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QHE, ASM

Vincent PasquierService de Physique ThéoriqueC.E.A. SaclayFrance

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From the Hall Effect to integrability

1. Hall effect.

2. Annular algebras.

3. deformed Hall effect and ASM.

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Most striking occurrence of a quantum macroscopic effect in the

real word• In presence of of a magnetic field, the

conductivity is quantized to be a simple fraction up to

• Important experimental fractions for electrons are:

• Theorists believe for bosons:

• Fraction called filling factor.

1010

12 pp

1pp

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Hall effect

• Lowest Landau Level wave functions

2

!)( l

zzn

n en

zz

nlr

n=1,2, …, 22 l

A

5

022n

l

A

Number of available cells alsothe maximal degree in each variable

mzzS nn )...( 1

1Is a basis of states for the systemLabeled by partitions

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N-k particles in the orbital k are

the occupation numbers

.........210 kNNNNCan be represented by a partition with N_0 particles in orbital 0,

N_1 particles in orbital 1…N_k particles in orbital k…..

There exists a partial order on partitions, the squeezing order

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Interactions translate into repulsionbetween particles.

mji zz

m universal measures the strength of the interactions.

Competition between interactions which spread electrons apart and highcompression which minimizes the degree n. Ground state is the minimal degreesymmetric polynomial compatible with the repulsive interaction.

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Laughlin wave functionoccupation numbers:

100100100

Keeping only the dominant weight of the expansion

1 particle at most into m orbitals (m=3 here).

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Important quantum numberwith a topological interpretation

• Filling factor equal to number of particles per unit cell:

• Number of variables

Degree of polynomial

3

1In the preceding case.

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ji

ji

ji

n

izi

zz

zzz

i

2

1

)(

Jack polynomials are eigenstates of the Calogero-Sutherland Hamiltonian on a circle with 1/r^2

potential interaction.

Jack Polynomials:

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Jack polynomials at

1

1

k

r

Feigin-Jimbo-Miwa-Mukhin

Generate ideal of polynomials“vanishing as the r power of the Distance between particles ( difference

Between coordinates) as k+1 particles come together.

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Exclusion statistics:

• No more than k particles into r consecutive

orbitals.

For example when

k=r=2, the possible ground states (most dense packings) are given by:

20202020….

11111111…..

02020202…..

Filling factor is

rkii

k

r

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Moore Read (k=r=2)

)(

1 Pfaff ji

ji

zzzz

When 3 electrons are put together, the wave function vanishes as:2

14

.

When 3 electrons are put together, the wave function vanishes as:

2x x

y

1 2

1

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With adiabatic time QHE=TQFT

One must consider the space P(x1,x2,x3|zi) of polynomials vanishing when zi-xj goes to 0.

Compute Feynman path integrals

x1 x2 x3

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Y3

1 23 4

0],[ ji YY

17

211

11 yTTy

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Non symmetric polynomials

• When additional degrees of freedom are present like spin, it is necessary to consider nonsymmetric polynomials.

• A theory of nonsymmetric Jack polynomials exists with similar vanishing conditions.

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Two layer system.

• Spin singlet projected system of 2 layers

mji

mji yyxx )()(

.

When 3 electrons are put together, the wave function vanishes as:m

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Exemples of wave functions• Haldane-Rezayi: singlet state for 2 layer system.

22

)()(

1Det ji

ji

yxyx

When 3 electrons are put together, the wave function vanishes as:

2x x

y

1 2

1

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Projection onto the singlet state

RVB-BASIS

- =

1 2 3 4 5 6

Crossings forbidden to avoid double countingPlanar diagrams.

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RVB basis:

Projection onto the singlet state

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q-deformation

• RVB basis has a natural q-deformation known as the Kazhdan Lusztig basis.

• Jack polynomials have a natural deformation Macdonald polynomials.

• Evaluation at z=1 of Macdonald polynomials in the KL basis have mysterious positivity properties.

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e=

e=2

)( 1 qq

iiii

ii

eeee

ee

1

2

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Razumov Stroganov Conjectures

1

2

3

4

5

6

6

1iieHAlso eigenvector of:

Stochastic matrix

I.K. Partition function:

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e =1

1 2

1

2

1

1

2

2 1 0 0 1

2 3 2 2 0

0 1 2 0 1

0 1 0 2 1

2 0 2 2 3

H=

1’ 2’

Stochastic matrix

If d=1

Not hermitian

1

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Transfer MatrixConsider inhomogeneous transfer matrix:

1 zz

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

z

zL

z

zLtrzzT n

i

L= +

11 zqzq

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)(),( 1 ni zzzzT

Transfer Matrix

0)(),( wTzT

Di Francesco Zinn-justin.

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Vector indices are patterns, entries are polynomials in z , …, z

1 n

ii

ii

yy

eeBosonic condition

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T.L.( Lascoux Schutzenberger)

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121

1221 )),(),(()(zz

qzqzzzzze

121 qzqz

e+τ projects onto polynomials divisible by:

e Measures the Amplitude for 2 electrons to be In the same layer

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Two q-layer system.

• Spin singlet projected system of 2 layers

m

jim

ji yqqyxqqx )()( 11

.

If i<j<k cyclically ordered, then 0),,( 42 zqzzqzzz kji

))(( 11jiji yqyqxqxq

Imposes for no new condition to occur 6qs

(P)

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Conjectures generalizing R.S.

• Evaluation of these polynomials at z=1

have positive integer coefficients in d:

.

,

,2

,1

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223

4

5

ddF

dFF

dF

F

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Other generalizations

• q-Haldane-Rezayi

))((

1Det

1jiji yqqxyx

Generalized Wheel condition, Gaudin Determinant

Related in some way to the Izergin-Korepin partition function?

Fractional hall effect Flux ½ electron

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Moore-Read

• Property (P) with s arbitrary.• Affine Hecke replaced by Birman-Wenzl-

Murakami,.• R.S replaced by Nienhuis De Gier in the

symmetric case.

ASxqqx ji

1

1 Pfaff

1)1(2 ksq

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Conclusions

• T.Q.F.T. realized on q-deformed wave functions of the Hall effect .

• All connected to Razumov-Stroganov type conjectures.

• Relations with works of Feigin, Jimbo,Miwa, Mukhin and Kasatani on polynomials obeying wheel condition.

• Understand excited states (higher degree polynomials) of the Hall effect.