[Paper Review] Vortex-bound solitons in topological superfluid $^3$He

J. T. Mäkinen, K. Zhang+1 more|arXiv (Cornell University)|Nov 30, 2022

Quantum, superfluid, helium dynamics4 citations

TL;DR

This paper investigates vortex-bound solitons in topological superfluid $^3$He, where topological defects arise from degenerate free energy minima in p-wave superfluid phases. Using nuclear magnetic resonance (NMR) techniques, the authors identify three distinct soliton-vortex structures—spin-mass vortices in the B phase, half-quantum vortices in polar and polar-distorted A phases, and a composite nexus with Kibble-Lazarides-Shafi walls in the polar-distorted B phase—demonstrating that solitons serve as key probes for topological order and Majorana modes.

ABSTRACT

The different superfluid phases of $^3$He are described by $p$-wave order parameters that include anisotropy axes both in the orbital and spin spaces. The anisotropy axes characterize the broken symmetries in these macroscopically coherent quantum many-body systems. The systems' free energy has several degenerate minima for certain orientations of the anisotropy axes. As a result, spatial variation of the order parameter between two such regions, settled in different energy minima, forms a topological soliton. Such solitons can terminate in the bulk liquid, where the termination line forms a vortex with trapped circulation of mass and spin superfluid currents. Here we discuss possible soliton-vortex structures based on the symmetry and topology arguments and focus on the three structures observed in experiments: solitons bounded by spin-mass vortices in the B phase, solitons bounded by half-quantum vortices in the polar and polar-distorted A phases, and the composite defect formed by a half-quantum vortex, soliton and the Kibble-Lazarides-Shafi wall in the polar-distorted B phase. The observations are based on nuclear magnetic resonance (NMR) techniques and are of three types: first, solitons can form a potential well for trapped spin waves, observed as an extra peak in the NMR spectrum at shifted frequency; second, they can increase the relaxation rate of the NMR spin precession; lastly, the soliton can present the boundary conditions for the anisotropy axes in bulk, modifying the bulk NMR signal. Owing to solitons' prominent NMR signatures and the ability to manipulate their structure with external magnetic field, solitons have become an important tool for probing and controlling the structure and dynamics of superfluid $^3$He, in particular half-quantum vortices with core-bound Majorana modes.

Motivation & Objective

• To establish a theoretical framework for vortex-bound solitons in topological superfluid $^3$He based on symmetry and topology.
• To identify and classify distinct soliton-vortex structures arising from degenerate free energy minima in different superfluid phases.
• To link soliton signatures in NMR spectra to the presence of topological defects, including half-quantum vortices and Majorana zero modes.
• To demonstrate that solitons serve as experimental probes for complex topological order and quantum phenomena in $^3$He.
• To provide a comprehensive analysis of NMR responses to solitons, enabling detection and manipulation of topological defects.

Proposed method

• Utilizes relative homotopy groups to classify topological and composite defects in p-wave superfluid $^3$He with broken symmetries.
• Analyzes the role of orienting energy terms (magnetic, spin-orbit) in stabilizing distinct order parameter configurations and defect structures.
• Applies nuclear magnetic resonance (NMR) techniques to detect soliton signatures: shifted spectral peaks, enhanced relaxation rates, and modified bulk NMR signals.
• Models the topological protection of solitons and vortices via the interplay of residual symmetries and order parameter topology.
• Examines the nexus defect—where a vortex terminates a soliton and also acts as a starting line for a Kibble-Lazarides-Shafi wall—using topological invariants.
• Correlates experimental NMR observations with theoretical predictions to validate the existence and structure of vortex-bound solitons.

Experimental results

Research questions

• RQ1 What types of vortex-bound soliton structures can emerge in topological superfluid $^3$He due to degenerate free energy minima?
• RQ2 How do NMR measurements detect the presence of solitons and their associated topological defects?
• RQ3 What is the role of half-quantum vortices and Majorana zero modes in the soliton-vortex composite structures?
• RQ4 How do symmetry-breaking patterns and orienting energy terms influence the stability and topology of soliton-vortex systems?
• RQ5 Can the Kibble-Lazarides-Shafi wall coexist with a vortex and soliton to form a composite defect, and how is it detected?

Key findings

• Three distinct vortex-bound soliton structures were experimentally observed: solitons bounded by spin-mass vortices in the B phase, half-quantum vortices in the polar and polar-distorted A phases, and a composite nexus in the polar-distorted B phase.
• NMR revealed solitons as potential wells for trapped spin waves, producing an extra peak in the NMR spectrum at a shifted frequency.
• Solitons were found to increase the relaxation rate of NMR spin precession, indicating strong coupling to quasiparticles.
• Solitons modify the boundary conditions for anisotropy axes in the bulk, leading to measurable changes in the bulk NMR signal.
• The polar-distorted A phase is the only known experimental platform hosting half-quantum vortices in a chiral superfluid, enabling studies of grand unified theories and Majorana modes.
• The ability to manipulate soliton structures with external magnetic fields establishes them as a tool for probing and controlling topological superfluid order.