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

Abstract

The ^Y-model on the one-dimensional lattice, infinitely extended to both directions, is studied by a method of C*-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*-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.

Approach to Equilibrium for a Class of Random Quantum Models of Infinite Range

“…By (25),H 0 is bounded below (in fact it is bounded) and (21) follows from a standard result in function theory (see,e.g., [10], 5.8).…”

Non-zero Entropy Density in the XY Chain Out of Equilibrium

“…The element T stems from the C * crossed product extension by Z 2 of the C * algebra S of the spins on Z for the two-sided chain (see [3,7]; for the one-sided chain, T can be dropped): The C * algebra O generated by S and the element T contains the CAR algebra A generated by the fermions b x , b * x on Z as a C * subalgebra. The extension to O of the * -automorphisms θ on S, defined as the rotation of the spins about the z-axis with angle π, θ(σ (x) j ) = −σ (x) j for j = 1, 2 and θ(σ (x) 3 ) = σ (x) 3 , yields the decomposition of S and A into even and odd parts with respect to parity, θ(A) = ±A, A ∈ O.…”

Bosonic Central Limit Theorem for the One-Dimensional Xy Model

“…To obtain CLT we use the method of [4]. We enlarge the algebra A to another algebraà adding a new selfadjoint unitary element T having the following property:…”

“…However, they are not physical equivalent in that the ergodic behavior of the time evolution is different. In [4], H. Araki introduced the crossed product formalism for the XY model to handle analytic aspects of the XY model and in [5] we have shown lack of ergodicity for the time evolution in the ground state representations when γ = 0 and |λ| < 1. We use the idea of H. Araki to show Bosonic central limit theorem for the XY model.…”