Emergent ergodicity at the transition between many-body localized phases
Rahul Sahay, Francisco Machado, Bingtian Ye, Chris R. Laumann, Norman Y. Yao
EEmergent ergodicity at the transition between many-body localized phases
Rahul Sahay, ∗ Francisco Machado, ∗ Bingtian Ye, ∗ Chris R. Laumann, and Norman Y. Yao
1, 3 Department of Physics, University of California, Berkeley, California 94720 USA Department of Physics, Boston University, Boston, MA, 02215, USA Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Strongly disordered systems in the many-body localized (MBL) phase can exhibit ground stateorder in highly excited eigenstates. The interplay between localization, symmetry, and topology hasled to the characterization of a broad landscape of MBL phases ranging from spin glasses and timecrystals to symmetry protected topological phases. Understanding the nature of phase transitionsbetween these different forms of eigenstate order remains an essential open question. Here, weconjecture that no direct transition between distinct MBL orders can occur; rather, a thermal phasealways intervenes. Motivated by recent advances in Rydberg-atom-based quantum simulation, wepropose an experimental protocol where the intervening thermal phase can be diagnosed via thedynamics of local observables.
Traditionally, the classification of phases of matterhas focused on systems at or near thermal equilibrium.Many-body localization (MBL) offers an alternative tothis paradigm [1–6]. In particular, owing to the presenceof strong disorder, MBL phases are characterized by theirfailure to thermalize [7–10]. This dynamical property im-poses strong constraints on the structure of eigenstates;namely, that they exhibit area-law entanglement and canbe described as the ground state of quasi-local Hamilto-nians [11, 12]. Perhaps the most striking consequenceis that such systems can exhibit order – previously re-stricted to the ground state – throughout their entiremany-body spectrum [12–17]. This offers a particularlytantalizing prospect for near-term quantum simulators:The ability to observe phenomena, such as coherent topo-logical edge modes, without the need to cool to the many-body ground state [18–22].The presence of eigenstate order in the many-body lo-calized phase also raises a more fundamental question:What is the nature of phase transitions between differenttypes of MBL order? This question highlights a delicatebalance between the properties of localization and phasetransitions. On the one hand, the stability of MBL iscontingent upon the existence of an extensive number ofquasi- local conserved quantities (“ ‘ -bits”) [11, 23]. Onthe other hand, the correlation length at a second-orderphase transition diverges [24]. Understanding and char-acterizing this interplay remains an outstanding chal-lenge. Indeed, while certain studies suggest the pres-ence of a direct transition between distinct MBL phases[16, 20, 25–28], others have found signatures of delocal-ization at the transition [29, 30].In this Letter, we conjecture that any transitionbetween distinct MBL phases is invariably forbiddenand that an intervening thermal phase always emerges(Fig. 1a). This conjecture is motivated by an extensivenumerical study of three classes of MBL transitions: (i) asymmetry-breaking transition, (ii) a symmetry-protectedtopological (SPT) transition, and (iii) a discrete timecrystalline transition (in a Floquet system). By sys- W J /W h
3. With increasing Γ, the size of thethermal region decreases until the system remain localizedfor all W J /W h . (inset) Schematic of the full phase diagramas a function of W J /W h , W V and Γ. tematically constructing the various phase diagrams, weshow that an intervening ergodic region emerges for allnumerically-accessible interaction strengths. Moreover,we demonstrate that this emergent ergodicity is inti-mately tied to the presence of a phase transition; a disorder-less , symmetry-breaking field suppresses the in-tervening ergodic phase. In addition to numerics, weanalyze two instabilities which could induce thermaliza-tion near the putative transition: (i) the proliferation oftwo-body resonances [2, 31, 32] and (ii) the run-away ofavalanches [33, 34]. We find that the latter is marginal.Finally, we propose and analyze an experimental plat-form capable of directly exploring the emergence of er-godicity at the transition between MBL phases. Ourproposal is motivated by recent advances in Rydberg-dressed, neutral-atom quantum simulators [35–42]; wedemonstrate that the phase diagram depicted in Fig. 1can be directly probed via quench dynamics of local ob- a r X i v : . [ c ond - m a t . d i s - nn ] A ug FIG. 2. (a-d) Characterization of the symmetry breaking model, Eqn. 1, for W V = 0 .
