Emergence of the Gibbs ensemble as a steady state in Lindbladian dynamics

TL;DR

This paper demonstrates how the Gibbs ensemble naturally emerges as a unique non-equilibrium steady state in Lindblad dynamics, under specific conditions on the system's algebra and jump operators, with examples from the XX and Fredkin models.

Contribution

It explicitly constructs the Gibbs NESS in Lindblad dynamics, linking it to the algebraic structure of the Hamiltonian and detailed balance conditions, and illustrates this with concrete models.

Findings

Gibbs NESS characterized by an effective Hamiltonian in the center of the commutant algebra.

Multiple steady states occur if the number of baths is less than the number of conserved charges.

Properly chosen jump operators fulfilling qDBC generate the Gibbs state as a steady state.

Abstract

We explicitly construct unique non-equilibrium steady state (NESS) of Lindblad master equation characterized by a Gibbs ensemble $\rho_{\text{NESS}} \propto e^{-\beta \tilde{H}}$, where the effective hamiltonian $\tilde{H}$ is an element in the center of the commutant algebra $\mathcal{C}$ of the original hamiltonian. Specifically, if $\mathcal{C}$ is Abelian, then $\tilde{H}$ consists only of $U(1)$ conserved charges of the original Hamiltonian. When the original Hamiltonian has multiple charges, it is possible to couple them with bathes at different temperature respectively, but still leads to an equilibrium state. Multiple steady states arise if the number of bathes is less than the number of charges. To access the Gibbs NESS, the jump operators need to be properly chosen to fulfill quantum detailed balance condition (qDBC). These jump operators are ladder operators for $\tilde{H}$ and jump process they generate form a vertex-weighted directed acyclic graph (wDAG). By studying the XX model and Fredkin model, we showcase how the Gibbs state emerges as an equilibrium steady state.