it was pointedout in Ref. 8, that localized eigenstates can sustain longrange order and undergo phase transitions that wouldnot occur in a finite temperature equilibrium ensemble.But a theory of such dynamical transitions is lacking.

In this paper we develop a theory of such a transitionin the non-equilibrium dynamics of random Ising spinchains with generic interactions

H =X

i

⇥Jz

i

�z

i

�z

i+1

+ hi

�x

i

+ Jx

i

�x

i

�x

i+1

+ . . .⇤

(1)

Here Jz

i

, hi

and Jx

i

are uncorrelated random variablesand . . . represents other possible interaction terms thatrespect the Z

2

symmetry of the model. For simplicityof the later analysis we take the distributions of coupling

constants to be symmetric around zero. Without the lastterm, Jx

i

, the hamiltonian can be mapped to a system ofnon-interacting Fermions. We include the coupling Jx

i

to add interactions between the fermions and therebymake the system generic. We shall assume throughoutthat the interactions are weak, so that almost alwaysJx

i

⌧ Jz

i

, hi

hi+1

.

The transverse field Ising model (1) undergoes aground state quantum phase transition controlled by aninfinite randomness fixed point [9]. The transition sep-arates between a quantum paramagnet obtained whenthe transverse field is the dominant coupling and a spinordered state established when the Ising coupling Jz isdominant. Recently, it was pointed out that this tran-sition can also occur in eigenstates with arbitrarily highenergy, provided that the system is in the many-bodylocalized phase. Here we develop a theory of the non-equilibrium transition, focusing on the universal singulare↵ects it has on the time evolution of the system in pres-ence of generic interactions.

We shall describe the time evolution of the systemstarting from initial states of arbitrarily high energy.Specifically, we take random Ising configurations of thespins in the Sz basis, such as |

in

i = | ""#", . . . ##" i .The theoretical analysis relies on the strong disorder realspace RG approach (SDRG) [10, 11], which we recentlyextended to address the quantum time evolution of ran-dom systems[7]. The properties of the transition are elu-cidated by tracking the time evolution of two quantities:spin correlations and entanglement entropy.

First, we show that the spin auto correlation functionC

z

(t) = h in

|Sz

i

(t)Sz

i

(0) | in

i decays as a power-law inthe paramagnetic phase, whereas it saturates to a posi-

| 0i =

Pick out largest couplings ⌦ = max (Jzi , hi)

Short times (t ≈1/Ω): System evolves according to Hfast

Other spins essentially frozen on this timescale.

Hfast

Longer times (t >>1/Ω): Eliminate fast modes (order Ω) perturbatively to obtain effective evolution for longer timescales.

R. Vosk and EA PRL (2013), PRL (2014)

RG flow in real time! reproduces the logarithmic growth of entanglement entropy

Complementary viewpoint: spectral RG (RSRG-X) Pekker, Refael, EA, Demler & Oganesyan PRX (2014)

Dynamical quantum phase transitions in random spin chains

Ronen Vosk and Ehud AltmanDepartment of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel

Quantum spin chains and related systems undergo interesting phase transitions in their groundstates. The transition of the transverse-field Ising model from a paramagnet to a magneticallyordered state is a paradigmatic example of a quantum critical point. On the other hand, quantumtime evolution of the same systems involves all energies and it is therefore thought to be muchharder, if at all possible, to have sharp transitions in the dynamics. In this paper we show thatthe non-equilibrium dynamics of random spin chains do exhibit phase transitions characterized byuniversal singularities. The sharpness of the transitions and integrity of the phases owes to many-body localization, which prevents thermalization in these systems. Using a renormalization groupapproach, we solve the time evolution of random Ising spin chains with generic interactions startingfrom initial states of arbitrary energy. As a function of the Hamiltonian parameters, the system istuned through a dynamical transition, similar to the ground state critical point, at which the localspin correlations establish true long range temporal order. In the state with dominant transversefield, a spin that starts in an up state loses its orientation with time, while in the ”ordered” stateit never does. As in ground state quantum phase transitions, the dynamical transition has uniquesignatures in the entanglement properties of the system. When the system is initialized in a productstate the entanglement entropy grows as log(t) in the two ”phases”, while at the critical point itgrows as log

↵(t), with ↵ a universal number. This universal entanglement growth requires generic(”integrability breaking”) interactions to be added to the pure transverse field Ising model.

