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Quantum Transport in Mesoscopic SystemsComplexity and Statistical Fluctuations. A Maximum Entropy Viewpoint$

Pier A. Mello and Narendra Kumar

Print publication date: 2004

Print ISBN-13: 9780198525820

Published to British Academy Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198525820.001.0001

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(p.361) APPENDIX A THE THEOREM OF KANE–SEROTA–LEE

(p.361) APPENDIX A THE THEOREM OF KANE–SEROTA–LEE

Source:
Quantum Transport in Mesoscopic Systems
Publisher:
Oxford University Press

Equation (4.124a) gives the conductivity tensor in RPA, which we reproduce here as follows:

(A.1)
σ α β ω ( r , r ) = 1 ω lim η 0 + 0 dτe i ω τ η τ < [ j ˜ α ( r , τ ) , j ˜ β ( r , 0 ) ] > sc + i e 2 n sc ( r ) m ω δ ( r r ) δ α β .
The Schrödinger equation arising from the Hamiltonian H sc is
(A.2)
( E N M H sc ) | N M > = 0 ,
where |NM> denotes the Mth eigenstate with N electrons and E NM is the corresponding energy. As explained in the text, the thermodynamic limit has to be taken (in the present case, this means making our system infinitely long in the x direction) before the limit as η → 0 is evaluated. In that limit, M becomes a continuous index (perhaps in combination with other discrete indices); for instance, for plane-wave states M would represent the collection of continuous indices {k 1, k 2, …, k N}. For simplicity of notation, however, and with the above understanding, in what follows we write summations over the index M instead of the corresponding integrals.

Taking the divergence on both sides of eqn (A.1) and using the continuity equation (4.127), we have

(A.3)
α σ α β ω ( r , r ) = 1 ω lim η 0 + 0 dτe i ω τ η τ ( 1 ) τ < [ j ˜ α ( r , τ ) , j ˜ β ( r , 0 ) ] > sc + i e 2 m ω β [ n s c ( r ) δ ( r r ) ] .
Let A denote the integral in the first line of this last equation. Integrating by parts, we find
(A.4)
A = < [ ρ ˜ ( r , 0 ) , j ˜ β ( r , 0 ) ] > sc + ( i ω η ) 0 d τ e i ω τ η τ < [ ρ ˜ ( r , τ ) , j ˜ β ( r , 0 ) ] > sc .
The commutator in the first line gives
(A.5)
[ ρ ˜ ( r , 0 ) , j ˜ β ( r , 0 ) ] = i e 2 n ^ ( r ) m x β δ ( r r ) .

(p.362) The equality (A.5) is proved as follows. At t = 0 the interaction and Schrödinger representations coincide, so that

2 m e 2 [ ρ ˜ ( r , 0 ) , j ˜ β ( r , 0 ) ] = 2 m e 2 [ ρ ^ ( r ) , j ˜ β ( r ) ] = i j [ δ ( r r i ) , δ ( r r j ) p j β + p j β δ ( r r j ) ] = i j { δ ( r r i ) , [ δ ( r r j ) p j β ] + [ δ ( r r i ) , p j β ] δ ( r r j ) } .
The commutator in the last line is
[ δ ( r r i ) , p j β ] = [ p j β , δ ( r r i ) ] = i x j β δ ( r r i ) = i δ i j x j β δ ( r r i ) = i δ i j x β δ ( r r i ) ,
so that
2 m e 2 [ ρ ˜ ( r , 0 ) , j ˜ β ( r , 0 ) ] = i i δ [ ( r r i ) δ ( r r i ) x β + δ ( r r i ) x β δ ( r r i ) ] = 2 i [ i δ ( r r j ) ] δ ( r r ) x β ,
from which the desired equality (A.5) follows.

The right-hand side of eqn (A.5), and thus the first term of (A.4), is purely imaginary, so that taking the real part of eqn (A.3) we find

