Hyperspherical theory of the quantum Hall effect: The role of exceptional degeneracy

Abstract

By separating the Schrödinger equation for N noninteracting spin-polarized fermions in twodimensional hyperspherical coordinates, we demonstrate that fractional quantum Hall (FQH) states emerge naturally from degeneracy patterns of the antisymmetric free-particle eigenfunctions. In the presence of Coulomb interactions, the FQH states split off from a degenerate manifold and become observable as distinct quantized energy eigenstates with an energy gap. This alternative classification scheme is based on an approximate separability of the interacting N -fermion Schrödinger equation in the hyperradial coordinate, which sheds light on the emergence of Laughlin states as well as other FQH states. An approximate good collective quantum number, the grand angular momentum K from K-harmonic few-body theory, is shown to correlate with known FQH states at many filling factors observed experimentally.

Universal few-body physics and cluster formation

“…The following development sketches one explicit version of this recasting of the Schrödinger equation into hyperspherical coordinates for an N-particle system in 3 dimensions. Note that a similar development for N 2D particles is presented by (Daily et al, 2015b), in the context of the quantum Hall effect. One first transforms the N laboratory frame position vectors { r i } in terms of a suitable set of N − 1 mass-weighted relative Jacobi coordinate vectors { ρ i }, plus the center of mass vector which is trivial and is therefore ignored throughout.…”

Hyperspherical Slater determinant approach to few-body fractional quantum Hall states

“…In a recent line of attack [1], a novel approach to the quantum Hall problem was presented that is based on the adiabatic hyperspherical representation [20,21,22], which originated in and has been extensively used in the context of few-body physics [22,23,24,25]. The hyperspherical approach not only provides complementary advantages and alternative qualitative pictures compared to previous methods, it is also more suitable for the discussion of few-body systems (i.e., cold atoms in rotating traps, electrons in a quantum dot [26,27,28,29,30]).…”

“…This reduction allows us to extend the hyperspherical method to 8 electrons in the 1/3 fractional quantum Hall region, with computational expenses comparable to our previous method implemented in Ref. [1], which could treat at most 6 electrons. This paper is organized as follows.…”

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In a recent study [1], a hyperspherical approach has been developed to study of few-body fractional quantum Hall states. This method has been successfully applied to the exploration of few boson and fermion problems in the quantum Hall region, as well as the study of inter-Landau level collective excitations [2,3].

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Few-body, hyperspherical treatment of the quantum Hall effect

“…In conventional treatments, the N-particle Hamiltonian is diagonalized in a single-particle defined Slater determinant basis, but in our treatment, we address the problem by transforming first to a set of collective coordinates and then transforming to a hyperspherical representation [4]. While its use in condensed matter physics has been limited, the adiabatic hyperspherical representation has been used successfully in a wide range of problems in few-body physics, including nuclear structure and reactivity, few-electron atoms, positron-electron systems, and Bose condensates.…”