Abstract
The magnetic skyrmion crystal is a periodic array of a swirling topological spin texture. Since it is regarded as an interference pattern by multiple helical spin density waves, the texture changes with the relative phase shifts among the constituent waves. Although such a phase degree of freedom is relevant to not only magnetism but also transport properties, its effect has not been elucidated thus far. We here theoretically show that a phase shift in the skyrmion crystals leads to a tetraaxial vortex crystal and a meronantimeron crystal, both of which show a staggered pattern of the scalar spin chirality and give rise to nonreciprocal transport phenomena without the spinorbit coupling. We demonstrate that such a phase shift can be driven by exchange interactions between the localized spins, thermal fluctuations, and longrange chirality interactions in spincharge coupled systems. Our results provide a further diversity of topological spin textures and open a new field of emergent electromagnetism by the phase shift engineering.
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Introduction
The skyrmion is a topological configuration of a continuous field. Although it was originally proposed to explain hadrons in the particle theory^{1,2}, it has turned out to be realized in various forms in condensed matter physics^{3}. One possible realization was discovered in magnets, in the form of skyrmionlike magnetic textures^{4,5,6,7,8}. The magnetic skyrmions often exist in their stable form, the socalled skyrmion crystal (SkX), which is a periodic array of the particlelike magnetic skyrmions. Importantly, such a SkX is approximately expressed as a superposition of multiple helical spin density waves, and hence, it can be regarded as an interference pattern by the multiple helices. It has attracted enormous attention since the swirling magnetic texture generates an emergent magnetic field through the Berry phase mechanism and results in peculiar transport phenomena, such as the topological Hall effect^{3,9,10}.
Similar to an isolated skyrmion, the SkX is characterized by three quantities: skyrmion number, vorticity, and helicity^{3}. However, as the SkX is regarded as an interference pattern, it has another degree of freedom, which has been overlooked in the previous researches, the phases of the constituent waves. This is exemplified for three scalar waves in Fig. 1a, b, where a phase shift in one of the three waves leads to a different interference pattern with different symmetry. Such a phase degree of freedom exists in all the interference phenomena, except for linearly independent waves in continuous space for which a phase shift is equivalent to a spatial translation. The SkX appears not in a continuous field but for spins on a discrete lattice, which leads to a further variety of the interference patterns by the discretization, even for the linearly independent waves. A shift of the relative phases changes not only magnetic textures but also emergent magnetic fields, and hence, transport properties, but such an interesting possibility has not been elucidated thus far.
In this study, we theoretically unveil the effect of phase shifts in the SkX and propose how to control the phase degree of freedom. Considering an itinerant electron model on a triangular lattice, we show that the SkX turns into a tetraaxial vortex crystal (TVX) or a meronantimeron crystal (MAX) by a phase shift of π/2. The phaseshifted states have distinct properties from the SkX: The SkX exhibits a net scalar chirality leading to the topological Hall effect, while the TVX and MAX exhibit a staggered one that does not lead to the topological Hall effect, but induces nonreciprocal transport phenomena that do not require the spinorbit coupling. We find that such a phase shift can be caused by several different mechanisms, such as exchange interactions between the localized spins, thermal fluctuations, and longrange chirality interactions. Our results open another route to a further variety of magnetic textures which have been overlooked in skyrmionhosting materials.
