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

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