Anomalous Cross Section Induced by Topological Quantum

Interference

De-Hone Lin Department of Physics, NSYSU 23 December 2004

phyiscs. mesoscopic inphenomena some ingunderstand in

useful be willand system potential general quite inappear toexpected is

casedisk hard ineffect nonlocal theuniversal, is particles charged the

onflux magnetic of influence nonlocal theSince examined. is

flux magnetic theand potential likedisk harda of process scattering The

d.establishe isflux magnetic B-A nonlocala and potential rangeshort

arbitray anfor problem scattering ldimensiona a two of theory wavePartial

phyiscs. mesoscopic in

phenomena other some ingunderstand in useful be willand system

potential general quite inappear toexpected iseffect nonlocal theuniversal,

is charged theonflux magnetic of ceinterferen quantum nonlocal theSince

effect. Hallquantum fractional thein model boson and fermion composite

explain helps chenergy whilow at flux magnetic quantized specific at the

revealed is section cross totalanomalous An d.establishe is

flux magnetic B-A nonlocala and potential rangeshort arbitrary an

for problem scattering dmensional a three of theory wavePartial

Fractional Quantum Hall States

•2-D electron system inside the GaAs/AlGaAs heterostructure

•High magnetic fields (B~10T)

•Low temperatures (T~0.1K)

bosons. composite to

theming transformelectrons, thequanta toflux magnetic of

attachment theas viewedbe can attachmentVortex (b)

repulsion. (Coulomb) electron-electron

reduces electron each onto vortices three Placing(a)

1/3. filling level- Landaufractionalat attraction vortex Electron

Coulomb forces flux quantum attachment

mK 100below turesat tempera

observed is 2/5

/at

plateau Hallfractional

rdenominato-even An

2ehxy

R. willett, J.P. Eisenstein,H.L. Stormer, D.C. Tsui, A.C. Gossard, andJ.H. English,PRL, vol. 59, 1776, 1989.

1998. 875, 71, RMP,

effect Hallquantum fractional The : LectureNobel Stormer, H.L.

pairs. fermion composite of formation the

is state thisof origin for they possibilit exciting An unclear.

remains state theof origin The allowed. benot should fraction

rdenominato even an suchat state FQHEA 5/2.at FQHE

Nature, Vol. 406, 863 (2000).

ctivity.supercondu etemperatur-high

---nsinteractio repulsivepurely from arising

pairing whichin phenomenonanother explain helpmay

arrives,it whena theory, such that possible isIt

pairing. fermion composite of theory BCSfulla not yet isit but

state, 2/5 theof ingunderstandour in forward stepimportant an is this

n,calculatio 1956 sCooper' Like1/2.not but

2/5at occurs pairing such why intoinsight some gives and

pairs,Cooper form can fermions composite ofsea a Fermi

thationdemonstrat beautifula is ncalculatio s'. Scarola

alet

N. Bonesteel, Nature, Vol. 406, 841 (2000).

A.K. Geim etc, Nature 407, 55, 2000.

field. magnetic of expulsion the toleads npenetratio whose

vortices,negative as wellas ,001.0 as little ascarry that vorticesobserved have We

. fromlly substantia differs alwaysflux that thefinding film, ctingsupercondu

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eh

Summary

Quantum interference of magnetic fluxQuantum interference of magnetic flux

Quantum interference in partial wave theory and anomalQuantum interference in partial wave theory and anomalous cross section in two dimensionsous cross section in two dimensions

Quantum interference in partial wave theory and anomalQuantum interference in partial wave theory and anomalous cross section in three dimensionsous cross section in three dimensions

Composite bonsons and fermionsComposite bonsons and fermions

Introduction

a for 0)(

and

afor 1

)(2

V

V

A charged particle

Radius

Phase shifts

Bound states therein

Charged particle Interferencepattern

D. Bohm and Y. Aharonov in 1959 found AB effect

Magnetic flux

)()()( )0(2

)0(1

)0( xxx

)()0(1 x

)()0(2 x

)()0( x

1

2

.)(exp)()(exp)(2

)0(21

)0(1

xxxdxA

c

iexxdxA

c

iex

xxdxA

c

iex

1

)0(1 )(exp)(

xxdxA

c

iex

2

)0(2 )(exp)(

)(x

1

2

)(x

.exp)()(

)(exp)()()(

)0(2

)0(1

)0(2

)0(1

c

iexx

xdxAc

iexxx

.2)/(by given is ceinterferen of cycle periodic The ec

.exp

dxAc

ie

The four-vector formulation of the non-integrable phase factor

is given by

C.N. Yang, and T.T. Wu, Phys. Rev. D 12, 2845 (1975).