7. (a) For W J /W h & χ increaseswith system size evincing the SG nature of the phase. In the PM phase, χ approaches a finite constant, albeit exhibitingtwo distinct behaviors (inset). (b) h r i -ratio as a function of W J /W h reveals an intervening thermal phase surrounded by twolocalized phases. The dash-dotted [dashed] line corresponds to the GOE [Poisson] expectation. (c) The half-chain entanglemententropy S L/ increases with system size for intermediate W J /W h , in agreement with the expected thermal volume-law. In thetwo localized phases, S L/ saturates to different values, highlighting the distinct nature of the underlying eigenstate order. (d)The variance of S L/ exhibits two distinct peaks in agreement with the presence of two distinct transitions. (e)[(f)] S L/ forthe SPT [DTC] model of Eqn. 2 [Eqn. 3] also demonstrates the presence of an intervening thermal phase. Each data pointcorresponds to averaging over at least 10 disorder realizations. servables within experimental decoherence time-scales.Let us start by considering the paradigmatic exampleof a disordered one dimensional spin chain, which hoststwo distinct MBL phases: H = X i J i σ zi σ zi +1 + X i h i σ xi + X i V i ( σ xi σ xi +1 + σ zi σ zi +2 ) , (1)where ~σ are Pauli operators and all coupling strengthsare disordered, with J i ∈ [ − W J , W J ], h i ∈ [ − W h , W h ],and V i ∈ [ − W V , W V ] [43]. We choose to work with thenormalization √ W J W h = 1 and perform extensive ex-act diagonalization studies up to system size L = 16[44]. In the absence of V i , the system reduces to the non-interacting, Anderson localized limit and for sufficientlystrong disorder (in J i and h i ), this localization persistsin the presence of interactions.The Hamiltonian (Eqn. 1) exhibits a Z symmetry cor-responding to a global spin-flip, G = Q i σ xi . In the many-body localized regime, two distinct forms of eigenstateorder emerge with respect to the breaking of this symme-try. For W h (cid:29) W J , W V , the transverse field dominatesand the system is in the MBL paramagnetic (PM) phase.The conserved ‘ -bits simply correspond to dressed ver-sions of the physical σ xi operators. For W J (cid:29) W h , W V ,the Ising interaction dominates and the eigenstates cor- respond to “cat states” of spin configurations in the ˆ z direction. Physical states break the associated Z sym-metry, the ‘ -bits are dressed versions of σ zi σ zi +1 , and thesystem is in the so-called MBL spin-glass (SG) phase[13, 16].These two types of eigenstate order can be dis-tinguished via the Edwards-Anderson order parameterwhich probes the presence of long-range Ising correla-tions in eigenstates | n i , χ = ⟪ L − P i,j h n | σ zi σ zj | n i ⟫ ,where ⟪ · · · ⟫ denotes averaging over disorder realizations[16, 28]. In the SG phase, this order parameter scales ex-tensively with system size, χ ∝ L , while in the PM phase,it approaches a constant O (1) value. Fixing W V = 0 . χ exhibits a clear transition from PM to SG as one tunesthe ratio of W J /W h (Fig. 2a). The finite-size flow of χ isconsistent with the presence of a single critical point at W J = 3 . , W h = 0 .
32 ( W J /W h ≈ h r i -ratio, a mea-sure of the rigidity of the many-body spectrum: h r i = ⟪ min { δ n , δ n +1 } / max { δ n , δ n +1 } ⟫ , where δ n = E n +1 − E n , E n is the n th eigenenergy and averaging is also doneacross the entire many-body spectrum [45, 46]. In theMBL phase, energy levels exhibit Poisson statistics with h r i ≈ .
39, while in the ergodic phase, level repulsionleads to the GOE expectation h r i ≈ .
53 [4, 6, 47]. Un-like χ , which exhibits a single transition, the h r i -ratioexhibits two distinct critical points, each characterizedby its own finite-size flow (Fig. 2b). This demarcatesthree distinct phases: two many-body localized phases(for W J /W h . . W J /W h &
10) separated by anintervening thermal phase. Interestingly, the location ofthe thermal-MBL transition at W J /W h ≈
10 matchesthe location of the spin-glass transition observed via χ .The fact that an additional thermal-MBL transition isobserved in the h r i -ratio, but not in χ , suggests that thePM regime has slightly more structure.In order to further probe this structure, we turn to thehalf-chain entanglement entropy, S L/ = − Tr[ ρ s log( ρ s )],where ρ s = Tr i ≤ L/ [ | n i h n | ]. The behavior of S L/ , illus-trated in Fig. 2c, allows us to clearly distinguish threephases: the MBL paramagnet, the thermal paramagnet,and the MBL spin-glass. For W J /W h (cid:28) .
1, the eigen-states are close to product states and the entanglemententropy S L/ is independent of L , consistent with a local-ized paramagnet. Near W J /W h ≈ S L/ increases withsystem size, approaching ( L log 2 − /
2, consistent witha thermal paramagnet. Finally, for W J /W h (cid:29)
10, thehalf-chain entanglement again becomes independent of L and, for very large W J /W h , approaches log 2, consistentwith the cat-state-nature of eigenstates in the MBL SGphase.A few remarks are in order. First, the variance of S L/ provides a complementary diagnostic to confirmthe presence of two distinct thermal-MBL transitions(Fig. 2d). Indeed, one observes two well-separated peaksin var( S L/ ), whose locations are consistent with thetransitions found in the h r i -ratio. Second, although χ only scales with system size in the SG phase, one expectsits behavior to be qualitatively different in the MBL ver-sus thermal paramagnet. In particular, in the MBL para-magnet, the ‘ -bits have a small overlap with σ zi σ zj andone expects χ >
1; meanwhile, in the thermal paramag-net (at infinite temperature) one expects χ → W V and W J /W h (Fig. 1a).Even for the smallest interaction strengths accessible, W V ∼ .
