Dr. Mauro M.Doria, professor at the Federal University of Rio de Janeiro, presented the description of topological insulators within the framework of the Abrikosov-Bogomolny equation. The presentation took place at the online seminar of the International Academic Cooperation Project
This is the first of two consecutive seminars on the problem of the Abrikosov-Bogomolny equations and their connection with topological phenomena. The seminar was devoted to the consideration of the Abrikosov-Bogomolny equations in the spinless (scalar) mode, which operate in 2nd-order superconductors near the upper critical field and in the crossover region between 2nd- and 1st-order superconductivity.The focus of the seminar is to obtain the Abrikosov-Bogomolny equations from the kinetic energy of a Schrödinger scalar field by decomposing this energy into three parts, which is known as a special case of the Laplace operator representation through the Lichnerowicz Laplace operator and leads to a local relation between the magnetic induction and the superconducting order parameter.
The Abrikosov-Bogoliubov (AB) equations provide a unique opportunity to understand the topological properties of a system. The simplest version of these equations involves complex numbers and describes superconductors in the region near the crossover between first- and second-order superconductivity, as well as in second-order superconductors in the vicinity of the upper critical field. These equations describe self-dual vortices in two-dimensional space, which means they can be applied to three-dimensional space with an order parameter that depends only on two spatial coordinates (in a plane perpendicular to the vortex lines).
The inclusion of spin makes the equations truly three-dimensional, but paradoxically, a more detailed study reveals that they describe surface quantum plasmons in conducting layers. Plasmons are collective excitations that propagate along interfaces and layers, and although their origin is two-dimensional, they exist in the surrounding three-dimensional space due to electromagnetic fields. They are found in many materials, such as metals, graphene, and layered superconductors. In this case, it has been discovered that plasmons are skyrmions and anti-skyrmions, whose stability is provided by their own magnetic field, which penetrates the layers and extends three-dimensionally around these layers. In this scenario, these objects are the fundamental quasiparticles of topological insulators.
