Bosonic Central Limit Theorem for the One-Dimensional Xy Model

Abstract

We prove the central limit theorem for Gibbs states and ground states of quasifree Fermions (bilinear Hamiltonians) and those of the off critical XY model on a onedimensional integer lattice.

Large Deviations in Quantum Lattice Systems: One-Phase Region

“…In the larger context of probabilistic results for quantum lattice systems we want to mention refs. 17,18,24,25, about the Central Limit Theorem and the algebra of normal fluctuations, and refs. 6, 7, on Shannon-McMillan type of theorems.…”

On the Algebra of Fluctuation in Quantum Spin Chains

“…We prove this limit theorem under the same condition of [12] . As we are handling a convergence of functionals rather that of measures, we had to add more argument and estimates which are absent both in our previous paper [12] and in commutative cases. Let us mention that Goderis, Verbeure and Vets have already considered the same limit theorem only for strictly local observables and again they could not prove their assumption for any non-trivial Gibbs states.…”

“…The advantages of our results in [12] are as follows. (i) We can prove our assumptions for several mixing states of one-dimensional systems.…”

“…(i) We can prove our assumptions for several mixing states of one-dimensional systems. The assumption of [12] is valid for Gibbs states for finite range interactions , ground states of the (massive) XY model, quasifree states of CAR algebras.…”

Central Limit Theorem for Locally Interacting Fermi Gas

“…However, these papers were a conceptual improvement and our construction owes much to them. The papers [Ma1]- [Ma2] had a similar spirit but, using less stringent ergodic conditions, gave non trivial application to space fluctuations of local observables in XY chains.…”