Critical Properties of the Many-Body Localization Transition

The transition from a many-body localized phase to a thermalizing one is a dynamical quantum phase transition that lies outside the framework of equilibrium statistical mechanics. We provide a detailed study of the critical properties of this transition at finite sizes in one dimension. We find that the entanglement entropy of small subsystems looks strongly subthermal in the quantum critical regime, which indicates that it varies discontinuously across the transition as the system size is taken to infinity, even though many other aspects of the transition look continuous. We also study the variance of the half-chain entanglement entropy, which shows a peak near the transition, and find substantial variation in the entropy across eigenstates of the same sample. Furthermore, the sample-to-sample variations in this quantity are strongly growing and are larger than the intrasample variations. We posit that these results are consistent with a picture in which the transition to the thermal phase is driven by an eigenstate-dependent sparse resonant “backbone” of long-range entanglement, which just barely gains enough strength to thermalize the system on the thermal side of the transition as the system size is taken to infinity. This discontinuity in a global quantity—the presence of a fully functional bath—in turn implies a discontinuity even for local properties. We discuss how this picture compares with existing renormalization group treatments of the transition.

Popular Summary

Many-body localization (MBL) phase transition is a new type of phase transition that occurs in the dynamics of certain quantum many-body systems that are fully isolated from any environmental dynamical degrees of freedom. “Thermalization” in an isolated quantum system with no external “bath” is a seemingly paradoxical phenomenon since the system always retains memory about its initial state for all times. Nevertheless, when such a system is large, it may be able to serve as a bath for itself such that local observables reach thermal equilibrium eventually, while the memory of initial conditions gets hidden in experimentally inaccessible global degrees of freedom. When this happens, the full system is said to be in the thermal phase. Researchers have, however, recently shown that there also exist classes of many-body systems with disorder and randomness that can get dynamically stuck and fail to thermalize, such that even local observables retain some memory of the initial state. Such systems are said to be in a “many-body localized,” or MBL phase. We explore the properties of one-dimensional systems that are transitioning between these two dynamical phases.

Our model relies on a finite-size scaling analysis of quantum entanglement properties. We find that the entanglement of the system’s eigenstates changes discontinuously at the transition, from “area-law” entanglement in the MBL phase to “volume-law” entanglement in the thermal phase, thereby supporting a view of this transition as a novel hybrid between continuous and discontinuous transitions. We use our data to develop a picture for how this transition is driven by the proliferation of an initially sparse network of quantum entanglement, and we develop an understanding of the effect of randomness in small systems.

These results further our understanding of this novel dynamical phase transition, which lies outside of the framework of statistical mechanics and all of our usual analysis tools.

Figure 1

Schematic depiction of the MBL-to-thermal phase transition as a function of disorder strength $W$ and system size $L$, showing the quantum critical regime at finite sizes. Exact diagonalization studies have only observed a crossover on the thermal side of the transition, with no observed crossover between the MBL and quantum critical regimes.

Figure 2

(a) Disorder-averaged half-chain entanglement entropy divided by the Page value $S_T$ for a random pure state as a function of $W$ and $L$. Note that $S/S_T$ approaches a step function at the transition going from zero in the MBL phase with area-law entanglement to one in the thermal phase. (b) Disorder-averaged level statistics ratio $\overline{r}$, which obeys a GOE distribution in the thermal phase and a Poisson distribution in the localized phase, respectively.

Figure 3

Disorder- and eigenstate-averaged entanglement entropy $S_1$ (in bits) computed in eigenstates of the full system, for a subsystem comprising one spin at the end of the chain. The (rounded) “plateaus” in $S_1$ for intermediate $L$ and $W$ are associated with the quantum critical regime, and they show strongly subthermal values of $S_1$.

Figure 4

The MBL-to-thermal phase transition and finite-size crossovers as a function of (a) $L/\xi$ and (b) $L_A/\xi$ (schematic). For the single-site entropy $S_1$, the only relevant scaling variable is $L/\xi$, and the curves at fixed $W$ in Fig. 3 correspond to the vertical lines in the crossover phase diagram (a), with the subthermal plateaus lying in the QC regime. Grover’s analysis [26] considers $L_A/\xi$ as the relevant scaling variable (b) and shows that if $S_A$ is continuous across $W_c$ then it must be thermal in the critical regime. The inconsistency between pictures (a) and (b) can be reconciled if there is a discontinuity in $S_A$ at $W_c$ in the limit $L\to \infty$ (yellow lines).

Figure 5

Standard deviation of the half-chain entanglement entropy ${\mathrm{\Delta }}_S$ divided by the random pure state value $S_T$, parsed by its contributions from cut-to-cut (dashed lines), eigenstate-to-eigenstate (solid lines), and sample-to-sample (dotted lines) variations.

Figure 6

Schematic depiction of two possible models of the crossover from the MBL phase to the thermal phase. (a) The picture from VHA’s RG [22] predicts contiguous thermal or localized blocks. At the crossover, a few long thermal blocks occupy a finite fraction of the system, giving a subthermal but volume-law half-chain EE. (b) An alternate picture for the transition with a sparse entangled backbone of small thermal blocks of spins with varying degrees of entanglement. The backbone is not contiguous but spans the entire system, and the thin blue lines denote entanglement between the blocks. In both pictures, the thermal clusters acquire just enough strength to thermalize the entire system on the thermal side of the transition in the limit $L\to \infty$.

Figure 7

Schematic illustration of our “picture” of the MBL phase transition showing the physical extent and entanglement properties of the typical longest entangled cluster in the different regimes.

Figure 8

Distribution of thermal block sizes for eigenstates with $0.45<S_{\mathrm{mid}}<0.55$. The exponential decrease in $N_{|\mathcal{B}|}$ with $|\mathcal{B}|$ suggests that the local entanglement structure in these states looks inhomogeneous with a network of small interspersed thermal and localized blocks. Inset: Distributions for $S_{\mathrm{mid}}$ near the crossover for different $L$’s and $W$’s.

Figure 9

Probability distributions of the end-spin entanglement entropy $S_1$ for different $W$’s and $L=10$, 14, 18 (blue, red, green curves). The distributions become extremely broad near the transition, with very little system size dependence, consistent with strongly subthermal mean values of $S_1$ in the quantum critical regime.