Closed systems evolving with Hamiltonian dynamics,are commonly thought to settle to a thermal equilibriumconsistent with the energy density in the initial state.Any sharp transition associated with the long time be-havior of observables must in this case correspond to clas-sical thermal phase transitions in the established thermalensemble. Accordingly in one dimension where thermaltransitions do not occur, dynamical transitions are notexpected either.

But systems with strong disorder may behave di↵er-ently. Anderson conjectured already in his original paperon localization, that closed systems of interacting parti-cles or spins with su�ciently strong disorder would fail tocome to equilibrium[1]. Recently, Basko et. al. [2] gavenew arguments to revive this idea of many-body localiza-tion, which has since received further support from the-ory and numerics[3–7]. An important point for our dis-cussion is that localized eigenstates, even at macroscopicenergies are akin to quantum ground states in their en-tanglement properties[7, 8]. In particular, it was pointedout in Ref. 8, that localized eigenstates can sustain longrange order and undergo phase transitions that wouldnot occur in a finite temperature equilibrium ensemble.But a theory of such dynamical transitions is lacking.

In this paper we develop a theory of such a transitionin the non-equilibrium dynamics of random Ising spinchains with generic interactions

H =X

i

⇥Jz

i

�z

i

�z

i+1

+ hi

�x

i

+ Jx

i

�x

i

�x

i+1

+ . . .⇤

(1)

Here Jz

i

, hi

and Jx

i

are uncorrelated random variablesand . . . represents other possible interaction terms thatrespect the Z

2

symmetry of the model. For simplicityof the later analysis we take the distributions of coupling

constants to be symmetric around zero. Without the lastterm, Jx

i

, the hamiltonian can be mapped to a system ofnon-interacting Fermions. We include the coupling Jx

i

to add interactions between the fermions and therebymake the system generic. We shall assume throughoutthat the interactions are weak, so that almost alwaysJx

i

⌧ Jz

i

, hi

hi+1

.

The transverse field Ising model (1) undergoes aground state quantum phase transition controlled by aninfinite randomness fixed point [9]. The transition sep-arates between a quantum paramagnet obtained whenthe transverse field is the dominant coupling and a spinordered state established when the Ising coupling Jz isdominant. Recently, it was pointed out that this tran-sition can also occur in eigenstates with arbitrarily highenergy, provided that the system is in the many-bodylocalized phase. Here we develop a theory of the non-equilibrium transition, focusing on the universal singulare↵ects it has on the time evolution of the system in pres-ence of generic interactions.

We shall describe the time evolution of the systemstarting from initial states of arbitrarily high energy.Specifically, we take random Ising configurations of thespins in the Sz basis, such as |

in

i = | ""#", . . . ##" i .The theoretical analysis relies on the strong disorder realspace RG approach (SDRG) [10, 11], which we recentlyextended to address the quantum time evolution of ran-dom systems[7]. The properties of the transition are elu-cidated by tracking the time evolution of two quantities:spin correlations and entanglement entropy.

First, we show that the spin auto correlation functionC

z

(t) = h in

|Sz

i

(t)Sz

i

(0) | in

i decays as a power-law inthe paramagnetic phase, whereas it saturates to a posi-

The TFIM makes a natural starting point for investigat-ing dynamical critical points in strongly disordered sys-tems. The TFIM, Eq. (1), however, can be mapped to a freefermion theory, making its dynamics fundamentally equiv-alent to that of a certain class of single-particle Andersonlocalization (with particle-hole symmetry). We avoid thisproblem by adding the J0 interaction term, which preservesthe Z2 symmetry, while making the model intrinsicallyinteracting. It has been shown that the dynamics in themany-body localized phase, i.e., in the presence ofinteractions, can be different than in the noninteractingcase [20–22,29,33].The paper is organized as follows. In Sec. II, we develop

the RSRG-X procedure and consider the flows it produces.The flows reveal the evolution of the many-body eigenstatestructure as temperature is varied, which allows us toidentify the dynamical phase transition and construct thephase diagram. We find that the temperature-tuned tran-sition is controlled by an infinite-randomness critical point,with the same scaling properties as the zero-temperaturequantum phase transition [25]. In Secs. III and IV, weexamine two dynamical observables: the low-frequencyspin autocorrelation function (and an associated Edwards-Anderson-type order parameter) and the frequency-dependent thermal conductivity. Using the RSRG-X, wefind that in the vicinity of the critical temperature, bothobservables show scaling behavior consistent with theinfinite-randomness critical point. The scaling behaviorbecomes nonanalytic in the infinite system-size limit.Finally, we discuss the implications of our results in Sec. V.

