Thermodynamic Equilibrium as a Symmetry of the Schwinger-Keldysh Action

TL;DR

The paper shows that quantum systems in thermal equilibrium possess a discrete, involutive symmetry T_beta of the Schwinger-Keldysh action, unifying static KMS and dynamical fluctuation-dissipation relations as Ward-Takahashi identities. This symmetry provides a practical criterion for equilibrium, constraining both unitary and dissipative terms, and clarifying how equilibrium emerges from the microscopic action. In the classical limit, T_beta reduces to a known Langevin-equilibrium symmetry, linking quantum and classical descriptions of equilibrium, while highlighting fundamental differences in microreversibility and detailed balance for quantum systems. The framework yields concrete results for two-time FDRs, reveals when non-equilibrium steady states violate the symmetry (e.g., Lindblad dynamics, baths at different temperatures), and offers a versatile tool for constructing symmetry-preserving approximations and deriving fluctuation relations. Overall, the work establishes T_beta as a central organizing principle for thermalization and equilibrium tests within the Schwinger-Keldysh formalism, with broad implications for quantum many-body dynamics and renormalization-group analyses.

Abstract

The time evolution of an extended quantum system can be theoretically described in terms of the Schwinger-Keldysh functional integral formalism, whose action conveniently encodes the information about the dynamics. We show here that the action of quantum systems evolving in thermal equilibrium is invariant under a symmetry transformation which distinguishes them from generic open systems. A unitary or dissipative dynamics having this symmetry naturally leads to the emergence of a Gibbs thermal stationary state. Moreover, the fluctuation-dissipation relations characterizing the linear response of an equilibrium system to external perturbations can be derived as the Ward-Takahashi identities associated with this symmetry. Accordingly, the latter provides an efficient check for the onset of thermodynamic equilibrium and it makes testing the validity of fluctuation-dissipation relations unnecessary. In the classical limit, this symmetry renders the one which is known to characterize equilibrium in the stochastic dynamics of classical systems coupled to thermal baths, described by Langevin equations.

Figures (1)

• Figure 1: (Color online) $(a)$ Schematic representation of the Schwinger-Keldysh partition function Kamenev2011Altland/Simons. The time evolution of the density matrix $\rho(t) = e^{-i H t} \rho e^{i H t}$ can be represented by introducing two time lines to the left and right of $\rho$. These time lines correspond to the $+$ and $-$ parts of the Schwinger-Keldysh contour, respectively. $(b)$ Schematic representation of the KMS condition for a four-time correlation function $\langle a_1(t_{a,1}) a_2(t_{a,2}) b_2(t_{b,2}) b_1(t_{b,1}) \rangle$ with $t_{a,1} < t_{a,2}$ and $t_{b,1} < t_{b,2}$, where $a_{1,2}$ and $b_{1,2}$ are bosonic field operators. As illustrated by the first equality (light blue box), this correlation function is properly time-ordered and therefore it can be directly represented within the Schwinger-Keldysh formalism with the operators $a_{1,2}$ and $b_{1,2}$ evaluated along the $-$ and $+$ contours, respectively. The thermal density matrix $\rho = e^{-\beta H}/\mathop{\mathrm{tr}}\nolimits e^{-\beta H}$ can be first split into the products of $e^{-\beta H/2}\times e^{-\beta H/2}$ and then these two factors can be moved in opposite directions along the two time lines, with the effect of adding $+i \beta/2$ and $-i \beta/2$ to the time arguments of $a_{1,2}$ and $b_{1,2}$, respectively. After these two factors have been moved to the end of the timelines, due to the cyclic property of the trace, they combine as represented by the second equality (orange box), where the time lines now take detours into the complex plane and the overall time order is effectively reversed as indicated by the arrows, which converge towards $\rho$ instead of departing from it as in the case of sketch $(a)$ or of the first equality of sketch $(b)$. The original time ordering can be then restored by means of the time-reversal operation $\mathsf{T}$, upon application of which operators are replaced by time-reversal transformed ones, $\tilde{\rho} = \mathsf{T} \rho \mathsf{T}^{\dagger}$ etc., and the signs of time variables are reversed. In addition, due to the anti-unitarity of $\mathsf{T}$ one has to take the Hermitian adjoint of the expression inside the trace. As a result, the order of operators is inverted and one obtains the third equality (green box) which is again properly time ordered. This construction can be generalized to arbitrary correlation functions, leading to Eq. \ref{['eq:52']}.