Nonreciprocal Transport Driven by Noncoplanar Magnetic Ordering with Meron–Antimeron Spin Textures

Abstract

Noncoplanar spin textures give rise not only to unusual magnetic structures but also to emergent electromagnetic responses stemming from scalar spin chirality, such as the topological Hall effect. In this study, we theoretically investigate nonreciprocal transport phenomena induced by noncoplanar magnetic orderings through microscopic model analyses. By focusing on meron–antimeron spin textures that exhibit local scalar spin chirality while maintaining vanishing global chirality, we demonstrate that the electronic band structure becomes asymmetrically modulated, which leads to the emergence of nonreciprocal transport. The present mechanism arises purely from the noncoplanar magnetic texture itself and requires neither net magnetization nor relativistic spin–orbit coupling. We further discuss the potential relevance of our findings to the compound Gd₂PdSi₃, which has been suggested to host a meron–antimeron crystal phase at low temperatures.

1. Introduction

Noncoplanar spin textures have garnered significant attention in condensed matter physics owing to their ability to generate emergent electromagnetic responses. These complex spin structures are stabilized through a variety of mechanisms, as extensively discussed in recent studies [1,2]. The microscopic key ingredient of these emergent electromagnetic responses lies in the scalar spin chirality, which is defined as a triple scalar product of spins, i.e., ${\chi }_{ijk}={\mathit{S}}_i·({\mathit{S}}_j\times {\mathit{S}}_k)$, where ${\mathit{S}}_i$ denotes the spin at site i, which naturally arises from noncoplanar spin textures [3,4,5,6]. Acting as an effective magnetic field on conduction electrons, the scalar spin chirality enables various ferromagnetic-structure-driven phenomena in systems without net magnetization or relativistic spin–orbit coupling. A prominent example is the topological Hall effect [7,8,9,10,11,12,13,14,15], which has been experimentally observed in materials hosting noncoplanar spin textures, such as skyrmion crystals [16,17,18,19,20,21,22] and other complex chiral spin states [23,24,25,26].

Previous investigations have primarily focused on noncoplanar spin textures accompanying a nonzero net scalar spin chirality, as these configurations naturally support the topological Hall effect. However, another class of noncoplanar spin textures exists that is characterized by finite local scalar spin chirality but zero net chirality over the entire system. A representative example is the meron–antimeron crystal, which is obtained by shifting a relative phase among the constituent spin density waves in a skyrmion crystal [27,28]; a meron corresponds to a topological spin texture with half-integer skyrmion number, while an antimeron is its counterpart with the opposite skyrmion number [29,30,31,32,33]. When arranged periodically, these textures form meron–antimeron crystals with vanishing global scalar spin chirality but finite local chirality. Such meron–antimeron crystals have been suggested or observed in compounds including Gd₂PdSi₃ [27], Co₈Zn₉Mn₃ [34], and GdRu₂Ge₂ [35].

Although the meron–antimeron crystal does not give rise to a topological Hall effect due to its zero net scalar spin chirality, it can exhibit a different transport phenomenon, nonlinear nonreciprocal transport, wherein the resistance depends on the direction of current flow without requiring conventional semiconducting junctions [36,37,38,39,40,41,42]. Recent studies have shown that such nonreciprocity can emerge when the spatial distribution of the local scalar spin chirality breaks both the spatial inversion and time-reversal symmetries [43,44]. This mechanism is fundamentally distinct from nonreciprocity induced by the interplay between relativistic spin–orbit coupling and the magnetic field in noncentrosymmetric chiral/polar materials [36,37], thereby broadening the range of candidate materials.

