M.I. Dyakonov

University of Montpellier II, CNRS, France

One-dimensional model for the Fractional Quantum Hall Effect

Outline: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/0209206

2|( ) exp( )

4!

|m

m

z zz

m

231

1( ... ) ( ) exp( )

4N i j ii j

z z z z z

Question about the ν = 2/3 state in Laughlin theory

32

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 (z1 – z2)3 .

(Only as (z1 – z2)1, like any antisimmetric function)

32

So, what property of Ψ2/3 makes it a good wavefunction?

One dimensional model for FQHEM.I. Dyakonov (2002)

),exp(2

1)(

ikk 1...,1,0 Mk

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

ksM

i

Ms

M

M

kks

2exp)(

1)

2()(

1

0

1...,1,0 Ms

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

2313/1 4

1exp)()...( ij

jiiN zzzzz

Laughlin wavefunction for ν = 1/3:

3

113/1 )]exp()[exp()...( ji

NjiN iiA

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)

12

11

22

21

02

01

32

313

212131 3)(),(zzzz

zzzz

zzzz

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:

04

03

02

01

34

33

32

31

44

43

42

41

54

53

52

51

14

13

12

11

24

23

22

21

44

43

42

41

54

53

52

51

432132 3),,,(

zzzzzzzzzzzzzzzz

zzzzzzzzzzzzzzzz

zzzz

exp( )z i(now

Simple and interesting answer within the 1D model

1. Write down Ψ2/3 in the same form:

3

21213/2 )]exp()[exp()...( ji

NjiN iiA

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 Φs(φ) as

3

113/1 )]exp()[exp()...( ji

NjiN iiA

)()...()...()...( 1)(

1131 1 Nsss

NN NssC

Nji

ssN

jiAssC1

31 )()...(

M

i 2expwhere with

Nji

ssN

jiBssD21

321 )()...(

Then the complementary function is

)()...()...()...( 21)(

212132 21 Nsss

NN NssD

where has 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