07, one observes a finite width region wherethe h r i -ratio increases with system size [44]. Althoughclearly indicative of a thermal intrusion, it is possible thatour analysis underestimates the size of the interveningergodic phase [48–50]. Extrapolating toward W V = 0,our phase diagram suggests the presence of a finite-widththermal region between the two MBL phases, which ter-minates at the non-interacting critical point (Fig. 1a).In order to verify that the presence of a phase transi-tion is indeed responsible for the intervening ergodic re-gion, one can explicitly break the Z symmetry in Eqn. 1.We do so by adding a disorder-less , on-site longitudinalfield, Γ P i σ zi . Despite the fact that the field is uni-
1, the Floquet system spontaneously breakstime-translation symmetry and is in the so-called DTCphase, while for h (cid:29)
1, the system is in a Floquet para-magnetic phase [20, 21, 29]. We analyze each of thesemodels using the four diagnostics previously described:(i) the order parameter, (ii) the h r i -ratio, (iii) the half-chain entanglement, and (iv) the variance, var( S L/ ). Weobserve the same qualitative behavior for both transi-tions across all diagnostics: An intervening ergodic phaseemerges which terminates at the non-interacting criticalpoint. This is illustrated in Figs. 2e,f for both the SPTmodel and the DTC model using S L/ ; all additional datafor the different diagnostics can be found in the supple-mental material [44]. We further analyze the finite-sizeeffects arising from small couplings [44], which we believeunderlie previous numerical observations of apparent di-rect transitions [20, 26–28]. Experimental Realization .—Motivated by recent ad-vances in the characterization and control of Rydbergstates, we propose an experimental protocol to di-rectly explore the emergence of ergodicity between MBLphases. Our protocol is most naturally implemented inone dimensional chains of either alkali or alkaline-earthatoms [35–42]. To be specific, we consider Rb withan effective spin-1/2 encoded in hyperfine states: |↓i = | F = 1 , m F = − i and |↑i = | F = 2 , m F = − i . Re-cent experiments have demonstrated the ability to gen-erate strong interactions via either Rydberg-dressing inan optical lattice (where atoms are typically spaced by ∼ . µ m) or via Rydberg-blockade in a tweezer array(where atoms are typically spaced by ∼ µ m) [35–42].Focusing on the optical lattice setup, dressing enablesthe generation of tunable, long-range soft-core Ising in-teractions, H ZZ = P i,j J ij σ zi σ zj , with a spatial profilethat interpolates between a constant at short distances(determined by the blockade radius) and a 1 /r van derWaals tail.A particularly simple implementation of a PM-SGHamiltonian (closely related to Eqn. 1) is to alternatetime evolution under H ZZ and H X = P i h i σ xi , with thelatter being implemented via a two-photon Raman tran-sition (Fig. 4a). In the high frequency limit, the dynamicsare governed by an effective Hamiltonian: H eff = τ τ + τ X i h i σ xi + τ τ + τ X ij J ij σ zi σ zj , (4)where H X is applied for time τ , H ZZ is applied fortime τ , and the Floquet frequency ω = 2 π/ ( τ + τ ) (cid:29) h i , J ij . This latter inequality ensures that both Flo-quet heating and higher-order corrections to H eff can besafely neglected on experimentally relevant time-scales[44, 52, 53]. Note that unlike the DTC model (Eqn. 3),here Floquet engineering is being used to emulate a staticMBL PM-SG Hamiltonian [54].A few remarks are in order. First, by applying theRydberg dressing to only one of the two hyperfine states(Fig. 4a), an additional longitudinal field H Z ∝ σ zi isnaturally generated. To restore the Z symmetry, onecan exactly cancel this field by embedding a spin echointo the Floquet sequence (Fig. 4b). In addition, vary-ing the spacing between the echo π -pulses (Fig. 4b) di-rectly controls the degree of cancellation, enabling one FIG. 4. (a) Schematic of the proposed experimental protocol.Within an optical lattice, neutral atoms are prepared alongtwo adjacent diagonals (i.e. with a gas microscope), defininga zig-zag spin chain configuration. Dressing with a Rydbergstate | r i leads to H ZZ + H Z , while a two-photon Raman tran-sition mediated by an excited state | e i leads to H X . (b) Bycombining rapid spin echo pulses with Floquet evolution un-der H X and H ZZ + H Z , one can engineer H eff (Eqn. 4). (c-e)Dynamics of σ xL/ (blue) and σ zL/ − σ zL/ (red) under H eff starting with initial states | ψ x i and | ψ zz i , respectively. Dif-ferent panels correspond to representative behaviors for thethree distinct phases (tuned via h ). (f) The height of the late-time plateau distinguishes between the three phases. Eachdata point corresponds to averaging over at least 10 disorderrealizations. to experimentally probe the effect of an explicit symme-try breaking field. Second, although our prior analysishas focused on eigenstate properties, these are inaccessi-ble to experiment. Fortunately, as we will demonstrate,the phase diagram can also be characterized via the dy-namics of local observables. The intuition behind this issimple: observables that overlap with an ‘ -bit exhibit aplateau at late times.To investigate this behavior, we use Krylov subspacemethods [55–58] to numerically simulate the dynam-ics of H eff with τ = τ = 1, J i,i +1 ∈ [ − , − J i,i +2 = 0 . J i,i +1 and h i ∈ [ h, h ]. We note that theratio of the nearest- to next-nearest-neighbor couplingstrength is chosen based upon the experimentally mea-sured Rydberg-dressing-interaction profile and a 1D zig-zag chain geometry (Fig. 4a) [37, 59, 60].For system sizes up to L = 20, we compute the dy-namics of initial states | ψ x i and | ψ zz i [61]; both statesare easily preparable in experiment, close to zero energydensity, and chosen such that h ψ x | σ xL/ | ψ x i = 1 and h ψ zz | σ zL/ − σ zL/ | ψ zz i = 1. Starting with | ψ x i as our ini-tial state and large h , we observe that h σ xL/ ( t ) i plateausto a finite value at late-times, indicating the system is inthe MBL PM phase (Fig. 4c). Analogously, for | ψ zz i andsmall h , we observe that h σ zL/ − ( t ) σ zL/ ( t ) i plateaus to afinite value at late-times, indicating the system is in theMBL SG phase (Fig. 4e). For h ∼ both observablesdecay to zero, indicating the system is the thermal phase(Fig. 4d). The plateau value of the two observables asa function of h clearly identifies the intervening ergodicregion (Fig. 4f).To ensure that one can observe the intervening ther-mal phase within experimental coherence times, we nowestimate the time-scales necessary to carry out our proto-col. Previous experiments using Rydberg dressing havedemonstrated coherence times T ∼ J i,i +1 ∼ (2 π ) ×
