PHYSICS 140B : STATISTICAL PHYSICS
HW ASSIGNMENT #3
(1) Consider an Ising ferromagnet where the nearest neighbor exchange temperature is
JNN /kB = 50 K and the next nearest neighbor exchange temperature is J NNN /kB = 10 K.
What is the mean field transition temperature T c if the lattice is:
(a) square
(b) honeycomb
(c) triangular
(d) simple cubic
(e) body centered cubic
Hint : As an intermediate step, you might want to show that the mean field transition
temperature is given by
kB TcMF = z1 JNN + z2 JNNN ,
where z1 and z2 are the number of nearest neighbors and next-nearest neighbors of a given
lattice site, respectively.
(2) Consider a three state Ising model,
Ĥ = −J
X
Si Sj − H
X
Si ,
i
hiji
where Si ∈ −1 , 0 , +1 .
(a) Writing Si = m + δSi and ignoring terms quadratic in the fluctuations, derive the
mean field Hamiltonian HMF .
(b) Find the dimensionless mean field free energy density, f = F MF /N zJ, where z is
the lattice coordination number. You should define the dimensionless temperature
θ ≡ kB T /zJ and the dimensionless field h ≡ H/zJ.
(c) Find the self-consistency equation for m = hS i i and show that this agrees with the
condition ∂f /∂m = 0.
(d) Expand f (m) to fourth order in m and first order in h.
(e) Find the critical temperature θ c .
(f) Find m(θc , h).
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(3) For the O(3) Heisenberg ferromagnet,
Ĥ = −J
X
Ω̂i · Ω̂j ,
hiji
find the mean field transition temperature T cMF . Here, each Ω̂i is a three-dimensional unit
vector, which can be parameterized using the usual polar and azimuthal angles:
Ω̂i = sin θi cos φi , sin θi sin φi , cos θi .
The thermodynamic trace is defined as
Tr A(Ω̂1 , . . . , Ω̂N ) =
Z Y
N
dΩi
A(Ω̂1 , . . . , Ω̂N ) ,
4π
i=1
where
dΩi = sin θi dθi dφi .
Hint : Your mean field Ansatz will look like Ω i = m + δΩi , where m = hΩi i. You’ll want
to ignore terms in the Hamiltonian which are quadratic in fluctuations, i.e. δΩ i · δΩj . You
can, without loss of generality, assume m to lie in the ẑ direction.
(4) A system is described by the Hamiltonian
Ĥ = −J
X
I(µi , µj ) − H
X
δµi ,A ,
i
hiji
where on each site i there are four possible choices for µ i : µi ∈ {A, B, C, D}. The interaction
matrix I(µ, µ0 ) is given in the following table:
I
A
B
C
D
A
+1
−1
−1
0
B
−1
+1
0
−1
C
−1
0
+1
−1
D
0
−1
−1
+1
(a) Write a trial density matrix
%(µ1 , . . . , µN ) =
N
Y
%1 (µi )
i=1
%1 (µ) = x δµ,A + y(δµ,B + δµ,C + δµ,D ) .
What is the relationship between x and y? Henceforth use this relationship to eliminate y in terms of x.
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(b) What is the variational energy per site, E(x)/N ?
(c) What is the variational entropy per site, S(x)/N ?
(d) What is the mean field equation for x?
(e) What value x∗ does x take when the system is disordered?
(f) Write x = x∗ + 34 ε and expand the free energy to fourth order in ε. The factor 34 should
generate manageable coefficients in the Taylor series expansion. You may want to use
a symbolic manipulator like Mathematica here.
(g) Sketch ε as a function of T for H = 0 and find T c . Is the transition first order or
second order?
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