Complexity at the Edge of Quantum Hall Liquids: Edge Reconstruction and Its Consequences

Kun YangNational High Magnetic Field Lab and Florida State Univ.

In Collaboration with

Xin Wan and Akakii Melikidze (NHMFL and FSU);

Edward Rezayi (Calstate LA)

Thanks to: Lloyd Engel and Dan Tsui

Quantum Hall Effect = Incompressibility or Gap for Charged Excitations.

Origin of incompressibility/charge gap:

Landau levels for IQHE (single electron);

Coulomb interaction for FQHE (many-body).

Why not an insulator?

Answer: Gapless chiral edge states.

Edge electrons form chiral Fermi/Luttinger liquid at the edge of anInteger/fractional quantum Hall liquid. (Halperin 82; Wen 90-92)

k

EF

Chiral Luttinger Liquid (CLL) Theory: Chirality => Universality in single electron and other properties.

Tunneling between QH edge and metal (Fermi liquid):

Cleaved edge overgrowth: atomic sharp edge

m = 3 = 2.7

Chang et al. (96) Grayson et al. (98)

/1

Chang et al. (01) Plateau?

55.0/2

Hilke et al. (01)

I ~ Vα; CLL predicts: = m, for Laughlin sequence with = 1/m; universal exponent also for Jain sequence that are maximally chiral.

No Universality in Tunneling Exponents!

Reason we propose in this work:

Electrostatics => Edge Reconstrucion

=> Additional, non-Chiral Edge Modes

=> Absence of Universality

Edge Reconstruction in IQH Regime

MacDonald, Eric Yang, Johnson (93); Chamon, Wen (94)

•Strong confining potential/weak Coulomb interaction

•Weaker confining potential/strong Coulomb interaction

<nk>

kkF

(x)

x

22 Bl

(x)

x

22 Bl

k

totalktotal KknK min

k

totalktotal KknK min

Edge reconstruction!

1

0

<nk>

k

1

0

ks1 ks2 ks3

Model for Numerical Study

• Competition: U vs. V

• Tuning parameter d

• In experiments

d ~ 10 lB or above

• N = 4-12 electrons at filling factor 1/3.

Background charge (+Ne)

dElectron

layer (-Ne)

Hard Wall

mmm

mmlnnlmmnl

lmn ccUccccVH

2

1

Coulomb interaction Confining potential

-8.5

-7.5

-6.5

-5.5

-4.5

-3.5

-2.5

-1.5

-0.5

15 30 45 60 75

M

E0

( in

un

its

of

e2 /

l B )

5.0

4.0

3.0

2.0

1.6

1.0

0.5

0.1

GGS

M = 45, same as Laughlin state

Numerical Evidence of Edge Reconstruction

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

r / lB

2l B

2 (r

)

d = 5.0

M = 65

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

r / lB

2l B

2 (r

)

Sharp Edge

Laughlin State d = 0.1

M = 45

Overlap > 95%

N = 6

Numerical Evidence of Edge Reconstruction

0.0

0.5

1.0

1.5

2.0

3 5 7 9N

dc 1.05.1 cd

In real samples: d > 10 lB; Edge Reconstructiondespite the cleaved edge!

Background charge (+Ne)

dElectron

layer (-Ne)

Hard Wall

Loss of Maximal Chirality due to Edge Reconstruction

Critical d ~ lB

• lB : fundamental in lowest LL

– Size of single electron w.f.

– Range of effective attraction between electrons due to exchange-correlation effects

– Length scale associated with edge reconstruction

• d : separation between electron and background layers

– Electrostatic: range of fringe field near edge

Electrostatic energy gain ~ exchange-correlation energy loss dc ~ lB

+ + + + + + + + + + + + + +

_ _ _ _ _ _ _ _ _ _ _

FF

d

2a

E(lB)

Evolution of Energy Spectra

μ < 0: vacuum of chiral bosons; no reconstruction.μ > 0: finite density of chiral bosons; edge reconstruction!

μ = 0: critical point of dilute Bose gas transition; ν = ½, z =2, etc. Thus δk ~ (d-dc)1/2, etc.

In the reconstructed phase, writeand integrate out fluctuations of n:

Reconstruction transition may also be first order, if thereis effective attraction between chiral bosons.

Chiral charge mode velocity v much larger than non-chiral neutral mode velocity vφ which leads to tunneling exponent:

Thus tunneling exponent non-universal and renormalized by a small amount from the original CLL prediction.

Detecting new modes: momentum resolved tunneling

Summary• Edge reconstruction occurs in a FQH liquid, even in the presence of

sharp edge confining potential (cleaved edges).• It leads to additional edge modes not maximally chiral, leading to non-

universality of tunneling exponent; presence may be detected through momentum resolved tunneling.

• Edge reconstruction is a quantum phase transition, in the universality class of 1D dilute Bose gas transition. Critical properties determined exactly.

• Finite temperature tends to suppress edge reconstruction. References: • X. Wan, K.Y., and E. H. Rezayi, Phys. Rev. Lett. 88, 056802 (2002).• X. Wan, E. H. Rezayi, and K. Y., Phys. Rev. B. 68, 125307 (2003).• K.Y., Phys. Rev. Lett. 91, 036802 (2003).• A. Melikidz and K.Y., Phys. Rev. B. 70, 161312 (2004); Int. J.

Mod. Phys B 18, 3521 (2004).• For closely related work see G. Murthy and co-workers, PRBs 04.