PHYSICS 140B : STATISTICAL PHYSICS
PRACTICE MIDTERM EXAM
Consider a four-state ferromagnetic Ising model with the Hamiltonian
X
X
X
Si Sj − H
Si ,
Si Sj − J2
Ĥ = −J1
hhijii
hiji
i
where the first sum is over all nearest neighbor pairs and the second
sum is over all next
nearest neighbor pairs. The spin variables Si take values in the set − 32 , − 12 , + 12 , + 32 .
(a) Making the mean field Ansatz Si = m + (Si − m), where m = hSi i is presumed
independent of i, derive the mean field Hamiltonian ĤMF . You may denote z1 as the
number of nearest neighbors and z2 as the number of next nearest neighbors of any
site on the lattice.
[15 points]
(b) Find the mean field free energy F (m, T, H).
[15 points]
ˆ
ˆ
(c) Adimensionalize, writing θ = kB T /J(0)
and h = H/J(0).
Find the dimensionless free
ˆ
energy per site f = F/N J(0).
[15 points]
(d) What is the self-consistent mean field equation for m?
[15 points]
(e) Find the critical temperature θc .
[15 points]
(f) For θ > θc , find m(h, θ) assuming |h| 1.
[15 points]
(g) What is the mean field result for |Si | ? Interpret your result in the θ → ∞ and
θ → 0 limits. Hint : We don’t neglect fluctuations from the same site.
[10 points]
1