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q-Q.H.E. and Topology

Vincent PasquierService de Physique ThéoriqueC.E.A. SaclayFrance

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From the Hall Effect to integrability

1. Hall effect.2. Transfer Matrix.3. Annular algebras.4. deformed Hall effect wave function as a

toy model for TQFT.5. Conclusion.

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Hall effect

• Lowest Landau Levelwave functions

2

!)( l

zzn

n enzz −=ψ

nlr =

n=1,2, …, 22 lAπ

4

022n

lA

=π

Number of available cells alsothe maximal degree in each variable

nnzz λλ ...1

1 Is a basis of states for the system

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Interactions

mji zz −

m measures the strength of the interactions.

Competition between interactions which spread electrons apart and highCompression which minimizes the degree n.

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With adiabatic time QHE=TQFT

• Bulk and edge.

Compute Feynman path integrals

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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:mε

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Projection onto the singlet state

RVB-BASIS

- =

1 2 3 4 5 6

Crossings forbidden to avoid double counting

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RVB basis:Projection onto the singlet state

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RVB Basise e e e

1354

1 5

4

1 2 3 4 5 6

1 23 45 6

3

11

Also with fluxes

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

))()((1 Perm jijijiji

yxyyxxyx

−−−⎥⎥⎦

⎤

⎢⎢⎣

⎡

− ∏

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

x x

y

1 2

1

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Moore Read

∏ −⎥⎥⎦

⎤

⎢⎢⎣

⎡

−)(1 Pfaff ji

ji

zzzz

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

No spin

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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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1 2 3 4

e2

16

e =1

1 2

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

12112

2 1 0 0 12 3 2 2 0

0 1 2 0 10 1 0 2 1

2 0 2 2 3

H=

1’ 2’

Stochastic matrixIf d=1

Not hermitian

1

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Transfer MatrixConsider inhomogeneous transfer matrix:

1−− zz

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

zzL

zzLtrzzT n

i =

L= +

11 −−− zqzq

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ΨΛ=Ψ )()(),( 11 nni zzzzzzT ……

Transfer Matrix

[ ] 0)(),( =wTzT

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I.K. Partition function

4

6

5

z 31

Total partition function can be expressed as a Gaudin Determinant

When d=q+1/q=1 symmetrical and given by a Schur function.

Stroganov

2

)( 1 nzzZ

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Hecke and Yang generators• Consider Hecke algebra generators:

• And Yang operators:

11

2

11211

1 zqqztztzY −

−

−−

=

0))((

1

11

212121

=+−

=−qtqt

tttttt Braid group relations

111

212121

==

YYYYYYYY Permutation relations

Or Yang-Baxter algebra

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T.L. (Jones)

22

Y4

1 2 3 4

0],[ =ji YY

2321

1 TyTy =−

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Content of Representation

Read eigenvalues of y i

Initial height

Final height

ut

u

ut2

u-1

u-1

1 2 3 4 …….

t

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Bosonic Ground State

,

,

1

ii

ii

yy

tt

Ψ=Ψ

Ψ=Ψ

⊗==Ψ∧

∑ ππ

Look for dual representation of AHA on polynomials

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AHA• Spin representation • Polynomial

representation

2223

1112

1

tytytyty

y

==

ini

ii

n

szzandzzwith

tty

==

=

+

+

: : 1

11

σ

σ

Triangular matrices

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Two q-layer system.

• Spin singlet projected system of 2 layers

mji

mji yqqyxqqx ∏∏ −− −− )()( 11

.

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

))(( 11jiji yqyqxqxq −− −−∏

Imposes for no new condition to occur 6qs =

(P)

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T.L. and Measure

21

121

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

qzqzzzzze−−

Ψ−Ψ=Ψ+−

τ

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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yi

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Flux seen by particles

i

Initial height

Final height

ut

u

ut2

u-1

u-1

1 2 3 4 …….

t

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k q-layer system

• Spin singlet projected system of k layers

mji

mji yqqyxqqx ∏∏ −− −− )()( 11

.

If i<j<k cyclically ordered, then

0),,( 22 ====Ψ zqzzqzzz kkji

∏=

−−1

1 )(a

aj

ai xqxq

Imposes for no new condition to occur)1(2 += kqs

(P)

1

2

k

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Other generalizations

• q-Haldane-Rezayi

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−− − ))((1Det 1

jiji 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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b

1

2

3

4

5

b

b

b

12

23

34

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Kasatani wheel conditions

11 ++ = aa

a

a b

i

i tqzz

111 =−+ rk qt

20

1

11

−≤

>⇒=

∑ +

++

rbiib

aa

aaaa

k and r

With r , Flux=1/r particle.

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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.

⎥⎥⎦

⎤

⎢⎢⎣

⎡

− −ji xqqx 1

1 Pfaff

1)1(2 =+ksq

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The case q root of unity

• When q+1/q=1, Hecke representation is no more semisimple and degenerates into a trivial representation.

• Stroganov Partition function (Schurfunction) can be recovered as the unique symmetrical polynomial of the minimal degree obeying (P).

• Other roots of unity?

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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. Proof of conjecture still missing.

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

• Excited states of the Hall effect.

cond-mat/0506075

math.QA/0507364