Due to the equivalence of the statistical ensembles, thermostatic properties of physical systems with short-range interactions can be calculated in different ensembles leading to the same physics. In particular, the ensemble equivalence holds for systems that undergo a continuous phase transition in the infinite volume limit with the result that the properties of the transition can also be investigated in the microcanonical approach. Considering as an example the spherical model, the ensemble equivalence is explicitly demonstrated by calculating the critical properties in the microcanonical ensemble and comparing them to the well known canonical results.
We consider the problem of whether the canonical and microcanonical ensembles are locally equivalent for short-ranged quantum Hamiltonians of
N spins arranged on a
d-dimensional lattices. For any temperature for which the system has a finite correlation length, we prove that the canonical and microcanonical state are approximately equal on regions containing up to
O(N1/(d+1)) spins. The proof rests on a variant of the Berry--Esseen theorem for quantum lattice systems and ideas from quantum information theory.