PHYSICS 140B : STATISTICAL PHYSICS
MIDTERM EXAM
9:00 am – 9:50 am / Friday Feb. 13 / WLH 2008
Consider a four-state ferromagnetic Ising model with the Hamiltonian
X
X
Si Sj − H
Si ,
Ĥ = −J
i
hiji
where the first sum is over all links of a lattice of coordination number z. The spin variables
Si take values in the set {−1 , 0 , 0 , +1}. Note that there are two distinct states, each with
Si = 0, and a total of four possible states on each site. Taking the trace for a single site
means we sum over the four independent states, one with S = +1, two with S = 0, and one
with S = −1.
(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 .
[15 points]
(b) Find the mean field free energy F (m, T, H).
[15 points]
(c) Adimensionalize, writing θ = kB T /zJ and h = H/zJ. Find the dimensionless free
energy per site f = F/N zJ.
[15 points]
(d) What is the self-consistent mean field equation for m?
[15 points]
(e) Find the critical temperature θc . Show that when h = 0 the graphical solution to the
mean field equation depends on whether θ < θc or θ > θc .
[15 points]
(f) For θ > θc , find m(h, θ) assuming |h| 1.
[15 points]
(g) What is the mean field result for hSi2 i? Interpret your result in the θ → ∞ and θ → 0
limits. Hint : We don’t neglect fluctuations from the same site.
[10 points]
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