Exact Matrix Product States for Quantum Hall Wave Functions

TL;DR

This paper demonstrates that fractional quantum Hall wave functions can be exactly represented as matrix product states, enabling efficient numerical calculations of observables, excitations, and entanglement spectra, with insights into their scaling behaviors.

Contribution

It introduces an exact MPS representation for quantum Hall wave functions and develops algorithms for computing observables, excitations, and entanglement spectra from these states.

Findings

Exact MPS representations for quantum Hall wave functions.

Efficient numerical computation of observables and excitations.

Scaling analysis of real-space entanglement spectra.

Abstract

We show that the model wave functions used to describe the fractional quantum Hall effect have exact representations as matrix product states (MPS). These MPS can be implemented numerically in the orbital basis of both finite and infinite cylinders, which provides an efficient way of calculating arbitrary observables. We extend this approach to the charged excitations and numerically compute their Berry phases. Finally, we present an algorithm for numerically computing the real-space entanglement spectrum starting from an arbitrary orbital basis MPS, which allows us to study the scaling properties of the real-space entanglement spectra on infinite cylinders. The real-space entanglement spectrum obeys a scaling form dictated by the edge conformal field theory, allowing us to accurately extract the two entanglement velocities of the Moore-Read state. In contrast, the orbital space spectrum is observed to scale according to a complex set of power laws that rule out a similar collapse.