Solution to Set 3

Part 1

(1) 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 m_A\; and m_B\;, and express them through the order parameters m\; and m^{\dagger}.

Part 2

(2) Assume that h = 0\;, so that m = 0\;, and solve the mean-field equations by expanding in m^{\dagger}. Determine the Neel (ordering) temperature, and calculate the order-parameter exponent.

Part 3

(3) 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 \chi = \partial m/ \partial h, as a function of temperature. Plot \chi\; as a function of temperature, and show that it has a cusp around T_N\;.

Part 4

(4) Imagine adding a ”staggered” external field h^{\dagger}, 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 \chi^{\dagger}= \partial m^{\dagger} / \partial h^{\dagger} . Show that \chi^{\dagger} blows up at the Neel temperature.