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Material Type: Exam; Class: Special Reading Topics:; Subject: Physics; University: University of Utah; Term: Fall 2008;
Typology: Exams
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(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.
ZV = exp{−
2 g
d~rd~r′ρ(~r)
− ln
|~r − ~r′| a
ρ(~r′)},
where ρ(~r) =
j qj^ δ(~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”.]
4 π
dxdτ N~ · ∂x N~ × ∂τ N .~
The identification is as follows: N~ → S~ (see eq.27) and d^2 x = dxdy, where y = cτ and c is the spin velocity.
Hxxz = J
n
{SxnSxn+1 + SnySyn+1 + ∆SnzSzn+1}.
The transformation is defined by the following non-local relation between spin S~ and fermion operators f :
S n+ = Sxn + iSny = K(n)f (^) n+ , S n− = fnK(n) , Snz = f (^) n+ fn −
Here K(n) is the “string” operator
K(n) = exp{iπ
n∑− 1
j=
f (^) j+ fj }.
The key step is to show that S n+ S n−+1 = f (^) n+ fn+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 =
k (k)f^
k fk, and derive their dispersion^ (k).