Altermagnetism Without Crystal Symmetry

Altermagnetism is a collinear magnetic order in which opposite spin species are exchanged under a real-space rotation. Hence, the search for physical realizations has focussed on crystalline solids with specific rotational symmetry. Here, we show that altermagnetism can also emerge in non-crystalline systems, such as amorphous solids, despite the lack of global rotational symmetries. We construct a Hamiltonian with two directional orbitals per site on an amorphous lattice with interactions that are invariant under spin rotation. Altermagnetism then arises due to spontaneous symmetry breaking in the spin and orbital degrees of freedom around each atom, displaying a common point group symmetry. This form of altermagnetism exhibits anisotropic spin transport and spin spectral functions, both experimentally measurable. Our mechanism generalizes to any lattice and any altermagnetic order, opening the search for altermagnetic phenomena to non-crystalline systems.

💡 Research Summary

The paper challenges the prevailing view that altermagnetism—a collinear magnetic order characterized by a combined spin‑flip and real‑space rotation symmetry (C₄·T)—requires a crystalline lattice with global rotational symmetry. The authors demonstrate that altermagnetism can emerge in amorphous solids where no such translational or rotational order exists. They construct a minimal t‑J‑like model on a three‑coordinated amorphous network, assigning each site two directional d‑orbitals (dxz and dyz). The kinetic term features orbital‑dependent nearest‑neighbor hopping: a strong σ‑type hopping t₁ along the orbital’s principal axis and a weaker π‑type hopping t₂ perpendicular to it. When t₁≠t₂ the band structure is split in orbital space, providing a natural anisotropy.

The interaction term is a ferromagnetic Heisenberg‑type spin‑orbital coupling that respects global SU(2)ₛₚᵢₙ × SU(2)ₒᵣᵦᵢₜ symmetry. For J>0 it energetically favors parallel spins within the same orbital and antiparallel spins between orthogonal orbitals. Because the kinetic part reduces the symmetry to SU(2)ₛₚᵢₙ × SO(2)ₒᵣᵦᵢₜ, the full Hamiltonian only retains a global spin rotation and a real‑space SO(2) rotation on average.

In a real‑space Hartree–Fock mean‑field treatment on a 400‑site amorphous lattice, the authors self‑consistently determine the local charge ⟨n_j⟩ and the “alter‑magnetization” ⟨m_j⟩ = ⟨σ_z τ_z⟩. The phase diagram as a function of electron filling n and interaction strength J (with t₁ = 1, t₂ = ½) reveals three distinct phases: (i) a trivial metal at low J where all four spin‑orbital states are degenerate and ⟨m⟩ = 0; (ii) a charge‑density‑wave (CDW) region near half‑filling where the interaction reduces to an effective nearest‑neighbor repulsion, leading to frustrated ordering on the non‑bipartite amorphous network; (iii) an altermagnetic phase at large J where ⟨m⟩ becomes finite. In this phase the system spontaneously selects a configuration in which, for example, the x‑orbitals host spin‑up electrons while the y‑orbitals host spin‑down electrons. This “spin‑orbital locking” yields anisotropic electronic dispersion: spin‑up carriers move preferentially along the x‑direction, spin‑down carriers along y. Consequently, the spin‑resolved band structure is split even though the net magnetization vanishes, fulfilling the defining property of altermagnets.

To probe this splitting, the authors compute a spin‑resolved spectral function A_s(ω,p) by projecting eigenstates onto plane‑wave states |p s μ⟩. Although crystal momentum is not a good quantum number in an amorphous system, the overlap is well‑defined near the Γ point, allowing a quasi‑Brillouin‑zone analysis. The spectral function shows a clear spin‑difference A_↑ − A_↓ in the altermagnetic regime, while it vanishes in the trivial metal. This quantity is directly measurable via spin‑resolved angle‑resolved photoemission spectroscopy (spin‑ARPES), which has already been applied to amorphous Bi₂Se₃ and related materials.

The authors also discuss transport signatures: the anisotropic spin conductivity tensor reflects the underlying spin‑orbital locking, offering an alternative experimental probe through non‑local spin‑Hall or spin‑Seebeck measurements. Notably, the altermagnetic phase is most robust at high fillings where the electronic states are strongly sensitive to the underlying disorder, suggesting that amorphous disorder can actually stabilize altermagnetism rather than suppress it.

Importantly, the mechanism relies only on local SU(2)ₛₚᵢₙ × SO(2)ₒᵣᵦᵢₜ symmetry and does not depend on any specific lattice geometry. Therefore it can be generalized to any non‑crystalline network, to higher‑order altermagnetic orders, and even to systems where the orbital degrees of freedom are represented by other anisotropic states (e.g., p‑orbitals, molecular orbitals). This opens a broad new research direction: searching for altermagnetic phenomena in glasses, amorphous thin films, and possibly liquid metals or strongly disordered alloys. Future work may explore material candidates (e.g., amorphous transition‑metal oxides, chalcogenide glasses), the role of spin‑orbit coupling, and external control via strain or electric fields. In summary, the paper establishes that global crystal symmetry is not a prerequisite for altermagnetism; instead, local spin‑orbital interactions and spontaneous symmetry breaking can generate the characteristic spin‑split band structure in fully disordered solids.