1 CH. 9: LINEAR RESPONSE THEORY AND THE POLARIZABILITY 1. The general Kubo formula 2. Application: - Dielectric properties of charged systems - Polarizability of the interacting e-gas Ch. 5,6, and 8 Bruus and Flensberg 9.1. LINEAR RESPONSE THEORY We start discussing Kubo’s linear response theory Ch. 5,6, and 8 Bruus and Flensberg The linear response theory is based on the idea that the response of a system to a weak perturbation is proportional to the perturbation itself. We thus need a formula for the proportionality constant (cf. r ind,e = cf ext )
Transcript
1
CH. 9: LINEAR RESPONSE THEORYAND THE POLARIZABILITY
1. The general Kubo formula2. Application:
- Dielectric properties of charged systems- Polarizability of the interacting e-gas
Ch. 5,6, and 8 Bruus and Flensberg
9.1. LINEAR RESPONSE THEORY
We start discussing Kubo’s linear response theory
Ch. 5,6, and 8 Bruus and Flensberg
The linear response theory is based on the idea that the responseof a system to a weak perturbation is proportional to the perturbation itself. We thus need a formula for the proportionalityconstant (cf. rind,e = c fext)
2
HOW TO EVALUATE NON -EQUILIBRIUM AVERAGES ?
t0 t
Thermal equilibrium state Non-equilibrium state
External perturbation begins to actextH
The general question one needs to answer is thus: Given an external perturbation, (e.g. external electric or magnetic field), what is the expectation value of a given observable to linear order in ?
extHA A
extH
)()(ˆˆ)(ˆ0tttt ext0 HHH0HH ˆ)(ˆ t
?)(ˆˆ)(ˆ tAtA rTr 0ˆ1
00
ˆˆˆˆ HeAZAA r TrTr
THE GENERAL KUBO FORMULA
)()(ˆˆˆ0ttt ext0(t) HHH
Suppose now that at time t0 an external perturbation isapplied to the system, driving it out of equilibrium
?)(ˆˆ)(ˆ tAtA rTr
We consider as a weak perturbation, so that it isconvenient to work in deviations from equilibrium
extH
)ˆ()](ˆ,[)](,ˆ[)](ˆ),(ˆ[)(ˆ 02extext0 HHHH Ot
it
itt
it rrrr
)(ˆˆ)(ˆ 0 tt rrr Solve the Liouville-von Neumann Eq. for
3
THE GENERAL KUBO FORMULA II
0
)',('ˆ)(ˆ)(ˆ0 t
RA ttCdtAtAtA
extH
where
0
)'(ˆ),(ˆ)'()',( ttAtti
ttC IIRAHext ext,H
)(ˆ tA• The inherent non-equilibrium quantity is expressed asa retarded correlation function of the system in equilibrium
Kubo formula
/)'(ˆ
0/)'(ˆ
0
0
0 ˆ),'(ˆ')(ˆ ttit
t
tti etedti
t HH H rr ext
use cyclic invarianceof the trace
Comments:
• (t-t´)expresses the causality of the solution retarded correlation function
responsefunction
proof onnext page
THE GENERAL KUBO FORMULA III
)](ˆ,[1
)](ˆ,ˆ[1
)(ˆ 0 ttti ext0 HH rrr
The differential Eq. is solved by expressing the left hand side as
)](ˆ,[1
)(ˆ 0
ˆˆˆˆ0000
teetedt
die
ti
ti
ti
ti
extHHHHH
rr
or
)](ˆ,[1
)](ˆ,[1
)(ˆ ,0
ˆ
0
ˆˆˆ0000
teteetedt
di I
ti
ti
ti
ti
extext HHHHHH
rrr
)]'(ˆ,[')(ˆ)(ˆ 00
0
tdti
tt ,I
t
t
II extHrrr
Proof:
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KUBO FORMULA IN FREQUENCY DOMAIN
)(ˆ)(ˆ tfBt extH
0
)'()'(ˆ),(ˆ'ˆ)(ˆ)(ˆ00 t II tftBtAdt
iAtAtA
Consider the case in which
B Time-independent operator
f(t) c-number
)(~
)(~
)(ˆ)(~ˆ fCtAedtA R
ABti
)'()'(ˆ),(ˆ)'()',(0
ttCtBtAtti
ttC RABII
RAB
with
It then follows, for cyclic invarianceof the trace
,0 t
COMPLEX FREQUENCIES
)()(~
tCedtC RAB
tiRAB
0
)'(ˆ),(ˆ)'()'( tBtAtti
ttC IIRAB
In order for the Fourier transform to be well defined, the integrand must decay for both plus and minus infinity.
