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

Anomalous Cross Section
Induced by Topological Quantum
Interference
De-Hone Lin
Department of Physics, NSYSU
23 December 2004
Partial wave theory of a two dimensiona l scattering problem for an arbitray
short range potential and a nonlocal A - B magnetic flux is establishe d.
The scattering process of a hard disk like potential and the magnetic flux
is examined. Since the nonlocal influence of magnetic flux on
the charged particles is universal, the nonlocal effect in hard disk case
is expected to appear in quite general potential system and will be useful
in understand ing some phenomena in mesoscopic phyiscs.
Partial wave theory of a three dmensional scattering problem for
an arbitrary short range potential and a nonlocal A - B magnetic flux
is establishe d. An anomalous total cross section is revealed
at the specific quantized magnetic flux at low energy whi ch helps explain
composite fermion and boson model in the fractional quantum Hall effect.
Since the nonlocal quantum interferen ce of magnetic flux on the charged is
universal, the nonlocal effect is expected to appear in quite general potential
system and will be useful in understand ing some other phenomena
in mesoscopic phyiscs.
Fractional Quantum Hall States
•2-D electron system inside the GaAs/AlGaAs heterostructure
•High magnetic fields (B~10T)
•Low temperatures (T~0.1K)
Coulomb forces
flux quantum attachment
Electron vortex attraction at fractional Landau - level filling   1/3.
(a) Placing three vortices onto each electron reduces
electron - electron (Coulomb) repulsion.
(b) Vortex attachment can be viewed as the attachment
of magnetic flux quanta to the electrons, transform ing them
to composite bosons.
An even - denominato r
fractional Hall plateau
h / e2
at  xy 
is observed
5/ 2
at tempera tures below 100 mK
R. willett, J.P. Eisenstein,
H.L. Stormer, D.C. Tsui
, A.C. Gossard, and
J.H. English,
PRL, vol. 59, 1776, 1989.
FQHE at   5/2. A FQHE state at such an even denominato r
fraction should not be allowed. The origin of the state remains
unclear. An exciting possibilit y for the origin of this state is
the formation of composite fermion pairs.
H.L. Stormer, Nobel Lecture : The fractional quantum Hall effect
RMP, 71, 875, 1998.
Nature, Vol. 406, 863 (2000).
Scarola et al.' s calculatio n is a beautiful demonstrat ion that
a Fermi sea of composite fermions can form Cooper pairs,
and gives some insight into why such pairing occurs at   5 / 2
but not   1/2. Like Cooper' s 1956 calculatio n,
this is an important step forward in our understand ing of the   5 / 2 state,
but it is not yet a full BCS theory of composite fermion pairing.
It is possible that such a theory, when it arrives,
may help explain another phenomenon in which pairing
arising from purely repulsive interactio ns - - high - temperatur e supercondu ctivity.
N. Bonesteel, Nature, Vol. 406, 841 (2000).
A.K. Geim etc, Nature 407, 55, 2000.
At first pointed out by Bardeen and Ginzburg in the early sixties, the amount
of magnetic flux carried by vortice s in supercondu cting materials depends on
their distance from the sample edge, and can be smaller th an one flux quantum,
  h / 2e. In bulk supercondu ctors, this reduction of flux becomes negligible
at submicrome tre distances from the edge, but in thin films the effect may survive
much farther into the material. 
Here we measure the amount of flux introduced by individual vortices in a
supercondu cting film, finding that the flux always differs substantia lly from  .
We have observed vortices that carry as little as 0.001 , as well as negative vortices,
whose penetratio n leads to the expulsion of magnetic field. 
Summary

Quantum interference of magnetic flux

Quantum interference in partial wave theory and
anomalous cross section in two dimensions

Quantum interference in partial wave theory and
anomalous cross section in three dimensions

Composite bonsons and fermions
Introduction
V () 
Bound states therein
1

and
V ()  0
2
for   a
for   a
Radius

Phase shifts
A charged particle
D. Bohm and Y. Aharonov in 1959 found AB effect
Interference
pattern
Charged particle
Magnetic flux

