The Higgs-Anderson mechanism is a general phenomenon that occurs when a system with a local symmetry undergoes a phase transition and acquires a mass gap. In superconductors, this mechanism explains the Meissner effect, which is the expulsion of magnetic fields from the interior of the material. The Higgs-Anderson mechanism also implies the existence of a collective excitation called the Higgs mode, which is related to the fluctuations of the order parameter.

Topological superconductors are a special class of superconductors that have non-trivial topological properties, such as edge states, Majorana fermions, and fractional statistics. These properties are protected by symmetries and are robust against local perturbations. Topological superconductors can be realized in various systems, such as spin-orbit coupled materials, heterostructures, and cold atoms.

The Higgs-Anderson mechanism in topological superconductors has some unique features that distinguish it from conventional superconductors. For example, the Higgs mode can couple to other excitations, such as magnons or surface plasmons, and exhibit novel phenomena, such as hybridization, amplification, or resonance. The Higgs mode can also be influenced by the presence of topological defects, such as vortices or domain walls, and reveal information about the underlying symmetry and topology of the system.

This is a very active and fascinating research area with potential applications in quantum information, metrology, and nanotechnology.

Superconductivity is a phenomenon in which a material can conduct electricity without any resistance below a certain critical temperature. Superconductors can be classified into different types based on their properties and mechanisms.

Weyl superconductors are a new class of superconductors that emerge from Weyl semimetals, which are materials that have linear dispersion around isolated points in the momentum space called Weyl nodes. Weyl nodes have a definite chirality, which means they act like sources or sinks of Berry curvature, a geometric quantity that measures the twisting of the electronic wave functions. Weyl semimetals also have exotic surface states called Fermi arcs, which connect the projections of the Weyl nodes on the surface.

When Weyl semimetals become superconducting, they exhibit novel features that are not found in conventional superconductors. For example, the superconducting gap function, which describes the energy cost of breaking a Cooper pair, can have point nodes that are protected by symmetry or topology. These point nodes can be manipulated by tuning the tilt of the Weyl nodes, which affects their separation and location in the momentum space. The point nodes also give rise to low-energy excitations called Bogoliubov-Weyl fermions, which are analogs of the Weyl fermions in the normal state. Moreover, the surface of Weyl superconductors can host chiral Majorana modes, which are exotic quasiparticles that are their antiparticles and obey non-Abelian statistics. These modes can be detected by various probes, such as tunneling spectroscopy, Josephson effect, or thermal transport34.

In this project, we aim to explore the different types of superconductivities in Weyl superconductors and their physical consequences. We will use theoretical methods, such as mean-field theory, Green’s function technique, and topological invariant calculation, to study the phase diagram, the electronic structure, the vortex physics, and the surface states of Weyl superconductors. We will also compare our results with existing and future experiments and propose new ways to realize and manipulate Weyl superconductivity in various systems.

A Josephson array is a network of superconducting islands connected by weak links, such as tunnel junctions or nanowires. A Josephson array can exhibit various collective phenomena, such as phase transitions, quantum fluctuations, and vortex dynamics. A vortex is a topological defect in the superconducting order parameter that carries a quantized magnetic flux.

Weyl superconductors are a new class of superconductors that emerge from Weyl semimetals, which are materials that have linear dispersion around isolated points in the momentum space called Weyl nodes. Weyl superconductors can have different types of pairing symmetries, such as scalar, pseudoscalar, vector, or axial vector. These pairing symmetries can be tuned by external parameters, such as magnetic field, strain, or doping.

In this project, we investigate the possibility of creating and manipulating half-vortices in a Josephson array made of Weyl superconductors. A half-vortex is a vortex that carries half of the quantum of flux and has a fractional winding number. Half-vortices can arise when the superconducting order parameter has both scalar and pseudoscalar components, which can be controlled by the flux biasing of the weak links. Half-vortices have exotic properties, such as fractional statistics, non-Abelian braiding, and the chiral Josephson effect.

We use theoretical methods, such as mean-field theory, renormalization group, and numerical simulation, to study the phase diagram, the stability, the transport, and the braiding of half-vortices in Weyl superconductor Josephson arrays. We also propose realistic experimental setups to realize and detect half-vortices in these systems.