# Matrix Product State Description and Gaplessness of the Haldane-Rezayi State

## TL;DR

This paper provides an exact matrix product state representation of the Haldane-Rezayi state, revealing its gapless nature and analyzing its topological properties through numerical methods on different geometries.

## Contribution

It introduces a novel exact matrix product state construction for the Haldane-Rezayi state based on non-unitary conformal field theory, capturing its ground state degeneracy and gapless behavior.

## Key findings

- Correlation length diverges in the thermodynamic limit.
- Topological entanglement entropy probes only the Abelian part of the theory.
- The state exhibits gapless behavior consistent with a non-unitary CFT.

## Abstract

We derive an exact matrix product state representation of the Haldane-Rezayi state on both the cylinder and torus geometry. Our derivation is based on the description of the Haldane-Rezayi state as a correlator in a non-unitary logarithmic conformal field theory. This construction faithfully captures the ten degenerate ground states of this model state on the torus. Using the cylinder geometry, we probe the gapless nature of the phase by extracting the correlation length, which diverges in the thermodynamic limit. The numerically extracted topological entanglement entropies seem to only probe the Abelian part of the theory, which is reminiscent of the Gaffnian state, another model state deriving from a non-unitary conformal field theory.