Unconventional magnons in collinear magnets dictated by spin space groups

Band representation theory for SSGs

Band representations are constructed from the atomic limit as introduced by Zak⁵². Building on Zak’s theory, the theory of topological quantum chemistry and symmetry indicators were used in electronic topological diagnosis, where thousands of materials with topological electronic bands have been successfully identified^{53,54,55,56,57,58,59}. Here we extend it to the SSG. First, we construct the spin site group \({G}_{{\rm{S}}}^{{\bf{q}}}\) of the SSG G_S, the corresponding character table and the irreps of orbit basis \({\rho }_{{S}^{\pm }}^{{\bf{q}}}\), where q is any point in the unit cell of magnetic lattice. Among these spin site groups, only 32 FM spin point groups can support non-zero magnetic moments and carry magnon. Second, to induce full band representations \({\rho }_{{S}^{\pm }}={\rho }_{{S}^{\pm }}^{{\bf{q}}}\uparrow {G}_{{\rm{S}}}\), we seek a coset decomposition of \({G}_{{\rm{S}}}^{{\bf{q}}}\). All orbits of the Wyckoff position \(\{{{\bf{q}}}_{\alpha }={g}_{\alpha }{{\bf{q}}}_{1}|{g}_{\alpha }\in {G}_{{\rm{S}}}\}\), α = 1, 2, …, n with multiplicity n of the Wyckoff position are derived. The SSG element g_α, combined with the translation \({\mathbb{T}}\), generate the decomposition of G_S with respect to the \({G}_{{\rm{S}}}^{{\bf{q}}}\):

The full band representation \({\rho }_{{S}^{\pm }}=({\rho }_{{S}^{\pm }}^{{\bf{q}}}\,\uparrow \,{G}_{{\rm{S}}})\) is induced from orbital representation \({\rho }_{{S}^{\pm }}^{{\bf{q}}}\), then we restrict it to band representations of k-little groups \({\rho }_{{S}^{\pm }}^{{\bf{k}}}={\rho }_{{S}^{\pm }}\downarrow {G}_{{\rm{S}}}^{{\bf{k}}}\) with ingredients:

where \({t}_{\alpha \alpha }=g{{\bf{q}}}_{\alpha }-{{\bf{q}}}_{\alpha }\). In this step, the character table of the unitary part of \({G}_{{\rm{S}}}^{{\bf{k}}}\) is also constructed. At last, we perform sum rules⁶⁰ to account for the introduction of antiunitary operations and the band representation \({\rho }_{{S}^{\pm }}^{{\bf{k}}}\) in spin Brillouin zone is determined. The bases of magnon bands S^± transform as the irreps of the spin site groups of the magnetic ions. For collinear magnets, the spin site group always has SO(2) spin rotation symmetry, S^± thus transform as m_s = ±1 irreps of SO(2) in spin space (m_s represents the spin angular momentum), whereas the real-space part only provides the identity irrep for S^±.

Two mechanisms of extra two-fold degeneracy provided by spin space

For a collinear AFM, G_S has the form of \((\{E| | {G}_{\uparrow }\}+\{{U}_{{\bf{n}}}({\rm{\pi }})| | A{G}_{\uparrow }\})\times {Z}_{2}^{{\rm{K}}}\, \ltimes \)\(SO(2)\). Here we briefly show how the combination of SO(2) spin symmetry with \(\{{U}_{{\bf{n}}}({\rm{\pi }})| | A\}\) or \(\{{T||A}\}\) will pair two one-dimensional irreps into a two-dimensional irrep for S^± in spin space.

Despite the pure space rotations (\({G}_{\uparrow }^{k}\)) and pure spin rotations from SO(2) symmetry in the little group \({G}_{{\rm{S}}}^{k}\), three symmetry operations account for the two-fold degeneracy in spin space, including \(\{{U}_{{\bf{n}}}({\rm{\pi }})| | A\}\), \(\{{T||A}\}\) and \(\{T{U}_{{\bf{n}}}({\rm{\pi }})| | E\}\). Here we show how these operations act with the \({U}_{{\bf{z}}}(\phi )\) in the \(\{{S}^{+},{S}^{-}\}\) basis.

The matrix representations of spin rotations and the time reversal are written as:

where K is the complex conjugation operator, and we use n = x.

For \(\{{U}_{x}({\rm{\pi }})| | A\}\):

where D_∞ provides two-dimensional irreps for S^±.

For \(\{{T||A}\}\):

where the time reversal binds two conjugated one-dimensional irreps in spin space into a two-dimensional co-irrep.

