Chiral Magnons and Cycloidal Phonons in Altermagnetic CuF$_{2}$ Monolayer

Abstract: Altermagnetism establishes momentum-dependent spin splitting through non-symmorphic crystal symmetries, yet whether these same symmetries simultaneously govern spin and lattice collective excitations remains open. Here we show, using first-principles calculations and linear spin-wave theory, that monolayer CuF$_2$ hosts both chirality-split magnons and cycloidal phonons controlled by the same $P2_1/c$ symmetry operations. The altermagnetic order drives strongly anisotropic magnon chirality via symmetric anisotropic exchange, with Dzyaloshinskii--Moriya interactions acting as a weak secondary modulation. Crucially, the phonon and magnon chiral responses are directionally complementary: cycloidal phonon angular momentum emerges precisely where magnon chirality is symmetry-suppressed, and vice versa. The magnon bands further carry quantized Chern numbers $C^M = \pm 2$, confirming non-trivial altermagnetic topology. These results establish monolayer CuF$_2$ as a platform where a single symmetry framework engineers magnonic, phononic, and topological responses, providing a direct connection between altermagnetism and spin-lattice chirality in two-dimensional materials.

• The paper demonstrates that monolayer CuF₂ hosts chirality-split magnons and cycloidal phonons governed by the same nonsymmorphic P2₁/c symmetries, with complementary momentum-space responses.
• Symmetric anisotropic exchange drives the magnon spectrum, producing shifts of about 208 meV compared with only 0.5 meV from DMI, while the calculated magnon Chern numbers are Cᴹ = ±2.
• Cycloidal phonon angular momentum appears along the spin-degenerate X–Γ–X′ direction, whereas magnon chirality is strongest along M′–Γ–M, suggesting directional control of spin–lattice excitations in two-dimensional altermagnets.

Overview and central claim

This paper addresses a specific open question in the study of altermagnetism: whether the non-symmorphic crystal symmetries responsible for momentum-dependent spin splitting also govern the collective spin and lattice excitations of the same material, and how magnonic and phononic chirality relate to one another in momentum space. Using first-principles calculations (DFT+$U$), density functional perturbation theory, and linear spin-wave theory (LSWT), the authors show that monolayer CuF$_2$ — a recently identified $d$-wave altermagnet — hosts both chirality-split magnons and cycloidal phonons whose directional properties are dictated by the same $P2_1/c$ spin space group operations. The central, and somewhat counterintuitive, finding is that these two chiral responses are directionally complementary: cycloidal phonon angular momentum appears precisely along Brillouin zone directions where magnon chirality is symmetry-suppressed, and vice versa. This contrasts with recent reports on helical altermagnets, where magnetic helicity enhances both spin and lattice chirality simultaneously.

Structural and electronic characterization

The calculations employ VASP with the PBE functional, a 500 eV plane-wave cutoff, and the Dudarev DFT+$U$ scheme with $U_{\mathrm{eff}} = 4$ eV, chosen to reproduce the insulating gap and local moment magnitude of Cu fluorides. The Cu$^{2+}$ ions carry a $3d^9$ configuration with a calculated moment of approximately $0.75\,\mu_B$ per Cu atom within a Wigner-Seitz radius of 1.16 Å. The authors note that while the gap magnitude varies with $U$, the $_2$0-wave altermagnetic spin splitting remains qualitatively robust across $_2$1–6 eV, which lends credibility to the choice.

A methodological point worth emphasizing concerns the symmetry of the monolayer. Earlier work proposed that the monolayer preserves the bulk crystal symmetry, but the authors show this conclusion depends on the symmetry-detection tolerance: strict tolerances reveal $_2$2 Å relaxation-induced distortions that lower the detected symmetry, whereas relaxed tolerances ($_2$3) recover the bulk $_2$4 structure with crystallographically equivalent Cu Wyckoff positions. The monolayer is therefore structurally very close to the bulk prototype, and the apparent symmetry reduction is a numerical artifact of relaxation rather than fundamental symmetry breaking.