Results
Let us start by classifying noncoplanar spin textures according to the type of constituent waves and the relative phases. First, we consider a superposition of spiral spin textures represented by \({{{{{{{{\bf{S}}}}}}}}}_{i}^{{{{{{{{\rm{spiral}}}}}}}}}=\mathop{\sum }\nolimits_{\nu = 1}^{3}\left(\sin {{{{{{{{\mathcal{Q}}}}}}}}}_{\nu }\cos {\phi }_{\nu },\sin {{{{{{{{\mathcal{Q}}}}}}}}}_{\nu }\sin {\phi }_{\nu },\cos {{{{{{{{\mathcal{Q}}}}}}}}}_{\nu }\right)\), where \({{{{{{{{\mathcal{Q}}}}}}}}}_{\nu }={{{{{{{{\bf{Q}}}}}}}}}_{\nu }\cdot {{{{{{{{\bf{r}}}}}}}}}_{i}+{\theta }_{\nu }\) and \({\phi }_{\nu }=\frac{2}{3}\pi (\nu 1)\); Q_{ν} and θ_{ν} are the wave vector and the phase of the νth spiral, respectively, and r_{i} is the position vector for site i. In the following analyses, as an archetype, we consider a twodimensional triangular lattice system with threefold rotationally symmetric wave vectors with spiral pitch Q: Q_{1} = (Q, 0), \({{{{{{{{\bf{Q}}}}}}}}}_{2}=(Q/2,\sqrt{3}Q/2)\), and \({{{{{{{{\bf{Q}}}}}}}}}_{3}=(Q/2,\sqrt{3}Q/2)\) satisfying ∑_{ν}Q_{ν} = 0. The spin texture \({{{{{{{{\bf{S}}}}}}}}}_{i}^{{{{{{{{\rm{spiral}}}}}}}}}\) is modulated by shifting the phases θ_{ν}. We demonstrate the situation for θ_{1} = θ_{2} = θ_{3} by changing the total phase Θ = ∑_{ν}θ_{ν}. Figure 1c, d displays the spin textures with Θ = 0 and π/2. In each figure, the upper and lower planes show the textures of spin and scalar spin chirality, respectively; the latter is defined by χ_{R} = S_{i} ⋅ (S_{j} × S_{k}), where R represents the position vector at the center of a triangle with sites i, j, k in the counterclockwise order. The case with Θ = 0 in Fig. 1c is a periodic array of skyrmions with the skyrmion number of one (n_{sk} = 1), which we call the SkX1. It retains sixfold rotational symmetry in both spin and chirality, and has a nonzero net chirality leading to the topological Hall effect. Meanwhile, the case with Θ = π/2 in Fig. 1d is a staggered arrangement of merons and antimerions (half skyrmions and antiskyrmions^{11}), which we call the MAX. In this state, the rotational symmetry is reduced to threefold. Moreover, the meron and antimeron carry the skyrmion number of +1/2 and −1/2, respectively, and hence, the total skyrmion number is zero in the MAX; accordingly, the net value of χ_{R}, \({\chi }^{\rm total}=\frac{1}{N}{\sum }_{{{{{{{{\bf{R}}}}}}}}}{\chi }_{{{{{{{{\bf{R}}}}}}}}}\) where N is the number of lattice sites, also vanishes and the MAX does not show the topological Hall effect. We note that the MAX was proposed as a candidate for the unidentified magnetic state next to the SkX1 found in a triangularlattice magnet Gd_{2}PdSi_{3}^{11}.
Next, we consider a superposition of sinusoidal waves^{12,13,14}, which is represented by \({{{{{{{{\bf{S}}}}}}}}}_{i}^{\sin }=\left(\cos {{{{{{{{\mathcal{Q}}}}}}}}}_{1},\cos {{{{{{{{\mathcal{Q}}}}}}}}}_{2},\cos {{{{{{{{\mathcal{Q}}}}}}}}}_{3}\right)\). Similar to \({{{{{{{{\bf{S}}}}}}}}}_{i}^{{{{{{{{\rm{spiral}}}}}}}}}\), different Θ gives different spin and chirality textures, as shown in Fig. 1e, f (the spin frame is rotated for better visibility). The spin texture with Θ = 0 in Fig. 1e is the other SkX called the SkX2, in which each skyrmion has the skyrmion number of two (n_{sk} = 2). In this state, while the spin texture has threefold rotational symmetry, the chirality χ_{R} is sixfold and the net value χ^{total} is nonzero, similar to the SkX1 in Fig. 1c. The phase shift by π/2 lowers the symmetry from sixfold to threefold, as shown in Fig. 1f; χ_{R} has a staggered configuration with no net scalar chirality, similar to the MAX in Fig. 1d. In this state, the spin texture is given by a periodic array of four types of vortices; the vortex axes, which are defined by the vorticity for xy, yz, and zx components of spins point to four corners of the tetrahedron (see Supplementary Information). Hence, we call the Θ = π/2 state the TVX.
Thus, in both cases, the phase shift changes not only the spin texture but also the symmetry and topology. In particular, the net value of the scalar chirality χ^{total} is sensitively dependent on Θ; the two types of SkXs at Θ = 0 have nonzero values and cause the topological Hall effect, while the MAX and TVX at Θ = π/2 have no net value and do not show the topological Hall effect. Interestingly, however, the breaking of sixfold rotational symmetry in χ_{R} in the MAX and TVX leads to Fermi surface deformations as discussed below, which can induce directiondependent nonreciprocal transport phenomena without the spinorbit coupling.
