7/28/2019 lec18.ps

1/3

Lecture XVIII 52

Lecture XVIII: Random Phase Approximation

Previously, we have seen that the quantum partition function of theweakly interacting electron gas can be written as eld integral

Z = Z 0 D e S [], S [] = 12 q=( m ,q )D

1

(q )

q 24 e2(q ) |q|2 + O(e4)where dielectric properties found to be controlled by density-density response function

(q ) =2

L 3k

1in k +

1in + im k + q +

To understand form of (q ), we have to digress and discuss

Matsubara Summations

Basic idea: by introducing auxiliary function g(z ) that has simple polesof strength unity at z = in , Cauchys theorem implies

n

f (in ) =1

2i C dz g(z )f (z )where contour C encloses only poles of g(z )

e.g. g(z ) =

exp(z ) 1, bosons

exp(z ) + 1

, fermions

Then, moving contour to innity

Poles at Matsubarafrequencies

z

c

z

Pole of f

z

n

f (in ) =

0

limR R2i 20 dg(Re i )f (Re i ) if simple

P

g(z P )f (z P )

12iP :f (zP )=0 dzg(z )f (z )

Lecture Notes October 2005

7/28/2019 lec18.ps

2/3

Lecture XVIII 53

Applied to (q ),

(q ) = 2

L 3k

g(k )im + k k + q

+g(k + q i 2m ) im k + k + q

=2

L3k

nF (k ) nF (k + q )im + k k + q

,

where nF () = 1e ( ) + 1is the Fermi distribution function

Finally, for |q | kF (2m)1/ 2 and kBT , k summation Lindhard function

(q ) 2 () 1 m

vF |q |tan 1

vF |q |m

where () is density of states at Fermi level

Static Limit: For |m | kF |q | /m , (0, q ) 2 (), i.e.

D(0, q ) 4e2

q 21

1 + 2 4e 2q 2 ()

Fourier transformed, static screened Coulomb interactione2

|r |e| r | / TF

where TF = 2 4e2 () Thomas-Fermi screening length

i.e. At long time scales (low frequencies), bare Coulomb interaction isrenormalised (screened) by collective charge uctuations

Physically, focusing a single electron, because it is negatively charged, other electrons will be repelled. As a result, a positively charged cloud of radius TF will form balancing the negative charge of the electron. When viewed from a distance larger than TF , the electron+cloud behaves as a neutral particle.

High Frequency Limit: For |n | kF |q |/m , (m , q ) q 2

m2mn,

where n = N/L 3 is the total number density (including spin)

D(q ) =4e2

q 21

1 + 4e 2 nm 2m

i.e. real time response ( im + i0) singular when p = 4 e2n/m Plasma frequency

In this case, there is a resonance which couples to the excitation mode where the pos-itively charged background and the negatively charged electrons are moving uniformly against each other.

Lecture Notes October 2005

7/28/2019 lec18.ps

3/3

Lecture XVIII 54

Ground State Energy

lim

Z e E g . s . .

In the RPA approximation, Z = Z 0 const .

det D 1/ 2 = Z 0 const .q

D(q) 1/ 2

i.e. E g.s. = E g.s.(e = 0) 1

2 q

ln D(q )

Lecture Notes October 2005