3Q3 State in Mn/Bismuthene/Ag(111)

• 3Q3 state is a noncoplanar triple‑q magnetic texture on a triangular Mn lattice, characterized by a tetrahedral spin arrangement with distinct Ising-like up and down variants.
• The state is stabilized by frustrated antiferromagnetic exchange, multi‐spin interactions, and spin–orbit-induced anisotropy imposed by the bismuthene overlayer.
• SP-STM experiments coupled with density-functional theory simulations confirm the p(2×2) periodicity and magnetic contrast swap under field reversal, validating its unique topological features.

Searching arXiv for the primary paper and closely related triple-$q$ references. Search query: (Chen et al., 17 Jul 2025) Magnetic Triple-q State in Antiferromagnetic Monolayer Interfaced with Bismuthene The 3Q3 state is a specific realization of a magnetic triple-$q$ state in a Mn monolayer covered by bismuthene and supported on Ag(111). In this setting it denotes a noncoplanar, three-dimensional antiferromagnetic spin texture on a two-dimensional triangular Mn lattice, with a $p(2\times 2)$ magnetic unit cell and a uniaxial anisotropy imposed by the bismuthene overlayer. Spin-polarized scanning tunneling microscopy (SP-STM) and density-functional theory identify a 3Q3-like texture as the magnetic ground state, with two Ising-like variants, 3Q3up and 3Q3down, distinguished by the sign of the out-of-plane spin component (Chen et al., 17 Jul 2025).

1. Triple-$q$ order and the meaning of “3Q3”

On a two-dimensional triangular or hexagonal lattice, a single-$q$ state is a spin spiral characterized by one wave vector $\mathbf q$. A triple-$q$ state is formed by superposing three symmetry-equivalent spirals with wave vectors $\mathbf q_1,\mathbf q_2,\mathbf q_3$ related by $120^\circ$ rotations in reciprocal space. In schematic form,

$$\mathbf S(\mathbf r) \sim \sum_{i=1}^{3} \mathbf m_i \cos(\mathbf q_i\cdot \mathbf r + \phi_i).$$

For a special choice of amplitudes and phases, this produces a three-dimensional spin structure on a two-dimensional lattice in which the four spins of the $q$0 magnetic cell point toward the corners of a regular tetrahedron in spin space. The pairwise angle is then the tetrahedral angle,

$q$1

In the Mn/bismuthene/Ag(111) system, the magnetic lattice is a triangular Mn monolayer with a $q$2 magnetic unit cell containing four Mn spins. The realized state is noncoplanar: each Mn moment has finite in-plane and out-of-plane components, and neighboring spins are neither parallel nor antiparallel. The texture is described as “3Q3-like” because it is closest to the canonical 3Q$q$3 orientation but is slightly distorted away from an ideal tetrahedral configuration by higher-order couplings and spin-orbit coupling.

The notation 3Q$q$4, 3Q$q$5, and 3Q$q$6 refers to three highly symmetric orientations built from the same $q$7 set and related by global spin rotations. In the construction used here, 3Q$q$83Q$q$9 is obtained by rotating all spins by $p(2\times 2)$0, and 3Q$p(2\times 2)$13Q$p(2\times 2)$2 by another rotation of the same angle. In a pure Heisenberg description these variants are symmetry-equivalent. In the present interface, however, spin-orbit coupling and lattice registry break that equivalence and select 3Q$p(2\times 2)$3.

The labels 3Q3up and 3Q3down denote two domains with the same in-plane pattern but opposite out-of-plane spin polarity. They are related by reversing the $p(2\times 2)$4-component of all spins in the 3Q3-like texture.

2. Structural realization on bismuthene-covered Mn/Ag(111)

The state is realized after evaporating Mn atoms onto $p(2\times 2)$5-Bi/Ag(111) at room temperature, which leads to interdiffusion and a reconstructed interface geometry. The resulting structure consists of a close-packed Mn monolayer pseudomorphically matching Ag(111), denoted $p(2\times 2)$6-Mn/Ag(111), covered by a single-layer bismuthene honeycomb with $p(2\times 2)$7 periodicity (Chen et al., 17 Jul 2025).

This geometry is central to the magnetic phenomenology. The Ag(111) substrate provides the triangular template and metallic environment. The Mn layer supplies the frustrated antiferromagnetic lattice. The bismuthene overlayer modifies both symmetry and spin-orbit physics. In particular, the bismuthene honeycomb breaks some rotational and translational equivalences that would otherwise relate different 3Q variants and different registries of the magnetic pattern.

