Relation between Statics and Dynamics in the Quench of the Ising Model to below the Critical Point

TL;DR

This paper investigates the relationship between statics and dynamics in the Ising model after a quench below the critical point, revealing a critical invariant state that depends on boundary conditions and extends to other models.

Contribution

It demonstrates that the asymptotic state after a quench is boundary-condition dependent and is critical, linking statics and dynamics in phase-ordering processes.

Findings

Asymptotic states are boundary-condition dependent.

The invariant state is critical and matches the equilibrium of antiperiodic boundary conditions.

The phenomenon extends to the spherical model with different constraints.

Abstract

The standard phase-ordering process is obtained by quenching a system, like the Ising model, to below the critical point. This is usually done with periodic boundary conditions to insure ergodicity breaking in the low temperature phase. With this arrangement the infinite system is known to remain permanently out of equilibrium, i.e. there exists a well defined asymptotic state which is time-invariant but different from the ordered ferromagnetic state. In this paper we establish the critical nature of this invariant state, by demonstrating numerically that the quench dynamics with periodic and antiperiodic boundary conditions are indistinguishable one from the other. However while the asymptotic state does not coincide with the equilibrium state for the periodic case, it coincides instead with the equilibrium state of the antiperiodic case, which in fact is critical. The specific example of the Ising model is shown to be one instance of a more general phenomenon, since an analogous picture emerges in the spherical model, where boundary conditions are kept fixed to periodic, while the breaking or preserving of ergodicity is managed by imposing the spherical constraint either sharply or smoothly.