Physics 216: Topics in many-body theory, spring 2002
Problem set 2: assigned 2/20/03, due 3/6/03
1. Green’s function: Define the retarded Green’s function through
iGr
αβ(t1, r1, t2, r2) = hΨα(t1, r1)Ψ†
β(t2, r2)+Ψ†
β(t2, r2)Ψα(t1, r1)iif t1−t2>0
0 if t1−t2<0. (1)
Calculate the Fourier-transformed Green’s function Gr(ω, p) for the free Fermi gas at temperature
T(i.e., the expectation value hi is taken with respect to the finite-temperature Fermi gas). Be sure
to specify where any poles occur. Check that contour integration of your result indeed gives zero
for negative t=t1−t2.
2. Consider a Cooper pair in He3, which is a p-wave superconductor, so the angular momentum
of the pair is ¯
h. Estimate the order of magnitude of the Cooper pair size as follows: assume that
the gap maximum is of the same order as Tc(of order 10−3K). Assume that the Cooper pair size,
of order ξ, is related to the gap through
ξ∼¯
hvF
∆.(2)
What is the order of magnitude of the rotational velocity of one He atom in a pair? You may wish
to use the fact that the effective mass is m∗= 3.1m.
Compare the rotational velocity to the Fermi velocity, which you can estimate by recalling that
the effective mass m∗= 3.1mand knowing that the Fermi momentum is pF=¯
h(0.8×108cm−1).
3. When discussing superconductivity, we diagonalized the low-energy excitations near the
Fermi surface in terms of new operators γk↑. Check the commutation relation of these operators.
Define a time-ordered Green’s function in terms of these operators and calculate it as a function of
(ω, p) for the BCS state.
Reversing the diagonalization, what does this say about the expectation values of pairs of the
original Ψ operators? In particular, can there be a nonzero expectation value in the BCS state of
“anomalous” combinations of operators like Ψ†(t1, r1)Ψ†(0,0) that vanish in the normal state?
4. Consider the following tight-binding-like model (one variant of the Hubbard model) on the
square lattice with one orbital per site:
H=−tX
hiji,σ
(ciσc†
jσ +h.c.)−X
i
Uini↑ni↓(3)
The first sum is over nearest-neighbor pairs.
Suppose that Uis positive (so that there is an attractive interaction whenever two particles
are on the same site) and weak compared to the hopping t. Also suppose that the system is at
half-filling (one electron per site on average). Start by solving the U= 0 problem. Then calculate
the Vkk0matrix elements induced by Uthat should appear in the pairing Hamiltonian as functions
of t. What do you expect to be the angular momentum of the Cooper pairs? (i.e., s-wave, p-wave,
etc.) Estimate the transition temperature in terms of the original parameters of the model.
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