Exact Diagonalization Of The Fractional Quantum Hall Many-Body Hamiltonian In The Lowest Landau Level

For a gaussian interaction V (x, y) = λ e− x2+y2 2r2 with long range r >> lB, lB the magnetic length, we rigorously prove that the eigenvalues of the finite volume Hamiltonian HN,LL = PLLHNPLL, HN = ∑N i=1 ( −i~∇xi − eA(xi) 2 + ∑ i,j; i6=j V (xi − xj), rotA = (0, 0, B), and PLL the projection onto the lowest Landau level, are given by the following set: Let M be the number of flux quanta flowing through the sample such that ν = N/M is the filling factor. Then each eigenvalue is given by E = E(n1, · · · , nN) = ∑N i,j=1;i6=j W (ni−nj). Here ni ∈ {1, 2, · · · ,M}, n1 < · · · < nN and the function W is given by W (n) = λ ∑ j∈Z e − 1 2r2 (L n M −jL)2 if the system is kept in a volume [0, L]. The eigenstates are also explicitely given. 1 e-mail: [email protected] In this paper we consider the two dimensional many electron system in finite volume in a constant magnetic field ~ B = (0, 0, B) described by the Hamiltonian HN = N ∑ i=1 ( ~ i ∇i − eA(~xi) 2 + ∑ i,j=1 i6=j V (~xi − ~xj) (1) We restrict to the completely spin polarized case and neglect the Zeemann energy. The electron-electron interaction is assumed to be a gaussian, V (x, y) = λ e− x2+y2 2r2 (2) which is long range, r >> lB, lB being the magnetic length. The only approximation we will use is (see (33,35) below) ∫ ds ds′ hn(s) hn′(s ′) e− lB 2r2 (s−s′)2 ≈ ∫ ds ds′ hn(s) hn′(s ′) (3) where hn(s) = cnHn(s) e − s 2 2 denotes the normalized Hermite function. With this approximation, the Hamiltonian PLLHNPLL, PLL being the projection onto the lowest Landau level, can be exactly diagonalized. There is the following Theorem: Let HN be the Hamiltonian (1) in finite volume [0, Lx]× [0, Ly ] (with magnetic boundary conditions (11), see below), let A(x, y) = (−By, 0, 0) and let the interaction be gaussian with long range, V (x, y) = λ e− x2+y2 2r2 , r >> lB . (4) Let PLL : FN → FLL N be the projection onto the lowest Landau level, where FN is the antisymmetric N-particle Fock space and FLL N is the antisymmetric Fock space spanned by the eigenfunctions of the lowest Landau level. Then, with the approximation (3), the Hamiltonian HN,LL = PLLHNPLL becomes exactly diagonalizable. Let M be the number of flux quanta flowing through [0, Lx]× [0, Ly] such that ν = N/M is the filling factor. Then the eigenstates and eigenvalues are labelled by N-tupels (n1, · · · , nN), n1 < · · · < nN and ni ∈ {1, 2, · · · ,M} for all i, HN,LLΨn1···nN = (ε0N + En1···nN )Ψn1···nN (5) where ε0 = ~eB/(2m) and En1···nN = N ∑ i,j=1 i6=j W (ni − nj) , W (n) = λ ∑ j∈Z e− 1 2r2 (Lx n M −jLx) 2 (6)