Exploring d-Wave Magnetism in Cuprates from Oxygen Moments

The antiferromagnetic parent phase of high-T$_c$ cuprates has been established as a Néel state of copper moments, but early work pointed out the important role of ligand oxygen orbitals. Using the three-orbital Emery model, we explore how, and under which conditions, doping-induced antiferromagnetic ordering of weak magnetic moments on the oxygen sites can lead to unconventional d-wave magnetism with spin-split electronic bands. The mechanism for forming such altermagnetic (AM) states in cuprates does not rely on a lowering of the crystal symmetry but rather on interaction-induced formation of magnetic moments on directional oxygen orbitals within the crystallographic unit cell. Therefore, we obtain two different types of AM, namely a (0,0)-AM and a ($π$,$π$)-AM. We explore different regimes and challenges for realizing oxygen AM supported by Hartree-Fock calculations and complementary exact diagonalization of small clusters. While the region of interacting parameters needed to realize these states may be difficult to achieve in known high-T$_c$ cuprates, we propose a scenario to realize AM induced by oxygen magnetic moments in a cuprate-based candidate compound using density functional theory and discuss experimental implications.

💡 Research Summary

The paper investigates the possibility of realizing altermagnetism (AM) – a magnetic state that combines spin‑split electronic bands with zero net magnetization – in the CuO₂ planes of high‑T_c cuprates, driven by magnetic moments on the oxygen ligands rather than on copper sites. Using the three‑orbital Emery model (Cu dₓ²₋y² and O pₓ, pᵧ orbitals) the authors explore how interaction‑induced moments on directional oxygen orbitals can generate two distinct AM phases: a (0,0)‑AM that does not break translational symmetry and a (π,π)‑AM that doubles the magnetic unit cell.

The Hamiltonian includes on‑site energies ε_d, ε_p, hopping amplitudes t_pd (Cu‑O) and t_pp (O‑O), and Hubbard repulsions U_d (Cu) and U_p (O). Parameter values realistic for La₂CuO₄ (ε_d = 0 eV, ε_p = 2.2 eV, t_pd = 1.3 eV, t_pp = 0.6 eV, U_d ≈ 8–10 eV, U_p ≈ 4–6 eV) are used as a baseline.

First, a Hartree‑Fock mean‑field treatment is performed. By introducing staggered magnetizations on Cu (m_d) and intra‑unit‑cell staggered magnetizations on the two oxygen orbitals (m_p), together with the Cu occupation n_d, a self‑consistent 12 × 12 Bloch Hamiltonian is solved. The phase diagram shows wide regions where (0,0)‑AM appears for sufficiently large U_p combined with moderate t_pp, while hole doping (δ > 0) lowers the critical U_p and expands the (π,π)‑AM region. The (0,0)‑AM originates from a direct exchange between oxygen sites, whereas the (π,π)‑AM requires redistribution of doped holes onto the oxygen orbitals, enhancing O‑Cu hybridization.

Second, exact diagonalization (ED) on a 4‑Cu + 8‑O cluster with periodic boundary conditions is carried out to capture correlation effects beyond mean‑field. With U_d = 8 eV and U_p = 4 eV, the authors map spin‑spin correlations ⟨S_i·S_j⟩ for Cu‑Cu and O‑O pairs across a range of t_pp and ε_p values, both at half‑filling and at δ = 0.25. The ED results reveal that the oxygen‑oxygen hopping t_pp is the dominant driver of AM, while the dependence on U_p is weak. Increasing |t_pp| first strengthens O‑O antiferromagnetic correlations, leading to a transition from conventional (π,π)‑AFM to (π,π)‑AM, and finally to (0,0)‑AM as t_pp grows. Lowering ε_p (making the charge‑transfer gap smaller) also promotes oxygen moment formation. These findings complement the Hartree‑Fock picture and highlight a hopping‑driven mechanism that is invisible in simple mean‑field theory.

Third, the authors propose a realistic material platform where oxygen‑based AM could be realized. Density‑functional theory calculations (GGA + U with U_d ≈ 8 eV, U_p ≈ 5 eV) on a cuprate‑derived heterostructure (e.g., CuO₂ layers interfaced with a transition‑metal oxide) show that the oxygen p‑orbitals can acquire a finite spin polarization, producing a d‑wave‑like spin splitting of the bands. The calculated band structure exhibits the hallmark of altermagnetism: spin‑dependent Fermi surfaces without net magnetization.

Finally, experimental signatures are discussed. Neutron scattering with spin polarization could detect intra‑unit‑cell staggered oxygen moments, while angle‑resolved photoemission spectroscopy (ARPES) with spin resolution should reveal the characteristic d‑wave spin splitting. Transport measurements may display anisotropic spin Hall or spin‑Nernst effects, reflecting the broken spin‑rotation symmetry despite the absence of a macroscopic magnetization.

In summary, the work demonstrates that, under realistic but somewhat enhanced interaction parameters, oxygen p‑orbitals in cuprates can host magnetic moments that generate altermagnetic order of d‑wave symmetry. It provides a comprehensive theoretical framework (mean‑field, exact diagonalization, DFT) and outlines concrete routes toward experimental verification, thereby extending the conventional copper‑centric view of magnetism in high‑T_c superconductors.