Universität Bonn
Physikalisches Institut
Prof. Dr. Johann Kroha, Johannes Rentrop and Ammar Nejati
10.12.2014
http://www.kroha.uni-bonn.de/teaching/lectures-and-seminars/wt2014/tcmp
Condensed Matter Theory I — WS14/15
Exercise 9
(Please return your solutions before 12:00 h Tue. 16.12.2014 [room PI 1.055].)
9.1 Green’s functions for phonons
(10 points)
Consider a one-dimensional chain of N atoms, each of mass M and coupled to one another via a
nearest-neighbour linear coupling of strength K. The Hamiltonian describing such a system is
"
#
N
X
p2j
1
2
H=
+ K(uj − uj−1 )
,
2M
2
j=1
where pj is momentum of the atom at site j and uj is the displacement of the atom at site j away
from the equilibrium position. As discussed in the lecture, the Hamiltonian can be diagonalized in
the form
X 1
1
2
(1)
pk p−k + M ωk uk u−k ,
H=
2M
2
k
where uk and pk are conjugate variables, and ωk is the dispersion for phonons.
a) Using the linear combination
s
~ †
b−k + bk
2M ωk
r
M ~ωk †
pk = i
b−k − bk ,
2
uk =
where b†k and bk are the phonon creation and annihilation operators respectively, write (1) in
the final, second quantized form
X
1
†
H=
~ωk bk bk +
.
2
k
Note that
[bk1 , b†k2 ]
= δk1 ,k2 .
b) We now consider the linear combinations
Ak := bk + b†−k
A†k := b†k + b−k .
Calculate the non-interacting phonon propagator Dk0 (ω) by obtaining
D n
oE
Dk0 (t − t0 ) ≡ − T Ak (t)A†k (t0 )
,
via the equations of motion and performing a Fourier transform with respect to t. You can
set t0 = 0 before the Fourier transform without loss of generality. Differentiate between the
cases t > 0 and t < 0.
8.2: Phonons in a metal: Kohn anomaly
(10 points)
In metals, it is expected that the harmonic interactions between the ions of a lattice are not only
between nearest neighbours, but long-ranged. The physical origin is that a displacement of an ionic
charge induces an electric charge polarization in the electron sea, that is long-ranged and oscillatory
in space. This oscillatory polarization then acts in its turn, on the other ions in the lattice.
Here, we consider a 1d linear chain of ions (with lattice constant a) immersed in an electron sea.
The force constant between an ion on the lattice site i and an ion on lattice site j is
κi−j = κ0
sin[2kF (i − j)a]
.
kF (i − j)a
a) Write down the classical Lagrangian for the system of ions and derive the equations of motion.
b) Make an ansatz of plane-wave solutions for the ion displacements, qj (t) = q0 exp[i(kxj − ωt)],
2
2
xj = ja, and derive an expression for ω(k)2 and ∂ω
∂k . Show that ω(k) has a cusp (divergent
slope) at k = 2kF . This effect is called the ‘Kohn anomaly’.
b) Draw the curve for ω(k).