Special Issue on Solitons in Quantum Physics

Broken symmetries in topological condensed matter systems have implications for the spectrum of Fermionic excitations confined on surfaces or topological defects. The Fermionic spectrum of confined (quasi-2D) ³He-A consists of branches of chiral edge states. The negative energy states are related to the ground-state angular momentum, $L_z = (N/2) \hbar$, for $N/2$ Cooper pairs. The power law suppression of the angular momentum, $L_z(T) \simeq (N/2)\,\hbar\,[1 - \frac{2}{3}(\pi T/\Delta)^2 ]$ for $0 \leqslant T \ll T_c$, in the fully gapped 2D chiral A-phase reflects the thermal excitation of the chiral edge Fermions. We discuss the effects of wave function overlap, and hybridization between edge states confined near opposing edge boundaries on the edge currents, ground-state angular momentum and ground-state order parameter of superfluid ³He thin films. Under strong lateral confinement, the chiral A phase undergoes a sequence of phase transitions, first to a pair density wave (PDW) phase with broken translational symmetry at $D_{c2} \sim 16 \xi_0$. The PDW phase is described by a periodic array of chiral domains with alternating chirality, separated by domain walls. The period of PDW phase diverges as the confinement length $D\rightarrow D_{c_2}$. The PDW phase breaks time-reversal symmetry, translation invariance, but is invariant under the combination of time-reversal and translation by a one-half period of the PDW. The mass current distribution of the PDW phase reflects this combined symmetry, and originates from the spectra of edge Fermions and the chiral branches bound to the domain walls. Under sufficiently strong confinement a second-order transition occurs to the non-chiral 'polar phase' at $D_{c1} \sim 9\xi_0$, in which a single p-wave orbital state of Cooper pairs is aligned along the channel.

Magnetic skyrmions have garnered considerable attention due to their topological properties and potential applications in information storage. These unique structures can be found in chiral magnets, including well-known compounds like MnSi and FeGe with a B20-type crystal structure. In this study, we utilized Lorentz transmission electron microscopy to investigate the influence of magnetic skyrmions on the Hall effect in FeGe under low magnetic fields. Additionally, we examined the magnetoresistance (MR) and Hall effect of FeGe under a high magnetic field of 28 T. Our findings reveal distinct mechanisms governing the MR at low and high temperatures. Notably, the anomalous Hall effect plays a significant role in the Hall resistivity observed at low magnetic fields. Meanwhile, the contribution of the skyrmion-induced topological Hall signal in the FeGe is ignorable. Furthermore, by employing a two-carrier model and fitting the carrier concentration of FeGe under high magnetic fields, we demonstrate a transition in the dominant carrier type from electrons to holes as the temperature increases. These results contribute to a deeper understanding of the intrinsic magnetic properties of FeGe.

The different superfluid phases of ³He 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 (HQVs) 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 ³He, in particular HQVs with core-bound Majorana modes.

This paper gives a pedagogical introduction to the mechanics of ferromagnetic solitons. We start with the dynamics of a single spin and develop all the tools required for the description of the dynamics of solitons in a ferromagnet.

We introduce topological phases of matter defined by skyrmions in the ground state spin—or pseudospin—expectation value textures in the Brillouin zone, the chiral and helical topological skyrmion phases of matter. These phases are protected by a symmetry present in centrosymmetric superconductors. We consider a tight-binding model for spin-triplet superconductivity in transition metal oxides and find it realizes each of these topological skyrmion phases. The chiral phase is furthermore realized for a parameter set characterizing Sr₂RuO₄ with spin-triplet superconductivity. We also find two types of topological phase transitions by which the skyrmion number can change. The second type occurs without the closing of energy gaps in a system described by a quadratic Hamiltonian without breaking the protecting symmetries when atomic spin–orbit coupling is non-negligible and there is a suitable additional degree of freedom. This contradicts the 'flat band' limit assumption important in use of entanglement spectrum and Wilson loops, and in construction of the ten-fold way classification scheme of topological phases of matter. We furthermore predict two kinds of bulk-boundary correspondence signatures—one for measurements which execute a partial trace over degrees of freedom other than spin, which yields quantized transport signatures—and a second resulting from skyrmions trapping defects with their own non-trivial topology that is discussed in a second work, which yields generalizations of unpaired Majorana zero-modes.

