Solution to Set 3

Ising antiferromagnet on a ”bipartite” lattice

This assignment corresponds to Section 8.3 and 8.4 in the Solid State Physics book. It deals with the topics of Magnetic Order.

Given Informatin

• Hamiltonian Failed to parse (syntax error): {\displaystyle H = \frac{J}{2} \sum_{<ij>} S_i S_j -− h\sum_i S_i } ,
• Bipartite lattice: a lattice with 2 sublattices, so that each spin on sublattice A interacts only with some spin on the other sublattice B. In an antiferromagnetic state, each sublattice assumes a uniform magnetization.
• Magnetization for Lattice A: $m_{A}=<S^{(A)}>$
• Magnetization for Lattice B: $m_{B}=<S^{(B)}>$
• Average magnetization: $m={\frac {1}{2}}(m_{A}+m_{B}),$
• "Staggered” magnetization: $m^{\dagger }={\frac {1}{2}}(m_{A}-m_{B}),$ (Note: It's the difference between the two sublattices)
• $m=1\;$ for perfect ferromagnetic order
• $m^{\dagger }=1$ for perfect antiferromagnetic order

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 $m^{\dagger }$.

First, Weiss mean field decoupling

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.