13 kHz and amicrowave-induced π -pulse duration ∼ µ s [37]. Takentogether, this leads to an estimate of ∼ µ s for theFloquet period (Fig. 4b). Crucially, within T (i.e. ∼ Discussion and outlook .—We conclude by discussingprevious analytical results and how they may shed lighton the origins of the intervening thermal phase. Inthe absence of interactions, the Hamiltonian transitionswe consider all fall into infinite-randomness universalityclasses characterized by both a divergent single-particledensity of states (DOS, D ( ε ) ∼ | ε log ε | − near zerosingle-particle energy ε ) and single-particle orbitals withdiverging mean and typical localization lengths ( ξ mean ∼| log ε | and ξ typ ∼ | log ε | respectively) [62–65]. Thesedivergences suggest that two-body resonances might di-rectly destabilize MBL upon the introduction of interac-tions; however, a simple counting of resonances in typicalblocks does not produce such an instability: In a block oflength l , there are lN ( ε ) “active” single particle orbitalswith ξ typ ( ε ) ≥ l , where N ( ε ) = R ε dε D ( ε ) is the inte-grated DOS [32, 44, 66]. These orbitals overlap in realspace and are thus susceptible to participating in pertur-bative two-body resonances. A perturbative instabilityof the localized state arises if lN diverges as ε →
0; evenfor arbitrarily small interactions, a large network of reso-nant pairs can be found at low enough energy. Using theDOS and localization lengths of the infinite-randomnesstransition, we find lN ∼ / | log ε | which vanishes slowlyas ε → average localization length controls this insta-bility [34]: for ξ > / log 2, thermal bubbles avalanche.However, this is within a model where the orbitals have asingle localization center. Near the infinite-randomnesstransition, the orbitals have two centers whose separa-tion is controlled by ξ mean but whose overlap onto a pu-tative thermal bubble is controlled by ξ typ . Thus, while ξ mean diverges logarithmically, the more appropriate ξ typ remains finite and this criterion does not produce an ab-solute instability [44]. We highlight that it is only a log-arithmic correction which causes the convergence of theaverage localization length; unaccounted channels mightprovide an additional logarithm leading to an absoluteavalanche instability. We leave this to future work.Finally, let us note that the direct numerical obser-vation of avalanche instabilities remains extremely chal-lenging [33, 69]; the presence of a robust intervening ther-mal region in our study suggests that an alternate mech-anism might be at the heart of our observations. Note added:
During the completion of this work, webecame aware of complementary work on the presence ofintervening thermal phases between MBL transitions [70]which will appear in the same arXiv posting.
Acknowledgements —We gratefully acknowledge dis-cussions with Ehud Altman, Anushya Chandran, Soon-won Choi, Phillip J. D. Crowley, Simon Hollerith,David Huse, Gregory D. Kahanamoku-Meyer and An-tonio Rubio-Abadal. We thank Immanuel Bloch for de-tailed comments on a draft. Krylov subspace numericsare performed using the dynamite package [55], a
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Mirlin, Rev. Mod. Phys. , 1355(2008).[63] D. S. Fisher, Phys. Rev. B , 6411 (1995).[64] L. Balents and M. P. A. Fisher, Phys. Rev. B , 12970 (1997).[65] B. M. McCoy and T. T. Wu, Phys. Rev. , 631 (1968).[66] N. Y. Yao, C. R. Laumann, S. Gopalakrishnan, M. Knap,M. M¨uller, E. A. Demler, and M. D. Lukin, Phys. Rev.Lett. , 243002 (2014).[67] D. J. Luitz, F. m. c. Huveneers, and W. De Roeck, Phys. Rev. Lett. , 150602 (2017).[68] T. Thiery, F. m. c. Huveneers, M. M¨uller, andW. De Roeck, Phys. Rev. Lett. , 140601 (2018).[69] I.-D. Potirniche, S. Banerjee, and E. Altman, PhysicalReview B , 205149 (2019), arXiv: 1805.01475.[70] S. Moudgalya, D. A. Huse, and V. Khemani, (to ap-pear). upplemental Material:Emergent ergodicity at the transition between many-body localized phases Rahul Sahay, ∗ Francisco Machado, ∗ Bingtian Ye, ∗ Chris R. Laumann, and Norman Y. Yao
1, 3 Department of Physics, University of California, Berkeley, California 94720 USA Department of Physics, Boston University, Boston, MA, 02215, USA Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
ADDITIONAL NUMERICAL DATA FOR ORDER PARAMETERS, LEVEL STATISTICS, ANDENTANGLEMENTSymmetry-Breaking Model
In the main text, we employed the order parameter χ , the h r i -ratio, the half-chain entanglement S L/ , and thevariance of the entanglement var( S L/ ) to diagnose the phase diagram of the symmetry breaking model (Eqn. 1) andreveal the existence of an intervening thermal phase. In this section, we provide additional data for these quantitiesat a variety of interaction strengths W V ∈ { . , . , . , . } (Fig. S1) to complement the data shown at W V = 0 . W V , signatures of the intervening thermal phase remain present. In the case of the h r i -ratioand S L/ (shown in the middle two columns of Fig. S1), this signature manifests itself in a distinct change in finitesize flow around W J /W h = 1. However, we comment that, at low interactions, due to both the small width of theintervening thermal phase and finite-size limitations [1–3], the thermal phase cannot be resolved in certain numericalprobes. For example, at W V = 0 .
