Quantum Scattering Theory and Applications

Home , Cross section (physics), Scattering, Scattering length, Scattering theory

A thesis presented

Adam LupuSax

The DepartmentofPhysics

in partial fulllment of the requirements

for the degree of

Do ctor of Philosophy

in the sub ject of

Physics

Harvard University

Cambridge Massachusetts

September

Adam LupuSax

Abstract

Scattering theory provides a convenient framework for the solution of a varietyof

problems In this thesis we fo cus on the combination of b oundary conditions and scattering

potentials and the combination of nonoverlapping scattering p otentials within the context

of scattering theory Using a scattering tmatrix approach we derivea useful relationship

between the scattering tmatrix of the scattering potential and the Green function of the

b oundary and the tmatrix of the combined system eectively renormalizing the scatter

ing tmatrix to account for the b oundaries In the case of the combination of scattering

potentials the combination of tmatrix op erators is achieved via multiple scattering the

ealsoderive metho ds primarily for numerical use for nding the Green function of ory W

arbitrarily shap ed b oundaries of various sorts

These metho ds can b e applied to b oth op en and closed systems In this thesis we

consider single and multiple scatterers in two dimensional strips regions which are innite

in one direction and b ounded in the other as well as two dimensional rectangles In D

strips both the renormalization of the single scatterer strength and the conductance of

disordered manyscatterer systems are studied For the case of the single scatterer we see

nontrivial renormalization eects in the narrow wire limit In the many scatterer case

wenumerically observe suppression of the conductance beyond that which is explained by

weak lo calization

In closed systems we fo cus primarily on the eigenstates of disordered many

scatterer systems There has b een substantial investigation and calculation of prop erties of

the eigenstate intensities of these systems We have for the rst time b een able to inves

tigate these questions numerically Since there is little exp erimental work in this regime

these numerics provide the rst test of various theoretical mo dels Our observations indicate

that the probability of large uctuations of the intensity of the wavefunction are explained

qualitatively byvarious eldtheoretic mo dels However quantitatively no existing theory

accurately predicts the probability of these uctuations

Acknowledgments

Doing the work which app ears in this thesis has been a largely delightful way to

sp end the last veyears The nancial supp ort for my graduate studies was provided bya

National Science Foundation Fellowship Harvard University and the HarvardSmithsonian

Institute for Theoretical Atomic and Molecular Physics ITAMP Together all of these

sources provided me with the wonderful opp ortunity to study without b eing concerned

ab out my nances

My advisor Rick Heller is a wonderful source of ideas and insights I b egan

working with Rick four years ago and I have learned an immense amount from him in that

One time From the very rst time wespokeIhave felt not only challenged but resp ected

particularly nice asp ect of having Rick as an advisor is his ready availability More than one

tricky part of this thesis has b een sorted out in a marathon conversation in Ricks oce I

cannot thank him enough for all of his time and energy

In the last veyearsIhave had the great pleasure of working not only with Rick

himself but also with his p ostdo cs and other students Maurizio Carioli was a p ostdo c

when I b egan working with Rick There is much I cannot imagine having learned so quickly

or so well without him particularly ab out numerical metho ds Lev Kaplan a student and

then p ostdo c in the group is an invaluable source of clear thinking and uncanny insight

He has also demonstrated a nearly innite patience in discussing our work My classmate

Neepa Maitra and I b egan working with Rick at nearly the same time and have b een partners

in this journey Neepas emotional supp ort and p erceptive comments and questions ab out

substantially easier Alex Barnett Bill Bies Greg my work have made my last ve years

Fiete Jesse Hersch Bill Hosten and Areez Mo dy all graduate students in Ricks group have

given me wonderful feedbackonthis and other work The substantial p ostdo c contingent

in the group Michael Haggerty Martin Naraschewski and Doron Cohen have b een equally

helpful and provided very useful guidance along the way

At the time I b egan graduate scho ol I was pleasantly surprised by the co op erative

spirit among my classmates Manyofusspentcountless hours discussing physics and sorting

out problem sets Among this crowd I must particularly thank Martin Bazant Brian

Busch Sheila Kannappan Carla Levy Carol Livermore Neepa Maitra Ron Rubin and

Glenn Wong for making many late nights b earable and oftentimes fun Imust particularly

thank Martin Carla and Neepa for remaining great friends and colleagues in the years that

followed Ihave had the great fortune to make go o d friends at various stages in my life and

I am honored to count these three among them

It is hard to imagine howIwould have done all of this without my ancee Kiersten

Conner Our up coming marriage has been a singular source of joy during the pro cess of

writing this thesis Her unagging supp ort and b oundless sense of humor have kept me

centered throughout graduate scho ol

My parents Chip Lupu and Jana Sax have both been a great source of supp ort

and encouragement throughout my life and the last veyears have b een no exception The

rest of my family has also b een very supp ortive particularly my grandmothers Sara Lupu

and Pauline Sax and my stepmother Nancy Altman It saddens me that neither of my

grandfathers Dave Lupu or N Irving Sax are alive to see this moment in my life but I

thank them b oth for teaching me things that have help ed bring me this far

Citations to Previously Published Work

Portions of chapter and App endix B have app eared in

Quantum scattering from arbitrary b oundaries MGE da Luz AS LupuSax

and EJ Heller Physical Review B no pages

Contents

Title Page

Abstract

Acknowledgments

Citations to Previously Published Work

Table of Contents

List of Figures

List of Tables

Intro duction and Outline of the Thesis

Intro duction

Outline of the Thesis

Quantum Scattering Theory in dDimensions

CrossSections

Unitarit y and the Optical Theorem

Green Functions

Zero Range Interactions

Scattering in two dimensions

Scattering in the Presence of Other Potentials

Multiple Scattering

Renormalized tmatrices

Scattering From Arbitrarily Shap ed Boundaries

Intro duction

Boundary Wall Metho d I

Boundary Wall Metho d I I

Perio dic Boundary Conditions

Green Function Interfaces

Numerical Considerations and Analysis

From Wavefunctions to Green Functions

Eigenstates

Contents

Scattering in Wires I One Scatterer

One Scatterer in a Wide Wire

The Green function of an empty p erio dic wire

Renormalization of the ZRI Scattering Strength

From the Green function to Conductance

Computing the channeltochannel Green function

One Scatterer in a Narrow Wire

Scattering in Rectangles I One Scatterer

Dirichlet b oundaries

Perio dic b oundaries

Disordered Systems

Disorder Averages

Mean Free Path

Prop erties of Randomly Placed ZRIs as a Disordered Potential

Eigenstate Intensities and the PorterThomas Distribution

Weak Lo calization

Strong Lo calization

Anomalous Wav efunctions in Two Dimensions

Conclusions

Quenched Disorder in D Wires

Transp ort in Disordered Systems

Quenched Disorder in D Rectangles

Extracting eigenstates from tmatrices

Intensity Statistics in Small Disordered Dirichlet Bounded Rectangles

Intensity Statistics in Disordered Perio dic Rectangle

Algorithms

Conclusions

Bibliography

A Green Functions

A Denitions

A Scaling L

A Integration of Energy Green functions

A Green functions of separable systems

A Examples

A The Gorkov bulk sup erconductor Green Function

B Generalization of the Boundary Wall Metho d

Contents

C Linear Algebra and NullSpace Hunting

C Standard Linear Solvers

C Orthogonalization and the QR Decomp osition

C The Singular Value Decomp osition

D Some imp ortant innite sums

D Identites from

D Convergence of Green Function Sums

E Mathematical Miscellany for Two Dimensions

E Polar Co ordinates

E Bessel Expansions

E Asymptotics as kr

E Limiting Form for Small Arguments kr

List of Figures

Transmission at normal incidence through a at wall via the Boundary Wall

metho d

APerio dic wire with one scatterer and an incident particle

Exp erimental setup for a conductance measurement The wire is connected

to ideal contacts and the voltage drop at xed current is measured

Reection co ecient of a single scatterer in a wide p erio dic wire

Numb er of scattering channels blo cked by one scatterer in a p erio dic wire of

varying width

Transmission co ecient of a single scatterer in a narrow p erio dic wire

CrossSection of a single scatterer in a narrow p erio dic wire

Comparison of Dressedt Theory with Numerical Simulation

PorterThomas and exp onential lo calization distributions compared

PorterThomasexp onential lo calization and lognormal distributions com

pared

Comparison of lognormal co ecients for the DOFM and SSSM

The wire used in op en system scattering calculations

Numerically observed mean free path and the classical exp ectation

Numerically observed mean free path after rstorder coherent backscattering

correction and the classical exp ectation

Transmission versus disordered region length for a diusive and b lo cal

ized wires

Typical low energy wavefunctions j j is plotted for scatterers in a

Dirichlet b ounded square Blackishighintensity white is low The scatterers

are shown as black dotsFor the top left wavefunction whereas

for the b ottom wavefunction increases from left to right

and top to b ottom whereas decreases in the same order

List of Figures

Typical medium energy wavefunctions j j is plotted for scatterers in a

Dirichlet b ounded square Black is high intensity white is low The

scatterers are shown as black dots For the top left wavefunction

whereas for the b ottom wavefunction increases from

left to right and top to b ottom whereas decreases in the same order

Intensity statistics gathered in various parts of a Dirichlet b ounded square

Clearly larger uctuations are more likely in at the sides and corners than in

the center The statistical error bars are dierent sizes b ecause four times

as muchdatawas gathered in the sides and corners than in the center

Intensit y statistics gathered in various parts of a Perio dic square torus

Larger uctuations are more likely for larger The erratic nature of the

smallest wavelength data is due to p o or statistics

Illustrations of the tting pro cedure We lo ok at the reduced as a function

of the starting value of t in the t top notice the logscale on the yaxis

then cho ose the C with smallest condence interval b ottom and stable

reduced In this case wewould cho ose the C from the t starting at t

Numerically observed lognormal co ecients tted from numerical data and

tted theoretical exp ectations plotted top as a function of wavenumb er k

at xed and b ottom as function of at xed k

Typical wavefunctions j j is plotted for scatterers in a p erio dic

square torus with The densityofj j is shown

Anomalous wavefunctions j j is plotted for scatterers in a p erio dic

square torus with The density of j j is shown We

note that the scale here is dierent from the typical states plotted previously

The average radial intensity centered on two typical peaks top and two

anomalous p eaks b ottom

Wavefunction deviation under small p erturbation for scatterers in a

p erio dic square torus

List of Tables

Comparison of lognormal tails of P t for dierentmaximum allowed singu

lar value

Comparison of lognormal tails of P t for strong and weak scatterers at xed

and

C Matrix decomp ositions and computation time

Chapter

Intro duction and Outline of the

Thesis

Intro duction

Scattering evokes a simple image We b egin with separate ob jects which are far

apart and moving towards each other After some time they collide and then travel away

from each other and eventually are far apart again We dont necessarily care ab out the

details of the collision except insofar as we can predict from it where and how the ob jects

will end up This picture of scattering is the rst one wephysicists learn and it is a b eautiful

example of the p ower of conservation laws

In many cases the laws of conservation of momentum and energy alone can b e

used to obtain imp ortant results concerning the prop erties of various mechanical

pro cesses It should be noted that these prop erties are indep endent of of the

particular typ e of in teraction b etween the particles involved

LD Landau Mechanics

Quantum scattering is a more subtle aair Even elastic scattering which do es

not change the internal state of the colliding particles is more complicated than its classical

counterpart

In classical mechanics collisions of two particles are entirely determined by

their velo cities and impact parameter the distance at whichtheywould pass if

they did not interact In quantum mechanics the very wording of the problem

must b e changed since in motion with denite velo cities the concept of the path

Chapter Introduction and Outline of the Thesis

is meaningless and therefore so is the impact parameter The purp ose of the

theory is here only to calculate the probability that as a result of the collision

the particles will deviate or as wesaybescattered through anygiven angle

LD Landau Quantum Mechanics

This socalled dierential crosssection the probability that a particle is scat

tered through a given angle is the very b eginning of any treatment of scattering whether

classical or quantum mechanical

However the crosssection is not the part of scattering theory up on which we

intend to build It is instead the separation between free propagation motion without

interaction and collision That this idea should lead to so much useful physics is at rst

surprising However the Schrodinger equation likeany other wave equation do es not make

this split particularly obvious It is indeed some worktorecover the b enets of this division

from the complications of wavemechanics

In fact the idea of considering separately the free or unp erturb ed motion of par

ticles and their interaction is usually considered in the context of p erturbation theory

Unsurprisingly then the very rst quantum mechanical scattering theory was Borns per

turbative treatment of scattering which he develop ed not to solve scattering problems

but to address the completeness of the new quantum theory

Heisenb ergs quantum mechanics has so far b een applied exclusively to the calcu

lation of stationary states and vibration amplitudes asso ciated with transitions

to the fact that all diculties of principle Bohr has already directed attention

asso ciated with the quantum approacho ccur in the interactions of atoms at

short distances and consequently in collision pro cesses I therefore attack the

problem of investigating more closely the interaction of the free particle ray

or electron and an arbitrary atom and of determining whether a description of

a collision is not p ossible within the framework of existing theory

M Born On The Quantum Mechanics of Col lisions

Later in the same note the connection to p erturbation theory is made clear One can then

show with the help of a simple p erturbation calculation that there is a uniquely determined

solution of the Schrodinger equation with a p otential V

Scattering theory has develop ed substantially since Borns note app eared Still

we will take great advantage of one common feature of p erturbations and scattering The

Chapter Introduction and Outline of the Thesis

division b etween p erturbation and unp erturb ed motion is one of denition not of physics

Much of the art in using p erturbation theory comes from recognizing just what division of

the problem will giveasolvable unp erturb ed motion and a convergent p erturbation series

In scattering the division between free motion and collision seems much more

natural and less exible However many of the metho ds develop ed in this thesis take

advantage of what little exibility there is in order to solve some problems not traditionally

in the purview of scattering theory as well as attack some which are practically intractable

by other means

Outline of the Thesis

In chapter we b egin with a nearly traditional development of scattering theory

The development deviates from the traditional only in that it generalizes the usual de

nitions and calculations to arbitrary spatial dimension This is done mostly b ecause the

applications in the thesis require two dimensional scattering theory but most readers will b e

familiar with the three dimensional version A generalized derivation allows the reader to

assume d andcheck that the results are what they exp ect and then use the d version

when necessary Ihaveasmuch as p ossible followed standard textb o ok treatments of each

piece of scattering theory I am condent that the ddimensional generalizations presented

here exist elsewhere in the literature For instance work using socalled hyp erspherical

co ordinates to solve fewb o dy problems certainly contains much of the same information

though p erhaps not in the same form

The nal two sections of chapter are a bit more sp ecic The rst section

deals with zero range interactions a to ol which will b e used almost constantly throughout

the remainder of the thesis It is our hop e that the treatment of the zero range interaction

in this section is considerably simpler than the various formalisms which are typically used

After this section follows a short section explicitly covering some details of scattering in two

dimensions

ve on to the central theoretical work in After this intro ductory material we mo

scattering theory Chapter covers two extensions of ordinary scattering theory The rst

is multiple scattering theory A system with two or more distinct scatterers can b e handled

by solving the problem one scatterer at a time and then combining those results This

is a nice example of the usefulness of the split between propagation and collision made

Chapter Introduction and Outline of the Thesis

ab ove Multiple scattering theory takes this split and some very clever b o okkeeping and

solves a very complex problem Our treatment diers somewhat from Fadeevs in order to

emphasize similarities with the techniques intro duced in section

A separation b etween free propagation and collision and its attendant b o okkeeping

have more applications than multiple scattering In section wedevelop the central new

theoretic to ol of this work the renormalized tmatrix In multiple scattering theory we

used the separation b etween propagation and collision to piece together the scattering from

multiple targets in essence complicating the collision phase With appropriate renormal

ization we can also change what wemeanby propagation We derive the relevant equations

and sp end some time exploring the consequences of the transformation of propagation The

sort of change wehave in mind will b ecome clearer as we discuss the applications

Both of the metho ds explained in chapter inv olve combining solved problems

and thus solving a more complicated problem The techniques discussed in chapter are

used to solve some problems from scratch In their simplest form they have b een applied

to mesoscopic devices and it is hop ed that the more complex versions might b e applied to

lo ok at dirty and clean sup erconductor normal metal junctions

We b egin working on applications in chapter where we explore our rst nontrivial

example of scatterer renormalization the change in scatterer strength of a scatterer placed in

a wire We b egin with a xed twodimensional zero range interaction of known scattering

amplitude We place this scatterer in an innite straight wire channel of nite width

Both the scatterer in free space and the wire without scatterer are solved problems Their

combination is more subtle and brings to b ear the techniques develop ed in Much of the

t on necessary applied mathematics but it concludes with the interesting chapter is sp en

case of a wire which is narrower than the crosssection of the scatterer which has zerorange

socantinany nite width wire This calculation could b e applied to a variety of systems

hydrogen conned on the surface of liquid helium for one

Next in chapter we treat the case of the same scatterer placed in a completely

closed b ox While a wire is still op en and so a scattering problem it is at rst hard to imagine

how a closed system could b e After all the dierential crosssection makes no sense in a

closed system Wonderfully the equations develop ed for scattering in op en systems are still

valid in a closed one and give in some cases very useful metho ds for examining prop erties

of the closed system As with the previous chapter much of the work in this chapter is

preliminary but necessary applied mathematics Here we rst confront the o ddity of using

Chapter Introduction and Outline of the Thesis

the equations of scattering theory to nd the energies of discrete stationary states With

only one scatterer and renormalization this turns out to b e mathematically straightforward

Still this idea is imp ortant enough to the sequel that we do numerical computations on

the case of ground state energies of a single scatterer in a rectangle with p erfectly reective

walls Using the metho ds presented here this is simply a question of solving one nonlinear

equation We compare the energies so calculated to numerically calculated ground state

energies of hard disks in rectangles computed with a standard numerical technique This

is intended b oth as conrmation that we can extract discrete energies from these metho ds

and as an illustration of the similarity between isolated zero range interactions and hard

disks

Having sp ent a substantial amount of time on examples of renormalization we

return multiple scattering to the picture as well We will consider in particular disordered

sets of xed scatterers motivated for example byquenched impurities in a metal Before

we apply these techniques to disordered systems we consider disordered systems themselves

in chapter Here we dene and explain some imp ortant concepts which are relevant to

disordered systems as well as discuss some theoretical predictions ab out various prop erties

of disordered systems

We return to scattering in a wire in chapter Instead of the single scatterer of

chapter we now place many scatterers in the same wire and consider the conductance of

the disordered region of the wire We use this to examine weak lo calization a quantum

eect present only in the presence of timereversal symmetry In the nal chapter we use

the calculations of this chapter as evidence that our disorder p otential has the prop erties we

would predict from a hard disk mo del as we explored for the one scatterer case in chapters

and

Our nal application is presented in chapter Here we examine some very sp ecic

prop erties of disordered scatterers in a rectangle These calculations were in some sense the

original inspiration for this work and are its most unique achievement Here calculations

are p erformed which are apparently out of reach of other numerical metho ds These

calculations b oth conrm some theoretical exp ectations and confound others leaving a rich

set of new questions At the same time it is also the most sp ecialized application we

consider and not one with the broad applicability of the previous applications

t some conclusions and ideas for future extensions of the In chapter we presen

ideas in this work This is followed after the bibliography byavarietyoftechnical app en

Chapter Introduction and Outline of the Thesis

dices which are referred to throughout the b o dy of the thesis

Chapter

Quantum Scattering Theory in

dDimensions

The metho ds of progress in theoretical physics have undergone a vast change

during the present century The classical tradition has been to consider the

world to be an asso ciation of observable ob jects particles uids elds etc

moving ab out according to denite laws of force so that one could form a mental

picture in space and time of the whole scheme This led to a physics whose aim

was to make assumptions ab out the mechanism and forces connecting these

observable ob jects to account for their b ehavior in the simplest p ossible way It

has b ecome increasingly evidentinrecenttimeshowever that nature works on

a dierent plan Her fundamental laws do not govern the world as it app ears in

our mental picture in an yvery direct way but instead they control a substratum

of whichwe cannot form a mental picture without intro ducing irrelevancies

PAM Dirac Quantum Mechanics

Nearly all physics exp eriments measure the outcome of scattering events This

ubiquity has made scattering theory a crucial part of any standard quantum text Not

surprisingly all the attention given to scattering pro cesses has led to the invention of very

powerful theoretical to ols many of which can b e applied to problems which are not tradi

tional scattering problems

After this chapter our use of scattering theory will involve mostly nontraditional

uses of the to ols of scattering theory However those to ols are so imp ortant to what follows

that wemust provide at least a summary of the basic theory

of quantum me There are nearly as many approaches to scattering as authors

chanics textb o oks As is typical we begin by dening the problem and the idea of the

Chapter Quantum Scattering Theory in dDimensions

scattering crosssection We then make the somewhat lengthy calculation which relates the

dierential crosssection to the potential of the scatterer We p erform this calculation for

arbitrary spatial dimension

At rst this mayseemlikemorework than necessary to review scattering theory

However in what follows we will frequently use two dimensional scattering theory While

we could have derived everything in two dimensions wewould then have lost the reassuring

feeling of seeing familiar three dimensional results The arbitrary dimension derivation gives

us b oth

We pro ceed to consider the consequences of particle conservation or unitarity and

derive the ddimensional optical theorem It is interesting to note that for b oth this calcula

tion and the previous one the dimensional dep endence enters only through the asymptotic

expansion of the plane wave

Once we have this machinery in hand we pro ceed to discuss p oint scatterers or

zero range interactions as they will playa largeroleinv arious applications whichfollow

