General spin-order theory via gauge Landau-Lifshitz equation

You-Quan Li, Ye-Hua Liu, and Yi Zhou

Phys. Rev. B 84, 205123 – Published 16 November 2011

Abstract

The continuum limit of the tilted SU(2) spin model is shown to lead to a gauge Landau-Lifshitz equation which provides a unified description for various spin orders. For a definite gauge with zero field strength, we find a double periodic solution, where the conical spiral, in-plane spiral, helical, and ferromagnetic spin orders become special cases. For another gauge with nonzero field strength, we obtain the skyrmion-crystal solution. By simulating the influence of magnetic field and temperature for our model, we find a spontaneous formation of a skyrmion-fragment lattice similar to the spin texture found in experiment very recently; we also obtain a wider range of skyrmion-crystal phase in the parameter space in comparison to the conventional Dzyaloshinsky-Moriya model.

Received 23 October 2011

DOI:https://doi.org/10.1103/PhysRevB.84.205123

Authors & Affiliations

You-Quan Li, Ye-Hua Liu, and Yi Zhou

Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University, Hangzhou 310027, People's Republic of China

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Issue

Vol. 84, Iss. 20 — 15 November 2011

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Images

Figure 1
Schematic illustration of the bundle ( $L, F, G$ ) for the two-dimensional system. The four lattice sites $b_{1}$ , $b_{2}$ , $b_{3}$ , and $b_{4}$ in sublattice $L_{B}$ (blue) and $a_{1} \in L_{A}$ (red) constitute the neighborhood $U^{a_{1}}$ . In this neighborhood, the tilting fields depict the relative orientation of the polar axes of the Bloch spheres attached on the five sites. The amplified balls above the lattice show, for example, the relative orientation between the ones at $a_{1}$ and $b_{2}$ that is depicted by the tilting $U_{b_{2}}^{a_{1}}$ , while the relative orientation between the ones at $a_{2}$ and $b_{2}$ is depicted by the tilting field $U_{b_{2}}^{a_{2}}$ that is defined on another neighborhood $U^{a_{2}}$ whose center is $a_{2}$ . The subscript of the tilting field is omitted for figure neatness.Reuse & Permissions
Figure 2
Schematic illustration of the spin order for the double periodic solution of the gauge Landau-Lifshitz equation.Reuse & Permissions
Figure 3
Schematic illustration for spin order of skyrmion-crystal solution (left panel) and the corresponding distribution of energy density in the zero magnetic field (right panel). The energy density is lower at the center of skyrmion because the local spin chirality is large there, which is favored by the DM interaction; it is larger in the boundaries of skyrmions but the spins there favor a perpendicular magnetic field.Reuse & Permissions
Figure 4
(a) Spin order of the skyrmion-fragment phase. Here is a patch of spin textures from our simulation for $K / J = \sqrt{2} \tan (2 π / 6)$ . Along the black dotted-line arrows, the spins form a helical order where each two successive spins have a relative $60^{\circ}$ difference in angle. (b) and (c) The spin $z$ - $z$ correlation in the 48 by 48 lattice for various phases in the zero magnetic field. Temperature is lowering, it goes from disordered to helical (b) and then to skyrmion-fragment (c) phases.Reuse & Permissions
Figure 5
Various spin textures result from Monte Carlo simulation. (a) helical. (b) helical $+$ skyrmion. (c) skyrmion lattice and (d) ferromagnetic $+$ skyrmion.Reuse & Permissions
Figure 6
Phase diagram of various spin orders (Sk, skyrmion lattice; H, helical; F, ferromagnetic; $+$ , coexistence) in the plane of the magnetic field versus temperature calculated in the 36 by 36 lattice for the DM model (left panel) and our covariant model (right panel). The color indicates the total number of skyrmions. The helical phase and the ferromagnetic phase have no skyrmions and the skyrmion lattice phase has many skyrmions. In our model the Sk phase occupies more area in the high field region because the intrinsic easy plane anisotropy competes with the magnetic field to prevent the spins from being polarized.Reuse & Permissions

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