Entanglement subspaces, trial wavefunctions, and special Hamiltonians in the fractional quantum Hall effect

TL;DR

This paper explores various methods to construct and relate trial wavefunctions in the fractional quantum Hall effect, establishing algebraic frameworks and explicit models that connect entanglement, vanishing properties, and conformal field theory.

Contribution

It unifies different approaches to fractional quantum Hall wavefunctions through an algebraic CFT framework and constructs explicit Hamiltonians for specific models.

Findings

Proves equivalence of wavefunction spaces in certain models

Constructs explicit zero-energy Hamiltonians for specific cases

Establishes a finite-size bulk-edge correspondence

Abstract

We consider spaces of trial wavefunctions for ground states and edge excitations in the fractional quantum Hall effect that can be obtained in various ways. In one way, functions are obtained by analyzing the entanglement of the ground state wavefunction, partitioned into two parts. In another, functions are defined by the way in which they vanish as several coordinates approach the same value, or by a projection-operator Hamiltonian that enforces those conditions. In a third way, functions are given by conformal blocks from a conformal field theory (CFT). These different spaces of functions are closely related. The use of CFT methods permits an algebraic formulation to be given for all of them. In some cases, we can prove that there is a ground state, a Hamiltonian, and a CFT such that, for any number of particles, all of these spaces are the same. For such cases, this resolves several questions and conjectures: it gives a finite-size bulk-edge correspondence, and we can use the analysis of functions to construct a projection-operator Hamiltonian that produces those functions as zero-energy states. For a model related to the N=1 superconformal algebra, the corresponding Hamiltonian imposes vanishing properties involving only three particles; for this we determine all the wavefunctions explicitly. We do the same for a sequence of models involving the M(3,p) Virasoro minimal models that has been considered previously, using results from the literature. We exhibit the Hamiltonians for the first few cases of these. The techniques we introduce can be applied in numerous examples other than those considered here.