1, the double peak structure of the variance of entanglement disappears; this is dueto the fact that the width of these peaks (at the system sizes accessible) exceeds the width of the intervening thermalphase. We remark that we see evidence of the intervening thermal phase to interaction strengths down to W V = 0 . FIG. S1. The order parameter χ , the h r i -ratio, the half-chain entanglement S L/ , and the variance of the entanglement var( S L/ )as a function of W J /W h for the model of Eqn. 1 of the main text. From the top row to the bottom, the interaction strengthsare chosen to bess W V ∈ { . , . , . } (from left to right). For panels depicting the h r i -ratio, the dash-dotted [dashed] linecorresponds to the GOE [Poisson] expectation. A minimum of 3 · disorder averages are performed for each quantity. a r X i v : . [ c ond - m a t . d i s - nn ] A ug W J /W h
07. The dash-dotted [dashed] linecorresponds to GOE [Poisson] value. Zooming into the region around criticality (inset), one can see evidence for an interveningthermal phase—at the largest system size considered ( L = 16) and around W J /W h = 1, the h r i begins increasing toward thethermal value, suggesting a flow toward the thermal value. A minimum of 10 disorder realizations were averaged over for the L = 12 ,
14 curves and 10 disorder realizations were averaged over for the L = 16 curve. L = 8
5. Observe that the behavior of the h r i -ratiois qualitatively the same for this parameterization as it is for the parameterization in the main text. A minimum of 5 · disorder realizations were averaged over for the L = 8 ,
10 curves and 20 disorder averages were performed for the L = 12 curve. and interactions chosen. Focusing on the behavior of the h r i -ratio, we consider the model: H = X i J i σ zi σ zi +1 + X i h i σ xi + X i V i σ zi σ zi +2 (S1)with fixed strength of the Ising coupling J i ∈ [1 / , / h i ∈ [ W h / , W h /
2] and interactionstrength V i ∈ [ W V / , W V / σ xi → σ zi σ zi +1 , σ zi → Q j
In the main text, we commented on the severity of finite-size effects in exact diagonalization numerics performedwith either weak interaction strength or disorder distributions strongly peaked at zero. In this section, we expandthis discussion by focusing on a single-parameter family of disorder distributions that emerge in the context of strongdisorder RG [4, 5]. E n e r g y
Minimum Coupling
14 (top to bottom). The middle column shows the V = 0 . V = 0 . disorder realization were averaged over for each data point. Heuristically, if there are links in the chain whose coupling is smaller than the many-body level spacing, we expectthe system to be ’cut’ across those links, producing apparent localization in any finite-size diagnostics. As the many-body level spacing decays as δ ∼ L/ L (possibly with additional sub-exponential corrections due to symmetries),this does not reflect the thermodynamic limit for any system with extensively many couplings sampled from an O (1)distribution. On the other hand, for system sizes accessible to exact diagonalization δ can remain larger than thesampled coupling strengths. This is particularly important when attempting to characterize the phase diagram ofthe model at weak interaction strength or when the disorder distribution is strongly peaked at zero. Consider thesymmetry breaking model of Eqn. 1 of the main text with a generalized disorder distribution: J i = JX Ji , h i = hX hi ,and V i = V X Vi where X Ji , X Ji , and X Vi are distributed according to P W ( X ) = 1 W (cid:18) X (cid:19) − /W (S2)with X ∈ (0 , W characterizes the strength of the disorder: the numerics reported in the main-text areperformed with uniform distributions ( W = 1), W > W → ∞ corresponds to the infinite-randomness fixed point. For a chain of length L , the typical minimumcoupling is of order J min ∼ L − W . Thus, at any given system size L , there is a large enough W such that J min istypically smaller than the level spacing δ , which is only weakly dependent on W .In Fig. S4 we compare the disorder averaged mean level spacing and minimum interaction coupling as a functionof W for J = h = 1 and V = 0 . .
3. Furthermore, we report the behavior of the h r i -ratio as a function of W forthese same parameters.For V = 0 . L = 8 ,
10, the mean level spacing δ is much greater than the average minimum coupling for all W suggesting strong finite-size limitations. By contrast, at larger system size ( L = 12 ,
14) and near the uniformdistribution case ( W ∼
1) the average minimum interaction coupling becomes larger than the mean level spacing.In parallel, the h r i -ratio appears to flow localized between system sizes L = 8 , ,
12 but starts to flow thermal forsystem sizes between L = 12 ,
14 for W ∼ V = 0 . L = 8) the mean level spacing is larger than the average minimumcoupling, but already for L &
10 and W . h r i -ratio to O
In order to demonstrate that the intervening ergodic region generically emerges between different MBL phases, weanalyzed two additional models distinct from the symmetry breaking model of Eqn. 1: a model with an SPT transtionand a model with an DTC transition (Eqs. 2 and 3; data shown in Fig. 2e,f). In this section, we provide additionalnumerical data for these models and construct the associated phase diagrams (analogous to the diagram in Fig. 1a).We begin with the SPT model in Eqn. 2. In order to diagnose the different phases of the SPT model, we computea string order parameter [6]: O = ⟪ L X i,j ; i +2 ≤ j − h n | σ zi σ yi +1 j − Y k = i +2 σ xk ! σ yj − σ zj | n i ⟫ (S3)where | n i is the eigenstate at the center of the many-body spectrum and ⟪ · · · ⟫ indicates averaging over disorderrealizations. This order parameter is the non-local analogue of the Edwards-Anderson order parameter for thesymmetry-breaking model and as such scales extensively in the MBL SPT phase and saturates to an O (1) constantin either the thermal or trivial MBL phases. The data for this order parameter, along with h r i -ratio, half-chainentanglement S L/ and the variance of the half-chain entanglement var( S L/ ) are shown in Fig. S5 for interactionstrength W V = 0 . , . A = 1 N T L N T X t =1 L X i =1 ( − t Tr { σ zi ( tT ) σ zi (0) } (S4)where T represents the period of one Floquet cycle. This order parameter is non-zero in the DTC phase and is zeroin the trivial Floquet MBL phase [7]. In our numerics, N T is set to 20. From our observations, this choice does notaffect the numerical value of the order parameter. Data for this order parameter, along with h r i -ratio and half-chainentanglement S L/ are shown in Fig. S6 for interaction strengths h z = 0 . , . h r i -ratio h r i
In this section, we detail the finite-size scaling analysis performed to extract the phase boundaries of different modelsconsidered in the main text (Fig. 1 and Fig. S7) . In particular, all phase boundaries are extracted via scaling collapsearound a crossing point of the h r i -ratio as a function some tuning parameter ∆ (log( W J /W h ) for the symmetry-breaking and SPT model, log( h x ) for the DTC model, and Γ in the explicit symmetry-breaking analysis). To performthe finite-size scaling analysis, we assume a standard h r i -ratio scaling ansatz for the thermal-MBL transition [8]: r L (∆) = ˜ f ( L /ν (∆ − ∆ c )) (S5)where r L (∆) is the disorder averaged h r i -ratio at ∆ and length L . To extract the critical point, we first linearlyinterpolate the simulated data points { r L (∆ i ) } as a function of ∆ yielding an interpolation ˜ r L (∆) which we can thenscale and sample from. The critical point ∆ c and critical exponent ν are extracted by numerically minimizing the FIG. S8. Left two columns: representative collapses for the h r i -ratio for the models of Eqn. 1, 2,and 3 of the main text atinteraction strengths W V = 0 . W V = 0 .