The structural registry matters because the 3Q texture can be shifted relative to the bismuthene lattice. The paper denotes these registries as 3Q$p(2\times 2)$8+x, 3Q$p(2\times 2)$9+y, and 3Q$q$0+xy, meaning that the spin pattern is shifted by half a $q$1 lattice vector in $q$2, $q$3, or both. This registry dependence is part of the mechanism by which the interface selects a unique 3Q3-like ground state.

The same interface also distinguishes this system from earlier Mn/Ag(111) work without Bi, where a $q$4 antiferromagnetic Néel state was observed on a Mn honeycomb lattice. Here the Mn instead forms a close-packed pseudomorphic monolayer, and the magnetic ground state becomes a noncoplanar triple-$q$5 texture.

3. Microscopic stabilization, anisotropy, and first-principles energetics

The paper does not present a closed-form microscopic Hamiltonian, but its physics is summarized by a frustrated-exchange model augmented by multi-spin terms and spin-orbit coupling,

$q$6

The central point is that the 3Q3-like state is not described primarily as a Dzyaloshinskii-Moriya-driven chiral spiral. The Dzyaloshinskii-Moriya interaction is not explicitly extracted or discussed. Instead, the analysis emphasizes frustrated antiferromagnetic exchange on the triangular Mn lattice, higher-order isotropic interactions, and spin-orbit-induced anisotropy (Chen et al., 17 Jul 2025).

Without spin-orbit coupling, density-functional calculations evaluate a continuous path connecting 1Q, 2Q, and 3Q textures. Along this path the two-spin Heisenberg contribution is constant, so any energy variation arises from four-spin and higher-order terms. Fitting the DFT energy variation with contributions up to 10th order shows a 4th-order term that stabilizes the 3Q state and a 6th-order term that destabilizes the ideal 3Q state and shifts the minima toward a distorted 3Q and a 2Q state. In this SOC-free limit, a distorted 3Q-like state and a 2Q state lie about $q$7–$q$8 meV per Mn below the ideal 3Q configuration.

Spin-orbit coupling, originating from the heavy Bi overlayer, changes this near-degenerate landscape decisively. It introduces magnetocrystalline anisotropy, differentiates the three canonical 3Q orientations, and also differentiates distinct registries of a given 3Q pattern under the Bi honeycomb. With SOC included, different 3Q$q$9+shift configurations differ in energy by up to $q$0 meV per Mn. Among the textures considered, 3Q3+xy is favored and lies below competing 1Q states even when the latter are aligned along their easy axis.

The same calculations show a uniaxial in-plane anisotropy for 3Q3+xy: when all spins are rotated uniformly around the $q$1-axis, the total energy has a single in-plane easy direction rather than threefold or continuous symmetry. This unidirectional magnetic anisotropy explains the absence of rotational domains in experiment.

The first-principles framework is based on full-potential linearized augmented plane wave calculations using the FLEUR code. Structural relaxations use GGA-PBE. Noncollinear calculations, including SOC self-consistently, use LSDA. Symmetric films with five Ag(111) layers are used for the geometry, while asymmetric films with six Ag layers are used for the noncollinear SOC calculations. Two structural models were compared. For a Mn honeycomb on BiAg$q$2/Ag(111), the magnetic ground state is a collinear AFM order on the Mn honeycomb, $q$3 meV/Mn below the ferromagnetic state. For bismuthene on $q$4-Mn/Ag(111), several magnetic orders were examined in the $q$5 Mn cell, and the 3Q state has the lowest energy, with

$q$6

4. SP-STM identification of the 3Q3-like texture

The experimental identification rests on SP-STM performed with two kinds of tips: bulk Cr tips and Fe-coated W tips. At zero field, a nonmagnetic STM image of the bismuthene-covered Mn monolayer shows a honeycomb lattice with $q$7 periodicity, reflecting the atomic structure. In contrast, SP-STM with a Cr tip reveals a checkerboard pattern with the same $q$8 period, establishing a magnetic contrast. After switching the apex magnetization of the Cr tip by a voltage pulse, the same area exhibits a stripe-like pattern, again with $q$9 periodicity (Chen et al., 17 Jul 2025).

This checkerboard/stripe duality is the decisive fingerprint. Simulated SP-STM images for a collinear antiferromagnetic state in the same $\mathbf q$0 cell do not reproduce the observed combination as cleanly. By contrast, simulated images for a 3Q state reproduce both the checkerboard pattern for one tip orientation and the stripe pattern for another. Line profiles along $\mathbf q$1 give a magnetic periodicity of $\mathbf q$2 in experiment and $\mathbf q$3 in simulation, confirming the $\mathbf q$4 magnetic unit cell.

