Interplay between inversion and translation symmetries in trigonal PtBi$_2$
The trigonal Weyl semimetal PtBi$_2$ presents an intriguing superconducting phase, previously reported to be confined to its topological Fermi arcs within a certain temperature range. This observation highlights the importance of a thorough understanding of its normal phase, particularly the roles that spin-orbit coupling (SOC) and inversion-symmetry breaking play in shaping its band structure. Our density-functional theory calculations reveal that the semimetallic nature of trigonal PtBi$_2$ can be interpreted as stemming from a noncentrosymmetric crystal distortion of a parent structure that drives a metal-to-semimetal transition. This distortion breaks inversion symmetry and, crucially, reduces translational symmetry. Due to its interplay with translational symmetry, inversion-symmetry breaking emerges as the dominant energy scale producing substantial asymmetries ($\sim$ 0.6\,eV) in certain short-range hopping amplitudes, superseding the effects of SOC, whose primary role is to define the characteristics of the low-energy nodal structure and of the topological Fermi arcs. This also applies to the formation of the Weyl nodes closest to the Fermi energy, which are found to exist even in the absence of SOC as a result of the orbital physics associated with the reduced translational symmetry.