On the Rank of the Reduced Density Operator for the Laughlin State and Symmetric Polynomials

Abstract: One effective tool to probe a system revealing topological order is to bipartition the system insome way and look at the properties of the reduced density operator corresponding to one partof the system. In this thesis we focus on a bipartition scheme known as the particle cut inwhich the particles in the system are divided into two groups and we look at the rank of thereduced density operator. In the context of fractional quantum Hall physics it is conjecturedthat the rank of the reduced density operator for a model Hamiltonian describing the system isequal to the number of quasi-hole states. Here we consider the Laughlin wave function as themodel state for the system and try to put this conjecture on a firmer ground by trying todetermine the rank of the reduced density operator and calculating the number of quasi-holestates. This is done by relating this conjecture to the mathematical properties of symmetricpolynomials and proving a theorem that enables us to find the lowest total degree ofsymmetric polynomials that vanish under some specific transformation referred to asclustering transformation.