Decomposition of fractional quantum Hall states: New symmetries and approximations

TL;DR

This paper uncovers new symmetry structures in fractional quantum Hall states, enabling approximations that achieve perfect overlap with exact states in the thermodynamic limit, even with significant Hilbert space omission.

Contribution

It introduces a differential equation method for spin-singlet FQH states and extends symmetry rules to build highly accurate approximations.

Findings

Overlaps with exact states reach unity in the thermodynamic limit.

The product rule applies to any FQH state expressed via parafermionic operators.

Approximate states omit over half of the Hilbert space but still achieve perfect fidelity asymptotically.

Abstract

We provide a detailed description of a new symmetry structure of the monomial (Slater) expansion coefficients of bosonic (fermionic) fractional quantum Hall states first obtained in Ref. 1, which we now extend to spin-singlet states. We show that the Haldane-Rezayi spin-singlet state can be obtained without exact diagonalization through a differential equation method that we conjecture to be generic to other FQH model states. The symmetry rules in Ref. 1 as well as the ones we obtain for the spin singlet states allow us to build approximations of FQH states that exhibit increasing overlap with the exact state (as a function of system size). We show that these overlaps reach unity in the thermodynamic limit even though our approximation omits more than half of the Hilbert space. We show that the product rule is valid for any FQH state which can be written as an expectation value of parafermionic operators.