Problem Set 3

Ising antiferromagnet on a ”bipartite” lattice

Hamiltonian:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = \frac{J}{2} \sum_{<ij>} S_i S_j −h\sum_i S_i }

Note that now the interaction between spins minimizes the energy when the spins anti-allign, i.e. for Failed to parse (syntax error): {\displaystyle S_i = −S_j} . 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

$m_{A}=<S^{(A)}>;\;\;m_{A}=<S^{(A)}>.$

The average magnetization then can be written as

$m={\frac {1}{2}}(m_{A}+m_{B}),$

and the so-called ”staggered” magnetization is defined by the difference between the two sublattices

$m^{\dagger }={\frac {1}{2}}(m_{A}-m_{B}),$

For perfect ferromagnetic order $m=1$, while for perfect antiferromagnetic order $m^{\dagger }=1$.

(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.