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Top view of the crystal structure for the monolayer CrGeTe3 . The honeycomb network is formed by the Cr atoms which are at the center of octahedrons. These Cr atoms carry effective spin 3/2. The abc and the xyz coordinate systems are shown on the bottom left and bottom right, respectively. The two-dimensional honeycomb lattice lies in the ab plane and c is perpendicular to the plane. The xyz coordinate systems are along the three Cr-Te bonds. In the abc coordinate system, x,y and z are given by x=−2/2,−6/6,3/3 , y=2/2,−6/6,3/3 , z=0,6/3,3/3 , respectively. The Kitaev interactions on the bonds in blue, green and red correspond to x,y and z Ising type, respectively.
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Skyrmions hold great promise in future spintronics applications since they are robust against local deformations. The meron, due to its topological equivalence to a half skyrmion, has been widely found to appear in pairs. Motivated by recent progresses in high-spin Kitaev magnets, here we investigate numerically a classical Kitaev-$\Gamma$ model wi...
Citations
... On the one hand, it has been proved that skyrmion can also appear in the centrosymmetric magnets with the help of other interactions [48][49][50][51][52][53][54][55][56][57][58][59], such as dipolar interaction [48,55], the single-ion anisotropy [49-54, 56, 57] or the bond-dependent interactions [52,60]. On the other hand, beyond skyrmion, there are many magnetic quasi-particles coming into the sight of researchers in recent years [38,61,62]. Whether there is topological magnon or even relevant phase transition exists in such spin textures is needed for further exploration. ...
... The topological magnon in noncollinear spin textures with a large magnetic unit cell at a weak magnetic field thus calls for an urgent study. It is revealed that the competition between the Kitaev and Γ interactions can generate many noncoplanar magnetic orders [30,41,[51][52][53][54][55][56][57][58], such as the (6+18) state [41,55,56], the nested zigzag-stripy order [54], and the C 3 -like [55] triple-meron crystal (TmX) [56]. Among them, the TmX is extremely alluring in that it has three merons within one magnetic unit cell and it occupies a large area in the phase diagram of the K-Γ model where K < 0 and Γ > 0 [55,56]. ...
... The topological magnon in noncollinear spin textures with a large magnetic unit cell at a weak magnetic field thus calls for an urgent study. It is revealed that the competition between the Kitaev and Γ interactions can generate many noncoplanar magnetic orders [30,41,[51][52][53][54][55][56][57][58], such as the (6+18) state [41,55,56], the nested zigzag-stripy order [54], and the C 3 -like [55] triple-meron crystal (TmX) [56]. Among them, the TmX is extremely alluring in that it has three merons within one magnetic unit cell and it occupies a large area in the phase diagram of the K-Γ model where K < 0 and Γ > 0 [55,56]. ...
... The topological magnon in noncollinear spin textures with a large magnetic unit cell at a weak magnetic field thus calls for an urgent study. It is revealed that the competition between the Kitaev and Γ interactions can generate many noncoplanar magnetic orders [30,41,[51][52][53][54][55][56][57][58], such as the (6+18) state [41,55,56], the nested zigzag-stripy order [54], and the C 3 -like [55] triple-meron crystal (TmX) [56]. Among them, the TmX is extremely alluring in that it has three merons within one magnetic unit cell and it occupies a large area in the phase diagram of the K-Γ model where K < 0 and Γ > 0 [55,56]. ...
... IV D 3. This leads to two, qualitatively different regions in parameter space, the ones where K and Γ have the same sign, and the ones where they have opposite signs. The ground states of the former regions are known exactly, whereas the ones of the latter are not fully understood [15,180,207,[214][215][216][217]. ...