The electronic structure confirms the $_2$5-wave altermagnetic character: spin splitting is present along M$_2$6–$_2$7–M and X$_2$8–Y–X, while the X–$_2$9–X$d$0 path preserves spin degeneracy even without spin-orbit coupling (SOC). The symmetry origin lies in four operations of the spin space group $d$1, with the key operation $d$2 combining spin reversal with a $d$3 rotation and fractional translation. Along $d$4–M, no operation maps $d$5 onto itself while exchanging spin sectors, permitting splitting; along $d$6–X, the non-symmorphic operations enforce degeneracy. This directional selectivity is shown to be the common symmetry origin of both the altermagnetic band structure and the phonon angular momentum.

Magnon sector: anisotropic exchange dominance and topology

The spin Hamiltonian — isotropic Heisenberg, Dzyaloshinskii–Moriya interaction (DMI), and symmetric anisotropic exchange — is parameterized via TB2J from OpenMX calculations, and the bosonic Bogoliubov–de Gennes problem is solved with Magnopy. The extracted couplings establish a clear hierarchy: $d$7 meV sets the antiferromagnetic scale, symmetric anisotropic exchange ($d$8 meV, ~7% of $d$9) selects the ordered plane, and DMI ($P2_1/c$0 meV, ~3%) produces weak ferromagnetic canting along $P2_1/c$1.

The term-resolved analysis is the strongest quantitative result in the magnonic sector. Switching off individual Hamiltonian terms shows that the symmetric anisotropic exchange — the microscopic counterpart of the phenomenological Anisotropic Altermagnetic Stiffness (AAS) of Gomonay et al. — produces a maximum spectral reconstruction of approximately 208 meV along M$P2_1/c$2–$P2_1/c$3–M, whereas DMI contributes only about 0.5 meV, localized near $P2_1/c$4. The ratio of maximum shifts, $P2_1/c$5, demonstrates that the altermagnetic magnon response is dominated by non-relativistic symmetric anisotropic exchange, with DMI acting as a weak secondary modulation. This confirms, at the ab initio level, the phenomenological picture that sublattice-dependent anisotropic stiffness — not DMI — drives chirality-split magnon spectra in collinear altermagnets.

The chiral inter-sublattice response, quantified through $P2_1/c$6 from the Bogoliubov amplitudes, is strongly anisotropic and concentrated along the antinodal M$P2_1/c$7–$P2_1/c$8–M path, coinciding with a reduced-gap region near $P2_1/c$9 where the relative phase structure of the magnon wavefunctions evolves most rapidly. Along X$U$0–Y–X, the high-energy branches remain nearly degenerate regardless of which anisotropic terms are retained, indicating a symmetry-constrained suppression of the anisotropic exchange projection onto these modes — consistent with the $U$1-wave phenomenology in which the AAS is maximal along antinodal and suppressed along nodal directions.

Including SOC reduces the spin space group to the Type-I collinear magnetic space group $U$2, placing the two Cu sites on distinct Wyckoff positions (1$U$3 and 1$U$4) and permitting anisotropic exchange. The Berry curvature, computed with the Fukui gauge-invariant method adapted to the bosonic metric via $U$5 link variables, is highly localized near avoided crossings and exhibits a dipolar pattern mirroring the $U$6-wave symmetry. Numerical integration yields quantized magnon Chern numbers $U$7, corresponding to a cumulative Berry-phase winding of $U$8. This is a notable result: the topological magnon phase arises in a fully compensated, collinear magnetic state, in contrast to conventional topological magnon insulators that require ferromagnetic order or noncollinear textures. The authors emphasize that SOC here does not generate the chirality itself but lowers the symmetry sufficiently to allow a net Berry flux. The bands are expected to support chiral magnon edge modes and a finite transverse thermal Hall conductivity, though these are predictions rather than computed quantities in this work.

Phonon sector: symmetry-allowed cycloidal motion

Phonon dispersions and eigenvectors are obtained from DFPT in a $U$9 supercell, and the mode-resolved phonon angular momentum is computed following Zhang and Niu, $U_{\mathrm{eff}} = 4$0, with all phonon calculations performed without SOC.

