Exploring many-body localization and thermalization using semiclassical methods

The discrete truncated Wigner approximation (DTWA) is a semiclassical phase-space method useful for the exploration of many-body quantum dynamics. In this work we investigate many-body localization (MBL) and thermalization using DTWA and compare its performance to exact numerical solutions. By taking as a benchmark case a one-dimensional random field Heisenberg spin chain with short-range interactions, and by comparing to numerically exact techniques, we show that DTWA is able to reproduce dynamical signatures that characterize both the thermal and the MBL phases. It exhibits the best quantitative agreement at short times deep in each of the phases and larger mismatches close to the phase transition. The DTWA captures the logarithmic growth of entanglement in the MBL phase, even though a pure classical mean-field analysis would lead to no dynamics at all. Our results suggest the DTWA can become a useful method to investigate MBL and thermalization in experimentally relevant settings intractable with exact numerical techniques, such as systems with long-range interactions and/or systems in higher dimensions.

Figure 1

Disorder average of scaled imbalance $I/N$ (initial state order parameter) for system size $N=30$. In the exact results, MBL is manifested when the initial order is partially preserved in the long term as an asymptotic nonzero value. Instead, in the ergodic phase it decays to zero. The cross-over is around $h\approx 3$. The semiclassical DTWA results approximately overlap the exact ones at intermediate times, with better agreement for both extremes of disorder strengths. Exact solutions were obtained either by exact diagonalization (bottom), or time-dependent density matrix renormalization group (t-DMRG) method (top). The bond dimension for the t-DMRG solution was adaptively chosen so that the truncation error never exceeded a tolerance of ${10}^{-9}$.

Figure 2

Average Rényi entropy $S_2$ among all possible two-site subsystems. System size $N=18$ (top) and $N=12$ (bottom). The MBL regime is characterized by logarithmic growth of $S_2$ even at long times, while the ergodic regime saturates faster. The cross-over is around $h\approx 3.5$. The ability of DTWA to reproduce tendencies in the transition regime is similar to that in Fig. 1, which is remarkable because this time the observable depends on two sites, and it is related to the entanglement between the subsystem and its surroundings. All exact solutions were obtained by numerical exact diagonalization.

Figure 3

Evolution of scaled QFI, as a measure of multipartite entanglement. Similar to Rényi entropy and entanglement entropy, the localized phase is characterized by ${\mathcal{F}}_I\sim lnt$, a growth that breaks down in the thermal regime. Even for this measure, DTWA demonstrates its suitability for exploring the MBL transition since it also predicts both a logarithmic growth and a breakdown due to thermalization. System size $N=30$ (top) and $N=12$ (bottom). Exact numerical solutions obtained as in Fig. 1.

Figure 4

Summary of the agreement between the exact solution and DTWA results as a function of disorder strength in terms of mean squared error (MSE, main figure) and coefficient of variation of the rms deviation (CVRMSD, inset). All MSE curves show a tendency to have bigger mismatches around the MBL transition and smaller errors deep in the localized and thermal phases.