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We study the spectral and transport properties of a Mott insulator driven bya static electric field into a non-equilibrium steady state. For thedissipation, we consider two mechanisms: Wide-band fermion reservoirs andphonons included within the Migdal approximation. The electron correlations aretreated via non-equilibrium dynamical mean field theory with an impurity solversuitable for strong correlations. We find that dissipation via phonons islimited to restricted ranges of field values around Wannier-Stark resonances.To cover the full range of field strengths, we allow for a small coupling withfermionic baths, which stabilizes the solution. When considering bothdissipation mechanisms, we find that phonons enhance the current for fieldstrengths close to half of the gap while lowering it at the gap resonance ascompared to the purely electronic dissipation used by Murakami and Werner[arXiv:1804.08257]. Once phonons are the only dissipation mechanism, thecurrent in the metallic phase is almost one order of magnitude smaller than thetypical values obtained by coupling to a fermionic bath. In this case, thetransport regime is characterized by an accumulation of charge in the upperHubbard band described by two effective chemical potentials.

Correlated Mott insulators in strong electric fields: Role of phonons in heat dissipation