... VI and VI), the bonds connecting the chains remain frustrated, which gives rise to a rich interplay of various complex phases. Figure 5 (ac) shows the classical phase diagrams for positive Γ and negative K, as proposed from various classical energy minimization approaches, including machine learning [180] (Fig. 5 (a)), Monte Carlo simulations [214,216] (Fig. 5 (b)), hybrid Monte Carlo and iterative variational minimizations [217] ( Fig. 5 (c)). Let us summarise the main features: i) In the region around the AFM Γ point, there is some consensus that a small negative K stabilizes a threefoldsymmetric order with 18 spin sublattices, whose stability region is varied across the different studies, see Fig. 5 (a-c). ...
... This state is dubbed 18C 3 order in Refs. [214,217], modulated S 3 ×Z 3 phase in Ref. [180], and triple-meron crystal in Ref. [216]. Applying the symmetries R x,y,z of Eq. (32) to the 18C 3 state delivers other ground states with 54 sublattices. ...
We review the recent advances and current challenges in the field of strong spin-orbit coupled Kitaev materials, with a particular emphasis on the physics beyond the exactly-solvable Kitaev spin liquid point. To that end, we give a comprehensive overview of the most relevant exchange interactions in $d^5$ and $d^7$ iridates and similar compounds, an exposition of their microscopic origin, and a systematic attempt to map out the most interesting correlated regimes of the multi-dimensional parameter space, guided by powerful symmetry and duality transformations as well as by insights from wide-ranging analytical and numerical studies. We also survey recent exciting results on quasi-1D models and discuss their relevance to higher-dimensional models. Finally, we highlight some of the key questions in the field as well as future directions.
... Meanwhile, the role played by the [001]-type and [111]-type SIAs in the spin-1 and spin-3/2 Kitaev honeycomb models has been studied extensively [42,43]. In the large-S limit, it is revealed that the SIA can stabilize an interesting triple-meron crystal consisting of three merons, leading to a finite topological number and a quantized topological Hall conductance [44]. These studies imply that Eq. (1) should also harbour a rich physics. ...
The Kitaev-type spin chains have been demonstrated to be fertile playgrounds in which exotic phases and unconventional phase transitions are ready to appear. In this work, we use the density-matrix renormalization group method to study the quantum phase diagram of a spin-1 Kitaev chain with a tunable negative single-ion anisotropy (SIA). When the strength of the SIA is small, the ground state is revealed to be a spin-nematic phase which escapes conventional magnetic order but is characterized by a finite spin-nematic correlation because of the breaking spin-rotational symmetry. As the SIA increases, the spin-nematic phase is taken over by either a dimerized phase or an antiferromagnetic phase through an Ising-type phase transition, depending on the direction of the easy axis. For large enough SIA, the dimerized phase and the antiferromagnetic phase undergo a ``Landau-forbidden" continuous phase transition, suggesting new platform of deconfined quantum critical point in spin-1 Kitaev chain.
The bond-dependent Kitaev interaction K is familiar in the effective spin model of transition
metal compounds with octahedral ligands. In this work, we find a peculiar non-coplanar magnetic
order can be formed with the help of K and next-nearest neighbor Heisenberg coupling J2 on the
triangular lattice. It can be seen as a miniature version of skyrmion crystal, since it has nine spins
and an integer topological number in a magnetic unit cell. The magnon excitations in such an order
are studied by the linear spin-wave theory. Of note is that the change in the relative size of J2 and K
produces topological magnon phase transitions although the topological number remains unchanged.
We also calculated the experimentally observable thermal Hall conductivity, and found that the signs
of thermal Hall conductivity will change with topological phase transitions or temperature changes in
certain regions.
