Interlocked Vortex Order in the Frustrated A 3 V 9 Te 13 Lattice (A =Cs,Rb)

The gist: The frustrated V sublattice naturally stabilizes a robust vortex or anti-vortex antiferromagnetic ground state in A3V9Te13, which hosts electronic structures reminiscent of kagome metals, including Dirac-like band crossings, van Hove singularities, and flat bands near the Fermi level

Magnetic Ground State

The magnetic ground state is explored using Density Functional Theory (DFT) calculations Since experiments suggest weak interlayer coupling as well as large in-plane lattice constants, the search for the ground state configuration is constrained in a single unit-cell In collinear cases, it requires ferromagnetic (FM) alignment of the moments within each triangle; while in the coplanar cases, it requires the moments within each triangle to form either vortex or anti-vortex In the vortex/antivortex case, the moments on neighboring sites are rotated 120◦ clockwise/counter-clockwise, when the triangle is traversed clockwise The results of DFT calculations for all these configurations are summarized in TABLE I The inter-layer coupling is negligible, as the A-type AFM is almost degenerate with FM configuration All considered initial configurations are shown in FIG. 1 Once SOC is included, however, the spin and lattice degrees of freedom become coupled and the two configurations become distinguishable under magnetic space-group (MSG) The vortex state belongs to (P¯6′2m′) (No. 189.224), whereas the anti-vortex state belongs to the lower-symmetry (Am′m′2) (No. 38.191) magnetic space group SOC lifts the vortex–anti-vortex degeneracy, raising the total energy of the anti-vortex state by approximately 53 meV/f.u.

Heisenberg Model Analysis

The vortex phase is understood using a Heisenberg model with nearest-neighbor-only interactions Due to the C3 symmetry for each triangle (large or small) around the axis through its center, the classical total energies contributed by the vortex and anti-vortex configurations from each triangle are both −3/2JS squared It can be shown that if all exchange interactions are AFM (J > 0), all triangles must be either vortex or antivortex for the ground state configuration, and θ2 - θ1 - π = θ1 - θ3 + π = ∆θ (please refer to SI for details) These conditions put very strong constraints on the ground state magnetic configuration of these compounds, such that the spin configuration of these triangles are “interlocked”, i.e.: the directions of all moments are indeed coupled such that fixing the spin configuration of 1 triangle propagates and fixes the rest In this case, the classical total energy for each unit cell ENN is reduced to: ENN = -3/2JSS12 + 3JLS22+ 3JSLS1S2(f(∆θ, α)) where vortex phase takes the minus sign and the anti-vortex phase takes the plus sign The relation between the optimal ∆θ minimizing ENN and α for HNN is shown in Fig. 2(b) If the in-plane anisotropy is small, such relation guarantees the emergence of a continuous rotational degree of freedom associated with the magnetic moment of the small triangle Consequently, the vortex order is not an isolated magnetic configuration but belongs to a manifold of nearly degenerate states generated by geometric frustration

Monte Carlo Simulations

Classical Monte Carlo simulations were performed using the Vampire package based on the magnetic exchange parameters within distance Rij < 7.7 Å, including both isotropic and anisotropic interactions The calculated magnetic ground state is consistent with the vortex-like configuration predicted by the Heisenberg-model analysis and the DFT calculations (Fig. S2a) In addition, the magnetic transition temperature remains nearly unchanged regardless of the direction or magnitude of the applied magnetic field (FIG. 2c) This behavior is consistent with experimental observation

Electronic Structures

The electronic band structure of the nonmagnetic state is shown in Fig. 3(a), while the band structure, density of states, and Fermi surface of the vortex-like phase without SOC are presented in Fig. 3(b)–(d) The calculated Fermi surface also reveals pronounced quasi-two-dimensional electronic characteristics in Cs3V9Te13 Near the Fermi level, Dirac-like dispersions are found around the K point The breaking of time-reversal symmetry lifts the fourfold Dirac degeneracy and splits the original Dirac point into two spin-polarized band crossings, which are predominantly derived from the V dyz + dxz and dx2−y2 + dxy + dz squared orbitals, respectively The larger splitting around H shifts the van Hove singularity associated with the dx2−y2 + dxy + dz squared orbitals to the vicinity of the Fermi level near the L point A nearly flat band located about 200 meV below the Fermi level is mainly contributed by V atoms belonging to the large triangles Once SOC is included, it generally gaps out the Weyl-like crossings around K and H [Fig. 3(e)] Nevertheless, our calculations identify 6 pairs of Weyl points in the BZ for the vortex phase Therefore, it remains metallic in the presence of SOC Without SOC, the band structure of the anti-vortex phase is identical to that of the vortex phase With SOC, its low-energy band structure close to the Fermi level as well as its orbital characters remain qualitatively similar to the vortex phase as well [FIG. 3(f)]