The optimal values of the phases θ_{ν} will be determined by multiple factors, such as lattice geometry and interactions between the spins. In the previous studies, the SkXs with Θ = 0 are stabilized, e.g., by the Dzyaloshinskii−Moriya (DM)^{6,15}, fourspin^{16,17,18,19}, frustrated^{20,21,22}, and spincharge interactions^{14,23,24,25} on various lattices. The key question addressed here is what is the relevant parameter to cause a phase shift that leads to switching of magnetic, topological, and transport properties. In the following, we unveil three different mechanisms for such a phase shift, by taking an archetypal model for itinerant magnets hosting SkXs, the Kondo lattice model on a triangular lattice where both the SkX1 and SkX2 appear in the ground state (see “Methods”).
We first demonstrate that a phase shift can be caused by introducing the exchange interactions between the localized spins described by \({{{{{{{{\mathcal{H}}}}}}}}}^{{{{{{{{\rm{loc}}}}}}}}}={\sum }_{ij}{J}_{ij}{{{{{{{{\bf{S}}}}}}}}}_{i}\cdot {{{{{{{{\bf{S}}}}}}}}}_{j}\) to the original Kondo lattice Hamiltonian \({{{{{{{\mathcal{H}}}}}}}}\). Considering the first, second, and thirdneighbor interactions, J_{1}, J_{2}, and J_{3} for J_{ij}, respectively, we perform a variational calculation to determine the groundstate phase diagram of the Hamiltonian \({{{{{{{\mathcal{H}}}}}}}}+{{{{{{{{\mathcal{H}}}}}}}}}^{{{{{{{{\rm{loc}}}}}}}}}\) at zero field (see “Methods”). Figure 2a, b shows the results on the J_{1}−J_{2} and J_{1}−J_{3} planes. While the SkX2 is stable in the wide range of parameters, we find two topological phase transitions: One is to the TVX while increasing J_{1} and decreasing J_{2} (increasing J_{3}) and the other is to the SkX1 while decreasing J_{1} and decreasing J_{2} (increasing J_{3}), as shown in Fig. 2a (Fig. 2b). The former transition from the SkX2 to the TVX is accompanied by the phase shift with Θ = π/2.
The phase transitions among these three states are understood from the higher harmonics in the spin structure. We show the spin structure factors in momentum (q) space for the SkX2, TVX, and SkX1 in the left panels of Fig. 2c (see “Methods”). Although the dominant peaks appear at Q_{1}, Q_{2}, and Q_{3} commonly in the three phases, subdominant peaks are found at different q among them: \({{{{{{{\bf{q}}}}}}}}={{{{{{{{\bf{Q}}}}}}}}}_{\nu }{{{{{{{{\bf{Q}}}}}}}}}_{\nu ^{\prime} }\) (\(\nu \,\ne\, \nu ^{\prime}\)) in the SkX2, q = 3Q_{ν} in the TVX, and q = 0 in the SkX1. Thus, considering that the Fourier transform of \({{{{{{{{\mathcal{H}}}}}}}}}^{{{{{{{{\rm{loc}}}}}}}}}\) is written as ∑_{q}J_{q}S_{q} ⋅ S_{−q}, the (anti)ferromagnetic interactions giving J_{0} < 0 (\({J}_{3{{{{{{{{\bf{Q}}}}}}}}}_{\nu }} < \, 0\)) tend to prefer the SkX1 (TVX).
It is noteworthy that the different superpositions of the constituent waves also give rise to the different peak positions in the qresolved scalar chirality shown in the right panels of Fig. 2c (see “Methods”). As mentioned above, both the SkX2 and SkX1 have the dominant peak at q = 0 reflecting nonzero χ^{total}, while the TVX has the dominant peaks at 2Q_{ν} with equal intensities and no weight at q = 0.