3, and h z = 0 . ν = 2, which is known analytically as thelower bound for critical exponent in generic disordered systems. square residual quality function: Q (∆ c , ν ) = X i,j X k (˜ r L i ((∆ k − ∆ c ) L /ν ) − ˜ r L j ((∆ k − ∆ c ) L /ν ) (S6)where i, j index the system sizes that we simulate (e.g. { , , , , } for the model of Eqn. 1) and k indexes theset of { (∆ k − ∆) L /ν } . The resulting scaling collapses for each model can be found in Fig. S8 and the scaling collapsesfor the symmetry-breaking analysis can be found in Figure S9. The errors in the critical point δ ∆ and the criticalexponent δν are estimated by adding a perturbation to each data point of the h r i -ratio which is sampled from auniform distribution with a width matching the standard error of that data point. The error reported is the standarddeviation of the extracted distribution of ∆ c and ν .While all extracted critical exponents violate known analytic bounds [9, 10], these results are consistent withprevious numerical studies [8, 11]. ADDITIONAL NUMERICAL DATA FOR THE EXPLICIT SYMMETRY BREAKING ANALYSIS
In the main text, we explored the effect of explicitly breaking the Z -symmetry of Eqn. 1 by introducing a disorder-less longitudinal field ∼ Γ P i σ zi . Interestingly, we observed that, upon increasing Γ, the intervening thermal phasedisappeared (Fig. 1b and Fig. 3). In this section, we provide additional numerical data highlighting the effects of thissymmetry-breaking field on the intervening thermal phase. First, we consider the h r i -ratio as a function of W J /W h for Γ ∈ { . , , . , } and V ∈ { . , . } in Fig. S10.For both interaction strengths, we find that, upon increasing Γ, the intervening thermal phase is suppressed andeventually disappears. Additionally, we show the h r i -ratio as a function of Γ for W J /W h = 1 and W V = 0 . FIG. S9. The left two panels depict representative collapses for the h r i -ratio as a function of W J /W h for Γ = 0 . W V = 0 . W J /W h at fixed Γ , W V . The right most panel depicts a collapse of the h r i -ratio as a function of Γ at W J /W h = 1 , W V = 0 .
3. This collapse extracts a critical point for the thermal MBL transition as we increase the symmetry-breaking field Γ.FIG. S10. The two rows exhibit the h r i -ratio as a function of W J /W h for interaction strengths W V ∈ { . , . } with Γ ∈{ . , . , . , . } in different columns. The dash-dotted [dashed] line corresponds to the GOE [Poisson] expectation of the h r i -ratio. Each data point corresponds to averaging over at least 3 · disorder realizations. h r i
In the main text, we proposed an experimental protocol which can be naturally implemented in one-dimensionalchains of neutral atoms trapped in an optical lattice. Here, optical dressing and Floquet engineering techniquesare used in order to simulate time-evolution under the Hamiltonian of Eqn. 4. In Fig. 4, we demonstrated that, byexamining the dynamics of local observables, one could diagnose the phase diagram of the aforementioned Hamiltonianand see evidence of an intervening thermal phase. The numerics presented were all in the “strong interaction” regime( ¯ J i,i +2 = 0 . J i,i +1 ) which was motivated by the “zig-zag” geometry of Fig. 4a. In this section, we provide additionalnumerics at “strong interaction” and further provide numerics in a “weak interaction” regime ( ¯ J i,i +2 = 0 . J i,i +1 )which can be realized via atoms arranged in a “linear” geometry (say a single row or column of a square opticallattice).In Fig. S12, we present the decay of h ψ x | σ xL/ | ψ x i ≡ h σ xL/ i and h ψ z | σ zL/ − σ zL/ | ψ z i ≡ h σ zL/ − σ zL/ i for the“strong interaction” regime (See [12] for the definition of | ψ x i and | ψ zz i ). Within 20 /J , all of the curves near h = 1approximately saturate to zero, indicating the presence of the intervening thermal phase. We note that late-timeplateaus for these curves are extracted by averaging each curve between t ∈ [19 /J, /J ]. These late-time plateaus areshown as a function of h in Fig. 4d. We remark that the non-zero late-time plateau for h σ xL/ ( t ) i [ h σ zL/ − σ zL/ i ] in theMBL SG [PM] phase is not a finite-time effect; the l -bit in these regimes has a finite overlap with the correspondingobservable.We conclude this section by presenting numerics in the “weak interaction” regime (with ¯ J i,i +2 = 0 . J i,i +1 ). In thetop panel of Fig. S13, we show the late-time plateaus of h σ xL/ ( t ) i and h σ zL/ − σ zL/ i as a function of h (analogous tothe “strong interaction” plateaus that we showed in Fig. 4d of the main text). Observe that in the predicted thermalregime (the yellow region), it appears that the plateau of h σ xL/ ( t ) i is finite, contrary to the thermal prediction. Thisapparent contradiction is explained by examining the decays in the bottom panel of Fig. S13. Here, for h = 1, we L = 16
