Correlated tunneling and the instability of the fractional quantum Hall edge Dror Orgad Oded Agam...
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Transcript of Correlated tunneling and the instability of the fractional quantum Hall edge Dror Orgad Oded Agam...
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