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

3

Fractional quantum Hall effect

Incompressibility (gap)

Landau levels

Interaction

Edges

4

Integer quantum Hall effect

Landau levels

Interaction

Edges

Landau levels ?

Wen’s theory

5

Chern-Simons theory

mean field

Composite fermions

Electron correlations built into the bulk are assumed to extend all the

way to the edge

6

Tunneling into the edge of a FQHE droplet

Landau levels

V

I

)A sharp cleaved 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

23

Summery

1. Instability due to correlated tunneling.

2. A similar behavior for and .

3. Edge reconstruction.

4. Universality of ?

37n = 2

3n =

Thank You!

a