Institute for Theoretical Condensed Matter Physics, Karlsruhe Institute of Technology, Germany.
«Topological and Tropical Superconductivity»
This short course presents superconductivity from a perspective based on off-diagonal long-range order (ODLRO) and macroscopic quantum coherence. Starting from the fundamental concepts of conventional superconductivity, the course develops a unified framework that naturally leads to unconventional, multi-component (“tropical”), and topological superconducting states. Special emphasis is placed on the role of symmetry, topology, vortices, and Majorana bound states, which have become central topics in contemporary condensed matter physics and quantum information science.
The first lecture introduces superconductivity as a form of macroscopic quantum order. It discusses broken U(1) symmetry, gauge invariance, the superconducting order parameter, and the concept of ODLRO, highlighting its physical consequences such as the Meissner effect, flux quantization, and vortex formation. The lecture also explores different superconducting phases, including conventional single-component condensates and multi-component or sign-changing order parameters, and introduces their connection to topology.
The second lecture focuses on the foundations of topological superconductivity. Using the Bogoliubov–de Gennes formalism, it explains how topology and symmetry can protect boundary and defect states. Simple model systems, such as the Kitaev chain and chiral superconductors, are used to introduce Majorana modes and the basic classification of topological superconducting phases. The lecture also discusses mechanisms through which electronic topology and spin textures can promote superconducting pairing.
Finally, the last lecture examines vortex physics in topological superconductors. After reviewing vortices and vortex-core states in conventional superconductors, it introduces Majorana zero modes bound to topological defects and discusses the factors governing their stability. The lecture concludes with coupled-vortex systems and vortex lattices, where interactions between Majorana modes can lead to emergent collective states and topological phase transitions.