Dynamics in many-body localized quantum systems without disorder

We study the relaxation dynamics of strongly interacting quantum systems that display a kind of many-body localization in spite of their translation-invariant Hamiltonian. We show that dynamics starting from a random initial configuration is nonperturbatively slow in the hopping strength, and potentially genuinely nonergodic in the thermodynamic limit. In finite systems with periodic boundary conditions, density relaxation takes place in two stages, which are separated by a long out-of-equilibrium plateau whose duration diverges exponentially with the system size. We estimate the phase boundary of this quantum glass phase, and discuss the role of local resonant configurations. We suggest experimental realizations and methods to observe the discussed nonergodic dynamics.

Figure 1

Left: Phase diagram at fixed strength of quantum fluctuations (hopping $t$) and energy density or (quasi)temperature $T\ge 0$, for one-dimensional models in which commuting interaction ($U$) and disorder ($W$) terms define a classical potential. Both ingredients lead to a rough energy landscape which suppresses quantum tunneling and transport. Right: The role of temperature differs crucially in the limits of disorder- and interaction-dominated localization: for weak interactions, the lower part of the spectrum is localized, whereas highly excited states are delocalized. The reverse happens when the interaction dominates. The dashed lines correspond to a cut at constant quantum fluctuations, disorder, and energy density. They suggest a reentrant localization in the many-body spectrum as interactions are increased.

Figure 2

Model: Two atomic species existing on commensurate lattices with different tunneling amplitudes. Heavy particles (green) impede the hopping of light particles (yellow). Those act as effective springs between the heavy particles and localize them by creating a complex energy landscape for them.

Figure 3

Examples of resonances. The configurations $C_1$ and $C_1^{\prime }$ (top) hybridize at first order of degenerate perturbation theory in $t$, while $C_2$ and $C_2^{\prime }$ (bottom) hybridize at second order. Moving the middle particle(s) to the right costs no energy.

Figure 4

Relaxation of inhomogeneity in the density, in the absence of resonances. Time is rescaled by the exponentially large sample-dependent $t_{\mathrm{eff}}^{-1}$. The solid line is the analytical result (10) for the thermodynamic limit. Inset: The same numerical data without rescaled time show that the density inhomogeneity persists for times which diverge with the system size.

Figure 5

Right: Logarithmically averaged ratio between the exact hopping $t_{\mathrm{eff}}^{\mathrm{exact}}$ and the naive estimate $t_{\mathrm{eff}}^{\left(1\right)}$, as a function of density $\rho$, for different numbers of particles. The ratio was found to scale as ${\rho }^{{\alpha }_N}$. Left: A plot of the fitted ${\alpha }_N$ against $N$ confirms that ${\alpha }_N=N-1$ (solid line).

Figure 6

Time evolution of the inhomogeneity for configurations containing first-order resonances. The samples of length $L=9,15$ have only one particle involved in the resonance; for $L=12$ two particles are involved. The presence of local resonances leads to partial relaxation processes at short time scales $\tau \approx O\left(t^{-1}\right)$. Moreover, by comparing the plot with the inset of Fig. 4, one sees that the global relaxation times $\sim t_{\mathrm{eff}}^{-1}$ are reduced by a factor of $t/U$ per particle involved in resonances. As resonances are rare, this effect does not alter the fact that $t_{\mathrm{eff}}^{-1}$ diverges exponentially in the thermodynamic limit.