Two universality classes for the many-body localization transition

TL;DR

This paper compares the many-body localization transition in spin chains with quasiperiodic and random fields, revealing two distinct universality classes and highlighting differences in finite-size effects and phase stability.

Contribution

It identifies two separate universality classes for the MBL transition and emphasizes the importance of quasiperiodic models in understanding the transition.

Findings

Two universality classes identified: quasiperiodic and random.

Finite-size effects differ significantly between models.

MBL phase is more stable in quasiperiodic systems.

Abstract

We provide a systematic comparison of the many-body localization transition in spin chains with nonrandom quasiperiodic vs. random fields. We find evidence that these belong to two separate universality classes: one dominated by "intrinsic" intra-sample randomness, and the second dominated by external inter-sample quenched randomness. We show that the effects of inter-sample variations are strongly growing, but not yet dominant, at the system sizes probed by exact-diagonalization studies on random models. Thus, the observed finite-size critical scaling collapses in such studies appear to be in a preasymptotic regime near the nonrandom universality class, but showing signs of the initial crossover towards the quenched-disorder dominated universality class. We also show that the MBL phase is more stable for the quasiperiodic model as compared to the random one, and the transition in the quasiperiodic model suffers less from certain finite-size effects. Altogether, our results motivate the need for a greater focus on quasiperiodic models in theoretical studies of the MBL transition.

Figures

Equations

$$H^{QP/R}$$

$$+\sum _{i=1}^{L}W\cos (2\pi ki+{\varphi }_i^{QP/R})S_i^z$$

$$+J^{\prime }\sum _{i=1}^{L-2}(S_i^x{S}_{i+2}^x+S_i^y{S}_{i+2}^y)$$

Full text

Many-body localization (MBL) generalizes the phenomenon of Anderson localization to the interacting setting Anderson (1958); Basko et al. (2006); Pal and Huse (2010); Oganesyan and Huse (2007); Nandkishore and Huse (2015); Altman and Vosk (2015). The dynamics in an MBL system fails to establish local thermal equilibrium, and even highly excited states can retain local memory of their initial conditions for arbitrarily late times. The transition between an MBL phase and a “thermalizing” one is not a thermodynamic phase transition and lies outside the framework of equilibrium statistical mechanics. Instead it is a novel eigenstate phase transition Huse et al. (2013); Pekker et al. (2014) across which thermal and “volume-law” entangled many-body eigenstates obeying the eigenstate thermalization hypothesis (ETH) Deutsch (1991); Srednicki (1994); Rigol et al. (2008) change in a singular way to non-thermal and area-law entangled eigenstates in the MBL phase.

Although the MBL transition has attracted much recent interest Kjäll et al. (2014); Luitz et al. (2015); Grover (2014); Serbyn et al. (2015); Agarwal et al. (2015); Devakul and Singh (2015); Torres-Herrera and Santos (2015); De Roeck et al. (2016); De Roeck and Huveneers (2016); Serbyn and Moore (2016); Gopalakrishnan et al. (2016); Zhang et al. (2016a); Yu et al. (2016); Khemani et al. (2016); Vosk et al. (2015); Potter et al. (2015); Zhang et al. (2016b); Dumitrescu et al. (2017), very little is definitively known about its properties. Phenomenological renormalization group (RG) treatments of the transition are approximate but can probe large system sizes, and such studies Vosk et al. (2015); Potter et al. (2015); Zhang et al. (2016b); Dumitrescu et al. (2017) find a continuous transition in one dimension with a finite-size critical scaling exponent $\nu_{FS}\sim 3$ satisfying rigorous Harris/CCFS/CLO scaling bounds Harris (1974); Chayes et al. (1986); Chandran et al. (2015) which require $\nu_{FS}\geq 2/d$ for transitions in $d$ dimensions in the presence of quenched randomness. On the other hand, most other studies of the transition use numerical exact diagonalization (ED) of spin-chains which is limited to system sizes $L\leq 22$. These ED studies observe an apparent scaling collapse near the transition, but with scaling exponents $\nu_{FS}\sim 1$ violating the CCFS/CLO bound Kjäll et al. (2014); Luitz et al. (2015). Strikingly, some aspects of this transition even look first-order-like in that quantities like the eigenstate entanglement entropy (EE) of small subsystems can vary discontinuously across the transition Khemani et al. (2016); Dumitrescu et al. (2017).

