Euler characteristic number of the energy band and the reason for its non-integer values

TL;DR

This paper clarifies why the Euler characteristic number of energy bands can be non-integer in trivial phases, linking it to the degeneracy of the quantum metric tensor and the validity of the Gauss-Bonnet theorem.

Contribution

It proves that the non-integer Euler number arises from the degeneracy of the quantum metric in trivial phases, providing a clear criterion for topological classification.

Findings

Quantum metric tensor is positive semi-definite.

Degeneracy of the metric leads to ill-defined Euler number.

Non-zero Berry curvature guarantees an even Euler number.

Abstract

The topological Euler characteristic number of the energy band proposed in our previous work (see Yu-Quan Ma et al., arXiv:1202.2397; EPL 103, 10008 (2013)) has been recently experimentally observed by X. Tan et al., Phys. Rev. Lett. \textbf{122}, 210401 (2019), in which a topological phase transition in a time-reversal-symmetric system simulated by the superconducting circuits is witnessed by the Euler number of the occupied band instead of the vanishing Chern number. However, we note that there are some confusions about the non-integer behaviors of the Euler number in the topological trivial phase. In this paper, we show that the reason is straightforward because the quantum metric tensor $g_{\mu \nu} $ is actually positive semi-definite. In a general two-dimensional two-band system, we can proved that: (1) If the phase is topological trivial, then the quantum metric must be degenerate (singular)~--- $\det {g_{\mu \nu} }=0$ in some region of the first Brillouin zone. This leads to the invalidity of the Gauss-Bonnet formula and exhibits an ill-defined ``non-integer Euler number''; (2) If the phase is topological nontrivial with a non-vanishing Berry curvature, then the quantum metric will be a positive definite Riemann metric in the entire first Brillouin zone. Therefore the Euler number of the energy band will be guaranteed an even number $\chi=2(1-g)$ by the Gauss-Bonnet theorem on the closed two-dimensional Bloch energy band manifold with the genus $g$, which provides an effective topological index for a class of nontrivial topological phases.