In the nal section we fo cus briey on two dimensions since two dimensional scattering

theory is the stage on which all the applications playout

CrossSections

At rst we will generalize to arbitrary spatial dimension a calculation from

pp relating the scattering crosssection to matrix elements of the p otential V

We consider a domain in which the stationary solutions of the Schrodinger equation

are known and we lab el these by For example in free space

ikr

In the presence of a potential there will be new stationary solutions lab eled by

lab els the asymptotic behavior of in terms of where sup erscript plus and minus

ddimensional spherical waves In particular

E E

E H

k k

and

ik r

r rf

k k

Chapter Quantum Scattering Theory in dDimensions

We assume the plane wave r is wide but nite so we may always go far enough away

that scattering at any angle but involves only the scattered part of the wave Since

the ux of the scattered waveisj Imf r g rjf j r the probability

scatt

p er unit time of a scattered particle to cross a surface element da is

v da v f d

But the current density ux p er unit area in the incidentwaveisv so

jf j

If unambiguous we will replace k and k by a and b resp ectively

We pro ceed to develop a relationship b etween the scattering function f and matrix

elements of the p otential V This will lead us to the denition of the socalled scattering

tmatrix

Consider two p otentials U r and U r b oth of which fall o faster r We will

show

h i

U U r dr r U r U r

a a

b b

h i

d d d

i k f f

We b egin with the Schrodinger equation for the s

r U E

b b

r U E

a a

We multiply by and the complex conjugate of by and then subtract

the latter equation from the former Since U r and U r are real we have dropping the

as and bs when unambiguous

i o h n

U U r r

Weintegrate over a sphere of radius R centered at the origin to get

i o h n

U U dr r r lim

m rR

Chapter Quantum Scattering Theory in dDimensions

For two functions of r and we dene

r r

and its integral over the ddimensional sphere

f g W j R d

R r R

r r da

Greens Theorem implies

h i

r r dr f g

and thus equation may b e written

n o

U U lim

Toevaluate the surface integral we substitute the asymptotic form of the s

n o ik r n o

ik r ik r ik r

b b

lim e e lim e f lim

R R

R R R

ik r ik r ik r

e e e

ik r

lim f lim f e f

d d d

R R

r r r

R R

z z

Since we are p erforming these integrals at large r we require only the asymptotic

form of the plane wave and only in a form suitable for integration We nd this form by

doing a stationary phase integral of an arbitrary function of solid angle against a plane

wave at large r That is

ikr

I lim e f d

r r

We nd the p oints where the exp onential varies most slowly as a function of the integration

variables in this case the angles in Since k r kr cos the stationary phase p oints

will o ccur at We expand the exp onential around each of these p oints to yield

f d I exp ik r

r r

exp ik r f d

r r

Chapter Quantum Scattering Theory in dDimensions

We p erform all the integrals using complex Gaussian integration to yield an asymptotic

form for the plane wave to b e used only in an integral

h i

ikr ik r d ik r

e e i e

r r

k k

ik r

where enforced by the second function means that r k

Well attack the integrals in equation one at a time b eginning with

o n

d iR k k r ik r ik r

a a

b b

iR k k r e d lim e e

Since k and k have the same length k k is orthogonal to k k Thus we can always

a a a

b b b

cho ose our angular integrals such that our innermost integral is exactly zero

Z Z

n o

ik r ik r ia sin ia sin ia sin

e e cos e d e d e

ia ia

n o

ik r ik r

Thus lim e e

We can do the second integral using the asymptotic form of the plane wave The

only contribution comes from the incoming part of the plane wave

ik r

d d d

ik r

lim e f k i f

We can do the third integral exactly the same way Again only the incoming part of the

plane wave contributes

ik r

d d d

ik r

lim f e k i f

The fourth integral is zero since b oth waves are purely outgoing Thus

n o

d d d

lim i k f f

which when substituted into equation gives the desired result

Lets apply the result to the case U V U Wehave

E D

d d d

V k i f

b b

a a

We also apply it to the case U U V yielding

E D

d d d

k i f V

a a

b b m

Chapter Quantum Scattering Theory in dDimensions

Finallywe apply to the case U V U V yielding

f f

Wenowhave

E D

m d

d d

k V f

b b

a a

Since the ddimensional density of states p er unit energy is

m dp m

d d d

E p hk hk

d d d

h dE h hk h

and the initial velo city is hk m we can write our nal result for crosssection in a more

useful form

D E

V E

b d

d hv

where all of the dimensional dep endence is in the density of states and the matrix element

For purp oses which will b ecome clear later it is useful to dene the socalled

tmatrix op erator t E suchthat

t E j i V

and our result may b e rewritten as

E D

E t E

d b

d hv

Unitarity and the Optical Theorem

The fact that particles are neither created nor destroyed in the scattering pro cess

forces a sp ecic relationship b etween the total crosssection d dd and f

It is to this relationship that wenowturn This section closely follows a calculation from

pp but as in the previous section generalizes it to arbitrary spatial dimension

Supp ose the incoming wave is a linear combination of plane waves as in

ikr

F e d

incoming

f f is simply a more So the asymptotic outgoing form of is where

b b

symmetric notation than used in the previous section

Z Z

ik r

ikr

F e F f d

Chapter Quantum Scattering Theory in dDimensions

For large r we can use the asymptotic for of the plane wave to p erform the rst

integral We then get

ik r i h

ik r d ik r

F e i F e d F f

ik r

We can write this more simply without the common factor ik

ik r ik r

e e

F i SF

d d

r r

where

S f

and f is an integral op erator dened by

fF F f d

S is called the scattering op erator or Smatrix Since the scattering pro cess

is elastic we must have as many particles going into the center as there are going out of

the center and the normalization of these twowaves must b e the same So S is unitary

S S

Substituting weget

d d

y y

i f f f i f

then divide through by i

d d

y y

f f f i f i i

We apply the denition and have

h i

d d

i f i f i f f d

the unitarity condition for scattering

For wehave

d d

o n

k k

jf j d f Im i

Chapter Quantum Scattering Theory in dDimensions

which is the optical theorem

Invariance under time reversal interchanging initial and nal states and the di

rection of motion of eachwave implies

S S

f f

which is called the recipro city theorem

Green Functions

The crosssection is frequently the end result of a scattering calculation However

for most of the applications considered here we are concerned with more general prop erties

of scattering solutions For these applications the machinery of Green functions is invalu

able and is intro duced in this section Much of the material in this section is covered in

greater detail in A more formal development some examples and some useful Green

function identities are given in app endix A

The idea of a Green function op erator is both simple and b eautiful Supp ose a

quantum system is initially at t in state j i What is the amplitude that the system

will b e found in state j i a time later This information is contained in the timedomain

Green function We can take the Fourier transform of this function with resp ect to and

get the energydomain Green function It is the energy domain Green function which we

explore in some detail b elow

We dene an energydomain Green function op erator for the Hamiltonian H via

z H iG z

is the identity op erator The i is used to avoid diculties when z is equal to where

an eigenvalue of H is always taken to zero at the end of a calculation We frequently use

the Green function op erator G z corresp onding to H H r

Consider the Hamiltonian H H V As in the previous section we denote the

eigenstates of H by j i and the eigenstates of H by j i These satisfy

o a

H j i E j i

o a a a

H E

a a

Chapter Quantum Scattering Theory in dDimensions

We claim

j i G E V

a a

o a a

The claim is easily proved by applying the op erator E H i to the left of both

a o

sides of the equation since E H i j i V j i E H i j i and

a o a o a

a a

E H iG E Using the tmatrix we can rewrite this as

a o a

j i G E t E j i

a a a a

a o

but we can also rewrite by iterating it inserting the right hand side into itself as

j itogive

h i

j i G E V j i G V j i

a a a a

a o o

From we get a useful expression for the tmatrix

t z V V G z V V G z V G z V

o o o

We factor out the rst V in each term and sum the geometric series to yield

i h

z t z V V G

n o

V G z V z G

o o

n o

V z H V i G z

VG z z H i

where is frequently used as the denition of t z

Wenow pro ceed to develop an equation for the Green function itself

z H V i G z

V z H i z H i

o o

z H i V z H i

o o

h i

G z V G z

o o

h i

We expand G z V in a p ower series to get

G z G z G z V G z G z V G z V G z

o o o o o o

Chapter Quantum Scattering Theory in dDimensions

whichwe can rewrite as

h i

G z G z G z V G z G z V G z

o o o o o

G z G z V G z

o o

and using and the denition of G z weget a tmatrix version of this equation

namely

G z G z G z t z G z

o o o

Green functions in the p osition representation

So far wehave lo oked at Green functions only as op erators rather than functions

in a particular basis Quite a few of the sp ecic calculations whichfollow are p erformed in

p osition representation and it is useful to identify some general prop erties of ddimensional

Green functions We b egin from the dening equation rewritten in p osition space

r z G r r z r r V r i

We b egin by considering an arbitrary p oint r and a small ball around it B r

o o

We can move the origin to r and then integrate b oth sides of over this volume

Z Z

z r V r r i G r r z dr r dr

o o

B B

We now consider the limit of this equation We assume that the potential is nite

and continuous at r r so V r rcan b e replaced by V r in the integrand We can

o o o

safely assume that

G r r z dr lim

B r

since if it werent the integral of the r G term would b e innite We are left with

lim r G r r z dr

We can apply Gausss theorem to the integral and get

G r r z d lim

So wehave a rst order dierential equation for G r r z for small jr r j

G z

h d

Chapter Quantum Scattering Theory in dDimensions

where

is the surface area of the unit sphere in ddimensions this is easily derived by taking the

pro duct of d Gaussian integrals and then p erforming the integral in radial co ordinates see

eg pp

In particular in two dimensions the Green function has a logarithmic singularity

on the diagonal where r r In d dimensions the diagonal singularitygoesas r

It is worth noting that in one dimension there is no diagonal singularityinG

As a consequence of our derivation of the form of the singularity in G we have

proved that as long as V r is nite and continuous in the neighb orho o d of r then

lim G r r r r z z G

This will prove useful in what follows

Zero Range Interactions

For the sake of generality up to now we have not mentioned a sp ecic p otential

However in what follows we will frequently be concerned with p otentials which interact

with the particle only at one point Such interactions are frequently called zero range

interactions or zero range p otentials

There is a wealth of literature ab out zero range in teractions in two and three

dimensions including their application to chaotic systems for example

and their applications in statistical mechanics including

In one dimension the Dirac delta function is just suchapointinteraction How

ever in two or more dimensions the Dirac delta function do es not scatter incoming particles

at all This is shown for two dimensions in In more than one dimension the wavefunction

can b e set to at a single p oint without p erturbing the wavefunction since an innitesimal

part of the singular solution to the Schrodinger equation can force to b e zero at a single

p oint Since there are no singular solutions to the Schrodinger equation in one dimension

the one dimensional Dirac function do es scatter as is well known

Trying to construct a p otential corresp onding to a zerorange interaction can be

quite challenging The formal construction of these interactions leads one to consider the

space where some condition must be satised at the p oint of Hamiltonian in a reduced

Chapter Quantum Scattering Theory in dDimensions

interaction Cho osing the wave function to be zero at the interaction point leads to the

mathematical formalism of selfadjoint extension theory so named b ecause the restriction

of the Hamiltonian op erator to the space of functions which are zero at a p oint leaves

a non selfadjoint Hamiltonian The family of p ossible extensions which would make the

Hamiltonian selfadjoint corresp ond to various scattering strengths

Much of this complication arises b ecause of an attempt to write the Hamiltonian

explicitly or to make sure that every p ossible zero range interaction is included in the

formalism Toavoid these details we consider a very limited class of zerorange interactions

namely zerorange swave scatterers

Consider a scatterer placed at the origin in two dimensions We assume the ph ysi

cal scatterer b eing mo deled is small compared to the wavelength l ambda E and thus

scatters only swaves So we can write the tmatrix for a general discussion of tmatrices

see e g

t z j i s z h j

If at energy E j i is incident on the scatterer we write the full wave incident

plus scattered as

j i G E t E j i

whichmay b e written more clearly in p osition space

rG r E s E r

At this p oint the scatterer strength s z is simply a complex constant We can

consider s E as it relates to the crosssection From equation with V j i replaced

D E

by t j i wehave since t z s z

a a

d d

E S k js E j

where S the surface area of the unit sphere in d dimensions is given by

We also consider another length scale akin to the threedimensional scattering

length Instead of lo oking at the asymptotic form of the wave function we lo ok at the s

wave comp onentofthewave function by using R r the regular part of the swave solution

to the Schrodinger equation as an incidentwave We then have

rR r G r E sE R

o o o

Chapter Quantum Scattering Theory in dDimensions

We dene an eective radius a as the smallest p ositiverealnumb er solution of

R xs E G x E

We can reverse this line of argument and nd s E for a particular a

R a

o e

s E

G a E

The p oint interaction accounts for the swave part of the scattering from a hard disk of

radius a From equation the cross section of a p ointinteraction with eective radius

a is

m R a

o a

d d

E S k

G a E

but this is exactly the swave part of the crosssection of a hard disk in ddimensions

Though zero range interactions have the crosssection of hard disks dep ending on the

dimension and the value of s E the p ointinteraction can b e attractive or repulsive

In three dimensions the E limit of a exists and is the scattering length as

dened in the mo dern sense It is interesting to note that other authors eg Landau

as and Lifshitz in their classic quantum mechanics text dene the scattering length

we have dened the eective radius namely as the rst no de in the swave part of the

wave function These denitions are equivalent in three dimensions but quite dierent in

two where the mo dern scattering length is not well dened but for any nite energy the

eective radius is

Scattering in two dimensions

While many of the techniques discussed in the following chapters are quite general

just as we will frequently use point interactions in applications due to their simplicitywe

will usually work in two dimensions either b ecause of intrinsic interest as in chapter or

umerical work is easier in two dimensions than three b ecause n

Since most people are familiar with scattering in three dimensions some of the

features of two dimensional scattering theory are at rst surprising For example

d m

jV j f

b b

a a

d k

Chapter Quantum Scattering Theory in dDimensions

implying that as E E whichisvery dierent from three dimensions Also

the optical theorem in two dimensions is dierent than its threedimensional counterpart

n o

Im e f

Wehave already mentioned the dierence in the diagonal b ehavior of the two and

three dimensional Green functions The logarithmic singularityin G prevents a simple idea

of scattering length from making sense in two dimensions This singularity comes from

the form of the freescattering solutions in two dimensions the equivalent of the three

dimensional spherical harmonics In two dimensions the free scattering solutions are Bessel

and Neumann functions of integer order The small argument and asymptotic prop erties of

these functions are summarized in app endix E

In two dimensions we can write the sp ecic form of the causal tmatrix for a zero

range interaction with eective radius a lo cated at r t E j r i s E h r j with

e s s s

J Ea

s E i

H Ea

where J x is the Bessel function of zero order and H x is the Hankel function of

zero order

Chapter

Scattering in the Presence of Other

Potentials

In chapter we presented scattering theory in its traditional form We computed

crosssections and scattering wave functions In this chapter we fo cus more on the to ols of

scattering theory and broaden their applicability Here we b egin to see the great usefulness

of the b o okkeeping asso ciated with tmatrices We will also b egin to use scattering theory

for closed systems an idea which is confusing at the outset but quite natural after some

practice

Multiple Scattering

Multiple scattering theory has b een applied to many problems from neutron scat

tering by atoms in a lattice to the scattering of electrons on surfaces In most applica

tions the individual scatterers are treated in the swave limit ie they can b e replaced by

zero range interactions of appropriate strength We b egin our discussion of multiple scat

tering theory with this sp ecial case before moving on to the general case in the following

section This is done for p edagogical reasons The general case involves some machinery

which gets in the way of understanding imp ortantphysical concepts

Chapter Scattering in the Presence of Other Potentials

Multiple Scattering of Zero Range Interactions

Consider a domain with given b oundary conditions and potential in which the

Green function op erator G z for the Schrodinger equation is known Into this domain we

place N zero range interactions lo cated at the p ositions fr g and with tmatrices ft z g

given by t z s z j r ihr j At energy E r is incidenton the set of scatterers and

i i

wewant to nd the outgoing solutions of the Schrodinger equation r in the presence

of the scatterers

We dene the functions r via

rr G r r E s E r

j j

i j j

j i

The number r represents the amplitude of the wave that hits scatterer i last That

i i

is r is determined by all the other r j i The full solution can b e written in

i j

terms of the r

r E r r E s rr G

i i

The expression gives a set of linear equations for the r This can be seen more

i i

simply from the following substitution and rearrangement

E s E r r r G r r

j j i i i

j j i

j i

We dene the N vectors a and b via a r andb r and rewrite

i i i i

as a matrix equation

i h

E a b t E G

where is the N N identity matrix tE is a diagonal N N matrix dened byt

s E and E is an odiagonal propagation matrix given by G

r r E for i j G

i j

G E

for i j

More explicitly t E G E isgiven by suppressing the E and

s G r r s G r r

B B N

B C

s Gr r s G r r

B C

B N

B C

s G r r s G r r

N B N N B N

Chapter Scattering in the Presence of Other Potentials

The odiagonal propagator is required since the individual tmatrices account for the di

agonal propagation That is the scattering events where the incidentwave hits scatterer i

propagates freely and then hits scatterer i again are already counted in t

We can lo ok at this diagrammatically We use a solid line to indicate causal

propagation and a dashed line ending with an to indicate scattering from the i

scatterer With this dictionary we can write the innite series form of t as

i i i i i i

so G G t G has the following terms

o o i o

i i i

Nowwe consider multiple scattering from two scatterers The Green function has the direct

term terms from just scatterer terms from just scatterer and terms involving b oth ie

The o diagonal propagator app earing in multiple scattering theory allows to add only the

terms involving more than one scatterer since the one scatterer terms are already accounted

for in each t

If at energy E t G is invertible we can solve the matrix equation for

a tG b

where the inverse is here is just ordinary matrix inversion We substitute into

to get

i h

E r r r E s E t E G rr G

j i

B B

We can dene a multiple scattering tmatrix

i h

j E t hr j E tE G E r i t

j ii i

and write the full solution in a familiar form

ji G E t E j i B

Chapter Scattering in the Presence of Other Potentials

An analogous solution can b e constructed for j i by replacing all the outgoing solutions

in the ab ove with incoming solutions sup erscript go es to sup erscript

We have shown that scattering from N zero range interactions is solved by the

inversion of an N N matrix As we will see b elow generalized multiple scattering theory

is not so simple It do es however rely on the inversion of an op erator on a smaller space

than that in which the problem is p osed

Generalized Multiple Scattering Theory

Wenow consider placing N scatterers not necessarily zero range with tmatrices

t z in a background domain with Green function op erator G z In what follows the

argument z is suppressed

We assume that each tmatrix is identically zero outside some domain C and we

further assume that the C do not overlap that is C C for all i j We dene the

i i j

C In the case of N zero range scatterers the scattering space is scattering space S

just N discrete p oints The denition of the scattering space allows a separation between

propagation and scattering events

As in the p oint scatterer case we consider the function rh r j i represen t

i i

ing the amplitude which hits the i scatterer last We can write a set of linear equations

for the

E E

ji G t

i j j

j i

where r is the incident wave As in the simpler case ab ove the full solution can be

written in terms of the via

ji G t

i i

The derivation b egins to get complicated here Since the scattering space is not

necessarily discrete we cannot map our problem onto a nite matrix We now b egin to

create a framework in which the results of the previous section can b e generalized

We dene the pro jection op erators P which are pro jectors onto the ith scatterer

that is

E D

f r if r C

f r P

if r C

P Also we dene a pro jection op erator for the whole scattering space P

Chapter Scattering in the Presence of Other Potentials

We can pro ject our equations for the r onto each scatterer in order to get

equations analogous to the matrix equation wehadfor r in the previous section

i i

E E

G t P ji P P

i i i

j i j

j i

and fur purely formal reasons we dene a quantity analogous to the vector a in the zero

range scatterer case

We note that r is nonzero on the scattering space only

With these denitions we can develop a linear equation for j i We begin by

summing over the N scatterers

X X

P ji P G t

j j

j i

Since t is unaected by multiplication by P we have t P t and t j i t

i i

i i i i i i

Thus we can rewrite as

X X

P ji P G P t

i j

j i

X X

P ji P G P t ji

i j

j i

Wecansimplify this equation if as in the zero range scatterer case we dene an

odiagonal background Green function op erator

X X X

G P G P PG P P G P

i j i i

B B B B

i i

j i

and a diagonal tmatrix op erator

t t

and note that

X X

t t G P P G

j i

B B

j i

We can now rewrite as

P ji G t B

Chapter Scattering in the Presence of Other Potentials

whichwemay formally solve for j i

i h

P ji t P G

h i

The op erator P G t is an op erator on functions on the scattering space S and the

b oldface sup erscript indicates inversion with resp ect to the scattering space only In the

case of zero range interactions the scattering space is a discrete set and the inverse is just

ordinary matrix inversion In general nding this inverse involves solving a set of coupled

linear integral equations

P is just the identity op erator on the scattering space We note that the pro jector

h i i h

P t G t G

B B

We can rewrite yielding

t ji G

Substituting into gives

h i

P ji ji G t G t

B B

The identity

B A AB A A

implies

h i

tG t ji ji G

Wenowdeneamultiple scattering tmatrix

h i

t t G t

which is zero outside the scattering space Our wavefunction can now b e written

ji G t ji

This derivation seems much more complicated than the sp ecial case presented rst

While this is true the underlying concepts are exactly the same The complications arise

from the more complicated nature of the individual scatterers Each scatterer now leads

to a linear integral equation rather than a linear algebraic equation what was simply a

Chapter Scattering in the Presence of Other Potentials

set of linear equations easily solved by matrix techniques b ecomes a set of linear integral

equations which are dicult to solve except in sp ecial cases

The techniques in this section are also useful formally We will use them later in

this chapter to eect an alternate pro of of the scatterer renormalization discussed in the

next section

Renormalized tmatrices

In the previous section we used the tmatrix formalism and some complicated

b o okkeeping to combine many scatterers into one tmatrix In this section wed like to

work with the propagation step Well start with a potential with known tmatrix in free

space We then imagine changing free space by adding b oundaries for example We then

nd the correct tmatrix for the same physical p otential for use with this new propagator

To begin we consider the scatterer in free space It has known tmatrix t z

which satises

z z t z G z G z G G

o o o s

where the subscript s is used to denote that this Green function is for the scatterer in free

space Now rather than free space we supp ose wehave a more complicated background but

one with a known Green function op erator G z We note that there exists a tmatrix

t z suchthat

G z G z G z t z G z

o o o

B B

z op erator in the algebra which follows but only as a formal to ol We will use the t

Frequently the division b etween scatterer and background is arbitrary we can often treat

the background as a scatterer or a scatterer as part of the background

with a zero range scatterer with eective radius a in As an example we begin

two dimensions In section we computed t z for this scatterer in free space We

place this scatterer into an innite wire with p erio dic transverse b oundary conditions The

causal Green function op erator G r r z can b e written as an innite sum and can be

calculated quite accurately using numerical techniques see chapter

Computing a tmatrix for the scatterer and b oundary together is quite dicult

Also such a tmatrix would b e nonzero not only on the scatterer but on the entire innite

length b oundary This lacks the simplicity of a zerorange scatterer tmatrix Instead wed

