Razumov-Stroganov

1. Razumov-Stroganov Vincent Pasquier Service de Physique Théorique C.E.A. Saclay France

2. Alternating sign matrices

4. 6 vertex model • Replace the 0 by type a,b vertices. • Replace +1,-1 by c vertices. • Put apropriate boundary conditions. • Give weight 1 to all vertices.

5. Can be computed exactly • Gaudin Izergin-Korepin determinant

6. In the case q =-1 3 • Partition function is symmetric in the variables qx, y. • Can be expressed as a Schur function • 00112233…. Wheel condition z,qz,q z 2

7. Evaluate Schur function at z=1:

8. e= e= 2

9. RVBbasis: Projection onto the singlet state

10. 1’ 2’ 1 e = 1 2 1 H= Stochastic matrix If d=1 Not hermitian

11. Razumov Stroganov Conjectures I.K.Partition function: 6 1 5 2 4 3 Also eigenvector of: Stochastic matrix

12. Consider inhomogeneous transfer matrix: Transfer Matrix L= +

13. Transfer Matrix Di Francesco Zinn-justin.

14. Vector indices are patterns, entries are polynomials in z , …, z 1 n Bosonic condition

15. T.L.( Lascoux Schutzenberger) e+τ projects onto polynomials divisible by:

16. 2 1 4 3 Y 3

18. How do we generate the ground state? • Start from an easy polynomial:

19. Wheel conditions

20. Evaluation?

21. Conjectures generalizing R.S. • Evaluation of these polynomials at z=1 have positiveinteger coefficients in d:

22. 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.

23. Hall effect • Lowest Landau Level wave functions n=1,2, …,

24. Number of available cells also the maximaldegree in each variable Is a basis of states for the system Labeled by partitions

25. N-k particles in the orbital k are the occupation numbers Can 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

26. Interactions translate into repulsionbetween particles. m universal measures the strength of the interactions. Competition between interactions which spread electrons apart and high compression which minimizes the degree n. Ground state is the minimal degree symmetric polynomial compatible with the repulsive interaction.

27. Laughlin wave functionoccupation numbers: Keeping only the dominant weight of the expansion 1 particle at most into m orbitals (m=3 here).

28. Important quantum numberwith a topological interpretation • Filling factor equal to number of particles per unit cell: Number of variables Degree of polynomial In the preceding case.

29. Jack Polynomials: Jack polynomials are eigenstates of the Calogero-Sutherland Hamiltonian on a circle with 1/r^2 potential interaction.

30. Jack polynomials at Feigin-Jimbo-Miwa-Mukhin Generateideal of polynomials“vanishing as the r power of the Distance between particles ( difference Between coordinates) as k+1 particles come together.

31. 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

32. Moore Read (k=r=2) When 3 electrons are put together, the wave function vanishes as:

33. . When 3 electrons are put together, the wave function vanishes as: x 1 2 x 1 y

34. 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.

35. Two layer system. • Spin singlet projected system of 2 layers . When 3 electrons are put together, the wave function vanishes as:

36. Exemples of wave functions • Haldane-Rezayi: singlet state for 2 layer system. When 3 electrons are put together, the wave function vanishes as: x 1 2 x 1 y

37. RVB-BASIS Projection onto the singlet state - = Crossings forbidden to avoid double counting Planar diagrams. 5 6 2 3 1 4

38. Two q-layer system. • Spin singlet projected system of 2 layers . (P) If i<j<k cyclically ordered, then Imposes for no new condition to occur

39. Other generalizations • q-Haldane-Rezayi Generalized Wheel condition, Gaudin Determinant Related in some way to the Izergin-Korepin partition function? Fractional hall effect Flux ½ electron

40. 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.