Energy distribution and local fluctuations in strongly coupled open quantum systems: The extended resonant level model

We study the energy distribution in the extended resonant level model at equilibrium. Previous investigations [Phys. Rev. B 89, 161306 (2014); Phys. Rev. B 93, 115318 (2016)] have found, for a resonant electronic level interacting with a thermal free-electron wide-band bath, that the expectation value for the energy of the interacting subsystem can be correctly calculated by considering a symmetric splitting of the interaction Hamiltonian between the subsystem and the bath. However, the general implications of this approach were questioned [Phys. Rev. B 92, 235440 (2015)]. Here, we show that, already at equilibrium, such splitting fails to describe the energy fluctuations, as measured here by the second and third central moments (namely, width and skewness) of the energy distribution. Furthermore, we find that when the wide-band approximation does not hold, no splitting of the system-bath interaction can describe the system thermodynamics. We conclude that in general no proper division subsystem of the Hamiltonian of the composite system can account for the energy distribution of the subsystem. This also implies that the thermodynamic effects due to local changes in the subsystem cannot in general be described by such splitting.