II. DYNAMICAL PHASE DIAGRAMOF THE hJJ0 MODEL

Our task is to develop a renormalization group proceduresuited for describing the excited states of the weakly

interacting hJJ0 model. Our approach is based on the coreideas of the RSRG methods developed for the ground statebehavior of random magnets [23–26,28], and for the TFIM,in particular [27]. In the ground state methods, at eachRSRG step we diagonalize the strongest local term of theHamiltonian [Eq. (1)] and fix the corresponding subsystemin the ground eigenstate of that term. We generate effectivecouplings between the subsystem and its neighboring spinsthrough second-order perturbation theory, allowing onlyvirtual departures from the chosen ground state of thestrong term. In the TFIM, when the nearest-neighborx-x interaction is dominant, the RSRG steps lead to theformation of a macroscopic ferromagnetic cluster thatbreaks the Ising symmetry. On the other hand, when thetransverse field is dominant, it pins most local spins to theþz direction. The addition of weak interactions, J0i ≪ Ji,hi, results in a shift of the critical point but cannot affect theuniversal zero temperature properties of the TFIM.

A. RSRG-XThe crucial difference between our RSRG-X method and

the ground state RSRG methods has to do with the choiceof the local eigenstates at each RG step. Instead of retainingonly the lowest energy states at each RG step, we canchoose either the low-energy or the high-energy manifoldof the local term. The two manifolds are separated by alarge gap, which controls the perturbation theory withwhich we generate the effective couplings. Thus, eachRSRG-X step corresponds to a binary branching of a tree,as illustrated in Fig. 2(a), where the leaves of the treecorrespond to the actual many-body spectrum, Fig. 2(b).Implementing the RSRG-X on the hJJ0 model requires

two types of RG decimation steps: site and bond decima-tions. Consider the nth RSRG-X step. If the largest gap inthe system is due to a field h2 [see Fig. 2(c)], the site could

(a) (b) (c) (d)

FIG. 2. (a) The RSRG-X tree for a six-site Ising chain. The leaves of the tree correspond to the many-body eigenstates of the spinchain. (b) The corresponding eigenspectrum found by exact diagonalization. (c),(d) RSRG-X rules for site (c) and bond (d) decimationin the hJJ0 model. In the site decimation rule, a local magnetic field (h2) is the dominant coupling; thus, the corresponding local spin iseither aligned (ground state) or antialigned (excited state) with it. Eliminating h2, we obtain a new set of couplings ~h1, ~J1, ~J1

0, and ~h3 tosecond order (see Appendix B for details). In the bond decimation rule, J2 is the dominant coupling, corresponding to the neighboringspins being either aligned (ground state) or antialigned (excited state, i.e., a domain wall) along the x axis.

HILBERT-GLASS TRANSITION: NEW UNIVERSALITY OF … PHYS. REV. X 4, 011052 (2014)

011052-3

hi = ⌦

JL JR

J̃ = JLJR/⌦

e.g. large transverse field:

Dynamical/spectral scheme: focus on low frequencies (energy differences)

Usual ground state scheme (Dasgupta-Ma): focus on low absolute energies

Outcome of RG: integrals of motion = (frozen spins)

Example: strong transverse field

�̃x

i

= Z�x

i

+ exponential tail

He↵ = e�iSHeiS

H =X

i

⇥Ji

�z

i

�z

i+1 + hi

�x

i

+ Vi

�x

i

�x

i+1

⇤hi = ⌦

JL JR

He↵ = ⌦�̃x

i

+ VL

�̃x

i

�x

L

+ VR

�̃x

i

�x

R

+JL

JR

⌦�z

L

�̃x

i

�z

R

J̃ = JLJR/⌦

In this RG scheme degrees of freedom are not eliminated but rather frozen into quasi-local integrals of motion:

Effective Hamiltonian (fixed point theory)

The fixed point model can serve as a useful phenomenological description of the many-body localized phase. Oganesyan & Huse (2013); Serbyn, Papic & Abanin (2013)

Vij

⇠ V e�|xi�xj |/⇠

HFP

=X

i

h̃i

�̃x

i

+X

ij

Vij

�̃x

i

�̃x

j

+X

ijk

Vijk

�̃x

i

�̃x

j

�̃x

k

+ . . .

Depends only on the quasi-local integrals of motion:

Note the analogy with Fermi-liquid theory!

