Physics 7910: take-home exam.
(Dated: December 10, 2008)
Exam is due December 17, 2008, in my office. Your task is to solve any two of the four problems
below. Do not hesitate to ask me for help.
1. Starting with eq.17 of the notes for lecture 19 (pdf file titled “Vortices in classical and quantum XY models”),
derive a Coulomb gas representation of the partition function ZV. This is done by integrating out field θ, using
momentum-space representation for it (see notes for lecture 18, where similar technique is used to calculate low-
temperature spin correlation function).
The end result should be
ZV= exp{− 1
2gZd~rd~r0ρ(~r)−ln |~r −~r0|
aρ(~r0)},
where ρ(~r) = Pjqjδ(~r −~rj) is the vortex “charge” density.
[This is part of the problem 9.10, p.588, from Chaikin and Lubensky “Principles of condensed matter physics”.]
2. Starting with eq.33 of the notes for lecture 19 (pdf file titled “Vortices in classical and quantum XY models”),
derive equations (34) and (35) of the notes.
To connect with our previous lectures, notice that topological charge Qin eq.33 is equivalent to our previous result
(lecture 17, p.3) :
Q=1
4πZdxdτ ~
N·∂x~
N×∂τ~
N.
The identification is as follows: ~
N→~
S(see eq.27) and d2x=dxdy, where y=cτ and cis the spin velocity.
3. Order-by-disorder phenomenon in two coupled antiferromagnetic planes.
Read and work through the paper “Failure of geometric frustration to preserve a quasi-two-dimensional spin fluid”
by M. Maltseva and P. Coleman, Phys. Rev. B 72, 174415 (2005). [download the paper from http://prb.aps.org
website or ask me for help with this.]
Your task is to derive equations (22 - 25) of the paper and check their correctness (it appears that there are misprints
in equations (24,25)).
You can do this following the notes for lecture 13.
4. Jordan-Wigner transformation for the S=1/2 XXZ spin chain.
This one-dimensional quantum problem is described by the Hamiltonian (here 0 ≤∆≤1)
Hxxz =JX
n
{Sx
nSx
n+1 +Sy
nSy
n+1 + ∆Sz
nSz
n+1}.
The transformation is defined by the following non-local relation between spin ~
Sand fermion operators f:
S+
n=Sx
n+iSy
n=K(n)f+
n, S−
n=fnK(n), Sz
n=f+
nfn−1
2.
Here K(n) is the “string” operator
K(n) = exp{iπ
n−1
X
j=1
f+
jfj}.
The key step is to show that S+
nS−
n+1 =f+
nfn+1, from which the rest easily follows.
Note that fermion operators satisfy antifommutation relations {fi, f+
j}=δij (if the meaning of this relation is not
clear to you, do not attempt this problem!).
Show also that in the case of XX model (when ∆ = 0), the spin Hamiltonian is equivalent to that of free non-
interacting fermions, Hxx =Pk(k)f+
kfk, and derive their dispersion (k).