On the particle entanglement spectrum of the Laughlin states

TL;DR

This paper provides a rigorous mathematical analysis of the particle entanglement spectrum of Laughlin states, connecting it to symmetric polynomial ideals and establishing bounds on polynomial degrees to understand topological phases.

Contribution

It reformulates the problem of the particle entanglement spectrum into symmetric polynomial ideals and provides explicit generators and degree bounds, advancing theoretical understanding.

Findings

Explicit generating family of symmetric polynomial ideals

Lower bounds on the total degree of polynomials in the ideal

Discussion of challenges in bounding individual variable degrees

Abstract

The study of the entanglement entropy and entanglement spectrum has proven to be very fruitful in identifying topological phases of matter. Typically, one performs numerical studies of finite-size systems. However, there are few rigorous results for finite-size systems. We revisit the problem of determining the rank of the "particle entanglement spectrum" of the Laughlin states. We reformulate the problem into a problem concerning the ideal of symmetric polynomials that vanish under the formation of several clusters of particles. We give an explicit generating family of this ideal, and we prove that polynomials in this ideal have a total degree that is bounded from below. We discuss the difficulty in proving the same bound on the degree of any of the variables, which is necessary to determine the rank of the particle entanglement spectrum.