On the nature of heat in strongly coupled open quantum systems

We study heat transfers in a single level quantum dot strongly coupled to fermionic reservoirs and subjected to a time-dependent protocol modulating the dot energy as well as the dot-reservoir coupling strength. The dynamics is described using nonequilibrium Greens functions (NEGFs) evaluated to first order beyond quasi-static driving. We show that any heat definition expressed as an energy change in the reservoir energy plus any fraction of the system-reservoir interaction is not an exact differential when evaluated along reversible isothermal transformations, except when that fraction is zero. However, even in that latter case the reversible heat divided by temperature, namely the entropy, does not satisfy the third law of thermodynamics and diverges in the low temperature limit. Our results cast doubts on the possibility to define a thermodynamically consistent notion of heat expressed as the expectation value of some Hamiltonian terms.