A finite phonon angular momentum emerges along the X–$U_{\mathrm{eff}} = 4$1–X$U_{\mathrm{eff}} = 4$2 direction — the same path where the electronic bands are spin-degenerate. The helicity $U_{\mathrm{eff}} = 4$3 vanishes, so the angular momentum has no longitudinal component and the modes are cycloidal rather than helical. Two control results support the symmetry interpretation: first, although X–$U_{\mathrm{eff}} = 4$4–X$U_{\mathrm{eff}} = 4$5 is spin-degenerate, these points are not time-reversal-invariant and time-reversal symmetry is broken by the Néel order, so finite $U_{\mathrm{eff}} = 4$6 is not forbidden; second, calculations for the ferromagnetic configuration yield $U_{\mathrm{eff}} = 4$7 identically, confirming that phonon angular momentum is suppressed in the centrosymmetric ferromagnetic phase. The Néel ground state's simultaneous breaking of time-reversal and inversion is therefore the necessary ingredient.

The complementarity mechanism

The directional complementarity follows from how the $U_{\mathrm{eff}} = 4$8 operations $U_{\mathrm{eff}} = 4$9 act on displacements versus spins. Both $^{2+}$0 and $^{2+}$1 reverse the orientation of the plane and hence flip $^{2+}$2. Along $^{2+}$3–X, $^{2+}$4 enforces the electronic spin-degenerate nodal line but leaves $^{2+}$5 locally unconstrained, while $^{2+}$6 only relates $^{2+}$7 at opposite momenta — so cycloidal phonons are symmetry-allowed. Along $^{2+}$8–M, the little group reduces to the identity: no symmetry forbids spin splitting, and indeed the altermagnetic response is maximal there, yet the phonon angular momentum is found to vanish, which the authors attribute to the global $^{2+}$9-wave alteraxial texture rather than to a local symmetry constraint. The authors are careful to note that the phonon angular momentum does not simply inherit the momentum-space structure of the altermagnetic splitting; both responses originate from the same symmetry operations but acquire distinct, complementary directional behavior.

This complementarity distinguishes CuF$3d^9$0 from helically ordered altermagnets, where magnetic helicity fosters coexistence and mutual enhancement of chiral magnons and phonons along the screw axis. CuF$3d^9$1 instead represents a symmetry-imposed regime in a collinear two-dimensional altermagnet where spin and lattice chirality occupy disjoint momentum-space sectors.

Limitations and open questions

Several caveats bear directly on the results. The magnon Chern numbers and predicted edge modes rely on SOC being included, and the topological phase is enabled by the symmetry reduction to $3d^9$2 — the authors do not compute the edge-state spectrum or thermal Hall conductivity explicitly. The claim that the monolayer retains bulk $3d^9$3 symmetry is tolerance-dependent, resting on the interpretation of $3d^9$4 Å relaxations as numerically insignificant; whether these small distortions have measurable physical consequences at low temperature is not addressed. The vanishing of phonon angular momentum along $3d^9$5–M is attributed to the global $3d^9$6-wave texture rather than derived from a local symmetry constraint, leaving the precise microscopic mechanism of that suppression an open question. The proposed Floquet-controlled transitions between topologically distinct magnon phases and chiral-phonon-assisted modulation schemes remain speculative proposals not demonstrated computationally. Finally, all results are for the ideal freestanding monolayer; substrate effects, finite temperature, and magnon–phonon hybridization beyond the harmonic and linear spin-wave approximations are not treated.

Conclusion

This work establishes monolayer CuF$3d^9$7 as a concrete realization of a unified symmetry framework in which the same non-symmorphic $3d^9$8 operations govern altermagnetic spin splitting, chirality-split magnons dominated by symmetric anisotropic exchange, and cycloidal phonons — while directing the spin and lattice chiral responses into complementary regions of momentum space. The quantized magnon Chern numbers $3d^9$9 identify a SOC-free route to topological magnon transport in a fully compensated collinear magnet. The demonstrated directional complementarity, together with the exchange hierarchy ($0.75\,\mu_B$0 ratio $0.75\,\mu_B$1), provides a symmetry-based design principle for coupled spin–lattice chirality in two-dimensional altermagnets and motivates analogous searches in related fluoride systems such as FeF$0.75\,\mu_B$2 and MnF$0.75\,\mu_B$3, where chiral magnons have recently been observed experimentally.