7/28/2019 lec17.ps

1/3

Lecture XVII 49

Lecture XVII: Weakly Interacting Electron Gas: Plasma Theory

How are the properties of an electron gas influenced by weak Coulomb interaction?

Qualitative considerations:

When is the intereaction weak? Defining r0 =1

n1/3as the average electron separation,

the typical p.e. e2

r0and k.e.

2

mr0lead to the dimensionless ratio, rs =

e2

r0

mr202

r0a0

, where

a0 is electron Bohr radius, from which one can infer that Coulomb effects dominate atlow density

At rs 35 there is (believed to be) a transition to an electron solid phaseknown as a Wigner crystal (cf. Mott-Hubbard insulator)

For most metals (2 < rs < 6), k.e. and p.e. comparable; fortunately (thanks toadiabatic continuity) weak coupling theory valid even for intermediate rs

Motivates consideration of weak coupling theory rs 1: -convention on spin

H =

ddr c(r)

p2

2mc(r) +

1

2

ddr

ddrc(r)c

(r

)e2

|r r|c(r

)c(r)

Aim: to explore dielectric properties and ground state energy of electron gas through...

Quantum partition function: using CSPI formulation

Z tr e(HN) =

(0)=()(0)=()

D(, )eS[,]

S[, ] =

0

d

ddr (r, )

+

p2

2m

(r, )

+1

2

ddr

ddr(r, )(r

, )e2

|r r|(r

, )(r, )

Expressed in Fourier basis: (r, ) =1L3

k,n

ei(krn)k,n,

S =

0

d

k

k() ( + k ) k() +1

2Ld

q=0

4e2

q2q()q()

where k =2k2

2mand q() =

ddr eiqr(r, )

k k()k+q,()

(N.B. neutralising background exclusion of q = 0 from sum)

With the action quartic in fermionic fields , Z can not be evaluated exactly

For weak interaction, rs 1, we could expand in Coulomb interaction: Feynman diagram expansion (cf. Gell-MannBruckner theory)

Lecture Notes October 2005

7/28/2019 lec17.ps

2/3

Lecture XVII 50

Alternative use field integral to isolate leading diagrammatic series expansion

known as the Random Phase Approximation (RPA)

General principle:

When confronted with interacting field theory, seek decomposition of interactionthrough introduction of auxiliary field which captures low-energy content of theory

In some cases, these fields are identified with the elementary particles that mediatethe interaction (see below); in others, these fields encode the low-energy collective modesof the system (e.g. superfluid, superconductor)

Decoupling facilitated using the Hubbard-Stratonovich transformation:

eR0 d

Pq=0

2e2

Ldq2q()q() =

D e

R0 d

Pq=0

q2

8 q()q()+ie

2Ld/2(q()q()+q()q())

Physically, represents (scalar) photon field which mediates Coulomb interactionN.B. real and periodic ( + ) = ()

Z =

D(, )

D exp

0

d

ddr

1

8()2 +

+

p2

2m + ie

Gaussian in Grassmann fields, field integral may be performed:

using identity D[, ] exp[M ] = detM = exp[ln detM]Z =

D exp

0

d

ddr

1

8()2 +

spin2 ln det

+

p2

2m + ie

Setting e = 0, photon field decouples from determinant;

recovers partition function of non-interacting electron gas

Perturbation Theory in e:

Define free particle Green function: G0 = [ +p2

2m ]1

and expand:ln(1 + x) =

n=1(x)

n/n

lndet

+

p2

2m + ie

trln

G10 + ie

= tr ln G10 + tr ln

1 + ieG0

= tr ln G10 tr

ieG0 +

1

2

ieG0

2+

First order term: for convenience, set k (k, n), etc.

2tr[G0] = 2k

G0(k) k|G0|k

1L3k=0 k||k =

2L3

k

1

in + k 0 = 0

Lecture Notes October 2005

7/28/2019 lec17.ps

3/3

Lecture XVII 51

0 = 0 due to neutralising background

Second order term:

2 e2

2tr[G0]

2 = e2k,q

G0(k) k|G0|k

1L3

q k||k + q

G0(k + q) k + q|G0|k + q

1L3

q k + q||k=

e2

2

q

(q)qq

where density-density response function,

(q) =2

L3 k1

in + k

1

in im + k+q

Combined with bare term, to leading order in e2 (Random Phase Approximation),

Z = Z0

D eS[], S[] =

1

2

q

D1(q) q2

4 e2(q)

|q|

2 + O(e4)

Z0 denotes partition function of non-interacting gas

Physically, D1(q) denotes dynamically screened Coulomb interaction

D1(q) = (q)q2

4, (q) = 1

4e2

q2(q)

where (q) is the energy and momentum dependent effective dielectric function

mq,

4e 2

q2

k,n

m

n

+

+k+q,

mq,( )

=

-1

mq,=

=

++ ...

+

Diagrammatic interpretation:

D(q) =4

q21

1 4e2

q2(q)

=4

q2

n=0

e2(q)

4

q2

n

Lecture Notes October 2005