Orbit Spaces in Superconductivity

TL;DR

This paper demonstrates a method using orbit space analysis and P-matrix techniques to classify possible superconducting phases based on crystal symmetry, providing a general approach applicable to various physical systems with compact symmetry groups.

Contribution

It introduces a simple example applying the P-matrix method to determine the orbit space and stratification in superconductivity, showcasing a broadly valid approach for symmetry-invariant problems.

Findings

P-matrix method effectively determines orbit space equations and inequalities.

Orbit space analysis clarifies possible low-symmetry superconducting phases.

Method applicable to all physical problems with compact symmetry groups.

Abstract

In the framework of Landau theory of phase transitions one is interested to describe all the possible low symmetry ``superconducting'' phases allowed for a given superconductor crystal and to determine the conditions under which this crystal undergoes a phase transition. These problems are best described and analyzed in the orbit space of the high symmetry group of the ``normal, non-superconducting'' phase of the crystal. In this article it is worked out a simple example concerning superconductivity, that shows the P-matrix method to determine the equations and inequalities defining the orbit space and its stratification. This approach is of general validity and can be used in all physical problems that make use of invariant functions, as long as the symmetry group is compact.