Order in 2D nodal superconductors

TL;DR

This thesis develops a topological theory for d-wave superconductors, highlighting how gapless excitations affect topological degeneracy and exploring magnetic impurities' influence on Casimir energy as a topological indicator.

Contribution

It introduces a topological framework for d-wave superconductors, extending concepts from s-wave cases and analyzing impurity effects on topological properties.

Findings

Gapless excitations destroy topological degeneracy.

Magnetic impurities influence Casimir energy, reflecting topology.

Work in progress with ongoing computations.

Abstract

A topological theory of d-wave superconductors is derived in this thesis. Ginzburg-Landau theory describes superconductivity by defining a complex order parameter and applying Landau's theory for phase transitions. However, there is no local, gauge invariant order parameter for a superconductor and classical order is no appropriate description. Topological order has proven to be a powerful tool for describing the Quantum Hall effect and it gives also an appropriate description of s-wave superconductors. For d-wave superconductors there are gapless excitations at four points on the Fermi surface. The topological theory for superconductors exhibits a similar structure for the ground state as found in the s-wave case. However, the gapless excitations destroy the topological degeneracy of the ground state and introduce an additional degeneracy in one of the flux sectors. In search for regularities which reflect the topological order, I include a magnetic point impurity and focus on the effects it has on the Casimir energy. Indeed, it is shown that the energy shift reflects the topology to some extend. However, this is work in progress and further computations will be done.