Physics 216: Topics in many-body theory, spring 2002
Problem set 3: assigned 4/3/03, due 4/22/03
1. To check our diagrammatic technique, calculate and interpret the first two terms for G(ω, k)
in momentum-space perturbation theory in the interaction strength V. The diagrammatic repre-
sentation of these terms is
2. Use bubble diagrams (“RPA”) to calculate simple Thomas-Fermi screening of the electron-
electron interaction (that is, ω= 0 and small k/kF). For reference, Abrikosov et al. (AGD) and
Mahan both have a very complete calculation of this (“Lindhard”) screening, but all you need to
do is the Thomas-Fermi limit. You may wish to start by reviewing the undergraduate calculation
of TF screening, as found in Ashcroft and Mermin or Ziman: the screened interaction is
˜
V(q) = 4πe
q2+k0
2, k0
2= 4πe2∂n0
∂µ .(1)
This problem does require a bit of algebra (why it wasn’t done in class), but doing one such
calculation is a good idea.
3. Show that in the rotationally symmetric gauge for A, the lowest Landau level eigenstates
in a constant magnetic field can be written as ψm∼zme−|z|2/4`2(you should also calculate the
normalization). Now suppose that the system of Nnoninteracting LLL electrons is put in a weak
radial confining potential V(r) = αr2, with αsufficiently small that no mixing occurs between
Landau levels. The ground state of the Nelectrons is now a Slater determinant of the single-
particle eigenstates m= 0, . . . , N −1.
Now consider the low-energy edge excitations of this state. For example, moving the last
electron out by angular momentum ¯
hgives an excited state. How many excitations within the LLL
are there of total momentum M, for M= 1, . . . , 5? This is sometimes referred to as the number of
“partitions” of the integer M. Give an estimate of the velocity of excitations at the edge. Finally,
show that after linearizing the E(m) relation at the edge, it is possible to write the spectrum of
edge excitations in terms of independent modes of angular momentum m= 1,2, . . .. What are the
total energy and angular momentum in terms of these modes? (As an optional exercise, you can
try to formally define bosonic annihilation and creation operators for these modes.)
Hint: it may be helpful to think of mode mas proportional to “move the moutermost electrons
out by ¯
hangular momentum”.
4. Use spin-wave theory to show, first, that even at zero temperature there is no long-range
order in 1D (as sketched in Lecture 18), and second, to estimate how much the antiferromagnet
moment is reduced by quantum fluctuations in 2D and 3D, for the cubic lattice, for S=1
2.
The known answer from Quantum Monte Carlo simulations in 2D is that for spin-half the
moment is reduced by about 40 percent from its classical value, which agrees well with neutron
scattering measurements.
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