A sensitive probe of the MBL transition is the standard deviation of the half-chain EE, $\Delta_{S}$, which peaks at the transition as the eigenstates change from area law to volume law entangled Kjäll et al. (2014). A careful parsing of $\Delta_{S}$ across inter- and intra- sample contributions near the transition reveals two notable features Khemani et al. (2016): (i) a sizeable volume-law scaling for $\Delta_{S}$ across eigenstates of the same sample, a property that none of the RG treatments capture and, (ii) a super-linear growth with $L$ for the sample-to-sample contribution to $\Delta_{S}$ at the system sizes studied by ED, a trend that is unsustainable in the large $L$ limit since the maximum possible EE scales as a volume law. This parsing indicates that the observed violations of CCFS/CLO bounds (which are derived from sample-to-sample variations) might result from a scenario in which the effect of quenched randomness across samples is not yet fully manifest, but growing strongly, at the sizes probed by ED Khemani et al. (2016). These data also suggest an intriguing scenario in which there might be two universality classes for transitions between MBL and thermal phases: one dominated by “intrinsic” eigenstate randomness within a given sample, and the second dominated by external quenched randomness across samples. In this scenario, the observed critical finite-size scaling collapses would appear to be in a preasymptotic regime near the first universality class (for which CCFS/CLO bounds do not apply), but showing the signs of the initial crossover towards the second external-randomness dominated universality class.

In this letter, we provide a more systematic analysis of the scenario above by studying the MBL transition in a quasiperiodic (QP) model with no quenched randomness. Following the work of Aubry and André Aubry and André (1980), the localization transition in non-interacting quasiperiodic models has been extensively studied. More recently, it was shown that interacting quasiperiodic models have an MBL phase Iyer et al. (2013), and signatures of this phase have been observed in cold-atomic experiments Schreiber et al. (2015); Bordia et al. (2016). However, compared to its non-interacting counterpart, the MBL transition in quasiperiodic models has received little theoretical attention. Nor have the points of similarity and difference between the MBL transition in quasiperiodic and random models been systematically studied. In this work, we provide a detailed finite-size scaling analysis of the QP-MBL transition, along with a comparison to the random MBL transition. We find that the MBL phase is more stable for the quasiperiodic model than for the random one, which is opposite to the trend for single-particle localization. This we attribute to the effects of locally thermal rare regions that destabilize MBL in the random system. The finite-size scaling we find suggests that there is a nonrandom universality class of the transition, and both models are governed by this universality class for the sizes accessible to ED. However, the random model is beginning to cross over towards the external-randomness-dominated universality class. Adding randomness to the quasiperiodic model is thus a “Harris-relevant” perturbation, causing this crossover. Fig 1 shows a schematic RG flow for this scenario. Altogether, our work not only advances our understanding of the global structure of quantum criticality in MBL systems, but also provides a concrete explanation for why numerical studies on random models see finite-size scaling collapse but obtain exponents violating Harris/Chayes bounds.

Model: We consider quasiperiodic/random spin chains of the form

[TABLE]

