Quantum quenches and many-body localization in the thermodynamic limit

We use thermalization indicators and numerical linked cluster expansions to probe the onset of many-body localization in a disordered one-dimensional hard-core boson model in the thermodynamic limit. We show that after equilibration following a quench from a delocalized state, the momentum distribution indicates a freezing of one-particle correlations at higher values than in thermal equilibrium. The position of the delocalization to localization transition, identified by the breakdown of thermalization with increasing disorder strength, is found to be consistent with the value from the level statistics obtained via full exact diagonalization of finite chains. Our results strongly support the existence of a many-body localized phase in the thermodynamic limit.

Figure 1

Exact diagonalization results for the averaged ratio of two consecutive energy gaps $r$ (see text) as a function of disorder strength in chains with $L=14$, 15, and 16 sites, and $V=2$. For $L=14$, the energy ratio was computed considering all $2^{14}=16384$ disorder field configurations (solid circles). We also show the energy ratio for $L=14$ (open circles), $L=15$ (open squares), and $L=16$ (open triangles) averaging over 9100 random samples. The error bars depict one standard deviation. They make apparent that the statistical errors are negligible at the scale of the figure.

Figure 2

Last order ($l=14$) of the NLCE calculation for the momentum distribution in the initial state with $T_I=2$, and in the DE and GE after quenches with four different values of the disorder strength $h$ (two below and two above the delocalization to localization transition). The inset depicts the last order of the NLCE for the kinetic energy $K$ after quenches as a function of $h$. Note that, for $h\lesssim 2.5<h_c$, the results in the DE and the GE are virtually indistinguishable.

Figure 3

Relative differences for the momentum distribution and the kinetic energy vs $l$ in the NLCE calculation for six values of $h$ and $T_I=2$. (a) $\delta {\left(m\right)}_l$, (b) $\delta {\left(K\right)}_l$, (c) $\Delta {\left(m^{\mathrm{DE}}\right)}_l$, and (d) $\Delta {\left(K^{\mathrm{DE}}\right)}_l$. Results for $\Delta {\left(m^{\mathrm{GE}}\right)}_l$ and $\Delta {\left(K^{\mathrm{GE}}\right)}_l$ are reported in Ref. [54]. For $h=4$ and 5, the results for $\delta {\left(m\right)}_l$ and $\delta {\left(K\right)}_l$ do not change with changing $l$, i.e., they have converged.

Figure 4

Relative differences for the momentum distribution vs $l$ in the NLCE calculation for $3.2\le h\le 3.8$. (a) $\delta {\left(m\right)}_l$ and (b) $\Delta {\left(m^{\mathrm{DE}}\right)}_l$. In (a), horizontal dashed lines correspond to the average value of last two orders of $\delta {\left(m\right)}_l$ for $h=3.6, 3.7$, and $3.8$.