On the Bott index of unitary matrices on a finite torus
Abstract
This article reviews the foundations of the theory of the Bott index of a pair of unitary matrices in the context of condensed matter theory, as developed by Hastings and Loring in J. Math. Phys. (51), 015214, (2010) and Annals of Physics 326 (2011) 1699-1759, providing a novel proof of the equality with the Chern number. The Bott index is defined for a pair of unitary matrices, then extended to a pair of invertible matrices and homotopic invariance of the index is proven. An insulator defined on a lattice on a 2-torus, that is a rectangular lattice with periodic boundary conditions, is considered and a pair of quasi-unitary matrices associated to this physical system are introduced. It is shown that their Bott index is well defined and the connection with the transverse conductance, the Chern number, is established proving the equality of the two quantities.
Keywords
Cite
@article{arxiv.1708.05912,
title = {On the Bott index of unitary matrices on a finite torus},
author = {Daniele Toniolo},
journal= {arXiv preprint arXiv:1708.05912},
year = {2023}
}
Comments
24 pages. 1 figure. In v4 more details are provided for the proof of theorem 10. Accepted by Letters in Mathematical Physics