Quantum Hall State ν = 1/3 and Antilexicographic Order of Partitions

Abstract

We focus on a certain aspect of trial wave function approach in the fractional quantum Hall effect. We analyze the role of partition orderings and discuss the possible numerical search for the partition determining the subspace of the Hilbert space containing a particular quantum Hall wave function. This research is inspired by analogical properties of certain polynomials which are the object of interest of the symmetric function theory, especially the Jack polynomials (related to the so-called "Jack states"). Presented method may be used in the search of candidate trial wave functions. We also justify (in certain cases) diagonalization of the Coulomb repulsion Hamiltonian restricted to certain subspaces. We focus on the states at filling factor ν = 1/3 in the lowest and second Landau level.

Emergence of Jack ground states from two-body pseudopotentials in fractional quantum Hall systems

“…The Jack polynomial [12][13][14][15]21,22,[28][29][30][31][32][33][34] , called simply a "Jack" and denoted by J α λ , is a symmetric polynomial indexed by the partition λ and the real number α. The partition is a sequence λ = (λ 1 , λ 2 , .…”

Cluster-cluster correlations beyond the Laughlin state

“…Analysis of quantum Hall wavefunctions in terms of the root partitions (or closely related notions) which are associated with the zeros seen by a particle on a cluster of n other particles has been attempted in the past 5,7,8,[28][29][30][31][32][33][34][35][36] . Here, we considered a straightforward generalization of this notion, wherein we ask about the number of zeros γ p,s seen by a p-particle cluster on an s particle cluster.…”

Ground States of Quantum Hall Three-Body ``Short-Range'' Repulsion and Mean Field Approximation: Correlation Functions and Overlaps

“…Surprisingly, even though Pf and APf are described entirely in terms of three-body interaction, they seem to capture many features of ground states of two-body Coulomb interaction Hamiltonians in half-filled first excited Landau level (LL1). Remarkably, the Moore-Read state can also be characterized as a Jack polynomial which makes it fall into the category of the Jack states and allows for application of tools known from the symmetric functions theory [14][15][16][17][18][19][20][21][22]. This is especially useful when one generates coefficients of Pf wave function * corresponding author for large systems, as the Jack states can be computed with relatively fast recursion formula [17,18,23,24].…”