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arXiv:2602.13146 · Low Temperature Physics

Mean-Force Hamiltonians from Influence Functionals

Gerard McCaul

quant-phSubmitted Feb 13, 2026• Announced Feb 16, 2026• 0 citations• Revised Feb 17, 2026 (v2)
v1 · Feb 13, 2026v2 · Feb 17, 2026
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Abstract

The Hamiltonian of mean force (HMF) provides the standard starting point for strong-coupling thermodynamics, yet explicit operator forms are known only in restricted settings. We present a quenched density framework that uses the Hubbard-Stratonovich transformation to rewrite the reduced equilibrium state as an average over local propagators in imaginary time. This approach rigorously separates the statistical definition of the environment from the algebraic structure of the system response. We apply this framework to the minimal case of a harmonic environment with a coupling commuting with the system Hamiltonian. In this scenario the correction to the HMF has an exact, closed-form expression. We validate this result against finite-bath trace-out calculations and stochastic imaginary-time sampling in a five-level projector-coupled model.

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The Hamiltonian of mean force (HMF) provides the standard starting point for strong-coupling thermodynamics, yet explicit operator forms are known only in restricted settings. We present a quenched density framework that uses the Hubbard-Stratonovich transformation to rewrite the reduced equilibrium state as an average over local propagators in imaginary time. This approach rigorously separates the statistical definition of the environment from the algebraic structure of the system response. We apply this framework to the minimal case of a harmonic environment with a coupling commuting with the system Hamiltonian. In this scenario the correction to the HMF has an exact, closed-form expression. We validate this result against finite-bath trace-out calculations and stochastic imaginary-time sampling in a five-level projector-coupled model.
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 α within [0,1], where α and 1-α 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 α 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 α 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 α→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.
We present a partition-free approach to the evolution of density matrices for open quantum systems coupled to a harmonic environment. The influence functional formalism combined with a two-time Hubbard-Stratonovich transformation allows us to derive a set of exact differential equations for the reduced density matrix of an open system, termed the extended stochastic Liouville--von Neumann equation. Our approach generalizes previous work based on Caldeira-Leggett models and a partitioned initial density matrix. This provides a simple, yet exact, closed-form description for the evolution of open systems from equilibriated initial conditions. The applicability of this model and the potential for numerical implementations are also discussed.
Based on the Hubbard-Stratonovich transformation, the dissipative interaction between the system of interest and the heat bath is decoupled and the separated system and bath thus evolve in common classical random fields. This manipulation allows us to establish a novel theoretical methodology by which the reduced density matrix is formulated as an ensemble average of its random realizations in the auxiliary white noise fields. Within the stochastic description, the interaction between the system and the bath is reflected in the mutually induced mean fields. The relationship between the bath-induced field and the influence functional in the path integral framework is revealed. As a demonstration of this approach, we derive the exact master equations for two model systems.
We introduce a numerical method to determine the Hamiltonian of mean force (HMF) Gibbs state for a quantum system strongly coupled to a reservoir. The method adapts the time evolving matrix product operator (TEMPO) algorithm to imaginary-time propagation. By comparing the real-time and imaginary-time propagation for a generalized spin-boson model, we confirm that the HMF Gibbs state correctly predicts the steady state. We show that the numerical dynamics match the polaron master equation at strong coupling. We illustrate the potential of the imaginary-time TEMPO approach by exploring reservoir-induced entanglement between qubits.
The exact dynamics of a system coupled to an environment can be described by an integro-differential stochastic equation for the reduced density. The influence of the environment is incorporated through a mean field which is both stochastic and nonlocal in time and where the standard two-time correlation functions of the environment appear naturally. Since no approximation is made, the presented theory incorporates exactly dissipative and non-Markovian effects. Applications to the spin-boson model coupled to a heat bath with various coupling regimes and temperature show that the presented stochastic theory can be a valuable tool to simulate exactly the dynamics of open quantum systems. Links with the stochastic Schrödinger equation method and possible extensions to "imaginary time" propagation are discussed.
We extend and benchmark the recently developed Effective-Hamiltonian (EFFH) method [PRX Quantum 4, 020307 (2023)] as an approximation to the equilibrium state (``mean-force Gibbs state'') of a quantum system at strong coupling to a thermal bath. The EFFH method is an approximate framework. Through a combination of the reaction-coordinate mapping, a polaron transformation, and a controlled truncation, it imprints the system-bath coupling parameters into the system's Hamiltonian. First, we develop a variational EFFH technique. In this method, the system's parameters are renormalized by both the system-bath coupling parameters (as in the original EFFH approach) and bath temperature. Second, adopting the generalized spin-boson model, we benchmark the equilibrium state from the EFFH treatment against numerically exact simulations and demonstrate a good agreement for both polarization and coherences using the Brownian spectral function. Third, we contrast the (normal and variational) EFFH approach with the familiar (normal and variational) polaron treatment. We show that the two methods predict a similar structure for the equilibrium state, albeit the EFFH approach offers the advantage of simpler calculations and closed-form analytical results. Altogether, we argue that for temperatures comparable to the system's frequencies, the EFFH methodology provides a good approximation for the mean-force Gibbs state in the full range of system-bath coupling, from ultraweak to ultrastrong.