Matrix Product State description of the Halperin States

TL;DR

This paper develops an exact Matrix Product State representation for Halperin states, a class of multicomponent fractional quantum Hall wavefunctions, enabling detailed analysis of their topological properties and correlations.

Contribution

The authors generalize MPS construction to multicomponent Halperin states involving multiple electronic operators, preserving symmetries and enabling topological characterization.

Findings

Exact MPS representation for Halperin states derived.

Topological entanglement entropy extracted.

Bulk correlation lengths evaluated and compared.

Abstract

Many fractional quantum Hall states can be expressed as a correlator of a given conformal field theory used to describe their edge physics. As a consequence, these states admit an economical representation as an exact Matrix Product States (MPS) that was extensively studied for the systems without any spin or any other internal degrees of freedom. In that case, the correlators are built from a single electronic operator, which is primary with respect to the underlying conformal field theory. We generalize this construction to the archetype of Abelian multicomponent fractional quantum Hall wavefunctions, the Halperin states. These latest can be written as conformal blocks involving multiple electronic operators and we explicitly derive their exact MPS representation. In particular, we deal with the caveat of the full wavefunction symmetry and show that any additional SU(2) symmetry is preserved by the natural MPS truncation scheme provided by the conformal dimension. We use our method to characterize the topological order of the Halperin states by extracting the topological entanglement entropy. We also evaluate their bulk correlation length which are compared to plasma analogy arguments.