Perturbative criteria for the ergodicity of interacting dissipative quantum lattice systems

TL;DR

The paper develops a perturbative ergodicity framework for quantum Markov semigroups describing interacting lattice systems of qudits and fermions. By treating the full dynamics as a perturbation $\mathcal{L}=\mathcal{L}_0+\mathcal{L}_1$ of a noninteracting dissipative dynamics, it proves existence of the infinite-volume Lindblad dynamics, a quantitative resolvent bound, and, when $M<\lambda_1$, a unique stationary state with exponential convergence of local observables. It further shows exponential decay of correlations and, under translation invariance, contraction in the specific quantum one-Wasserstein distance, providing a robust ergodicity criterion with explicit constants. The results are illustrated through quantum spin systems, classical-spin conjugations, the XYZ model with site dissipation, and interacting fermions with a fermionic gradient structure, yielding concrete small-coupling thresholds for ergodicity. Overall, the work advances quantitative mixing and stability analyses for open quantum lattice systems and informs decoherence and transport phenomena in many-body settings.

Abstract

We introduce a class of quantum Markov semigroups describing the evolution of interacting quantum lattice systems, specified either as generic qudits or as fermions. The corresponding generators, which include both conservative and dissipative evolutions, are given by the superposition of local generators in the Lindblad form. Under general conditions, we show that the associated infinite volume dynamics is well defined and can be obtained as the strong limit of the finite volume dynamics. By regarding the interacting evolution as a perturbation of a non-interacting dissipative dynamics, we further obtain a quantitative criterion that yields the ergodicity of the quantum Markov semigroup together with the exponential convergence of local observables. The analysis is based on suitable a priori bounds on the resolvent equation which yield quantitive estimates on the evolution of local observables.

Theorems & Definitions (37)

• Lemma 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 3.1
• proof
• proof : Proof of Lemma \ref{['lem:LX']}
• Lemma 3.2
• proof
• Lemma 3.3