Splitting obstructions and $\mathbb{Z}_2$ invariants in time-reversal symmetric topological insulators

TL;DR

This paper analyzes the Fu-Kane-Mele $bZ_2$ index in time-reversal symmetric topological insulators, providing a decomposition theorem, a homotopy classification, and linking the index to the Chern number of decomposed factors.

Contribution

It introduces a decomposition theorem for projection-valued maps with time-reversal symmetry, classifies their topological structure, and relates the $bZ_2$ index to the Chern number of a factor.

Findings

Any projection-valued map admits a time-reversal related splitting.

Complete homotopy classification of these maps is established.

The $bZ_2$ index is connected to the Chern number of a decomposed component.

Abstract

The Fu-Kane-Mele $\mathbb{Z}_2$ index characterizes two-dimensional time-reversal symmetric topological phases of matter. We shed some light on some features of this index by investigating projection-valued maps endowed with a fermionic time-reversal symmetry. Our main contributions are threefold. First, we establish a decomposition theorem, proving that any such projection-valued map admits a splitting into two projection-valued maps that are related to each other via time-reversal symmetry. Second, we provide a complete homotopy classification theorem for these maps, thereby clarifying their topological structure. Third, by means of the previous analysis, we connect the Fu-Kane-Mele index to the Chern number of one of the factors in the previously-mentioned decomposition, which in turn allows to exhibit how the $\mathbb{Z}_2$-valued topological obstruction to constructing a periodic and smooth Bloch frame for the projection-valued map, measured by the Fu-Kane-Mele index, can be concentrated in a single pseudo-periodic Kramers pair.