PHYSICS 140B : STATISTICAL PHYSICS
HW ASSIGNMENT #4
(1) The Dieterici equation of state is
p (v − b) = RT exp −
a
vRT
.
(a) Find the critical point (pc , vc , Tc ) for this equation of state
(b) Writing p̄ =
p/pc , v̄ = v/vc , and T = T /Tc , rewrite the equation of state in the form
p̄ = p̄ v̄, T .
(c) For the brave only! Writing p̄ = 1 + π, T = 1 + t, and v̄ = 1 + ǫ, find ǫliq (t) and ǫgas (t)
for 0 < (−t) ≪ 1, working to lowest nontrivial order in (−t).
(2) Consider a ferromagnetic spin-1 triangular lattice Ising model . The Hamiltonian is
Ĥ = −J
X
Siz Sjz − H
hiji
X
Siz ,
i
where Siz ∈ {−1 , 0 , +1} on each site i, H is a uniform magnetic field, and where the first
sum is over all links of the lattice.
(a) Derive the mean field Hamiltonian ĤMF for this model.
(b) Derive the free energy per site F/N within the mean field approach.
(c) Derive the self consistent equation for the local moment m = hSiz i.
(d) Find the critical temperature Tc (H = 0).
(e) Assuming |H| ≪ kB |T − Tc | ≪ J, expand the dimensionless free energy f = F/6N J
in terms of θ = T /Tc , h = H/kB Tc , and m. Minimizing with respect to m, find
an expression for the dimensionless magnetic susceptibility χ = ∂m/∂h close to the
critical point.
(3) Consider the antiferromagnetic quantum Heisenberg model on a bipartite lattice:
H=J
X
Si · Sj
hiji
where J > 0 and the sum is over the links of the lattice.
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(a) Break up the lattice into a dimer covering (a dimer is a pair of nearest neighbor sites)1 .
Denote one sublattice as A and the other as B. You are to develop a mean field theory
of interacting dimers in the presence of a self-consistent staggered field
hSA i = −hSB i ≡ mẑ .
The mean field Hamiltonian then breks up into a sum over dimer Hamiltonians
Hdimer = JSA · SB + (z − 1)J hSB i · SA + (z − 1)J hSA i · SB
= JSA · SB − Hs (SAz − SBz )
where the effective staggered field is Hs = (z − 1)Jm, and z is the lattice coordination
number. Find the eigenvalues of the dimer Hamiltonian when S = 21 .
(b) Define x = 2Hs /J. What is the self-consistent equation for x when T = 0? Under what conditions is there a nontrivial solution? What then is the self-consistent
staggered magnetization?
Hint: Write your mean field Hamiltonian as a 4 × 4 matrix using the basis states | ↑↑ i, | ↑↓ i,
| ↓↑ i, and | ↓↓ i. It should be block diagonal in such a way which allows an easy calculation
of the eigenvalues.
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There are exponentially many such dimer coverings, i.e. the number grows as eαN where N is the number
of lattice sites. Think about tiling a chessboard with with dominoes. The mathematical analysis of this problem
was performed by H. N. V. Temperley and M. E. Fisher, Phil. Mag. 6, 1061 (1961).
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