where $S_{i}^{\{x/y/z\}}$ are spin 1/2 degrees of freedom on site $i$, $J=J^{\prime}=J_{z}=1$, and $k=\frac{\sqrt{5}-1}{2}$ is an irrational wavenumber. For the quasiperiodic model, $\phi_{i}^{QP}=\phi\in[-\pi,\pi)$ is an arbitrary global phase offset such that the on-site fields are periodic with a period that is incommensurate with the lattice. This choice with $J^{\prime}=J_{z}=0$ is the non-interacting Aubry-André model which is localized for $W>1$ Aubry and André (1980). For comparison, we will also study a random model in which the phase is chosen randomly and independently on each site, $\phi_{i}^{R}=\in[-\pi,\pi)$. We choose this form for the random fields instead of the more conventional uniform distribution Pal and Huse (2010); Luitz et al. (2015) to keep the distribution of the on-site fields constant between the random and QP models, which enables a more direct comparison between the two. Both models are many-body localized for large field amplitudes $W>W_{c}^{QP/R}$. We add the next-nearest neighbor terms with strength $J^{\prime}$ to break the integrability of the models in the limit $W\rightarrow 0$, which allows the system to thermalize more completely within the thermal phase even for relatively small system sizes.

Fig. 2 benchmarks the location of the MBL transition(s) in (1) using the half-chain entanglement entropy, $S$, and the level statistics ratio, $r$. Fig. 2(a)/(b) shows $S$ divided by $S_{T}=0.5(L\log(2)-1)$, which is the Page Page (1993) value for a random pure state, in the quasiperiodic/random models respectively. The data is averaged over $1000-10^{5}$ disorder samples depending on $L$ (in the quasiperiodic model, the averaging is over different choices for the global phase shift $\phi^{QP}$), and over the middle quarter of the eigenstates in the $S^{z}_{\rm tot}=0$ sector for each sample (for $L=16,18$ we average over the middle 200 eigenstates). In both models, $S/S_{T}$ as a function of $W$ approaches a step function with increasing $L$, going from zero in the MBL phases with area-law entanglement to one in the thermal phase.

Fig. 2(c)/(d) shows the level statistics ratio Oganesyan and Huse (2007) $r\equiv\min\{\Delta_{n},\Delta_{n+1}\}/\max\{\Delta_{n},\Delta_{n+1}\}$, where $\Delta_{n}=E_{n}-E_{n+1}$ is the spacing between eigenenergy levels, in the quasiperiodic/random models respectively. This ratio approaches the GOE (Gaussian Orthogonal Ensemble) value $r\cong 0.53$ in the thermal phase and the Poisson value $r\cong 0.39$ in the localized phase for both models.

A few points of note. First, the location of the crossing in the entropy/level statistics data drifts towards larger $W$ with increasing $L$ in both models, as is typical of all ED studies. However, the drifting of the crossing is stronger in the random model as compared to the QP one, suggesting that the QP model suffers less from this finite-size effect so the behavior we are seeing may be closer to the true asymptotic large-$L$ regime. Second, as a related point, the transition is sharper (narrower in width) in the QP model. Third, despite the functional similarities in the choice of potentials between the two models, $W_{c}^{QP}<W_{c}^{R}$, where we estimate $W_{c}^{QP}\gtrsim 4.25$ and $W_{c}^{R}\gtrsim 5.5$ (these are estimated as lower bounds since, as always, there is no observed crossover on the MBL side of the transition Khemani et al. (2016)). This means that the QP model remains localized down to a smaller value of $W$, which is most likely due to the absence of rare Griffiths events which can disrupt localization in the random model. Indeed, within the MBL phase, the mean entanglement is larger in the random model than in the QP one (for comparable $W/W_{c}$), and distributions of the EE in the random model have longer tails to large entanglement reflecting rare events (see Supplement)

Variance of the half-chain entanglement entropy: We now study the standard deviation of the half-chain entanglement entropy $\Delta_{S}$ which peaks at the MBL transition, while it tends to zero deep in the MBL/ETH phases Kjäll et al. (2014). Following the prescription in Ref. Khemani et al., 2016, we parse the contributions to $\Delta_{S}$ due to fluctuations from sample-to-sample ($\Delta_{S}^{\rm samples}$) from eigenstate-to-eigenstate within a given sample ($\Delta_{S}^{\rm states}$) and from different entanglement cuts within a given eigenstate ($\Delta_{S}^{\rm cuts}$); see Fig 3. We use all cuts that produce a contiguous subsystem of length $L/2$. Since $S/S_{T}$ lies between [math] and $1$, $\Delta_{S}/S_{T}$ can be at most 0.5.

