# Topological phase transitions in multi-component superconductors

## TL;DR

This paper investigates the nature of phase transitions between trivial and topological superconducting states in multi-component systems, revealing intermediate phases, symmetry breaking, and topological defects with potential material applications.

## Contribution

It provides a comprehensive analysis of phase transitions in multi-component superconductors, including symmetry considerations, intermediate phases, and Majorana fermion defects, with analytical solutions.

## Key findings

- Intermediate phases with coexisting pairing components
- Spontaneous time-reversal symmetry breaking in inversion-symmetric systems
- Existence of Majorana fermions bound to topological defects

## Abstract

We study the phase transition between a trivial and a time-reversal-invariant topological superconductor in a single-band system. By analyzing the interplay of symmetry, topology and energetics, we show that for a generic normal state band structure, the phase transition occurs via extended intermediate phases in which even- and odd-parity pairing components coexist. For inversion-symmetric systems, the coexistence phase spontaneously breaks time-reversal symmetry. For noncentrosymmetric superconductors, the low-temperature intermediate phase is time-reversal breaking, while the high-temperature phase preserves time-reversal symmetry and has topologically protected line nodes. Furthermore, with approximate rotational invariance, the system has an emergent $U(1) \times U(1)$ symmetry, and novel topological defects, such as half vortex lines binding Majorana fermions, can exist. We analytically solve for the dispersion of the Majorana fermion and show that it exhibit small and large velocities at low and high energies. Relevance of our theory to superconducting pyrochlore oxide Cd$_2$Re$_2$O$_7$ and half-Heusler materials is discussed.