Correlation diagrams: an intuitive approach to correlations in quantum Hall systems

Abstract

A trial wave function Ψ(1, 2,..., N) of an N electron system can always be written as the product of an antisymmetric Fermion factor F{zij} = Πi<jzij , and a symmetric correlation factor G{zij}. F results from the Pauli principle, and G is caused by Coulomb interactions. One can represent G diagrammatically [1] by distributing N points on the circumference of a circle, and drawing appropriate lines representing correlation factors (cfs) zij between pairs. Here, of course, zij = zi – zj, where zi is the complex coordinate of the ith electron. Laughlin correlations for the ν = 1/3 filled incompressible quantum liquid (IQL) state contain two cfs connecting each pair (i,j). For the Moore-Read state of the half-filled excited Landau level (LL), with ν = 2 + 1/2, the even value of N for the half-filled LL is partitioned into two subsets A and B, each containing N/2 electrons [2]. For any one partition (A,B), the contribution to G is given by GAB = Πi<j∈Azij2 Πk<ι∈B zkι2. The full G is equal to the symmetric sum of contributions GAB over all possible partitions of N into two subsets of equal size. For Jain states at filling factor ν = p/q < 1/2, the value of the single particle angular momentum ι satisfies the equation 2ι = ν-1N - Cν, with Cν = q + 1 - p. The values of (2ι, N) define the function space of G{zij}, which must satisfy a number of conditions. For example, the highest power of any zi cannot exceed 2ι + 1 - N. In addition, the value of the total angular momentum L of the lowest correlated state must satisfy the equation L = (N/2)(2ι+1 - N) - KG, where KG is the degree of the homogeneous polynomial generated by G. Knowing the values of L for IQL states (and for states containing a few quasielectrons or a few quasiholes) from Jain’s mean field CF picture allows one to determine KG. The dependence of the pair pseudopotential V(L2) on pair angular momentum L2 suggests a small number of correlation diagrams for a given value of the total angular momentum L. Correlation diagrams and correlation functions for the Jain state at ν = 2/5 and for the Moore-Read states will be presented as examples. The generalizations of the method of selecting G from small to larger systems will be discussed.