Quasi-Many-Body Localization in Translation-Invariant Systems

We examine localization phenomena associated with generic, high entropy, states of a translation-invariant, one-dimensional spin ladder. At early times, we find slow growth of entanglement entropy consistent with the known phenomenology of many-body localization in disordered, interacting systems. At intermediate times, however, anomalous diffusion sets in, leading to full spin polarization decay on an exponentially activated time scale. We identify a single length scale which parametrically controls both the spin transport times and the apparent divergence of the susceptibility to spin glass ordering. Ultimately, at the latest times, the exponentially slow anomalous diffusion gives way to diffusive thermal behavior. We dub the intermediate dynamical behavior, which persists over many orders of magnitude in time, quasi-many-body localization.

Figure 1

Schematic of the one-dimensional spin-$1/2$ ladder. $\sigma$ spins reside in the top chain and $S$ spins in the bottom. Along each chain, the spins are coupled by nearest neighbor $XY$ (flip-flop) interactions with strength $J^{\prime }$ and $J$, respectively. Across each rung, spins are coupled via Ising interactions of strength $J_z$. The green dotted line indicates the position of the cut used to divide the ladder when evaluating the entanglement entropy.

Figure 2

(a) Growth of entanglement entropy for $L=4$ sites, with $J^{\prime }={10}^{-3}$ and $J_z=10$. The entanglement entropy is averaged over 30 random initial product states. (b) Analogous data for $L=8$ averaged over 100 random initial product states. (c) Schematic short-time entanglement entropy behavior. At $t_1\sim 1/J$, a single-particle-localized (SPL) plateau is observed. At time scales $t_1^{\prime }\sim 1/J^{\prime }$, interactions set in and a logarithmic growth of entanglement begins. At $t_2\sim e^L/J^{\prime }$, this growth saturates for short chains unless it is preempted by the final “plateau” at $t_3\sim 1/J_{\mathrm{eff}}=(J^{\prime 2}/J_z{)}^{-1}$ [52]. (d) $S_{\text{ent}}(t)$ for Heisenberg coupling along the ladders with $L=4$, $J_z=10$, and $J^{\prime }={10}^{-2}$, ${10}^{-3}$, ${10}^{-4}$, ${10}^{-5}$. Since there is no single-particle limit, the initial dynamics are $J^{\prime }$ independent, but $t_3$ continues to scale as $1/J^{\prime 2}$.

Figure 3

A typical time trace of the decay of spin polarization. The blue line shows $D_s(t)$, while the red line shows $D_{\sigma }(t)$. The black dashed line indicates the position of $t_3$ as determined from the entanglement entropy, and the solid black line or dot depicts $\tau$, the time at which all polarization has fully decayed. The inset depicts $\tau (k=2\pi /L)$ as a function of the system size for $J_z=5$, 10, 20 and filling fractions ${\nu }_s=1/2$, ${\nu }_s\approx 1/3$, $1/4$. For ${\nu }_s\approx 1/3$, $1/4$, commensuration effects at finite size prevent the choice of identical filling across sizes. For ${\nu }_s\approx 1/3$, the sizes are $L=6$, 8, 10 with ${\nu }_s=1/2$, $3/8$, $3/10$, respectively. For ${\nu }_s\approx 1/4$, the sizes are $N=8$, 10 with ${\nu }_s=1/4$, $1/5$, respectively [52]. The data at ${\nu }_s\approx 1/3$, $1/4$ are qualitatively consistent with saturation at strong effective disorder.

Figure 4

$\tau$ as a function $k$. Data are obtained at $J_z=5$, $J^{\prime }=0.01$. (a) Red circles correspond to $L=10$ and filling ${\nu }_s=1/2$, while (b) blue circles correspond to $L=10$, ${\nu }_s=1/5$. To confirm the $L$ independence of the decay time, we also show data for $L=6$, 8, 12 at a variety of $k$. The time scale for diffusive decay (as extrapolated from larger $J^{\prime }$ data) is shown as the black dashed line in (a); for small systems, ${\tau }_{\text{diffusion}}$ is so slow that the relaxation is limited by $e^{1/(k\xi )}$. However, for the largest system sizes, we observe the exponential quasi-MBL behavior to be cut off by an extremely slow diffusion.

Figure 5

(a) $\chi$, the susceptibility to spin glass ordering as a function of system size $L$ for $\nu =1/2$ and $J_z=5$, 10, 20, 40. The inset shows the raw $\rho (W)$ data used to generate the main figure. The value of $\chi$ is taken to be that of $\rho$ at $W={10}^{-6}$. The slope of unity on the log-log plot demonstrates that we are clearly in the linear response regime.