First, note that the peak value of $\Delta_{S}^{\rm states}/S_{T}$ is independent of $L$ in both the QP (Fig. 3a) and random (Fig. 3b) models indicating a volume law scaling, $\Delta_{S}^{\rm states}\sim L$, in both and thus a substantial variance in $S$ across eigenstates of the same sample. This property has not been included by any of the phenomenological RG approaches to the transition, and it indicates that the network of resonances driving the transition varies substantially across eigenstates of a given sample. Also note that the peak value of $\Delta_{S}^{\rm cuts}/S_{T}$ decreases with increasing $L$ (Figs. 3a, 3b), indicating subvolume law scaling for $\Delta_{S}^{\rm cuts}$ in both models. This subvolume law scaling limits the spatial inhomogeneity of the resonant network of entanglement at the transition Khemani et al. (2016). Together, these data indicate that the intra-sample critical variations across eigenstates and entanglement cuts look qualitatively similar between the random and QP models. In RG terms, this suggests that, for these sizes, the intra-sample finite-size critical behavior of the two models are perhaps governed mostly by the same fixed point (c.f. Fig. 1).

On the other hand, the two models look strikingly different when considering inter-sample variations. In the quasiperiodic model, $\Delta_{S}^{\rm samples}$ is far sub-dominant to the intra-sample contributions and is not growing systematically with $L$ (Fig. 3c). This indicates that the different quasiperiodic samples are quantitatively similar in their entanglement properties, and sample-to-sample fluctuations are not the dominant source of the finite-size critical rounding in the quasiperiodic model at these sizes.

By contrast, in the random model, the peak value of $\Delta_{S}^{\rm samples}/S_{T}$ grows strongly with $L$ which naively indicates that $\Delta_{S}^{\rm samples}$ scales super-linearly with $L$ (Fig. 3d), a trend that is not sustainable in the asymptotic large $L$ limit. This indicates that effects of inter-sample quenched randomness are not yet fully manifest but growing strongly at these small sizes. In RG terms, we interpret this as an indication of an RG flow, due to the external randomness, that is away from the fixed point that governs the nonrandom quasiperiodic model and is towards the infinite-randomness Harris/Chayes obeying fixed point that will asymptotically govern the transition for this random model (c.f. Fig. 1).

Two universality classes: We now turn to the finite-size critical scaling properties of the MBL transition in the two models. Figure 4 shows scaling collapse for $S/S_{T}$ and $\Delta_{S}^{\rm states}/S_{T}$, where both quantities are fit to a form $g[(W-W_{c})L^{1/\nu}]$, where $W_{c}$ denotes the critical disorder strength and $\nu$ is the finite-size scaling exponent.

We see a scaling collapse in the quasiperiodic model with $W_{c}\sim 4.25$ and $\nu\sim 1$ (Fig 4 a,c). First, note that quasiperiodic models without quenched randomness are not subject to the CCFS/CLO bound which require $\nu\geq 2/d$. Instead, such models fall under the purview of the Harris-Luck criterion Luck (1993) which imposes the weaker bound $\nu\geq 1/d$ 111 Luck generalizes the Harris bound on the relevance of disorder near a clean equilibrium phase transition to the more general aperiodic case Luck (1993). This work, however, still refers to a clean critical point and the equivalent of the CCFS Chayes et al. (1986) scaling bounds which do not refer to a clean transition do not exist for the general aperiodic case to the best of our knowledge.. The observed scaling exponents are certainly already quite close to obeying this bound, considering the small sizes studied. This, combined with our observations of finite-size drifts in the discussion surrounding Fig 2, suggests that the critical behavior in the quasiperiodic model might be close to its asymptotic large-$L$ form even at these sizes. If the scaling exponent continues to be $\nu\sim 1$ even in the asymptotic limit, then it is clear that the MBL transition in quasiperiodic models belongs to a different universality class from the transition in models with quenched randomness which must obey the CCFS/CLO bound—this would make the external randomness Harris-relevant when added to the quasiperiodic model (c.f. Fig 1).

