Einstein Relation for a Driven Disordered Quantum Chain in the Subdiffusive Regime

A quantum particle propagates subdiffusively on a strongly disordered chain when it is coupled to itinerant hard-core bosons. We establish a generalized Einstein relation (GER) that relates such subdiffusive spread to an unusual time-dependent drift velocity, which appears as a consequence of a constant electric field. We show that GER remains valid much beyond the regime of the linear response. Qualitatively, it holds true up to strongest drivings when the nonlinear field effects lead to the Stark-like localization. Numerical calculations based on full quantum evolution are substantiated by much simpler rate equations for the boson-assisted transitions between localized Anderson states.

Figure 1

Results of full time propagation on systems with $L=14$ for energy increase $\mathrm{\Delta }E(t)$ in (a), and $R(t)$ in (b) and (c). In (d) $\gamma$ vs $W$ is shown, as obtained from fitting $R(t)\propto t^{\gamma }$ for $t\gtrsim 1$ in case of ED (circles) and using FGR (squares). In the latter case $\gamma$ was extracted from $⟨x^2(t){⟩}_0\propto t^{\gamma }$; see Ref. [54]. Note that unless otherwise specified, we have used ${\omega }_0=g=1$, and $t_g=0.5$.

Figure 2

In (a) we show $R(t)$ for $W=4$ for different $F>0$, up to maximal $F=6$ using ED on $L=14$ sites, and in (b) the spread $⟨x^2(t){⟩}_F$, using FGR for $L=400$.

Figure 3

Cumulative distributions $I(\mathrm{\Gamma })$ in panels (a) and (b) for different $F$ were obtained from Eqs. (5)–(6), based on FGR. In panel (c) we show results for the simplified (toy model) rates as given by, Eq. (12). The dashed lines in all panels represent the threshold for normal diffusion, $I(\mathrm{\Gamma })\propto \mathrm{\Gamma }$.