On the $XY$-Model on Two-Sided Infinite Chain

Abstract

The $XY$-model on the one-dimensional lattice, infinitely extended to both directions,is studied by a method of $C^{\ast }$-algebras. Return to equilibrium is found for any vector state in the cyclic representation of the equilibrium state. A known relation between the algebras of Pauli spins and the algebra of canonical anticommutation relations (CARs) is used to obtain an explicit solution. However the $C^{\ast }$-algebras generated by the two sets of operators become dissociated in the thermodynamic limit of an infinite one-dimensional lattice extending in both directions (in contrast to onesided chain) and this causes a mathematical complication. In particular, we find three features different from the case of one-sided infinite chain: (1) There are no non-trivial constant observables. (2) The (twisted) asymptotic abelian property holds only partially and not in general. (3) Return to equilibrium occurs for all values of the parameter γ and is proved by a method different from the case of one-sided chain.