PHYSICS 140B : STATISTICAL PHYSICS
HW ASSIGNMENT #5
(1) Consider the S = 1 Ising Hamiltonian
Ĥ = −J
X
Siz Sjz − H
Siz + ∆
i
hiji
where Siz ∈ {−1, 0, +1}.
X
X
Siz
i
2
,
(a) Making the mean field approximation in the first term, but treating the other terms
exactly, find the corresponding mean field Hamiltonian ĤMF .
(b) Defining θ = kB T /zJ, h = H/zJ, and δ = ∆/zJ, find the free energy f = F/N zJ as
a function of θ, δ, h, and m.
(c) Write the mean field equation for this model.
Sjz , where the
individual spin polarizations take values Siz ∈ {−S, . . . , +S}. Find an expression for the
mean field value of Tc . Hint: Expand the free energy for small m = hSiz i, and find what
value of T makes the coefficient of the quadratic term vanish.
(2) Consider a spin-S Ising model. The Hamiltonian is Ĥ = −J
P
z
hiji Si
(3) Consider the O(2) model,
Ĥ = − 12
X
Jij n̂i · n̂j − H ·
i,j
X
n̂i ,
i
where n̂i = cos φi x̂ + sin φi ŷ. Consider the case of infinite range interactions, where
Jij = J/N for all i, j, where N is the total number of sites.
(a) Show that
"
#
Z
P
βJ X
N βJ
2
exp
n̂i · n̂j =
d2 m e−N βJm /2 eβJm· i n̂i .
2N
2π
i,j
(b) Using the definition of the modified Bessel function I0 (z),
Z2π
dφ z cos φ
e
,
I0 (z) =
2π
0
show that
−β Ĥ
Z = Tr e
=
Z
d2 m e−N A(m,h)/θ ,
where θ = kB T /J and h = H/J. Find an expression for A(m, h).
(c) Find the equation which extremizes A(m, h) as a function of m.
(d) Look up the properties of I0 (z) and write down the first few terms in the Taylor
expansion of A(m, h) for small m and h. Solve for θc .
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