2009/10/28 by M. B. Hastings, T. A. Loring · 1 citation
Physics and Astronomy · Mathematics · #math-ph #math.MP #math.OA #quant-ph
paper · pdf · doi:10.1063/1.3274817
published as J. Math. Phys. 51, 015214 (2010). · 35 pages, 2 figures
arxiv created 2009/10/28 · arxiv updated 2015/05/14
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 Z2 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.