Interpreting non-semielliptical complex bands

Complex band structure (CBS) emerges when translational symmetry is broken and material states with complex wavevectors become admissible. The resulting complex bands continuously connect conventional bands and their shapes are directly related to measurable physical quantities. To date, interpretations of complex bands usually assume they are semielliptical because this is the shape produced by the Su-Schrieffer-Heeger (SSH) model. However, numerous studies have reported CBSs with distinctly non-semielliptical shapes, including loops (essentially deformed, asymmetric semiellipses), spikes, and vertical lines. The primary goal of this work is to explore the phenomenology of these shapes such that deeper physical insight can be obtained from a qualitative inspection of a material's CBS. By using several variations on the SSH model, we find that (i) vertical lines are unphysical numerical artifacts, (ii) spikes indicate perfectly evanescent states in the material that couple adjacent layers but do not transfer amplitude, and (iii) asymmetric loops result from hybridization. Secondarily, we also develop a strategy for eliminating any unphysical vertical lines from calculations, thereby improving computational techniques for CBS.