Gibbs states of a one dimensional quantum lattice

Abstract

A one dimensional infinite quantum spin lattice with a finite range interaction is studied. The Gibbs state in the infinite volume limit is shown to exist as a primary state of a UHF algebra. The expectation value of any local observables in the state as well as the mean free energy depend analytically on the potential, showing no phase transition. The Gibbs state is an extremal KMS state.

Complex-time singularity and locality estimates for quantum lattice systems

“…More than four decades ago, Araki [1] established that, for a general class of one-dimensional interactions, the infinite volume time-evolution of a local observable is entire analytic in the time variable. In fact, Araki's work established a locality principle for the complex-time evolution of local observables: the support of a local observable stays bounded (up to a small correction) as it evolves in complextime no matter how large the system is.…”

Generalized zeta-functions for Axiom 𝐴 basic sets

“…First suppose that (A, /) is a subshift of finite type (see [13]). Then the theorem can be proved by the "transfer matrix" method of statistical mechanics (see [7], [1], [12], [9], [10]). The general case reduces to that one: using a Markov partition for A (see [11], [2]) one can, by a combinatorial lemma of Manning [6], write where K m consists of the f periodic points of prime period m. [5] ).…”

On the Algebra of Fluctuation in Quantum Spin Chains

“…The following invariance of F θ under the time evolution is a corollary of results due to H.Araki in [1] …”