Lecture
1:
Linear
Response
Theory
Last
semester
in
8.511,
we
discussed
linear
response
theory
in
the
context
of
charge
screening
and
the
freefermion
polarization
function.
This
theory
can
be
extended
to
a
much
wider
range
of
areas,
however,
and
is
a
very
useful
tool
in
solid
state
physics.
We’ll
begin
this
semester
by
going
back
and
studying
linear
response
theory
again
with
a
more
formal
approach,
and
then
returning
to
this
like
superconductivity
a
bit
later.
1.1
Response
Functions
and
the
Interaction
Representation
In
solid
state
physics,
we
ordinarily
think
about
manybody
systems,
with
something
on
the
order
of
10
23
particles.
With
so
many
particles,
it
is
usually
impossible
to
even
think
about
a
wave
function
for
the
whole
system.
As
a
result,
it
is
often
more
useful
for
us
to
think
in
terms
of
the
macroscopic
observable
behaviors
of
systems
rather
than
their
particular
microscopic
states.
One
example
of
such
a
macroscopic
property
is
the
magnetic
susceptibility
χ
H
≡
∂M
,
which
∂H
is
a
measure
of
the
response
of
the
net
magnetization
M
of
a
system
to
an
applied
magnetic
Feld
H
(
r,
t
).
This
is
the
type
of
behavior
we
will
be
thinking
about:
we
can
mathematically
probe
the
system
with
some
perturbing
external
probe
or
Feld
(
e.g.
H
(
r,
t
)),
and
try
to
predict
what
the
system’s
response
will
be
in
terms
of
the
expectation
values
of
some
observable
quantities.
Let
ˆ
H
be
the
full
manybody
Hamiltonian
for
some
isolated
system
that
we
are
interested
in.
We
spent
most
of
8.511
thinking
about
how
to
solve
for
the
behavior
of
a
system
governed
by
ˆ
H
.
As
interesting
as
that
behavior
may
be,
we
will
now
consider
that
to
be
a
solved
problem.
That
ˆ
is,
we
will
assume
the
existence
of
a
set
of
eigenkets
{
n
}
that
diagonalize
H
with
associated
eigenvalues
(energies)
E
n
.
In
addition
to
ˆ
H
,
we
now
turn
on
an
external
probe
potential
V
ˆ
,
such
that
the
total
Hamil
tonian
H
T
ot
satisFes:
ˆ
ˆ
H
T
ot
=
H
+
V
ˆ
(1.1)
In
particular,
we
are
interested
in
probe
potentials
that
arise
from
the
coupling
of
some
external
scalar
or
vector
Feld
to
some
sort
of
“density”
in
the
sample.
±or
example,
the
external
Feld
can
be
an
electric
potential
U
(
r,
t
),
which
couples
to
the
electronic
charge
density
ρ
ˆ(
r
)
such
that
ˆ
V
=
dr
ˆ
ρ
(
r
)
U
(
r,
t
)
(1.2)
V
where
the
electron
density
operator
ˆ
ρ
(
r
)
is
given
by
N
±
ˆ
ρ
(
r
)
=
δ
(
r
−
r
i
)
(1.3)
i
=1
1
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View Full DocumentResponse
Functions
and
the
Interaction
Representation
2
In
frst
quantized
language,
with
r
i
being
the
position
oF
electron
i
the
N
electron
system.
In
second
quantized
notation,
recall
ρ
ˆ(
r
)
=
Ψ
†
(
r
)
Ψ
(
r
)
(1.4)
where
Ψ
†
(
r
)
and
Ψ
(
r
)
are
the
electron
feld
creation
and
annihilation
operators,
respectively.
The
momentum
space
version
oF
the
electron
density
operator,
ˆ
ρ
(
q
),
is
related
to
ρ
r
)
through
the
±ourier
transForms:
i
ρ
r
) =
e
q
·
r
ρ
q
)
(1.5)
q
i
r
Ψ
(
r
) =
e
k
·
c
(1.6)
k
k
such
that
e
−
i
r
ρ
q
) =
q
·
(1.7)
r
=
c
†
c
(1.8)
k
−
k
q
k
Equation
(1.7)
is
the
frst
quantized
Form
oF
ˆ
ρ
(
q
),
and
equation
(1.8)
is
the
second
quantized
Form
with
c
†
the
creation
operator
For
an
electron
with
momentum
1
k
−
q
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 Fall '09
 PatrickLee
 Charge, Polarization, electron density, ˆr, linear response theory, interaction representation

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