The noncollinear spin textures provide promising avenues to stabilize exotic magnetic phases and excitations. They have attracted vast attention in the past decades due to their nontrivial band topology. Distinct from the conventional route of involving the Dzyaloshinskii-Moriya interaction in a honeycomb magnet, the interplay of bond-dependent Kitaev and Γ interactions, originating from the spin-orbit coupling and octahedra crystal field in real materials, has demonstrated to be another source to generate noncollinear spin textures with multiple spins in a magnetic unit cell. Notably, earlier works have revealed a triple-meron crystal (TmX) consisting of 18 spins in the frustrated Kitaev-Γ model. Aligning with previous efforts, here we attempt to identify that the TmX hosts several peculiar features with the help of the linear spin-wave theory. To begin with, the symmetric anisotropic exchanges are beneficial for the existence of nonreciprocal magnons, which are stabilized by an external magnetic field. Further, within the regime of TmX, successive topological phase transitions occur, accompanied by the changes of Chern number in value and thermal Hall conductivity in sign. In addition, the topological nature of magnons is also verified by the onset of chiral edge modes in a nanoribbon geometry. Our findings pave the way to study topological phenomena of noncollinear spin textures in potential Kitaev materials.
We review the recent advances and current challenges in the field of strong spin-orbit coupled Kitaev materials, with a particular emphasis on the physics beyond the exactly-solvable Kitaev spin liquid point. To this end, we present a comprehensive overview of the key exchange interactions in candidate materials with a specific focus on systems featuring effective $J_{\rm eff}\!=\!1/2$ magnetic moments. This includes, but not limited to, $5d^5$ iridates, $4d^5$ ruthenates and $3d^7$ cobaltates. Our exploration covers the microscopic origins of these interactions, along with a systematic attempt to map out the most intriguing correlated regimes of the multi-dimensional parameter space. Our approach is guided by robust symmetry and duality transformations as well as insights from a wide spectrum of analytical and numerical studies. We also survey higher spin Kitaev models and recent exciting results on quasi-one-dimensional models and discuss their relevance to higher-dimensional models. Finally, we highlight some of the key questions in the field as well as future directions.
We propose a theoretical framework to investigate elementary excitations at finite temperatures within a localized electron model that describes the interactions between multiple degrees of freedom, such as quantum spin models and Kugel-Khomskii models. Thus far, their excitation structures have been mainly examined using the linear flavor-wave theory, an SU(N) generalization of the linear spin-wave theory. This technique introduces noninteracting bosonic quasiparticles as elementary excitations from the ground state, thereby elucidating numerous physical phenomena, including excitation spectra and transport properties characterized by topologically nontrivial band structures. Nevertheless, the interactions between quasiparticles cannot be ignored in systems exemplified by S=1/2 quantum spin models, where strong quantum fluctuations are present. Recent studies have investigated the effects of quasiparticle damping at zero temperature in such models. In our study, extending this approach to the flavor-wave theory for general localized electron models, we construct a comprehensive method to calculate excitation spectra with the quasiparticle damping at finite temperatures. We apply our method to the Kitaev model under magnetic fields, a typical example of models with topologically nontrivial magnon bands. Our calculations reveal that chiral edge modes undergo significant damping in weak magnetic fields, amplifying the damping rate by the temperature increase. This effect is caused by collisions with thermally excited quasiparticles. Since our approach starts from a general Hamiltonian, it will be widely applicable to other localized systems, such as spin-orbital coupled systems derived from multi-orbital Hubbard models in the strong-correlation limit.
Kitaev-type spin chains have been demonstrated to be fertile playgrounds in which exotic phases and unconventional phase transitions are ready to appear. In this work, we use the density-matrix renormalization-group method to study the quantum phase diagram of a spin-1 Kitaev chain with a tunable negative single-ion anisotropy (SIA). When the strength of the SIA is small, the ground state is revealed to be a spin-nematic phase, which escapes conventional magnetic order but is characterized by a finite spin-nematic correlation because of the breaking spin-rotational symmetry. As the SIA increases, the spin-nematic phase is taken over by either a dimerized phase or an antiferromagnetic phase through an Ising-type phase transition, depending on the direction of the easy axis. For large enough SIA, the dimerized phase and the antiferromagnetic phase undergo a “Landau-forbidden” continuous phase transition, suggesting new platform of deconfined quantum critical point in spin-1 Kitaev chain.