Chapter Scattering in the Presence of Other Potentials

like to nd a tmatrix T z for the scatterer such that the full Green function G z

may b e written

G z G z G z T z G z

B B B

Well call T z the renormalized tmatrix This name will b ecome clearer b elow

Lets start with a guess What can happ en in our rectangle that couldnt happ en

in freespace The answer is simple amplitude may scatter o of the scatterer hit the walls

and return to the scatterer again That is there are multiple scattering events b etween the

background and the scatterer Diagrammatically this is just like the two scatterer multiple

scattering theory considered in the previous section where scatterer instead of being

another scatterer is the background see equation

Naively wewould exp ect to add up all the scattering events dropping the z s

T t t G t t G t G t t G t

B B B B

This is p erhaps clearer if we consider the p osition representation

T r r t r r dr d r r tr r r t r r G

and since our scatterer is zerorange

t rr s r r r r

s s

which simplies

r r r T r r s r r r r s G r r r

s s s s s s

Summing the geometric series yields

T r r r r r r

s s

s G r r

s s

as an op erator equation

T t

G t

This is not quite right With multiple scattering we had to dene an odiagonal Green

function since the diagonal part was already accounted for by the individual tmatrices

Something similar is needed here or we will b e double counting terms which scatter prop

agate without hitting the b oundary and then scatter again

Chapter Scattering in the Presence of Other Potentials

in the previous derivation We can correct this by subtracting G from G

G t G t G G t t G T t t G t

o o o

B B B

G t G

For future reference well state the nal result

h i

T z t z

t z G z z G

We havent proven this but we can derive the same result more rigorously in at least two

ways both of which are shown below The rst pro of follows from the expression

for the tmatrix derived in section This is a purely formal derivation but it has the

advantage of b eing relativ ely compact Our second derivation uses the generalized multiple

scattering theory of section While this derivation is algebraically quite tedious it

emphasizes the arbitrariness of the split between scatterer and background by treating

them on a completely equal fo oting

We note that the freespace Green function op erator G z could b e replaced by

any Green function for which the tmatrix of the scatterer is known That should b e clear

from the derivations b elow

Derivation

Formal Derivation

Supp ose wehave H H H H where H is the Hamiltonian of the back

o B s B

reasonable p otential ground and H is the scatterer Hamiltonian which may be any

There is a tmatrix for the scatterer without the background

t z H G z z H i

s o

and for the scatterer in the presence of the background where the background is treated as

part of the propagator

T z H G z z H H i

s o B

This yields an expression for the full Green function G z op erator

G z G z G z T z G z

B B B

Chapter Scattering in the Presence of Other Potentials

z solves where G

z H H iG z

o B

We wish to nd T z in terms of t z G z and G z Formallywe can use

to solvefor H

h i

G z H t z z H i t z G z z H H i

s o o s

s o

We substitute this into

h i

T z t z G z G z G z z H H i

o B

o s

whichwe can rewrite

h i

T z t z G z G z G z t z G z

o o o o

h ih i

G z G z T z G z G z

B B B B

and canceling a few inverses wehave

i h i h

z t z G z T z T z t z G

We rewrite this as

i h i h

z t z T z t z G z z t z G t z G

o o

whichwe solv eforT z

i h i h

z t z z t z G G z t z T z t z G

o o

To pro ceed well need the op erator identity

AB A A B A

Whichwe apply to yield

h i h i

z t z G z t z T z t z G z t z G

o o

h i h i

t z G z t z G z t z G z t z

o o

n o

t t z G z t z G z z

h i

t z

t z G z G z

o B

Chapter Scattering in the Presence of Other Potentials

Thus wehaveveried our guess alb eit in an exceedingly formal way

Derivation Using Generalized Multiple Scattering Theory

Consider a situation with two scatterers in free space One the background with

a scattering tmatrix t which is identically zero outside the domain C and the other

a scattering tmatrix t which is zero outside the domain C They may each be point

scatterers or extended scatterers We assume that the scatterers do not overlap ie C

C The scattering space S is simply the union of C and C S C C From

this point on in the derivation we drop the sup erscript since we carried it through

the previous derivation and it should b e clear here that there is a sup erscript onevery

tmatrix and every Green function We also drop the argumentz

Nowwe apply the generalized multiple scattering theory of section where the

background Green function op erator is just G Wehave an explicit form for t

h i

t t

i ij

and G

i j

G i j

According to our derivation of section the tmatrix may b e written

h i

t tG t

Painful as it is lets write out all of the terms in the ab ove expression for t We

drop the hats on all the op erators since everything in sight is an op erator First wehave

to invert tG and wehave to do it carefully b ecause none of these op erators necessarily

commute

t G

t G t G t G t G t G

o o o o o

t G t G t G t G t G

o o o o o

and thus

t G t G t t G t G t G t

o o o o o

t tG

t G t G t G t t G t G t

o o o o o

Chapter Scattering in the Presence of Other Potentials

So in detail

G G

G t G t G t G G t G t G t G

o o o o o o o o

G t G t G t G t G G t G t G t G t G

o o o o o o o o o o

Wewant to rewrite this in the form

G G G T G

where T is the renormalized tmatrix op erator for scatterer in the presence of scatterer

First we add and subtract G t G

o o

G G G t G

o o o

G t G G t G t G t G G t G t G t G

o o o o o o o o o o

G t G t G t G t G G t G t G t G t G

o o o o o o o o o o

t G t G t G G G G t G G

o o o o o o o

G t G t G t G

o o o o

G t G t G t G t G G t G t G t G t G

o o o o o o o o o o

but

h i

t G t G t G t G t G t G

o o o o o o

t G t G t G t G t G G G G t G G

o o o o o o o o o

G t G t G t G

o o o o

G t G t G t G t G G t G t G t G t G

o o o o o o o o o o

We use the op erator identity to rewrite G

G G G t G G t G t G t G t G t G

o o o o o o o o o

G t G t G t G

o o o o

G t G G t G t G t G t G G t G t G t

o o o o o o o o o o

Several terms now have the common factor t G t G This allows us to collapse

o o

several terms

t G G t G G G G t G G G t

o o o o o o o o

t G t G

o o

Chapter Scattering in the Presence of Other Potentials

but G G t G G is the Green function for scatterer alone Also t G t G

o o o o o

t G G so wehave

t G G G G

t G G

Nowwe see that equation has exactly the same form as equation So we

have derived restoring the andthez

h i

T z t z

t z G z G z

which is identical to equation as was to be shown In the notation of the previous

derivation T z T z t z tz and G z G z

Consequences

Free Space Background

What happ ens if G z G z Our formula should reduce to T z t z

And it do es

T z t z t z

z t z G z G

Closed Systems

Supp ose G comes from a nite domain eg a rectangular domain in two dimen

sions Then wehave

t t T

t G G t G t G

o o o

B B

It is not obvious from this that T do esnt dep end on the choice if incoming or outgoing

solutions in the ab ove equation However it is clear from physical considerations that a

closed system only has one class of solutions In fact the ab ove equation is indep endentof

the choice of incoming or outgoing solutions for the free space quantities We can show this

in a nonrigorous wayby observing that

h i

t G G

and that

t H G

s o

Chapter Scattering in the Presence of Other Potentials

T H G

To show this rigorously would require a careful denition of what these various inverses

mean since many of the op erators can b e singular We need to prop erly dene the space on

which these op erators act This would be similar to the denition of the scattering space

used in section

The Green function op erator of a closed system has poles at the eigenenergies

o o

That is t z and G z have p oles at z E for n f g For z near E

B B

n n

t z

z E

and

G z R G z

o n o

G z

z E

z has no p oles though it mayhave other sorts of singularities So we dene G

n n

R G E R G E

o o n o o

Since we have added a scatterer in general none of the p oles of G z should

where be p oles of Gz This is something we can check explicitly Supp ose z E

Then jj

R R R

n n n

Gz GE t z

t z R

which is easily simplied

t z R

We assume that t E and we know that R b ecause E is a simple p ole of

n n

G z So there exists such that

t z R

So wehave

t z R

and thus

O Gz

t z

Therefore the p oles of G z are not p oles of Gz unless tE

B n

Chapter Scattering in the Presence of Other Potentials

So Where are the Poles in Gz

As we exp ect the analytic structure of Gz and T z coincide at least for the

most part see p More simply since the p oles of G z are not poles of Gz

only p oles of T z willcontribute p oles to Gz

Recall

h i

T z t z

t z G z G z

so p oles of T z occuratE E satisfying

i h

G E G E t E

B n n n

i h

G E G E t E

B n n n

E G E G E t

o o

n n n

This is a simple equation to use when G comes from scatterers added to free space so t is

known When G is a Green function given apriori eg the Green function of an innite

wire in dimensions the ab ove equation b ecomes somewhat more dicult to evaluate

Well address this issue in a later chapter ab out scattering in dimensional wires

Perturbatively small scatterers

For small tz the only way to satisfy equation is for G z tobevery large

But G z is large only near a p ole E of G z This is a nice result It implies that for

B B

small tz ie a small scatterer the p oles E of Gz are close to the p oles E of G z as

n B

we might exp ect This idea has b een used see to explore the p ossibility that an atomic

force microscop e used as a small scatterer can prob e the structure of the wavefunction of

a quantum dot

Renormalization of a zero range tmatrix

A zero range tmatrix provides a simple but subtle challenge for the application

of this renormalization Since

t z s z j r ihr j

s s

wehave

h i

T z jr ihr j

s s

G r r z G r r z

B s s s s

s z

Chapter Scattering in the Presence of Other Potentials

But as we have already discussed in section in more than one dimension the Green

function G r r z for the Schrodinger equation is singular in the r r limit So we

havetodeneT z a bit more formally

h i

j r ihr j T z lim

s s

r r

G r r z G r r z

B s s

s z

This limit can be quite dicult to evaluate Four particular cases are dealt with in chap ters and

Chapter

Scattering From Arbitrarily

Shap ed Boundaries

Intro duction

In the previous chapter wedevelop ed some p owerful to ols for solving complicated

scattering problems All of them were built up on one or more Green functions In this chap

ter we consider a varietyoftechniques for computing Green functions in various geometries

The techniques discussed in this chapter are useful when wehave a problem whichinvolves

scattering on a surface of codimension one one dimension less than the dimension of the

system for example scattering from a set of one dimensional curves in two dimensions

We b egin by computing the Green function of an arbitrary numb er of arbitrarily

shap ed smo oth Dirichlet b oundaries placed in freespace The metho d is con

structed by nding a p otential which forces to satisfy the Dirichlet b oundary condition

The technique is somewhat more general It can enforce an arbitrary linear combination

of Dirichlet and Neumann b oundary conditions The more general case is dealt with in

app endix B

We then rederive the fundamental results by considering certain expansions of

rather than a potential This lends itself nicely to the generalizations which follow in the

next two sections We can use expansions of to simply match b oundary conditions The

rst generalization is a small but useful step from Dirichlet b oundary conditions to p erio dic

b oundary conditions

Chapter Scattering From Arbitrarily Shaped Boundaries

We next consider scattering from a b oundary between two regions with dierent

known Green functions This cannot b e handled as a b oundary condition but nonetheless

all the scattering takes place at the interface This metho d could be used to scatter from

a p otential barrier of xed height that was the original motivation for its development It

could also be used to scatter from a sup erconductor emb edded in a normal metal or vice

versa since each has its own known Green function see A This idea is b eing actively

pursued

Boundary Wall Metho d I

Consider

V r ds s r rs

where the integral runs over the surface C Here we will assume a pragmatic p oint of view

by supp osing that our mathematical problem is well p osed ie there do es exist a solution

for the Schrodinger equation satisfying the b oundary conditions considered Obviously the

metho d has no meaning when this is not so The b oundary condition

emerges as the limit of the p otentials parameters For nite the p otential

has the eect of a p enetrable or leaky wall A similar idea has b een used to incorp orate

Dirichlet b oundary conditions into certain classes of solvable potentials in the context of

the path integral formalism Here we use the delta wall more generally resulting

in a widely applicable and accurate pro cedure to solve b oundary condition problems for

arbitrary shap es

Consider the Schrodinger equation for a ddimensional system H r rEr

with H H V As is well known the solution for r is given by

rr dr G r r V r r

H Hereafter where rsolves H rrErandG r r is the Green function for

for notational simplicitywe will suppress the sup erscript E in G

Now weintro duce a typ e potential

V r ds r rs C

Chapter Scattering From Arbitrarily Shaped Boundaries

where the integral is over C a connected or disconnected surface rs is the vector p osition

of the p oint s on C we will call the set of all such vectors S and is the potentials

strength Clearly V r for r S

In the limit the wavefunction will satisfy with s as shown

b elow For nite awave function sub ject to the p otential will satisfy a leaky form

of the b oundary condition

Inserting the potential into the volume integral is trivially p erformed

with the delta function yielding

Z Z

rr ds G r rs rs r ds G r rs T rs

C C

Thus if rs T rs is known for all s the wave function everywhere is obtained

from by a single denite integral For r rs some p ointof S

rs rs ds G rs rs rs

whichmay b e abbreviated unambiguously as

s s ds G s s s

We can formally solve this equation getting

h i

I G

where stand for the vectors of ss and ss on the b oundaryand I for the identity

op erator The tildes remind us that the free Green function op erator and the wavevectors

are evaluated only on the b oundary

We dene

h i

T I G

and then it is easy to see that T in is given from by

T rs ds T s s s

In order to make contact with the standard tmatrix formalism in scattering the

ory wenotethataT op erator for the whole space may b e written as

tr r ds ds r rs T s s r rs

i i

f f

Chapter Scattering From Arbitrarily Shaped Boundaries

Finallywe observe that can b e written as

i h

IG T

i h

Inserting this into wehave For the op erator T converges to G

i h

G I G

So satises a Dirichlet b oundary condition on the surface C for

Boundary Wall Metho d II

We can simplify the previous derivation considerably if we assume that there exists

an op erator

tE ds ds jrs i T s s E rs

such that the solution of our scattering problem j i may b e written

j i ji G E t E j i

or in p osition space

rr dr dr G r r E tr r E r

whichwe can simplify to

r r ds ds G r rs E T s s E rs

The solution is a bit simpler if wemake a notational switch

ts ds T s s E rs

Now we enforce the dirichlet b oundary condition rs That gives us a Fredholm

integral equation of the rst kind

rs E ts rs ds G r

whichwemaysolve for ts eg using standard numerical metho ds Formallywe can

solve this with

ts ds G s s E s

where the new notation reminds us that the inverse is calculated only on the b oundary

That is G s s satises

s rs E G s s E s s ds G r

o o

Chapter Scattering From Arbitrarily Shaped Boundaries

Perio dic Boundary Conditions

We can generalize this for other sorts of b oundary conditions In this section well

deal with p erio dic b oundary conditions As with DirichletNeumann b oundary conditions

we may write the solution in terms of an expansion in freespace solutions to the wave

equation However rst we should prop erly characterize our p erio dic b oundaries Well

consider a kind of generalized p erio dic b oundaries

Consider two surfaces C and C b oth parameterized by functions of a generalized

parameter s We insist that the surfaces are parameterized such that both parameter

functions have the same domain We dene p erio dic b oundary conditions as any set of

conditions of the form

r s r s

nr s nr s

where denotes the partial derivativeinthedirection normal to the curve rs We

nrs

note that dierent parameterizations of the surfaces may yield dierent solutions

For example consider the unit square in twodimensions Standard p erio dic

b oundary conditions sp ecify that

y y y

x x

x x x

y y

Wecho ose the surface C as the left side and top of the square and C as the right side and

b ottom of the b ox Wemaycho ose a real parameter s where s parameterizes

the sides of the box from top to bottom and s parameterizes the top and bottom

from right to left However twisted p erio dic b oundaries may also b e considered

y y y

x x

x x x

y y

Chapter Scattering From Arbitrarily Shaped Boundaries

In our language this simply means cho osing a dierent parameterization of the two pieces

of the b ox

To solve this problem wenow expand in terms of the freespace Green function

on the b oundary

rr Gr r s E f sGr r s E f s ds

We insert the expansion into equations

r s Gr s r s E f s Gr s r s E f s ds

Gr s r s E f s Gr s r s E f s ds r s

r s

nr s

i h

Gr s r s E f s Gr s r s E f s ds

nr s nr s

r s

nr s

i h

s Gr s r s E f s ds Gr s r s E f

nr s nr s

This is a set of coupled Fredholm equations of the rst typ e Tomake this clearer

we dene

as r s r s

a s r s r s

nr s nr s

G s s E G r s r s E G r s r s E

o o

G s s E G r s r s E G r s r s E

o o

G r s r s E G r s r s E G s s E

o o

nr s nr s

G s s E G r s r s E G r s r s E

o o

nr s nr s

and then rewrite the equations at the b oundary

G s s f s G s s f s ds as

G s s f s G s s f s ds a s

Chapter Scattering From Arbitrarily Shaped Boundaries

whichwe can write at least schematically as a matrix equation

a f G G

A A A

f a G G

where the G s are linear integral op erators in the space of functions on the b oundary and

the f s and as are vectors functions in that space

Formallywecansolvethisequation

f G G a

A A A

f G G a

While we cannot usually in vert this op erator analytically we can sample our

b oundary at a discrete set of points We then construct and invert this op erator in this

nite dimensional space and using this nite basis construct an approximate solution to

our scattering problem

Green Function Interfaces

In this section well deal with scattering from an arbitrarily shap ed p otential step

Though this is not purely a b oundary condition the problem is solved as in the previous

two cases if sums of solutions to freespace equations satisfy certain conditions on a

b oundary

We consider an ob ject with p otenial V emb edded in freespace we could consider

Green function as an ob ject with one Green function emb edded in a space with another

long as the asymptotic solutions of the wave equation are known in b oth regions but well

b e a bit more sp ecic where the b oundary b etween the two regions is parameterized by s

via rs as the Dirichlet b oundary was in section

It is immediately clear that an expansion of the form cannot work for the

wavefunction inside the dielectric though it maywork for the wavefunction outside since

the Green function used in the expansion does not solve the wave equation inside Weneed

a separate expansion for the inside wavefunction This is no surprise since at a p otential

step b oundary we have twice as many b oundary conditions continuity of and its rst

e We expand the inside and outside solutions separately to get derivativ

r r G r rs E t s ds

out out out out out in

Chapter Scattering From Arbitrarily Shaped Boundaries

r r G r rs E t s ds

out

in in in in in

where r r r

out

The incoming wave inside the p otential step is puzzling at rst Because the basis

in whichwe are expanding the solutions is not orthogonal wehave some freedom in cho osing

i We can cho ose r which corresp onds to the entire inside wavefunction

b eing pro duced at the b oundary sources This is mathematically correct but a little awk

When the step is ward when the p otential step is small compared to the incident energy

small the wavefunction inside and outside will b e much like the incoming wave whichleads

us to

ii A more physical but harder to dene choice is to cho ose r to b e a solution

to the wave equation inside the step and which has the prop erty

lim rs rs

out

Though this seems ad ho c it is mathematically as valid as choice i and has the nice

prop erty that

lim t s

inout

which is app ealing

Nowwe write our b oundary conditions

rs rs

out

rs rs

ns ns

out

rs G rs rs E t s ds

out out out

rs G rs rs E t s ds

in in in

rs G rs rs E t s ds

ns ns

out out out

rs G rs rs E t s ds

ns ns

in in in

where is the normal derivative at the b oundary p oint rs

Chapter Scattering From Arbitrarily Shaped Boundaries

The ab ove is a set of coupled Fredholm equations of the rst typ e To make this

clearer we dene

as rs rs

out

a s rs rs

ns ns

out

v s t s

out

w s t s

G s s E G rs rs E

out

G s s E G rs rs E

out

s s E G rs rs E G

Nowweha ve the following system of integral equations

G s s E v s G s s E w s ds as

o i

s s E w s ds a s s s E v s G G

i o

with all the Gs and s given

Wemayschematically represent this as a matrix equation

v a G G

o i

A A A

w a G G

whichwemay formally solve

v G G a

o i

A A A

w a G G

This formal solution is not much use except p erhaps in a sp ecial geometry However it do es

lead directly to a numerical scheme Simply discretize the b oundary by breaking it into N

pieces fC g of length Lab el the center of each piece by s and change all the integrals in

i i

the integral equations to sums over i Now the schematic matrix equation actually b ecomes

Chapter Scattering From Arbitrarily Shaped Boundaries

a N N matrix problem which can be solved by LU decomp osition techniques or the

We mightalsoworry ab out mutliple step edges or dierent steps inside each other

All this will work as well but we will get a set of equations for eachinterface so the problem

may get quite costly This would not be a sensible way to handle a smo othly varying

potential However as noted at the b eginning the formalism here works for any known

G and G and so certain smo oth potentials may be handled if their Green functions

out

are known

Numerical Considerations and Analysis

Discretizing The Boundary Wall Equations

As discussed in Section the key idea in our metho d is to calculate T andor T

on C and then to p erform the integral Unfortunately in the great ma jority of cases

the analytical treatment is to o hard to b e applied In such cases we consider the problem

numerically

We divide the region C into N parts fC g Then we approximate

j j N

r r dsG r rs rs

r ds G r rs rs

with s the middle p oint of C and r rs Now considering r r we write r

j j j j i i

r M r for M see discussion below If r r and

i ij j N

wehave M and thus T with T I M r r

which is the discrete T matrix So

h i

I M T

i i j

and

r r G r r T

j j j

where wehave used a mean value approximation to the last integral in and dened

olume of C the v

j j

Chapter Scattering From Arbitrarily Shaped Boundaries

It follows from that

M ds G r rs

ij i

We can approximate

M G r r

ij i j j

However G r r may diverge for i j eg the free particle Green functions in two or

i j

more dimensions We discuss these approximations in detail in Section

If we consider it is easy to show from the ab ove results that

r r G r r M

j j j

Equation is then the approximated wave function of a particle under H interacting

with an imp enetrable region C

The Many Scatterer Limit

The b oundary wall metho d is a sort of multiple scattering approach to building

b oundaries In the many scatterer limit we can make this connection explicit

Recall that the inverse of the multiple scattering tmatrix for p oint scatterers has

the form

T E if i j

G r r E if i j

o i j

spacing We assume that the discretization of the b oundary wall metho d is uniform with

l i If for the b oundary wall M matrix we dene B l M we get a simpler

version of the discretized equation

r r G r r B

j j

we notice that B has the same odiagonal elements as M The diagonal elements of B

have the form where k E

Z Z

l l

kl kl

ln ln G r rs x E dx lnkx dx B

i i ii

l e

Chapter Scattering From Arbitrarily Shaped Boundaries

If we want to identify this with a multiple scattering problem we must have T E

ln whichisthelow energy form of the p ointinteraction tmatrix discussed in section

for a scatterer of scattering length le

Thus in the many scatterer limit kl the Dirichlet b oundary wall metho d

b ecomes the multiple scattering of many pointlike scatterers along the b oundaries where

each scatterer has scattering length le

Quality of the Numerical Metho d

The numerical solution approaches the solution as N In prac

tice we cho ose N to be some nite but large number In this section we explain how to

cho ose N for a given problem and how the approximation aects this choice

In order to analyze the p erformance of the numerical solution we must dene

some measure of the quality of the solution We measure how well a Dirichlet b oundary

blo cks the ow of current directed at it Thus we measure the current

j f rr rg

b ehind a straightwall of length l To simplify the analysis weintegrate j n over a detector

lo cated on one side of a wall with a normally incident plane wave on the other side We

divide this integrated currentby the current whichwould have b een incident on the detector

wall present We call this ratio T the transmission co ecient of the wall without the

