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

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.

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

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  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 andAkakii Melikidze(NHMFL and FSU); Edward Rezayi (Calstate LA) Thanks to: Lloyd Engel andDan 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. EF 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 => Universality in single electron and other properties.

  4. Cleaved edge overgrowth: atomic sharp edge Chang et al. (01) Plateau? m = 3 a = 2.7 Hilke et al. (01) Grayson et al. (98) Chang et al. (96) Tunneling between QH edge and metal (Fermi liquid): I ~ Vα; 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 => Edge Reconstrucion => Additional, non-Chiral Edge Modes => Absence of Universality

  6. <nk> 1 0 k ks1 ks2 ks3 Edge Reconstruction in IQH Regime MacDonald, Eric Yang, Johnson (93); Chamon, Wen (94) • Strong confining potential/weak Coulomb interaction <nk> r(x) 1 0 k x kF • Weaker confining potential/strong Coulomb interaction r(x) x Edge reconstruction!

  7. Hard Wall F = NF0 / n Background charge (+Ne) d Electron layer (-Ne) Coulomb interaction Confining potential 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 > 95%

  9. Hard Wall F = NF0 / n Background charge (+Ne) d Electron layer (-Ne) Numerical Evidence of Edge Reconstruction In real samples: d > 10 lB; Edge Reconstruction despite the cleaved edge!

  10. Loss of Maximal Chirality due to Edge Reconstruction

  11. 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 2a + + + + + + + + + + + + + + d _ _ _ _ _ _ _ _ _ _ _ F F DE(lB) Critical d ~ lB

  12. Evolution of Energy Spectra

  13. 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. 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:

  14. Chiral charge mode velocity vmuch 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.

  15. Detecting new modes: momentum resolved tunneling

  16. 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.

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