We study solitons in a zig-zag lattice of magnetic dipoles. The lattice comprises two sublattices of parallel chains with magnetic dipoles at their vertices. Due to orthogonal easy planes of rotation for dipoles belonging to different sublattices, the total dipolar energy of this system is separable into a sum of symmetric and chiral long-ranged interactions between the magnets where the last takes the form of Dzyaloshinskii–Moriya (DM) coupling. For a specific range of values of the offset between sublattices, the dipoles realize an equilibrium magnetic state in the lattice plane, consisting of one chain settled in an antiferromagnetic (AF) parallel configuration and the other in a collinear ferromagnetic fashion. If the offset grows beyond this value, the internal DM field stabilizes two Bloch domain walls at the edges of the AF chain. The dynamics of these solitons is studied by deriving the long-wavelength lagrangian density for the easy axis antiferromagnet. We find that the chiral couplings between sublattices give rise to an effective magnetic field that stabilizes the solitons in the antiferromagnet. When the chains displace respect to each other, an emergent Lorentz force accelerates the domain walls along the lattice.

We derive a unified theory of spin and charge degrees of freedom in a ferromagnet. The spin-transfer torque and spin electromotive force are examined from the coarse-grained perspective of collective coordinates. The resulting equations of motion reflect a balance of conservative, gyroscopic (Berry-phase), and dissipative forces. We then expand the space of collective coordinates by adding the electric charge. The adiabatic spin-transfer torque and spin electromotive force (emf) turn out to be a gyroscopic force; their nonadiabatic counterparts are a dissipative force.

Here we revisit the topic of stationary and propagating solitonic excitations in self-repulsive three-dimensional (3D) Bose–Einstein condensates by quantitatively comparing theoretical analysis and associated numerical computations with our experimental results. Motivated by numerous experimental efforts, including our own herein, we use fully 3D numerical simulations to explore the existence, stability, and evolution dynamics of planar dark solitons. This also allows us to examine their instability-induced decay products including solitonic vortices and vortex rings. In the trapped case and with no adjustable parameters, our numerical findings are in correspondence with experimentally observed coherent structures. Without a longitudinal trap, we identify numerically exact traveling solutions and quantify how their transverse destabilization threshold changes as a function of the solitary wave speed.

We present a theory of optimal topological textures in nonlinear sigma-models with degrees of freedom living in the Grassmannian $\mathrm{Gr}(M,N)$ manifold. These textures describe skyrmion lattices of N-component fermions in a quantising magnetic field, relevant to the physics of graphene, bilayer and other multicomponent quantum Hall systems near integer filling factors ν > 1. We derive analytically the optimality condition, minimizing topological charge density fluctuations, for a general Grassmannian sigma model $\mathrm{Gr}(M,N)$ on a sphere and a torus, together with counting arguments which show that for any filling factor and number of components there is a critical value of topological charge $d_\mathrm {c}$ above which there are no optimal textures. Below $d_\mathrm {c}$ a solution of the optimality condition on a torus is unique, while in the case of a sphere one has, in general, a continuum of solutions corresponding to new non-Goldstone zero modes, whose degeneracy is not lifted (via a order from disorder mechanism) by any fermion interactions depending only on the distance on a sphere. We supplement our general theoretical considerations with the exact analytical results for the case of $\mathrm{Gr}(2,4)$, appropriate for recent experiments in graphene.

In a liquid of superconducting vortices, a longitudinal thermal gradient generates a transverse electric field. This Nernst signal peaks at an intermediate temperature and magnetic field, presumably where the entropy difference between the vortex core and the superfluid environment is largest. There is a puzzling similarity of the amplitude of this peak across many different superconductors. This peak can be assimilated to a minimum in the viscosity to entropy density ratio of the vortex liquid. Expressed in units of $\frac{\hbar}{k_\mathrm{B}}$, this minimum is one order of magnitude larger than what is seen in common liquids. Moreover, the entropy stocked in the vortex core is not identical to the entropy bound to a moving magnetic flux line. Due to a steady exchange of normal quasi-particles, entropy can leak from the vortex core. A slowly moving vortex will be peeled off its entropy within a distance of the order of a superconducting coherence length, provided that the $\frac{\Delta}{E_\mathrm{F}}$ ratio is sufficiently large.