Interplay of Tilt and Axion Fields in Topological Superconductors: Anisotropy in the Meissner Effect

Abstract

Topological superconductors host gapless surface states that fundamentally alter their electromagnetic response through the axion field term $θ\vec{E}\cdot\vec{B}$, 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 $\vecζ$ 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 $\vecζ$, 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.

Figures (8)

• Figure 1: Penetration profiles of topological surface states for varying tilt magnitudes $\zeta_z$. The axion field $\theta$ exhibits a $2\pi$-periodicity with $\theta=0$ and $\theta=2\pi$ both corresponding to trivial (non-topological) superconducting phases.
• Figure 2: Magnetic field profile perpendicular to the tilt direction as a function of $z$, the distance from the surface inside the superconductor, for varying initial values of $B_\perp$ at the surface. Here, $z$ is scaled with $\lambda_B$, the penetration depth of the magnetic field in trivial case, and $B_\perp$ is scaled with $B_{E_0}=e^2 E_0/\pi$, where $E_0$ is the applied electric field at the surface. The contribution of the axion-mediated electrically induced magnetic field increases with rising $E_0$. This enhancement transforms the magnetic field profile from an exponential decay function to a hypergeometric function with a broad peak near the surface.
• Figure 3: Magnetic field profile perpendicular to the tilt direction as a function of $z/\lambda_B$ for varying tilt magnitudes. In (a) the initial surface value of $B_\perp$ is $B_0=B_{E0}$, while in (b) $B_0=(1/6) B_{E0}$. (c) All data from the panel (b), and additional data for a system with $B_0/B_{E_0}=1/6$, fall into a universal curve for $(1-\zeta^2)B_\perp/B_{E_0}$ upon the scaling $z\rightarrow \sqrt{1-\zeta^2}~ z$ and $\lambda_{B(\theta)}\rightarrow\sqrt{1-\zeta^2} ~ \lambda_{B(\theta)}$.
• Figure 4: Magnetic field inside the superconductor as a function of $\zeta$ at $z=0$ and $z=\lambda_B$, where $B_\perp$ reaches its maximum value. Various magnitudes of the applied magnetic field are considered. Increasing the applied magnetic field reduces the contribution of the surface states, while tilting enhances this contribution.
• Figure 5: Magnetic field behavior near the surface, parallel and perpendicular to the tilt direction. All vertical axes show $z/lambda_B$. Magenta solid curves represent $B_\perp$, while gray dashed curves denote $B_\parallel$. (a)–(c): $B_\perp$ and $B_\parallel$ in a TWS with fixed $\zeta=0.5$ and varying $\alpha$. (d)–(f): $B_\perp$ and $B_\parallel$ in a TWS with fixed $\zeta=0$ and varying $\alpha$. (g) Trivial superconductor case ($\theta=0$), where both $B_\perp$ and $B_\parallel$ exhibit identical exponential decay, independent of $\zeta$. (h) TWS case revealing anisotropic penetration: $B_\perp$ shows tilt-dependent modification, while $B_\parallel$ remains unaffected by $\zeta$.