Correlation Diagrams: An Intuitive Approach to Correlations in Quantum Hall Systems

A trial wave function \f (1,2,...,N) of an N electron system can always be written as the product of an antisymmetric Fermion factor F {Zij }= Tii<jZij , and a symmetric correlation factor G {Zij }. F results from Pauli principle, and G is caused by Coulomb interactions. One can represent G diagrammatically ( I J 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, and Zi is the complex coordinate of the i111 electron. Laughlin correlation for the v=l/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 v=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[21.

For any one partition(A,B)the contribution to G is given by GAB = Tiiz\Tik<Ii;sZ2kt · The full G is equal to the symmetric sum of contributions GAB over all possible partitions of N into two equal subsets. For Jain states at filling factor v=p/ q < 1/ 2 , the  value  of  the  single  particle angular momentum e satisfies the equation  20=v- 1N-Cv, with Cv = q + 1 - p. The values of (2 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 2e+ 1-N. In addition, the value of the total angular momentum L of the lowest correlated state must satisfy the equation L=(N / 2) (2e+ 1-N)-Ka, where Ka 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 Ka. 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 v=2 /S and for the Moore-Read stated will be presented as example.

[1] J.J. Quinn, Waves in random and complex media (2014) 898867

[2] S.B. Mulay, J.J . Quinn, and M.A. Shattuck, submitted to J. Math. Phys. (2014)