Magnetic penetration depth in topological superconductors: Effect of Majorana surface states and application for UTe$_2$

Abstract

In this study, we examine how orbital degrees of freedom and Majorana surface states influence the magnetic penetration depth in the superconductor UTe$_2$. Using a two-orbital model, we analyze pairing states belonging to the irreducible representations of the $D_{2h}$ crystal symmetry: $A_u$, $B_{1u}$, $B_{2u}$, and $B_{3u}$. For bulk nodal states such as $B_{2u}$, we find that the penetration depth for screening currents along the antinodal direction and the cylindrical axis scales as $T^2$, in strong contrast to the conventional $T^4$ law. This behavior originates from quasiparticles near the point nodes contributing to the interorbital paramagnetic current. We further show that Majorana surface states can dominate the low-temperature response. The fully gapped $A_u$ state hosts Majorana cones, which produce a $T^3$ dependence of the penetration depth when the ratio of penetration depth to coherence length ($\kappa$) is small. In contrast, the other pairing states exhibit Majorana Fermi arcs: the exponent is $n=2$ along the dispersive direction, while along the dispersionless direction it depends on whether the arcs terminate at endpoints. The exponent $n=2$ in the dispersive direction is robust, while it in the dispersionless direction relies on the presence or absence of the endpoints of the arcs and deviates from $n=2$ when endpoints are absent. Our results demonstrate that penetration-depth measurements provide a direct probe of Majorana surface states in low-$\kappa$ superconductors. For larger $\kappa$, the surface contribution becomes negligible and the temperature dependence is governed by bulk quasiparticles.