Effective spin model in momentum space: Toward a systematic understanding of multiple-$Q$ instability by momentum-resolved anisotropic exchange interactions

Abstract

Multiple-$Q$ magnetic states, such as a skyrmion crystal, become a source of unusual transport phenomena and dynamics. Recent theoretical and experimental studies clarify that such multiple-$Q$ states ubiquitously appear under different crystal structures in metals and insulators. Toward a systematic understanding of the formation of the multiple-$Q$ states in various crystal systems, in this theoretical study, we present a low-energy effective spin model with anisotropic exchange interactions in momentum space. We summarize specific six symmetry rules for nonzero symmetric and antisymmetric anisotropic exchange interactions in momentum space, which are regarded as an extension of Moriya's rule. According to the rules, we construct the effective spin model for tetragonal, hexagonal, and trigonal magnets with crystal- and momentum-dependent anisotropic exchange interactions based on magnetic representation analysis. Furthermore, we describe the origin of the effective anisotropic exchange interactions in itinerant magnets by perturbatively analyzing a multi-band periodic Anderson model with the spin-orbit coupling. We apply the effective spin model to an itinerant magnet in a $P6/mmm$ crystal and find various multiple-$Q$ states with a spin scalar chirality in the ground state. Our results provide a foundation of constructing effective phenomenological spin models for any crystal systems hosting the multiple-$Q$ states, which will stimulate further exploration of exotic multiple-$Q$ states in materials with the spin-orbit coupling.

27 pages, 15 figures, 7 tables

References in corpus (48)

• Spontaneous Skyrmion Ground States in Magnetic Metals
• A new class of chiral materials hosting magnetic skyrmions beyond room temperature
• Nanometric square skyrmion lattice in a centrosymmetric tetragonal magnet
• Skyrmion fractionalization and merons in chiral magnets with easy-plane anisotropy
• Skyrmion Qubits: A New Class of Quantum Logic Elements Based on Nanoscale Magnetization
• Fractional antiferromagnetic skyrmion lattice induced by anisotropic couplings
• Bose Hubbard Models with Synthetic Spin-Orbit Coupling: Mott Insulators, Spin Textures and Superfluidity
• Néel-type skyrmion lattice in tetragonal polar magnet VOSe${}_2$O${}_5$
• Antiferromagnetic skyrmion crystals: generation, topological Hall and topological spin Hall effect
• Meron, skyrmion, and vortex crystals in centrosymmetric tetragonal magnets
• Theory of helical spin crystals: phases, textures and properties
• Imaging the coupling between itinerant electrons and localised moments in the centrosymmetric skyrmion magnet GdRu2Si2
• Room-temperature antiskyrmions and sawtooth surface textures in a noncentrosymmetric magnet with $S_4$ symmetry
• Square skyrmion crystal in centrosymmetric itinerant magnets
• Topological spin crystals by itinerant frustration
• Multiple-Q Instability by (d-2)-dimensional Connections of Fermi Surfaces
• Nanometric skyrmion lattice from anisotropic exchange interactions in a centrosymmetric host
• Thermodynamically stable skyrmion lattice in tetragonal frustrated antiferromagnet with dipolar interaction
• Noncoplanar multiple-$Q$ spin textures by itinerant frustration: Effects of single-ion anisotropy and bond-dependent anisotropy
• Anomalous Hall resistivity and possible topological Hall effect in the EuAl${}_4$ antiferromagnet
• Zoology of multiple-Q spin textures in a centrosymmetric tetragonal magnet with itinerant electrons
• Unconventional Topological Hall Effect in Skyrmion Crystals Caused by the Topology of the Lattice
• Vortices, skyrmions, and chirality waves in frustrated Mott insulators with a quenched periodic array of impurities
• Multiple skyrmion crystal phases by itinerant frustration in centrosymmetric tetragonal magnets
• Field-Direction Sensitive Skyrmion Crystals in Cubic Chiral Systems: Implication to $4f$-Electron Compound EuPtSi
• Degeneracy Lifting of Néel, Bloch, and Anti-Skyrmion Crystals in Centrosymmetric Tetragonal Systems
• Charge-Density-Wave Order and Multiple Magnetic Transitions in Divalent Europium Compound EuAl${}_4$
• Skyrmion crystal and spiral phases in centrosymmetric bilayer magnets with staggered Dzyaloshinskii-Moriya interaction
• Mean-field approach for skyrmion lattice in hexagonal frustrated antiferromagnets
• Helicity locking of square skyrmion crystal in a centrosymmetric lattice system without vertical mirror symmetry
• Hedgehog-lattice spin texture in classical Heisenberg antiferromagnets on the breathing pyrochlore lattice
• Locking of skyrmion cores on a centrosymmetric discrete lattice: onsite versus offsite
• Spin excitation spectra in helimagnetic states: proper-screw, cycloid, vortex crystal, and hedgehog lattice
• Replica symmetry breaking in the RKKY skyrmion crystal system
• Generic nearest-neighbour kagome model: XYZ & Dzyaloshinskii-Moriya, and comparison to pyrochlore
• Controlled Creation of Quantum Skyrmions
• Skyrmion crystal in the RKKY system on the two-dimensional triangular lattice
• Thermodynamic insights into the intricate magnetic phase diagram of EuAl${}_4$
• Phase transitions between helices, vortices, and hedgehogs driven by spatial anisotropy in chiral magnets
• Magnetic field-temperature phase diagrams for multiple-$Q$ magnetic orderings: Exact steepest descent approach to long-range interacting spin systems
• Double-$Q$ Chiral Stripe in the $d$-$p$ Model with Strong Spin-Charge Coupling
• New Microscopic Magnetic Hamiltonian for Exotic Spin Textures in Metals
• Essential model parameters for nonreciprocal magnons in multisublattice systems
• Antiferromagnetic Skyrmions and Skyrmion Density Wave in Rashba Hund's Insulator
• Phase degree of freedom and topology in multiple-$Q$ spin textures
• Magnetic Chirality Induced from Ruderman-Kittel-Kasuya-Yosida Interaction at an Interface of a Ferromagnet/Heavy Metal Heterostructure
• Skyrmion crystal under $D_{3h}$ point group: Role of out-of-plane Dzyaloshinskii-Moriya interaction
• Antiskyrmions and Bloch Skyrmions in Magnetic Dresselhaus Metals

