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

Abstract

We study a quantum spin system on the $d$-dimensional hypercubic lattice $Λ$ with $N=L^d$ sites with periodic boundary conditions. We take an arbitrary translation invariant short-ranged Hamiltonian. For this system, we consider both the canonical ensemble with inverse temperature $β_0$ and the microcanonical ensemble with the corresponding energy $U_N(β_0)$. For an arbitrary self-adjoint operator $\hat{A}$ whose support is contained in a hypercubic block $B$ inside $Λ$, we prove that the expectation values of $\hat{A}$ with respect to these two ensembles are close to each other for large $N$ provided that $β_0$ is sufficiently small and the number of sites in $B$ is $o(N^{1/2})$. 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 (here restricted to the case of the canonical and the microcanonical ensembles), but we prove improved estimates in an elementary manner. We also review and prove standard results on the thermodynamic limits of thermodynamic functions and the equivalence of ensembles in terms of thermodynamic functions. The present paper assumes only elementary knowledge on quantum statistical mechanics and quantum spin systems.

26 pages, versions 2 and later include appendix in which we present a (hopefully readable) proof of the standard results on the equivalence of ensembles. The results in versions 1 and 2 were mathematically correct, but the discussion about the width of the MC ensemble was confused. Now I present correct interpretations of the rigorous results