Quantum spin chain dissipative mean-field dynamics

TL;DR

This paper investigates the complex dynamics of quantum spin chains in the infinite volume limit, revealing how different operator algebras evolve under dissipative mean-field dynamics, with implications for hybrid quantum-classical systems.

Contribution

It introduces a detailed analysis of the emergent behavior of various operator algebras in dissipative quantum spin chains, highlighting non-Lindblad generators and hybrid quantum-classical features.

Findings

Mean-field operators become time-dependent scalar averages.

Quasi-local operators evolve unitarily despite dissipation.

Collective fluctuations exhibit non-linear, dissipative dynamics with hybrid features.

Abstract

We study the emergent dynamics resulting from the infinite volume limit of the mean-field dissipative dynamics of quantum spin chains with clustering, but not time-invariant states. We focus upon three algebras of spin operators: the commutative algebra of mean-field operators, the quasi-local algebra of microscopic, local operators and the collective algebra of fluctuation operators. In the infinite volume limit, mean-field operators behave as time-dependent, commuting scalar macroscopic averages while quasi-local operators, despite the dissipative underlying dynamics, evolve unitarily in a typical non-Markovian fashion. Instead, the algebra of collective fluctuations, which is of bosonic type with time-dependent canonical commutation relations, undergoes a time-evolution that retains the dissipative character of the underlying microscopic dynamics and exhibits non-linear features. These latter disappear by extending the time-evolution to a larger algebra where it is represented by a continuous one-parameter semigroup of completely positive maps. The corresponding generator is not of Lindblad form and displays mixed quantum-classical features, thus indicating that peculiar hybrid systems may naturally emerge at the level of quantum fluctuations in many-body quantum systems endowed with non time-invariant states.