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Correlated tunneling and the instability of the fractional quantum
Hall edge
Dror Orgad Oded Agam
July 21, 2009
PRL 100,156802 (2008)
2
Outline
The system
Historical overview of theory and experiments
The model
A toy model
Solution & implications
5
Chern-Simons theory
mean field
Composite fermions
Electron correlations built into the bulk are assumed to extend all the
way to the edge
7
Tunneling into the edge of a FQHE droplet:
Wen’s theory I V aµ
a
1n-
1
2
3
4
5
1 2 3 4
Tunneling into the edge of a FQHE droplet: Experimental
results
3a =
1 12 3
n< <
for
8
I V aµ
Tunneling into the edge of a FQHE droplet: Experimental
results
Chang et al., PRL 1996 2.7a ; for13
n =
a
1n-
1
2
3
4
5
1 2 3 4
Grayson et al., PRL 1998:
11.16 0.58a n- -; for 1 4n< <
9
I V aµ
Tunneling into the edge of a FQHE droplet: back to Theory
a
1n-
1
2
3
4
5
1 2 3 4
1a n-:
Conti & Vinagle, 1998
Han & Thouless, 1997
Zülicke & MacDonald, 1999
Hydrodynamical Theory
The nature of the underlying quasiparticles is ignored
Alexeev et al., 2000Tunneling via impurity states
sharply located at the Fermi levelLee & Wen, 1998
Lopez & Fradkin, 1999Non-propagating modes
10
Tunneling into the edge of a FQHE droplet: additional
experiments I V aµ a
1n-
1
2
3
4
5
1 2 3 4
Chang et al., 2001
Tunneling into the edge of a FQHE droplet: Theory again
Levitov, Shytov & Halperin,1998, 2001
Smearing of Wen’s original result due to finite value of xxr
11
I V aµ a
1n-
1
2
3
4
5
1 2 3 4
Tunneling into the edge of a FQHE droplet: More
experiments
Hilke et al., 2001
12 0.55a n-» - 1 1.75n< <for
Tunneling into the edge of a FQHE droplet: Numerics
Mandal & Jain, 2002 12.43 for 321.96 for 531.74 for 7
a n
a n
a n
= =
= =
= =
12
The edge tunneling puzzle:
Non-universality!?
Wen’s theory - is it complete?
We show:
“Correlated tunneling” may lead to an edge instability towards a new configuration with reconstructed edge.
Similar behavior has been observed in the numerical studies of Tsiper & Goldman (2001), and Wan ,Yang & Rezayi, (2002/3)
13
Landau levels of Composite Fermions2
5n =
The interaction Hamiltonian:
( ) ( ) ( ) ( ) ( )2 2 † †int
, , ,
12 i j k l
i j k l
H d rd r r r V r r r ry y y y¢ ¢ ¢ ¢= -å ò
123
Hartree term , i j k l= =
Fock term , i l k j= =Correlated tunneling terms but i j k l= ¹ or while i j k l¹ =
( ) ( ) ( ) ( ) ( ) ( )† † †1 1 2 2 1 2( ) .V r r r r r r r r C cy y y y y yé ùé ù¢ ¢ ¢- + +ê úë ûë û
Edge states
14
( )2
1 2
2T
C
N N NH
C
+ -=
Correlated tunneling: A toy model
( )† †1 2 1 2 2 1( )CTH N N bb bbl= + +
Correlated tunneling
CH H= CTH+
0l = Ground state 1 2 TN N N+ =0l ¹ ( )11 22
b b b± = ±
( )21 2 1( )1
1 1 22 2T
T
C NN N NH N N N
C NC C
l
l++ -
+ --
æ öæ ö+ ÷ ÷ç ç÷ ÷ç ç= - + +÷ ÷ç ç÷ ÷-ç ç÷ ÷è øè ø
Eigenvalues :
2 21 1 42
CC
l± +
1 2 TN N N+ - ® ±¥N- ® ±¥
15
Landau levels of Composite Fermions2
5n =
The Chiral Luttinger Model for the edge states:
123
( )10
, 1,2 , 1,2
14 x i ij i x i ij x j i ij j
i j i j
S dxd i K V d NV NLt
pt ff ff t
p-
= =
= ¶ ¶ +¶ ¶ +å åò ò3 2
2 3
v gK V g v
æ ö æ ö÷ç ÷ç÷ç= = ÷ç÷ç ÷ç÷ ÷çç ÷ è øè ø
( )† †1 1 2
1,2
1: : . .
