Local topology for periodic Hamiltonians and fuzzy tori

TL;DR

This paper develops local index formulas for quantum Hamiltonians with periodic boundary conditions across various dimensions and symmetry classes, utilizing spectral localizers and fuzzy tori to establish a general invariant theory.

Contribution

It introduces new local index formulas for periodic quantum Hamiltonians, incorporating fuzzy tori and spectral localizers across multiple dimensions and symmetry classes.

Findings

Constructed local index formulas for various symmetry classes.

Developed a general invariant theory for fuzzy tori.

Extended the framework to multiple spatial dimensions.

Abstract

A variety of local index formulas is constructed for quantum Hamiltonians with periodic boundary conditions. All dimensions of physical space as well as many symmetry constraints are covered, notably one-dimensional systems in Class DIII as well as two- and three-dimensional systems in Class AII. The constructions are based on several periodic variations of the spectral localizer and are rooted in the existence of underlying fuzzy tori. For these latter, a general invariant theory is developed.