The second mechanism to cause the phase shift is thermal fluctuations. We study the finitetemperature behavior of the Kondo lattice Hamiltonian \({{{{{{{\mathcal{H}}}}}}}}\) by performing the Langevin dynamics simulations with the kernel polynomial method (see “Methods”)^{26,27,28}. Figure 3a shows the results at zero magnetic field H = 0 where the ground state is the SkX2^{14}. We find two phase transitions at T_{1} ≃ 0.0055 and T_{2} ≃ 0.009. The transition at T_{1} is characterized by the onset of χ_{0}, suggesting that the SkX2 remains stable up to T_{1}. Note that the true magnetic longrange order is limited to zero temperature in the present twodimensional system due to the Mermin−Wagner theorem^{29}; the state for 0 < T < T_{1} is a chiral spin liquid having the SkX2 spin texture with a finite correlation length (but much longer than the system size). The realspace spin configuration at T = 0 is shown in Fig. 3c, which corresponds to that in Fig. 1e. Meanwhile, the transition at T_{2} appears to be signaled by the onset of the higher harmonics \({\chi }_{2{{{{{{{{\bf{Q}}}}}}}}}_{\nu }}\) as plotted in Fig. 3a. A snapshot of the realspace spin configuration at T = 0.006 is shown in Fig. 3d, which well reproduces the spin texture for the TVX in Fig. 1f. From these results, we conclude that the low and intermediatetemperature phases are chiral spin liquids with the SkX2 and TVX spin textures, respectively, and the transition at T_{1} is associated with the phase shift of π/2 between them.
The appearance of the TVX at finite temperature is explained by an effective chirality interaction as follows. At the meanfield level, the entropic contributions are in general given in the form of n thorder magnetic interactions as \(T{\sum }_{{{{{{{{{\bf{q}}}}}}}}}_{1}\cdots {{{{{{{{\bf{q}}}}}}}}}_{n}}({{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{{\bf{q}}}}}}}}}_{1}}\cdot {{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{{\bf{q}}}}}}}}}_{2}})\cdots ({{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{{\bf{q}}}}}}}}}_{n1}}\cdot {{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{{\bf{q}}}}}}}}}_{n}})\delta ({{{{{{{{\bf{q}}}}}}}}}_{1}+\cdots +{{{{{{{{\bf{q}}}}}}}}}_{n})\)^{30,31}. Among them, the lowestorder contribution to the phase shift appears in the sixth order. By considering \({{{{{{{{\bf{S}}}}}}}}}_{i}^{\sin }\), the dominant entropic contribution is given as \(T{{{{{{{\rm{Re}}}}}}}}[({{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{{\bf{Q}}}}}}}}}_{1}}\cdot {{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{{\bf{Q}}}}}}}}}_{1}})({{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{{\bf{Q}}}}}}}}}_{2}}\cdot {{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{{\bf{Q}}}}}}}}}_{2}})({{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{{\bf{Q}}}}}}}}}_{3}}\cdot {{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{{\bf{Q}}}}}}}}}_{3}})]=T{{{{{{{\rm{Re}}}}}}}}[{\{{{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{{\bf{Q}}}}}}}}}_{1}}\cdot ({{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{{\bf{Q}}}}}}}}}_{2}}\times {{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{{\bf{Q}}}}}}}}}_{3}})\}}^{2}]\propto T\cos 2{{\Theta }}\). Thus, the sixthorder entropic term, which has the form of chiralitychirality interactions, depends on Θ and tends to stabilize the TVX at finite temperature.
Next, let us discuss the case of SkX1. Figure 3b represents the results under a magnetic field, where the ground state is the SkX1. Similar to the zerofield case in Fig. 3a, two phase transitions occur at \({T}_{1}^{\prime}\simeq 0.0045\) and \({T}_{2}^{\prime}\simeq 0.0085\). The lowtemperature state below \({T}_{1}^{\prime}\) is the SkX1 (with quasilongrange order in the xy components), whose spin configuration is shown in Fig. 3e^{14}. Recalling the phase shift of π/2 from the SkX2 to the TVX in the zerofield case, one may expect that the SkX1 changes into the MAX by raising temperature, but we find that the intermediate phase for \({T}_{1}^{\prime} \, < \, T \, < \, {T}_{2}^{\prime}\) is a different superposition of three sinusoidal waves which has χ^{total} = 0. This is because the Zeeman energy gain in the MAX is not sufficient to overcome the obtained state. We note that the threefold rotational symmetry is broken in the intermediate phase; hence, we call this phase the anisotropic 3Q state (see Supplementary Information). This phase transition from the SkX1 to the anisotropic 3Q state is also accounted for by the sixthorder entropic contribution, similar to that from the SkX2 to the TVX at zero field, as will be shown below.
