Physics 6311: Statistical Mechanics - Homework 13

due date: Tuesday, Dec 2, 2025

_________________________________________________________________________________________

Problem 1: Mean-field theory of an Ising antiferromagnet (40 points)

Consider an Ising model ( $S_{i} = \pm 1$ ) given by a Hamiltonian

H = - J \sum_{⟨ ij ⟩} S_{i} S_{j} - μB \sum_{i} S_{i}

with a negative exchange interaction $J$ on a square lattice.

a): Describes the ground state in the absence of a field ( $B = 0$ )? Find the ground state energy per spin. (You can neglect boundary terms)
b): What is the ground state for large field $B$ ? What is its energy per spin?
c): At what field value $B_{c}$ does the ground state switch between the states you found in parts a) and b).
d): Now consider nonzero temperatures and derive the mean-field theory for this model. Motivated by the state you found in part a), you will need to consider two sublattices, each with its own average spin value.
e): Find the critical temperature for the onset of antiferromagnetism (the so-called Neel temperature) in the absence of a field, $B = 0$ .