1
due date: May 4th, 5pm.
Ph 127c - Problem set 2
1. Debye screening in a vortex gas.
In class we derived the interactions between vortices, and showed that the free energy is given by:
1X
Fv =
2πχpi pj ln (Λ|~xi − ~xj |)
2
(1)
i6=j
(a) The vortex interaction can be written as:
Z
1X
1
Fv =
pi V (~xi ) =
d2 ~xρ(~x)V (~x)
(2)
2 i
2
P
where V (~xi ) is a potential energy, and ρ(~x) =
x − ~xi ) is a charge density. What is the linear
i pi δ(~
differential equation that determines V ? Hint: think about electrostatics.
(b) In class we defined the vortex fugacity as ζ = e−Ec /T . For a dilute gas of vortices, ζ is roughly the vortex
−1
density per Λ−2 area, where Λ = ξSF
is the UV cutoff of the theory. In the presence of other vortices, the
density at point ~x gets modified by the local potential V (~x). What is the charge density at point ~x as a
function of V (~x), ζ, ξ, and T ? Hint - approximate the Λ−2 square as a single statistical-mechanical system
in a grand-canonical ensemble of particles with fugacity ζ. A better hint - the answer should contain a
sinh.
(c) The results of items 1a, 1b can be combined to the self-consistent Debye equation for an interacting plasma.
The density from 1b produces screening of the logarithmic interaction, which can be seen by including the
screening density in the field equation of 1a. What is the potential of a point charge, with charge p = 1
placed at the origin and surrounded by a plasma with the parameters χ, ζ, T and Λ all given?
2π
R∞ d2 k ikx
R 1/|x|
R kdkdθ
Particularly, using the approximation
e
≈
2
2
k +a
k2 +a2 show that the logarithmic interaction
−∞
0
0
is cut off at a distance ξD . Find ξD .
(d) The knowledge of the Debye screening length ξD (ζ) allows us to construct a self-consistent equation for ζ
of a single vortex in a free plasma. To do this, first write the bare fugacity ζ = e−(Ec +Eself )/T of a single
vortex, where Eself is the self interaction of a particle in the plasma through the field V =V (Λ−1 ), which
is too a function of ζ.
The resulting equation is a self-consistent equation for ζ. At what value of χ and T does it admit a stable
non-zero solution for ζ?
2. Dual sine-Gordon model for a vortex gas
In this problem you will show that the vortex part of the partition function of a superfluid can be transformed
to the following path integral:
P
pi pj ln |~
xi −~
xj |Λ
K
∞
2n
R
P
Q
1 2n
2
i>j
ZV =
ζ
d
~
x
e
i
n!2
(3)
n=0
i=1
R
R 2 1
2
= C D[θ(~x)] exp − d ~x 2 λ(∇θ(~x)) − 2ζ cos θ(~x) .
with C a constant, and λ an interaction parameter that we will determine in the end of the mapping. Eq. (3)
Is extremely helpful in the analysis of interacting systems, and repeats itself in many physical system.
(a) The parameter ζ will turn out to be the fugacity of the vortices. As a first step, expand the expression in
Eq. (3) to a power series of ζ.
Show that the result can be written as:
∞
n
R
P
P R Q
ζn
C
d2 ~xi D[θ(~x)]
n!
n=0
{pi =±1} i=1
(4)
R 2 1
2
exp − d ~x 2 λ(∇θ(~x)) − iθ(~x)ρ(~x)
where:
ρ(~x) =
X
i
pi δ(~x − ~xi )
(5)
2
(b) By completing the square, and integrating the θ field out, show that the expression reduces to:
C0
∞
P
n=0
ζn
n!
P
{pi =±1}
n
R Q
d2 ~xi
(6)
i=1
R
R
exp − 21 d2 ~x d2 ~x0 ρ(~x)K(~x − ~x0 )ρ(~x0 )
What is the Fourier transform of the kernel K(~x − ~x0 )? Regard the mention of ’Fourier transform’ as a
hint, as well as the term ’Parseval identity’.
(c) By identifying K(~x − ~x0 ) with the logarithmic vortex -vortex interaction, find what λ is in terms of K.
3. K-T transition in superconducting films.
Superconducting films are quite well described by the free energy of a U(1) order parameter, or an x-y magnet,
with the following form:
Z
1 πh̄D
T − Tc
1 1
2
4
2
2
F [φ] = d · NF d x
|∆|
(7)
|∇∆| +
|∆| +
2 4T
Tc
2 π2 T 2
with ∆ = |∆|eiφ being a complex order parameter, usually connected with the gap in the density of states of a
superconductor. d here is the thickness of the film, NF = 4πkF2 /h̄vF is the density of states of the fermions at
the Fermi-surface, D is the diffusion constant in the metal, and Tc is its critical temperature.
(a) Superfluid films always support vortices. What would be the core size of a vortex in terms of the parameters
in Eq. (7)? Assume that the temperature is well below Tc .
(b) What is the stiffness of the film, χ, as a function of temperature? Strictly speaking, the definition of the
stiffness is:
χ≡
Lx ∂ 2 F
Ly ∂∆φ2
(8)
where F is the free energy of the film with the phase φ having boundary conditions along the x = 0 and
x = Lx lines with φ(Lx ) − φ(0) = ∆φ. Lx , Ly are the dimensions of the film in the x and y directions
respectively.
(c) Superconducting films also undergo a Kosterlitz-Thouless transition. At what temperature does the K-T
transition occur for a film described by Eq. (7)? Derive your result parametrically, and use:
kF =
2π
4Å
D = 5 · 10−3 m2 /s
vF = 106 m/s
d = 10nm
Tc = 4K
(9)
What is this temperature relative to the ciritcal temperature?
(d) For a square film of side Lx = Ly = L = 1µm, with the parameters above, what is the width of the critical
region? By critical region, we denote the temperature range in which the correlation length ξKT > L,
which curtails the critical behavior of the sample.
4. Feynman relation - Optional (a bit of a field theory rerun of problem 5 from PS1)
(This question refers to both superconductors, and superfluids. For conciseness the analog terms for the superconductor will be mentioned in brackets)
A rotating 2d superfluid (a superconducting film in a magnetic field) is described by the following free energy:
"
#
2
Z
1 h̄
u 4
2
2
F = d d ~x
∇−α
~ (~x) ψ − r|ψ| + |ψ|
(10)
2m i
2
Where:
(
∇×α
~=
~ SF
2mΩ
~ SC
2eB
~ = ẑΩ is the angular velocity (B
~ = ẑB is the magnetic field).
where Ω
In the following assume that ψ = ψ0 eiφ and that ψ0 is constant in space, except for in vortex cores.
(11)
3
(a) Show that the low energy field theory for φ is given by:
Z
F =d
2
1
1
~ (~x) .
d2 ~x χ ∇φ − α
2
h̄
(12)
What is χ?
(b) Show that the neutrality condition for a non-rotating (zero magnetic field) superfluid is replaced by:
( mΩ
πh̄ SF
ρ+ − ρ− =
(13)
2eB
SC
h
The above connection between vortex density and angular rotation is called the Feynman relation.
It implies that rotating superfluids, and superconductors in a magnetic field must
have a finite density of vortices.
These usually form an Abrikosov Lattice.
See, e.g.,
http://cua.mit.edu/ketterle group/Projects 2001/Vortex lattice/Vortex.htm.