We note that the usual definition of the Fourier transform (FT) is
For retarded functions as
which are zero at negative times, only can pose a problem.
It is then usual to define the FT as
,)()(~
tCeedtC RAB
ttiRAB
positive infinitesimal 0
5
POSITION DEPENDENT PERTURBATION
When the external perturbation is position dependent,
),()(ˆ)(ˆ trfrBrdt
extH
one readily finds
),(~
)(~
)(~ˆ
)( rfCrdA RrAB
9.2 APPLICATION OF KUBO FORMULA:DIELECTRIC PROPERTIES
When dealing with systems containing charged particles, ase.g. the interacting e-gas, one is often interested in dielectricproperties, and in particular in the linear response properties
i) When such a system is subjected to an externalelectromagnetic perturbation, the charge is redistributed,and the system gets polarized.
ii) This in turn affects the measurement, an effect known asscreening
6
KUBO FORMULA FOR THE DIELECTRIC FUNCTION
The (nonlocal) dielectric function or permittivityyields the proportionality between the external and total potential:
)',';,( trtr
)','()',';,(''),( trtrtrdtrdtr
totext
Our purpose is to find assuming linear response theory:
External perturbation )(ˆ),(ˆ rtrrd
eextext rH
Induced charge: 0ˆˆ eeinde, rrr
)','()',('' )'()(0
trttCdtrd Rrrt
extinde, eer rr
0,,)'()( )','(ˆ),,(ˆ)'()',( trtrtt
ittC II
Rrr
eeee
rrrr charge-charge correlation function
DIELECTRIC FUNCTION II
)','()',';,(''),(0
trtrtrdtrdtr R
t
extinde, cr
0,,)'()( )','(ˆ),,(ˆ)'()',()',';,( trtrtt
ittCtrtr II
Rrr
R
eeeerrc rr
),'()'('),( trrrrdtr
indeeind r u
Once the induced charge is known, the induced potential follows:
)',''()','';,'()'(''''),(),(0
trtrtrrrdtrdrdtrtr tR
t
exeeexttot c
u
)',';,''()''('')'()'()',';,(1 trtrrrrdttrrtrtr R c eeu
polarizability function
Coulomb interaction
S.I.unitsee '4
1)'(
0 rrrr
u
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POLARIZABILITY OF TRANSLATIONAL INVARIANT SYSTEMS
)','()',';,( ttrrtrtr RR cc
Translational invariant systems in space and time
),(~),(~),(~ 1 qqq
exttot or ),(~),(~),(~ qqq
totext
with
),(~)(~1),(~ 1 c qqq R eeu
2ee
1
qq
0
)(~
u
POLARIZATION OF ELECTRON GAS
)','()',';,( ttrrtrtr RR ccE-gas:
')(
02,,
')'(
0
2,1,
0,,
)'(
2
2
21
21
)',(ˆ),,(ˆ)'(
)',(ˆ),,(ˆ)'(
)','(ˆ),,'(ˆ)'(
)'()'()',(~
rqqiII
q
rqirqirrqiII
qq
rqiII
rrqiRR
etqtqtti
eetqtqrdtti
etrtrrrdtti
errrrdttq
ee
ee2
ee
1
1
rr
rr
rr
cc
V
V
0,, )',(ˆ),,(ˆ)'()',(~ tqtqtt
ittq II
R
ee
1 rrcV
rrr
'
result must be independent of r´
8
POLARIZATION OF e-GAS II
0,, )',(ˆ),,(ˆ)'()',(~ tqtqtt
ittq II
R
ee
1 rrcV
),(~)(~1),(~ 1 c qqq R eeu
Hence, in general, for the e-gas in linear response to ext it holds:
Note that here refers to the Hamiltonian in the absence ofthe perturbation ext, but it can include e.g. the Coulombinteraction, if the interacting e-gas is considered.
