Problem:A classical antiferromagnet in two dimensions

This problem is an illustration of antiferromagnetic order.

We imagine a two-dimensional square lattice with lattice constant \(a\) and one magnetic atom on each lattice point with spin \(S\). The magnetic interactions is of the Heisenberg type from the Elastic magnetic scattering page, \(H_{ij} = - J_{ij} {\bf S}_i \cdot {\bf S}_j \).

The interaction constant is \(J<0\) between nearest neighbours, and zero otherwise.

Question 1

Calculate the energy of the antiferromagnetic state with alternating spin directions between nearest neighbours, as illustrated in this figure (middle) from the Elastic magnetic scattering page.

Hint

Since the system is symmetric, you need only to look at magnetic bonds from one spin - and then scale with the total number of spins.

Hint

When counting bonds, remember that a bond connects two spins, but should only be counted once.

Solution

Each of the \(N\) spins interact with 4 nearest neighbours. Since each magnetic coupling involve two spins, there are \(2N\) couplings in total. Each pair of nearest neighbour spins point in different directions, giving an energy contribution of \(JS^2\). Thus, the total energy is \(E_0 = 2 N J S^2\).