PHYSICS 140B : STATISTICAL PHYSICS
HW ASSIGNMENT #5
(1) Find in the scientific literature two examples of phase transitions, one first order and
one second order. Describe the systems and identify the order parameter in each case.
Reproduce a plot of the order parameter versus temperature showing how the transition
is discontinuous (first order) or continuous (second order). If the transition is first order,
is a latent heat reported? If second order, are any critical exponents reported? Some
examples of phase transitions, which might help you in your searches: liquid-gas transitions,
Curie transitions, order-disorder transitions, structural phase transitions, normal fluid to
superfluid transitions (3 He and 4 He are two examples), metal-superconductor transitions,
crystallization transitions, etc. An excellent search engine for scholarly publications is
available at scholar.google.com.
(2) Consider a two-state Ising model, with an added dash of flavor from what you have
learned in your Physics 130 (Quantum Mechanics) sequence. You are invited to investigate
the transverse Ising model , whose Hamiltonian is written
X
X
Ĥ = −J
σix σjx − H
σiz ,
i
hiji
where the σiα are Pauli matrices:
0 1
x
σi =
1 0 i
σiz
,
=
1 0
.
0 −1 i
You may find the material in §6.15 of the notes useful to study before attempting this
problem.
(a) Using the trial density matrix,
%i =
1
2
+ 12 mx σix + 12 mz σiz
ˆ ≡ f (θ, h, mx , mz ), where θ = k T /J(0),
ˆ
compute the mean field free energy F/N J(0)
B
ˆ
and h = H/J(0). Hint: Work in an eigenbasis when computing Tr (% ln %).
(b) Derive the mean field equations for mx and mz .
(c) Show that there is always a solution with mx = 0, although it may not be the solution
with the lowest free energy. What is mz (θ, h) when mx = 0?
(d) Show that mz = h for all solutions with mx = 0.
(e) Show that for θ ≤ 1 there is a curve h = h∗ (θ) below which mx 6= 0, and along which
mx vanishes. That is, sketch the mean field phase diagram in the (θ, h) plane. Is the
transition at h = h∗ (θ) first order or second order?
(f) Sketch, on the same plot, the behavior of mx (θ, h) and mz (θ, h) as functions of the
field h for fixed θ. Do this for θ = 0, θ = 21 , and θ = 1.
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(3) Consider the U(1) Ginsburg-Landau theory with
Z
F =
ddx̃
h
2
1
2 a |Ψ|
i
˜ 2 .
+ 41 b |Ψ|4 + 12 κ |∇Ψ|
Here Ψ(x̃) is a complex-valued field, and both b and κ are positive. This theory is appropriate for describing the transition to superfluidity. The order parameter is hΨ(x̃)i. Note
that the free energy is a functional of the two independent fields Ψ(x̃) and Ψ∗ (x̃), where
Ψ∗ is the complex conjugate of Ψ. Alternatively, one can consider F a functional of the real
and imaginary parts of Ψ.
(a) Show that one can rescale the field Ψ and the coordinates x̃ so that the free energy
can be written in the form
Z
h
i
F = ε0 ddx ± 21 |ψ|2 + 14 |ψ|4 + 12 |∇ψ|2 ,
where ψ and x are dimensionless, ε0 has dimensions of energy, and where the sign on
the first term on the RHS is sgn (a). Find ε0 and the relations between Ψ and ψ and
between x̃ and x.
(b) By extremizing the functional F [ψ, ψ ∗ ] with respect to ψ ∗ , find a partial differential
equation describing the behavior of the order parameter field ψ(x).
(c) Consider a two-dimensional system (d = 2) and let a < 0 (i.e. T < Tc ). Consider the
case where ψ(x) describe a vortex configuration: ψ(x) = f (r) eiφ , where (r, φ) are
two-dimensional polar coordinates. Find the ordinary differential equation for f (r)
which extremizes F .
(d) Show that the free energy, up to a constant, may be written as
"
ZR
2
f2
F = 2πε0 dr r 21 f 0 + 2 +
2r
#
1
4
1−f
2 2
,
0
where R is the radius of the system, which we presume is confined to a disk. Consider
a trial solution for f (r) of the form
f (r) = √
r2
r
,
+ a2
where a is the variational parameter. Compute F (a, R) in the limit R → ∞ and
extremize with respect to a to find the optimum value of a within this variational
class of functions.
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