We display the Fermi surfaces in the SkX2, TVX, and SkX1 in the inset of Fig. 3c–e, respectively. The Fermi surface in the TVX in Fig. 3d is threefold rotationally symmetric, meaning the breaking of the sixfold rotational symmetry of the triangular lattice, as expected from the above discussion. This leads to a nonreciprocal transport in itinerant electrons. Notably, there appear threefold rotationally symmetric Fermi surfaces even in the SkX2 in Fig. 3c (very weakly broken) and SkX1 in Fig. 3e. This is not due to the shift of Θ but by phaselocking at (θ_{1}, θ_{2}, θ_{3}) = (π/3, −π/3, 0) (any permutation is allowed) so that the skyrmion cores avoid the lattice sites (the values of θ_{ν} depend on Q; see Supplementary Information). Thus, the results indicate that the individual phase θ_{ν}, as well as the total phase Θ, are relevant in the actual discrete lattice systems.
The third mechanism is higherorder spin interactions, inferred from the above entropic mechanism. In general, the kinetic motion of itinerant electrons induces effective spin interactions, which can be explicitly derived by perturbation expansion in terms of the spincharge coupling in the Kondo lattice model. The lowestorder contribution is a bilinear interaction called the Ruderman−Kittel−Kasuya−Yosida interaction^{32,33,34}, and the next fourthorder biquadratic interaction was shown to be relevant to stabilize the SkXs^{35}. We here consider a higherorder sixspin contribution given by \(\tilde{L}[{\{{{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{{\bf{Q}}}}}}}}}_{1}}\cdot ({{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{{\bf{Q}}}}}}}}}_{2}}\times {{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{{\bf{Q}}}}}}}}}_{3}})\}}^{2}+{{{{{{{\rm{H.c.}}}}}}}}]\) (see “Methods” and Supplementary Information). It is worthy to note that this has a similar form to the above sixspin entropic term, and it is the lowestorder contribution whose energy depends on Θ under ∑_{ν}Q_{ν} = 0 in the perturbation expansion. This chirality interaction is different from those discussed in the previous studies that stabilize noncoplanar spin states but appear to be irrelevant to the phase shift^{36,37}.
To clarify the effect of the chirality interaction, we investigate the groundstate phase diagram of the effective spin Hamiltonian \({{{{{{{{\mathcal{H}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}\) by variational calculations and the simulated annealing (see “Methods”). We find that the SkX2 gives the lowest energy for 0 ≤ L < 1, while the TVX does for larger L, as shown in Fig. 4a. The optimal θ_{ν} are obtained as (θ_{1}, θ_{2}, θ_{3}) = (π/3, −π/3, 0) for the former and (π/3, π/6, 0) for the latter (any permutation is allowed), both of which are consistent with those in the Kondo lattice model above. Thus, the chirality interaction prefers the Θ = π/2 states to the Θ = 0 ones, namely, it brings about the phase shift in SkXs.
Furthermore, we find that the value of L necessary for the phase shift can be largely reduced by combining the first mechanism by the exchange interactions for the localized spins. Figure 4b shows the groundstate phase diagram of the Hamiltonian \({{{{{{{{\mathcal{H}}}}}}}}}^{{{{{{{{\rm{eff}}}}}}}}}+{{{{{{{{\mathcal{H}}}}}}}}}^{{{{{{{{\rm{loc}}}}}}}}}\) with J_{1} only obtained by the simulated annealing. While increasing J_{1}, the phase boundary between the SkX2 and TVX is shifted to a smaller L rapidly. We note that the SkX2 turns into the SkX1 while decreasing J_{1}, whose tendency is similar to the results in Fig. 2a, b. In addition, we obtain the phase transition from the SkX1 to the anisotropic 3Q state while increasing L, similar to the result in Fig. 3b, which also indicates that L plays a similar role to the temperature.
Discussion
We have theoretically guided a new direction of exploring further exotic magnetic states beyond the SkXs by using the phase degree of freedom among the constituent spin density waves. The phase shift turns the SkXs into other states represented by the TVX, which are characterized by staggered emergent magnetoelectric fields and breaking of the lattice rotational symmetry. We unveiled three microscopic mechanisms which drive such a phase shift in itinerant magnets: the exchange interactions between the localized spins, thermal fluctuations, and the longrange chirality interactions.