In the present study, we theoretically examine the emergence of nonreciprocal transport arising from meron–antimeron spin textures via a microscopic model analysis. Using a tight-binding model that hosts the meron–antimeron crystal on a two-dimensional triangular lattice, we demonstrate that the electronic band structure under the meron–antimeron crystal exhibits asymmetric dispersions in terms of the wave vector. Subsequent calculations of the Drude-type nonlinear conductivity reveal the presence of nonlinear nonreciprocal transport along the y direction, with no such behavior along the x direction. We emphasize that this nonreciprocal transport is purely triggered by the noncoplanar magnetic structure, requiring neither net magnetization nor relativistic spin–orbit coupling. Finally, we discuss the experimental implications of our findings for the skyrmion-hosting compound Gd₂PdSi₃, where the nature of the zero-field phase—whether it is a single-Q spiral state or a meron–antimeron crystal—is still under debate. Our results serve as a promising platform to identify the underlying magnetic phase in Gd₂PdSi₃.

The remainder of this paper is organized as follows: In Section 2, we introduce the tight-binding model and define the meron–antimeron spin textures on the triangular lattice. Then, after presenting the asymmetric band structure in the absence of the spin–orbit coupling in Section 3, we show the behavior of nonlinear nonreciprocal transport in Section 4. We discuss the experimental relevance of our results, with emphasis on Gd₂PdSi³, in Section 5. Finally, we conclude our results in Section 6.

2. Model

We consider a s-d-type tight-binding model comprising itinerant electrons and localized spins on a two-dimensional triangular lattice. The s-d Hamiltonian is given by

$$\mathcal{H}=-t_1\sum _{\langle i,j\rangle ,\sigma }{c}_{i\sigma }^{†}{c}_{j\sigma }+J\sum _i{\mathit{s}}_i·{\mathit{S}}_i,$$

(1)

where $c_{i\sigma }^{†}$ and $c_{i\sigma }$ are the creation and annihilation operators, respectively, for an itinerant electron with spin $\sigma$ at the lattice site i, and ${\mathit{S}}_i$ represents an ith classical localized spin of unit magnitude, i.e., $|{\mathit{S}}_i|=1$. The first term in Equation (1) describes the kinetic energy of itinerant electrons, including nearest-neighbor hopping with amplitude $t_1$. The second term in Equation (1) describes the onsite exchange interaction between the spin of itinerant electrons, expressed as ${\mathit{s}}_i=(1/2){\sum }_{\sigma ,{\sigma }^{\prime }}{c}_{i\sigma }^{†}{\mathbf{\sigma }}_{\sigma {\sigma }^{\prime }}{c}_{i{\sigma }^{\prime }}$, and the localized spin ${\mathit{S}}_i$, mediated by the coupling constant J. Here, $\mathbf{\sigma }=({\sigma }_x,{\sigma }_y,{\sigma }_z)$ denotes the vector of Pauli matrices. The sign of J is inconsequential in the present context, since the localized spins ${\mathit{S}}_i$ are treated as classical vectors. For the numerical analyses presented in Section 3 and Section 4, we set $t_1=1$ as the unit of energy and choose $J=1$. The different choices of $t_1$ and J do not alter the following results at the qualitative level. The s-d model has been widely employed in the study of transport phenomena in systems with topological spin textures [45,46,47,48,49,50,51,52], owing to its capacity to accommodate and stabilize a variety of spin configurations, such as the skyrmion crystal [53,54,55,56,57,58,59,60], hedgehog crystal [61], and other noncoplanar magnetic states [62,63,64,65], through large-scale numerical simulations.

Here, we consider a fixed spin configuration of localized spins. Specifically, we employ the meron–antimeron crystal spin texture, which consists of a superposition of triple-Q spin density waves, expressed as [66]

$$\begin{array}{c}\hfill {\mathit{S}}_i=N_i\left(\begin{array}{c}sin{\mathcal{Q}}_1-{\displaystyle \frac{1}{2}}sin{\mathcal{Q}}_2-{\displaystyle \frac{1}{2}}sin{\mathcal{Q}}_3\\ {\displaystyle \frac{\sqrt{3}}{2}}sin{\mathcal{Q}}_2-{\displaystyle \frac{\sqrt{3}}{2}}sin{\mathcal{Q}}_3\\ -(cos{\mathcal{Q}}_1+cos{\mathcal{Q}}_2+cos{\mathcal{Q}}_3)\end{array}\right),\end{array}$$

