One dimensional nexus objects, network of Kibble-Lazarides-Shafi string walls, and their spin dynamic response in polar distorted B-phase of $^3$He

TL;DR

This paper explores the topological network of Kibble-Lazarides-Shafi string walls in superfluid $^3$He, revealing their connection to spin solitons, and matches theoretical predictions with experimental NMR data.

Contribution

It introduces the concept of 1D nexus objects connecting spin solitons and KLS domain walls in superfluid $^3$He, providing a detailed topological and energetic analysis.

Findings

Identification of KLS string walls as descendants of HQVs.

Numerical calculation of equilibrium spin textures and surface energies.

Exact match of NMR spectral shifts with experimental data.

Abstract

The Kibble-Lazarides-Shafi (KLS) domain wall problem in the axion solution of CP violation in QCD has condensed-matter based analogy in the nafen-distorted superfluid $^3$He. Recent experiment in rotating superfluid $^3$He produced the network of KLS string walls in human controllable system. In this system, KLS string wall appears in two-step symmetry break transition from normal phase to polar-distorted B phase and turn out to be the descendants of HQVs of polar phase. Here we show the KLS string wall smoothly connects to spin solitons when the spin orbital coupling is taken into account. This means HQVs are 1D nexus which connects the spin solitons and the KLS domain walls. This is because the subgroup $G=\pi_{1}(S_{S}^{1},\tilde{R}_{2})$ of relative homotopy group describing the spin solitons is isomorphic to the group describing the half spin vortices -- the spin textures KLS string wall. In the nafen-distorted $^3$He system, 1D nexus objects and the spin solitons with topological invariant $2/4$ form two different types of network, which are named as pseudo-random lattices of inseparable and separable spin solitons. These two types lattices correspond to two different representations of $G$. We discuss the condition under which pseudo-random lattices model works. The equilibrium configuration and surface densities of free energies are calculated by numeric minimization. Based on the equilibrium spin textures of different pseudo-random lattices, we calculated their transverse NMR spectrum, the resulted frequency shifts and $\sqrt{\Omega}$-scaling of ratio intensity exactly coincide with the experimental measurements. We also discussed the mirror symmetry in the presence of KLS domain wall and the influence of the explicitly break of this discrete symmetry. Our discussions and considerations can be applied to the composite defects in other condensed matter and cosmological system.