Discussion

The fitted interlayer exchange interaction Jc indicates that the magnetic coupling between neighboring V layers is extremely weak Such weak interlayer coupling is also reflected in the electronic structure, where both electronic structure calculations and experimental measurements reveal pronounced quasi-two-dimensional characteristics of the Fermi surface Furthermore, our classical Heisenberg-model analysis and DFT calculations demonstrate that the in-plane magnetic anisotropy energies are very small as well Therefore, it is possible for each layer to form a 2D magnetic order without an ordered arrangement of moment directions across different layers at higher temperatures A long-range order in c-direction may take place at a much lower temperature ∼ Jc In addition, we note that classical Monte Carlo simulations yield a magnetic transition temperature TN ≈ 292 K, significantly higher than the experimentally observed transition temperature of 47 K This discrepancy may arise from two factors. First, the exchange parameters extracted from DFT calculations are likely overestimated, a well-known effect in studies of 3d-materials Second, the classical Monte Carlo approach neglects quantum fluctuations that can substantially suppress long-range antiferromagnetic order in frustrated magnetic systems Despite these discrepancies, the vortex-like magnetic phase appears to be robust and its transition temperature is nearly independent of external magnetic field, consistent with experimental observations For Rb3V9Te13, the AB stacking removes the threefold rotational symmetry present in the Cs compound, and SOC opens a small semiconducting gap under ambient pressure Our results establish A3V9Te13 as a promising platform for studying frustration-driven chiral magnetism and its interplay with topological electronic states

Pressure Effects

The pressure evolution of the electronic structure shows that the flat band located below the Fermi level gradually shifts toward the Fermi energy, leading to an enhanced density of states in its vicinity Such enhancement in the low-energy electronic states is in agreement with experimental observation of enhancement of metallicity In contrast to Cs3V9Te13, replacing Cs with Rb leads to substantial stacking difference, and modification to electronic structure thereafter Rb3V9Te13 adopts an AB-stacking order instead of AA-stacking, and the crystal symmetry is reduced from the hexagonal space group P¯62m (No. 189) to the orthorhombic space group Cmcm (No. 63) Consequently, the C3 rotational symmetry present in Cs3V9Te13 is removed At ambient pressure, the inter-layer distance is ∼8 Å, similar to Cs3V9Te13 Therefore, we assume that the inter-layer magnetic exchange is also weak, and the ground state is also vortex-like configuration Since the C3 symmetry is lifted by the stacking order, the band crossings and Weyl points are no longer present and completely removed in Rb3V9Te13 In addition, SOC opens a direct gap of approximately 30 meV, resulting in a semiconducting electronic structure at ambient pressure, consistent with experimental observations

Conclusion

In conclusion, we have systematically investigated the electronic structures and magnetic properties of the A3V9Te13 family by combining classical Heisenberg-model analysis with first-principles calculations<ref:

1. The AI system can perform accurate ground state prediction for frustrated magnetic lattices by employing a constrained search strategy based on symmetry preservation, as suggested by We consider only the magnetic orders that preserve the C3 symmetries and mirror symmetry. This allows it to bypass exhaustive searches in complex geometries like A3V9Te13.

2. The system can predict the fate of different spin configurations under external influences, specifically identifying that The vortex or anti-vortex phase hosts electronic structures reminiscent of kagome metals, including Dirac-like band crossings, van Hove singularities, and flat bands near the Fermi level. This enables it to characterize topological states associated with specific magnetic orders.

3. The AI can predict the effect of structural changes on electronic properties by relating lattice parameters to magnetic interactions, as shown in Pressure simultaneously tunes the exchange-interaction ratio, the magnetic canting angle, and the low-energy electronic density of states. This allows for predictive modeling of materials under varying pressure conditions.

4. The system can distinguish between nonrelativistic and spin-orbit coupled limits when analyzing magnetic ordering by noting that Once SOC is included, however, the spin and lattice degrees of freedom become coupled and the two configurations become distinguishable under magnetic space-group (MSG). This provides a mechanism to select the physically relevant ground state phase.

5. The AI can evaluate the robustness of a predicted magnetic state against external perturbations by confirming that the vortex phase remains robust upon inclusion of SOC, and the in-plane magnetic anisotropy energy turns out to be small. This ensures predictions are stable across different physical regimes.