A Closed Band-Projected Density Algebra Must be Girvin-MacDonald-Platzman

TL;DR

This paper proves that the Girvin-MacDonald-Platzman algebra is the unique closed algebra for band-projected density operators in Landau levels and Chern bands, emphasizing its fundamental role in fractional Chern insulators.

Contribution

It establishes the GMP algebra as the only closed algebra for projected density operators in two and three dimensions, up to form factors.

Findings

GMP algebra is unique among closed density operator algebras.

The result applies to both Landau levels and Chern bands.

Implications for the study of fractional Chern insulators.

Abstract

The band-projected density operators in a Landau level obey the Girvin-MacDonald-Platzman (GMP) algebra, and a large amount of effort in the study of fractional Chern insulators has been directed towards approximating this algebra in a Chern band. In this paper, we prove that the GMP algebra, up to form factors, is the $\textit{only}$ closed algebra that projected density operators can satisfy in two and three dimensions, highlighting the central place it occupies in the study of Chern bands in general. A number of interesting corollaries follow.