0
DIELECTRIC LOSS FUNCTION
dielectric loss function
0,, )','(ˆ),,(ˆ)'()','( trtrtt
ittrr II
R
ee rrc
Note: A relation exists between the imaginary part of a correlation function and the excitations of the system.Explicitly seen using the Lehmann representation.
),(~Im)(~),(~Im 1 c qqq R eeu
9
9.3 POLARIZABILTIY OF FREE e-GAS
0,,0 )',(ˆ),,(ˆ)'()',(~ tqtqtt
ittq II
R
ee
1 rrcV
ti
qkk
kekk
kk
qkkeccetqcc)(
,,
,,,
,ˆˆ),(ˆˆˆˆ
r0H
c
,
)')((
')(
',',
)(
0',''',
0
)]()([)'(
ˆˆ,ˆˆ)'(
)',(~
''
k
tti
qkk
ti
kk
ti
qkkqkk
R
qkk
qkkqkk
effe
tti
eecccce
tti
ttq
V
V
2
2
','',''' ˆˆˆˆˆˆ,ˆˆ cccccccc )(ˆˆ
0 kkkfcc Fermi
function
For simplicity, we start to look to -1 to first order in uee, whichmeans that we evaluate for the non-interacting e-gas:
0 0~~ cc
POLARIZABILITY OF FREE e-GAS
c
,
)')((
0 )]()([)'()',(~
k
tti
qkk
R qkkeff
ett
ittq
V
2
c
,
,
)(
0
0
/)(
)]()([
)]()([),(~
k qkk
qkk
k
i
qkk
iR
i
ffe
eeffe
edi
q qkk
V
V2
2
Lindhard function
))/)(()]()([),(~Im,
0 c
qkk
kqkk
R ffe
qV
2
and
10
EXCITATIONS OF THE NON-INTERACTING e-GAS
))/)(()]()([),(~Im,
2
0 c
qkk
kqkk
R ffe
qV
yields informations about the excitations, i.e., we analyzefor which and isq
.0),(~Im c qR
Consider )()( kfk
F0 kT
0
0
0)()(
FF
FF
and
and
kk
kk
qkk
or
qkk
ifffqkk
i)
ii) ),(~Im),(~Im 00 cc qq RR Consider only > 0
EXCITATIONS OF THE NON-INTERACTING e-GAS II
iii) Hence:
FF
F
kqqvm
q
qvm
q
m
qk
m
q
2,2
22
0 2
min
2
max2
q/kF
/F
1
1 2
2hole
ejectedelectron
incomingphoton
, the range of possible excitations is
11
9.4 POLARIZABILITY WITHIN THE RPA
Dielectric properties of the interacting e-gascan only be described properly if one is able to treatinteraction effects to all orders in perturbation theory.
To this extent different approximation schemes have beendeveloped. Here we summarize some of the outcomes of thesecalculations:
• renormalization of the Coulomb interaction (screening)which removes the divergence at q=0.
