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1. Assume that the Landau free energy is given by
FL =
Z
d
3
x
h
atm2 + bm6
i
,
where a and b are constants and t = (T − Tc)/Tc. Find the critical exponent β.
2. Consider a two-dimensional system that undergoes a phase transition and has
a Landau-Ginzburg free energy near the critical point given by
FL =
Z
d
2
x
h
tψ~ · ψ~ + u(ψ~ · ψ~ )
2 − 2u
0ψ
2
xψ
2
y + g(ψ~ · ψ~ )
3 +
κ
2
(∇ψx)
2 +
κ
2
(∇ψy)
2 −~h · ψ~
i
where ψ~ = (ψx, ψy) is a two-component order parameter, t = (T − Tc)/Tc, Tc
is the critical temperature of the system, and u, g, and κ are constants. ~h is an
external ordering field.
(a) Find the most probable values of ψ~ when the system is uniform, u = u
0 =
0,~h = 0, and g > 0 for both t > 0 and t < 0.
(b) Find the critical exponent γx for this model with ∂ψx/∂hx ∼ |t|
−γx when
u = u
0 > 0 and g = 0.
(c) What is the order of the phase transition if u < 0, u
0 = 0, and g > 0?
(d) What kind of spontaneous symmetry breaking does this system exhibit if
u = u
0 > 0 and g = 0?
(e) What kind of spontaneous symmetry breaking does this system exhibit if
u
0 = 0, u > 0, and g > 0?
(f) Identify the Goldstone modes for this system when u
0 = 0 and find their
contribution to the free energy. Use the finite Fourier transform
f(~x) = 1
√
A
∑
~q
e
i~q·~x
f~q
and assume that
Z
d
2x
A
e
i(~q+~q
0
)·~x = δ~q,~q
0
where A is the area of the system.
3. Assume the correlation function for a system with a scalar order parameter
has the form
Γ(~r) = hm(~r)m(0)i − hm(0)i
2 = Cr2−D−η
exp(−r/ξ), with ξ = ξ0t
−ν
1
(a) Find the susceptibility in terms of C, ξ, η and the dimensionality D. (You
do not need to explicity calculate the angular part of the integral, i.e.
just write your answer with ΩD defined by R
d
Dr =
R
dΩ
R
drrD−1 ≡
ΩD
R
drrD−1
)
(b) Find the critical exponent γ in terms of η and ν.
4. The impact of a gradient term on the liquid-gas phase boundary: Assume the
free energy per unit length is given by f(ρ, T) + κ
2
(∂xρ)
2 where ρ(x) is the
local density (and assume a planar geometry). Here, the density distribution
can be thought of as a Landau-Ginzburg field. The number of particles in is
N = A
R xl
xg
dxρ(x). Here xl,g are points sufficiently far into the liquid and gas
phases that the gradient term goes to zero and A is the surface area between
the two phases. Assume throughout the system is connected to an external
particle reservoir with chemical potential µ0.
(a) Write down an integral expression for the contribution to the Grand Potential of this system between the points xl and xg. Express it in terms of
the local pressure P(x) and local chemical potential µ(x). Note that the
fundamental thermodynamic relation gives −P(ρ(x), T) = f(ρ(x), T) −
µ(ρ(x), T)ρ(x).
(b) Find an integral expression for ∆Ω, the difference between this Grand
Potential and the Grand Potential between xl and xg when κ = 0.
(c) Transform the integral over x to an integral density from ρl
to ρg and
vary ∆Ω wrt the function h(ρ) = ∂xρ to find a condition that minimizes
the Grand Potential over the liquid-gas transition region (i.e. think about
minimizing the integrand wrt h). Plug this back into the expression for
∆Ω to find
∆Ω =
√
2κA
Z ρl
ρg
dρ
q
P(ρg) − P(ρ) + (µ(ρ) − µ(ρg))ρ ≡ Aσ
where σ is the surface tension.
2
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