The simple cubic Ising model with competing interactions

The simple cubic Ising model with competing interactions - a histogram Monte Carlo study

Diplom thesis in Physics by Ralf K. Heilmann
FAU Erlangen-Nürnberg (1991)
(Thesis language: German)

Abstract

We have studied the simple cubic Ising model with antiferromagnetic nearest (Jnn) and next-nearest (Jnnn) interactions. The Hamiltonian can be written as -ßH = K(S1 + aS2), where ß = 1/kT, K = ßJnn, and a = Jnnn/Jnn. S1 and S2 are the sums over all nearest and next-nearest-neighbor spin pairs, respectively. Utilizing Monte Carlo simulations, finite-size scaling, and recently developed histogram techniques we map the T/Jnn-a phase diagram.

There are two ground states for T/Jnn > 0:

the AFI-state with alternating spins in all directions (ABAB-sequence of antiferromagnetic planes) at a < 0.25
the "layered" AFII-state (AAAA-sequence of antiferromagnetic planes) at a > 0.25.

The sequence of layers is arbitrary at a = 0.25.

For the AFI-state we find a second-order transition into disorder. Critical exponents are of the 3d-nn-Ising class and appear not to change from a = 0 to 0.245. The AFII-disorder phase boundary is determined to be first order, as well as the AFI-AFII transition. The three critical lines meet at an umbilical point close to T/Jnn = 1.15 and a = 0.25. The exact location and nature of that critical point could not be determined due to a combination of low temperature, critical slowing down, and high degeneracy of the ground state. We assume that strong crossover effects complicate finite-size analysis in the vicinity of this critical point. Extensive simulations hint towards the point being located at a value of a slightly greater than 0.25.