Exact Diagonalization of the Fractional Quantum Hall Many-Body Hamiltonian in the Lowest Landau Level

Abstract

For a gaussian interaction V(x,y)=λe^{-(x^2+y^2)/r^2} with long range r>>l_B, l_B the magnetic length, we rigorously prove that the eigenvalues of the finite volume Hamiltonian H_{N,LL}=P_{LL} H_N P_{LL}, H_N=\sum_{i=1}^N [-i\hbar \nabla_{x_i}-eA(x_i)]^2+\sum_{i,j; i\ne j} V(x_i-x_j), \rotA=(0,0,B), and P_{LL} 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(n_1,...,n_N)=\sum_{i,j=1;i\ne j}^N W(n_i-n_j). Here n_i\in {1,2,...,M}, n_1<...<n_N and the function W is given by W(n)=λ\sum_{j\in Z} e^{-{1/r^2}(L{n/M}-jL)^2} if the system is kept in a volume [0,L]^2. The eigenstates are also explicitely given.