D-Algebra Structure of Topological Insulators

TL;DR

This paper generalizes the GMP algebra to higher-dimensional topological insulators using D-algebra, revealing connections to topological invariants and differences between even and odd dimensions.

Contribution

It introduces a D-algebra framework for topological insulators, extending the GMP algebra to higher dimensions and exploring its relation to topological invariants and symmetries.

Findings

D-algebra generalization of GMP algebra in higher dimensions

Isotropic D-algebra in even dimensions related to D/2-Chern number

Non-isotropic algebra in odd dimensions includes weak topological insulator index

Abstract

In the quantum Hall effect, the density operators at different wave-vectors generally do not commute and give rise to the Girvin MacDonald Plazmann (GMP) algebra with important consequences such as ground-state center of mass degeneracy at fractional filling fraction, and W_{1 + \infty} symmetry of the filled Landau levels. We show that the natural generalization of the GMP algebra to higher dimensional topological insulators involves the concept of a D-algebra formed by using the fully anti-symmetric tensor in D-dimensions. For insulators in even dimensional space, the D-algebra is isotropic and closes for the case of constant non-Abelian F(k) ^ F(k) ... ^ F(k) connection (D-Berry curvature), and its structure factors are proportional to the D/2-Chern number. In odd dimensions, the algebra is not isotropic, contains the weak topological insulator index (layers of the topological insulator in one less dimension) and does not contain the Chern-Simons \theta form (F ^ A - 2/3 A ^ A ^ A in 3 dimensions). The Chern-Simons form appears in a certain combination of the parallel transport and simple translation operator which is not an algebra. The possible relation to D-dimensional volume preserving diffeomorphisms and parallel transport of extended objects is also discussed.