[HFL, n̂k] = 0HFL =X

k⇠kF

✏kn̂k +X

k,k0

fkk0 n̂kn̂k0

Results of the effective theory

•  Persistence of local coherence: spin echos Bahri, Vosk, EA, Vishwanath (2013); Serbyn et. al. (2013)

•  Simple explanation for log growth of entanglement entropy

•  Anomalous relaxation of observables (related to log growth) Vasseur et. al. (2014), Serbyn et. al. (2014)

tent ⇠ V �1ij ⇠ V �1elij/⇠ Sent ⇠ l ⇠ ⇠ log(V t)

Vij

There can be distinct localized states at high energy

E

Glass

Paramagnet

Huse et. al. (2013)

log(h/Jz)

Paramagnet: integrals of motion are �̃x

i

Broken symmetry in individual eigenstates but not in thermal ensemble

Eigenstate-glass: integrals of motion are �̃z

i

| Ei ⇡

Dynamical critical points fully characterized using the RG approach. Vosk and EA 2013; Pekker, Refael, EA, Demler, Oganesyan 2013;

Can also have topological localized states with Protected coherent edge modes at high energy ! Bahri, Vosk, EA and Vishawanth (2013)

Limitation of the RG scheme: resonances

Resonances between decimated sites can generate a slow mode that is not accounted for by the RG

Ω-δΩΩ Jeff

If Jeff>δΩ

Resonances do not proliferate in MBL phase! (Irrelevant in RG sense).

(Vosk and EA PRL 2013)

This RG scheme is limited to the MBL phase. Cannot address the transition to a thermal phase!

How to describe the many body localization transition?

Thermalizing

Energy eigenstates are highly entangled:

SA ⇠ Ld

Many-body localized

Eigenstates have low entanglement

SA ⇠ Ld�1

? (area law) (volume law)

Localization transition: fundamental change in entanglement pattern. More radical than in any other phase transition we know !

Coarse Grained Model

Spin chain fragmented into puddles of different types: incipient insulators and incipient metals. Modeled as random matrices:

�i,�i gi

��1i = ⌧i Time for entanglement to spread across the block

�i Mean level spacing in the block

gi = �i/�i Number of correlated levels in the random matrix

gi ⌧ 1 gi � 1 “thermalizing block” “insulating block”

(Wigner-Dyson statistics) (Poisson level statistics)

Digression: entanglement vs. transport time

To entangle two sides of the block it is enough to make transitions which change the energy by order 1.

l

Need l such transitions to transport extensive energy (order l) from one side of the block to the other.

⌧tr = l⌧

Starting point for RG: chain of coupled blocks

1Γ

1g2Γ

2g3Γ

3g4Γ

4g5Γ

5g

12Γ 12g 23Γ 23g 34Γ 34g 45Γ 45g

Γij = far end-to-end entanglement rate of adjacent blocks (Γ of the two blocks if they were considered as a single block)

Δij = Mean level spacing of the two block system

gij >> 1 ‘effective’ link (‘thermalizing’)

gij << 1 ‘ineffective’ link (‘insulating’)

�34 g34Meaning of the link variables:

�ij ⇠ 2�lij

Schematics of the RG

1Γ

1g2Γ

2g3Γ

3g4Γ

4g5Γ

5g

12Γ 12g 23Γ 23g 34Γ 34g 45Γ 45g

1Γ

1g2Γ

2g5Γ

5g

12Γ 12g

�34 g34

�̃R g̃Rg̃L�̃L

Join blocks which entangle with each other on the fastest scale. Then compute renormalized couplings to the left and right.

Computing the flow will tell us whether we end up with one big thermalizing matrix (g>>1) or a big insulator (g<<1) at large scales

RG scheme

1Γ

1g2Γ

2g3Γ

3g

12Γ 12g 23Γ 23g

12Γ

12g3Γ

3g

(i) Two ‘insulating’ links, i.e. g12<<1 and g23<<1

The simplest limits:

�R =�12�23

�2

log ⌧tot

= log ⌧12 + log ⌧23 � log ⌧2

Can be derived for insulators from first principles. But also simply understood by taking a log of the two sides:

ltot

= l12 + l23 � l2

RG scheme

1Γ

1g2Γ

2g3Γ

3g

12Γ 12g 23Γ 23g

12Γ

12g3Γ

3g

(ii) Two thermalizing links, i.e. g12, g23 >>1

The simplest limits:

3

lax an extensive energy imbalance across the block. Onthe end-to-end entanglement time-scale ⌧

i

the amount ofenergy transported across the block remains of order themicroscopic energy scale, so is not extensive. To relaxan extensive energy imbalance requires transporting anextensive (in l

i

) amount of energy, so requires of orderli

entanglement times. Hence ⌧tr

⇠ li

⌧i

. Note that theentanglement time ⌧

i

is well defined even in a system sub-ject to external periodically time dependent fields, suchas a Floquet system, where total energy is not conservedand there is no extensive quantity that can be trans-ported, so the transport time is meaningless.

