Statistical Mechanics - Physics at Oregon State University

190 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE.
The positions of the sites on a lattice are given by so-called Bravais lattice
vectors Ri. The Ising model can be generalized to include interactions between
all spins. Typically, the strength of the interaction depends only on the distance
between the spins:
E {σi} = − 1
2
J(| Ri − Rj|)σiσj − h
i=j
Calculate Tc in the mean field approximation.
Problem 8.
Consider the following generalization of the Ising model. The value of the
spin parameter Si on lattice site i can be ±1 or 0. The energy of the configuration
{Si} is
E {Si} = −J
SiSj − h
Use the density operator approach. Assume that the fluctuations in the
spins are independent, hence
< S1, S2, · · · |ρ|S ′ 1, S ′ 2, · · · >=< S1|ρ|S ′ 1 >< S2|ρ|S ′ 2 > · · ·
Derive a self-consistency equation for the average moment M on each site.
Show that Tc is proportional to qJ.
Problem 9.
Consider the following generalization of the Ising model. The value of the
spin parameter Si on lattice site i can be ±1 or 0. The energy of the configuration
{Si} is
E {Si} = −J
SiSj − h
Using the mean field approximation, derive a self-consistency equation for
the average moment M on each site.
Problem 10.
Consider a two-dimensional triangular lattice (q=6). A cluster in this lattice
is formed by a triangle of three sites. The interactions between the atoms in
this cluster are treated exactly. The effect of the rest of the lattice is treated in
the mean field approach. Evaluate Tc in this case. Compare your result to the
results of the mean-field and Bethe cluster approach.
i
i
Si
Si
i
σi