(2)

where ${\mathcal{Q}}_{\nu }={\mathit{Q}}_{\nu }·{\mathit{r}}_i+{\theta }_{\nu }$; we choose three ordering wave vectors connected by threefold rotational symmetry of the triangular lattice, where ${\mathit{Q}}_1=Q(1,0)$, ${\mathit{Q}}_2=Q(-1/2,\sqrt{3}/2)$, and ${\mathit{Q}}_3=Q(-1/2,-\sqrt{3}/2)$, with $Q=\pi /3$ does not exhibit loss of generality (the lattice constant of the triangular lattice is set as unity). ${\theta }_{\nu }$ denotes the relative phase among triple-Q spin density waves; we set ${\theta }_1=5\pi /6$, ${\theta }_2=\pi /6$, and ${\theta }_3=\pi /2$ so that a noncoplanar spin texture has neither a net magnetization nor a scalar spin chirality, as detailed below [67]. The normalization constant $N_i$ ensures the unit magnitude of spin, $|{\mathit{S}}_i|=1$, at every site.

Figure 1a illustrates the spin configuration of the meron–antimeron crystal. The spin configuration includes approximately three distinct types of magnetic vortices. The first is an in-plane vortex, characterized by the absence of a z-component in the spin, centered around regions where $S_i^z=0$. The second is a meron-like vortex, exhibiting a skyrmion number of $-1/2$, centered around regions where $S_i^z=-1$. The third is an antimeron-like vortex, possessing a skyrmion number of $+1/2$, centered around regions where $S_i^z=+1$. The number of spins pointing in the positive z direction is equal to that of those pointing in the negative z direction, leading to cancelation of the net magnetization along the z axis.

Figure 2 shows the spatial distribution of the scalar spin chirality, which reflects the local topological nature of the spin configuration. The scalar spin chirality, which is defined as ${\chi }_{\mathit{R}}={\mathit{S}}_i·({\mathit{S}}_j\times {\mathit{S}}_k)$ ($i,j$, and k are in a counterclockwise order), is calculated for each triangle plaquette $\mathit{R}$. By comparing with the spin configuration in Figure 1a, one can identify that the meron-like (antimeron-like) vortices correspond to regions with negative (positive) scalar spin chirality, while the in-plane vortices yield zero scalar spin chirality. By summing the scalar spin chirality around each meron (antimeron)-like vortex, the topological number of $-1/2$ ($1/2$) is obtained by following the formula in Ref. [68], whose expression is explicitly given by

$$\begin{array}{c}\hfill n_{\mathrm{sk}}={\displaystyle \frac{1}{4\pi N_{\mathrm{m}}}}〈\sum _{\mathit{R}}{\Omega }_{\mathit{R}}〉,\end{array}$$

(3)

where $N_{\mathrm{m}}$ is the number of magnetic unit cells. ${\Omega }_{\mathit{R}}$ is given by

$$\begin{array}{c}\hfill tan\left({\displaystyle \frac{{\Omega }_{\mathit{R}}}{2}}\right)={\displaystyle \frac{{\mathit{S}}_i·({\mathit{S}}_j\times {\mathit{S}}_k)}{1+{\mathit{S}}_i·{\mathit{S}}_j+{\mathit{S}}_j·{\mathit{S}}_k+{\mathit{S}}_k·{\mathit{S}}_i}}.\end{array}$$

(4)

Consequently, both the net scalar spin chirality and net skyrmion number vanish in the meron–antimeron crystal structure. The stabilization mechanism of this meron–antimeron spin configuration on the triangular lattice has been clarified only in noncentrosymmetric systems with the Dzyaloshinskii–Moriya interaction [66]; the stability in the centrosymmetric case has not been clarified yet. Since we focus on the qualitative behavior of nonlinear nonreciprocal transport under the meron–antimeron crystal, we assume the transport property under the fixed spin configuration for simplicity.