4 i ii
S dxd hct l l y y y yp =
æ ö÷ç ÷= + +ç ÷ç ÷çè øåò
Can be diagonalized exactly.0 1S S S= +
16
Diagonalization
0S S= 1S+
( )10
, 1,2
14 x i ij i x i ij x j
i j
S dxd i K Vtt ff ffp
-
=
= ¶ ¶ +¶ ¶åò, 1,2
i ij ji j
d NV NLp
t=
+ åò
( )1 2
110cff f= + ( )1 2
12nff f= -
( )( )
( )( )
2
0
2
141
4
x c c c x c
x n n n x n
S dxd i v
dxd i v
t
t
t ff fp
t ff fp
= ¶ ¶ + ¶
+ ¶ ¶ + ¶
ò
ò
12
+ +
+ +
- -
- -
+ -
+-
+-
+ -
( )5cv v g= +
nv v g= -
0x =
25n =Tunneling density of
states:
17
Tunneling density of states:
( )10
, 1,2
14 x i ij i x i ij x j
i j
S dxd i K Vtt ff ffp
-
=
= ¶ ¶ +¶ ¶åò
( )1 2
110cff f= + ( )1 2
12nff f= -
( )( )
( )( )
2
0
2
141
4
x c c c x c
x n n n x n
S dxd i v
dxd i v
t
t
t ff fp
t ff fp
= ¶ ¶ + ¶
+ ¶ ¶ + ¶
ò
ò
12
0x =( ) ( ) ( ) ( )
( ) ( )[ ] ( ) ( )[ ]
1 10, 0,0†1 1
5 10, 0,0 0, 0,0
2 2
10, 0,0
2
12
c c n n
i t i
i t i t
t e ea
e ea
ff
ff ff
y yp
p
-
- -
=
=
52
1
t: 1
2
1
t: 3
1t
: 3a =
25n =0S S=
18
Diagonalization
( )( )2 20 0 0 0
14
naux x n x
vS dxd i v d N
Lt
pt ff f t
p= ¶ ¶ + ¶ +ò ò
0S S= auxS+1S+ 25n =
1 .Transformation to new bosonic fields:
1 1 12 2 020
1 1 12 21 12
1 102 21010
fj
j f
j f
-
-
æ öæ öæ ö ÷ ÷ç ç÷ç ÷ ÷ç ç÷ ÷ç ÷ç ç÷ ÷ ÷ç ç÷ ç÷ ÷ç =÷ ç ç÷ ÷ç ÷ ç ç÷ ÷ç ÷ ç ÷ç ÷ç ÷ ÷ç ÷ç÷ç ÷ ÷ç ç ÷è ø ÷ç è øè ø
0 0
1 1
22
1 1 11
1 1 12
1 1 1
N
N
N
æ ö æ öæ ö-÷ ÷÷ç çç÷ ÷÷ç çç÷ ÷÷ç çç÷ ÷÷ç çç÷ ÷÷= -ç çç÷ ÷÷ç ç÷ ç ÷÷ç ÷ ç ÷ç ÷÷ç ÷÷çç÷ - ÷÷ç çç ÷÷ç è øè øè ø
N
N
N
0 0 1 1 0 2 2 1 2 F F F F F F= = =F F F
2 .Refermionization
2exp
2i
i i ii x ia L
px j
pé ù
= +ê úê úë û
FN
19
Diagonalization
( )( )2 20 0 0 0
14
naux x n x
vS dxd i v d N
Lt
pt ff f t
p= ¶ ¶ + ¶ +ò ò
0S S= auxS+1S+ 25n =
1 .Transformation to new bosonic fields:
2 .Refermionization
2exp
2i
i i ii x ia L
px j
pé ù
= +ê úê úë û
FN3 .Transformation to new fermionic fields
( )0 1
12
ni xvel
x x x±
± = ±4 .Bosonization
( ) ( )2 2 2, , ,x j x j± ± ±® ®N N5 .Diagonalization
1 10 2 2
cos cossin1 2 2
sin sin2cos2 2 2
0g g
g
g gg
q j
q j
jq
+-
- -
--
æ öæ ö æ ö÷ç÷ç ÷ ÷çç÷ç ÷ ÷ç÷ çç ÷ ÷÷ çç ÷ ÷ç ÷ çç= ÷ ÷ç ÷ çç ÷ ÷ç ÷ çç ÷ ÷ç ÷ ç÷ ÷ç÷ç ÷ ÷çç÷ç ÷ è ø÷çè ø è ø
1 10 2 2
cos cossin1 2 2
sin sincos2 22 2
0Q
Q
Q
g gg
g gg
+
-- -
--
æ öæ öæ ö ÷ç ÷÷ çç ÷ ÷ç÷ çç ÷ ÷÷ ç çç ÷ ÷÷ ç ç÷ç ÷÷ ç= ÷çç ÷÷ ç ÷çç ÷÷ ç ÷ç ÷ç ÷ ÷ç ÷ç÷ç ÷ç ÷÷ çç ÷ ÷ç÷çè ø è øè ø
N
N
N
20
The diagonalized action:
0S S= 1S+ auxS+ 25n =
( )( )2 2
2 2 20 0
0 0
14 x i i i x i i i
i i
S dxd i u d uQ uQLt
pt q q q t
p = =
æ ö÷ç= ¶ ¶ + ¶ + + ÷ç ÷÷çè øå åò ò
0q Is the new rotated auxiliary field with velocity 0 nu v=
22
1,2 2 2 5c n c nv v v v
ulp
æ ö+ -æ ö ÷÷ çç= + ÷÷ çç ÷ ÷çè ø è øm
Instability:when becomes
negative, i.e.1u
5 c nvvl p>
2
2
0.05c nev v
e
e
l e
: :
:
Neguyen, Joglekar & Murthy, 2004))
21
Regularization
( ) ( ) ( )22 422 41
4 2 2u x x x xS dxd i ut
h ht q q q q q
p= ¶ ¶ + ¶ + ¶ + ¶ò
Edge dispersion : ( ) 3E k uk kh= +( )E k uk=
functions of h
Two additional )counter propagating) edge states
22
Comments:
( )E k uk k kh= +Benjamin-Ono type regularization:
Extreme cases: Wigner Crystal – Fermi liquid
Noise measurements (Misha Reznikov)
37n = 2
3n =and
3 c nvvl p ¶>and718 c nvvl p> respectively