The two-block parameters, �ij

, �ij

and gij

= �ij

/�ij

,are defined as the block parameters that would ensue ifthe two adjacent blocks are treated as a single block.For instance �

ij

= W2�(li+lj)p

li

+ lj

⇠= �i

�j

/W . Wecall the link between these adjacent blocks i and j “ef-fective” if g

ij

� 1 and “ine↵ective” if gij

⌧ 1. A gen-eral requirement to be met by the initial distributionsand retained throughout the RG flow is that the small-est block rate min

i

�i

is larger than the largest two-blockrate ⌦ = max

ij

�ij

. ⌦, the largest two-block rate, servesas the running RG frequency cuto↵ scale. In this wayall the fast rates (� > ⌦) are intra-block, while the slowrates, below the cuto↵ scale, are inter-block.

We now frame the RG as a strong disorder schemeoperating on the chain in real space. At each RG stepthe cuto↵ scale ⌦ is reduced by joining the two blockswith the largest inter-block rate �

ij

. Thus the old two-block parameters become the new one-block parametersof this new larger block. The non trivial part of therenormalization is to determine the new two-block pa-rameters �

L

and �R

, which connect the new block toits left and right neighbors. To compute these rates wehave to solve for the entanglement rate of three coupledblocks. This calculation cannot be done microscopicallyin the most general case, but the structure of the solutionis rather constrained by the known behavior in limitingcases. These constraints allow us to formulate a closedand self consistent RG scheme. Modifying details of theRG scheme within the allowed constraints does not sig-nificantly change the outcome.

Suppose we are now joining blocks 1 and 2 with thefastest two-block rate �

12

and want to find the new rate�R

of the three block system 1, 2, 3. There are two limitsin which we can obtain simple reliable expressions forthis rate. First, if both links are ine↵ective, g

12

⌧ 1 andg23

⌧ 1, then we can compute �R

by straight forwardperturbation theory in the weak dimensionless couplings(see appendix A) to obtain

�R

= �12

�23

/�2

. (1)

This case describes the process of making a bigger insula-tor out of two insulating links. When applied repeatedlyto a long insulating chain this rule indeed leads to theexpected exponential increase of the entanglement timewith the length of the insulator.

Second, if both links lead to e↵ective coupling, g12

� 1

and g23

� 1, then the entanglement spreads sequentiallythrough the three block chain and we must add the en-tanglement times G�1 = ��1:

1

�R

=1

�12

+1

�23

� 1

�2

. (2)

In a system with energy conservation the above formulais simply Ohm’s law for the thermal resistances.The two RG rules given above lead to the correct scal-

ing of length and time in insulating regions (l ⇠ log ⌧)and fully conducting regions (l ⇠ ⌧ ⇠ ⌧

tr

/l). To com-plete the RG scheme we have to determine the behaviorof boundaries between insulating and conducting regions.There we expect to encounter three block systems withone e↵ective link g

12

� 1 and one ine↵ective link g23

⌧ 1.We have to distinguish the case in which the e↵ective linkis a link between two metallic blocks from the case whenit is a link between a metallic and an insulating block.Joining an insulator and a conductor, even if the link

ultimately turns out to be e↵ective, leads to exponentialsuppression of the relaxation rate with the length of theinsulator (see appendix A). Coupling this structure to yetanother insulator (i.e. the ine↵ective link g

23

) would leadto further exponential suppression and hence insulating-like scaling of �

R

as prescribed in the RG rule (1). Inappendix A we justify this formula using resummationof the perturbation theory. On the other hand, if thee↵ective link g

12

is between two conductors, then theexponential suppression of the rate �

R

is only the resultof the transport through the ine↵ective link g

23

whichconstitutes the bottleneck for entanglement spread. Inthis case we should use the second RG rule (2), whichessentially adds the time of this bottleneck to the fastertimescale ��1

12

.We did not give expressions for intermediate regimes

where gij

⇠ 1. Our approach will rely on having sucha wide distribution of g’s at the interesting fixed point,that the probability of having g ⇠ 1 on a link vanishes.In practice we thus treat any g > 1 as g � 1, and anyg < 1 as g ⌧ 1.

