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

1
Complexity at the Edge of Quantum Hall Liquids
Edge Reconstruction and Its Consequences

• Kun Yang
• National 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

2
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?
3
Answer Gapless chiral edge states.
k
Edge electrons form chiral Fermi/Luttinger liquid
at the edge of an Integer/fractional quantum Hall
liquid. (Halperin 82 Wen 90-92)
Chiral Luttinger Liquid (CLL) Theory Chirality
gt Universality in single electron and other
properties.
4
Tunneling between QH edge and metal (Fermi
liquid)
I Va CLL predicts a m, for Laughlin
sequence with n 1/m universal exponent also
for Jain sequence that are maximally chiral.
5
No Universality in Tunneling Exponents!

• Reason we propose in this work
• Electrostatics gt Edge Reconstrucion
• gt Additional, non-Chiral Edge Modes
• gt Absence of Universality

6
Edge Reconstruction in IQH Regime

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

• Strong confining potential/weak Coulomb
  interaction

ltnkgt
r(x)
1
0
k
x
kF

• Weaker confining potential/strong Coulomb
  interaction

r(x)
x
Edge reconstruction!
7
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.

8
Numerical Evidence of Edge Reconstruction
N 6
Overlap gt 95
9
Numerical Evidence of Edge Reconstruction
In real samples d gt 10 lB Edge
Reconstruction despite the cleaved edge!
10
Loss of Maximal Chirality due to Edge
Reconstruction
11
(No Transcript)
12
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

13
Evolution of Energy Spectra
14
Evolution of Energy Spectrum
N 9
k0 0.15/lB
15
µ lt 0 vacuum of chiral bosons no reconstruction.
µ gt 0 finite density of chiral bosons edge
reconstruction!
µ 0 critical point of dilute Bose gas
transition ? ½, z 2, etc. Thus dk
(d-dc)1/2, etc.
Reconstruction transition may also be first
order, if there is effective attraction between
chiral bosons.
In the reconstructed phase, write and integrate
out fluctuations of n
16
Chiral charge mode velocity v much larger than
non-chiral neutral mode velocity vf which leads
to tunneling exponent
Thus tunneling exponent non-universal and
renormalized by a small amount from the original
CLL prediction.
17
Detecting new modes momentum resolved tunneling
18
(No Transcript)
19
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.