ISING CHAIN IN A QUASIPERIODIC MAGNETIC FIELD

Abstract

This paper is devoted to the study of the ground state properties of an Ising chain in a magnetic Held of the form hi=h sin (ki+ϕ). The ground state energy is exactly computed in various situations. For a given h>2, the ground state energy E(h, k, ϕ) presents local minima as a function of k. This is a mode locking. If h<2, and only for k close enough. to π, the ground state is purely ferromagnetic, the transition being of the first order. As a general feature, the various physical quantities (magnetization, ground state energy…) are shown to be discontinuous at any rational value of k when the ground state is not ferromagnetic. Finally, the rigidity of the ground state under small displacement is also studied. All these results are compared to the ones obtained in a quite similar model: the Frenkel-Kontorova (FK) model. For instance, in our model which is shown to reduce to a constrained FK model, one can observe a lock-in transition, and the critical magnetic field hc(k) is computed, as opposed to the critical potential for the defectible/undefectible transition in the FK case. The hull function is also exactly computed. All these results are illustrated by means of numerical simulations.