Correlation Lengths and Topological Entanglement Entropies of Unitary and Non-Unitary Fractional Quantum Hall Wavefunctions

TL;DR

This paper investigates the topological and correlation properties of various fractional quantum Hall wavefunctions, demonstrating that non-unitary states like the Gaffnian are gapless and differ fundamentally from unitary states, with implications for their physical relevance.

Contribution

It introduces a Matrix Product State formalism for non-abelian FQH states and compares the topological entanglement entropy and correlation lengths of unitary and non-unitary states, revealing key differences.

Findings

Gaffnian state exhibits infinite correlation length, indicating gaplessness.

Moore-Read state has finite correlation length, consistent with a gapped phase.

Topological entanglement entropy matches CFT predictions for unitary states.

Abstract

Using the newly developed Matrix Product State (MPS) formalism for non-abelian Fractional Quantum Hall (FQH) states, we address the question of whether a FQH trial wave function written as a correlation function in a non-unitary Conformal Field Theory (CFT) can describe the bulk of a gapped FQH phase. We show that the non-unitary Gaffnian state exhibits clear signatures of a pathological behavior. As a benchmark we compute the correlation length of Moore-Read state and find it to be finite in the thermodynamic limit. By contrast, the Gaffnian state has infinite correlation length in (at least) the non-Abelian sector, and is therefore gapless. We also compute the topological entanglement entropy of several non-abelian states with and without quasiholes. For the first time in FQH the results are in excellent agreement in all topological sectors with the CFT prediction for unitary states. For the non-unitary Gaffnian state in finite size systems, the topological entanglement entropy seems to behave like that of the Composite Fermion Jain state at equal filling.