III. FIXED POINTS AND FLOWS

Application of these RG rules to a chain with a randomdistribution of coupling constants leads to a flow of thosedistributions. Instead of solving the rather complicatedintegro-di↵erential equations for the scale-dependent dis-tributions we simply simulate the RG process on an en-semble of chains, each with up to 105 or more initialblocks. Each block in the initial state is taken to be a100 ⇥ 100 matrix with uniform � = W/100 and g = 1,so the initial block lengths are l

0

= log2

(100). This im-mediately implies also a uniform �

ij

. The randomness isintroduced in the distribution of the inter block couplingconstant g

ij

taken from a log normal distribution withmean hlog(g

0

)i and standard deviation �g

= 1. the ini-tial distribution of g

ij

needs to be stated more precisely,

ΓR cannot be derived perturbatively in this case. But we know: energy transport is diffusive and (therefore) entanglement propagates ballistically.

RG scheme

1Γ

1g2Γ

2g3Γ

3g

12Γ 12g 23Γ 23g

12Γ

12g3Γ

3g

These rules capture scaling in big insulating (i) or conducting regions (ii).

The same rules can be applied when the three sites sit at interfaces provided all individual blocks and links have extreme values of g, i.e. g>>1 or g<<1.

Outcome of the RG flow

1Γ

1g2Γ

2g3Γ

3g4Γ

4g5Γ

5g

12Γ 12g 23Γ 23g 34Γ 34g 45Γ 45g

hloggi

MBL

hlog g0i = 1

-2

critical

How does diffusion disappear ?

d log g

dL

hlog g0ic ⇡ �1.2

L

RG results – dynamical scaling exponent for transport Relation between transport time τtr and length l of blocks:

Surprise! The transition is from localized to anomalous diffusion. Seen also in recent ED studies: Bar-Lev et al 2014 ; Agarwal et al 2014

(bare coupling) hlog g0i

ltr ⇠ t↵ltr ⇠ log t

ltr ⇠pDt

⌧ = ⌧tr/lScaling relation between transport and entanglement spreading:

SE ⇠ t↵

1�↵

4 hlog g0i

log

l

log τtr

0

Dyn

amic

al e

xpon

ent (α

)

Anomalous diffusion = Griffith phase

Exponentially rare insulating puddles in the metal

Exponentially long delay

⌧(l) = ⌧0el/l0

P (l) ⇠ l�10 e�l/⇠

l � ⇠ � lo

⇠ ! 1Infinite randomness but thermal critical point at

All “insulating” puddles ultimately thermalize but at broadly distributed times! Broad distribution of times:

P (⌧) = ⌧�10

⇣⌧0⌧

⌘1+l0⇠

(bare coupling) hlog g0i

ltr ⇠ t↵ltr ⇠ log t

ltr ⇠pDt

Entanglement scaling in eigenstates

P (S,L, g0) =1

LP̃

S

L,

L

⇠(g0)

�

Near critical point expect distribution of S to scale:

In particular all moments:

µS(L, g0) = Lfµ

L

⇠(g0)

�

g12 ~ # of 2-block product states in an eigenstate of the coupled system SE(L/2) ⇠ log2 [g(L) + 1]

See Frank-Pollmann’s talk in the workshop (Kjall et. al. PRL 2014). Identified δS as a good scaling variable to identify transition in ED

6

FIG. 3. should it be labelled as dividing by Sth = the ther-mal entropy, instead of L? If we did all the logs base two andthe entropy in bits, then dividing by L would be the sameas by the entropy (a) Entanglement entropy of eigenstatesdivided by the thermal entropy (solid lines) and its fluctu-ations (dashed) near the many-body localization transitioncomputed from the RG as described in the text for di↵er-ent chain lengths L. (b) Data collapse of hSi/L and �S/L

obtained with the fitted correlation length exponent ⌫ ⇠= 2.8.

less coupling g and the entanglement entropy in eigen-states. Suppose we renormalized the chain all the waydown to the point where we have only two blocks re-maining in the system. If these two blocks were decou-pled then the exact eigenstates would be non-entangledproduct states of the two blocks. The rate �

ij

rep-resents the lifetime of the product states due to weakcoupling between the blocks (relative to intra-block cou-pling). The true eigenstates are then a superpositionof the ⇠ g

12

+ 1 = 1 + �12

/�12

� 1 product statesnearest in energy (one is added to correctly match thedecoupled limit g

12

= 0, where the superposition stillcontains one state, the original product state). HenceS12

= log(1 + g12

) has the meaning of a “diagonal” en-tropy associated with a single energy eigenstate when thecorresponding density matrix is expressed in the basis ofproduct states. This entropy is related to entanglemententropy, but is defined without tracing out part of thesystem; it can be as large as the full thermal entropy ofthe two blocks.