Our results indicate that the skyrmionbased physics arising from nonzero net chirality can be switched on and off by changing the relative phases among the constituent waves. Furthermore, the lowering of the rotational symmetry by the phase shift induces nonreciprocal transport even in centrosymmetric systems without the spinorbit coupling. These features open a new direction of emergent electromagnetism by the phase shift engineering. This would be realized in centrosymmetric skyrmionhosting materials where the multiplespin interactions rooted in the spincharge coupling might play an important role^{11,38,39}. While it is not straightforward to identify the phase shift by diffraction techniques such as the neutron scattering and the resonant xray scattering, our findings suggest that the angleresolved photoemission spectroscopy and transport measurements will give good probes to detect the phase shift and new phases like the TVX.
Furthermore, the concept of the phase shift is not limited to the field of skyrmionics but ubiquitously useful for a variety of topological spin crystals, such as vortex crystals and hedgehog crystals, since they are characterized by the multipleQ spin density waves. Our results suggest that the overlooked phase degree of freedom can induce further interesting topological phase transitions, unconventional electronic structures, topological properties, and conductive phenomena, which will stimulate future exploration of functional spintronics materials in both experiment and theory.
Methods
Kondo lattice model
We consider the Kondo lattice model on a triangular lattice, whose Hamiltonian is given by
The first term represents the kinetic energy of itinerant electrons, where \({c}_{i\sigma }^{{{{\dagger}}} }\) (c_{iσ}) is a creation (annihilation) operator of an itinerant electron at site i and spin σ. The second term represents the exchange coupling between itinerant electron spins \({{{{{{{{\bf{s}}}}}}}}}_{i}=(1/2){\sum }_{\sigma ,\sigma ^{\prime} }{c}_{i\sigma }^{{{{\dagger}}} }{{{{{{{{\boldsymbol{\sigma }}}}}}}}}_{\sigma \sigma ^{\prime} }{c}_{i\sigma ^{\prime} }\) [σ = (σ^{x}, σ^{y}, σ^{z}) is the vector of Pauli matrices] and classical localized spins S_{i} with ∣S_{i}∣ = 1. The third term represents the Zeeman coupling to an external magnetic field H. In the calculations, we take the model parameters common to those in ref. ^{14}: the nearest and thirdneighbor hoppings, t_{1} = 1 and t_{3} = −0.85, respectively, J_{K} = 1, and the chemical potential μ =−3.5, which gives the characteristic wave vectors at Q_{ν} (ν = 1, 2, 3) with Q = π/3 in the main text (we take the lattice constant unity). In this parameter set, the ground state at zero field becomes the SkX2 in Fig. 1e, and that in a field becomes the SkX1 in Fig. 1c^{14}. Note that the SkX2 is stable for other values of Q_{ν} while changing the hopping parameters and the electron filling^{14}; our arguments on the phase shift in the main text are not limited to Q = π/3 but can be applied to such other cases. To identify the spin and chirality structure, we compute the spin structure factor
and the chirality structure factor
respectively, where μ = (u, d) represent upward and downward triangles, respectively.
Variational calculation for the Kondo lattice model
In the variational calculations in Fig. 2a, b, we compare the energy of the following spin textures as the variational states: the triple spiral and sinusoidal crystals in Fig. 1c–f while varying three θ_{ν} under the constraint ∣S_{i}∣ = 1 at each site, the singleQ spiral state characterized by \({{{{{{{{\bf{S}}}}}}}}}_{i}=(\cos {{{{{{{\bf{q}}}}}}}}\cdot {{{{{{{{\bf{r}}}}}}}}}_{i},\sin {{{{{{{\bf{q}}}}}}}}\cdot {{{{{{{{\bf{r}}}}}}}}}_{i},0)\) where q = (0, 0), (Q_{1}, 0), (4π/3, 0) denoted as 1Q K and \((0,2\pi /\sqrt{3})\) denoted as 1Q M, and the conical (1Q C) state characterized by \({{{{{{{{\bf{S}}}}}}}}}_{i}=(1/{N}_{m})(\cos {{{{{{{{\bf{Q}}}}}}}}}_{1}\cdot {{{{{{{{\bf{r}}}}}}}}}_{i},\sin {{{{{{{{\bf{Q}}}}}}}}}_{1}\cdot {{{{{{{{\bf{r}}}}}}}}}_{i},{a}_{z})\) where N_{m} is the normalization factor and a_{z} is the variational parameter. We assume Q = π/3 in the variational calculations, since it was shown that the spin states with Q = π/3 give the lowest grand potential by performing the unbiased Langevin dynamics simulations with the kernel polynomial method when the exchange interactions between the localized spins are zero^{14}. The phase diagram is obtained for the system size N = 96^{2} under the periodic boundary condition.
