  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.