For comparison, we also examine a single-Q spiral state, whose spin configuration is defined by

$$\begin{array}{c}\hfill {\mathit{S}}_i=\left(\begin{array}{c}sin{\mathcal{Q}}_1\\ 0\\ cos{\mathcal{Q}}_1\end{array}\right).\end{array}$$

(5)

The real-space spin configuration of the single-Q spiral state is shown in Figure 1b. Due to its coplanar spin texture, this spin configuration does not exhibit any local scalar spin chirality on individual triangular plaquettes, in stark contrast to the meron–antimeron crystal.

3. Electronic Band Structure

Under the spin configurations in Figure 1a,b, we calculate the corresponding Fermi surfaces at the chemical potential $\mu =-1$. Figure 3a displays the Fermi surfaces in the first magnetic Brillouin zone for the meron–antimeron crystal. While the overall shape retains an approximate sixfold symmetry, subtle symmetry breaking is evident. Specifically, the Fermi surfaces remain invariant under the mirror operation $k_x\leftrightarrow -k_x$ but not under $k_y\leftrightarrow -k_y$, indicating an asymmetric structure along the $k_y$ direction. Meanwhile, the threefold rotational symmetry is preserved in the Fermi surfaces. To satisfy such a transformation, the lowest-order functional form of the asymmetric band deformation in the $\mathit{k}\to \mathbf{0}$ limit is given by the third-order function of $\mathit{k}=(k_x,k_y)$ rather than the linear order of $\mathit{k}$. In this case, the functional form is given by $k_y(3k_x^2-k_y^2)$, which plays a key role in the nonlinear nonreciprocal transport discussed in Section 4. It is noted that this asymmetric band modulation is purely driven by noncoplanar magnetic textures of the meron–antimeron crystal, as the present model excludes the effect of the relativistic spin–orbit coupling [43,44,69]. In contrast, the Fermi surfaces for the single-Q spiral state in Figure 3b exhibit a twofold symmetry and are symmetric under both $k_x\leftrightarrow -k_x$ and $k_y\leftrightarrow -k_y$, indicating the absence of asymmetric modulations and nonlinear transport.

From a symmetry viewpoint, asymmetric deformations of the Fermi surfaces in the meron–antimeron crystal are attributed to the simultaneous breaking of spatial inversion and time-reversal symmetries. Indeed, the spin configuration in Figure 1a is not invariant under both symmetry operations. This symmetry breaking is clearly manifested in the spatial distribution of the scalar spin chirality, as shown in Figure 2; their color pattern is reversed under either operation. Furthermore, the breaking of sixfold rotational symmetry is important to obtain the asymmetric band modulations in the $k_x-k_y$ plane. Similar asymmetric modulations have also been obtained in the case of electronic systems [70,71,72,73,74,75] and magnonic systems [76,77,78,79,80] once the symmetry conditions in terms of the spatial inversion and time-reversal symmetries are satisfied.

In addition to the asymmetric modulations of the electronic bands, the system under the meron–antimeron spin texture exhibits antisymmetric spin splitting. As shown in Figure 4a–c, the spin polarizations in each component are characterized as a momentum functional form of $k_y{\sigma }_x$, $k_x{\sigma }_y$, and $k_x(k_x^2-3k_y^2){\sigma }_z$, respectively. The first two terms correspond to Rashba-type spin splitting, $k_y{\sigma }_x-k_x{\sigma }_y$, typically associated with a polar field along the z direction, as observed at surfaces and interfaces. The third term corresponds to Ising-type spin splitting, commonly arising under a trigonal crystal field, as found in materials such as MoS₂ [81,82]. However, the mechanism required to induce antisymmetric spin splitting in the present system is different from that in the above studies [81,82]; the noncoplanar magnetic ordering provides an internal mean field that gives rise to spin splitting, even in the absence of relativistic spin–orbit coupling [43]. In other words, this mechanism is rooted in the magnetic texture itself rather than in relativistic effects, which is conceptually aligned with the emerging field of noncentrosymmetric noncollinear altermagnetism, which explores systems exhibiting antisymmetric spin splitting without conventional spin–orbit interactions [83,84,85,86,87]. In contrast, the single-Q spiral state exhibits antisymmetric spin polarization only in the y-spin component, following the functional form $k_x{\sigma }_y$. This effect originates from the vector spin chirality in the $zx$ plane [84,88], and lacks the antisymmetric spin polarization in the x- and z-spin components.

