Einstein relation for subdiffusive relaxation in Stark chains

We investigate chains of interacting spinless fermions subject to a finite external field $F$ (also called Stark chains) and focus on the regime where the charge thermalization follows the subdiffusive hydrodynamics. First, we study reduced models conserving the dipole moment and derive an explicit Einstein relation which links the subdiffusive transport coefficient with the correlations of the dipolar current. This relation explains why the decay rate ${\Gamma }_q$ of the density modulation with wave vector $q$ shows $q^4$ dependence. In the case of the Stark model, a similar Einstein relation is also derived and tested using various numerical methods. They confirm an exponential reduction of the transport coefficient with increasing $F$. On the other hand, our study of the Stark model indicates that upon increasing $q$ there is a crossover from subdiffusive behavior, ${\Gamma }_q\propto q^4$, to the normal diffusive relaxation, ${\Gamma }_q\propto q^2$, at the wave vector $q^{*}$ which vanishes for $F\to 0$.