Clustering of higher order connected correlations in C$^*$ dynamical systems

TL;DR

This paper demonstrates that in $C^*$ dynamical systems, higher order connected and free correlation functions vanish under clustering conditions, with applications to high temperature quantum spin models and their space-time correlations.

Contribution

It establishes the vanishing of higher order and free correlation functions under clustering in $C^*$ systems and extends ergodicity to higher correlations, with applications to quantum spin models.

Findings

Higher order connected correlations vanish at the same rate as two-point functions.

Clustering extends to higher order correlations and ergodicity.

Exponential decay of correlations outside the Lieb-Robinson light cone in high temperature states.

Abstract

In the context of $C^*$ dynamical systems, we consider a locally compact group $G$ acting by $^*$-automorphisms on a C$^*$ algebra $\mathfrak{U}$ of observables, and assume a state of $\mathfrak{U}$ that satisfies the clustering property with respect to a net of group elements of $G$. That is, the two-point connected correlation function vanishes in the limit on the net, when one observable is translated under the group action. Then we show that all higher order connected correlation functions (Ursell functions, or classical cumulants) and all free correlation functions (free cumulants, from free probability) vanish at the same rate in that limit. Additionally, we show that mean clustering, also called ergodicity, extends to higher order correlations. We then apply those results to equilibrium states of quantum spin lattice models. Under certain assumptions on the range of the interaction, high temperature Gibbs states are known to be exponentially clustering w.r.t. space translations. Combined with the Lieb-Robinson bound, one obtains exponential clustering for space-time translations outside the Lieb-Robinson light-cone. Therefore, by our present results, all the higher order connected and free correlation functions will vanish exponentially under space-time translations outside the Lieb-Robinson light cone, in high temperature Gibbs states. Another consequence is that their long-time averaging over a space-time ray vanishes for almost every ray velocity.