## Topological Superconductivity

Matter comes in many shapes and forms. Conventional matter such as magnets and crystals is defined by broken symmetries. In contrast, topological states of matter show distinct physical properties from conventional matter with the same broken symmetries such as the presence of robust surface metallicity and quantized responses to electromagnetic fields. Topological superconductors are an elusive class of topological matter whose underlying superconductivity makes them fundamentally richer than their non-superconducting counterparts. We are interested in finding platforms for topological superconductivity and tapping their revolutionary technological potential.

## SUSY and non-local Majorana fermions in superconductor vortices

Superconductor vortices host discrete energy levels that carry critical information about the parent metal. Weyl semimetals have two distinct types of metallic states in their spectrum -- Weyl fermions in the bulk and Fermi arcs on the surface -- that are intricately coupled and cannot be captured by standard Hamiltonians. We ask, "what is the vortex spectrum of a superconductor that descends from a Weyl semimetal?'' We find that the spectrum is determined by semiclassical quantization of cyclotron-like closed orbits consisting of Fermi arcs on opposite surfaces connected by one-way bulk conduits. Miraculously, merely tilting the vortices can transmute them between bosonic, fermionic and supersymmetric, with the last class hosting a pair of non-local Majorana fermions. Thus, we propose a tabletop and tunable realization of a long sought-after system that displays supersymmetry, namely, an equivalence between bosons and fermions. At tilts that we dub "magic angles", the vortex spectrum becomes independent of the slab thickness. In many models and materials, non-local Majorana fermions and supersymmetry exist precisely at these tilts. Learn even more

Closed semiclassical orbits govern the vortex spectrum and endow it with characteristic dependences on the Berry phases and penetration depths of the Fermi arcs, bulk Weyl node locations, vortex orientation and sample thickness.

When the surface Fermi arcs (red curves) plus the projections (black solid lines) of the geodesics between bulk Weyl nodes in the same kz-plane (black dashed lines) form M closed loops, the vortex is gapped and has (lacks) end Majorana modes if M is odd (even). If the Fermi-arc + geodesic-projections form open contours, the vortex is gapless.

## New route to Majorana fermions

The search for Majorana fermions - particles that are their own anti-particles that were first proposed but never found in high-energy physics - has driven fierce activity in condensed matter, where they occur as zero energy states in topological defects such as vortex cores of type-II superconductors. Among 3D materials, criteria for finding Majorana fermions at the ends of superconducting vortices in time-reversal symmetric insulators and metals are well-known. We derive analogous criteria for the third and last generic type of 3D band structure, namely, time-reversal symmetric Weyl semimetals. The criteria we find depend only on the bulk and surface band structure of the semimetal, which are outcomes of routine photoemission experiments and first-principles simulations, and allow tuning the Majorana fermions in and out of existence by merely tilting an external magnetic field. Learn even more

## Topological superconductivity in β-PdBi2

While numerous 3D topological insulators have been found, their superconducting counterparts have been hard to come by, with superfluid He-3B the only confirmed example and no confirmed examples in solids. In an experimental collaboration, we found evidence that the superconducting states of undoped and K-doped beta-PdBi2 may be trivial and topological superconductors, respectively. If confirmed, this could be the first known bulk material that undergoes a topological phase transition inside the superconducting phase. Learn even more

In the high-T normal state, K-doping exposes the surface states of the topological metal. In the superconductor at low-T, it seems to change the pairing symmetry from even to odd; the latter symmetry is known to yield a topological superconductor, which can be diagnosed by the T-dependence of the Meissner effect.

In a type-I topological superconductor, the surface Majorana fluid (blue curve) leaks into the bulk beyond the London penetration depth and enhances the bulk penetration of an external magnetic field, depicted by red arrows. The length scales are reversed in a type-II topological superconductor so the enhancement is less striking, but still exists.

## Meissner effect in topological superconductors

The expulsion of magnetic fields -- known as the Meissner effect -- is a fundamental property of superconductors. While type-I superconductors expel magnetic fields as long as they are superconducting, type-II superconductors show the Meissner effect below a critical field Hc1. We establish that the surface states of a topological superconductor give rise to an unusual Meissner effect that has a characteristic, power-law temperature dependence. Interestingly, in a type-I superconductor, the magnetic field can "ride on" the surface states and penetrate much deeper into the bulk than the bulk superconductivity would have allowed. Learn even more