Polar phase of superfluid $^3$He: Dirac lines in the parameter and momentum spaces

TL;DR

This paper explores the polar phase of superfluid helium-3, revealing the existence of Dirac nodal lines in both the quasiparticle spectrum and the parameter space, highlighting novel topological features.

Contribution

It identifies and characterizes Dirac lines in the fermionic and bosonic spectra of superfluid helium-3's polar phase, including a unique bosonic Dirac loop in parameter space.

Findings

Fermionic spectrum has a Dirac loop in momentum space.

Bosonic (magnon) spectrum has a Dirac loop in magnetic field parameter space.

Bosonic Dirac system is at the boundary between type-I and type-II.

Abstract

The time reversal symmetric polar phase of the spin-triplet superfluid $^3$He has two types of Dirac nodal lines. In addition to the Dirac loop in the spectrum of the fermionic Bogoliubov quasiparticles in the momentum space $(p_x,p_y,p_z)$, the spectrum of bosons (magnons) has Dirac loop in the 3D space of parameters -- the components of magnetic field $(H_x,H_y,H_z)$. The bosonic Dirac system lives on the border between the type-I and type-II.