Boundary Conditions for Quantum Lattice Systems

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. Mathematics Subject Classification (1991): 82B10, 46L60, 82B20, 1 Inst. Theor. Fysica, Universiteit Leuven, B-3001 Heverlee, Belgium. Email: [email protected] 2 Onderzoeksleider, NFWO Belgium 3 Fachbereich Physik, Universität Osnabrück, 49069 Osnabrück, Germany. Email: [email protected]