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QHE, ASM Vincent Pasquier Service de Physique Théorique C.E.A. Saclay France 1 From the Hall Effect to integrability 1. Hall effect. 2. Annular algebras. 3. deformed Hall effect and ASM. 2 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 10 10 • Important experimental fractions for p electrons are: 2 p 1 • Theorists believe for bosons: p p 1 • Fraction called filling factor. 3 Hall effect • Lowest Landau Level wave functions n z n ( z) e n! zz 2 l r l n n=1,2, …, A 2l 2 4 A n 0 2 2l 1 n Number of available cells also the maximal degree in each variable S ( z1 ... zn ) m Is a basis of states for the system Labeled by partitions 5 N-k particles in the orbital k are the occupation numbers N0 N1N 2 .... N k ..... 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 6 Interactions translate into repulsion between particles. z z m i j 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. 7 Laughlin wave function occupation numbers: Keeping only the dominant weight of the expansion 100100100 1 particle at most into m orbitals (m=3 here). 8 Important quantum number with a topological interpretation • Filling factor equal to number of particles per unit cell: • 1 3 Number of variables Degree of polynomial In the preceding case. 9 Jack Polynomials: n i 1 ( zi z ) 2 i i j zi z j zi z j Jack polynomials are eigenstates of the CalogeroSutherland Hamiltonian on a circle with 1/r^2 potential interaction. 10 Jack polynomials at r 1 k 1 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. 11 Exclusion statistics: • No more than k particles into r consecutive orbitals. i i k r For example when k=r=2, the possible ground states (most dense packings) are given by: 20202020…. 11111111….. 02020202….. r Filling factor is k 12 Moore Read (k=r=2) When 3 electrons are put together, the wave function vanishes as: 2 1 Pfaff ( zi z j ) zi z j 13 . When 3 electrons are put together, the wave function vanishes as: x 1 2 2 x y 1 14 With adiabatic time QHE=TQFT One must consider the space P(x1,x2,x3|zi) of polynomials vanishing when zi-xj goes to 0. x1 x2 x3 Compute Feynman path integrals 15 1 2 3 4 Y3 [Yi , Y j ] 0 16 1 y1T1 T1 y2 17 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. 18 Two layer system. • Spin singlet projected system of 2 layers m m ( x x ) ( y y ) i j i j . When 3 electrons are put together, the wave function vanishes as: m 19 Exemples of wave functions • Haldane-Rezayi: singlet state for 2 layer system. 1 Det 2 ( xi y j ) ( xi y j ) 2 When 3 electrons are put together, the wave function vanishes as: x 1 2 2 x y 1 20 RVB-BASIS Projection onto the singlet state - = Crossings forbidden to avoid double counting Planar diagrams. 1 2 3 4 5 6 21 RVB basis: Projection onto the singlet state 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 e= 2 1 e= (q q ) 2 ei ei ei ei 1ei ei 24 Razumov Stroganov Conjectures I.K. Partition function: 6 1 5 2 4 3 6 Also eigenvector of: Stochastic matrix H ei i 1 25 1’ 2’ 1 e= 1 1 H= Stochastic matrix If d=1 Not hermitian 2 3 2 2 0 2 2 1 2 0 1 0 1 1 0 2 1 0 1 0 2 2 3 2 2 1 0 0 1 2 1 26 Transfer Matrix Consider inhomogeneous transfer matrix: zn z1 T ( z , zi ) tr ( L( )....L( )) z z L= + 1 1 q z q z z z 1 27 Transfer Matrix T ( z, zi )( z1 zn ) T ( z), T (w) 0 Di Francesco Zinn-justin. 28 Vector indices are patterns, entries are polynomials in z , …, z 1 n ei ei yi yi Bosonic condition 29 T.L.( Lascoux Schutzenberger) z1 q z2 q 1 (e ) ( ( z1 , z2 ) ( z2 , z1 )) z1 z2 e+τ projects onto polynomials divisible by: z1q z2 q 1 e Measures the Amplitude for 2 electrons to be In the same layer 30 Two q-layer system. • Spin singlet projected system of 2 layers (qx q i 1 x j ) m (qyi q 1 y j ) m . (q x q i 1 1 x j )(q yi q y j ) (P) If i<j<k cyclically ordered, then Imposes s q6 ( zi z, z j q 2 z, z k q 4 z) 0 for no new condition to occur 31 Conjectures generalizing R.S. • Evaluation of these polynomials at z=1 have positive integer coefficients in d: F5 1, F4 2d , F3 F2 d , 2 F1 d 3 d. 32 Other generalizations • q-Haldane-Rezayi 1 Det 1 ( xi y j )( qxi q y j ) Generalized Wheel condition, Gaudin Determinant Related in some way to the Izergin-Korepin partition function? Fractional hall effect Flux ½ electron 33 Moore-Read • Property (P) with s arbitrary. • Affine Hecke replaced by Birman-WenzlMurakami,. • R.S replaced by Nienhuis De Gier in the symmetric case. sq 2( k 1) 1 1 Pfaff AS 1 qxi q x j 34 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. 35