4. Nonlinear Nonreciprocal Transport

Reflecting the asymmetric band modulation along the $k_y$ direction in the meron–antimeron crystal, nonlinear nonreciprocal transport along the y direction is anticipated [41,89]. To quantify this response, we calculate the second-order nonlinear conductivity ${\sigma }_{yyy}$, defined via the relation $J_y={\sigma }_{yyy}{E}_y^2$, where $J_y$ and $E_y$ denote the electric current and electric field in the y direction, respectively. The calculation is based on the Drude-type mechanism by focusing on the intraband process, where the conductivity is expressed as [90,91]

$${\sigma }_{yyy}=-{\displaystyle \frac{e^3{\tau }^2}{2{\hslash }^3{N}_s{N}_{\mathit{k}}}}\sum _{\mathit{k},n}{f}_{n\mathit{k}}{\partial }_y{\partial }_y{\partial }_y{\epsilon }_{n\mathit{k}},$$

(6)

with ${\epsilon }_{n\mathit{k}}$ being the energy eigenvalue and $f_{n\mathit{k}}$ the Fermi distribution function for the band index n, obtained by the exact diagonalization of the Hamiltonian in Equation (1) incorporating the meron–antimeron crystal spin texture. ${\partial }_y$ represents the derivative in terms of the wave vector $k_y$. Here, e, $\tau$, ℏ, $N_s$, and $N_{\mathit{k}}$ represent the electron charge, the relaxation time, the reduced Planck constant, the number of sites within the magnetic unit cell (=48), and the number of supercells, respectively, where we set $e=\tau =\hslash =1$ and $N_{\mathit{k}}=600\times 600$. In the following calculation, we set the temperature to $T=0.01$. Although we consider the DC component here, qualitatively similar results to $J_y\left(2\omega \right)={\sigma }_{yyy}^{\prime }(\omega ,\omega )E_y\left(\omega \right)E_y\left(\omega \right)$ and $J_y\left(0\right)={\sigma }_{yyy}^{″}(\omega ,-\omega )E_y\left(\omega \right)E_y(-\omega )$ are expected for the AC component with the frequency $\omega$.

Figure 5a shows a contour plot of ${\sigma }_{yyy}$ as a function of the electron filling per site $n_{\mathrm{e}}$ (with $n_{\mathrm{e}}=2$ denoting full filling) and the molecular field $h=JS/2$. The results reveal that ${\sigma }_{yyy}$ becomes nonzero in almost all of the parameter regions, ranging from positive (red) to negative (blue) values. The parameter dependence on filling and the molecular field is generally nontrivial, as exemplified in the case of $n_{\mathrm{e}}=0.1$ in Figure 5b, due to multi-band effects stemming from the multi-sublattice structure. This suggests that both electron filling and field strength are key tuning parameters for optimizing the nonlinear response: the former can be controlled via electron or hole doping, while the latter can be varied through temperature. Under the present model parameters, the maximum absolute value of ${\sigma }_{yyy}$ is found near $n_{\mathrm{e}}=1.02$ and $h=4.9$. Notably, ${\sigma }_{yyy}$ in the low- and high-filling regions behaves in a relatively monotonic way, since the shape of the Fermi surface becomes simple in these regimes.