Field-dependent measurements with Fe-coated W tips resolve the up/down domain character. With the tip magnetization aligned out of plane by an external field, images taken at $\mathbf q$5 show coexisting checkerboard and stripe domains. Reversing the field to $\mathbf q$6 reverses the out-of-plane tip magnetization and causes the two contrast types to swap in the same spatial regions: areas that were checkerboard become stripe, and vice versa, while the $\mathbf q$7 periodicity and spatial landmarks are unchanged.

This contrast interchange demonstrates that the two domain types share the same in-plane texture and differ mainly by the sign of $\mathbf q$8. Simulations of a 3Q3-like texture with a canted tip magnetization reproduce the same behavior: for fixed tip orientation, 3Q3up yields one contrast pattern and 3Q3down the other, and inverting the tip’s out-of-plane component swaps the images. That correspondence identifies the actual magnetic state as a 3Q3-like triple-$\mathbf q$9 texture with two Ising-like variants.

5. Magnetic-field response, physical significance, and broader magnetic context

A perpendicular field $q$0 plays a dual role. It aligns the Fe-coated tip magnetization out of plane, and it also biases the Mn spin system. The fields used, $q$1, are not strong enough to destroy the underlying 3Q3 order. The main experimental signature is therefore not a collapse of the triple-$q$2 texture but the interchange of SP-STM contrast between 3Q3up and 3Q3down domains. The paper notes that some domains might switch between the two variants as Zeeman energy and anisotropy compete, but the essential observation is the field-reversal-induced contrast swap consistent with an Ising-like degree of freedom in the out-of-plane spin component (Chen et al., 17 Jul 2025).

The uniaxial anisotropy imposed by bismuthene is crucial for this response. It locks the 3Q3-like texture to a single in-plane axis and suppresses rotational domain multiplicity. This differentiates the present system from more isotropic realizations of 3Q magnetism, where several 3Q orientations may remain nearly degenerate.

The paper situates the state within a broader family of 3Q and multi-$q$3 magnets that includes Mn/Cu(111), Mn/Re(0001), CoM$q$4S$q$5, and theoretical triangular Heisenberg antiferromagnets. What is distinctive here is the coexistence of a noncoplanar antiferromagnetic Mn monolayer and a strong-SOC bismuthene overlayer. A plausible implication is that the interface combines noncoplanar spin chirality in the Mn layer with strong spin-orbit physics in the bismuthene layer, making it a promising platform for topological magnetotransport, magnetic proximity effects in bismuthene, spin-orbit torque phenomena, and spintronic control of multi-$q$6 antiferromagnets. The paper does not explicitly calculate scalar spin chirality or transport coefficients, so such implications remain prospective rather than established.

A common misconception is to treat every noncoplanar surface spin texture as primarily DMI-driven. In this case, the paper points instead to frustrated exchange, substantial higher-order interactions, and SOC-induced anisotropy as the principal ingredients selecting the 3Q3-like ground state.

6. Terminological scope and disambiguation

The label “3Q3 state” is highly context-dependent across the literature. In the present usage it denotes the SOC-selected magnetic triple-$q$7 ground state of bismuthene-covered Mn/Ag(111). In unrelated fields, however, the same label has been interpreted in markedly different ways: as high-filling $q$8 fractional quantum Hall states (Liu et al., 2011), as a three-dot/three-qubit operating regime in triple quantum dots (Granger et al., 2010), as three-state phase-matching or coherent-state quantum key distribution schemes (Duo et al., 2020, Bradler et al., 2017, Schiavon et al., 2016), as a three-state non-Hermitian system with an exceptional-point-induced flip-of-states (Bhattacherjee et al., 2018), as pure symmetric three-qubit states characterized by $q$9 (Meill et al., 2017), as tripartite qutrit states classified through commutative Frobenius algebras (Honda, 2012), as a superconducting qutrit detector based on three macroscopic flux states (Shnyrkov et al., 2011), as a general pure three-qubit channel for controlled remote state preparation (Zhang et al., 2015), as a purification-based classification of three-qubit pure states (Faujdar et al., 2014), and as the spin-$\mathbf q_1,\mathbf q_2,\mathbf q_3$0 symmetric sector of three qubits (Albertini et al., 2021).

Within condensed-matter magnetism, by contrast, the term has a narrower and more specific meaning. It refers to a noncoplanar triple-$\mathbf q_1,\mathbf q_2,\mathbf q_3$1 antiferromagnetic texture on a triangular Mn monolayer, specifically the third canonical orientation of the 3Q family, rendered unique and slightly distorted by the lattice registry and strong spin-orbit coupling of the bismuthene/Ag(111) interface.