Beyond the Kubo formula: unification of adiabatic and isothermal response in quantum systems

Abstract

The static limit of the Kubo susceptibility describes the adiabatic response of an isolated quantum system, in which energy-level occupations remain unchanged, and generally differs from the isothermal susceptibility, where these occupations readjust to thermal equilibrium. In this work, we introduce a single dynamical framework in which the adiabatic and isothermal responses arise naturally as two distinct limiting regimes. This can be done by considering explicitly an open system version of the Kubo approach in which the dissipation is governed by a typical thermalization rate of the system. By exchanging the order of the static limit and of the vanishing thermalization rate limit, the protocol yields either the adiabatic or isothermal response. This protocol has a solid physical motivation in terms of the thermalization dynamics and is valid for generic systems. We apply it to the case of the magnetic response of the electron gas and reproduce the expected results.

The paper addresses the difference between the zero-frequency Kubo susceptibility of an isolated quantum system, where instantaneous energy-level populations remain fixed, and the isothermal susceptibility, where populations readjust to equilibrium. It introduces an explicit thermalization rate $\kappa$ alongside the probe frequency $\omega$, interpreting the two static responses as different orders of limits of an open-system susceptibility $\chi_{AB}(\omega,\kappa)$. Using thermal-state expressions and a Lehmann representation, it obtains $\chi_{AB}^{T}=\chi_{AB}^{S}-\beta\operatorname{Cov}(A_0,B_0)$, where $A_0,B_0$ are energy-basis diagonal components. For self-response under the convention $H=H_0+\lambda A$, this gives $\chi_{AA}^{T}\leq\chi_{AA}^{S}\leq0$. Taking $\kappa\to0$ before $\omega\to0$ yields the isolated response; reversing the order yields the isothermal response. The explicit application is a free electron gas coupled by particle exchange to a thermal fermionic bath. Keldysh Green-function calculations give $\chi_M(\omega,\kappa)\simeq\chi_P\,2\kappa/(2\kappa-i\omega)$ at low temperature and $\omega,\kappa\ll\varepsilon_F$, recovering zero uniform isolated spin susceptibility and the Pauli isothermal susceptibility. At nonzero momentum, the two static limits coincide in this example; subsequently taking momentum to zero recovers the Pauli result. The bath calculation assumes a flat, effectively momentum-independent bath spectrum and constant broadening. Extensions to other relaxation mechanisms require their corresponding self-energies and consistency with conservation laws. Isolated many-body thermalization at conserved entropy, distinct from fixed-population quantum adiabatic evolution, is explicitly left for future investigation.