Topological Invariant of Velocity Field in Quantum Systems

Abstract

AbstractA velocity field approach is proposed to characterize the topological invariants of quantum states based on the Poincar–Hopf theorem. It is found that the zero modes of the velocity field flow play the roles of effective topological charges to dominate the topological invariants against the parameter deformation. The validity of this approach is demonstrated by using the quantum sphere and torus models. These results are consistent with the mathematical results of the vector fields in the manifolds of the sphere and torus, Euler characteristic for sphere and for torus. The non‐Hermitian quantum torus model and differences in the topological invariants obtained using the velocity field and Chern number methods are discussed. 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.