Boundary conditions for quantum lattice systems

TL;DR

This paper investigates the limitations of classical boundary condition theories when extended to quantum lattice systems, demonstrating their failure and proposing an alternative local data-based characterization.

Contribution

It shows that classical boundary condition results do not hold in quantum systems and introduces a new approach using inequalities with local data for quantum states.

Findings

Classical boundary condition theory fails in quantum lattice systems.

Counterexamples show failure even with classical subsystems in quantum environments.

Proposes inequalities involving local data as an alternative characterization.

Abstract

For classical lattice systems, the Dobrushin-Lanford-Ruelle theory of boundary conditions states that the restriction of a global equilibrium state to a subsystem can be obtained as an integral over equilibrium states of the subsystem alone. The Hamiltonians for the subsystem are obtained by fixing a configuration for the variables in the complement of the subsystem, or more generally, by evaluating the full interaction Hamiltonian with respect to a state for the complement. We provide examples showing that the quantum mechanical version of this statement is false. It fails even if the subsystem is classical, but embedded into a quantum environment. We suggest an alternative characterization of the local restrictions of global equilibrium states by inequalities involving only local data.