Second-order mean force Gibbs state of fermionic open systems

Abstract

Thermodynamics typically assumes that a system in contact with a thermal reservoir relaxes to the Gibbs state, or to the grand canonical Gibbs state when it also exchanges particles. However, this only holds when the coupling is very weak. Beyond this, the expected equilibrium state is the mean force Gibbs (MFG) state, obtained by tracing the reservoir out of the thermal state of the combined system. Despite its relevance, general analytic results for the MFG state have so far been obtained almost exclusively for bosonic reservoirs. Here we derive the grand canonical MFG state to second order in the coupling for a generic fermionic system coupled to a thermal reservoir at arbitrary chemical potential, using the concept of the mode reduced state in order for the fermionic MFG state to be well defined. The correction to the bare Gibbs state separates into particle and hole channels weighted by the empty and occupied reservoir states, related through the Kubo-Martin-Schwinger condition. We apply the result to the resonant level model, two modes coupled to a shared reservoir, and the single impurity Anderson model. For the two modes, the reservoir mediates coherence between modes that do not interact directly, something the bare Gibbs state cannot capture. We verify our results against numerically exact calculations, including real-time dynamics for the interacting model, and find agreement between the static prediction and the dynamical relaxation.