Effect of magnetic field on whirling-anti-whirling order in icosahedral-quasicrystal approximant

Recent neutron measurement in the icosahedral quasicrystal approximant Au-SM-Tb (SM=Al, Ga) has revealed unique noncollinear magnetic order ``whirling-anti-whirling states’’. Here, we report theoretical analysis on the magnetic-field-direction dependence on the whirling-anti-whirling order in the 1/1 approximant crystal. By performing exact-diagonalization calculation for the effective model taking into account the uniaxial magnetic anisotropy arising from the crystalline electric field, we show the metamagnetic transition takes place simultaneously with the topological transition under the magnetic field along the (111) direction. After the metamagnetic transition, the emergent fictious magnetic field induced by the chirality of noncoplanar magnetic moments appears, the analysis of which concludes that the topological Hall effect is expected to be observed in the electrical conductivity $σ_{xy}$ and $σ_{yz}$ for the applied field direction from (111) to (001).

💡 Research Summary

The paper investigates the magnetic-field response of the “whirling‑anti‑whirling” non‑collinear spin order recently observed by neutron diffraction in the icosahedral quasicrystal approximant Au‑SM‑Tb (SM = Al, Ga). Using an effective spin Hamiltonian that incorporates nearest‑ and next‑nearest‑neighbor exchange (J₁ > 0, J₂) and a uniaxial anisotropy axis set by the crystalline electric field (CEF) with an anisotropy angle θ≈90°, the authors perform exact‑diagonalization on the 24‑site unit cell of the 1/1 approximant.

At zero field, for J₂/J₁ ≥ 7.5 the ground state is the whirling‑anti‑whirling configuration: each icosahedron (IC) hosts a whirling spin texture (topological charge n = +3) while the neighboring IC carries the opposite anti‑whirling texture (n = ‑3). The total scalar chirality χ_T vanishes because the contributions from the two ICs cancel, implying no emergent fictitious magnetic field and thus no topological Hall effect.

When a magnetic field is applied along the three‑fold (111) axis, the system undergoes a metamagnetic transition at a critical field h_M ≈ 0.47 (in units of J₁). Above h_M the spins tilt uniformly, the topological charge on each IC becomes zero (a topological transition coincident with the metamagnetic one), and a finite total scalar chirality χ_T = (‑5.64, 0, 3.76) appears. This non‑zero χ_T acts as an emergent magnetic field that couples to conduction electrons, giving rise to a topological Hall conductivity σ_μν ∝ ε_μνρ χ_T^ρ.

The authors further explore the dependence on field direction by rotating the field from the two‑fold (001) axis to the three‑fold (111) axis while keeping its magnitude fixed (|h| = 0.5). For h ∥ (001) the post‑transition state is doubly degenerate; the x‑component of χ_T cancels, leaving only a z‑component, which would generate a Hall signal in σ_xy. As the field is tilted away from (001), the degeneracy is lifted, and a unique ground state with χ_T = (‑5.64, 0, 3.76) persists for angles 0° < θ_h < 54.7°. At the exact (111) orientation (θ_h = 54.7°) a triple degeneracy re‑emerges, producing three symmetry‑related χ_T vectors whose components partially cancel, but a net y‑component remains, predicting observable Hall responses in both σ_xy and σ_yz (and σ_zx).

These theoretical results explain the experimentally observed whirling‑anti‑whirling order and metamagnetic behavior in Au‑Al‑Tb and Au‑Ga‑Tb approximants, and they predict that a topological Hall effect should be detectable in the same materials. The paper suggests that low‑temperature transport measurements, especially of σ_xy and σ_yz under controlled field orientations, can directly test the emergence of the fictitious magnetic field associated with the scalar chirality. Moreover, the presence of domain structures due to the high‑symmetry field directions implies that single‑crystal or domain‑aligned samples will be advantageous for isolating the Hall signals.

In summary, the study demonstrates that (i) a magnetic field along a high‑symmetry axis can simultaneously trigger a metamagnetic and a topological transition in a quasicrystal approximant, (ii) the resulting non‑coplanar spin texture generates a finite scalar chirality acting as an emergent magnetic field, and (iii) this emergent field leads to a measurable topological Hall effect, with its tensorial components dictated by the field direction. The work provides a concrete theoretical framework for exploring topology‑driven transport phenomena in complex aperiodic magnetic materials.