Two-dimensional Ising model with Einstein site phonons

Abstract

We consider a simple Ising magnetic model in two dimensions with Einstein site phonons and study it using Monte Carlo simulations that take into account both degrees of freedom simultaneously. In non-frustrated systems, like the square lattice with ferromagnetic and antiferromagnetic interactions, we find that the coupling of the magnetic to the elastic degrees of freedom gradually lowers the magnetic ordering transition until it is completely suppressed at a critical value of the coupling constant. Above this the system suffers a simultaneous magnetic and structural transition into a dimerised state with lower crystalline symmetry and ferromagnetic clusters antiferromagnetically aligned. In the case of the Kagomé lattice with antiferromagnetic interactions, which is frustrated, we show that a similar ordered state takes place when the coupling constant is above a critical value.

Simple microscopic model for magneto-electric coupling in type-II antiferromagnetic multiferroics

“…Ref. 25) In a previous paper [26] two of the authors have studied a magneto-elastic two-dimensional Ising model in which three main phases were in competition: a ferro (FM) or antiferromagnetic (AFM) phase on the undistorted square lattice, a so-called plaquette or checkerboard phase (CB), and the stripe phase (ST). The latter corresponds to an E-type uudd magnetic ordering along the two principal crystal directions.…”

“…The angle θ is treated like in a clock model of 360 equally spaced angles, and the displacement ρ is chosen randomly in a distribution from 0 to a temperature dependent maximum δ max (T ). The use of the latter has no impact on the results obtained from the simulation; it is introduced as a way to optimise speed by avoiding the proposal of extremely unlikely moves at low temperatures [26].…”

“…Each step of the simulation is split into elastic and magnetic moves. Our assumption, similar to the Born-Oppenheimer approximation, translates into the fact that each elastic move is done with a relaxed magnetic configuration [26].…”

“…Results for γ i = 0: purely magnetoelastic system.-The magneto-elastic phase diagram of the model in the absence of any polarization effects has been studied in Ref. 26. As the magneto-elastic coupling, α, is increased, the critical temperature of the ordered FM or AFM Ising phase decreases steadily, reaching T = 0 at a critical coupling α c = K e /2, above which the system goes through a simultaneous magnetic and structural transition.…”

“…We use the order parameter for the checkerboard phase as defined in Ref. 26. For this we use a unit-cell like the one shown in figure S1.…”