Problem Set 3

Ising antiferromagnet on a ”bipartite” lattice

Hamiltonian:

Failed to parse (syntax error): {\displaystyle H = \frac{J}{2} \sum_{<ij>} S_i S_j -− h\sum_i S_i } ,

with the sum is over all nearest neighbor sites i and j. 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

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

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 $m^{\dagger }=1$.

(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 $m^{\dagger }$.

(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 $\beta$ .

(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 $T_{N}\;$.

(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.