Physics 216: Topics in many-body theory, spring 2002
Problem set 4: assigned 5/8/03, due 5/16/03
This final problem set and the two-page project writeup are due in 549 Birge before 5 p.m. on
Friday May 16. No extensions can be granted as course grades will be turned in over that weekend.
1. Prove the “fluctuation-dissipation theorem” relating the response function R(q, ω) to the
spin structure factor S(q, ω):
S(q, ω) = −2
1−e−ω/kT ImR(q, ω ).(1)
Recall the definitions of these two quantities: Sis the correlation function hs(x1, t1)s(x2, t2)i, while
Rwas defined in class as the response of an observable to a small perturbation in the Hamiltonian.
Note that the sign convention used in the notes (that of Auerbach, say) differs by a −from what
you may find elsewhere.
2. Use the method of resolvent Green’s functions introduced in class to find the phase shift and
induced density of states for the impurity level in the noninteracting Anderson model
H=X
σ
dc†
d,σcd,σ +X
k,σ
c†
k,σck,σ +X
k,σ
(Vkc†
d,σck,σ +h.c.).(2)
for a flat band with constant density of states ρ0over the energy range −D≤E≤Dand no k
dependence of V. That is, rederive following the steps in class (Lecture 22)
T r G+() = X
k
G+
k,k() + G+
d,d()
=X
k
1
+iη −k
+∂
∂ log +iη −d−X
k
|Vk|2
+iη −k!.(3)
Then use this expression to calculate the phase shift η() and hence the change in the density of
states induced by the impurity, which should be Lorentzian (you may assume that dand ˜dare
near the band center, and hence not within the level width ∆ of a band boundary).
3. (from Sachdev) Show that the single O(2) quantum rotor discussed in class
HQ=−∆∂2
∂θ2−˜
hcos θ(4)
describes the scaling limit of the classical 1D XY model. Hint: Write the XY model partition
function as
Z=Z2π
0
M
Y
i=1
dθi
2πhθ1|T|θ2ihθ2|T|θ3ihθ3|T|θ4i. . . hθM|T|θ1i
= TrTM(5)
with
hθ|T|θ0i= exp Kcos(θ−θ0) + h
2(cos θ+ cos θ0),(6)
and show exp(−aHQ)≈T. Find ∆ and ˜
hin terms of the coupling K, the lattice spacing a, and
the classical field h,
1