Mechanism and Modulation of Chiral Semimetal with Coexistence Surface State of Z₂ and Chern under Vector Field

In this paper, we investigated the surface states of the chiral semimetal under the three-dimensional (3D) vector external field. Firstly, we introduced a Lorentz rotation transformation to transform the 3D vector field into a one-dimensional scalar field to calculate the surface state of the system. We found that there were two types of surface states as drumhead surface state and coexistence surface state of Z₂ and Chern. And the topological invariant of the coexistence state with mirror symmetry breaking was calculated by a mirror Chern number. Then, by theoretical analysis and numerical calculation of the zero mode under the 3D external field we found that the coexistence surface state still existed when the emergent symmetries were broken, which demonstrated the coexistence is robust to the emergent symmetries. This work provided a convenient method for the experimental study of 3D vector external field. As well as the calculation of the precise control of surface states by a 3D vector field can be a feasible method for field measurement in the surface engineering field.