Almost Everywhere Ergodicity in Quantum Lattice Models

Abstract

We rigorously examine, in generality, the ergodic properties of quantum lattice models with short range interactions, in the $$C^*$$ C ∗ algebra formulation of statistical mechanics. Ergodicity results, in the context of group actions on $$C^*$$ C ∗ algebras, assume that the algebra is asymptotically abelian, which is not generically the case for time evolution. The Lieb-Robinson bound tells us that, in a precise sense, the spatial extent of any time-evolved local operator grows linearly with time. This means that the algebra of observables is asymptotically abelian in a space-like region, and implies a form of ergodicity outside the light-cone. But what happens within it? We show that the long-time limit of the n-th moment of a ray-averaged observable, along space-time rays of almost every speed, converges to the n-th power of its expectation in the state (i.e. its ensemble average). Thus ray averages do not fluctuate in the long time limit. This is a statement of ergodicity, and holds in any state that is invariant under space-time translations and that satisfies weak clustering properties in space. The ray averages can be performed in a way that accounts for oscillations, showing that ray-averaged observables cannot sustainably oscillate in the long time limit. We also show that in the GNS representation of the algebra of observables, for any KMS state with the above properties, the long-time limit of the ray average of any observable converges (in the strong operator topology) to the ensemble average times the identity, again along space-time rays of almost every speed. This is a strong version of ergodicity, and indicates that, as operators, observables get “thinner” almost everywhere within the light-cone. A similar statement holds under oscillatory averaging.