Our experimental proposal hinged on implementing the Hamiltonian of Eqn. 4 via a two-stage Floquet drive (Fig. 4bin the main text). In this section, we present an analysis of the conditions required to ensure faithful engineeringFloquet.In the main text, our driving protocol approximated the evolution under the time-independent Hamiltonian ofEqn. 4 as exp ( − iH eff ( τ + τ )) ≈ exp ( − iH X τ ) exp ( − iH ZZ τ ) . (S7)While in the infinite-frequency limit, this sequence yields the correct effective Hamiltonian (Eqn. 4 of the main text),the leading order finite frequency corrections is linear in the inverse frequency of the drive. Fortunately, such linearterm can be cancelled by symmetrizing the Floquet sequence [13] (without requiring an additional π -pulse) as shownin the top of Fig. S14:exp ( − iH eff ( τ + τ )) ≈ exp ( − iH X τ /
2) exp ( − iH ZZ τ ) exp ( − iH X τ / . (S8)This enables us to consider larger values of τ and τ while keeping the simulation error small. In particular, to leadingorder, the trotterization error is controlled by the small parameter¯ h i ¯ J i,i +1 τ τ · max { ¯ h i τ , ¯ J i,i +1 τ } (cid:28) . (S9)With this Floquet sequence in hand, we seek to demonstrate that faithful Floquet evolution can be realized inexperimentally relevant parameter regimes. To this end, we choose two typical sets of parameter values, (¯ h i = 5,¯ J i,i +1 = 1, ¯ J i,i +2 = 0 . J i,i +1 and τ = τ ) and (¯ h i = 1, ¯ J i,i +1 = 1, ¯ J i,i +2 = 0 . J i,i +1 and τ = τ ), corresponding to theMBL PM and thermal phase respectively. Using such parameters, we compare trotterized and effective Hamiltonianevolution for a single disorder realization (Figs. S14 and S15). In particular, for both parameter sets, we compute0 FIG. S14. Trotterized evolution in the MBL PM phase (¯ h i = 5, ¯ J i,i +1 = 1, ¯ J i,i +2 = 0 . J i,i +1 and τ = τ )—In the top row,we show a single disorder realization of h σ xL/ ( t ) i and h σ zL/ − ( t ) σ zL/ ( t ) i under evolution by the effective Hamiltonian (points)and the trotterized evolution (line). The bottom row displays the expectation of the effective Hamiltonian per lattice site. the energy density with respect to the effective Hamiltonian h H eff i /L , and the two local observables used to diagnosedifferent phases, h σ xL/ i and h σ zL/ − σ zL/ i . As we decrease the Floquet period τ , both the Floquet heating effectsand the discrepancy in local observables are quickly suppressed. For the MBL PM phase, when ¯ J i,i +1 τ . .
14 ,the trotterized evolution very well approximate evolution under the time-independent Hamiltonian. In contrast, inthe thermal regime, faithful evolution can be achieved with much longer pulse timings with ¯ J i,i +1 τ . .
4. Both τ ’s are accessible in current experimental setups for an appropriate choice of local energy scale ¯ J i,i +1 . For the MBLPM case ( τ = τ = 0 . / ¯ J i,i +1 ), the decay of local observables start to saturate around 3 / ¯ J i,i +1 , which correspondsapproximately 20 Floquet cycles. Similarly, for the thermal case ( τ = τ = 0 . / ¯ J i,i +1 ), local observables saturate inabout 5 / ¯ J i,i +1 which is less than 20 Floquet cycles.We end by noting that, even though H eff is exactly conserved from the start of the evolution, for most choices of τ = τ it remains flat. This suggests that observed dynamics arise from higher order corrections to H eff that are notincluded, rather than Floquet heating. TWO-BODY RESONANCE COUNTING AT INFINITE-RANDOMNESS
In this section, we expand on the resonance counting criterion for the stability of localization of a non-interactingchain at infinite randomness against perturbative interactions. We consider a non-interacting Anderson-localized chaincharacterized by its density of single-particle states (DOS) D ( ε ) and the localization length ξ ( ε ) of single-particle1 FIG. S15. Trotterized evolution in the thermal phase (¯ h i = 1, ¯ J i,i +1 = 1, ¯ J i,i +2 = 0 . J i,i +1 and τ = τ )—In the top row,a single disorder realization of h σ xL/ ( t ) i and h σ zL/ − ( t ) σ zL/ ( t ) i is shown under both evolution by the effective Hamiltonian(points) and the true trotterized evolution (line). The bottom row displays the expectation of the effective Hamiltonian perlattice site. orbitals (with energy ε ). At the infinite randomness fixed point, the localization length and DOS both diverge as | (cid:15) | → α and an exponentially scaling envelope ψ ∗ α ∼ √ ξ ( ε α ) e −| x − α | /ξ ( ε α ) determined by its energy ε α . The presence of multiple centers (e.g. two in a typical state produced by the strongdisorder renormalization group treatment of the Ising model) does not parametrically modify the estimates below.Similarly, the presence of ‘pairing’ terms in the fermionization of the Ising model is not parameterically important.We consider a generic local interaction, which we schematically model by a density-density operator ∼ V R dx ˆ n ( x )ˆ n ( x ).Writing it in terms of the non-interacting orbitals, we have V αβγδ = V Z dx ψ α ( x ) ψ β ( x ) ψ ∗ γ ( x ) ψ ∗ δ ( x ) ∼ V p ξ α ξ β ξ γ ξ δ Z dx e − ( | x − α | /ξ α + | x − β | /ξ β + | x − γ | /ξ γ + | x − δ | /ξ δ ) . (S10)Two-body resonances occur when V αβγδ > | ( ε α − ε δ ) − ( ε γ − ε β ) | . In general, any small finite strength of interactionsproduces some density of resonances, but this need not modify the ergodic properties of the system; instead it can“dress” the local conserved quantities to be many-particle operators—this is at the heart of MBL. However, if thenumber of resonances in a localization volume becomes sufficiently large, then the local character of the