Problem Set 3

Ising antiferromagnet on a ”bipartite” lattice

Hamiltonian:

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Note that now the interaction between spins minimizes the energy when the spins anti-allign, i.e. for Si = −Sj . A bipartite lattice is one that has two sublattices, so that each spin on sublattice A interacts only with some spin on the other sublattice B. In this case, in an antiferromagnetic state, each sublattice assumes a uniform magnetization. We can introduce the magnetization for each sublattice mA = DS(A) i E; mB = DS(B) i E.

The average magnetization then can be written as m = 1 2 (mA + mB) , and the so-called ”staggered” magnetization is defined by the difference between the two sublattices m† = 1 2 (mA − mB) . For perfect ferromagnetic order m = 1, while for perfect antiferromagnetic order m† = 1. (a) Use Weiss mean-field decoupling to replace one of the spins in the Hamiltonian by its thermal average. The Weiss field experienced by a given spin is then proportional to the sublattice magnetization on the other sublattice. Write down self-consistent equations for mA and mB, and express them through the order parameters m and m†. (b) Assume that h = 0, so that m = 0, and solve the mean-field equations by expanding in m†. Determine the Neel (ordering) temperature, and calculate the order-parameter exponent �. (c*) Now consider a small external field h > 0, so that both order parameters can assume a nonzero value (Note: m will be small). By keeping only the leading terms in h and m, calculate the uniform spin susceptibility � = @m/@h, as a function of temperature. Show that � has a cusp around TN. (d*) Imagine adding a ”staggered” external field h†, which would be positive on sublattice A, but would be negative on sublattice B. Concentrate on the system with no uniform field (h = 0), and determine the behavior of the staggered susceptibility �† = @m† @h† . Show that �† blows up at the Neel temperature.