Instead of T as a function of N we consider T vs where Nlk is the number of

b oundary pieces p er wavelength

We consider three metho ds of constructing the matrix M for eachvalue of The

rst is the simplest approximation

ds G r rs i j

G r r i j

i j

whichwe call the fullyapproximated M The next is a more sophisticated approximation

with

ds G r rs js s j

i i j

G r r js s j

i j i j k

Chapter Scattering From Arbitrarily Shaped Boundaries

whichwe call the bandintegrated M because we p erform the integrals only inside a band

of wavelengths Finallywe consider

ds G r rs ij M

i ij

whichwe call the integrated M

Numerically the bandintegrated and integrated M require far more compu

tational work than the fullyapproximated M which requires the fewest integrals All

The calculation of T or T from M scales as metho ds of calculating M scale as O N

and the calculation of r O N for a particular r from a given T scales as O N

Which of these various calculations dominates the computation time dep ends on what sort

of computation is b eing p erformed When computing wavefunctions computation time is

typically dominated by the large number of O N vector multiplications However when

calculating r in only a small numb er of places eg when p erforming a ux calculation

computation time is often dominated by the O N construction of T

In Figure we plot log T vs for the three methods above and

We see that all three metho ds blo ck more than of the currentfor However it is

clear from the Figure that the integrated M and to a lesser extent the bandintegrated

M strongly outp erform the fullyapproximated M for all plotted

From Wavefunctions to Green Functions

We pause in the development of these various b oundary conditions to explain how

to get Green functions whichwell need to use these b oundary conditions in more complex

scattering problems from the wavefunctions weve b een computing

All the ab ove metho ds compute wavefunctions from given b oundary conditions

and incoming waves In several cases the idea of an incoming wave is somewhat strange

For example what is an incoming wave on a periodic b oundary However we include

e only to allow the computation of Green functions as we outline b elow the incoming wav

When wehave an eigenstate of the b oundary condition the incoming wavemust b e b ecome

unimp ortantandwewillshow that b elow

One simple way to form a Green function from a general solution for wavefunctions

is to recall that Gr r E is the wavefunction at r given that the incoming wave is a free

space p ointsource at r rG r r E This yields an expression for the Green function o

Chapter Scattering From Arbitrarily Shaped Boundaries 25 integrated 20 band-integrated fully-approximated 15 p for different approximations to M 2 10 Behavior of |t| 5 -1 -2 -3 -4 -5

-0.5 -1.5 -2.5 -3.5 -4.5 -5.5

10 |t| log

Figure Transmission at normal incidence through a at wall via the Boundary Wall metho d

Chapter Scattering From Arbitrarily Shaped Boundaries

everywhere as a linear function of G

Example I Dirichlet Boundaries

For Dirichlet b oundaries wehad

rr G r rs E T s s rs ds ds

where T s s G s s So

Gr r E G r r E G r rs E T s s G rs r E ds ds

o o o

Example II Perio dic Boundaries

For periodic boundaries we had

rr G r r s E f sG r r s E f s ds ds

o o

where f and f are determined from

G G f a

A A A

G G f a

If we dene

F s r f s given rG r r E

wehave

E F s r G r r s E F s r ds Gr r E G r r E G r r s

o o o

Though this lo oks likewehavetosolvemany more equations than just to get the wavefunc

tion we note that the op erator inverse whichwe need to get the wavefunction is sucientto

get the Green function just as in the Dirichlet case We simply apply that inverse to more

vectors Thus for all b oundary conditions the Green function requires extra matrixvector

multiplication work but the same amount of matrix inversion work

Chapter Scattering From Arbitrarily Shaped Boundaries

Eigenstates

It is also useful to be able to use the ab ove metho ds to identify eigenenergies

and eigenstates if they exist of the ab ove b oundary conditions This is actually quite

simple All of the various cases involved inverting some sort of generalized Green function

op erator on the b oundary This inverse is a generalized tmatrix and its p oles corresp ond

to eigenstates Poles of t corresp ond to linear zero es of G and so we may use standard

techniques to check for a singular op erator If the op erator we are inverting is singular its

nullspace holds the co ecients required to form the eigenstate A more concrete explanation of this can b e found in section

Chapter

Scattering in Wires I One

Scatterer

In this section we consider the renormalization of the scatterer strength due to

the presence of innite length b oundaries The picture we have in mind is that of two

dimensional wire one dimensional free motion with p erio dic b oundary conditions in the

transverse direction and a single scattering center This will b e our rst example of scatterer

renormalization by an external b oundary

One Scatterer in a Wide Wire

It seems intuitively clear that the scattering o a small ob ject in the middle of a

big wire should be much like scattering in free space After all if the conning walls are

further aw ay than any other length scale in the problem they ought to play a small role

In this section we attempt to make this similarity explicit by computing the transmission

co ecient for a wide wire with one small scatterer in its center We will simply b e making

the connection b etween crosssection and transmission in the ballistic limit

Chapter Scattering in Wires I One Scatterer

W θ

Figure APerio dic wire with one scatterer and an incident particle

We begin with a simple classical argument Supp ose a particle in the wire is

incident with angle with resp ect to the walls as in gure What is the probability

that such a particle scatters For a small scatterer the probability is approximately P

Of course this must break down b efore P but for W this will a

W cos

small range of

Wenowneedtoknowhowthevarious incident angles are p opulated in a particular

scattering pro cess For this we must think more carefully ab out the physical system we

have in mind In our case this is a two prob e conductance measurement on our wire

as pictured in gure Our physical system involves connecting our wire to contacts

of scattering particles eg electrons running a xed current which serve as reservoirs

I through the system and then measuring the voltage V Theoretically it is the ratio

of current to voltage which is interesting thus we dene the unitless conductance of this

systemg he IV

Chapter Scattering in Wires I One Scatterer

I W

Figure Exp erimental setup for a conductance measurement The wire is connected

to ideal contacts and the voltage drop at xed current is measured

In such a setup all of the transverse quantum channels are p opulated with equal

probability Since the quantum channels are uniformly distributed in momentum wehavefor

the probability density of nding a particular transverse wavenumb er k dk dk

y y y

E sin and together these give the density of incoming angles in We also knowthat k

cos Well also assume the scattering is isotropic the plane of the scatterer d

half of the scattered wave scatters backward So wehave

R P d

E The maximum cross section of a zero range interaction in two dimensions is

see and corresp onds to scattering all incoming swaves If we put this into we get

R Interestingly this is exactly one quantized channel of reection The wire has

and the reection co ecient as manychannels as half wavelengths t across it N

is simply N indicating that one channel of reection in free space the swave channel

is also one channel of reection in the wire though no longer one sp ecic channel

We can check this conclusion numerically the numerical techniques will be dis

cussed later in the chapter as a check on the renormalization technique and the numerical

metho d In gure we plot the numerically computed reection co ecient and the the

oretical value of N for wires of varying widths from half wavelengths to half

wavelengths The exp ected behavior is nicely conrmed although there is a noticeable

Chapter Scattering in Wires I One Scatterer

discrete jump at a couple of widths where new channels have just op ened

0.07 Numerical Data Quasi-Classical

0.06

0.05

0.04 R

0.03

0.02

0.01

0 20 30 40 50 60 70 80 90 100

Number of Open Channels

Figure Reection co ecient of a single scatterer in a wide p erio dic wire

As the wire b ecomes narrower at the left of the gure we see that the agreement

between the measured value and the quasiclassical theory is p o orer This is no surprise

since our quasiclassical argument is b ound to break down as the height of the wire b ecomes

comparable to the wavelength and scatterer size This is a hint of what we will see in

section where the limit of the narrow wire is considered Before we consider that

problem we develop some necessary machinery First we compute the Green function of

the empty p erio dic wire we then consider the renormalization of the scattering amplitude

in a wire and the connection b etween Green functions and transmission co ecients

It is interesting to watch the transition from wide to narrow in terms of scattering

ve we saw that the scattering from one scatterer in a wide wire can be channels Ab o

understo o d classically As we will see later in the chapter scattering in the narrow wire

W a is much more complex If we shrink the wire from wide to narrow we can

watch this transition o ccur This is shown in gure

Chapter Scattering in Wires I One Scatterer

0.8

0.6

0.4 Channels Blocked by Scatterer

0.2

0 0 10 20 30 40 50 60

Number of Open Channels

Figure Number of scattering channels blo cked by one scatterer in a p erio dic wire of

varying width

The Green function of an empty p erio dic wire

Wenow pro ceed to do some tedious but necessary mathematical work necessary to

apply the metho ds of chapter to this system We b egin by computing the Green function

of the wire without the scatterer

The state jk ai dened by

ik x

hx y jk a i e y

where y satises

y y

a a a

a a

d d

a a

dy dy

y y W

y dy

Chapter Scattering in Wires I One Scatterer

is an eigenstate of the innite ordered p erio dic wire We can write the Green function of

the innite ordered wire as see eg or app endix A

j k aihk aj

G z dk

z k i

In order to p erform the diagonal subtraction required to renormalize the single scatterer t

matrices see section we need to compute the Green function in p osition representation

Equations are satised by

y with

a a

sin y with y

W W W

a a

y cos y with

W W W

where the cos and sin solutions are degenerate for each a

Since the eigenbasis of the wire is a pro duct basis the system is separable we can

apply the result of app endix A section A and wehave

G jaihajg z E

or in the p osition representation we will switch between the vector r and the pair x y

frequently in what follows

x x E y g G r r E G x y x y z y

a a a

B B

where the one dimensional free Green function is

ik xx

dk g x x z

z k i

p p

exp i z jx x j if Imf z g Re fz g

exp jjjx x j if z jj

When doing the Green function sum we have to sum over all of the degenerate

states at each energy Thus for all but the lowest energy mo de which is nondegenerate

the y part of the sum lo oks like

a a a ay y a

y sin y cos y cos y cos sin

W W W W W

Chapter Scattering in Wires I One Scatterer

which is sensible since the Green function of the periodic wire can dep end only on y y

So at this p oint wehave

ay y a

G x y x y z g x x z cos g x x z

o o

W W W W

As nice as this form for G is we need to do some more work To renormalize

free space scattering matrices we need to p erform the diagonal subtraction discussed in sec

tion In order for that subtraction to yield a nite result G must have a logarithmic

diagonal singularity The next bit of work is to make this singularity explicit

It is easy to see where the singularity will come from Since g x x jj

for E real there exists an M suchthat

G x y x y E const

whichdiverges Wenow pro ceed to extract the singularity more systematically

We b egin by substituting the denition of g and explicitly splitting the sum into

twoparts The rst part is a nite sum and includes energetically allowed transverse mo des

op en channels For these mo des the energy argument to g is p ositive and waves can

propagate down the wire The rest of the sum is over energetically forbidden transverse

mo des closed channels For these mo des waves in the x direction are evanescent We

N W k E a W and dene N the greatest integer such that E

a W E and have

ik jxx j E jxx j i

ay y e i ie

cos G x y x y E

W W k

W E

jxx j

ay y e

cos

W W

In order to extract the singularity we add and subtract a simpler innite sum

see D

exp jx x j

ay y

cos

W a

exp x x W

cosh x x h cos y y W

Chapter Scattering in Wires I One Scatterer

to G This gives

i E jxx j

ie i ay y

G x y x y E cos

W W

W E

ik jxx j

jx x j W exp a

k i a

jxx j

jx x j W exp a

ay y e

cos

W W a

exp x x W

cosh x x W cos y y W

This is an extremely useful form for numerical work For x x or y y we have

transformed a slowly converging sum into a much more quickly converging sum This is

dealt with in detail in D

In this form the singular part of the sum is in the logarithm term and the rest

of the expression is convergent for all x x y y In fact the remaining innite sum is

uniformly convergent for all x x as shown in D Wecannow p erform the diagonal

subtraction of G We b egin by considering the x x y y limit of G

i W i

lim G x y x y E

xx y y

W k i a

W E

W a

exp x x W

lim ln

xx y y

cosh x x W cos y y W

We can use equation D to simplify the limit of the logarithm

exp x x W

lim ln

xx y y

cosh x x W cos y y W

h i

x x y y lim ln

xx y y

ln lim ln jr r j

Since from A

Y ln ln jr r j E lim G r r z

Chapter Scattering in Wires I One Scatterer

wehave

x y x y E G r r E x y E lim G G

B B

i i W

W k i a

W E

W i

ln Y

W a

W E

which is indep endent of x and y as it must be for a translationally invariant system and

nite as proved in section

The case of a Dirichlet b ounded wire is very similar and so theres no need to

rep eat the calculation For the sake of later calculations we state the results here Wehave

G x y x y E

ajxx j

ik jxx j

i a a e e

sin y sin y

W W W k i a

ajxx j

jxx j

e e a a

sin y sin y

W W W a

xx y y

sinh sin

W W

xx y y

sinh sin

W W

and

x y E G

X X

i a a

y sin y sin

W W k i a W W a

a a

y i

ln sin ln Y

W E

Renormalization of the ZRI Scattering Strength

Recall that the full Green function may b e written

E G E G E T E G E G

B B B

where T E is computed via the techniques in chapter namely renormalization of the free space tmatrices

Chapter Scattering in Wires I One Scatterer

We b egin with a single scatterer in free space at x y The tmatrix of that

scatterer is see section

t E s E j ihj

which is renormalized by scattering from the b oundaries of the wire

T E S E j ihj

where

h i h i

s E G E S E s E s E G E

B B

Thus for a p erio dic wire

W i i

S E s E

W k i a

W E

ln Y

W a

W E

and in the Dirichlet b ounded wire

W i a i

sin S E s E

W y k i a

W E

a W

sin

W y a

ln sin ln Y

h E

So wehave

x y E x y E S E G x y x y E G G x y x y E G

B B B

with S E dened ab ove

From the Green function to Conductance

Since we nowhave the full Green function in the p osition representation we can

compute it between any two p oints in the wire We can use this to compute the unitless

conductance g of the wire via the FisherLee relation

g TrT T

Chapter Scattering in Wires I One Scatterer

where T is the transmission matrix i e T is the amplitude for transmitting from

channel a in the left lead to channel b in the right lead T is constructed from G r r E

via

T iv G x x E expik x k x

a a

ab b

where

D E

G x x E x a G E x b y G x y x y E y dy dy

is the Green function pro jected onto the channels of the leads Since the choice of x

are arbitrary we can cho ose them large enough that all the evanescent mo des are and x

arbitrarily small and thus we can ignore the closed channels So there are only a nite

number of propagating mo des and the trace in the FisherLee relation is a nite sum We

note that the prefactor v k k is there simply b ecause wehave normalized our channels

a a

via j y j dy rather than to unit ux More detail on this is presented in and

even more in the review

Computing the channeltochannel Green function

We write the Green function of the innite ordered wire as see eg

j k aihk aj

G z dk

z k i

D ik xx E

dk x a G z x b

z k i

Since

z z T z G z G G z G

B B B

and T z S z j r ihr j wehave

s s

x x z g x x z G

g x x z y S z y g x x z

s a a s s s

b b

o o

If the tmatrix comes from the multiple scattering of many zero range interactions

the tmatrix can b e written

tz jr i tz h r j

i j

ij ij

Chapter Scattering in Wires I One Scatterer

In that case wewillhave a slightly more complicated expression for the channeltochannel

Green function

G x x z g x x z

g x x z y tz y g x x z

i a a i j j

b b

o o

One Scatterer in a Narrow Wire

Whereas the wide wire can be mo deled classically the narrow wire is essentially

quantum mechanical In fact what we mean by a narrow wire is a wire with fewer than

one halfwavelength across it

When the wire b ecomes narrow the renormalization of the scattering amplitude

due to the presence of the wire walls b ecomes signicant This is immediately apparent when

we consider the case of a wire whichisnarrower than the scattering length of the scatterer

In essence we consider reducing a twodimensional scattering problem to a onedimensional

one by shrinking the width of the wire

Wed like to consider the case where the wavelength is much larger than the

width of the wire This cannot b e done for a wire with Dirichlet b oundary conditions since

there the lowest energy with one op en channel gives a half wavelength across the wire We

will instead use a wire with p erio dic b oundary conditions to consider this case If the wide

wire result remains valid we would exp ect the transmission co ecient to be zero when

there is only one op en channel since in the wide wire our scatterer reected exactly one

channel However as we already saw in gure this is not the case As we see in more

detail in gure the transmission co ecient has a surprisingly nontrivial b ehavior when

the width of the wire shrinks b elowa wavelength

Chapter Scattering in Wires I One Scatterer

1 Numerical Data a) Renormalized t-matrix Theory 0.8

0.6 T 0.4

0.2

0 0 0.5 1 1.5 2

Number of half wavelengths across wire

Figure Transmission co ecient of a single scatterer in a narrow p erio dic wire

It is p ossible to dene a sort of crosssection in one dimension for instance via

the d optical theorem from section In gure we plot the crosssection of the

scatterer rather than the transmission co ecient

Chapter Scattering in Wires I One Scatterer

2.2

1.8

1.6

1.4

1.2

1 Cross Section (1D)

0.8

0.6

0.4

Cross Section 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Half Wavelengths Across Wire

Figure CrossSection of a single scatterer in a narrow p erio dic wire

There are various unexp ected features of this transmission In particular why

should the transmission go to at some nite wire width Also why should the transmission

go to for a zero width wire A less obvious feature but one which is p erhaps more

interesting is that for very small widths the behavior of the transmission co ecient is

indep endent of the original scattering strength

These features are consequences of the renormalization of the scattering amplitude

Toseehow this transmission o ccurs we compute the full channelto channel Green function

for one scatterer in the center x of a narrowh p erio dic wire Since the wire is

narrow there is only one op en channel so the channel to channel Green function has only

one term

lim G x y x y E g x x E

p p

lim S E g x E g x E

o o

W W

Chapter Scattering in Wires I One Scatterer

The small W limit of S E is straightforward Using the result wehave

i W a

S E ln

W W a

W E

For small W

W EW

a a

and so the closedchannel sum may b e approximated

X X

EW EW EW EW

a a a

a a

EW a i

W E

S E ln

W E

Using the denition of g and factoring out the large iW E in S wehave

i E jxx j

lim G x y x y E e

E jxj i

p p

E W

p p

W E W E E a W

i i W i

i E jx j

p p

W E

which after the dust settles can b e rewritten as

i E jxx j

y E x y x lim G e

W a E W i E

E jxjjx j i

ln e

i W i

Since weareinterested only in transmission we can assume xandx sojx x j

jxj jx j With this caveat wehave

h a EW

i E jxjjx j

lim G x y x y E e ln

Finally from wehave

E W E a W

ln T i

Chapter Scattering in Wires I One Scatterer

since in units whereh m v E We have plotted this small W approximation

jT j in gure We see that there is go o d quantitative agreement for widths of fewer

than wavelength What is p erhaps more surprising is the reasonably go o d qualitative

agreement for the entire range of one wavelength

Chapter

Scattering in Rectangles I One

Scatterer

In this section we b egin the discussion of closed systems and eigenstates As with

the scatterer in a wire problem we have quite a bit of preliminary work to do We rst

compute the Background Green function G and the diagonal dierence h r jG G j ri

B B o

for the Dirichlet rectangle

We then p erform a simple numerical check on this result by using it to compute

the ground state energy of a square with a single zero range interaction at the center

