Two Universality Classes for the Many-Body Localization Transition

We provide a systematic comparison of the many-body localization (MBL) transition in spin chains with nonrandom quasiperiodic versus random fields. We find evidence suggesting that these belong to two separate universality classes: the first dominated by “intrinsic” intrasample randomness, and the second dominated by external intersample quenched randomness. We show that the effects of intersample quenched randomness 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 external-randomness-dominated universality class. Our results provide an explanation for why exact-diagonalization studies on random models see an apparent scaling near the transition while also obtaining finite-size scaling exponents that strongly violate Harris-Chayes bounds that apply to disorder-driven transitions. 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.

Figure 1

Schematic RG flow for a one-dimensional system displaying an MBL transition. In the absence of external randomness, the critical fixed point is dominated by intrinsic intrasample variations and is not constrained by Harris-Chayes bounds (pink star). The addition of external quenched randomness is a Harris-relevant perturbation that causes the nonrandom fixed point to flow towards an “infinite randomness” disorder dominated fixed point (blue star). The “detuning” parameter quantifies the ratio of off-diagonal to diagonal couplings in the most local basis for the coarse grained model. The MBL phase is more stable in the nonrandom model and thus the critical flow is towards higher detuning. We propose that the effects of external randomness are not yet fully apparent at the sizes probed by ED studies, and the transition in these systems is mostly still governed by the nonrandom fixed point while beginning to cross over towards the random fixed point (shaded oval).

Figure 2

[(a) and (b)] Average half-chain eigenstate EE divided by the Page value $S_T$ for the quasiperiodic (a) and random (b) models. $S/S_T$ approaches a step function at the transition, going from 0 in the MBL phase to 1 in the thermal phase. Insets show that the location of the crossings drifts towards larger $W$ with increasing system size, but the finite-size drift is stronger in the random model. [(c) and (d)] Level statistics ratio $\overline{r}$ which obeys GOE (Poisson) distributions in the thermal (localized) phases, respectively, in the quasiperiodic (c) and random (d) models. Both diagnostics show that the MBL phase in the quasiperiodic model is stable down to a lower value of $W$ as compared to the random one.

Figure 3

Standard deviation of the half-chain EE ${\mathrm{\Delta }}_S$ divided by the Page value $S_T$, parsed by its contributions from eigenstate-to-eigenstate (solid, circles), cut-to-cut (dashed, stars), and sample-to-sample (dotted, triangles) variations in the quasiperiodic [(a) and (c)] and random [(b) and (d)] models. The intrasample variations look qualitatively similar between the two models [(a) and (b)], suggesting that these are mostly governed by the same fixed point at these sizes. However, the intersample variations are growing strongly with $L$ in the random model as it begins to cross over towards its asymptotic disorder dominated fixed point (d), while they are subdominant with no systematic $L$ dependence in the quasiperiodic model (c).

Figure 4

Finite-size critical scaling collapse for $S$ [(a) and (b)] and ${\mathrm{\Delta }}_S^{\text{states}}$ [(c) and (d)] data in the quasiperiodic and random models. We see that $\nu \sim 1$ for both models, again suggesting that the transition in both models is mostly governed by the same nonrandom fixed point at these sizes. This exponent is in violation of CCFS/CLO bounds, which must asymptotically constrain the random model—note that $\nu$ is slightly larger for the random model consistent with the suggestion that the effects of quenched randomness are growing but not yet fully apparent at these sizes. The critical $W_c$ is larger in the random model.