Majorana bound states in anisotropic and tilted Dirac and Weyl systems

Abstract

Topological superconductors host Majorana boundary modes whose robustness is protected by the nontrivial topology of the bulk Bogoliubov quasiparticle spectrum. While Majorana bound states associated with Dirac and Weyl quasiparticles have been extensively investigated, much less is known about how anisotropic quasiparticle velocities and tilted band structures modify their microscopic properties. Here, we develop an analytical framework for Majorana bound states in effective two-dimensional (2D) Bogoliubov--de Gennes Dirac theories describing the surface quasiparticles of topological superconductors and extend the analysis to three-dimensional (3D) tilted Weyl systems by establishing a microscopic connection to the same low-energy description through superconducting pairing. For anisotropic 2D Dirac systems, we derive closed analytical expressions for the continuum topological invariant, Majorana wave function, localization length, propagation velocity, and finite-size minigap for arbitrary interface orientations. We show that the chirality of the Majorana channel is determined by the sign of the velocity-matrix determinant, while its localization and dispersion are governed jointly by the velocity tensor and the interface geometry. For tilted 2D Dirac cones, we demonstrate that the tilt leaves the spinor eigenstates, Berry phase, and projected pairing symmetry unchanged, but strongly suppresses the Majorana propagation velocity and finite-size minigap as the Lifshitz transition between type-I and type-II regimes is approached. Finally, we consider superconducting tilted 3D Weyl systems and show that the projection of a conventional spin-singlet $s$-wave pairing interaction onto the low-energy Weyl bands naturally generates an effective chiral $p_x\pm ip_y$ pairing symmetry, providing a microscopic Bogoliubov--de Gennes description that supports localized Majorana surface states.