## Monique Combescot and Shiue-Yuan Shiau

Print publication date: 2015

Print ISBN-13: 9780198753735

Published to Oxford Scholarship Online: March 2016

DOI: 10.1093/acprof:oso/9780198753735.001.0001

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# (p.465) Appendix A Some Mathematical Results

Source:
Excitons and Cooper Pairs
Publisher:
Oxford University Press

# A.1 Kronecker symbol and delta function

The Kronecker symbol and the delta function are fundamentally the same: they both force the quantity at hand, whether quantized or continuous, to be equal. The associated prefactors however differ when switching from a discrete sum over k quantized in 2π/L to an integral

(A.1)
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These prefactors can be $(2π)D$, LD, or ND, where D is the space dimension, L is the sample size, and N is the number of terms in the 1D discrete sum. While these prefactors can be easily guessed from dimensional arguments, they of course follow from hard algebra.

## A.1.1 For R and Q quantized in a size L = Na sample

1. (i) For 1D vectors R = na and $Q=p(2π/L)$, with L = Na, and $(n,p,N)$ integers, the sum

(A.2)
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(p.467) reduces to N for Q = 0 while, for $Q≠0$,

(A.3)
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since $Q=p(2π/L)$. So,

(A.4)
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For 3D vectors, we split the R sum through $Q⋅R=QxRx+QyRy+QzRz$ and then use Eq. (A.4) for $NxNyNz=N3$. This gives

(A.5)
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with N3 replaced by N2 for 2D vectors. So, for space dimension D, we end with

(A.6)
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2. (ii) In the same way, for 1D vectors R = na and $Q=p(2π/L)$, with L = Na,

(A.7)
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The above sum reduces to N for R = 0 while, for $R≠0$, it reads $1−eiRN2π/L/1−eiR2π/L$. As the numerator cancels for R = na with n integer, the sum in Eq. (A.7) reduces to $NδR,0$, which extends in D dimensions as

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## A.1.2 For r continuous and Q quantized in 2π‎/L

1. (i) In 1D, the integral $∫0LdreiQr$ is equal to L for Q = 0, while for $Q≠0$, it reads $(eiQL−1)/iQ$, which cancels for Q quantized in 2π/L. This result extends in D dimensions as

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2. (ii) In 1D, the discrete sum

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(p.468) is equal to N for r = 0, while for $r≠0,$ it reads

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We then note that, for $|r|≪L=Na$,

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where $δa(r)=(πr)−1sin(πr/a)$ is a possible definition of the delta function when the width a goes to 0. By noting that $S(r=0)=N=L/a$ while $δa(0)=1/a$, we end with

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This result extends in D dimensions as

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## A.1.3 For r and Q continuous in an infinite sample

The delta distribution is mathematically defined, in 1D, through

(A.15)
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for any nonsingular function f(r). A possible representation of δ‎(r) is

(A.16)
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A simple way to show that the above integral has the (A.15) property is to add a convergence factor to the exponential. We then find that

(A.17)
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For η‎ very small, $η/(r2+η2)$ is very much peaked on r = 0; so, for f(r) nonsingular, we can replace $f(r)$ with f(0) in the product $f(r)2η/(r2+η2)$. We are left with

(A.18)
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which leads to Eq. (A.15) for δ‎(r) given by Eq. (A.16).

(p.469) Extension to D dimensions gives

(A.19)
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and, in the same way,

(A.20)
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# A.2 Fourier transform and series expansion

## A.2.1 Fourier transform

Let r be a continuous vector, and $|r〉$ be the set of states normalized by

(A.21)
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Their closure relation reads in D dimensions

(A.22)
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as seen from $I2=I$.

