Mean-Force Hamiltonians from Influence Functionals

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.

Figures (4)

• Figure 1: Variance scaling. The sample variance $\sigma^2_\xi$ of the stochastic field grows linearly with inverse temperature $\beta$ (a) and linearly with the reorganisation energy $\lambda \propto g^2$ (b). Symbols are simulation results; dotted lines are the theoretical prediction $2\beta\lambda_{\mathrm{disc}}$.
• Figure 2: Temperature dependence and scaling. (a) Population of the coupled state ($|4\rangle$) vs $\beta$ for various couplings $g$. Circles (ED) match solid lines (analytic). (b) The extracted reorganisation energy $\lambda_{\mathrm{est}}$ collapsed by $g^2$ vs $\beta$. All curves hover near the theoretical thermodynamic limit (dashed lines), verifying that the bath's effect reduces to a single reorganisation energy $\lambda$.
• Figure 3: Strong-coupling benchmark. (a) Population $p_4$ vs $g^2$ at fixed $\beta$, showing excellent agreement over the full range. (b) Coefficients in the system basis vs $g^2$. The projector component $P_4=|4\rangle\langle 4|$ scales as $-\lambda$, while off-diagonal coherences are exponentially suppressed. This strong-coupling projector dominance is consistent with ultrastrong-coupling analyses of the mean-force Gibbs state cresserWeakUltrastrongCoupling2021a. The bare system Hamiltonian $H_S$ remains unrenormalised (dotted), confirming $H_{\mathrm{MF}} = H_S - \lambda f^2$.
• Figure 4: Microscopic validation. (a) Detailed population snapshot at fixed $\beta, g$: "Analytic HMF" (blue) corrects the "Bare Gibbs" (grey) to match ED (orange). (b) Effective energy shifts $\Delta_i$ for each level; only the coupled level ($|4\rangle$) shifts, by exactly $-\lambda$. Non-coupled levels remain unperturbed.