Bilayer Triple-Q Magnetic State

• Bilayer triple-Q state is a noncollinear, noncoplanar magnetic configuration in ultrathin Mn-based films, characterized by tetrahedral spin arrangements.
• It arises from superposing three symmetry-equivalent spin spirals with a distinct 2π/3 phase shift, enforcing a 109.5° separation between spins.
• Stabilized by higher-order exchange and anisotropic interactions, this state supports finite topological orbital magnetization despite having zero net spin.

The bilayer triple-$Q$ state is a noncollinear, noncoplanar magnetic ground state stabilized in certain ultrathin transition-metal bilayers and multilayers, notably in Mn-based films on heavy-element substrates. This state is characterized by a superposition of three symmetry-equivalent spin spirals within each atomic layer, with interlayer coupling that enforces tetrahedral angles both within and between layers. The resulting configuration features zero net spin moment but can support a finite “topological” orbital magnetization due to nontrivial spin chirality. The stabilization and orientation of the bilayer triple-$Q$ state depend sensitively on higher-order magnetic interactions and anisotropic exchange couplings.

1. Spin Structure and Tetrahedral Geometry

In a single Mn atomic layer on a hexagonal lattice, the triple-$Q$ (“3$Q$”) state is formed by superposing three spin spirals with propagation vectors $\mathbf{Q}_1$, $\mathbf{Q}_2$, and $\mathbf{Q}_3$ pointing along the three symmetry-equivalent $\overline{\Gamma}{-}\text{M}$ directions of the Brillouin zone. This generates a four-sublattice magnetic unit cell, with the spin directions on the sublattices given by the corners of a tetrahedron in spin space: $$\begin{align*} \hat{\mathbf n}_1 &= \frac{(+1,+1,+1)}{\sqrt{3}} \ \hat{\mathbf n}_2 &= \frac{(+1,-1,-1)}{\sqrt{3}} \ \hat{\mathbf n}_3 &= \frac{(-1,+1,-1)}{\sqrt{3}} \ \hat{\mathbf n}_4 &= \frac{(-1,-1,+1)}{\sqrt{3}} \end{align*}$$ The nearest-neighbor spins thus subtend tetrahedral angles $\arccos(-1/3) \approx 109.5^\circ$. The spin at site $i$ is assigned as $\mathbf{S}_i = S\,\hat{\mathbf n}_{p(i)}$, $p(i)\in\{1,2,3,4\}$.

In the bilayer extension, each layer forms its own 3$Q$ texture. The ideal “bilayer triple-$Q$” state ($3Q_{\rightleftarrows}$, Editor's term) features a relative lateral ($2\pi/3$) phase shift between the two layers, ensuring that vertical nearest-neighbor spins across the two layers also subtend the tetrahedral angle. Both intra-layer and interlayer neighbor pairs have $109.5^\circ$ spin separation. The vector structure of the bilayer 3$Q$ state can be written as: $$\begin{aligned} \mathbf{S}^T(\mathbf{r}^T) &= \frac{1}{\sqrt{3}} \sum_{n=1}^3 \hat{\mathbf{e}}_n\,\cos(\mathbf{Q}_n \cdot \mathbf{r}^T) \ \mathbf{S}^B(\mathbf{r}^B) &= \frac{1}{\sqrt{3}} \sum_{n=1}^3 \hat{\mathbf{e}}_n\,\cos\bigl(\mathbf{Q}_n \cdot \mathbf{r}^B + \tfrac{2\pi}{3}\bigr) \end{aligned}$$ with $\hat{\mathbf{e}}_n$ being mutually coplanar or tetrahedral unit vectors.

2. Microscopic Energetics and Exchange Hamiltonians

The stability of the bilayer triple-$Q$ state depends on a balance of magnetic interactions:

• Heisenberg exchange: Dominant nearest-neighbor intralayer couplings, e.g., $J_1^\parallel = -36.78$ meV for hcp-Mn/Ir(111), favor antiferromagnetism.
• Dzyaloshinskii–Moriya interaction (DMI): In multilayer systems, the DMI may be zero or nonzero depending on the substrate and stacking. In Mn/Ir(111), $D_1^\parallel \approx 2.82$ meV.
• Anisotropic symmetric exchange (ASE): For example, in Pd/Mn/Re(0001), $J_\mathrm{ASE}(1\textrm{NN}) = -0.30$ meV is the leading spin–orbit term coupling the triple-$Q$ state to the lattice (Nickel et al., 2023).
• Higher-order exchange interactions (HOI): Both intralayer and interlayer, including biquadratic, three-site, and four-site terms. In Mn/Ir(111), interlayer “odd” HOI (e.g., $Y_1^{\perp,\textrm{odd}}\approx -5.0$ meV) are decisive for the energetic preference for the triple-$Q$ state (Beyer et al., 5 Jun 2025).

The general atomistic Hamiltonian incorporates these terms: $$\begin{aligned} H = &-\sum_{i\neq j} J_{ij} (\mathbf{S}_i \cdot \mathbf{S}_j) -\sum_{i\neq j} \mathbf{D}_{ij} \cdot (\mathbf{S}_i \times \mathbf{S}_j) \ &\phantom{{}={}} - \text{HOI terms} - K_u \sum_i (S_i^z)^2 \end{aligned}$$

The inclusion of interlayer higher-order terms is crucial: pairwise interlayer exchange tends to favor collinear RW-AFM ($1Q$) states. Only when interlayer “odd” higher-order terms are included does the system stabilize the ideal bilayer 3$Q$ ground state.

