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.
Highlights
The composite objects formed by topological defects with different dimensions, such as the Kibble-Lazarides-Shafi (KLS) string wall [1,2], play significant roles in Grand unified theories and cosmological models
We show under the low angular velocity limit, the randomly distributed spin soliton network can be mapped to models of regular lattices consisting of spin solitons
We calculate the group which describes the spin degree of freedom of KLS string wall in the region ξH < r < ξD and the relative homotopy group of spin solitons when r > ξD. We prove the former is isomorphic to the subgroup of the latter, the spin soliton is smoothly connected to the KLS domain wall by half quantum vortices (HQVs)
Summary
The composite objects formed by topological defects with different dimensions, such as the Kibble-Lazarides-Shafi (KLS) string wall [1,2], play significant roles in Grand unified theories and cosmological models. In the vicinity of the second time symmetry breaking, it is clear that the HQVs formed in the polar phase turn to be string-wall composite topological objects described by the relative homotopy group π1(R1, R2 ), i.e., π1(RP ) = π1(R1, R2 ),. Because the geometric size of KLS string wall is around dipole length, the spin orbital coupling (SOC) energy further reduces the vacuum manifold of PdB to discrete sets. This gives rise to spin solitons, which are described by relative homotopy group [46]. We discuss the possible planar spin solitons attached on string monopole networks in PdB phase
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