On the Algebra of Fluctuation in Quantum Spin Chains

TL;DR

This paper proves a central limit theorem for non-commuting operators in quantum spin chains, establishing a foundation for understanding fluctuations and their time evolution in quantum statistical mechanics.

Contribution

It provides a novel proof of the CLT for quasi-local operators in mixing quantum spin chains and constructs the algebra of normal fluctuations for Gibbs states.

Findings

Central limit theorem established for non-commuting operators

Constructs the algebra of normal fluctuations for Gibbs states

Shows the fluctuation algebra satisfies the β-KMS condition

Abstract

We present a proof of the central limit theorem for a pair of mutually non-commuting operators in mixing quantum spin chains. The operators are not necessarily strictly local but quasi-local. As a corollary we obtain a direct construction of the time evolution of the algebra of normal fluctuation for Gibbs states of finite range interactions on a one-dimensional lattice. We show that the state of the algebra of normal fluctuation satisfies the $\beta$-KMS condition if the microscopic state is a $\beta$-KMS state.