Lehrstuhl für Theoretische Nanophysik
Prof. Dr. L. Pollet
Dr. T. Barthel
6th Exercise Sheet Many-Body Physics
Will be discussed on Fri June 08.
Exercise 1: Particle scattering off random impurities
Consider an electron in a disordered potential,
Z
X †
H=
k ck ck + d3 xU (x)ψ † (x)ψ(x)
(1)
k
with the external one-body potential generated by a random set of Ni impurities located at positions Ri ,
X
U (x) =
U(x − Rj ).
(2)
j
There is no interaction between the electrons. The energy of the electron
is thus conserved. We will be interested in properties that are averaged
over the impurity positions rather than the properties of a single impurity
position realization. We will hence take the quenched average (meaning that
the impurity average takes place after the thermodynamic average),
Z
1
hAi = Πj d3 Rj hA[{Rj }]i
(3)
V
We are interested in the impurity average of the Green function. The average of the disorder distribution leads to an unimportant shift in chemical
potential, but the fluctuations of the disorder distribution will scatter the
electrons (and lead to an effective interaction between them). If the fluctuations in the impurity potentials are delta-correlated in time, and form white
noise in space, then one shows that
Z
0
0
δU (x)δU (x ) = e−iq·(x−x ) |u(q)|2 = ni u20 δ(x − x0 ),
(4)
q
where the second equality follows from the white noise condition (ie, neglecting higher order moments) leaving the fluctuations in the impurity scattering potential
local, ni = Ni /V is the concentration of impurities
R entirely
3
and u(q) = d xU(x)e−iq·x the Fourier transform of the impurity scattering potential.
Derive the Feynman rules for the Green function by expanding ψ † (x) =
R † −ik·x
R
such that Vdisorder = k,k0 c†k ck0 δU (k − k0 ). Show in particular
k ck e
that momentum has to be conserved after disorder averaging.
Ignoring vertex correctios, show that the Green function can be written in
the form
1
G(k, z) =
(5)
z − k + isgnIm(z)/(2τ )
with τ −1 = 2πni u20 the electron elastic scattering rate. In order to do so,
write down the lowest order disorder-averaged diagram for the selfenergy,
replace the integral over momenta by an integration over the density of
states. Argue that the real part of the selfenergy is unimportant, and that
the density of states can be replaced with a constant. What happens to the
scattering rate if you calculate the first order diagram for the selfenergy
selfconsistently?