Nucleation of Ergodicity by a Single Mobile Impurity in Supercooled Insulators

We consider a disordered Hubbard model and show that, at sufficiently weak disorder, a single spin-down mobile impurity can thermalize an extensive initially localized system of spin-up particles. Thermalization is enabled by resonant processes that involve correlated hops of the impurity and localized particles. This effect indicates that Anderson localized insulators behave as “supercooled” systems, with mobile impurities acting as ergodic seeds. We provide analytical estimates, supported by numerical exact diagonalization, showing how the critical disorder strength for such mechanism depends on the particle density of the localized system. In the $U\to \infty$ limit, doublons are stable excitations, and they can thermalize mesoscopic systems by a similar mechanism. The emergence of an additional conservation law leads to an eventual localization of doublons. Our predictions apply to fermionic and bosonic systems and are readily accessible in ongoing experiments simulating synthetic quantum lattices with tunable disorder.

Figure 1

Spectral-statistics parameter $r$ (bottom panels) for a system with a Hamiltonian (1) with one spin-down impurity, shown as a function of disorder strength $W/t$ and renormalized energy $ϵ=(E-E_{\min })/(E_{\max }-E_{\min })$. Panel (a) illustrates the case of moderate interaction $U=t$, while (b),(c) show large-$U$ values, when a doublon can form. The upper panels show the many-body density of states (DOS) at $W=1.6t$, which exhibits a clear separation between sectors with and without doublon at large $U$. At all values of $U$, a transition of Wigner-Dyson statistics (blue regions) at moderate disorder to Poisson statistics (yellow regions) is observed. At large $U$, this transition occurs in the doublon band, while the singlon band is AL. Averaging was performed over 2000 different disorder realizations, $L=12$ sites, $N_{\uparrow }=6$, and $N_{\downarrow }=1$.

Figure 2

(a) Finite-size behavior of the averaged parameter $r$ as function of disorder strength $W/t$, for $U=t$. Disorder averages were performed over 4000, 2000, 1000, 100 realizations for $L=8$, 10, 12, and 14. (b) Critical disorder $W_C$, at which $r=r_C$, as a function of ${\rho }_{\uparrow }=N_{\uparrow }/L$. The labels on the $x$ axis denote the values of $N_{\uparrow }$ and $L$ considered to evaluate $W_C$, following the procedure described in the main text. The curve is consistent with the estimate (4), valid in the thermodynamic limit. For $L=14$, 16, and 18 averaging is performed over 100 to 1000 realizations, depending on $N_{\uparrow }$. (c),(d) Same as above, but for $U=100t$ and energies in the middle of the many-body band involving one doublon.

Figure 3

Left: time evolution of the imbalance $\mathcal{I}$ for the initial state sketched in the inset. Different interaction strengths are considered ($U=0,t,100t$), and periodic boundary conditions are chosen. Right: the evolution of the local spin-down density at site zero, $n_{\downarrow ,0}$. Simulations are performed for $L=8$, 10, 12 (light to dark lines), $N_{\uparrow }=L/2$, $W=1.5t$ and averaged over 1000 disorder realizations. The horizontal gray lines correspond to $1/L$.

Figure 4

(a) Effective correlated hopping processes of doublon and particles. (b) Typical states connected to leading order in $t$. Doublon hopping is accompanied in the change of occupation of localized orbitals and the displacement of impenetrable holes.