3. Lattice Symmetry, State Selection, and Spin-Orbit Effects

Without anisotropic exchange, the 3$Q$ tetrahedron is free to rotate in spin space. ASE uniquely selects the 3$Q^1$ orientation, with one spin out-of-plane and three in-plane $120^\circ$ apart, as the minimum-energy state. For $J_\mathrm{ASE}<0$ (e.g., Pd/Mn/Re(0001)), this selects the $3Q^1_{\textrm{M}}$ domain, ensuring a single magnetic domain with threefold symmetry and a robust pattern in spin-polarized STM. Spin–orbit interactions such as magnetocrystalline anisotropy (MAE) and DMI cancel due to symmetry for the ideal tetrahedral 3$Q$ state, leaving ASE as the only relevant spin–orbit term (Nickel et al., 2023).

Modifications from ideality, such as increased sixth-order HOI or lattice stacking variations (e.g., hcp-stacking), can distort the 3$Q$ angles, reducing the symmetry and introducing rotational domains. For example, in hcp-stacked Mn/Re(0001), the tetrahedral angle can increase to $\alpha\approx55^\circ$ (Nickel et al., 2023), yielding a distorted 3$Q$.

4. Energetic Hierarchy and DFT Results

Density functional theory and atomistic modeling yield the following energetic landscape for representative systems (Beyer et al., 5 Jun 2025):

State	$E$ (meV/Mn)	$\Delta E$ (meV, reference)
Bilayer ideal $3Q_{\rightleftarrows}$	$-23.3$	$-23.3$
Bilayer non-ideal $3Q_{\rightrightarrows}$	$+7.8$	$+7.8$
Bilayer $1Q$ RW-AFM$_{\rightleftarrows}$	$0$	$0$

The energy gain of the ideal bilayer $3Q_{\rightleftarrows}$ over the collinear $1Q$ state ($\approx23$ meV/Mn) arises only when interlayer HOI are included. In Pd/Mn and Rh/Mn on Re(0001), analogous stabilization energies $\Delta E_{3Q-1Q}\approx 20$–$40$ meV/Mn are observed (Nickel et al., 2023).

5. Topological Spin Textures and Orbital Magnetization

The noncoplanar spin texture of the triple-$Q$ state gives rise to a finite scalar spin chirality: $$\chi_{ijk} = \mathbf{S}_i \cdot (\mathbf{S}_j \times \mathbf{S}_k)$$ This chirality generates a “topological” orbital moment per Mn via the real-space Berry-phase mechanism, even in the absence of net spin moment:

• In Pd/Mn bilayers (almost ideal 3$Q$), $|m_\mathrm{orb}|\approx 0.21\,\mu_B$ per Mn.
• In Rh/Mn bilayers (closer to perfect tetrahedral order), $|m_\mathrm{orb}|\approx 0.36\,\mu_B$ (Nickel et al., 2023).
• In the model Mn/Ir(111) bilayer 3$Q_{\rightleftarrows}$, $M_\mathrm{orb}^z\approx 0.02\,\mu_B$ per Mn layer ($0.04\,\mu_B$ per two Mn), with vanishing net spin magnetization (Beyer et al., 5 Jun 2025).

Distortions from the ideal 3$Q$ angles reduce the orbital moment, as quantified numerically.

6. Variants, Competing Phases, and Robustness in Magnetic Bilayers

Alternative triple-$Q$ states and related multi-$Q$ textures arise in models with additional interactions or external fields. For example, in bilayer systems with staggered DMI, multiple triple-$Q$ phases emerge depending on interlayer coupling and field strength. Low-field triple-$Q$ states can display uniform nonquantized scalar chirality of sign opposite to skyrmion crystals, while high-field triple-$Q$ states approach the symmetric limit with nearly equal amplitude modulations in both layers (Hayami, 2021). In all cases, whether the state achieves nontrivial integer topological charge depends on the specific amplitude and phase relations of the $Q$-vector superpositions.

7. Experimental Signatures and Implications

Experimental realization and imaging of the bilayer triple-$Q$ state have been achieved using spin-polarized STM in Pd/Mn and Rh/Mn bilayers on Re(0001), directly confirming the predicted orientation, lack of rotational domains, and expected magnetic supercell contrast amplitudes ($\delta z\approx\pm10$ pm in Pd/Mn) (Nickel et al., 2023). DFT-based STM simulations reproduce the hexagonal and stripe contrasts observed under tip-magnetization rotation.

The theoretical and computational evidence supports that the bilayer triple-$Q$ state is stabilized by a precise interplay of lattice symmetry, inter- and intralayer higher-order exchange, and, where relevant, anisotropic exchange couplings. Its zero net spin moment but large topological orbital magnetization make it a promising prototype for antiferromagnetic spintronic and orbitronic applications in ultrathin multilayers, leveraging emergent phenomena tied to the topology of noncoplanar spin textures (Beyer et al., 5 Jun 2025).