Physics 582, Fall Semester 2008
Professor Eduardo Fradkin
Problem Set No. 5:
Due Date: November 9, 2008, 9:00 pm
1
Femions in one dimension
In this problem we will consider an application of the Dirac theory to a problem
in condensed matter physics: polyacetylene.
Polyacetylene is a long polymer
chain of the type (
CH
)
n
. The motion of the conduction electrons in polyacety
lene can be described by the following model due to Su, Schrieffer and Heeger.
In this model, one considers a linear chain of carbon atoms (
C
) with classical
equilibrium positions at the regularly spaced sites
{
x
(
n
)

x
(
n
) =
na
0
}
(where
a
0
is the lattice constant).
The carbon atoms
share
their
π
orbital electrons,
one per carbon atom. These electrons are allowed to hop from site to site. This
hopping process is
modulated
by the lattice vibrations. Since the mass
M
of the
atoms is
much larger
than the mass of the electrons (or, what is the same, the
tunneling hopping (kinetic) energy
t
of the electrons is
much larger
than the
kinetic energy of the atoms), we can give an
approximate
description by treat
ing the atoms classically while treating the electrons as quantum mechanical
objects. The Hamiltonian , for a lattice with
N
(even) sites, is
H
=
−
N
2
summationdisplay
n
=
−
N
2
+1
summationdisplay
σ
=
↑
,
↓
[
t
−
α
(
x
(
n
)
−
x
(
n
+ 1))]
bracketleftbig
c
†
σ
(
n
)
c
σ
(
n
+ 1) + h
.
c
.
bracketrightbig
+
N
2
summationdisplay
n
=
−
N
2
+1
bracketleftbigg
P
2
n
2
M
+
D
2
(
x
(
n
)
−
x
(
n
+ 1))
2
bracketrightbigg
(1)
where
c
†
σ
(
n
) and
c
σ
(
n
) are
fermion
operators which create and destroy a
π

electron with spin
σ
at the
n
th
site of the chain, the
x
(
n
)’s are the coordi
nates of the carbon atoms (measured from their classical equilibrium positions),
P
(
n
) are their momenta,
M
is the carbon mass,
D
is the elastic constant,
t
is
the electron hopping matrix element (for the undistorted lattice) and
α
is the
electronphonon coupling constant. Polyacetylene has one electron per carbon
atom and, hence, it is a
halffilled
system and there are
N
electrons in a chain
with
N
atoms.
The study of this problem is greatly simplified by considering a continuum
version of the model.
If the coupling constant
α
is not too large, the only
physical processes which are important are those which mix nearly degenerate
states,
i.e.,
the only electronic states that will matter are those within a narrow
1
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band of width 2
E
c
centered at the Fermi energy
E
F
= 0.
In this limit, the
single particle dispersion law becomes
E
(
p
)
≈
v
F
(
p
±
p
F
).
These states are
right moving
electrons ( with
p
≈
p
F
) and
left moving
electrons ( with
≈ −
p
F
).
Here
v
F
is the Fermi velocity. These considerations motivate the following way
of writing the electron operators
c
σ
(
n
) =
e
ip
F
n
R
σ
(
n
) +
e
−
ip
F
n
L
σ
(
n
)
(2)
Likewise, since the only processes in which phonons mix electrons near
±
p
F
have momentum
q
≈
0 (forward scattering) or
q
≈
2
p
F
(backward scattering),
it is also natural to split the phonon fields into two terms
x
(
n
) =
δ
(
n
) +
e
2
ip
F
n
Δ
+
(
n
) +
e
−
2
ip
F
n
Δ
−
(
n
)
(3)
where
p
F
=
π
2
a
0
.
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 Fall '08
 Leigh
 Physics, Energy, ground state, carbon atoms, Boundary conditions, ground state energy

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