The above definition might not reflect a bulk entropyin cases where the last decimated link is a very weak linkwhich happens to be located far from the center and closeto one of the ends of the chain. To avoid this issue we

use a slightly modified definition of the entropy. We keeptrack of the coupling g associated with the block thatspans the middle of the original chain at each stage ofthe RG and record its maximum over the entire flow. Wedenote the outcome as g

max

and define S = log(1+gmax

).The need for taking g

max

rather than the last surviv-ing g is particularly important when there is a very weaklink somewhere in the chain. As a toy example considera chain of three blocks, where blocks 1 and 2 are coupledand together span the interface, whereas blocks 2 and 3are completely disconnected (i.e �

23

= 0). In this casewe will first join blocks 1 and 2 to get a new block withg12

> 0, which spans the interface. Obviously there is en-tanglement across the interface, which S

12

= log(1+g12

)represents. However if we now continue to renormalize wewould obtain g = 0 for the last remaining block, which ofcourse represents only the absence of entanglement acrossthe disconnected link.

The RG scheme is repeated on a large number of dis-order realizations allowing to obtain a full distributionof the the entanglement entropy. Examples of entropydistributions found in the di↵erent states, including thelocalized state, the critical point, the Gri�ths phase andthe di↵usive regime are shown in appendix D. Here in Fig.3(a) we present the average and standard deviation ofs = S/L as a function of the bare coupling hlog g

0

i calcu-lated for varying system sizes L. we are still sloppy abouta factor of log 2 As expected, the curves S(log g

0

)/Land �S(log g

0

)/L sharpen with increasing size in a waywhich suggests the existence of a critical point in thelimit L ! 1. In this case we anticipate that near thecritical point the functions S(g

0

, L)/L and �S(g0

, L)/Lshould all collapse on scaling functions of a single vari-able (g

0

� g0c

)L1/⌫ , where the critical value g0c

and theuniversal critical exponent ⌫ are fitting parameters (seeFig. 3(b)).

The correlation-length critical exponent extractedfrom the data collapse, ⌫ ⇠= 2.8, satisfies the Harris in-equality ⌫ � 2/d required for stability of the criticalpoint24,27. It is interesting that a much smaller exponent,which violates this inequality, was found in recent finitesize scaling analysis of exact diagonalization data16,28.This, as well as other di↵erences from our scaling formmay be due to the small system sizes studied in Refs.[16,28], L < 18, which are likely to be too small to ap-proach the scaling limit. Indeed in our case, although westart from a coarse grained model, system sizes of 50 ormore blocks are needed. We do make some assumptionsin choosing some of the details of our RG rules. Therethus remains the possibility that the precise value of thisexponent ⌫ is sensitive to such details, so this estimateof ⌫ should be viewed as possibly approximate, and thequestion of whether the correct exponent ⌫ is larger thanor “only” equal to the Harris bound of 2 remains to beanswered.

The entanglement scaling functions shown in Fig. 3(b)expose important properties of the subdi↵usive Gri�thsphase. A finite size system in the subdi↵usive Gri�ths

6

FIG. 3. should it be labelled as dividing by Sth = the ther-mal entropy, instead of L? If we did all the logs base two andthe entropy in bits, then dividing by L would be the sameas by the entropy (a) Entanglement entropy of eigenstatesdivided by the thermal entropy (solid lines) and its fluctu-ations (dashed) near the many-body localization transitioncomputed from the RG as described in the text for di↵er-ent chain lengths L. (b) Data collapse of hSi/L and �S/L

obtained with the fitted correlation length exponent ⌫ ⇠= 2.8.

less coupling g and the entanglement entropy in eigen-states. Suppose we renormalized the chain all the waydown to the point where we have only two blocks re-maining in the system. If these two blocks were decou-pled then the exact eigenstates would be non-entangledproduct states of the two blocks. The rate �

ij

rep-resents the lifetime of the product states due to weakcoupling between the blocks (relative to intra-block cou-pling). The true eigenstates are then a superpositionof the ⇠ g

12

+ 1 = 1 + �12

/�12

� 1 product statesnearest in energy (one is added to correctly match thedecoupled limit g

12

= 0, where the superposition stillcontains one state, the original product state). HenceS12

= log(1 + g12

) has the meaning of a “diagonal” en-tropy associated with a single energy eigenstate when thecorresponding density matrix is expressed in the basis ofproduct states. This entropy is related to entanglemententropy, but is defined without tracing out part of thesystem; it can be as large as the full thermal entropy ofthe two blocks.

