Interplay of tilt and axion fields in topological superconductors: Anisotropy in the Meissner effect

Topological superconductors host gapless surface states that fundamentally alter their electromagnetic response through the axion field term $\ensuremath{\theta}\mathbit{E}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{B}$, arising from the topological magnetoelectric effect. In this work, we investigate the electromagnetic properties of a three-dimensional topological Weyl superconductor by leveraging its theoretical mapping to a four-dimensional topological insulator with $s$-wave superconducting boundaries. By incorporating the tilt of Weyl cones into this model, we demonstrate that the tilt vector $\mathbit{\ensuremath{\zeta}}$ anisotropically modifies the axion field profile near the surface, leading to a tilt-enhanced Meissner effect and anomalous magnetic penetration depths. We show that the magnetic field component perpendicular to the tilt direction exhibits a non-exponential, hypergeometric decay dictated by the interplay between the axion term and $\mathbit{\ensuremath{\zeta}}$, while the parallel component remains largely tilt-insensitive---a hallmark of axion-mediated anisotropy absent in trivial superconductors. Remarkably, all tilt-dependent electromagnetic responses follow a universal scaling law, revealing a fundamental symmetry in the system's behavior. Furthermore, we predict a tilt-dependent planar Hall current at the surface, directly tied to the topological surface states.