 ( x)
1
( 0)
1

 ( x)
( 0)
2

 ( x)
( 0)
2

( 0) 
( 0) 
 ( x )  1 ( x )  2 ( x )
( 0)

x  


ie

1( 0) ( x ) exp   A( x )  dx
 c 1

1


 (x )
2


 ie x   
 ( x ) exp   A( x )  dx 
 c 2

( 0)
2

 (x )


x  
x  




ie
ie



( 0)
(0)
 1 ( x ) exp   A( x )  dx  2 ( x ) exp   A( x )  dx.
 c 1

 c 2







ie

( 0)
(0)
 ( x )  1 ( x )  2 ( x ) exp   A( x )  dx
 c

 ie 
( 0) 
(0) 
 1 ( x )  2 ( x ) exp   .
 c 
The periodic cycle of interferen ce is given by   (c / e)2 .
The four-vector formulation of the non-integrable phase factor
is given by
 ie

exp   A  dx .
 c

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.
Aharonov-Bohm magnetic flux
V () 
Bound states therein
1

and
V ()  0
2
for   a
for   a
Radius

Phase shifts
A charged particle
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
Potential and an Aharonov-Bohm Flux


    ( 0 )  
2 
 E  H 0  x , i  G x , x ; E    ( x  x ),



2


2
where H 0  
  V ( x ).
2
In polar coordinates for the cylindrically symmetric system:
(0)  
G x , x ; E  

1 im(  ')
G  ,  ' ; E  e

2
m  
(0)
m
Magnetic field exists in the system, then

 


 
ie

(0)
Gx , x ; E   G x , x ; E  exp   A(r )  dr 
x

c



x
For the Aharonov-Bohm Flux
 
 yeˆ x  xeˆ y
A( x )  2 g
,
2
2
x y
the magnetic field
B3  4g ( x ),
and the magnetic flux
g   / 4 .


 2  d 2
1 d  2 
 2   V (  ) G (  ,  ; E )   (    )
 E    2 
 d  


 2   d
Where
  m  0 is a real number wit h 0  2eg / c   /  0
The corresponding radial wave equation reads


 2  d 2
1 d  2 
 2   V (  )  Rk (  )  0
 E    2 
 d  


 2   d
with k  2E / .
The solution in exterior region
a
Rk (  )  k cos   (k ) J  (k )  sin   (k ) N (k )
The general solution of a scattering particle reads

k ( x ) 


m  
k cos   (k ) J  (k )  sin   (k ) N  (k )e im

At x  


 


i
 ie     
k ( x )  Asymp exp ik  x exp   A( x )  dx    f ( )
exp ik 

 c


The scattering amplitude
1
f ( ) 
2k

e
m  
i (   / 4 )
2i sin   e
The total cross section
4 
2
 t   sin  
k m
im
For identical bosons (fermions) carrying the magnetic flux, the differenti al cross
section is given by  ( )  f ( )  f (   ) .
2

Thus the total cross section reads   ( )d which yield
-
 t (bosons) 

16

 t (fermions) 

2

m   ,even
16

 sin
2

 sin
m   ,odd
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。
Total cross sections for identical bosons carrying the magnetic flux.
The cross section  t  0 at the quantized magnetic flux   ( 2n  1) 0 .
Periodic structures of cross sections of bosons carrying the magnetic flux.
 t  0 when the magnetic flux is quantized at   (2n  1) 0 for ka  0.5.
Total cross sections of identical fermions carrying the magnetic flux.
 t  0 when the magnetic flux is quantized at   2n 0 for ka  0.5.
Periodic structures of total cross sections for identical fermions carrying the magnetic flux.
 t  0 when the magnetic flux is quantized at   2n 0 for ka  0.5.
Quantum Interference and
Anomalous Cross Section in
Three Dimensions
Plane wave

 
exp{ik  r }  
l 0
l

~  l
m
4i l jl (kr)Ylm*~ ( k ,  k )Ylm~ ( ,  )
Quantum interference of magnetic flux leads to
 
 
r


ie
exp{ik  r } exp{  A( r )  dr }
c P


q 0

*
~
C
j
(
kr
)
Z
 q ,m l
qm ( k ,  k ) Z qm ( ,  )
m  
~
~
~
l
where l  q  m  0 , and Cq ,m  i (2l  1).
The angular part is defined as
Z q,m ( ,  ) 
~
m 
( q  1)( l  m   0  1) 

  0 ( m  0 , m  0 )
im
cos
sin
P
(cos

)
e


~
q
2
2
 2 ( l  1)

( m  0 , m  0 )
q
where P
(cos  ) is the Jacobi function.
The general solution for a charged particle moving in a short
range potential, and an Aharonov-Bohm magnetic flux is found to be


k (r )  
q 0


m  
u ~l (r )
r
Z ( k ,  k ) Z qm ( ,  )
*
qm
At large distance, we expect it to become like
 

r


  
ie
exp{ikr}

k ( r )  F  exp{ik  r } exp{  A( r )  dr }  f ( ,  )
c P
r


The asymptotic behaviorufor
~ (u
r~l)(r ) is given by
l
~
 C ( k ) 

l
u~l ( r )  
sin  kr 
  ~l 
2
 k


r 
1
~
l
~

~
l
 sin  ~l sin( l  ) sin  kr 
2

 



