$\mathbb{Z}_k^{(r)}$-Algebras, FQH Ground States, and Invariants of Binary Forms

TL;DR

This paper explores the relationship between $ ext{Z}_k^{(r)}$-algebras and fractional quantum Hall (FQH) ground states, proposing generalized Hamiltonians and using invariant theory to compute correlations, with evidence for certain cases.

Contribution

It introduces a generalized projection Hamiltonian framework and an invariant-based method for calculating correlations in $ ext{Z}_k^{(r)}$-algebra-based FQH states, advancing understanding of their ground states.

Findings

Proposes a generalized projection Hamiltonian for $ ext{Z}_k^{(r)}$-algebra FQH states.

Develops an invariant theory-based ansatz for correlation functions.

Provides evidence that the Hamiltonian yields a unique ground state when r=2.

Abstract

A prominent class of model FQH ground states is those realized as correlation functions of $\mathbb{Z}_k^{(r)}$-algebras. In this paper, we study the interplay between these algebras and their corresponding wavefunctions. In the hopes of realizing these wavefunctions as a unique densest zero energy state, we propose a generalization for the projection Hamiltonians. Finally, using techniques from invariants of binary forms, an ansatz for computation of correlations $\langle\psi(z_1)\cdots\psi(z_{2k})\rangle \prod_{i<j}(z_i-z_j)^{2r/k}$ is devised. We provide some evidence that, at least when $r=2$, our proposed Hamiltonian realizes $\mathbb{Z}_k^{(2)}$-wavefunctions as a unique ground state.