Cited by in corpus (28)

• Stabilization mechanisms of magnetic skyrmion crystal and multiple-$Q$ states based on momentum-resolved spin interactions
• Widely-sweeping magnetic field-temperature phase diagrams for skyrmion-hosting centrosymmetric tetragonal magnets
• Distorted triangular skyrmion lattice in a noncentrosymmetric tetragonal magnet
• Single helicity of the triple-$q$ triangular skyrmion lattice state in cubic chiral helimagnet EuPtSi
• Anisotropic spin model and multiple-$Q$ states in cubic systems
• Ferrochiral, antiferrochiral, and ferrichiral skyrmion crystals in an itinerant honeycomb magnet
• Checkerboard bubble lattice formed by octuple-period quadruple-$Q$ spin density waves
• Hybrid skyrmion and anti-skyrmion phases in polar $C_{4v}$ systems
• Magnetic bubble crystal in tetragonal magnets
• Antiferro skyrmion crystal phases in a synthetic bilayer antiferromagnet under an in-plane magnetic field
• Anisotropic skyrmion crystal on a centrosymmetric square lattice under an in-plane magnetic field
• Chern insulating state with double-$Q$ ordering wave vectors at the Brillouin zone boundary
• Orbital moiré and quadrupolar triple-q physics in a triangular lattice
• Spin-lattice couplings in $3d$ ferromagnets: analysis from first-principles
• Bimeron Crystals by a Linearly Polarized AC Electric Field in Frustrated Magnets
• Magnetic instability under ferroaxial moment
• Four-Spin Interactions as a Route to Multiple-Q Topological Magnetic Order
• Field-induced transformation between triangular and square skyrmion crystals in a tetragonal polar magnet
• Three-sublattice antiferro-type and ferri-type skyrmion crystals in centrosymmetric magnets
• Helicity Selection of the Cycloidal Order in Noncentrosymmetric EuIrGe${}_3$
• Field-direction-dependent skyrmion crystals in noncentrosymmetric cubic magnets: A comparison between point groups $(O,T)$ and $T_{\mathrm{d}}$
• Square skyrmion lattice in multiorbital $f$-electron systems
• Emergent topological quasiparticle kinetics in constricted nanomagnets
• Double-$Q$ and quadruple-$Q$ instabilities at low-symmetric ordering wave vectors under tetragonal symmetry
• Spin-orbit coupled periodic Anderson model: Kondo-Dirac semimetal and orbital-selective antiferromagnetic semimetal
• Effects of crystal field and momentum-based frustrated exchange interactions on multiorbital square skyrmion lattice
• Skyrmion generation via Laguerre-Gaussian beam irradiation in frustrated magnets
• Symmetry Rules on Multipole Interactions under Crystallographic Point Groups and Application to Multiple-$Q$ Multipole States