Phase degree of freedom and topology in multiple-$Q$ spin textures

Abstract

A periodic array of topological spin textures, such as skyrmions and hedgehogs, is called the multiple-$Q$ spin texture, as it is represented by a superposition of multiple spin density waves. Depending on the way of superposition, not only the magnetic but also the topological properties are modified, leading to a variety of quantum transport and optical phenomena caused by the emergent electromagnetic fields through the Berry phase. Among others, the phase degree of freedom of the superposed waves is potentially important for such modifications, but its effect has not been fully elucidated thus far. Here we perform systematic theoretical analyses of magnetic and topological properties of the multiple-$Q$ spin textures with the phase degree of freedom. By introducing a hyperspace with an additional dimension corresponding to the phase degree of freedom, we establish a generic framework to deal with the phase shift in the multiple-$Q$ spin textures. Applying the framework to the two-dimensional 3$Q$ spin textures, we clarify the complete topological phase diagram while changing the phase and magnetization, which depends on the types of the superposed waves. We also study the three-dimensional 4$Q$ spin textures and clarify even richer topological phase diagrams. In particular, we find novel topological phase transitions associated with the previously unidentified Dirac strings on which the hedgehogs and antihedgehogs cause pair creation and fusion. Moreover, we demonstrate that phase shifts are caused by an external magnetic field in both 3$Q$ and 4$Q$ cases by analyzing the numerical data in the previous studies. Our results illuminate the topological aspects of the skyrmion and hedgehog lattices with the phase degree of freedom, which would be extended to other multiple-$Q$ textures and useful for the exploration of topologically nontrivial magnetic phases and exotic quantum phenomena.

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• Triple-$\mathcal{Q}$ partial magnetic orders induced by quadrupolar interactions: Triforce order scenario for UNi${}_4$B
• Skyrmion and vortex crystals in the Hubbard model
• Unconventional Surface State Pairs in a High-Symmetry Lattice with Anti-ferromagnetic Band-folding
• Hidden Topological Transitions in Emergent Magnetic Monopole Lattices
• Phase shifts, band geometry and responses in triple-Q charge and spin density waves
• Current-induced motion of nanoscale magnetic torons over the wide range of the Hall angle
• Theory of collective excitations in the quadruple-$Q$ magnetic hedgehog lattices
• Engineering a skyrmion crystal in ferromagnetic/antiferromagnetic bilayers free from the DM interaction
• Field-direction-dependent skyrmion crystals in noncentrosymmetric cubic magnets: A comparison between point groups $(O,T)$ and $T_{\mathrm{d}}$
• Single-$q$ Cycloid and Double-$q$ Vortex Lattices in Layered Magnetic Semimetal EuAg${}_4$Sb${}_2$
• A $C^{\ast }$-algebraic view on the interaction of real- and reciprocal space topology in skyrmion crystals
• Stacking-dependent nonreciprocal transport in magnetic skyrmions