Finitetemperature calculation for the Kondo lattice model
We adopt the Langevin dynamics simulation with the kernel polynomial method^{27} to study the finitetemperature properties of the Kondo lattice model in Fig. 3. In the kernel polynomial method, we expand the density of states by up to 2000th order of the Chebyshev polynomials with 16^{2} random vectors. In the Langevin dynamics, we use a projected Heun scheme for 1000−5000 steps with the time interval Δτ = 2. The simulations are done for N = 60^{2}, 72^{2}, 96^{2}, and 120^{2} sites, and the thermal averages are taken for 100−800 samplings after the thermalization. In the main text, we show the results for N = 96^{2} and 120^{2}, as those for N ≥ 72^{2} are convergent within the error bars. The data at different temperatures are obtained independently starting from different random states. In the simulations, we compute the q components of the scalar chirality \({\chi }_{{{{{{{{\bf{q}}}}}}}}}=\sqrt{{S}_{\chi }({{{{{{{\bf{q}}}}}}}})/N}\) at q = 0, 2Q_{1}, 2Q_{2}, and 2Q_{3}, as plotted in Fig. 3a.
Effective spin model
An effective spin model, which is derived from the Kondo lattice model in equation (1), is given by^{35}
where \({{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{{\bf{Q}}}}}}}}}_{\nu }}=(1/\sqrt{N}){\sum }_{i}{{{{{{{{\bf{S}}}}}}}}}_{i}{e}^{i{{{{{{{{\bf{Q}}}}}}}}}_{\nu }\cdot {{{{{{{{\bf{r}}}}}}}}}_{i}}\). The first two terms describe bilinear and biquadratic interactions, which are derived by second and fourthorder perturbation expansions in terms of the spincharge coupling, respectively^{35}; J > 0 and \(\tilde{K}=K/N \; > \; 0\), and N denotes the number of sites. The J and K terms provide a minimal effective model for the Kondo lattice model, stabilizing the SkX2 at zero field (Fig. 3c) and the SkX1 at finite fields (Fig. 3e)^{35}. Meanwhile, the third term with \(\tilde{L}=L/{N}^{2}\) represents an interaction between the scalar spin chirality composed of \({{{{{{{{\bf{S}}}}}}}}}_{{{{{{{{{\bf{Q}}}}}}}}}_{\nu }}\) (see Supplementary Information).
Variational calculation and simulated annealing for the effective spin model
In the variational calculations in Fig. 4a, we compare the energy of the four states in Fig. 1c–f while varying three θ_{ν} under the constraint ∣S_{i}∣ = 1 at each site. The results are for J = 1 and K = 0.4 while changing L for the system with N = 96^{2} sites. In the simulated annealing in Fig. 4a, b, our simulations are carried out with the standard Metropolis local updates in real space, by reducing the temperature successively, from T = 0.1−1.0 to ≃ 10^{−4} with the cooling rate of 0.99995−0.99999. The final temperature is typically T = 10^{−4}. Each phase is identified by its spin and chirality configurations. We also start the simulations from the spin structures obtained at low temperatures to determine the phase boundaries between different magnetic states.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The codes used for this study are available from the corresponding author upon reasonable request.
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Acknowledgements
The authors thank R. Ozawa for the fruitful discussions. This research was supported by JSPS KAKENHI Grants Numbers JP18H04296 (JPhysics), JP18K13488, JP19K03752, JP19H01834, 19K03740, JP19H05822, JP19H05825, and JP21H01037, JST CREST (JPMJCR18T2), and JST PRESTO (JPMJPR1912 and JPMJPR20L8). This work was also supported by the Toyota Riken Scholarship. T.O. acknowledges support by the Endowed Project for Quantum Software Research and Education, The University of Tokyo (https://qsw. phys.s.utokyo.ac.jp/). Parts of the numerical calculations were performed in the supercomputing systems in ISSP, the University of Tokyo.
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S.H. and Y.M. conceived the project. S.H. performed the numerical and analytical calculations. All the authors interpreted the results and wrote the manuscript.
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Hayami, S., Okubo, T. & Motome, Y. Phase shift in skyrmion crystals. Nat Commun 12, 6927 (2021). https://doi.org/10.1038/s41467021270830
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DOI: https://doi.org/10.1038/s41467021270830
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