Let us briefly estimate the quantitative value of ${\sigma }_{yyy}$. By supposing the linear conductivity to be ${\sigma }_{yy}\sim {10}^{-1}$ and ${\sigma }_{yyy}\sim {10}^{-1}$, the ratio ${\sigma }_{yyy}/{\left({\sigma }_{yy}\right)}^2$ [m/A], which is independent of the relaxation time in the clean limit, is estimated to be ${10}^{-4}$ for $t_1\sim 0.2$ [eV], where the dimensions of ${\sigma }_{yy}$ and ${\sigma }_{yyy}$ are given by [A/V] and [Am/V²], respectively. This value is comparable to the values in the two-dimensional material systems, such as a few-layer WTe₂ and strained WSe₂ [92].

5. Experimental Relevance

Finally, we comment on the experimental relevance of our results to the compound Gd₂PdSi₃, in which the skyrmion crystal was discovered under external magnetic fields despite the centrosymmetric crystal structure [27,93,94,95,96]. However, the nature of the low-field magnetic phase remains unresolved, even after detailed resonant X-ray scattering (RXS) measurements. Two main candidates have been proposed: the meron–antimeron crystal and the single-Q spiral state. Distinguishing between these two is particularly challenging using RXS due to the presence of multi-domain structures in the sample, which obscure the signatures of each magnetic configuration [27].

The present results offer two promising approaches for identifying the low-field magnetic phase in Gd₂PdSi₃. One approach involves measuring the second-order nonlinear conductivity. As shown in our analysis, the meron–antimeron crystal yields finite nonlinear nonreciprocal conductivity characterized by a nonzero ${\sigma }_{yyy}$, while the single-Q spiral state exhibits vanishing nonlinear conductivity due to the presence of effective time-reversal symmetry. Another promising method is spin-resolved angle-resolved photoemission spectroscopy, which enables direct observation of antisymmetric spin polarization in the band structure. In the meron–antimeron crystal, the antisymmetric spin polarization manifests in a pattern described by $k_y{\sigma }_x-k_x{\sigma }_y+c_1{k}_x(k_x^2-3k_y^2){\sigma }_z$ ($c_1$ is the numerical constant), whereas it simplifies to $k_x{\sigma }_y$ in the single-Q spiral state when the effect of the relativistic spin–orbit is small. In particular, even in the multi-domain structure, the single-Q state is not expected to show antisymmetric spin polarization along the z-axis, in contrast to the case of the meron–antimeron crystal.

6. Conclusions

In conclusion, we have theoretically demonstrated that nonlinear nonreciprocal transport can emerge purely from noncoplanar magnetic textures, specifically those forming a meron–antimeron crystal structure, without requiring net magnetization or relativistic spin–orbit coupling. By employing a microscopic tight-binding model on the two-dimensional triangular lattice, we showed that the meron–antimeron spin configuration induces asymmetric band deformations and antisymmetric spin splitting in the electronic structure, both of which are qualitatively distinct from those observed in the single-Q spin configuration. This asymmetry is a consequence of the simultaneous breaking of spatial inversion and time-reversal symmetries, driven by the spatial distribution of the local scalar spin chirality. We further calculated the second-order nonlinear conductivity tensor ${\sigma }_{yyy}$ based on the Drude-type mechanism, and revealed its strong dependence on both the electron filling and the strength of the molecular field. Our results suggest that nonlinear nonreciprocal transport can be maximized under specific model parameter conditions, which can be tuned via temperature or carrier doping. Lastly, we discussed the potential experimental relevance of our findings to Gd₂PdSi₃, a centrosymmetric compound that has been suggested to host the meron–antimeron crystal phase at a zero magnetic field. Our proposed method of using nonlinear nonreciprocal transport and/or spin-resolved angle-resolved photoemission spectroscopic measurements enables us to distinguish the meron–antimeron crystal from the single-Q spiral state, which could be helpful in determining a low-field magnetic phase diagram in Gd₂PdSi₃.