Many-Body Localization in Imperfectly Isolated Quantum Systems

We use numerical exact diagonalization to analyze which aspects of the many-body localization phenomenon survive in an imperfectly isolated setting, when the system of interest is weakly coupled to a thermalizing environment. We show that widely used diagnostics (such as many-body level statistics and expectation values in exact eigenstates) cease to show signatures of many-body localization above a critical coupling that is exponentially small in the size of the environment. However, we also identify alternative diagnostics for many-body localization, in the spectral functions of local operators. Diagnostics include a discrete spectrum and a hierarchy of energy gaps, including a universal gap at zero frequency. These alternative diagnostics are shown to be robust, and continue to show signatures of many-body localization as long as the coupling to the bath is weaker than the characteristic energy scales in the system. We also examine how these signatures disappear when the coupling to the environment becomes larger than the characteristic energy scales of the system.

Figure 1

Evolution of the spectrum of a ($p$-bit) spin flip operator as coupling $g$ to the bath increases in a typical disorder realization with disorder strength $w$. Results are for a system with periodic boundary conditions containing $N=7$ spins coupled to a bath with $N_b=7$ spins, and are averaged over all spins in the system. The top figure shows the spectral function in the low disorder delocalized phase, whereas the other figures show the spectral function of a system that would be localized if perfectly isolated, but which is coupled to a bath with a coupling $g$. The inhomogeneity of the spectral function and the existence of a hierarchy of gaps (especially the zero frequency gap) are diagnostics of localization. As $g$ is increased [(b)–(d)], the structure in the spectral function is gradually washed out, giving a crossover to thermalization. Note that the eigenstates become effectively thermal for $g>0.09$, according to the finite size scaling (Figs. 3, 4), but the local spectral functions retain signatures of localization until $g$ becomes comparable to the characteristic energy scales in the system ($g\approx 1$).

Figure 2

Crossover from Poisson to Gaussian statistics in the $l$-bit Hamiltonian as $g$ is increased. Results are for a system with $N=8$ spins and bath with $N_b=7$ spins averaged over $\sim 50000$ eigenstates obtained from several disorder configurations. The dark blue solid line is the Poisson distribution expected for localized systems, and the light blue dashed line is the GOE distribution expected for thermalizing systems.

Figure 3

(a) The average of the ratio of adjacent energy gaps $⟨\overline{r}⟩$ (defined in the text) in the $l$-bit Hamiltonian as $g$ is increased for system sizes $N=4$, 5, 6, 7, 8 and $N_b=N-1$. Data are averaged over $\sim 50000$ eigenstates obtained from several disorder configurations. (b) Collapse of data in (a) is in good agreement with analytic arguments for the finite size scaling presented in the main text, and depends only on $N_b$.

Figure 4

(a) Increasing thermalization of the states in the center of the band of the $l$-bit Hamiltonian as $g$ is increased for system sizes $N=4$, 5, 6, 7, 8 and $N_b=N-1$. $⟨m⟩$ as defined in the text is measured at the site of the central spin. Data are averaged over $\sim 50000$ eigenstates obtained from several disorder configurations. (b) Collapse of data in (a) agrees with analytical estimates of finite size scaling for $N_b=N-1\ge 5$. For a finite size system with $N_b$ spins in the bath, the eigenstates become effectively thermal for $g>\exp (-0.35N_b)$, implying that eigenstates in the thermodynamic limit become thermal for infinitesimal $g$.

Figure 5

Broadening of a spectral line as a function of $g$ for a system of $p$ bits with $N=4$ and $N_b=8$, 9, 10 averaged over more than 38 000 eigenstates obtained from several disorder configurations at $w=10$. $\ln (g_c)=-0.345N_b<-2.76$ for the sizes shown here. The mean and the median of the probability distribution of the linewidth $\mathrm{\Gamma }$ are extracted from the data as discussed in the Supplemental Material. The dotted lines are linear fits to the data. The solid lines are fits to the theoretical prediction.