(A.6)
Re [ α σ α β ω ( r , r ) ] = Re [ lim η 0 + i ω η ω 0 d τ e i ω τ η τ < [ ρ ˜ ( r , τ ) , j ˜ β ( r , 0 ) ] > sc ] .
We now calculate the commutator occurring in eqn (A.6) to be
(A.7)
< [ ρ ˜ ( r , τ ) , j ˜ β ( r , 0 ) ] > sc = < ρ ˜ ( r , τ ) , j ˜ β ( r , 0 ) j ˜ β ( r , 0 ) , ρ ˜ ( r , τ ) > sc = N M N M P ( N , E N M ) × [ < N M | e ( i/ ) H sc τ ρ ^ ( r ) e ( i/ ) H sc τ | N M > < N M | j ^ β ( r ) | N M > < N M | j ^ β ( r ) | N M > < N M | e ( i/ ) H sc τ ρ ^ ( r ) e ( i/ ) H sc τ | N M > ] = N M M P ( N , E N M ) × [ e ( i/ ) ( E N M E N M ) τ < N M | ρ ^ ( r ) | N M > < N M | j ^ β ( r ) | N M > e ( i/ ) ( E N M E N M ) τ < N M | j ^ β ( r ) | N M > < N M | ρ ^ ( r ) | N M > ] ,
(p.363) where
(A.8)
P ( N , E N M ) = e β ( E N M μ N ) Z ( β , μ )
denotes the statistical mechanical weight of the state |NM>. We have used the fact that ρ^(r) and ĵβ(r′) conserve the particle number. In eqn (A.6) we also need an integral of the type
0 d τ e i Ω τ η τ = i Ω +i η .
Substituting this last equation and eqn (A.7) into (A.6), we find
(A.9)
Re [ α σ α β ω ( r , r ) ] = Re { lim η 0 + i ω ( i ω η ) N M M P ( N , E N M ) × [ < N M | ρ ^ ( r ) | N M > < N M | j ^ β ( r ) | N M > ω + ( i / ) ( E N M E N M ) + i η < N M | j ^ β ( r ) | N M > < N M | ρ ^ ( r ) | N M > ω + ( i / ) ( E N M E N M ) + i η ] } = Re { lim η 0 + ω + i η ω N M M [ P ( N , E N M ) P ( N , E N M ) ] × < N M | ρ ^ ( r ) | N M > < N M | j ^ β ( r ) | N M > ω + E N M E N M + i η } .
We will find it convenient to introduce, in the right-hand side of this last equation, two energy integrals of the form (p.364)
(A.10)
Re [ α σ α β ω ( r , r ) ] = Re { lim η 0 + ω +i η ω N M M d E d E δ ( E E N M ) δ ( E E N M ) × P ( N , E ) P ( N , E ) ω + E E + i η < N M | p ^ ( r ) | N M > < N M | j ^ β ( r ) | N M > } .
We need a limit of the type
(A.11)
lim η 0 + ω +i η Ω + i η = lim η 0 + ( ω +i η ) Ω i η Ω 2 + η 2 = lim η 0 + ( ω +i η ) [ Ω i η Ω 2 + η 2 i π η / π Ω 2 + η 2 ] = ω [ p 1 Ω i π δ ( Ω ) ] .
Thus
(A.12)
Re [ α σ α β ω ( r , r ) ] = Re N M M d E d E δ ( E E N M ) δ ( E E N M ) × [ P ( N , E ) P ( N , E ) ] [ p ω + E E i π δ ( ω + E E ) ] × < N M | p ^ ( r ) | N M > < N M | j ^ β ( r ) | N M > .
In the d.c. limit, i.e., as ω → 0 (with the understanding that the thermodynamic limit has been taken first, even before the η → 0+ limit, so that, according to the comment made immediately after eqn (A.2), the summation ∑MM′ really stands for an integral over continuous indices), we have
(A.13)
lim η 0 + Re [ α σ α β ω ( r , r ) ] = Re N M M d E d E δ ( E E N M ) δ ( E E N M ) × p P ( N , E ) P ( N , E ) E E < N M | p ^ ( r ) | N M > < N M | j ^ β ( r ) | N M > + Re N M M i π d E d E δ ( E E N M ) δ ( E E N M ) δ ( E E ) × [ P ( N , E ) P ( N , E ) ] < N M | p ^ ( r ) | N M > < N M | j ^ β ( r ) | N M > .

(p.365) The last two lines in the above equation do not contribute, due to the delta function δ (EE′), so that

(A.14)
lim η 0 + Re [ α σ α β ω ( r , r ) ] = Re N M M d E d E δ ( E E N M ) δ ( E E N M ) × p P ( N , E ) P ( N , E ) E E < N M | p ^ ( r ) | N M > < N M | j ^ β ( r ) | N M > .

The matrix element of the electron density operator ρ^(r) between the states Φ(r 1, …, r N) and Ψ(r 1, …, r N) is

(A.15)
( Φ , p ^ ( r ) Ψ ) = i = 1 N d 3 r 1 d 3 r N δ ( r i r ) Φ * ( r 1 , , r N ) Ψ ( r 1 , , r N ) .
The matrix element of the current operator ĵβ(r) between the same two states is given by eqn (4.10) as
(A.16)
( Φ , j ^ β ( r ) Ψ ) = e 2 i m i = 1 N d 3 r 1 d 3 r N δ ( r i r ) × { Φ * ( r 1 , , r N ) [ x i β Ψ ( r 1 , , r N ) ] [ x i β Φ * ( r 1 , , r N ) ] Ψ ( r 1 , , r N ) } .
We now assume that the system under study is time-reversal invariant. Thus, we can always find a basis of real eigenfunctions of the Hamiltonian H sc. Equations (A.15) and (A.16) then show that the matrix elements of the density and the current operators between two such eigenstates are real and purely imaginary, respectively. Thus the right-hand side of eqn (A.14) vanishes and, as a result,
(A.17)
lim ω 0 Re [ α σ α β ω ( r , r ) ] = 0.
Similarly, one finds
(A.18)
lim ω 0 Re [ α σ α β ω ( r , r ) ] = 0 ,
which is eqn (4.144) in the text (see [90]).