Unbounded Growth of Entanglement in Models of Many-Body Localization

An important and incompletely answered question is whether a closed quantum system of many interacting particles can be localized by disorder. The time evolution of simple (unentangled) initial states is studied numerically for a system of interacting spinless fermions in one dimension described by the random-field $XXZ$ Hamiltonian. Interactions induce a dramatic change in the propagation of entanglement and a smaller change in the propagation of particles. For even weak interactions, when the system is thought to be in a many-body localized phase, entanglement shows neither localized nor diffusive behavior but grows without limit in an infinite system: interactions act as a singular perturbation on the localized state with no interactions. The significance for proposed atomic experiments is that local measurements will show a large but nonthermal entropy in the many-body localized state. This entropy develops slowly (approximately logarithmically) over a diverging time scale as in glassy systems.

Figure 1

(a) Entanglement growth after a quench starting from a site factorized $S^z$ eigenstate for different interaction strengths $J_z$ (we consider a bipartition into two half chains of equal size). All data are for $\eta =5$ and $L=10$, except for $J_z=0.1$ where $L=20$ is shown for comparison. The inset shows the same data but with a rescaled time axis and subtracted $J_z=0$ values. (b) Saturation values of the entanglement entropy as a function of $L$ for different interaction strengths $J_z$. The inset shows the approach to saturation.

Figure 2

(a) Growth of the particle number fluctuations of a half chain after a quench starting from a site factorized $S^z$ eigenstate for different interaction strength $J_z$. All data are for $\eta =5$ and $L=10$ except for $J_z=0.1$ where $L=20$ is shown for comparison. The inset shows the same data but with a rescaled time axis and subtracted $J_z=0$ values. (b) Saturation values of the particle number fluctuations as a function of $L$ for different interaction strengths $J_z$. The inset shows the approach to saturation.