For \(\{T{U}_{{\bf{x}}}({\rm{\pi }})| | E\}\):

\(\{T{U}_{{\bf{x}}}({\rm{\pi }})| | E\}\) cannot contribute to the new degeneracy.

Therefore, we can conclude that both \(\{{U}_{{\bf{n}}}({\rm{\pi }})| | A\}\) and \(\{{T||A}\}\) can lead to the emergence of two-dimensional irreps in spin space. More details on band representations of collinear SSGs and the construction of magnonic band representation for SSG \(I{}^{\bar{1}}a{}^{\bar{1}}\bar{3}{}^{\infty m}1\) (Cu₃TeO₆ case) can be found in Supplementary Information section 1.6.

Database for collinear SSG symmetry

Starting from 1,421 collinear SSGs, we establish the position-space description of collinear SSGs, where the spin site groups for all spin Wyckoff positions can be defined and characterized by 90 collinear spin point groups. Among these groups, only 32 FM spin point groups support non-zero magnetic moments, resulting in a reduction of the total number of spin Wyckoff positions from 12,481 to 6,368. Meanwhile, we introduce a momentum-space description for collinear SSGs, exhausting the spin Brillouin zones and high-symmetry k points using 24 mapped centrosymmetric symmorphic space groups. Subsequently, by employing band representation theory for SSGs as stated earlier, we derive the irreducible little co-representations of magnons in collinear SSGs and enumerate all band representations hosting unconventional magnons. Finally, we construct k·p effective models around the degenerate points and evaluate the node topology characterized by chiral charge or monopole charge, identifying all unconventional magnons beyond the scope of MSGs. The general positions, spin Wyckoff positions, spin site groups, spin Brillouin zones, k-little co-groups and magnon band representations for all collinear SSGs are available in our online database FINDSPINGROUP.

Density functional theory calculations

All density functional theory (DFT) calculations herein were performed using the projector augmented wave method, implemented in the Vienna Ab initio Simulation Package (VASP)^61,62. The generalized gradient approximation of the Perdew–Burke–Ernzerhof-type exchange-correlation potential⁶³ was adopted. For all candidate materials, we used a cut-off energy of 500 eV, which typically leads to numerical convergence. We used Γ-centred Monkhorst–Pack meshes⁶⁴, with the standard for each direction being the product of the number of k points and a lattice length greater than 45 Å. For the d– and f-electron magnetic atoms, the initial magnetic moments were set to 5 μ_B and 7 μ_B, respectively. To include the effect of electron correlation, the DFT + U approach within the rotationally invariant formalism⁶⁵ was performed with the U_eff values based on the reported value from literature, which are provided in Supplementary Information section 5 for each material. To get an accurate determination of the exchange interactions, a self-consistency convergence within 10⁻⁷ eV was achieved in our calculations. Tight-binding models were constructed from DFT bands using the WANNIER90 package^66,67, and then the TB2J code⁶⁸ was used to extract the magnetic exchange parameters. The spin-exchange cut-off distance was set to truncate when the absolute value of the remaining Heisenberg exchange coefficients J is one-thousandth of the largest J value or numerically less than 0.001 meV. Detailed parameters of Heisenberg exchange interactions for calculating the magnon band structure can be found in Supplementary Information section 6.

Magnon band structure calculations

The magnon band structures were all calculated using the linear spin wave theory and the Heisenberg spin Hamiltonian. The ground-state spin Hamiltonian can be changed into quadratic Hamiltonian by performing the Holstein–Primakoff transformation and Fourier transformation:

where \({\psi }^{\dagger }(k)={({a}_{k1}^{\dagger }\ldots {a}_{{km}}^{\dagger }{a}_{-k1}\ldots {a}_{-{km}})}^{T},H(k)=\left(\begin{array}{cc}h(k) & g(k)\\ {g(k)}^{\dagger } & {h(-k)}^{T}\end{array}\right)\)

where δ_ab is the Kronecker delta, and R and τ represent the lattice translation vector and the position of magnetic ions in the lattice basis, respectively. When S_a is parallel to S_b, α_ab = 1, γ_ab = 1 and λ_ab = 0; when S_a is antiparallel to S_b, α_ab = 0, γ_ab = −1 and λ_ab = −1.

On the basis of the above, the eigenvalues and eigenvectors of magnon Hamiltonian can be calculated by diagonalizing H(k)· I₋, where \({I}_{-}=\left(\begin{array}{cc}{I}_{m} & 0\\ 0 & -{I}_{m}\end{array}\right)\), and I_m is the m-directional identity matrix, where m represents the number of magnetic ions in a primitive cell under SSG.