1. It is non-local in the sense that it exists even when the interfering beams pass through a field free region and is associated with the entire closed curve C.

2. It is topological in the sense that the phase shift is unaffected

when is deformed within the field free region.

3. It is geometrical in the sense that the above phase factor

represents parallel transport (holonomy transformation) around

with respect to the electromagnetic connection gauge.

a for 0)(

and

afor 1

)(2

V

V

A charged particle

Radius

Phase shifts

Bound states therein

Aharonov-Bohm magnetic flux

The system is very important in understanding the quantum Hall effect, superconductivity, and the transport properties of nano structures.

Quantum interference in partial wave theory and anomalous cross

section in two dimensions

Partial Wave Method for a Short Range Partial Wave Method for a Short Range Potential and an Aharonov-Bohm FluxPotential and an Aharonov-Bohm Flux

).(2

where

),(;,,

22

0

2)0(0

xVH

xxExxGi

xHE

m

imm eEGExxG )'()0()0(

2

1;',;,

In polar coordinates for the cylindrically symmetric system:

Magnetic field exists in the system, then

x

xrdrA

c

ieExxGExxG

)(exp;,;, )0(

For the Aharonov-Bohm Flux

,ˆˆ

2)(22 yx

exeygxA yx

),(43 xgB

the magnetic field

.4/ g

and the magnetic flux

)();,()(1

2 2

2

2

22

EGV

d

d

d

dE

000 //2 hnumber wit reala is cegm

Where

0)()(1

2 2

2

2

22

kRVd

d

d

dE

The corresponding radial wave equation reads

./2 with Ek

)()(sin)()(cos)( kNkkJkkR k

The general solution of a scattering particle reads

im

mk ekNkkJkkx

)()(sin)()(cos)(

x

At

iki

fxdxAc

iexkixk exp)()(expexpAsymp)(

aThe solution in exterior region

im

m

i eiek

f

sin22

1)( )4/(

The total cross section

m

t k 2sin4

The scattering amplitude

yield which)( reads section cross total theThus

.)()()(by given is section

cross aldifferenti theflux, magnetic thecarrying (fermions) bosons identicalFor

-

2

d

ff

even,

2sin16

(bosons)m

t

odd,

2sin16

(fermions)m

t

包含 AB effect 的分波散射理論所繪的短範圍位能相互作用的散射截面圖，圖一橫軸是能量的大小，縱軸是散射截面的大小，可看出低能量時散射截面產生驚人的下降現象 ;圖二橫軸是磁通的大小，可看到散射截面隨著磁通以週期性變化的神奇現象。這些效應對於納米量子傳輸系統和納米量子光電系統有許多重要的應用。

.)12(flux magnetic quantized at the 0 section cross The

flux. magnetic thecarrying bosons identicalfor sections cross Total

0 nt

.5.0for 1)(2nat quantized isflux magnetic the when0

flux. magnetic thecarrying bosons of sections cross of structures Periodic

0 kat

.5.0for 2nat quantized isflux magnetic the when0

flux. magnetic thecarrying fermions identical of sections cross Total

0 kat

.5.0for 2nat quantized isflux magnetic the when0

flux. magnetic thecarrying fermions identicalfor sections cross totalof structures Periodic

0 kat

Quantum Interference and Anomalous Cross Section in

Three Dimensions

Plane wave

),(),()(4}exp{ ~*~

~0

mlkkmlll

l

lml

YYkrjirki

mqmkkqmlmq

q

r

P

ZZkrjC

rdrAc

ierki

),(),()(

})(exp{}exp{

*~,

0

Quantum interference of magnetic flux leads to

).1~

2( and ,~ where

~

,0 liCmql lmq

The angular part is defined as

immm

q

m

mq ePl

mlqZ )(cos

2sin

2cos

)1~

(

)1~

()1(),( ),(

2

0,

00

0

function. Jacobi theis )(cos where ),( 00 mmqP

The general solution for a charged particle moving in a shortrange potential, and an Aharonov-Bohm magnetic flux is found to be

m

qmkkqml

qk ZZ

r

rur ),(),(

)()( *

~

0

At large distance, we expect it to become like

r

ikrfrdrA

c

ierkiFr

r

Pk

}exp{),(})(exp{}exp{)(

)(~ rul

2

~sin)