We compare these energies with those computed by successiveoverrelaxation a standard

lattice technique for a hard disk of the same eective radius as sp ecied for the scat

terer This provides the simplest illustration of the somewhat subtle pro cess of extracting

discrete sp ectra from scattering theory

We then compute the background Green function and diagonal dierence for the

p erio dic rectangle a torus as we will need that in

Dirichlet b oundaries

The Background Green Function

We consider a rectangular domain DinR dened by

D l Wfx y j x l y Wg

Chapter Scattering in Rectangles I One Scatterer

We wish to solveSchrodingers equation

r E r

in the domain D sub ject to the b oundary condition

r D r

where D is the b oundary of D

As in the previous chapter weset Our equation reads

E x y

x y

The eigenfunctions of L in the ab ove domain with given b oundary con

x y

dition are

n x m y

x y sin sin

l W

which satisfy

n m

x y x y

nm nm

x y l W

We can thus write downabox Green function in the p osition representation

m y m y

n x n x

sin sin sin sin

l l W W

G x y x y E

n m

l W

which satises

z G x y x y z x x y y

x y

Resumming the Dirichlet Green Function

As in the previous chapter wed like to rewrite this double sum as a single sum We

can do this by rearranging the sum and recognizing a Fourier series First we rearrange

as follows

m y m y

sin sin

X X

n x n x

h W

sin sin G x y x y E

m n

l l l W

n m

l W

Chapter Scattering in Rectangles I One Scatterer

We dene

k E

E N

where x is the greatest integer less than equal or equal to x and then apply a standard

trigonometric identity to the pro duct of sines in the inner sum to yield

m y y m y y

cos cos

X X

n x n x W

W W

G x y x y E sin sin

l l l

k m

n m

WenowneedtheFourier series see eg

cos a x cos nx

a n a sin a a

for x So wehave

n x n x

G x y x y E sin sin

l l l

cos k W y y cos k W y y

n n

k sin hk

n n

and after applying trigonometric identities wehave

n x n x

y E G x y x sin sin

l l l

sin k W y sin k y

n n

k sin k W

n n

Wenow rewrite our expression for G yet again

n x n x

sin sin sink y sink y W

n n

l l

G x y x y E

l k sin k W

n n

n x n x

sin sin sinh y sinh y W

n n

l l

l sinh W

n n

Chapter Scattering in Rectangles I One Scatterer

Prop erties of the Resummed G

As we will show below the sum in is not uniformly convergent as y y

However this limit is essential for calculation of renormalized tmatrices We can however

add and subtract something from each term so that we are left with convergent sums and

singular sums with limits we understand

Since the sum is symmetric under y y this is obvious from the physical sym

metry as well as the original double sum equation we may cho ose yy We dene

y y and rewrite the sum as follows

n x n x

sin sink y sin k y W sin

n n

l l

G x y x y k

l k sin k W

n n

where

n x n x sinh y sinh W y

n n

u sin sin

l l sinh W

n n

We rewrite u to simplify the following analysis

n x n x

u sin sin

l l

h i

y W W y

n n n n

e e e e

We dene

n x n x

g sin sin e

l l

We observe that for M n

exp W

j g j exp

n n

exp W

exp

n exp W El

exp exp

l n

n exp W El

exp exp

M l M

Chapter Scattering in Rectangles I One Scatterer

and therefore

exp W El M

g exp exp

M l

exp

which converges for but not uniformly Thus we cannot take the limit inside

the sum However if we can subtract o the diverging part we may be able to nd a

uniformly converging sum as a remainder With that in mind wedene

n n x l n x

sin exp h sin

l l n l

We rst note that the sum h may b e p erformed exactly see app endix D yielding

h i

sin sinh

l l

h i

h ln

sin sinh

l l

Wenow subtract h fromg toget

n n

g h

n n

n x l n x

sin sin

l l n

n n El El

exp exp e

n l n l

As weshow inappendixDwe can place a rigorous upp er b ound on g h

n n

for suciently large M

More preciselygiven

l l El

M max E E ln

then

El l M

g h h exp

n n

M l

and g h is uniformly convergent in W

n n

Wenowhave

G x y x y E

n x n x

sin sink y sin k y W sin

n n

l l

k sin k W

n n

Chapter Scattering in Rectangles I One Scatterer

h h y h W h W y

n n n n

h h y h W h W y

n n n n

g h

n n

g y h y

n n

g h h h

n n

g W y h W y

n n

The innite sums of hs can b e p erformed see app endix D so wehave

G x y x y E

n x n x

sin sink y sink y W sin

n n

l l

k sin k W

n n

h h y h W h W y

n n n n

h i h i

xx xx y

sinh sinh sin sin

l l l l

h i h i

ln ln

xx xx y

sinh sinh sin sin

l l l l

h h i i

xx xx W W y

sin sin sinh sinh

l l l l

h i h i

ln ln

xx W xx W y

sin sinh sin sinh

l l l l

g h

n n

g y h y

n n

g h h h

n n

g h y h h y

n n

When using these expressions we truncate the innite sums at some nite M The analysis

in app endix D allows us to b ound the truncation error With this in mind we dene a

Chapter Scattering in Rectangles I One Scatterer

quantity whichcontains the nonsingular part of G truncated at the M term

S x y x y E

n x n x

sin sink y sink y W sin

n n

l l

k sin k W

n n

h h y h W h W y

n n n n

h h i i

xx y xx W

sin sinh sin sinh

l l l l

i i h h

ln ln

y W xx xx

sinh sinh sin sin

l l l l

i h

W y xx

sinh sin

l l

i h

W y xx

sinh sin

l l

g h

n n

g y h y

n n

g h h h

n n

g h y h h y

n n

i h i h

y y xx

sinh sin

l l

i h i h

ln G x y x y E S x y x y E

B M

y y xx

sinh sin

l l

Wenowhave an approximate form for G whichinvolves only nite sums and straightfor

ward function evaluations This will b e analytically useful when we subtract G from G

o B

It will also provenumerically useful in computing G

Explicit Expression for G r k

Within the dressedt formalism we often need to calculate G r r z G r r z

B o

This quantity deserves sp ecial attention b ecause it involves a very sensitive cancellation of

innities Recall that

ln k jr r j Y lim G r r k lim

rr rr

Chapter Scattering in Rectangles I One Scatterer

where Y is the regular part of Y as dened in equation A of section A

If G G is to be nite we need a canceling logarithmic singularity in G

B o B

Equation makes it apparent that just such a singularity is also present in G The

logarithm term in that equation has a denominator whichgoesto as r r We carefully

manipulate the logarithms and write an explicitly nite expression for G G

B o

G r E

n x

sin

S x y x y E

i x

ln kl Y ln ln sin

Ground State Energies

One nice consequence of equation is that it predicts a simple formula for the

ground state energy of one scatterer with a known Green function background Sp ecically

implies that p oles of T E o ccur when

G r E

B s

From section wehave

Y ka

sE J ka

for ka wehave

i i

ln Y ln kl ln

G r E so We dene F E

Y ka

F E ln kl Y

J ka

n x

sin

S X Y X Y

ln ln sin X

We can use the ka expansion of the Neumann function to simplify F abit

h i

R R

F E ln al Y ka Y

o o

Chapter Scattering in Rectangles I One Scatterer

n x

sin

S x y x y

ln ln sin

In gure we compare the numerical solution of the equation F E with a

numerical simulation p erformed with a standard metho d SuccessiveOver Relaxation

Perio dic b oundaries

The Background Green Function

We consider a rectangular domain inR dened by

l Wfx y j xl y Wg

We wish to solveSchrodingers equation

r r E

in the domain sub ject to the b oundary conditions

y l y

x y x y

x x

x xl

x x W

x y x y

y y

y y W

Weset Our equation reads

E x y

x y

in the ab ove domain with given b oundary con The eigenfunctions of L

x y

dition are

n x m y

x y sin sin

l W

m y n x

cos x y cos

l W lW

Chapter Scattering in Rectangles I One Scatterer

Succesive OverRelaxation and DressedtMatrix Ground State Energies

SOR

Dressedt Matrix

Scattering Length radius

Numerical Simulation and Dressedt Theory for Ground State Energy Shifts small

Simulation

Full Theory

Scattering Length

Figure Comparison of Dressedt Theory with Numerical Simulation

Chapter Scattering in Rectangles I One Scatterer

n x m y

x y sin cos

l W

m y n x

x y sin cos

l W

all of whichsatisfy

m n

x y x y

nm nm

x y l W

Note that the m and n is p ermissible but only the cosine state survives so there is

no degeneracy Wecanthus write down a b ox Green function in the p osition representation

n jxx j m jy y j

cos cos

l W

G x y x y E

n m

lW E lW

l W

where trigonometric identities have b een applied to collapse the sines and cosines into just

two cosines We note that this Green function dep ends only on jx x j and jy y j as it

must

Resumming G

As with the Dirichlet case wed like to resum this Green function to make it a

single sum and to create an easier form for numerical use and the G G subtraction

o B

We b egin by reorganizing G as follows

m jxx j

cos

X X

n jy y j

G x y x y E cos

n m

lW E lW W

n m

l W

and then apply the following Fourier series identity see eg to the inner sum

cos kx cos a x

a k a sin a a

for x

We dene

E N

k E

Chapter Scattering in Rectangles I One Scatterer

X jx x j

Y jy y j

where x is the greatest integer equal to or less than x Wenowhave

i h i h

l l

E X cos nY cos k X cos

G X Y E

p p

l l

h E sin E k sin k

n n n

h i

cos cosh nY X

sinh

n n

Wenowfollow a similar derivation to the one for Dirichlet b oundaries Wecho ose

an M N such that we may approximate We can then approximate G by a

n B

nite sum plus a logarithm term arising from the highest energy terms in the sum That

sum lo oks like

nY e cos

We can sum this using see eg where X X mo d

k x

We dene

h i h i

l l

cos cos X nY cos k X E

X Y E S

p p

l l

h E sin E k sin k

n n n

h i

nY cosh X cos

sinh

n n

and write our approximate G as

M X X

W W

ln e Y e G X Y E S X Y E cos

cos nY e

Chapter Scattering in Rectangles I One Scatterer

Explicit Expression for G r E

With this resummed version of G we can write an explicitly nite expression for

G r E G r r E G r r E

B B o

though b oth G r r E and G r r E are logarithmically innite

B o

As with the Dirichlet case all we need to do is carefully manipulate the logarithm

in G in the X Y limit Our answer is

kW i

M R

G r E S E ln Y X

p o

Chapter

Disordered Systems

Disorder Averages

Averaging

When working with disordered systems we are rarely interested a particular real

ization of the disorder but instead in average prop erties The art in calculating quantities

in such systems is cleverly approximating these averages in ways which are appropriate for

sp ecic questions In this section we will consider only the average Green function of a dis

ordered system hGi rather than for instance higher moments of G Go o d treatments of

these approximations and more p owerful approximation schemes may b e found in

Supp ose at xed energy we have N ZRIs with individual tmatrices t E

s jr ihr j Any prop erty of the system dep ends at least in principle on all the variables

i i j

s r We imagine that there are no correlations between dierent scatterers lo cation or

i i

strength and that each has the same distribution of lo cations and strengths Thus we can

dene the ensemble average of the Green function op erator GE

hGi dr ds r s G

i i r i s i

Wewilltypically use uniformly distributed scatterers and xed scattering strengths

In this case r where V is the volume of the system and s s s s where

r i s i o i o

s is the xed scatterer strength

We can write G as G G TG and since G is indep endent of the disorder

o o o o

hGi G G h T i G

o o o

Chapter Disordered Systems

Thus wemust compute hT i This is such a useful quantity there is quite a bit of machinery

develop ed just for this computation

SelfEnergy

The selfenergy is a sort of average p otential though it is not hV i dened via

h Gi G G hGi

o o

and thus

h Gi G

This last equation explains whywecallE the selfenergy

G E H

Within the rst two approximations we discuss the selfenergy is just prop ortional to the

identity op erator so it can b e thought of as just shifting the energy

We can also use to nd in terms of h T i

hT i G h T i

or for hT i in terms of

hT i G

Thus knowledge of either hT i orisequivalent

Recall that G G G TG G G VG G VG VG means that the

o o o o o o o o o

amplitude for a particle to propagate from one p oint to another is the sum of the amplitude

for it to propagate from the initial p oint to the nal point without interacting with the

the initial p oint to the p otential potential and the amplitude for it to propagate from

interact with the p otential one or more times and then propagate to the nal p oint We

can illustrate this diagrammatically

where solid lines represent free propagation G and a dashed line ending in an in

represents an dicates an interaction with the impurity potential V Each dierent

interaction with the impurity p otential at a dierent impurity When multiple lines connect

Chapter Disordered Systems

to the same interaction vertex the particle has interacted with the same impuritymultiple

times

An irreducible diagram is one which cannot b e divided into two subdiagrams just

by cutting a solid line a free propagator The selfenergy is equivalentto a sum over

only irreducible diagrams with the incoming and outgoing free propagators removed

which is enough to evaluate G since we can build all the diagrams from the irreducible ones

by adding free propagators

h Gi G G G G G G G G

o o o o o o o o

There are a variety of standard techniques for evaluating the selfenergy The

simplest approximation used is known as the Virtual Crystal Approximation VCA

This is equivalent to replacing the sum over irreducible diagrams by the rst diagram in the

sum ie h V i Since we dont use the p otential itself this approximation is actually

more complicated to apply then the more accurate average tmatrix approximation ATA

We note that in a system where the impuritypotential is known hV i is just a real number

and so the VCA just shifts the energy by the average value of the p otential

The ATA is a more sophisticated approximation that replaces the sum bya

sum of terms that involve a single impurity

but this is up to averaging the same as the single scatterer tmatrix t Thus the ATAis

equivalenttoh t i This approximation neglects diagrams like

which involve scattering from two or more impurities We note that scattering from two

or more impurities is included in G just not in Of course while scattering from several

impurities is accounted for in Ginterference b etween scattering from various impurities is

Chapter Disordered Systems

neglected since diagrams which scatter from one impurity than other impurities and then

the rst impurity again are neglected That is

is included but

is not At low concentrations such terms are quite small However as the concentration

increases these diagrams contribute imp ortant corrections to G One such correction comes

from coherentbackscattering whichwell discuss in greater detail in section

We will use the ATAbelowto show that the classical limit of the quantum mean

free path is equal to the classical mean free path For N uniformly distributed xed strength

scatterers the average is straightforward

N N

X Y X

t dr ds s s s r r

i i i i o i i

i i i

For each term in the sum the r delta function will do one of the volume integrals and the

rest will simply integrate to canceling the factors of V out front The s delta functions

will do all of the s integrals leaving

t s ns

i o o

Thus the self energy is simply prop ortional to the scattering strength multiplied by the

concentration We note that s is in general complex and this will make the poles of the

Green function complex implying an exp onential decay of amplitude as a wave propagates

We will interpret this decayasthewavemechanical mean free time

Mean Free Path

Classical Mechanics

Consider a domain of volume V in ddimensions with reectivewalls V has units

of length Supp ose we place N scattering centers in this domain at random uniformly

cross section has units of length Now supp ose distributed each with classical

Chapter Disordered Systems

wehaveapoint particle that has just scattered o of one of the scattering centers It now

p oints in a random direction What is the probability that it can travel a distance without

scattering again

If the particle travels a distance x without scattering then there must b e a tub e

of volume x which is empty of scattering centers The probability of that is given by the

pro duct of the chances that eachoftheN scatterers without the reectivewalls this would

be N but since we will take N large and wedohave reectivewalls well leaveitasN

is not in the volume x That chance is so

P x

More precisely P x is the probability that the free pathlength is less than or equal

to x We dene n NV the concentration So

x n

P x

We taketheN while n const limit whichisvalid for innite systems and

a good approximation when the mean free path is smaller than the system size Wehave

n x N

xe P x lim P

and thus the quasiclassical mean free path is

Z Z

hxi x P x dl P x dl

x n

Quasiclassical indicated by the subscript qc here means that the transp ort between

scattering events is classical but the crosssection of each scatterer is computed from quan

tum mechanics

Quantum Mechanics

The mean free path is not so simple to dene for a quantum mechanical system

After all a particle no longer has a tra jectory or a well dened distribution of path lengths

What then do we mean by the mean free path in a quantum mechanical system

One p ossibility is simply to replace the classical crosssection in the expression

with the quantum mechanical crosssection well refer to this as the quasiclassical

Chapter Disordered Systems

mean free path In what follows well show that this is equivalent to the lowdensity

weakscattering approximation to the selfenergy discussed ab ove

We b egin by noting that the free Green function takes a particularly simple form

in the momentum representation

p p

G p p E

E E

From this and the lowdensityweakscattering approximation to wehave

p p

Gp p E

E E

If we write iwehave

p p

Gp p E

E i E

Nowwe consider the Fourier transform of this Green function with resp ect to energy which

gives us the timedomain Green function in the momentum representation we are ignoring

the energy dep endence of only for simplicity

iE t

Gp p t Gp p E dE e

p p

iE t

dE e

E i E

i E t t

i p p e e

which implies an exp onential attenuation of the wave if Imfg is negative

For the ATA wehave ns whichforatwo dimensional ZRI with scattering

iJ Ea

length aisn and thus

H Ea

J Ea

H Ea

which is manifestly negative

We can asso ciate the damping with a mean free time via Since at

E we have for the mean xed energy the velo city in units where h m is v

free path

Ea H

n n

J Ea

o repro ducing the quasiclassical result

Chapter Disordered Systems

Prop erties of Randomly Placed ZRIs as a Disordered

Potential

In this section we will be concerned with averages of T in the momentum rep

resentation As in we will assume the validity of the lowdensity weakscattering

approximation to T

T s j r ihr j

i i i

In momentum space this can b e written

i kk r

T k k s e k j T j k

Using the average wehave

T k k N h si f k k

where

iqr

f q e dr

The function f hasatwo imp ortant prop erties Firstly f which implies as wesaw

in that h T i N h si Also when the b ounding region is all of space wehave

q f q

Together these prop erties imply that the average ATA tmatrix cannot change the momen

tum of a scattered particle except insofar as the system is nite A nite system will givea

region of low momentum where momentum transfer can o ccur but for momenta larger than

L momentum transfer will still b e suppressed

Wenow consider the second momentofT or more sp ecically

D E

T k k T k k

Since

ikk r ikk r

j i

T k k s s e e

wehave

N N

D E D E

X X

T k k jhsij f k k jsj

i j

D E

f k k N jsj N N jh sij

Chapter Disordered Systems

Thus

i E E hD D

jhsij f k k N jsj T k k T k k

We note that if s s i wehave

i o

D E h i

T k k T k k N js j f k k

In this case

E D

jT k kj jhT k kij

At this point it is worth considering our geometry and computing f explicitly

Well assume we are placing scatterers in a rectangle with length xdirection a and width

ydirection b Then wehave up to an arbitrary phase

q x q y

x y

f q sinc sinc

a b

where

sin x

sinc x

Thus jf k k j is zero for k k and then grows to for larger momentum transfer

The zero momentum transfer hole in the second momentofT is an artifact of a p otential

made up of a xed number of xed size scatterers To make contact with the standard

condensed matter theory of disordered p otentials we should allow those xed numb ers to

varythus making a more nearly constant second momentofT We can do this easily enough

byallowing the size of the scatterers to varyaswell Then jsj j sj In fact

j sj jsj we should cho ose a distribution of scatterer sizes such that jsj

Of course the scatterer strength s is not directly prop ortional to the scattering

length For example if the scattering length varies uniformly over a small range a It is

straightforward to show that for small a

ka ds

jsj j sj

dka

Eigenstate Intensities and the PorterThomas Distribu

tion

One interesting quantity in a b ounded quantum system with discrete sp ectrum

eg a Dirichlet square with random scatterers inside is the distribution of wavefunction

Chapter Disordered Systems

intensities t at xed energy E

t j r j P t E E

where is the mean level spacing and r is a sp ecic p oint in the system Since P t is a

probability distribution wehave P t dt and since the integrated square wavefunc

tion is normalized to wealsohave tP t dt

Though computing this quantity in general is quite hard we can as a baseline

compute it in one sp ecial case As a naive guess at the form of the wavefunction in a

disordered system we conjecture that the wavefunction is a Gaussian random variable at

eachpoint in the system That is we assume that the distribution of values of is

bj j

where a and b are constants to b e determined We note that in a b ounded system we can

make all the wavefunctions real We pro ceed to determine the constants a and b First we

x dx whichgives use the fact that is a probability distribution so

a e dx a

implying b a Wealsoknow that the normalization of implies x x dx

whichimplies

a x

a x e dx

So wehave implying a

j j

From this we can compute P tvia

p p

Z Z

x t x t

x dx t x x dx P t

jxj

We use the delta functions to do the integral and have

P t

After substituting our previous result for wehave

P t e

Chapter Disordered Systems

which is known as the PorterThomas distribution

If timereversal symmetry is broken eg by a magnetic eld the wavefunction

will in general b e complex In that case we can use the same argument where the real part

and imaginary part of are each Gaussian random variables In that case we get

P te

This dierence b etween timereversal symmetric systems and their nonsymmetric counter

parts is a recurring motif in disordered quantum systems

A derivation of the ab ove results from Random Matrix Theory using the as

sumption that the Hamiltonian is a random matrix with the symmetries of the system is

available many places for example In this language the Hamiltonian of timereversal

invariant systems are part of the Gaussian Orthogonal Ensemble GOE whereas Hamil

tonians for systems without timereversal symmetry are part of the Gaussian Unitary

Ensemble GUE We will adopt this bit of terminology in what follows

Of course most systems do not b ehave exactly as the appropriate random matrix

ensemble would indicate These dierences manifest themselves in a variety of prop erties

of the system In the numerical simulations which follow in chapters and we will see

these departures quite clearly

For disordered systems GOE behavior is exp ected when there are many weak

scattering events in every path whichtraverses the disordered region guaranteeing diusive

transp ort without signicant quantum eects from scattering phases More preciselytosee

GOE b ehavior we exp ect to need a mean free path whichismuch smaller than the system

size L but much larger than the wavelength We will see this limit emerge

in wavefunction statistics in chapter

Lo calization Weak

In the previous section we discussed one consequence of the assumption of a

random wavefunction Of course the wavefunction is not random it shows the consequences

of the underlying classical dynamics In the next few sections we explore some consequences

of the underlying dynamics for the quantum prop erties

Weak lo calization is a generic name for enhanced probability of nding a particle

in a given region due to short time classical return probability In quantized classically

Chapter Disordered Systems

chaotic but not disordered systems wavefunction scarring is the b est known form

of weak lo calization In disordered systems the most imp ortant consequence of of weak

lo calization is the reduction of conductance due to coherent backscattering

It is not dicult to estimate the coherent backscattering correction to the con

ductance We b egin by noting the conductance we exp ect for a wire with no coherent

backscattering Sp ecically when L we exp ect the DC conductivity of a

disordered wire to satisfy the Einstein relation

e D

where is the conductivity e is the charge of the electron is the ddimensional density

of states p er unit volume and D is the classical diusion constant

The DC conductivity is prop ortional to P r r the probability that a particle

starting at p oint r on one side of the system reaches r on the other side Quantum

mechanically this quantity can b e evaluated semiclassically bya sum over classical paths

P r r A

tegral of the classical action over the path The where A jA j e and S is the in

p p p

quantum probability diers from the classical in the interference terms

P r r P r r A A

classical

p p

Typically disorder averaging washes out the interference term However when

r r the terms arising from paths which are timereversed partners will have strong

interference even after averaging since they will always have canceling phases Since every

path has a time rev ersed partner wehave

h P r ri P r r

classical

But this enhanced return probability implies a suppressed conductance since P r r dr

must b e smaller by a factor of P r r by conservation of probability Thus

classical

due to this interference eect

But P r r is something we can compute straightforwardly If we dene

classical

R t to be the probability that a particle which left the p oint r at time t returns at

