On the local equivalence between the canonical and the microcanonical distributions for quantum spin systems

Hal Tasaki

Published 2016-09-22Version 1

We study a quantum spin system on the $d$-dimensional hypercubic lattice $\Lambda$ with $N$ sites with translation invariant short-ranged Hamiltonian under periodic boundary conditions. For this model, we consider both the canonical ensemble with inverse temperature $\beta_0$ and the microcanonical ensemble with the corresponding energy $U_N(\beta_0)$. Take an arbitrary self-adjoint operator $\hat{A}$ whose support is contained in a hypercubic block $B$ inside $\Lambda$. When $\beta_0$ is sufficiently small, we prove that the expectation values (with respect to these two ensembles) of $\hat{A}$ are close to each other provided that the number of sites in $B$ is $o(N^{1/2})$ and $N$ is large. This establishes the equivalence of ensembles on the level of local states in a large but finite system. The result is essentially that of Brandao and Cramer (with a more restrictive setting and a bigger block), but our proof is elementary.