Accurate Interpolation of Bands in Brillouin Zone

Band crossings are essential to the topological behavior of materials, yet they are often difficult to predict because first-principles calculations provide energies only at discrete k-points. We present a method to build continuous band structures across the Brillouin zone from such discrete data, enabling precise identification of crossing points. The approach uses spline interpolation with a consistent band ordering at every k-point, preserving smooth and meaningful dispersions even near degeneracies. Accurate interpolation is also critical for numerical integration of properties such as the density of states and transport coefficients, which depend on fine details of the dispersion. We apply the method to analytic model bands and to ab initio data for one-, two-, and three-dimensional materials, showing that it reproduces expected dispersions and captures crossings faithfully. This provides a practical way to obtain reliable, continuous band structures for exploring electronic and topological properties.