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

Introduction Laughlin wavefunction for ν = 1/3: One-particle wavefunctions at the lowest Landau level (disk geometry): 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/

Question about the ν = 2/3 state in Laughlin theory 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 ? NOBODY KNOWS …. However we know that will certainly NOT go to zero as (z 1 – z 2 ) 3. (Only as (z 1 – z 2 ) 1, like any antisimmetric function) So, what property of Ψ 2/3 makes it a good wavefunction?

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

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 !

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

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

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

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

Conclusions MERCI ! * 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 ν and 1- ν * Understanding this might help to better understand FQHE * 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