Dynamics of magnetic vortices

We study a two-dimensional system of spin-polarized fermions on the kagome lattice at filling fraction $f=1/3$ interacting through a nearest-neighbor interaction $V$. Above a critical interaction strength ${V}_{c}$ a charge-density wave with a broken ${Z}_{3}$ symmetry is stabilized. Using the unrestricted mean-field approximation, we present several arguments showing that elementary topological point defects in the order parameter bind a fractional charge. Our analysis makes use of two appealing properties of the model: (i) For weak interaction, the low-energy degrees of freedom are described by Dirac fermions coupled to a complex-valued mass field (order parameter). (ii) The nearest-neighbor interaction is geometrically frustrated at filling $f=1/3$. Both properties offer a route to fractionalization and yield a consistent value $\ifmmode\pm\else\textpm\fi{}1/2$ for the fractional charge as long as the symmetry between the up and the down triangles of the kagome lattice is preserved. If this symmetry is violated, the value of the bound charge varies continuously with the strength of the symmetry-breaking term in the model. In addition, we have numerically computed the confining potential between two fractionally charged defects. We find that it grows linearly at large distances but can show a minimum at a finite separation for intermediate interactions. This indicates that the polaron state, formed upon doping the charge-density wave, can be viewed as a bound state of two defects.

Fractionally charged topological point defects on the kagome lattice

Nonlinear solitonlike excitations in a spin chain with a biquadratic anisotropic exchange interaction are investigated using the coherent-state method combined with the Holstein-Primakoff bosonic representation of the spin operators. It is argued that the modified terms of the nonlinear Schr\"odinger equation are strongly restricted by the relation between the continuum approximation (\ensuremath{\eta}=d/\ensuremath{\lambda}, the degree of the long-wavelength approximation, where d is the lattice constant and \ensuremath{\lambda} is the characteristic wavelength of excitation) and the semiclassical approximation (\ensuremath{\epsilon}=1/ \ensuremath{\surd}S , the degree of the truncation of the operator expansion, where S is the spin length). When assuming that \ensuremath{\eta} and \ensuremath{\epsilon} have the same order (\ensuremath{\eta}\ensuremath{\sim}\ensuremath{\epsilon}) after retaining terms of equivalent order O(${\mathrm{\ensuremath{\epsilon}}}^{6}$), the motion of the Bose operator for the anisotropic case satisfies the nonlinear Schr\"odinger equation with cubic nonlinearity, and solitonlike excitations are obtained. The other two cases of the relations between \ensuremath{\eta} and \ensuremath{\epsilon}(\ensuremath{\eta}\ensuremath{\sim}${\mathrm{\ensuremath{\epsilon}}}^{3/2}$ and \ensuremath{\eta}\ensuremath{\sim}${\mathrm{\ensuremath{\epsilon}}}^{2}$) are also discussed.

Solitonlike excitations in a spin chain with a biquadratic anisotropic exchange interaction.

The direct and inverse scattering problem is solved for the elliptic sinh-Gordon equation. It is shown that the inverse scattering transform may be useful in the analysis of localised singular solutions. As an example, a cylindrically symmetric singular solution is discussed in detail.

https://iopscience.iop.org/article/10.1088/0266-5611/6/4/006/pdf

Direct and Inverse Scattering Problem Associated with the Elliptic Sinh-Gordon Equation

The integration procedure based on the inverse scattering method is developed for a 2D elliptic sine-Gordon equation with a periodic-wave type asymptotic behaviour on one space variable The soliton solutions are found in explicit form

https://iopscience.iop.org/article/10.1088/0266-5611/5/6/006/pdf

Inverse problem for an elliptic sine-Gordon equation with an asymptotic behaviour of the cnoidal-wave type

Multi-vortex-like solutions of the sine-Gordon equation

Vortices in the sine-Gordon system and solution of the boundary value problem by the inverse scattering transform

Solitons in a ferrimagnet

Two-dimensional solutions of the Landau-Lifshitz equation

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