Almost Commuting Matrices, Localized Wannier Functions, and the Quantum Hall Effect

TL;DR

This paper explores topological obstructions to approximating almost commuting matrices with commuting ones, revealing implications for localized Wannier functions and the quantum Hall effect, with numerical and theoretical insights into these obstructions.

Contribution

It establishes $K$-theoretic and $Z_2$ index obstructions for approximating almost commuting matrices, linking them to physical phenomena like the quantum Hall effect and time-reversal symmetry.

Findings

Numerical demonstration of $K$-theoretic obstructions in quantum Hall models.

Identification of a $Z_2$ index obstruction for self-dual matrices.

First quantitative result showing the index as the sole obstruction on the sphere.

Abstract

For models of non-interacting fermions moving within sites arranged on a surface in three dimensional space, there can be obstructions to finding localized Wannier functions. We show that such obstructions are $K$-theoretic obstructions to approximating almost commuting, complex-valued matrices by commuting matrices, and we demonstrate numerically the presence of this obstruction for a lattice model of the quantum Hall effect in a spherical geometry. The numerical calculation of the obstruction is straightforward, and does not require translational invariance or introducing a flux torus. We further show that there is a $Z_2$ index obstruction to approximating almost commuting self-dual matrices by exactly commuting self-dual matrices, and present additional conjectures regarding the approximation of almost commuting real and self-dual matrices by exactly commuting real and self-dual matrices. The motivation for considering this problem is the case of physical systems with additional antiunitary symmetries such as time reversal or particle-hole conjugation. Finally, in the case of the sphere--mathematically speaking three almost commuting Hermitians whose sum of square is near the identity--we give the first quantitative result showing this index is the only obstruction to finding commuting approximations. We review the known non-quantitative results for the torus.