Effects of crystal field and momentum-based frustrated exchange interactions on multiorbital square skyrmion lattice

TL;DR

This work demonstrates that a square-shaped skyrmion lattice can be stabilized in Ce-based, centrosymmetric, tetragonal magnets by a cooperative mechanism combining crystal-field–induced multiorbital effects and momentum-dependent frustrated exchange, including higher-harmonic couplings. Using self-consistent mean-field calculations on a $6\\times6$ lattice, the authors show robust S-SkL formation across broad ranges of the crystal-field splitting $\\Delta$ and orbital content parameter $\\alpha$, with interorbital coupling playing a decisive role. Decomposition of internal energies confirms sustained contributions from both $\\Gamma_{t7}$ doublets and a persistent interorbital channel, while tuning the off-diagonal coupling $\\gamma$ reveals how multiorbital effects shape nearby double-$Q$ states. When higher-harmonic wave vectors are suppressed (small $\\xi$), the S-SkL collapses, underscoring the essential role of higher harmonics in stabilizing this topological texture. Overall, the results extend skyrmion-host design principles to $f$-electron systems beyond Gd- and Eu-based magnets and provide concrete experimental avenues for Ce-based materials via inelastic neutron scattering and X-ray spectroscopy to detect the predicted states.

Abstract

Motivated by recent theoretical predictions of a square-shaped skyrmion lattice (S-SkL) in centrosymmetric tetragonal Ce-based magnets [Yan Zha and Satoru Hayami, Phys. Rev. B 111, 165155 (2025)], we perform a comprehensive theoretical investigation on the role of multiorbital effects, magnetic anisotropy, and momentum-based frustrated exchange interactions in stabilizing such topologically nontrivial magnetic textures. By employing self-consistent mean-field calculations over a broad range of model parameters, we demonstrate that the cooperative interplay among multiorbital effects, frustrated exchange interactions at higher-harmonic wave vectors, and crystal-field anisotropy is crucial for the stabilization of the S-SkL. Furthermore, the competition between the easy-plane intraorbital coupling and the easy-axis interorbital coupling leads to a significant enhancement of the S-SkL stability region. We also identify a plethora of multi-$Q$ states, including magnetic bubble lattice and double-$Q$ phases with a local/global scalar chirality. Our findings elucidate the microscopic mechanism responsible for the emergence of S-SkLs in Ce-based magnets and provide a route toward realizing skyrmion lattices in a broader class of $f$-electron materials beyond conventional Gd- and Eu-based systems lacking orbital angular momentum.

Figures (11)

• Figure 1: (a) Dependence on the superposition coefficient $\alpha$ ($0 \le \alpha \le 1$) of $\abs{\bra{\Gamma_{t7 \pm}^{(1)}}J^z\ket{\Gamma_{t7 \pm}^{(1)}}}$, $\abs{\bra{\Gamma_{t7 \pm}^{(2)}}J^z\ket{\Gamma_{t7 \pm}^{(2)}}}$, and $\abs{\bra{\Gamma_{t7\pm}^{(\iota)}} J^{x(y)}\ket{\Gamma_{t7\mp}^{(\iota)}}}$ with $\iota \in \{1,2\}$. The first two correspond to the diagonal elements of $J^z$ for the two Kramers doublets, while the last corresponds to the off-diagonal elements of $J^x$ (and $J^y$). (b) $\alpha$ dependence of $\abs{\bra{\Gamma^{(\iota)}_{t7\pm}}J^z \ket{\Gamma^{(\kappa)}_{t7\pm}}}$ and $\abs{\bra{\Gamma^{(\iota)}_{t7\pm}}J^{x(y)} \ket{\Gamma^{(\kappa)}_{t7\mp}}}$ with $\iota,\kappa\in\{1,2\},\ \iota\neq\kappa$. These off-diagonal matrix elements of $J^z$ and $J^{x(y)}$ encode the interorbital coupling between the two $\Gamma_{t7}$ Kramers doublets.
• Figure 2: (a)--(e) $\Delta$--$h$ phase diagrams for $\alpha = 0.3$, $0.38, 0.408, 0.6124$, and $0.65$ at low temperature ($T = 0.05$) in the square-lattice system with the relatively strong higher-harmonic wave-vector contribution ($\xi = 0.875$). The magnetic field $h$ and the crystal-field splitting $\Delta$ (between the two $\Gamma_{t7}$ Kramers doublets) are shown on the vertical and horizontal axes, respectively. The negative-$\Delta$ region corresponds to a rough reversal of the energy-level hierarchy of the two doublets. Each colored region represents a distinct low-temperature magnetic configuration, including the 1$Q$ CS state, 1$Q$ VS state, various 2$Q$ states (labeled I--XIV), the S-MBL I and R-MBL I states, the S-SkL and S-SkL$'$ states, and the fully polarized ferromagnetic (Ferro) state.
• Figure 3: Real-space and momentum-space characteristics of the magnetic phases stabilized in the positive-$\Delta$ region, or simultaneously in both the positive- and negative-$\Delta$ regions. Panels (a)--(c) show (a) the three-dimensional magnetic-moment configurations in a $6 \times 6$ unit cell, where the magnetic moments are drawn at each site and normalized in length for clarity; the color scale represents the normalized polar angle $\theta' = \theta / \pi$, varying continuously from 0 (north pole) to 1 (south pole); (b) the structure-factor distributions $J(\bm{q})$ in momentum space, where $n_x$ and $n_y$ denote multiples of $2\pi / 6$ with $-3 \le n_x, n_y \le 3$ in the first Brillouin zone; and (c) the local scalar chirality $\chi_i$ within the $6 \times 6$ unit cell.
• Figure 4: Real-space and momentum-space characteristics of the magnetic phases stabilized only in the negative-$\Delta$ region. Panels (a)--(c) show (a) the three-dimensional magnetic-moment configurations in a $6 \times 6$ unit cell, with normalized moment lengths and a color scale indicating the normalized polar angle $\theta' = \theta / \pi$; (b) the corresponding structure-factor distributions $J(\bm{q})$ in momentum space, with $n_x$ and $n_y$ representing multiples of $2\pi / 6$ in the first Brillouin zone ($-3 \le n_x, n_y \le 3$); and (c) the local scalar chirality $\chi_i$ in the $6 \times 6$ unit cell.
• Figure 5: Magnitude of the summed structure factor components (vertical axis), $J^{xy}_{\bm{Q}_1} + J^{xy}_{\bm{Q}_2}$, $J^{z}_{\bm{Q}_1} + J^{z}_{\bm{Q}_2}$, $J^{xy}_{\bm{Q}'_1} + J^{xy}_{\bm{Q}'_2}$, and $J^{z}_{\bm{Q}'_1} + J^{z}_{\bm{Q}'_2}$, for (a) $\alpha = 0.6124$ and (b) $\alpha = 0.65$ at fixed magnetic field $h = 1$, plotted as functions of the crystal-field splitting $\Delta$ (horizontal axis).