~sin(sin ~

lkrl

l

by given is )(for behavior asymptotic The ~ rul

ll

r

l

lkr

k

kCru ~

1~

~2

~sin

)()(

),(),(~sinsin1

~cossin

)1~

2(1

),( ,*,

~)

~(

2~

0~

~

mqkkmq

l

li

l

i

mq

ZZle

lel

kf

l

l

The Scattering amplitude is found to be

)(cossin)12(1

)(0

ll

i

q

Pelk

f l

At the quantized values of flux,the result reduces to the well-known amplitude

x

y

e

Magnetic flux

e

)0,2/( kk ),( dt

where

22

2,

42

02 sinsin)cos(sinsin21

~)(cossin)12(4 mq

mqt

Z

k

.)1

~()1(

)2/1~

()2/1(~ and ),2( 2

,0

lq

lqZmq mq

reads ),( section cross total the,0 ,2/ i.e. flux,

magnetic thelar toperpendicu directionincident theof case theIn

k dtk

22

2,

42

,02 sinsin)cos(sinsin21

~)(cossin)12(16

)( mq

evenmqt

Z

kbosons

22

2,

42

,02 sinsin)cos(sinsin21

~)(cossin)12(16

)( mq

oddmqt

Z

kfermions

),(),(),( ft ),(),(),(by given areflux magnetic thecarrying

(fermions) bosons identical for the sections cross totalThe

ft

Hard Sphere Potential

The phase shift is given byThe phase shift is given by

)(/)(tan kankaj

Accordingly, the total cross sections is given by

)()()sin(2)()(

~)()(cos)12(16

2/12/12

2/12

2/1

2,

22/1

2

02 kaJkaJkaJkaJ

ZkaJ

kmq

mqt

20 2 flux with magnetica and sphere, harda

by scattering particle chargeda for section-cross totalThe

a

.,2,1,0 ,2/)12(flux of valuesquantized

at theappear Anomalous axis.-z thealongflux magnetica and sphere,

hardfor sections-cross totalscattering theof structure periodic The

0 nn

).2( )12(flux magnetic quantized

at the )(disappearappear Anomalous flux. magnetic the

carrying bosons identical for the sections-cross totalThe

00 nn

.5.0for )12(at quantized isflux the whenzero

approaches section cross The flux. magnetic thecarrying bosons identical

of sections-cross for the noscillatio AB of stucture periodic The

0 kan

).)12((

2flux magnetic quantized at the )(disappearappear Anomalous

flux. magnetic theCarrying bosons identical for the sections-cross totalThe

0

0

n

n

.5.0for 2at quantized isflux the whenzero

approaches section cross The flux. magnetic thecarrying fermions identical

of sections-cross for the noscillatio AB of stucture periodic The

0 kan

Conclusions:

(1) The total cross section is drastically decreased in the long wave length limit and(or) sufficient short range potential. This phenomenon may ascribed to the magnetic flux induced transparency (FIT).

(2) The cross section is symmetric around magnetic flux with the oscillating period ,where n is the positive integer, and is the fundamental magnetic flux quantum .

2/0 n

0ec /2

(3) For identical “Bosons” (“Fermions”), there exists the phenomenon of FIT only for odd (even) number multiple of , and the cross section is symmetric around the odd (even) number multiple of with the oscillating period . Such effect is similar to the picture of the composite Boson (Fermion) in two dimensional fractional quantum Hall effect, and is useful in the question on pinning force in superconductor.

0

02

Thank you!

Composite Particles in Fractional Quantum Hall Effect

Nature, Vol. 406, 863 (2000).

ctivity.supercondu etemperatur-high

---nsinteractio repulsivepurely from arising

pairing whichin phenomenonanother explain helpmay

arrives,it whena theory, such that possible isIt

pairing. fermion composite of theory BCSfulla not yet isit but

state, 2/5 theof ingunderstandour in forward stepimportant an is this

n,calculatio 1956 sCooper' Like1/2.not but

2/5at occurs pairing such why intoinsight some gives and

pairs,Cooper form can fermions composite ofsea a Fermi

thationdemonstrat beautifula is ncalculatio s'. Scarola

alet

N. Bonesteel, Nature, Vol. 406, 841 (2000).