Phase Transition in Ferromagnetic Ising Models with Non-Uniform External Magnetic Fields

TL;DR

This paper investigates how non-uniform external magnetic fields influence phase transitions in ferromagnetic Ising models, establishing conditions under which phase transitions occur or are suppressed.

Contribution

It demonstrates that summable magnetic fields induce a first-order phase transition, while bounded fields with positive lim inf prevent phase transitions.

Findings

Summable magnetic fields lead to a first-order phase transition.

Positive bounded magnetic fields with lim inf > 0 prevent phase transitions.

The results clarify the role of field summability and bounds in phase transition phenomena.

Abstract

In this article we study the phase transition phenomenon for the Ising model under the action of a non-uniform external magnetic field. We show that the Ising model on the hypercubic lattice with a summable magnetic field has a first-order phase transition and, for any positive (resp. negative) and bounded magnetic field, the model does not present the phase transition phenomenon whenever $\liminf h_i> 0$, where ${\bf h} = (h_i)_{i \in \Z^d}$ is the external magnetic field.