The above definition might not reflect a bulk entropyin cases where the last decimated link is a very weak linkwhich happens to be located far from the center and closeto one of the ends of the chain. To avoid this issue we

use a slightly modified definition of the entropy. We keeptrack of the coupling g associated with the block thatspans the middle of the original chain at each stage ofthe RG and record its maximum over the entire flow. Wedenote the outcome as g

max

and define S = log(1+gmax

).The need for taking g

max

rather than the last surviv-ing g is particularly important when there is a very weaklink somewhere in the chain. As a toy example considera chain of three blocks, where blocks 1 and 2 are coupledand together span the interface, whereas blocks 2 and 3are completely disconnected (i.e �

23

= 0). In this casewe will first join blocks 1 and 2 to get a new block withg12

> 0, which spans the interface. Obviously there is en-tanglement across the interface, which S

12

= log(1+g12

)represents. However if we now continue to renormalize wewould obtain g = 0 for the last remaining block, which ofcourse represents only the absence of entanglement acrossthe disconnected link.

The RG scheme is repeated on a large number of dis-order realizations allowing to obtain a full distributionof the the entanglement entropy. Examples of entropydistributions found in the di↵erent states, including thelocalized state, the critical point, the Gri�ths phase andthe di↵usive regime are shown in appendix D. Here in Fig.3(a) we present the average and standard deviation ofs = S/L as a function of the bare coupling hlog g

0

i calcu-lated for varying system sizes L. we are still sloppy abouta factor of log 2 As expected, the curves S(log g

0

)/Land �S(log g

0

)/L sharpen with increasing size in a waywhich suggests the existence of a critical point in thelimit L ! 1. In this case we anticipate that near thecritical point the functions S(g

0

, L)/L and �S(g0

, L)/Lshould all collapse on scaling functions of a single vari-able (g

0

� g0c

)L1/⌫ , where the critical value g0c

and theuniversal critical exponent ⌫ are fitting parameters (seeFig. 3(b)).

The correlation-length critical exponent extractedfrom the data collapse, ⌫ ⇠= 2.8, satisfies the Harris in-equality ⌫ � 2/d required for stability of the criticalpoint24,27. It is interesting that a much smaller exponent,which violates this inequality, was found in recent finitesize scaling analysis of exact diagonalization data16,28.This, as well as other di↵erences from our scaling formmay be due to the small system sizes studied in Refs.[16,28], L < 18, which are likely to be too small to ap-proach the scaling limit. Indeed in our case, although westart from a coarse grained model, system sizes of 50 ormore blocks are needed. We do make some assumptionsin choosing some of the details of our RG rules. Therethus remains the possibility that the precise value of thisexponent ⌫ is sensitive to such details, so this estimateof ⌫ should be viewed as possibly approximate, and thequestion of whether the correct exponent ⌫ is larger thanor “only” equal to the Harris bound of 2 remains to beanswered.

The entanglement scaling functions shown in Fig. 3(b)expose important properties of the subdi↵usive Gri�thsphase. A finite size system in the subdi↵usive Gri�ths

Entanglement scaling in eigenstates

g12 ~ # of 2-block product states in an eigenstate of the coupled system SE(L/2) ⇠ log2 [g(L) + 1]

P (S,L, g0) =1

LP̃

S

L,

L

⇠(g0)

�

Near critical point expect distribution of S to scale:

In particular all moments:

µS(L, g0) = Lfµ

L

⇠(g0)

�

Griffiths phase is thermal! S/L flows to the thermal value at long scales, fluctuation flows to zero.

⌫ = 2.81

�S/L

S/L

Thermalizing

Volume law entanglement

Many-body localized

Area law entanglement

“Classical” dynamics Quantum coherent dynamics

Localized fixed-point

Dynamical RG Random matrix RG

diffusive sub-diffusive

Summary

Thermalizing

Volume law entanglement

Many-body localized

Area law entanglement

“Classical” dynamics Quantum coherent dynamics

Localized fixed-point

Dynamical RG Random matrix RG

SA broadly distributed at crit. point

diffusive sub-diffusive

Outlook

•  Generalizations to higher dimensions? •  Are there other paradigms of non-thermal states?

–  Localization in trans. Invariant systems? –  Non-thermal delocalized states?

•  Numerical approaches for quantum dynamics?

•  Experiments: –  cold atoms, –  NV centers in diamond –  Conventional solid state probed at finite frequency?