Chapter Disordered Systems

time twehave

P r r R tdt

classical

The lower cuto v is there since our particle must scatter at least once to return and

that takes a time of order the mean free time The upp er cuto is present since we have

only a nite disordered region and so after a time t L D the particle has diused out

and will not return For a square sample L is ambiguous up to a factor of The upp er

cuto can also b e provided by a phase coherence time If particles lose phase coherence

for instance byinteraction with a nite temp erature heat bath only paths whichtake less

time than will interfere In this case the expression for the classical return probabilityis

slightly mo died

P r r e R tdt

classical

The return probability R tdt can b e estimated for a diusive system Of all the

tra jectories that scatter only those that pass within a volume v dt of the origin contribute

The probability that a scattered particle falls within that volume is just the ratio of it

to the total volume of diusing tra jectories Dt V where d is the eective number of

dimensions the numb er of dimensions of the disordered sample andV d

is the volume of the unit sphere in ddimensions this is easily calculated using pro ducts of

Gaussian integrals see eg pp and D v d So

v dt

R tdt

Dt V

With this expression for R tdt in hand wecandotheintegral and get

t d

P r r

lnt d

classical

D V

t d

For future reference we state the sp ecic results for one and two dimensions

Rather than state the result of the estimation ab ove we give the correct leading order

results computed by diagrammatic p erturbation theory see eg These results

have the same dep endence on and L but slightly dierent prefactors than our estimate

L d

ln d

Chapter Disordered Systems

Strong Lo calization

Strong lo calization characterized by the exp onential decay of the wavefunction

around some lo calization site has dramatic consequences for b oth transp ort and wavefunc

tion statistics Strong enough disorder can always exp onentially lo calize the wavefunction

Weak disorder can sometimes lo calize the wavefunction but this dep ends on the strength

of the disorder and the dimensionalityof the system Below we sketch out an argument

originally due to Thouless which claries how strong lo calization occurs as well as its di

mensional dep endence

The idea is simple As a wavepacket diuses we consider at each instant of time

a ddimensional box which surrounds it At each moment of volume Cr Dt

we can pro ject the wavepacket onto the eigenstates of the surrounding b ox The average

energy level spacing in that b oxis

d d

E E Dt

However the wavepacket has b een diusing for a time t and so we can use its auto correlation

function At dened by

iE t

Ath j t i h j n i e

to lo ok at the sp ectrum of the wavepacket If we can completely resolve the sp ectrum no

more dynamics o ccurs except the phase evolution of the eigenstates of the box Thus the

We wavepacket has lo calized After a time t we can resolve levels with spacing E

dene the Thouless Conductance g via

d d

g hD E t

If g we can resolve the levels of the wavepacket and it lo calizes Conversely if g

we cannot resolve the eigenstates making up the wavepacket and diusion continues

The rst conclusion we can draw from this argument is the dimensional dep endence

of the t limit of g In one dimension it is apparent that lim g and thus all

states in a weakly disordered one dimensional system lo calize In two dimensions this

argument is inconclusive and seems to dep end on the strength of the disorder In fact it is

disordered two dimensional systems lo calize as well but b elieved that all states in weakly

Chapter Disordered Systems

with exp onentially large lo calization lengths For d lim g and we exp ect the

states to b e extended

When measuring conductance the dierence b etween lo calized and extended states

in the disordered region is dramatic If the state is exp onentially lo calized in the disordered

region it will not couple well to the leads and the conductance will b e suppressed We can

lo ok for this eect by lo oking at the conductance of a disordered wire as a function of the

length of the disordered region If the states are extended we exp ect the conductance to

varyasL whereas if the states are lo calized we exp ect the conductance to vary as e

where is the lo calization length

The eect of exp onential lo calization on wavefuntion statistics is equally dramatic

For instance in two dimensions since we knowthat

jrr j

r e

we can compute

Z Z

rdr t j rj rdr t e P t

In gure we plot the PorterThomas distribution and the exp onential lo calization dis

p p

tribution for a lo calization length and so we can see just how stark

this eect is

Anomalous Wavefunctions in Two Dimensions

The description ab ove do es not explain much ab out how strong lo calization oc

curs either in the timedomain or as a function of disorder This transition is particularly

interesting in two dimensions since even the existence of strongly lo calized states in weakly

disordered systems is subtle

While it is not at all obvious that large uctuations in wavefunction intensity in

extended states is related to strong lo calization it seems an interesting place to lo ok While

this was p erhaps the original reason these large uctuations were studied they have spawned

an indep endent set of questions The simplest of those questions is the one answered by

P t namelyhow often do these large uctuations o ccur

Various eldtheoretic techniques have b een broughtto b ear on this problem

All predict the same qualitativebehavior of P t Namely that for large values of

Chapter Disordered Systems 30 25 Porter-Thomas 20 t 15 Strong localization, localization length = L/10 Strong localization, localization length = L/100 10 5 0 1 0.1 0.01

0.001 1e-05 1e-06 1e-07 1e-08

0.0001

P(t)

Figure PorterThomas and exp onential lo calization distributions compared

Chapter Disordered Systems

t P t has a lognormal form

C ln t

P t e

Various workers have argued for dierent forms for C which dep ends on the energy E the

mean free path the system size L and the symmetry class of the system existence of

timereversal symmetry The lognormal distribution lo oks strikingly dierent from either

the PorterThomas distribution or the strong lo calization distribution In gure we plot

all of these distributions for an example choice of for t We only consider large

values of t b ecause for small values of t P t will have a dierent form For small enough

values of t these calculations predict that P t has the PorterThomas form This allows

them to b e trivially distinguished from the stronglo calization form whichisvery dierent

from PorterThomas for small t

We will fo cus in particular on two dierent calculations The rst app earing

in uses the direct optimal uctuation metho d DOFM and predicts

lnF kL

where D is an O constant Another calculation app earing in and using the sup er

symmetric sigma mo del SSSM predicts

lnF L

where for timereversal invariant systems and for systems without timereversal

symmetry and D is an O constant

in gure we plot b oth of these In order to see the dierences between C

co ecients versus b oth wavelength and mean free path for various values of D and D

Conclusions

In this chapter we reviewed material on quenched disorder in op en and closed

metallic systems In the chapters that followwe will often compare to these results or try

to verify them with numerical calculations

Chapter Disordered Systems 100 90 Porter-Thomas 80 Log-normal form 70 t 60 Strong localization, localization length = L/100 50 40 30 20

1e-06 1e-08 1e-10 1e-12 1e-14 1e-16 1e-18 1e-20 1e-22 1e-24

0.0001

P(t)

Figure PorterThomasexp onential lo calization and lognormal distributions compared

Chapter Disordered Systems

DOFM

SSSM

DOFM

SSSM

Figure Comparison of lognormal co ecients for the DOFM and SSSM

Chapter

Quenched Disorder in D Wires

We consider an innite wire of width W with a disordered segment of length L as

in Fig This wire will b e taken to have p erio dic b oundary conditions in the transverse

y direction Leads Disordered Region W

Figure The wire used in op en system scattering calculations

To connect with the language of mesoscopic systems we may think of the dis

ordered region as a mesoscopic sample and the semiinnite ordered regions on each side

as p erfectly conducting leads For example one can imagine realizing this system with an

AlGaAs quantum dot

We can measure many prop erties of this system For instance wehave used renor

malized scattering techniques to mo del a particular quantum dot Typical quantum

Chapter Quenched Disorder in D Wires

dot exp eriments involve measuring the conductance of the quantum dot as a function of

various system parameters eg magnetic eld dot shap e or temp erature Thus we should

consider how to extract conductance from a Green function

In this section we discuss numerically computed transp ort co ecients This allows

us to verify that our disorder p otential has the prop erties that we exp ect classically Since

we are interested in intensity statistics and how they dep end on transp ort prop erties it is

imp ortantto compute these prop erties in the same mo del we use to gather statistics For

instance as discussed in the previous chapter the ATA breaks down when coherent back

scattering contributes a signicantweak lo calization correction to the diusion co ecient

In this regime it is useful to verify that the corrections to the transp ort are still small enough

to use an expansion in When the disorder is strong enough strong lo calization o ccurs

and a dierent approximation is appropriate

Of course transp ort in disordered systems is interesting in its o wn right Our

metho d allows a direct exploration of the theory of weak lo calization in a disordered two

dimensional systems

Transp ort in Disordered Systems

Diusion from Conductance

Computing the conductance of a disordered wire is not much dierentthancom

puting the conductance of the one scatterer system We use to compute the channel

tochannel Green function G from the multiple scattering tmatrix nd the transmission

matrix T from and compute the conductance e h using the FisherLee

relation

In chapter we assumed the classical transp ort to and from the scatterer was

ballistic and derived an expression whichrelated the intensity reection co ecient which

observed conductance to the scattering cross section of an can be trivially related to the

obstacle in the wire When many scatterers are placed randomly in the wire this assumption

is no longer valid Most scattering events o ccur after other scattering events and thus the

probability of various incident angles is uniform rather than determined by the channels

of the wire Also there will typically b e many scattering events So when we have many

randomly placed scatterers we will assume that the transp ort is diusive This will give

Chapter Quenched Disorder in D Wires

us a dierent relationship between the reection co ecient and the crosssection of each

obstacle

It is instructive to compute the exp ected reection co ecient as a function of the

concentration and crosssection under the diusion assumption and compare that result

computed under the ballistic assumption R toR

D B

We b egin from the relation between the intensity transmission co ecient T

and the number of open channels N

ie the transmission co ecient is just the unitless conductance p er channel From this we

see that do es not go to for an emptywireaswemight exp ect Only a nite amountof

ux can b e carried in a wire with a nite numberofopenchannels and this gives rise to a

socalled contact resistance Wethus split the unitless conductance into a disordered

region dep endentpart anda contact part

e h h

e N

s c

where the contact part is chosen so that lim

T s

At this p oint we invoke the assumption of diusive transp ort This allows us to

use the Einstein relation to relate the conductivity of the sample to the diusion

constantvia

e D

where is the density of states p er unit volume in two dimensions The LW

in frontof relates conductivity to conductance in two dimensions

Wenowhave

h hW D N L

e hW D N LN hW D

c c

If we substitute this into wehave

hW D

LN hW D

As in the previous chapters we cho ose units where h and m so h m

We also cho ose units where the electron charge e WenowuseD vd k and

N kW and get

Chapter Quenched Disorder in D Wires

and

R T

D D

Finallywe use l LW Ntoget

For very few scatterers we usually have N W and thus

R N

Compare this to the ballistic result and we see that they are related by R R This

B D

factor arises from the dierent distributions of the incoming angles In practice there can

b e a large crossover region b etween these twobehaviors when the nonuniform distribution

of the incoming angles can make a signicant dierence in the observed conductance

We note that can b e rearranged to yield

L T

R R

D D

which we will use as a way to compare numerical results with the assumption of quasi

classical diusion

Numerical Results

As our rst numerical calculation we lo ok at the mean free path as a function of

scatterer concentration at xed energy In gure we plot vs n for a p erio dic wire of

width with a length disordered region There are ab out wavelengths across the wire

is and thus the wire has op en channels We use scatterers with maximum swave cross

section of We consider concentrations from to scatterers p er unit area At each

concentration we compute the conductance of realizations of the scatterers and average

them This range of concentrations corresp onds to a range of quasiclassical mean free path

from to We plot b oth the quasiclassical mean free path and the numerical result

computed via

From the gure we see that the numerical result is noticeably smaller than the

quasiclassical result though it diers by at most over the range of concentrations

studied This dierence is partially explained by two things At low concentrations the

numerically observed mean free path is smaller than the quasiclassical b ecause the transp ort

Chapter Quenched Disorder in D Wires 700 600 quasi-classical numerical (diffusive) 500 400 concentration 300 200 100

0.5 0.4 0.3 0.2 0.1

0.55 0.45 0.35 0.25 0.15 0.05

mean free path free mean

Figure Numerically observed mean free path and the classical exp ectation

Chapter Quenched Disorder in D Wires 700 600 quasi-classical 500 numerical (corrected diffusive) 400 concentration 300 200 100

0.5 0.4 0.3 0.2 0.1

0.55 0.45 0.35 0.25 0.15 0.05

mean free path free mean

Figure Numerically observed mean free path after rstorder coherent backscattering correction and the classical exp ectation

Chapter Quenched Disorder in D Wires

is not completely diusive As wesaw at the b eginning of this chapter ballistic scattering

leads to a larger reection co ecient than diusive ones This leads to an apparently smaller

mean free path More interestingly there is coherent backscattering at all concentrations

see section though its eect is larger at higher concentration since the change in

conductance due to weak lo calization is prop ortional to We correct the conductance

via to rst order in and then plot the corrected mean free path and the classical

exp ectation in gure The agreement is clearly b etter though there is clearly some

other source of reduced conductance At the lowest concentrations there is still a ballistic

correction as noted ab ove but this cannot accountforthelower than exp ected conductance

at thigher concentrations where the motion is clearly diusive

As increases the dierence b et ween the classical and quantum b ehavior do es

as well For large enough this will lead to lo calization In order to verify that the

transp ort is still not lo calized we compute the transmission co ecient vs the length of

the disordered region for xed concentration If the transp ort is diusive T will satisfy

which predicts T L for large L If instead the wavefunctions in the disordered

region are exp onentially lo calized T will fall exp onentially with distance ie T e

for two dierent concentrations and energies In b oth plots In gure we plot T versus L

realizations of the disorder p otential are averaged at each point In gure a there are

wavelengths across the width of the wire as in the previous plot and the concentration

is scatterers p er unit area T is clearly more consistent with the diusive exp ectation

than the strong lo calization prediction

We compare this to gure b where the wavelength and mean free path are

comparable and the wire is only a few wavelengths wide What we see is probably quasi

one dimensional strong lo calization Consequently T do es not satisfy but rather has

an exp onential form We note that the data in gure b is rather erratic but still much

of more strongly consistent with strong lo calization than diusion Numerical observation

exp onential lo calization in a true two dimensional system would b e very dicult since the

two dimensional lo calization length is exp onentially long in the mean free path

Numerical Considerations

Numerical computation using the techniques outlined in chapters and is quite

simple For each set of parameters computation pro ceeds as follows

Chapter Quenched Disorder in D Wires

1 Numerical a) Diffusive 0.9 Strong localization (fitted)

0.8

0.7

0.6

T 0.5

0.4

0.3

0.2

0.1

0 0 0.5 1 1.5 2 L

1 Numerical b) Diffusive 0.9 Strong localization (fitted)

0.8

0.7

0.6

T 0.5

0.4

0.3

0.2

0.1

0 0 0.5 1 1.5 2

Figure Transmission versus disordered region length for a diusive and b lo calized wires

Chapter Quenched Disorder in D Wires

Compute the random lo cations of N scatterers

Compute the renormalized tmatrix of each scatterer

Compute the scattererscatterer Green functions for all pairs of scatterers

Construct the inverse multiple scattering matrix T

Invert T giving T using the Singular Value Decomp osition

Use formula to compute the channeltochannel Green function

Use and the FisherLee formula to nd the conductance

Rep eat for as many realizations as required to get hi

The b ottleneck in this computation can b e either the O N SVD or the O N N

application of dep ending the concentration and the energy The computations ap

p earing ab ove were done in one to four hours on a fast desktop workstation DEC Alpha

There are faster matrix inversion techniques than the SVD but few are as stable

when op erating on nearsingular matrices Though that is crucial for the nite system

eigenstate calculations of the next chapter it is nonessential here If one were to switch

to an LU decomp osition or a QR decomp osition see app endix C we could sp eed up the

inversion stage by a factor of four or two resp ectively For most of the calculations we

p erformed the computation of the channeltochannel Green function computation was

more time consuming and so such a switchwas never warranted

Chapter

Quenched Disorder in D

Rectangles

While analytic approaches to disordered quantum systems ab ound certain typ es

of exp erimental or numerical data is dicult to come by There is a particular dearth of

wavefunction intensity statistics for D disordered systems

Exp erimentallywavefunctions are dicult quantities to measure The application

of atomic force microscopy to quantum dot systems is a promising technique An

atomic force microscop e AFM can be capacitively coupled to a quantum dot in such a

way that the at the p ointinthe twodimensional electron gas b elow the tip electrons are

eectively excluded The size of the excluded region dep ends on a variety of factors but is

typically of the order of one wavelength square One scenario for measuring the wavefunction

involves moving the tip around and measuring the shift in conductance of the dot due to

the lo cation of the tip For a small enough excluded region this conductance shift should

be prop ortional to the square of the wavefunction Even for a wavelength sized excluded

region this technique should be capable of resolving most of the no dal structure of the

wavefunction from which the wavefunction could b e reconstructed Preliminary numerical

tal group is calculations suggest that this technique could work and at least one exp erimen

working on this scheme or a near variant

One successful exp erimental approach has been using microwave cavities as an

analog computer to solve the Helmholtz equation in disordered and chaotic geometries

This technique has led to some of the only data available on disordered D systems How

Chapter Quenched Disorder in D Rectangles

ever exp erimental limitations make it dicult to consider enough ensembles to do prop er

averaging Instead the statistics are gathered from states at several dierent energies

Since the mean free path and wavelength b oth dep end on the energy the mixing of dier

ent distributions makes analysis of the data dicult Still the data do es suggest that large

uctuations in wavefunction intensities are p ossible in two dimensional weakly disordered

systems

Numerical metho ds which would seem a natural way to do rapid ensemble av

eraging have not been applied or at least not been successful at reaching the necessary

parameter regimes and sp eed requirements to consider the tails of the intensity distribution

There are some results for the tightbinding Anderson mo del However the nature of

the computations required for the Anderson mo del makes it similarly dicult to gather

sucient intensity statistics to t the tails of the distribution It is the purp ose of the

section to illustrate how the techniques discussed in this thesis can provide the data which

is so sorely needed to move the theory see section forward

We begin with an abstract treatment of the extraction of discrete eigenenergies

and corresp onding eigenstates from tmatrices This is a topic worth considering carefully

since it is the basis for all the calculations which follow The tmatrix has a pole at the

eigenenergies so the inverse of the tmatrix is nearly singular

we discuss the diculties of studying Once the preliminaries are out of the way

intensity statistics in the parameter regime and with the metho ds of In particular

we consider the impact of Dirichlet b oundaries on systems of this size We consider the

p ossibility that dynamics in the sides and corners of a Dirichlet rectangle can have a strong

eect on the intensity statistics and p erhaps mask the eects predicted by the FieldTheory

We also discuss eigenstate intensity statistics for a p erio dic rectangle a torus

with various choices of disorder p otential ie various and Well briey discuss the

tting pro cedure used to show that the distributions are well describ ed b y a lognormal

form and extract co ecients Well then compare these co ecients to the predictions of

eld theory The computation of intensity statistics in the diusiveweak disorder regime is

the most dicult numerical work in this thesis It also pro duces results which are at o dds

with existing theory Thus we sp end some time exploring the stability of the numerics and

p ossible explanations for the dierences b etween the results and existing theory

Finally as in the last chapter well discuss the numerical techniques more sp eci

cally and outline the algorithm Here some less than obvious ideas are necessary to gather

Chapter Quenched Disorder in D Rectangles

statistics at the highest p ossible sp eed

Extracting eigenstates from tmatrices

We want to apply the results of Chapter to nite systems i e systems with

discrete sp ectrum In some sense this is very similar to section but in this case we

have many scatterers instead of one what was simply ro ot nding b ecomes linear algebra

Supp ose t z has a nontrivial nullspace when z E Dene P and P as

n N R

pro jectors onto the nullspace and range of t E resp ectively Since we know that E is

n n

a p ole of T z weknow that for jj andv S wehave

T E j v iT E P jv i P j v i

n n R N

We dene a pseudoinverse on the range of T B via

t E B P jv i P jv i

n R R R

which exists since t E is explicitly nonnull on P jv i Wecannowinvert t z in the

n R

neighborhood of z E

tE jv iB P j P j v i v i

n R R N

Thus the residue of tz atz E pro jects anyvector onto the nullspace of t E

n n

Recall that the full wavefunction is written

P ji j iji G t ji lim G

N B B

Since the tmatrix term has a p ole the incidentwave term is irrelevantandthewavefunction

is up to a constant G P ji

B N

When the state at E is nondegenerate which is generic in disordered systems

there exists a vector j i the pro jector may b e written P jihj and thus

j i N G E j i

n B n

where N is a normalization constant In p osition representation

rN G r r E r dr

n B n S

Chapter Quenched Disorder in D Rectangles

If the state is mfold degenerate P j ih j and we have the solutions

N j j

N G j i

j B j

Thus the task of nding eigenenergies of a multiple scattering system is equivalent

to nding E such that t E has a nontrivial null space Finding the corresp onding

n n

eigenstates is done by nding a basis for that null space

Supp ose that our tmatrix is generated by the multiple scattering of N zero range

scatterers see In this case the pro cedure outlined ab ove can be done numerically

alue Decomp osition SVD We have using the Singular V

t j r i A hr j

i ij j

We can decomp ose A via A UV where U and V are orthogonal and is diagonal

The elements of are called the singular values of A We can detect rank deciency

the existence of a nullspace in A by lo oking for zero singular values Far more detail on

numerical detection of rankdeciency is available in

Once zero singular values are found the vector where jal phai jr i

i i

needed to apply sits in the corresp onding column of V It is imp ortant to note

that wehave extracted the eigenstate without ever actually inverting A whichwould incur

tremendous numerical error Thus our wavefunction is written

rN G r r E

B i n i

We normalize the wavefunction by sampling at many p oints and determining N numerically

