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This paper establishes rigorous conditions under which nonequilibrium steady states in infinite quantum spin systems coincide with their finite-system thermodynamic limits, addressing a critical gap in the theoretical understanding of open quantum systems. By linking these properties to the condition number of the Liouvillian and spectral gaps, the work provides essential mathematical tools for analyzing long-time dynamics in strongly correlated quantum matter. Such foundational advances are vital for developing accurate models of decoherence and thermalization required for future quantum simulation architectures.
Nonequilibrium steady state in Lindblad dynamics for infinite quantum spin systems
arXiv:2508.07448v2 Announce Type: replace-cross Abstract: We consider Lindblad dynamics of quantum spin systems on infinite lattices and define a nonequilibrium steady state (NESS) and a time-averaged nonequilibrium steady state (TANESS) on the basis of $C^*$-algebraic formalism. Generically, the NESS on an infinite system does not equal the thermodynamic limit of NESSs on finite systems. We give a sufficient condition that they coincide with each other, in terms of both a condition number, which quantifies the normality of a Liouvillian, and some spectral gaps on finite subsystems. To appreciate the importance of the condition number, we provide an example in which the spectral gaps have nonzero lower bounds uniformly for any finite subsystems but a thermodynamic limit and a long-time limit (or a long-time average) do not commute with each other.
Abstract:We consider Lindblad dynamics of quantum spin systems on infinite lattices and define a nonequilibrium steady state (NESS) and a time-averaged nonequilibrium steady state (TANESS) on the basis of $C^*$-algebraic formalism. Generically, the NESS on an infinite system does not equal the thermodynamic limit of NESSs on finite systems. We give a sufficient condition that they coincide with each other, in terms of both a condition number, which quantifies the normality of a Liouvillian, and some spectral gaps on finite subsystems. To appreciate the importance of the condition number, we provide an example in which the spectral gaps have nonzero lower bounds uniformly for any finite subsystems but a thermodynamic limit and a long-time limit (or a long-time average) do not commute with each other.
52 pages, 1 figure. v2: the definition of the condition number revised to improve the main theorems. Added new subsection 4.5 showing that the main theorems are applicable to onsite Liouvillians. Typos fixed
Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); Strongly Correlated Electrons (cond-mat.str-el); Quantum Physics (quant-ph)
arXiv:2508.07448 [math-ph]
(or arXiv:2508.07448v2 [math-ph] for this version)
Full text under the publisher's open licence · 179 words
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