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M.I. Dyakonov University of Montpellier II, CNRS, France

One-dimensional model for the Fractional Quantum Hall Effect. M.I. Dyakonov University of Montpellier II, CNRS, France. Outline: The FQHE problem Laughlin function Unresolved questions One-dimensional model Interesting but strange result More questions. Introduction.

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M.I. Dyakonov University of Montpellier II, CNRS, France

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  1. One-dimensional model for the Fractional Quantum Hall Effect M.I. Dyakonov University of Montpellier II, CNRS, France • Outline: • The FQHE problem • Laughlin function • Unresolved questions • One-dimensional model • Interesting but strange result • More questions

  2. Introduction One-particle wavefunctions at the lowest Landau level (disk geometry): Laughlin wavefunction for ν = 1/3: It is well established that this is a good wavefunction. For other fractions like ν = 2/5 the situation is not so clear. For a review see: M.I. Dyakonov, Twenty years since the discovery of the Fractional Quantum Hall Effect: current state of the theory, arXiv:cond-mat/0209206

  3. ν = 2/3 electron state is the ν = 1/3 hole state ! In terms of hole coordinates it should have the Laughlin form. Question: what does look like in terms of electron coordinates ? However we know that will certainly NOT go to zero as (z1– z2)3 . (Only as (z1– z2)1, like any antisimmetric function) Question about the ν = 2/3 state in Laughlin theory NOBODY KNOWS …. So, what property of Ψ2/3 makes it a good wavefunction?

  4. One dimensional model for FQHE M.I. Dyakonov (2002) Consider M degenerate one-particle states on a circle: There are N < M spinless fermions with a repulsive interaction (e.g. Coulomb) Problem: find the ground state for a given filling factor ν = N/M

  5. Crystal-like state in this model Wannier (localized) states: At ν = 1/3 these states can be filled to form a crystal However a Laughlin-like state presumably is preferable !

  6. Proposed wavefunction for 1D model: Laughlin-like wavefunction for one dimensional model Laughlin wavefunction for ν = 1/3: The normalization constant A is known (was calculated by Dyson a long time ago)

  7. 2. Decompose it as a superposition of determinants: i.e. for M=6, N=2 (now 3. Replace each determinant by the complimentary determinant with M–N states: How to construct the electron ν = 2/3 wavefunction 1. Take the ν = 1/3 hole wavefunction (N coordinates)

  8. 1. Write downΨ2/3 in the sameform: Simple and interesting answer within the 1D model 2. This function contains powers of exp(iφ) greater thanM 3. Take these powers modulo M ! This procedure gives the correct answer !!! Isn’t this bizarre?

  9. where with Then the complementary function is where has the same form as C! Another presentation of same thing can be rewritten in the basis of Wannier functions Φs(φ) as The modulo M rule now works automatically !

  10. Conclusions * I believe that the essential properties of the FQHE energy spectrum can be reproduced whenever one has M degenerate states filled by N fermions with a repulsive interaction * Like in the case of FQHE, only exact numerical calculations with small numbers of electrons can tell whether the proposed wavefunction is the true ground state * There must be some interesting math behind the observed beautiful relation between wavefunctions for νand1-ν * Understanding this might help to better understand FQHE MERCI !

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