• converging expression for the ground state energy• screening of external potentials• collective excitations (plasmons)
PREDICTION OF THE RANDOM PHASEAPPROXIMATION
One non-perturbative approximation scheme is the random phase approximation (RPA), which enables to select an infinite ensemble of relevant processes for the interacting e-gas in the high-density limit rs 0
),(~)(~),(~)(~),(~
)(~),(00
0
ccc
qqqq
qqq
RR
R
eeee
ee1
RPA 1
1
11
uuu
),(~ c qR RPA
• The expression for shows that it is a solution of a Dyson eq. with self-energy
• To first order in , the previous result for -1 is reproduced
• Poles of the denominator provide new excitations beside the e-h ones !!!
ee )(~ qu
),(~ c qR RPA
ee )(~ qu
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EVALUATION OF POLARIZATION
),(~0 c qR
Evaluation of
i) Start from the Lindhard form
c
i
ffdk
dke
i
ffeq
qkk
qkk
k qkk
qkkR
/)(
)]()([)2(
)2(
2
/)(
)]()([),(~
0
1
1
23
2
,
2
0 V
Low temperature limit FBTk
• Low T: )()( kfk
Fk
qkqmqkkqkk
22
22
cos
and observe
•
REAL PART
),(~Re 0 c qR ii) Focus on
• Define dimensionless frequencies, momenta:FF
xk
qx
4,
2 0
• Use standard logarithmic integrals
c
mkqqmkqqfdk
dke
q
k
k
R
F
2/)2(
1
2/)2(
1)(
2
),(~Re
220
1
1
22
2
0
P
])ln()[(1
)ln();ln(11
axbaxbaxa
baxdxbaxabax
dx
x
xxfxxfeq F
R
8
),(),(
2
1)(2),(~Re 002
0 c D
with0
20
22
00 ln1),(
xxx
xxxx
x
xxxf
13
IMAGINARY PART
)/)(()]()([)2()2(
),(~Im
0
1
1
23
0
c
qkkqkk
R
ffdkdke
q22
0,0
0,2
,18
)(
0
20
0
20
2
2
0
x
xxxx
x
xxxxxxx
x
x
F
other for
for
for
D
FF
xk
qx
4,
2 0
DIELECTRIC FUNCTION IN RPA
),(~)(~1),( 0 c qqq RRPA
eeu
Let us investigate the frequency dependent dielectric function
We look at:
FB
F
F
Tk
xkq
xxq
)1(
)(v 0
or
or high frequencies
long wavelength
low temperatures
,0),(~Im 0 c qR 2
2
2
2
0
v
5
31),(~Re e
qq
mq FR
c en
0
22
2
2
,v
5
311),(
m
eqq p
Fp eRPA
n
plasmafrequency
no dispersion
14
PLASMA OSCILLATIONS AND PLASMONS
The plasma frequency is an important parameterof the interacting e-gas lying in the ultraviolet range:
F
F
kq
qv
,
0
22
2
2
,v
5
311),(
m
eqq p
Fp eRPA
n
• Example: Aluminum
p Metal becomes transparent to radiation when
p • Incoming radiation with is reflected
p 162.27 10 Hz
• Plasma frequency is related to collective charge density oscillations, called plasmons. They are solution of the eq.
0),( q
RPA
PLASMA OSCILLATIONS AND PLASMONS II
22F
2
222 v
10
3)(
v
5
310),( qq
qq
pp
p
Fp
RPA
• Plasmons are solution of the equation 0),( q
RPA
Meaning, due to , that thesystem can sustain a finite tot even for vanishing small ext
),(~),(~),(~ 1 qqq
exttot
F
x
40
Fk
qx
2
1
plasmons
q/kF1 2
2
dampedplasmon
electron-holecontinuum
propagatingplasmon
0~Im 0 Rc
15
SUMMARY FOR INTERACTING e-GAS
),(
),(~~;
),(
)(~
),(~)(~1
)(~),(~
RPARPA0
c
k
k
k
k
kku
k k
R
extRPAtot,
ee
ee
eeRPA
uuu
1. Screening Renormalization of i) Coulomb interaction,ii) external potential
• static screening
• dynamic screening
0
2
002
)(1
)0,0(~,1)0,(
c eqk
q
kq F
R D2
TF
2TF
RPA
2. Collective (plasmons) + particle-hole excitations3. Lowering of ground state energy due to interactions