In practice wemay use this pro cedure in a varietyofways Perhaps the simplest

is lo oking at parts of the sp ectra of sp ecic congurations of scatterers We dene S E

as the smallest singular value of t E standard numerical techniques for computing the

value as Computing S E is O N Then we SVD give the smallest singular

NN N

use standard numerical techniques eg Brents metho d see to minimize S E We

then check that minimum found is actually zero within our numerical tolerance of zero

to be precise These standard numerical techniques are more ecient when the minima

are quadratic whichiswhywe square the smallest singular value Wehave to b e careful to

consider S E at many energies p er average level spacing so we catch all the levels in the

sp ectrum Since weknow the average level spacing see is

V E m V

Chapter Quenched Disorder in D Rectangles

we know approximately how densely to search in energy The generic level repulsion in

disordered systems helps here since the p ossibility of nearby levels is smaller

Intensity Statistics in Small Disordered Dirichlet Bounded

Rectangles

Our rst attempt to gather intensity statistics will b e p erformed on systems with

relatively few scatterers and thus large mean free paths This regime is chosen for

comparison with the exp erimental results of

When gathering intensity statistics we need to nd eigenstates near a particular

energy This is most simply done by cho osing an energy and then hunting for a zero

singular value in some window ab out that energy For windows smaller than the average

level spacing we will frequently get no eigenstate at all This metho d works but is very

inecient since we frequently p erform the timeconsuming SVD on matrices which will not

yield a state However for this number of scatterers this ineciency is tolerable In the

next section we will consider improvements on this technique

In gure we plot j j on a grayscale black for high intensity and white for

dots for a set of low in a scatterer system along with the scatterers plotted as black

typical low energy wavefunctions in order to give the reader a picture of the sort of scatterer

densityandwavelength of the system we are considering

In the exp eriments of higher energies than those depicted in gure are

used Some sample wavefunctions in the relevant energy range are shown in gure In

the original work this energy range was chosen b ecause must b e signicantly smaller than

the system size for various eldtheoretic techniques to b e applicable Unfortunately there

are not enough scatterers in the system to make the motion diusive This is clear from the

large mean free paths in the states in where L varies from to This problem

was overlo oked in the original work due to a confusion ab out the relevant mean free path

So rather than L we have L This means that we exp ect

o o

the b oundaries to play a large role in the dynamics of the system In gure we plot the

intensity statistics for a square with Dirichlet b oundaries and In

these and subsequent intensity statistics plots we nd P t by computing wavefunctions

accumulating a histogram of values of j j and then dividing the numb er of counts in each

Chapter Quenched Disorder in D Rectangles

Figure Typical low energy wavefunctions j j is plotted for scatterers in a

Dirichlet b ounded square Black is high intensity white is low The scatterers are shown as

black dotsFor the top left wavefunction whereas for the

b ottom wavefunction increases from left to right and top to b ottom whereas decreases in the same order

Chapter Quenched Disorder in D Rectangles

Figure Typical medium energy wavefunctions j j is plotted for scatterers in a

Dirichlet b ounded square Black is high intensity white is low The scatterers are shown as

black dots For the top left wavefunction whereas for the

b ottom wavefunction increases from left to right and top to b ottom whereas decreases in the same order

Chapter Quenched Disorder in D Rectangles 30 sides center 25 corners Porter-Thomas 20 t 15 10 5 0 1 0.1 0.01

0.001 1e-05 1e-06 1e-07

0.0001

P(t)

Figure Intensity statistics gathered in various parts of a Dirichlet b ounded square

Clearly larger uctuations are more likely in at the sides and corners than in the center

The statistical error bars are dierent sizes b ecause four times as muchdatawas gathered

in the sides and corners than in the center

Chapter Quenched Disorder in D Rectangles

bin by the total numb er of counts and the width of each bin

As is clear from the gure anomalous p eaks are more likely near the sides and

most likely near the corners This large b oundary eect makes it unlikely that existing

theory can b e t to data which is eectivelyanaverage over all regions of the rectangle

Intensity Statistics in Disordered Perio dic Rectangle

In order to minimize the eect of b oundary conditions we change to p erio dic

b oundary conditions and decrease L while holding L constant While p erio dic b ound

o o

ary conditions wont eliminate nite size system eects they do remove the enhanced lo cal

ization near the Dirichlet b oundary

Decreasing L at constant L comes at a cost Since the numb er of scatterers

o o

required to hold the concentration constant scales as the volume of the system a larger

system means more scatterers and a larger multiple scattering matrix The metho d used in

the previous section will not work quickly enough to be of practical use here at least not

on a workstation This leads us to some very sp ecic numerical considerations whichwe

address b elow

Numerical Considerations

The most serious ineciency in computing wavefunctions are all the wasted SVDs

which do not result in a state However we can overcome this b ottleneckby lo oking more

closely at the small singular values We will see that small singular values corresp ond to

states of a system with slightly dierent size scatterers

First we recall that the renormalized scattering strength of a ZRI in a p erio dic

rectangle is indep endent of p osition and thus the scattering strengths of all the scatterers are

the same just as they were in freespace This means that the denominator of the multiple

scattering tmatrix S E G is symmetric since S E the real scalar renormalized

scattering strength is the same for all scatterers

If there is an eigenstate at energy E there exists some vector w such that

w S E G

If we have no zero eigenvalue but instead a small one then there exists

Chapter Quenched Disorder in D Rectangles

vector w such that

S E G w w

whichimplies

w G w

S E

and so

S E

w G

Thus there exists a nearby S E S E suchthatwedohave an eigenstate at E

Since S E parameterizes the renormalized scatterer strength S E corresp onds to a nearby

scatterer strength Thus in order to include states corresp onding to small eigenvalues we

must cho ose an initial scattering strength such that S E is still a p ossible scattering

strength

As discussed in section C for a symmetric matrix the singular values are up to

a sign the eigenvalues Thus wemay use the columns of V corresp onding to small singular

values to compute eigenstates

This provides a huge gain in eciency We pick a particular energy cho ose a

realization of the potential nd the SVD of the tmatrix and then compute a state from

every vector corresp onding to a small singular value Small here means small enough that

the resultant scatterer strength is in some sp ecied range We frequently get more than one

state p er realization

With this technique in place there is little we can do to further optimize each use

of the SVD The other b ottleneck in state pro duction is the computation of the background

Green function Each Green function requires p erforming a large sum of trigonometric

and hyp erb olic trigonometric functions Thus for mo derate size systems eg scatter

ers lling the inverse multiple scattering matrix takes far longer than inverting it Green

function computation time is also a b ottleneck when computing the wavefunction from the

multiple scattering tmatrix

We could try to improve the convergence of the sums and thus require fewer terms

p er Green function That would bring a marginal improvement in p erformance Of course

when function computation time is a b ottleneck a frequent approach is to tabulate function

values and then use lo okups into the table to get function values later Anaive application

of this is of no use here since our scatterers move in every realization and thus tabulated

Green functions for one realization are useless for the next There is a simple way to get

Chapter Quenched Disorder in D Rectangles

the advantages of tabulated functions and changing realizations Instead of tabulating the

Green functions for one realization of scatterers we tabulate them for a larger number of

scatterers and then cho ose realizations from the large number of precomputed lo cations

For example if we need scatterers p er realization we precompute the Green functions

for scatterers and cho ose of them at a time for each realization

In order to check that this do esnt lead to any signicant probability of getting

physically similar ensembles we sketch here an argument from Consider a random

potential of size L with mean free path A particle diusing through this system typically

undergo es L collisions The probability that a particular scatterer is involved in one

of these collisions is roughly this number divided by the total number of scatterers nL

where n is the concentration and d is the dimension of the system Thus a shift of one

scatterer can eg shift an energy level by ab out a level spacing when

E L

d d

nL n L

is of order one That is wemust move n L scatterers In particular in two dimensions

wemust move n n scatterers to completely change a level

There are ways to cho ose N scatterers from M p ossible lo cations and

ways to indep endently cho ose two sets Of those pairs of sets

M M N n

A A

N n n

o o

N n or more scatterers in common The rst factor is the number of ways to cho ose have

the common N n scatterers and the second is the number of ways to cho ose the rest

indep endently Thus the probability p that two indep endently chosen sets of scatterers

have N n or more in common is

M N n M

A A

n N n

o o

A N

Chapter Quenched Disorder in D Rectangles

We will be lo oking at thousands or millions of realizations The probability that no pair

among R realizations shares as manyasn scatterers is

p p

For large M and N this probability is extremely small For example the fewest scatterers

we use will b e whichwell cho ose from p ossible lo cations Our simulations are in

a square and the mean free path is implying that n The chance that all

but scatterers are common between any of one million realizations scatterers chosen

from a p o ol of is less than Thus we need not really concern ourselves with the

p ossibilityofo versampling a particular conguration of the disorder p otential

The combination of getting one or more states from nearly very realization and

precomputing the Green functions leads to an improvement of more than three orders of

magnitude over the metho d used for the smaller systems of This allows us to consider

systems with more scatterers and thus get closer to the limit L

With this improvement we can pro ceed to lo ok at the distribution of scatterer

intensities for various values of and in a L square

Observed Intensity Statistics

In gure we plot the numerically observed P t for various with in

a square L It is clear that large values of j j are more likely at larger

In order to compare these distributions with theoretical calculations we must t

them to various functional forms Since we are counting indep endentevents the exp ected

distribution of events in each bin is Poisson This needs to be taken into account when

tting the numerically computed P ttovarious analytic forms

While there may be many forms of P t which the data could t among the

various ones motivated by existing theory only some pro duct of lognormal and p owerlaw

t reasonably well That is we will t P tto

C ln tC ln t C

where C is just a normalization constant C is the p ower in the powerlawandC is the

lognormal co ecient We note that the exp ectation is that lognormal b ehavior will o ccur

in the tail of the distribution In order to account for this weteachnumerically computed

Chapter Quenched Disorder in D Rectangles lambda=.063 lambda=.031 lambda=.0016 Porter-Thomas t 10 0.1 0.01

0.001 1e-05 1e-06 1e-07 1e-08 1e-09

0.0001

P(t)

Figure Intensity statistics gathered in various parts of a Perio dic square torus Larger

uctuations are more likely for larger The erratic nature of the smallest wavelength data is due to p o or statistics

Chapter Quenched Disorder in D Rectangles

10000

1000

100

10 Reduced ChiSquared

0.1 2 4 6 8 10 12 14 16 18 20 Minimum t

-0.8

-0.9

-1

-1.1

-1.2 and 95% Confidence Interval 2

C -1.3

-1.4

-1.5 2 4 6 8 10 12 14 16 18 20

Minimum t

Figure Illustrations of the tting pro cedure We lo ok at the reduced as a function

of the starting value of t in the t top notice the logscale on the yaxis then cho ose the

C with smallest condence interval b ottom and stable reduced In this case wewould

cho ose the C from the t starting at t

Chapter Quenched Disorder in D Rectangles

P t b eginning at various values of t We then lo ok at the reduced D N

where there are D tting parameters and N data p oints for eacht A plotofatypical

sequence of values is plotted in gure top Once settles down to a near constant

value we cho ose the t with the smallest condence interval for the tted C A typical

sequence of C s and condence intervals for one t is plotted in gure b ottom The

b ehavior of is consistent with the assumption that P t do es not have the form

until wereach the tail of the distribution

As discussed in section there are two eld theoretic computations whichgive

two dierent forms for C

lnF kL

and

lnF L

We can attempt to t our observed C s to these two forms We nd that neither

form works very well at all In gure we compare these ts to the observed values of C

as wevary k at xed top and vary at xed k

Thus while the numerically computed intensity statistics are well tted by a log

normal distribution as predicted by theory the co ecients of the lognormal do not seem

to b e explained by existing theory

Anomalous Wavefunctions

It is interesting to lo ok at the anomalous wavefunctions themselves In gures

we plot twotypical wavefunctions and in we plot two anomalously p eaked wavefunctions

For eachweshow a contour plot of and a densityplotof j j The scale is dierent on each

wavefunction so that each is shown with maximum visual dynamic range This necessarily

obscures the fact that the typical wavefunction have a maximum j j of as opp osed to

or for the anomalous states

It is dicult to determine much information from the entire wavefunction In

order to simplify things a bit we lo ok at the average scaled intensity at various distances

from the p eak r via

j r r x cos y sin j d R r

j r j r

Chapter Quenched Disorder in D Rectangles

numerical 4

(1) Fitted C2

3.5 (2) Fitted C2

2.5 2 C

1.5

0.5 100 150 200 250 300 350 400 k

14 numerical

12 (1) Fitted C2

(2) Fitted C2 10

8 2 C

0 0.1 0.2 0.3 0.4 0.5 0.6

Figure Numerically observed lognormal co ecients tted from numerical data and

tted theoretical exp ectations plotted top as a function of wavenumber k at xed

and b ottom as function of at xed k

Chapter Quenched Disorder in D Rectangles

Figure Typical wavefunctions j j is plotted for scatterers in a p erio dic

square torus with The densityof j j is shown

Chapter Quenched Disorder in D Rectangles

Figure Anomalous wavefunctions j j is plotted for scatterers in a p erio dic

square torus with The density of j j is shown We note that the

scale here is dierent from the typical states plotted previously

Chapter Quenched Disorder in D Rectangles

Typical Peaks 1 Scaled Average Radial Wavefunction (peak=19.3) 0.9 Scaled Average Radial Wavefunction (peak=22.4) 0.8

0.7

0.6 > 2 0.5 <|psi| 0.4

0.3

0.2

0.1

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 kr/(2 pi)

Anomalous Peaks 1 Scaled Average Radial Wavefunction (peak=34.1) 0.9 Scaled Average Radial Wavefunction (peak=60.1) 0.8

0.7

0.6 > 2 0.5 <|psi| 0.4

0.3

0.2

0.1

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

kr/(2 pi)

Figure The average radial intensitycentered on twotypical p eaks top and two anoma lous p eaks b ottom

Chapter Quenched Disorder in D Rectangles

In gure we plot R r for twotypical p eaks one from eachwavefunction in gure

and the two anomalous p eaks from the wavefunctions in gure Here we see that each set

of p eaks haveavery similar b ehavior in their average decay and oscillation The anomalous

p eaks have a more quickly decaying envelop e as they must to reach the same Gaussian

random background value This is predicted in although we have not yet conrmed

the quantitative prediction of those authors Again wenote

Numerical Stability

Throughout this work wehavechecked the accuracy and stability of our numerics

in several ways In chapter we checked that the numerical pro cedure we use gives the

correct classical limit crosssection for one scatterer In chapter we checked that for an

appropriate choice of parameters many scatterers in a wire give the classically exp ected

mean free path

In chapter we checked that the analytic structure of renormalized tmatrices

correctly predicts ground state energies in Dirichlet b ounded squares However wedonot

us we do the have a direct way to check higher energy eigenenergies or eigenstates Th

next b est thing and check the stability of the numerical pro cedure which pro duces them

That is wecheck that small p erturbations of the input to the metho d scatterer lo cations

scatterer strengths Green functions etc do not result in large changes to the computed

wavefunction

As a simple p erturbation we will consider random changes in scatterer lo cation

We consider one realization of the scattering p otential compute the wavefunction and

then pro ceed to move the scatterers eachby some small amount in a random direction We

then compute the new wavefunction and compare the twowavefunctions We rep eat this

for ever larger p erturbations of the scatterers until we have completely decorrelated the

wavefunctions We consider two measures for comparison We dene r r r

and we consider b oth

Vhjrj i

where weaverage over r and

maxf g

maxfj j g

The former is a standard measure of the dierence of functions and the latter weexpectto

b e more sensitivetochanges in anomalously large p eaks

Chapter Quenched Disorder in D Rectangles

In order to see if a small p erturbation in scatterer lo cations pro duces a small

change in the wavefunction we need an appropriate scale for the scatterer p erturbation If

Random Matrix Theory applies weknow that a single state can b e shifted by one level and

thus completely decorrelated bymoving one scatterer with crosssection approximately the

wavelength by one wavelength From that argument and that the fact that the motion of

each scatterer is uncorrelated with the motion of the others we can see that the appropriate

parameter is approximately

where x is average distance moved by a single scatterer That is if the error in the

wavefunction is comparable to we can assume it comes from physical changes in the

wavefunction not numerical error

In gure we plot our two measures of wavefunction deviation against on a

loglog scale for several between and for a state with an anomalously large

p eak Since our deviations are no larger then exp ected from Random Matrix Theorywe

can assume that the numerical stabilityissuciently high for our purp oses

Though not exactly a source of numerical error we might worry that including

states that result from small but nonzero singular values has an inuence on the statistics

If this were the case wewould need to carefully cho ose our singular value cuto in order to

match the eld theory However the inuence on the statistics is minimal as summarized

in table

Maximum Singular Value

Table Comparison of lognormal tails of P t for dierent maximum allowed singular

value

Conclusions

In contrast to the well understo o d phenomena observed in disordered wires we

have observed some rather more surprising things in disordered squares While the various

theoretical computations of the exp ected intensity distribution app ear to correctly predict

the shap e of the tail of the distribution none of them seem to correctly predict the dep en

dence of that shap e on wavelength or mean free path

0.0001

Figure Wavefunction deviation under small p erturbation for scatterers in a

p erio dic square torus

Chapter Quenched Disorder in D Rectangles

We have considered a variety of explanations for the discrepancies between the

eld theory and our numerical observations There is one way in which our p otential diers

drastically from the potential which the eld theory assumes We have a nite number

of discrete scatterers whereas the eld theory takes the concentration to innity while

taking the scatterer strength to zero while holding the mean free path constant Thus it

seems p ossible that there are pro cesses which can cause large uctuations in j j which

dep end on there b eing nite size scatterers In order to explore this p ossibilitywe consider

the dep endence of P t on the scatterer strength at constant wavelength and mean free

path Sp ecically at two dierent wavelengths and and xed

we double the concentration and halve the crosssection of the scatterers The results are

summarized in table While changes in concentration and scatterer strength do inuence

the co ecients of the distribution they do not do so enough to explain the discrepancy with

the eld theory

Strong Scatterers Weak Scatterers

Table Comparison of lognormal tails of P t for strong and weak scatterers at xed

and

Algorithms

We have used several dierent algorithms in dierent parts of this chapter We

have gathered sp ectral information ab out particular realizations of scatterers gathered

statistics in small systems where we only used realizations with a state in a particular

energy window gathered statistics from nearly every realization by allowing the scatterer

size to change and computed wavefunctions for particular realizations and energies Below

wesketch the algorithms used to p erform these various tasks

particular realization at a We will frequently use the smallest singular value of a

E where i lab els the realization When only one particular energy which we denote S

realization is involved the i will b e suppressed To compute S E we do the following N

Chapter Quenched Disorder in D Rectangles

Compute the renormalized tmatrix of each scatterer

Compute the scattererscatterer Green functions for all pairs of scatterers

Construct the inverse multiple scattering matrix T

Find the singular value decomp osition of T and assign S E the smallest singular

value

In order to nd sp ectra from E to E for a particular realization of scatterers

Load the scatterer lo cations and sizes

cho ose a E less than the average level spacing

Set E E

a If EE nd S E S E E andS E E otherwise end

N N N

b If the smallest singular value at E E is not smaller than for E and E E

increase E by E and rep eat from a

c Otherwise apply a minimization algorithm to S E in the region near E E

Typically minimization algorithms will b egin from a triplet as wehave calculated

ab ove

d If the minimum is not near zero increment E and rep eat from a

e If the minimum coincides with an energy where the renormalized tmatrices are

extremely small it is probably a spurious zero broughtonby a state of the empty

background Increment E and rep eat from a

f The energy E at which the minimum was found is an eigenenergy Save it

increment E by E and rep eat from a

The b ottleneck in this computation is the lling of the inverse multiple scattering

matrix and computation of the O N SVD Performance can be improved by a clever

choice of E but to o large a E can lead to states b eing missed altogether The optimal E

can only b e chosen by exp erience or by understanding the width of the minima in S E

The origin of the width of these minima is not clear

The simpler metho d for computing intensity statistics by lo oking for states in an

energy window of size E ab out an energy E go es as follows

Chapter Quenched Disorder in D Rectangles

Cho ose a realization of scatterer lo cations

Use the metho d ab ovetocheck if there is an eigenstate in the desired energy window

ab out E If not cho ose a new realization and rep eat

If there is an eigenstate use the singular vector corresp onding to the small singular

value to compute the eigenstate Make a histogram of the values of j j sampled

on a grid in p osition space with spacing approximately one halfwavelength in each

direction

Combine this histogram with previously collected data and rep eat with a new real

ization

The b ottlenecks in this computation are the same as the previous computation

In this case a clever choice of window size can improv e p erformance

The more complex metho d for computing intensity statistics by lo oking only at

energy E is a bit dierent

Randomly cho ose a numb er of lo cations approximately twice the numb er of scatter

ers

Compute and store all renormalized tmatrices and the Green functions from each

scatterer to each other scatterer

Cho ose a grid on which to compute the wavefunction

Compute and store the Green function from each scatterer to each lo cation on the

wavefunction grid

Cho ose a subset of the precomputed lo cations construct the inverse multiple scatter

ing matrix T and nd the SVD of T

For each singular value smaller than some cuto weve tried b oth and compute

the asso ciated eigenstate on the grid count the values of j j and combine with

previous data Cho ose a new subset and rep eat

For this computation the b ottlenecks are somewhat dierent The O N SVD is

one b ottleneck as is computation of individual wavefunctions whichisO SN where S is

the number of points on the wavefunction grid

Chapter Quenched Disorder in D Rectangles

For all of these metho ds near singular matrices are either sought frequently en

countered or b oth This requires a numerical decomp osition which is stable in the presence

of small eigenvalues The SVD is an ideal choice The SVD is usually computed via

transformation to a symmetric form and then a symmetric eigendecomp osition Since the

matrix we are decomp osing can be chosen to be symmetric we could use the symmetric

eigendecomp osition directly We imagine some marginal improvement might result by the

substitution of such a decomp osition for the SVD

Chapter

Conclusions

Scattering theory can b e applied to some unusual problems and in some unexp ected

ways Several ideas of this sort have been develop ed and applied in this work All the

metho ds develop ed here are related to the fundamental idea of scattering theory namely the

separation b etween propagation and collision The applications range from the disarmingly

simple single scatterer in a wire to the obviously complex problem of intensity statistics in

weakly disordered two dimensional systems

These ideas allow calculation of some quantities which are dicult to compute

other ways for example the scattering strength of a scatterer in a narrowtwo dimensional

wire as discussed in section It also allows simpler calculation of some quantities which

have been computed other ways eg the eigenstates of one zero range interaction in a

rectangle also known as the Seba billiard

The metho ds develop ed here also lead to vast numerical improvements in calcu

lations which are p ossible but dicult other ways for example the calculation of intensity

statistics in closed weakly disordered two dimensional systems as demonstrated in sec

tion

The results of these intensity statistics calculations are themselves quite interest

ing They app ear to contradict previous theoretical predictions ab out the likeliho o d of

large uctuations in the wavefunctions in such systems At the same time some qualitative

features of these theoretical predictions havebeenveried for the rst time

There are a variety of foreseeable applications of these techniques One of the

most exciting is the p ossible application to sup erconductor normalmetal junctions For

instance a disordered normal metal region with a sup erconducting wall will have dierent

