Download Anyons in the FQHE

Anyons in the FQHE
Aut.: Jernej Mravlje
Adv.: Anton Ramsak
Fermions, bosons, and … anyons
 (1,2)   (1) (2)  (2) (1)  (2,1)   (1,2)
 (1,2)   (1) (2)  (2) (1)  (2,1)   (1,2)

fermions
bosons

2D: anyons

Crucial for understanding the
properties of the FQHE!

 (2,1)  exp (i ) (1,2)
Spin…
l , max. proj. m  l
S 
 n
m  m
m  n  m  l  n 2
Spin is quantized in half integers. But …
S   S x  iS y [Si , S j ]  i ijk Sk
2D – any spin
Spin-statistics theorem
s
Pauli (1940)]

2
2D: any spin -> any statistics?
… and statistics
 (2,1,3,4,..., N )  exp(i  ) (1,2,..., N )
 (1,2, ,..., N )  exp(i2  ) (1,2,..., N )
leads to statistics
  n
if the wave-function is single-valued
f ( )  

2D vs. 3D
3D : 2  0
2D : 2  0
Configurational space
Redundancy in notation

x  ( x1 , x2 ,..., xn )
Leinaas & Myrheim (1977)
xi  X
ex. : xi  ( x, y, z, s z )i
Configurational space of identical particles
Credundant  X N
Ccorrect  X N S N
ex.: 1D
( x1 , x2 )  ( x2 , x1 )
Anyons in 1D
2 2 2
2 2 2 2
H 
(
 )

2m x12  22
4m x 2 m z 2
x
z
Boundary conditions:
 ( x,0)  0

( x,0)  0
z

 *
j z   ( x,0)
( x,0)  ( x,0)
( x,0)  0
z
z
*

( x,0)   ( x,0)
z

 k ( x, z )  exp( ix)(cos kz  sin kz)
k
Model for anyon


B
 E  
t
 

  B  dS

 
 (  E )  dS   E  ds  E 2r   t  


q
c
lz  M z  qEr  

2
2
C 2
q
l z  

4
4
0 
h

 2
e
e
Addition of  0 is equal to a boost for  2 in angular momentum
of our model particle. Boson to fermion and vice-versa!
Arbitrary flux -> anyon.
Two particle anyonic wavefunction




( p1  qA1 ) ( p2  qA2 )
H

2m
2m


P 2 ( p  qArel )
H

4m
m



r   
A1, 2  
zˆ  2 ; r  r1  r2
2
r
 (r ,   )   (r , )
 

A  A  ; ( ) 
2
P2 p2
H

4m m
 (r , )  exp( iq) (r , )  exp( 
 (r,   )  exp( iq  2) (r,)
iq
) (r ,  )
2
anyonic!
HE and QHE

 
eE  ev  B
neE  jB
B
U
I
ne
RH  B ne
• HE: Hall (1879)
charge and density of carriers
• QHE: Von Klitzing et al. (1980)
standard for resistance
h
(Von Klitzing const. RK  2  25812.807449(86) )
e
measurement of fine structure
constant
IQHE: explanation
Landau levels
 2
1 
H
( p  eA)
2m
 B
A  ( y , x )
2
H nj  c (n  1 2) nj
J nj  j nj
(r l B ) 2
 nj (r ,  )  (r lB ) exp( ij ) L (r lB ) exp( 
)
2
j
j
n
jmax  R 2 lB2  B S  0    0
Filling factor:
  N Z  N jmax
z  x  iy
 0 j ( z )  z exp( 
j
z
2
2
)
FQHE
discovered in 1982
(Tsui et al. )
Stormer (1992)
Laughlin ground state
noninteracting
z
 0 j ( z )  z exp( 
j

2
2
)

 ( z1 , z2 )  ( z1 ) ( z2 )  ( z2 ) ( z1 ) exp( 
0
1
0
1
z1  z2
2
4
2
z1  z2
2
)  ( z1  z2 ) exp( 
filled Landau level
Z
 ( z1 , z2 ,..., z N )   ( zi  z j ) exp( 
i j
2
1
zj )

4 j
Laughlin ground state at filling:   1 m
N
 ( z1 , z2 ,..., z N )   ( zi  z j ) m exp( 
i j
Laughlin(1983)
1
2
z1 )

4 j
- a guess
- shows excellent overlap with numerical results
4
2
)
Fractional charge and fractional
statistics of Laughlin excitations
 0 : j  j  1  j   j 1
excitation of ground state m
z0 ( z1 , z2 ,..., z N )   ( zi  z0 )m
i
fractionally charged
C 2
q
l z  

4
4
anyonic!
Measurements of frac. charge and frac.
statistics

Shot noise measurements

Resonant tunneling
experiments
Saminadayar & Glattli (1997)
Quantum computation
Averin & Goldman (2001)
Conclusion




Anyons in 2D
Fractional charge of excitations of
FQHE ground-state
Fractional statistics -> anyons
Possible use for quantum computation
which is robust against decoherence