We now introduce the states $|q〉$ in the dual space, defined as

(A.23)
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They are normalized as

(A.24)
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(A.25)
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By writing the function $f(r)$ of the continuous variable r as $〈r|f〉$, we find that

(A.26)
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In the same way, the Fourier transform $fq$ of $f(r)$ reads in terms of $f(r)$ as

(A.27)
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(p.470) This compact approach to Fourier transform leads to an equal splitting of the prefactor $(1/2π)D$ between $f(r)$ and $fq$. It however fails to address the fundamental restriction on $f(r)$ or $fq$ convergence, as illustrated in Section A.3 for the Coulomb potential: $4πe2/q2$ decreases fast enough at large q to have a Fourier transform, but $e2/r$ does not.

## A.2.2 Series expansion

We now consider a function of the continuous variable r in a sample volume LD and with periodic boundary conditions, $φ(r)=φ(r+L)$. Because of this periodicity, we can restrict $φ(r)=〈r|φ〉$ to r inside the LD volume. The closure relation for the relevant $|r〉$ states then reads

(A.28)
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A possible complete basis for such periodic functions is made of $|Q〉$ states with wave function

(A.29)
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The states $|Q〉$ have the $L$ periodicity, $〈r|Q〉=〈r+L|Q〉$ for $eiQ⋅L=1$, that is, for Q quantized in 2π/L. These states are normalized as

(A.30)
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(A.31)
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as seen from $I2=I$.

The periodic function $φ(r)$ expands on the $|Q〉$ states as

(A.32)
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or in the dual space as

(A.33)
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For $φ(r)$ equal to a constant ϕ0, this series expansion has one term only, $φQ=δQ,0φ0LD/2$, while the average value of $φ(r)$ is related to $φQ=0$ as

(A.34)
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# (p.471) A.3 Coulomb scatterings

## A.3.1 Bulk systems

We here show that, in a 3D sample volume L3, the Coulomb scattering associated with nonzero momentum transfer Q reads

(A.35)
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Although commonly done, dropping the sample volume by setting $L3=1$ is unwise. Indeed, keeping the volume factor L3 is necessary to check homogeneity in the obtained results: potential scatterings must scale as an energy, that is, as $e2/[L]$. Keeping L3 also allows us to identify overextensive terms, which must cancel out exactly, and underextensive terms, which can be neglected in extensive quantities like the system energy.

Equation (A.35) follows from calculating

(A.36)
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The integral over q being equal to $π/2$, we end up with

(A.37)
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At this stage, we can note that discrete sums over Q quantized in 2π/L are commonly calculated by turning to an integral according to

(A.38)
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(A.39)
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with $VQ=4πe2/L3Q2$. However, this procedure is not fully satisfactory, because $4πe2/L3Q2$ is infinite for Q = 0. So, the Q = 0 term has to be excluded from the Q sum. This singularity can be directly seen from

(A.40)
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which diverges for Q = 0 because of the long-range character of Coulomb potential.

It is possible to give a meaning to Eq. (A.40) by restricting the integral to a finite volume, namely,

(A.41)
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(p.472) For Q = 0, this gives $V0(3)=2πe2/L$ while, for $Q≠0$, Eq. (A.37) leads to

(A.42)
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For q continuous, the integral over r gives $(2π)3δ(q−Q)$, as shown in Eq. (A.20). So, we end with

(A.43)
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in agreement with Eq. (A.35).

## A.3.2 Two–dimensional quantum wells

Addressing problems dealing with semiconductor quantum wells requires considering Coulomb scatterings in 2D. Following the same procedure, we first calculate

(A.44)
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The integral over θ‎ gives $(π/2)J0(qr)$, while the integral of the Bessel function $J0(x)$ from 0 to ∞ gives 1. So, we end up with

(A.45)
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Since the Fourier transform of $e2/r$ is infinite when calculated in an infinite volume, we introduce, as for 3D,

(A.46)
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For Q = 0, we again find that $V0(2)=2πe2/L$ while, for $Q≠0$, Eq. (A.45) leads to

(A.47)
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For q continuous, the integral over r gives $(2π)2δ(q−Q)$. So, we end with

(A.48)
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Note that the factors L3 or L2 makes the $VQ≠0(D)$ scatterings scale as $e2/[L]$; so, these scatterings are energy-like quantities, as required.