Chapter Conclusions

dynamics b ecause of the Andreev reection from the sup erconductor The sup erconductor

energy gap can be used to prob e various features of the dynamics in the normal metal

Also the eld of quantum chaos has so far been fo cused on systems with an obvious

chaotic classical limit Systems with purely quantum features very likely have dierent

and interesting behavior A particlehole system like the sup erconductor is only one such

example

A dierent sort of application is to renormalized scattering in atomic traps The

zero range interaction is a frequently used mo del for atomatom interactions in such traps

The trap itself renormalizes the scattering strengths as do es the presence of other scatterers

Some of this can be handled at least in an average sense with the techniques develop ed

here

Wewould also liketo extend some of the successes with p oint scatterers to other

shap es There is an obvious simplicity to the zero range interaction which will not be

shared with any extended shap e However other shap es eg nite length line segments

have simple scattering prop erties which can b e combined in muchthewaywehave combined

single scatterers in this work

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App endix A

Green Functions

A Denitions

Green functions are solutions to a particular class of inhomogeneous dierential

equations of the form

A z Lr G r r z r r

G is determined by A and b oundary conditions for r and r lying on the surface S of

the domain Here z is a complex variable while Lr is a dierential op erator which is

timeindep endent linear and Hermitian Lr has a complete set of eigenfunctions f rg

which satisfy

Lr r r A

n n n

Eachofthe r satisfy the same b oundary conditions as G r r z The functions f rg

n n

are orthonormal

r r A

n nm

and complete

r r r r A

In Dirac notation we can write

z LGz A

Lj i j i A

n n n

h j i A

n m nm

j ih j A

n n

Appendix A Green Functions

In all of the ab ove sums over n maybeintegrals in continuous parts of the sp ectrum

For z we can formally solve equation A to get

Gz A

z L

Multiplying by A weget

X X X

j ih j

n n

A Gz j ih j j ih j

n n n n

z L z L

n n n

Werecover the rrepresentation bymultiplying on the left by h rj and on the rightby jr i

r r

Gr r z A

In order to nd A we had to assume that z When z we can write

n n

a limiting form for Gz

A G z

z Li

and its corresp onding form in p osition space

r r

G r r z lim A

z i

where G z is called the retarded or causal Green function and G z is called the

advanced Green function

These names are reections of prop erties of the corresp onding timedomain Green

functions Consider the Fourier transform of A with resp ect to z

r r

iE th

e dE A G r r t t lim

E i

We switch the order of the energy sums integrals and have

r r

iE h

G r r E ie dE A lim

We can p erform the inner integral with contour integration by closing the contour in either

the upp er or lower half plane We are forced to cho ose the upp er or lower half plane by

the sign of For we must close the contour in the lower halfplane so that the

exp onential forces the integrand to zero on the part of the contour not in the original

integral However if the con tour is closed in the lower half plane only p oles in the lower

r r is zero for tau and half plane will be picked up by the integral Thus G

therefore corresp onds only to propagation forward in time G r r on the other hand

is zero for and corresp onds to propagation backwards in time G is frequently useful in formal calculations

Appendix A Green Functions

A Scaling L

We will nd it useful to relate the Green function of the op erator L to the Green

function of L where is a complex constant

Supp ose

h i

G z A z L

We note that L and L have the same eigenfunctions but dierent eigenvalues ie

Lj i j i A

n n n

X X

j ih j j ih j z

n n n n

G z G A

n n

So wehave

G z G A

A Integration of Energy Green functions

Claim

Gr r z Gr r z dr Gr r z A

Pro of

Z Z

I Gr r z Gr r z dr hr j Gz j rihrj Gz j r i dr A

I hr j Gz jrihrj dr Gz j r i A

but

jrihrj dr A

I h r j Gz Gz j r i A

We expand the Green functions

jnihnj

A Gz

z E

n n

Appendix A Green Functions

X X

jnihnj j mihmj

I h r j j r i A

z E z E

n m

But since hn jm i

jnihnj

I h r j j r i A

z E

whichisvery like the Green function except that the denominator is squared We can get

around this by taking a derivative

jnihnj d

j r i A I h r j

dz z E

Wemove the derivative outside the sum and matrix elementtoget

d d

I Gr r z A hr j Gz j r i

dz dz

as desired

A Green functions of separable systems

Supp ose wehave a system which is separable That is the eigenstates are pro ducts

of functions in each of the domains into which the system can separated For example a

system in which an electron can move freely in the xdirection but is conned to the region

jy j h has eigenfunctions which are pro ducts of plane waves in x and quantized mo des

in y The Green function of such a system is written

X X

jij ihjh j

A G z

z i

We can move either of the sums inside to get

X X

j ih j

G z j ihj A

z i

but now the inner sum is just another Green function alb eit of lower dimension so we

have

X X

j ih jG z A z G z jihjG

Appendix A Green Functions

Consider the example mentioned at the b eginning of this section Well lab el the

eigenstates in the xdirection by their wavenumber k and lab el the transverse mo des bya

channel index n So wehave

G z jk ihk jG E k i dk jnihnjg E A

n n

where g z is the one dimensional free Green function In the p osition representation

this latter equality is rewritten as

G x y x y z y y g x x E A

n n n

A Examples

Below we examine two examples of the explicit computation of Green functions

We b egin with the mundane but useful free space Green function in two dimensions Then

we consider the more esoteric Gorkov Green function the Green function of a single electron

in a sup erconductor

A The FreeSpace Green Function in Two Dimensions

In twodimensions we can derive the freespace Green function of the op erator

h m using symmetry L r corresp onding to D free quantum mechanics with

and limiting prop erties See eg The Green function G satises

z r G r r z r r A

By translational symmetry of L G r r z is a function only of jr r j For

G z satises the homogeneous dierential equation

z r G z A

Recall that

r r A

so wemay rewrite A as

Z Z

z G d r G z d A

o o

Appendix A Green Functions

In radial p olar co ordinates

r A

A r f

Thus

Z Z

r G z d G z d A

o o

where the last equality follows from Gauss Theorem in this case the fundamental theorem

of calculus So wehave

z G d A

which as gives

G z

G z ln const A

Also

lim G z A

General solutions of A are linear combinations of Hankel functions of the rst

and second kind of the form See eg

h i

p p

A H zB H z e A

n n

Since we are lo oking for a indep endent solution we must have n Since

So H zblows up as B The b oundary condition A xes A

wehave

G r r z H z jr r j A

is the Hankel function of zero order of the rst kind where H

H xJ xiY x A

o o

where J x is the zero order Bessel function and Y x is the Neumann function of zero

o o

order

It will be useful to identify some prop erties of Y x for use elsewhere We will

often b e interested in Y x for small x As x

J xlnx A Y xY x

o o

Appendix A Green Functions

R R

Y x is called the regular part of Y x We note that Y Ordinarily

o o

the sp ecic value of this constant is irrelevant since it is overwhelmed by the logarithm

However we will have o ccasion to subtract the singular part of G and the constant Y

will b e imp ortant

A The Gorkov bulk sup erconductor Green Function

In this app endix we will derive the Green Function for particles and holes in

a sup erconductor We plan to use this Green function to simulate Andreev scattering

systems We will nd an explicit solution in the case of a uniform bulk sup erconductor

Andreev scattering takes place at normalmetal sup erconductor b oundaries NS

When an electron in the normal metal hits the sup erconductor b oundary it can scatter as

a hole with the timereversed momentum of the electron That is the electron disapp ears

at the b oundary and a hole app ears which go es back along the electrons path at the

same sp eed as the electron came in This typ e of scattering leads to a class of tra jectories

whichinterfere strongly even in chaoticdisordered systems and pro duce weaklo calization

eects

The simplest way to deal with the coexistence of particles and holes is to solvea

set of coupled Schrodingerlike equations known as the Bogoliub ov equations see eg

h i

j f i H j f i j g i A ih

i h

ih H jg i jf i A jg i

where H is the single particle Hamiltonian jri r h rj is the p ossibly p osition

dep endent chemical potential and jri r h rj is the p ossibly p osition dep endent

sup erconductor energy gap In the case we have the Schrodinger equation for jf i

ersed Schrodinger equation for jg i the hole state the electron state and the timerev

If we form the spinor

jf i

ji A

j g i

Appendix A Green Functions

we can write the Bogoliub ov equations as

ih j i Hji A ji

In order to compute the Green function op erator Gz for this system weform

z H

G z z H A

z H

which at rst lo oks dicult to inv ert However there are some nice techniques we can

apply to a matrix with this form To understand this we need a brief review of

quantum mechanics

Recall the Pauli spin matrices are dened

and that the set fI g is a basis for the vector space of complex matrices

The Pauli matrices satisfy

i j i j j i

ij k k

f g I A

i j i j j i ij

where is the threesymbol is if ij k is a cyclic p ermutation of if ij k is a

ij k ij k

noncyclic p ermutation of and otherwise and is the Kronecker delta

To simplify the later manipulations we rewrite G

G z g z Gz gz A

where

z H

g z A

z H o

Appendix A Green Functions

Wenow write

g z aI ib A

where

H a

b Im Re iz

Lets make the additional assumptions that

h i

a b i

h i

b b i j

i j

For our problem as long as Iconst we satisfy these assumptions That is we are

in a uniform sup erconductor

a ib ib ib

A g z

ib ib a ib

Since all the op erators commute wecaninvert this likea matrix of scalars

a ib ib ib

gz a Iib

a b b b a I b

ib ib a ib

We clarify this by explicit multiplication

I I

gz g z A a I b a Iib a I ib

a b a b

Nowwe use the relation A to simplify b

X X

b b b b b f g b I A

i i j j i j i j

i j

So wehave

I I

gz g z a I b a I b I I A

a b a b

Appendix A Green Functions

At this p ointwehave an expression for gz but its not obvious howweevaluate

We use another trick and factor gz asfollows dening b jb j

a b

I s

gz a Iib A

a b a isb

Why do es this help Weve replaced the problem of inverting a b with the problem of

invertinga ib We recall that a H andb b b b jj z So

a ib H i jj z A

which means

a ib G i jj z A

where

A G z

H z

We note that if E const G z G E z Wedene

fermi fermi

z jj

f z

z jj

and then write

h i

G f z G f z G G

h i

g z

G G G f z G f z

So nallyweha ve a simple closed form expression for Gz

h i

G f z G f z G G

h i

gz Gz

G G G f z G f z

Various Limits

Appendix A Green Functions

When wehave

f z

and thus

Gz A

as wewould exp ect for a normal metal

App endix B

Generalization of the Boundary

Wall Metho d

Handling the general b oundary condition

B s rs s rs

C C

requires a more complex p otential

o n

B V r ds s r rs s s

and thus is somewhat more dicult than the case of Dirichlet b oundary conditions con

sidered in section First we assume ns a unit vector normal to C at each point s

and

f rs ns rf rs B

Second we insert B into to get

o n

rs B rr ds s G r rs s s

whichthenwe consider at a p oint rs onC with the same notational abbreviation used

in Section

n o

s s ds s G s s s s s B

As it stands B is not a linear equation in To x this we m ultiply both sides by

n o

s s and dene

n o

s s s s

Appendix B Generalization of the Boundary Wal l Method

n o

s s s s

o n

G s s B G s s s s

This yields

B B B B

s s ds s G s s s B

a linear equation in andsolved by

h i

B B B

I G B

where again the tildes emphasize that the equation is dened only on C The diagonal

op erator is

f ssf s B

We dene

h i

B B

T I G B

es the original problem that solv

rr ds G r rs T rs B

for

B B B

T rs ds T s s s B

h i

B B

As in Section in the limit s T converges to G which

when inserted into B gives

o h i n

B B B B

s s I G G s B s s

the desired b oundary condition B

For completeness we expand T in a p ower series

i h

B B

B T G

B B j

T s s s s s s B T s s

where

B j B B

T s s ds ds G s s s G s s s s s B

j j j

allowing one at least in principle to compute T s s and thus the wavefunction every where

App endix C

Linear Algebra and NullSpace

Hunting

In this app endix we deal with the linear algebra involved in implementing various

metho ds discussed ab ove We b egin with the standard techniques for solving Ax v typ e

equations when A is of full rank We do this mostly to establish notation

In many of the techniques ab ove we had a matrix whichwas a function of a real

parameter usually a scaled energy At and wewanted to lo ok for t such that Atx

has nontrivial solutions x The standard technique for handling this sort of problem

is the Singular Value Decomp osition SVD Well discuss this in C

There are other metho ds to extract nullspace information from a matrix and they

are typically faster than the SVD Well discuss one such metho d the QR Decomp osition

QRD is a wonderful reference for all that follows

C Standard Linear Solvers

A system of N linear equations in N unknowns

a x b for j fN g C

ij i j

may b e written as the matrix equation

Ax b C

Appendix C Linear Algebraand Nul lSpace Hunting

where A a This implies that a formal solution is available if the matrix inverse A

ij ij

exists Namely

x A b C

Most techniques for solving C do not actually invert A but rather decomp ose

A in a form where we can compute A b eciently for a given b One such form is the

LU decomp osition

A LU C

where L is a lower triangular matrix and U is an upp ertriangular matrix Since it is simple

to solve a triangular system see section we can solve our original equations with

atwo step pro cess We nd a y which solves Uy b and then nd x whichsolves Lx y

This is an abstract picture of the familiar pro cess of Gaussian elimination Essentially

there exists a pro duct of unit diagonal lower triangular matrices whichwillmake A upp er

triangular Each of these lower triangular matrices is a gauss transformation which zero es all

the elements b elow the diagonal in A one column at a time The LU factorization returns

the inverse of the pro duct of the lower triangular matrices as Land the resulting upp er

triangular matrix in U It is easy to show that the pro duct of a lowerupp er triangular

matrices is lowerupp er triangular and the same for the inverse That is L represents a

sequence of gauss transformations and U represents the result of those transformations For

N ops oating p oint op erations to large matrices the LUD requires approximately

compute

The LUD has several drawbacks The computation of the gauss transformations

involves division by a as the i column is zero ed This means that if a is zero for any i

ii ii

the computation will fail This can happ en two ways If the leading principal submatrix

of A is singular ie detA i i then a will be zero when the i column is

zero ed If A is nonsingular pivoting techniques can successfully nd an LUD of a row

p ermuted version of A Row p ermutation of A is harmless in terms of nding the solution

However if A is singular then we can only chase the small pivots of A down r columns

where r rankA Atthispointwe will encounter small pivots and numerical errors will

destroy the solution Thus we are led to lo ok for metho ds which are stable even when A is singular

Appendix C Linear Algebraand Nul lSpace Hunting

C Orthogonalization and the QR Decomp osition

Supp ose we wanted to keep the basic idea of the LU decomp osition zeroing the

columns of A but avoid the pivot problem Wehave to b e careful b ecause the inverse of the

transformation matrix the L in LU has to b e easy to invert or the decomp osition wont b e

much help One class of transformations which mightwork are orthogonal transformations

Orthogonal transformations include reections and rotations Orthogonal transformations

do not rescale so they are stable in the presence of small pivots and they are easy to

invert the inverse of an orthogonal transformation is its transp ose The simplest such

metho d is the QR decomp osition

A QR C

where Q is orthogonal and R is upp ertriangular Performing this decomp osition is straight

forw ard Consider the rst column of A A rotation or reection ab out a correctly chosen

plane will zero all but the rst element simply rotate a basis vector into the rst column

of A and then invert that rotation Now go to the second column To zero everything

b elowthe diagonal it is sucientto consider a problem of dimension smaller and rotate

in that smaller space This leaves the previously zero ed elements alone since those vectors

are null in the smaller space We do this one column at a time and accomplish our goal

Awonderful consequence of the QRD is that the eigenvalues of A are the diagonal

elements of R Since Q is orthogonal it do es not change eigenvalues it just rotates the

eigenvectors This makes it ideal for nullspace hunting since we can do QRDs and lo ok for

are in generated in order of their small eigenvalues Typically the eigenvalues in a QRD

absolute value as a side eect of the transformation

If A is singular with rank N a typical QRD algorithm will leave the zero eigen

value in the last row of R That is the last row of R is all zero es A mo died back

substitution algorithm will immediately return a vector in the null space Since the null

space is one dimensional this vector spans it A similar approach will work with multi

dimensional nullspaces but requires a bit more work

The QRD is not magic It do es not provide a solution where noneexists While

e the Ax b will still fail for singular A the algorithm is stable the attempt to solv

and nonzero b it will simply do so in the backsubstitution phase rather than during

decomp osition

There is a x for this backsubstitution problem We can pivot the columns of

Appendix C Linear Algebraand Nul lSpace Hunting

A as we do the QRD and then though we still cannot solve an illp osed problem we can

extract a least squares solution from this column pivoted QRD QRD CP

For large N the QRD requires approximately N ops twice as many as LU

and QRD CP requires N ops to compute

C The Singular Value Decomp osition

It is p ossible to further reduce R by p ostmultiplying it by a sequence of orthogonal

transformations The SVD is one such complete orthogonalization It reduces R to a

diagonal matrix with p ositiveentries That is the SVD gives

A UV C

where U and V are orthogonal and diag In this formulation zero singular

values corresp ond to zero eigenvalues and the corresp onding vectors may b e extracted from

V Since the SVD is computed entirely with orthogonal transformations it is stable even

when applied to singular matrices

The SVD of a symmetric matrix is an eigendecomp osition That is the singular

values of a symmetric matrix are the absolute values of the eigenvalues and the singular

vectors are the eigenvectors If a symmetric matrix has two eigenvalues with the same

absolute value but opp osite sign the SVD cannot distinguish these eigenvalues and may

mix them For nondegenerate eigenvalues the sign information is enco ded in UV which

is a diagonal matrix made up of All of this follows from a careful treatment of the

uniqueness of the decomp osition

S the The SVD can be applied to computing only the singular values SVD

singular values and the matrix V SVD SV and the singular values the matrix V and the

matrix U SVD USV The op count is dierent for these three versions we include LUD

and QRD for comparison and we summarize this in table C

Appendix C Linear Algebraand Nul lSpace Hunting

Algorithm ops seconds

SVD USV N

SVD SV N

SVD S N

QRD CP N

QRD N

LUD N NA

Table C Comparison of op counts and timing for various matrix decomp ositions

The timing test was the decomp osition of a matrix on a DEC Alpha

workstation

We include this table to p oint out that choice of algorithm can have a dramatic

eect on computation time For instance when lo oking for a t such that At is singular

wemay use either the SVD S or the QRD to examine the rank of At However using the

S QRD will b e at least times faster than using the SVD

App endix D

Some imp ortant innite sums

D Identites from

Recall that

ln for jxj D

n x

Thus

in n

e e

for real D ln

n e

e cos ne

Re ln ln D

n expi cosh cos

and

sinne e sin

Im ln D arctan

n expi e cos

Since

sin n sin n cos n cos n D

wend

e sin n sin n e

e ln ln

n cosh cos cosh cos

cosh cos

Appendix D Some important innite sums

cosh cos

sin sinh

sinh sin

We also note that for

i i

ln e e e ln

cosh cos

ln D

since in the small argument expansion of the exp onentials the constant and linear terms

cancel Similarly for j j

sinh sin

ln ln ln sin D

sinh sin

D Convergence of Green Function Sums

D Dirichlet Boundary Green Function Sums

Consider

a sinnx sinnx exp h n l E exp D

n n n

where E and

n l

exp D b sinnxsinnx

n l

In this case wemay p erform b for using the identities ab ove section D

We need to showthat a b converges and that it converges uniformly

n n

with resp ect to for all Since j sinnx sinnx j wehave

ja b j

n n

l El n n El

exp exp e

n n l n l

and exp xa expxaso For x

ja b j

n n

Appendix D Some important innite sums

El n n l El

e exp exp exp

n n l n l

n El El El l

exp e exp

n l n n n

Since

n E implies

n El n

exp exp D

l n l

x n E implies and since x

El El

n n

ln l

E implies n

h h

n n

e e D

x e x

ln C l l

E wehave for nmax E

l El n El

ja b j exp Ce D

n n

n l n n

l El

Further for n E ln

l n El El

D ja b j exp

n n

n l n n

l l El

E E Therefore for M max ln

M El El

a b exp D

n n

l M exp M

Appendix D Some important innite sums

l l

But and is a monotonically increasing function of x so

exp exp h

l l

thus

El h M El

D a b exp

n n

l M exp M h

Thus a b converges for all

n n

El n El

Further since exp

n l

f D ja b j

n n n

P P

and f converges Therefore a b converges uniformly with resp ect

n n n

nM nM

to for

D Perio dic Boundary Green Function Sums

Consider

exp D a cos njx x j n l E

n n

where E and

l n

b cosnjx x j exp D

n l

b for using the identities ab ove section D In this case wemay p erform

We need to showthat a b converges and that it converges uniformly

n n

with resp ect to for all Since j cos njx x jj wehave

ja b j

n n

El El n n l

exp exp

n n l n l

For x and exp xa expxaso

ja b j

n n

El El n n l

exp exp exp

n n l n l

El n El l El

exp exp

n l n n n Since

Appendix D Some important innite sums

n E implies

n n El

exp exp D

l n l

E implies x n and since x

El El

n n

x e x pro of

wehave for n E

El n El l

exp D ja b j

n n

n l n n

n El El l

D exp ja b j

n n

n l n n

Therefore for M E

El El M

D a b exp

n n

l M exp M

l l

But and is a monotonically increasing function of x so

h exp exp

l l

thus

M El El h

a b exp D

n n

h l M exp M

Thus a b converges for all

n n

n El El

exp Further since

n l

f D ja b j

n n n

P P

and f converges Therefore a b converges uniformly with resp ect

n n n

nM nM

to for

App endix E

Mathematical Miscellany for Two

Dimensions

E Polar Co ordinates

r r E

r r r r

E Bessel Expansions

l ik r

i J kr E e

ik y ik r sin il

e e J kre E

ik x ik r cos l

e e i J krcosl E

sin ky sinkr sin J kr sin l E

sin kx sin kr cos J krcosl E

Appendix E Mathematical Miscel lany for Two Dimensions

E Asymptotics as kr

J kr cos kr E

Y kr sin kr E

i kr

H kr e E

i kr

e H E kr H kr

n n

E Limiting Form for Small Arguments kr

for n E J kr

Y kr ln kr E

H kr i ln kr E

for Refng E n Y kr

i kr

H kr for Refng E n