Note on "Experimental Measurement of Quantum Metric Tensor and Related Topological Phase Transition with a Superconducting Qubit"

TL;DR

This paper clarifies the calculation of the Euler characteristic number for a two-band model, showing it is not suitable for detecting topological phase transitions due to boundary effects and its invariance across critical points.

Contribution

It corrects the computation of the Euler characteristic number by considering boundary contributions, revealing its limitations in characterizing topological phase transitions.

Findings

Euler characteristic number is not an integer for |h|>1 without boundary correction

The Euler characteristic number remains unchanged across the critical point |h|=1

Boundary effects are crucial in correctly determining topological invariants

Abstract

In the paper [X. Tan et al., Phys. Rev. Lett. 122, 210401 (2019)], we have studied the Euler characteristic number $\chi$ for a two-band model; However, the calculated $\chi$ there is not an integer when the parameter $|h|>1$ and then may not be considered as a suitable topological number for the system. In this note, we find that the Bloch vectors of the ground state for $|h|>1$ do not cover the whole Bloch sphere and thus the Euler characteristic number should be effectively associated with the manifold of a disk, rather than a sphere. After taking into account the boundary contribution, we derive the correct Euler characteristic number. Unfortunately, the Euler characteristic number does not change when crossing the critical points $|h|=1$ and thus can not be used to characterize the topological phase transition of the present model.