conservedquantity is lost and we expect delocalization. Counting the number of perturbative resonances, induced by interactions,can then identify instabilities to thermalization.Owing to the localized nature of the single-particle orbitals, the matrix element will only be large whenever allfour orbitals overlap. Without loss of generality, we can take α to be the orbital with smallest localization length ξ α < ξ β , ξ γ , ξ δ . For ease of notation let ε = ε α . This suggests the following organization of our counting: given suchan orbital, first we compute how many other orbitals (labeled orbital δ ) exist within a block of size ‘ = ξ α around α and with energy δε around ε ; second, given the energy difference between orbital α and δ , what is the number of pairsof orbitals β and γ that have an energy difference within V αβγδ of the initial pair. Under this organization, one musthave that both estimates diverge: the first ensures that there is always an initial pair that can transition, while the2second ensures that, given a particular pair of orbitals, additional pairs can resonantly transition. While the formercan be simply estimated as δεD ( ε ) ‘ , the latter requires a more careful analysis. Fixing the pair of resonances α and δ , we must find the number of pairs of orbitals β and γ that satisfy three conditions: (1) the within ‘ distance fromorbital α , (2) their localization length is larger than ‘ , and (3) their energy difference close to the energy differencebetween α and γ (where close is given by the strength of the matrix element). The number R of such pairs can beestimated as follows: given an orbital γ within the block ‘ , we need to find another orbital β whose energy is in awindow of size V αβγδ around ε γ − δε . At some energy ε γ < ε , the number of such orbitals γ is ∼ ‘D ( ε γ ) dε γ and thenumber of corresponding orbitals β is ∼ ‘D ( ε β ) V αβγδ , where ε β = ε γ − ( δε ). Integrating yields the total number ofresonances: R = Z ε dε γ ‘D ( ε γ ) ‘D ( ε β ) V αβγδ ∼ Z ε dε γ ‘D ( ε γ ) ‘D ( ε β ) V ‘ p ξ α ξ β ξ γ ξ δ . (S11)We make progress under the following approximation: take δε = Cε with a small C . Physically, this means that theinitial orbitals have similar energies, and thus similar localization lengths, ξ δ ≈ ξ α = ‘ .We can check that this counting argument reproduces previous work on interaction instabilities of localized systemsin Ref. [14]. There, D ( ε ) remains a constant, while the localization length diverges as a power-law, ξ ( ε ) ∼ ε − ν . Thetwo conditions are then: CεD ( ε ) ξ ( ε ) ∼ ε − ν (S12) R ∼ V ‘ Z ε dε | ε + Cε | ν/ | ε | ν/ ∼ ‘ | ε | ν/ | ε | ν/ ∼ ε − ν (S13)Both quantities diverge when 1 − ν <
0, which agrees with previous estimates, ν > /d where d = 1, using adiagramatic approach.We can turn to the infinite randomness fixed point, which is characterized by a Dyson singularity with D ( ε ) ∼ [ ε log ε ] − and ξ ( ε ) ∼ log ε . We note that the ξ ( ε ) corresponds to the typical localization length. Owing to thebi-locality of the free fermion wave functions [4, 5], the average localization length captures the distance between thetwo localization centers while the typical localization length captures the spread around each center—the latter isresponsible for the mixing between orbitals and thus controls the matrix element.In such systems we have: Cε ε log ε log ε ∼ ε → R ∼ V ‘ Z ε dε | log( ε + Cε ) | − − / | log( ε ) | − − / | ε + Cε ||| ε | & V ‘ | log( ε ) | − − / | ε | Z ε dε | log( ε ) | − − / | ε |∼ V ‘ | log( ε ) | − − / | ε | | log( ε ) | − − / ∼ ε | log( ε ) | − (S15)While the latter condition diverges as ε →
0, the former does not. This means that, within a block of size ‘ we arenot guaranteed to find an appropriate orbital to start the resonance process. ∗ These authors contributed equally to this work.[1] R. K. Panda, A. Scardicchio, M. Schulz, S. R. Taylor, and M. ˇZnidariˇc, EPL (Europhysics Letters) , 67003 (2020).[2] D. A. Abanin, J. H. Bardarson, G. D. Tomasi, S. Gopalakrishnan, V. Khemani, S. A. Parameswaran, F. Pollmann, A. C.Potter, M. Serbyn, and R. Vasseur, (2019), arXiv:1911.04501.[3] Z. Papi, E. M. Stoudenmire, and D. A. Abanin, Annals of Physics , 714 (2015).[4] D. S. Fisher, Phys. Rev. B , 6411 (1995).[5] F. Evers and A. D. Mirlin, Rev. Mod. Phys. , 1355 (2008).[6] Y. Bahri and A. Vishwanath, Phys. Rev. B , 155135 (2014).[7] D. V. Else, B. Bauer, and C. Nayak, Phys. Rev. Lett. , 090402 (2016). [8] D. J. Luitz, N. Laflorencie, and F. Alet, Phys. Rev. B , 081103 (2015).[9] A. B. Harris, Journal of Physics C: Solid State Physics , 1671 (1974).[10] A. Chandran, C. R. Laumann, and V. Oganesyan, “Finite size scaling bounds on many-body localized phase transitions,”(2015), arXiv:1509.04285.[11] J. A. Kj¨all, J. H. Bardarson, and F. Pollmann, Phys. Rev. Lett. , 107204 (2014).[12] Here, | ψ x i is a state that is spin polarized in the + y -direction except for sites L/ − L/ x and − x directions respectively. Similarly, | ψ z i is a state that is similarly spin polarized along + y except for sites L/ − L/ z direction with the pattern ↑↑↓↓ .[13] J. Choi, H. Zhou, H. S. Knowles, R. Landig, S. Choi, and M. D. Lukin, Phys. Rev. X , 031002 (2020).[14] R. Nandkishore and A. C. Potter, Phys. Rev. B90