Generalized Hamiltonian of mean-force approach to open quantum systems coupled to finite baths in thermoequilibrium

The prevailing method for addressing strong-coupling thermodynamics typically involves open quantum systems coupled to infinite baths in equilibrium through the Hamiltonian of mean force (HMF). However, its applicability to finite baths remains limited. In this work, we transcend this limit by considering the impacts of system-bath coupling which is not only on the system but also on the finite bath feedback. When the bath and system sizes are comparable, they act as each other's effective 'bath'. We introduce a parameter $\ensuremath{\alpha}$ within $[0,1]$, where $\ensuremath{\alpha}$ and $1\ensuremath{-}\ensuremath{\alpha}$ act as weighting factors to distribute the system-bath coupling between the system and the bath. Our approach innovatively incorporates the effect of coupling on the respective effective 'bath' into both the system and bath in the statistical factor form. We generalize the HMF and propose quantum Hamiltonians of mean forces to handle the distributed coupling of the system to finite baths. The additional parameter $\ensuremath{\alpha}$ is determined by minimizing the joint free energy density and is a new thermodynamic quantity representing the finite bath's influence. Examining the damped quantum harmonic oscillator with a finite bath, we find that $\ensuremath{\alpha}$ affects both states, leading to a complex phase diagram with unique phenomena at critical temperatures ${T}_{L}$ and ${T}_{R}$, including valleys, peaks, negative values, and discontinuities in entropy and specific heat. Notably, these anomalies disappear when $\ensuremath{\alpha}\ensuremath{\rightarrow}1$, both at high temperatures and in the thermodynamic limit, indicating the negligible influence of coupling on the bath states and reverting to the HMF. Thus, studying finite-sized baths holds substantial significance in small quantum systems.