It is an interesting curiosity that the non-interacting Aubry-Andre transition also has $\nu=1$, so one might be tempted to believe that the critical properties of the interacting quasiperiodic transition belong to the same universality class as the non-interacting one. However, a careful analysis (not shown) reveals that properties like the volume law scaling of $\Delta_{S}^{\rm states}$ across the many-body eigenstates are absent in the non-interacting model.

Turning to the random model, we see a scaling collapse with a larger critical disorder strength $W_{c}\sim 5.5$ (Fig 4 b,d) which is consistent with the presence of rare Griffiths effects in the random model which can aid with thermalization. The scaling exponent $\nu\sim 1$ confirms our earlier observation that the transition in the random model looks in many respects like it belongs to the quasiperiodic universality class at these sizes which are too small to feel the full effects of the quenched randomness. Also note that the scaling exponent is consistently slightly larger for the random model as compared to the quasiperiodic one, which is congruent with the theory that the random model is “en-route” to crossing over to a different disorder dominated scaling regime with $\nu\geq 2$ at larger system sizes.

Summary and outlook: We systematically examined the MBL transition in random and quasiperiodic models, and found that the MBL phase is stable down to a smaller disorder strength in the quasiperiodic case. Moreover, finite-size scaling analysis near the transition strongly suggests that the quasiperiodic model asymptotically belongs to a different universality class from the random one. We find scaling exponents $\nu\sim 1$ for both models; however while this exponent may be close to its asymptotic value for the quasiperiodic model (and in agreement with the Harris-Luck bound), we know that the asymptotic scaling exponent in the disordered model must satisfy $\nu\geq 2/d$ because the width of the finite-size scaling window is constrained to be greater than $\sim L^{-d/2}$ due to sample-to-sample fluctuations from the quenched randomness. Indeed, the sample-to-sample standard deviation of the entanglement entropy in the random model clearly shows that the effects of randomness are not fully apparent, but growing strongly, at the sizes studied, and many critical properties of the random models at these sizes look similar to those of quasiperiodic models. In RG terms, the transition in both the random and quasiperiodic models appears to be governed by the same nonrandom fixed point for the sizes accessible to ED, but the random model is starting to crossover towards the disorder dominated fixed point.

Additionally, the entanglement structure at the critical fixed points in RG studies Vosk et al. (2015); Potter et al. (2015) indicates that the asymptotic disorder-dominated regime in these random models might only be apparent in samples larger than $\sim$ 100 spins  Khemani et al. (2016), which will most likely remain inaccessible to both experimental and numerical work. Our work indicates that there should be a greater focus on quasiperiodic models in finite-size studies of the MBL transition, since the asymptotic scaling regime of the transition is likely more accessible in such models. Further, it is possible that the MBL phase in quasiperiodic models is more stable even in higher dimensions and for longer-ranged interactions since the recent arguments De Roeck and Huveneers (2016) on the instability of MBL due to rare, thermal inclusions arising from disorder fluctuations don’t apply to quasiperiodic models. Of course, a renormalization group study of the transition in a quasiperiodic model, if possible, would be a helpful next step for better understanding the properties of this new universality class. It is also intriguing to ask whether the two cases studied in the present work cover all universal possibilities for MBL transitions, or if there are further classifications — say for example in the case of a transition to an MBL phase accompanied by the simultaneous development of spontaneous symmetry breaking Huse et al. (2013); Pekker et al. (2014); Vasseur et al. (2016), or for MBL transitions in models with correlated disorder with varying degrees of correlation.

Acknowledgements: We thank Anushya Chandran, Chris Laumann, Subroto Mukerjee, Siddharth Parameswaran, Andrew Potter and Shivaji Sondhi for stimulating discussions, and Liangsheng Zhang for preliminary work on related models. This work was supported by the Harvard Society of Fellows (VK) and NSF grant DMR-1408560 (DS).