The Scattering amplitude is found to be

1
f ( ,  )  
k q 0
i ~l
2~

e sin  ~l cos l   *
~
(2l  1) 
~

~  Z q ,m ( k ,  k ) Z q ,m ( ,  )
i ( ~l  l  )
1  e
m  
sin  ~l sin l  

At the quantized values of flux,
the result reduces to the well-known amplitude

1
i l
f ( )   (2l  1)e sin  l Pl (cos  )
k q 0
Magnetic flux


y

e
x

e
In the case of the incident direction perpendicu lar to the magnetic
2,  ktotal
 0)cross section   d ( ,  ) reads
flux, i.e.  k   / 2, (kk  0,/ the
t

 t   d ( ,  )
4
t  2
k


q 0
m  
 
~2
2
4


(2  1) sin   cos (  ) Z q ,m


2
2
1  2 sin   sin  cos(     )  sin   sin  
where
~2
  (2q  m  0 ), and Z q ,m
~
(q  1 / 2)( l  1 / 2)

.
~
(q  1)( l  1)
The total cross sections for the identical bosons (fermions)
 t ( , )  f ( , )  (   ,   )
carrying the magnetic flux are given by  t ( ,  )  f ( ,  )  (   ,    )
 t (bosons ) 
16
k2


q 0
16
 t ( fermions)  2
k


q 0
~


(2  1) sin 2   cos 4 (  ) Z q2,m



2
2
m   ,even 
1  2 sin   sin  cos(     )  sin   sin  

~


(2   1) sin 2   cos 4 (  ) Z q2,m



2
2
1

2
sin

sin

cos(



)

sin

sin

m   ,odd 






Hard Sphere Potential
The phase shift is given by
tan    j (ka) / n (ka)
Accordingly, the total cross sections is given by
16
t  2
k


~
(2  1) cos 2 (  ) J 2 1/ 2 (ka) Z q2,m
q 0
m  
J 2 1/ 2 (ka)  J 2 1/ 2 (ka)  2 sin(  ) J  1/ 2 (ka) J   1/ 2 (ka)
 
The total cross - section for a charged particle scattering by
a hard sphere, and a magnetic flux with  0  2a 2
The periodic structure of the scattering total cross - sections for hard
sphere, and a magnetic flux along the z - axis. Anomalous appear at the
quantized values of flux   (2n  1) 0 / 2, n  0,1,2, .
The total cross - sections for the identical bosons carrying
the magnetic flux. Anomalous appear (disappear ) at the
quantized magnetic flux   (2n  1) 0 (  2n 0 ).
The periodic stucture of AB oscillatio n for the cross - sections of
identical bosons carrying the magnetic flux. The cross section approaches
zero when the flux is quantized at   (2n  1) 0 for ka  0.5.
The total cross - sections for the identical bosons Carrying the magnetic flux.
Anomalous appear (disappear ) at the quantized magnetic flux   2n 0
(  (2n  1) 0 ).
The periodic stucture of AB oscillatio n for the cross - sections of
identical fermions carrying the magnetic flux. The cross section approaches
zero when the flux is quantized at   2n 0 for ka  0.5.
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   n 0 / 2
with the oscillating period  0 ,where n is the positive integer, and is
the fundamental magnetic flux quantum 2c / e.
(3) For identical “Bosons” (“Fermions”), there exists the
phenomenon of FIT only for odd (even) number multiple of  0 , and
the cross section is symmetric around the odd (even) number
multiple of with the oscillating period 2 0 . 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.
Thank you!
Composite Particles in Fractional
Quantum Hall Effect
Nature, Vol. 406, 863 (2000).
Scarola et al.' s calculatio n is a beautiful demonstrat ion that
a Fermi sea of composite fermions can form Cooper pairs,
and gives some insight into why such pairing occurs at   5 / 2
but not   1/2. Like Cooper' s 1956 calculatio n,
this is an important step forward in our understand ing of the   5 / 2 state,
but it is not yet a full BCS theory of composite fermion pairing.
It is possible that such a theory, when it arrives,
may help explain another phenomenon in which pairing
arising from purely repulsive interactio ns - - high - temperatur e supercondu ctivity.
N. Bonesteel, Nature, Vol. 406, 841 (2000).