Topological charges H(k)

We characterize the topology of the degenerate point by calculating the topological charge. For the nodal point, we calculate the Wilson loops on a sphere enclosing the nodal point^69,70:

where θ is the polar angle of the sphere, and \(A(k)={\rm{i}}\langle \psi (k)| \nabla | \psi (k)\rangle \) is the Berry connection.

Now we show the different symmetry operations on A(k):

As a result, the topological charge is zero at P-symmetric k points as PW(θ) = W(π − θ), where Wilson loop is symmetric about the θ = π/2 plane. This conclusion can be generalized to PC_n-invariant k point, where C_n is any proper space rotation. However, the time reversal T and spin rotation U does not give any constrains on the topological charge.

For collinear FMs, when the k-little group is chiral, it can host a non-zero topological charge. For collinear AFMs, the magnon Hamiltonian can be separated into two spin channels with spin angular momentum S = ±1, the two degenerate spin channels should be connected by TA or UA. If the k-little sublattice group \({G}_{\uparrow }^{k}\) is chiral, it can host a non-zero topological charge C_↑ in the spin-up channel. The two spin channels can have identical or opposite topological charges when the two sublattices are connected by proper or improper A. In the former, the topological charge will be doubled as C = 2C_↑, whereas C = 0 in the latter. For the compensated charge with improper A, we can define a monopole charge C₂ = (C_↑ − C_↓)/2.

Therefore, the nodal points of two degenerate branches have opposite (identical) topological charges for type II (IV) SSGs owing to the PT (Uτ) symmetry in the whole-spin Brillouin zone. For type III SSGs, if \({G}_{\uparrow }^{k}\) is chiral and the k-little group G^k contain TA and UA operation, the two magnon branches will degenerate with doubled or compensated topological charge when A is proper or improper. However, when the k-little group does not contain TA and UA operations, the two magnon branches will split and the sign of topological charge in two channels is irrelevant.

Detailed information about the non-zero topological charges of magnonic irreps in collinear SSGs at symmetry-protected degeneracies is provided on our online program FINDSPINGROUP.

Band topology and nonlinear magnons

For collinear magnets, there exists an effective time-reversal symmetry TU_n(π) that squares to one. Consequently, magnons in collinear magnets fall into the symmetry class AI of the Altland–Zirnbauer tenfold classification for topological insulators and superconductors, which does not support strong topological insulating phase in three dimensions^71,72. Recent studies have shown that SOC-free weak topological insulators could exist in altermagnets⁷³. In contrast, under nonlinear spin wave theory, the introduction of magnon–magnon interaction does not break the SO(2) symmetry in collinear magnets. Thus, it does not change the band degeneracy, although it may cause band renormalization⁷⁴. However, adding a SOC term to the collinear spin Hamiltonian typically opens a small gap, sometimes rendering the emergence of symmetry class AII and \({{\mathbb{Z}}}_{2}\) gapped topological phase. For example, the introduction of a Dzyaloshinskii–Moriya interaction in pyrochlore, honeycomb and kagome FMs can transform Dirac magnons into topological magnon bands^75,76,77. In this case, the description of SSGs is still useful in that it can be used to search the SOC-induced small gap, which is often the perquisite of magnon topological materials, such as Chern insulators and axion insulators. Furthermore, the incorporation of the Dzyaloshinskii–Moriya interaction and magnon–magnon interactions can cause multiple topological phase transitions^78,79. Overall, understanding the evolution from node topology to band topology with the introduction of the SOC effect and its impact on magnon transport will be valuable for the development of magnon-based spintronic devices, and is left for future studies.

Comparison between electrons and magnons in collinear SSGs

Importantly, we emphasize that although our focus is on magnon systems, the main results of unconventional quasiparticles originating from band degeneracies can be straightforwardly applied to electronic systems. In the framework of the (magnetic) space group, the double-valued representation of spin-1/2 fermions requires the so-called double group to describe an additional −1 phase of 2π rotation. In sharp contrast, the particularity of collinear SSGs renders that the dimension of the band representations for both fermions and bosons remains invariant within 4π rotation. This is because the infinite spin-only group SO(2) has the double-covering group of itself, and can thus be regarded as either a single group or a double group. Therefore, SO(2) always provides one-dimensional irreps labelled by spin angular momentum m_s = ±1/2 for electrons and m_s = ±1 for magnons. The only difference between fermions and bosons in collinear SSGs occurs solely within the phase in irrep matrices. Detailed information on the comparison between band representations for electrons and magnons in collinear SSGs is provided in Supplementary Information section 1.6.4.