Brillouin torus decomposition for two-dimensional topological insulators

Two-band Chern insulators are topologically classified by the Chern number, $c$, which is given by the integral of the Berry curvature of the occupied band over the Brillouin torus. The curvature itself comes from the imaginary part of a more basic object, the quantum geometric tensor, $Q$. On the other hand, the integral over the Brillouin torus of the real part of $Q$ gives rise to another magnitude, the quantum volume, ${v}_{g}$, that like $c$, jumps when the system undergoes a topological phase transition and satisfies the inequality ${v}_{g}\ensuremath{\ge}2|c|$. The information contained in ${v}_{g}$ about the topology of the system has been investigated recently. In this paper we present new results regarding the underlying geometric structure of two-dimensional two-band topological insulators. Since a generic model describing the system can be characterized by a map, the classifying map, from the Brillouin torus to the two-sphere, we study its properties at the geometric level. We present a procedure for splitting the Brillouin torus into different sectors in such a way that the classifying map when restricted to each of them is a local diffeomorphism. By doing so, in the topological phases we are able to isolate a region contained in the Brillouin torus whose volume is the minimal one, ${v}_{\mathrm{min}}=2|c|$, and the integral of the Berry curvature on it is $c$. For cases in which ${v}_{g}>2|c|$, the regions contributing to the excess of volume, ${v}_{\mathrm{ex}}={v}_{g}\ensuremath{-}2|c|$, are found and characterized. In addition, the present work makes contact with, and clarifies, some interpretations of the quantum volume in terms of the Euler characteristic number that were done in the recent literature. We illustrate our findings with a careful analysis of some selected models for Chern insulators corresponding to tight-binding Hamiltonians.