## 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

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)
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The Schrödinger equation arising from the Hamiltonian H sc is
(A.2)
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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)
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Let A denote the integral in the first line of this last equation. Integrating by parts, we find
(A.4)
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The commutator in the first line gives
(A.5)
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(p.362) The equality (A.5) is proved as follows. At t = 0 the interaction and Schrödinger representations coincide, so that

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The commutator in the last line is
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so that
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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)
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We now calculate the commutator occurring in eqn (A.6) to be
(A.7)
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(p.363) where
(A.8)
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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
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Substituting this last equation and eqn (A.7) into (A.6), we find
(A.9)
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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)
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We need a limit of the type
(A.11)
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Thus
(A.12)
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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)
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(p.365) The last two lines in the above equation do not contribute, due to the delta function δ (EE′), so that

(A.14)
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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)
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The matrix element of the current operator ĵβ(r) between the same two states is given by eqn (4.10) as
(A.16)
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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)
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Similarly, one finds
(A.18)
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which is eqn (4.144) in the text (see [90]).