Equivalence of ensembles in quantum lattice systems: States

Ricardo Lima

Carlos E. Fiore, Cláudio José DaSilva

Yu. R. Dashyan

Huzihiro Araki, Patrick Ion

Fernando G. S. L. Brandão, M. Cramer

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 $O(N^{1/(d+1)})$ spins. The proof rests on a variant of the Berry--Esseen theorem for quantum lattice systems and ideas from quantum information theory.

Hallmann Óskar Gestsson, Charlie Nation, Alexandra Olaya-Castro

Abstract We develop a formalism for mapping the exact dynamics of an ensemble of disordered quantum systems onto the dynamics of a single particle propagating along a semi-infinite lattice, with interactions determined by the probability distribution of disorder realisations of the original heterogeneous quantum ensemble. This provides a geometric interpretation of the loss of coherence when averaging over the ensemble and allows computation of the observables of the ensemble in a single simulation. Conversely, the reverse map implies lattice particle dynamics to be equivalent to the dynamics of an ensemble of disordered quantum systems. The potential of the equivalence is demonstrated by computing the exact disorder-averaged density matrix dynamics for a qubit and a dimer ensemble, and interpreting dynamics of a tight-binding chain with constant and staggered hoppings in terms of an ensemble of disorder realisations.