Topological invariant of velocity field in quantum systems

TL;DR

This paper introduces a velocity field approach to characterize topological invariants in quantum systems, linking zero modes to topological charges and using the Euler characteristic for global properties, validated on models like the quantum sphere and torus.

Contribution

The paper proposes a novel velocity field method to identify topological invariants, offering an alternative to Chern number calculations and connecting to the Poincaré-Hopf theorem.

Findings

Velocity field zero modes act as topological charges.

Topological invariants are consistent with Euler characteristics.

Velocity field invariants differ from Chern number methods.

Abstract

We introduce the velocity field of the Bloch electrons and propose the velocity field approach to characterize the topological invariants of quantum states. We find that the zero modes of the velocity field flow play the roles of effective topological charges or defects. A key global property of the zero modes is topological invariant against the parameter deformation. These can be characterized by the Euler characteristic based on the Poincar\'{e}-Hopf theorem. We demonstrate the validity of this approach by using the quantum sphere and torus models. The topological invariants of the velocity field in the quantum sphere and torus are consistent with the mathematical results of the vector fields in the manifolds of the sphere and torus, Euler characteristic $\chi=2$ for sphere and $\chi=0$ for torus. We also discuss the non-Hermitian quantum torus model and compare differences in the topological invariants obtained using the velocity field and Chern number methods. The topological invariant characterized by the velocity field is homeomorphic in the Brillouin zone and the subbase manifold of the SU(2)-bundle of the system, whereas the Chern number characterizes a homotopic invariant that is associated with the exceptional points in the Brillouin zone. These results enrich the topological invariants of quantum states and provide novel insights into the topological invariants of quantum states.