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

Understand

We study a quantum spin system on the $d$-dimensional hypercubic lattice $\Lambda$ 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 $\beta_0$ and the microcanonical ensemble with the corresponding energy $U_N(\beta_0)$.
• For an arbitrary self-adjoint operator $\hat{A}$ whose support is contained in a hypercubic block $B$ inside $\Lambda$, 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 $\beta_0$ is sufficiently small and the number of sites in $B$ is $o(N^{1/2})$.