The interplay between correlated electrons, lattice structure, and magnetism in 2D materials with open d- or f-shells leads to fascinating physical phenomena. Reduced dimensionality in these materials enhances electron correlation due to suppressed screening. Recent discoveries of magnetic 2D van der Waals (vdW) materials have broadened the applications of 2D materials to include magnetic functionalities. The magnetism in these materials is highly tunable via pressure, stacking arrangements, and external fields, opening possibilities for energy-efficient spintronic devices. Even in bulk form, the reduced coordination numbers in quasi-2D lattices restrict electron hopping, increasing the importance of Coulomb interactions and directly influencing exchange interactions. A critical challenge is to accurately model and understand magnetic excitations in these systems. Despite considerable research, accurate ab initio descriptions of spin excitations and a complete understanding of magnetic interactions in magnetic 2D vdW materials (m2Dv) remain elusive. Spin-orbit coupling (SOC) plays a complex role, and the presence and influence of the Dzyaloshinskii-Moriya interaction (DMI) – crucial for topological magnons and skyrmions – are not fully understood. 2D honeycomb ferromagnets are analogous to graphene, and earlier theoretical work suggested that next-nearest-neighbor DMI could induce a spin-wave (SW) gap at Dirac points. Inelastic neutron scattering (INS) experiments on CrI₃ showed a gap along high-symmetry lines, indicating a possible sizable DMI or Kitaev interaction. However, ab initio studies have shown that both DMI and Kitaev interactions in pristine CrI₃ are insufficient to explain the observed gap size. Thus, the role of relativistic magnetic interactions remains unclear. A reinvestigation of exchange interactions beyond density functional theory (DFT), to address electron-correlation effects, is necessary.

Existing theoretical investigations of magnetic interactions in m2DVs primarily rely on mapping total energies of various collinear spin configurations onto the Heisenberg model, often using DFT + U. However, the accuracy and applicability of this method for CrI₃ are questionable, with DFT overestimating exchange couplings and U corrections further exacerbating the discrepancy with experimental findings. The magnetic force theorem (MFT) approach, while more suitable for small spin deviations from the ground state, yields varying and inconclusive results. Discrepancies likely arise from the details of tight-binding Hamiltonian constructions and Green's functions, often assisted by the Wannier function approach. Direct calculation of dynamic transverse spin susceptibility (DTSS), while ideal, presents computational challenges and is often limited to simple systems due to its computational demands. Previous studies have mostly focused on simple systems, and complexities arise in the evaluation of the exchange-correlation kernel and ensuring the Goldstone theorem beyond mean-field schemes.

This study uses CrI₃, a well-studied m2Dv, to determine the role of electron correlations on magnetic interactions and excitations. The dynamic transverse spin susceptibility (DTSS), which characterizes spin excitations, was calculated and compared to SW spectra from INS experiments. Electron correlation effects beyond DFT were included using the quasiparticle self-consistent GW (QSGW) method, incorporating on-site and off-site nonlocal potentials. DFT + U methods were also employed for comparison. The QSGW method explicitly calculates nonlocal potentials, including both on-site and long-range off-site components, which directly influences the relative positions of cation-3d and anion-p bands and thus the indirect exchange interactions of Cr ions via iodine sites. The bare transverse spin susceptibility χ₀(r, r′, q, ω) was calculated using a linear response method, and the full transverse susceptibility χ was calculated within the random phase approximation (RPA). Because magnetic moments and excitations are largely confined to Cr sites, χ₀ was calculated on a product basis and projected onto the local spin densities of Cr pairs. This discretization allowed for determination of the exchange-correlation kernel using a sum rule, mapping χ onto the Heisenberg model to extract pair exchange parameters, and reduced computational effort. The Goldstone magnon mode at q=0 and ω=0 was ensured by determining the kernel to satisfy the sum rule. High-resolution SW spectra were obtained by calculating the real-space bare susceptibility on a dense R-mesh and performing a Fourier transform. The study considered both the low-temperature rhombohedral (R-CrI₃) and high-temperature monoclinic (M-CrI₃) structures of CrI₃, noting the sensitivity of interlayer Cr coupling to stacking arrangement and the need for explicit treatment of interlayer exchanges. Exchange parameters for a Heisenberg model (H = -Σ_ij J_ijê_i·ê_j) were extracted from the inverse of the susceptibility matrix using a Fourier transform. The linear SW theory was employed to recalculate SW spectra with these parameters. The effects of electron correlations were investigated using DFT + U and QSGW. The study also used QSGW + U to investigate the effects of additional on-site interactions.

DFT calculations overestimated SW energies, particularly along the interlayer direction. DFT + U reduced the SW energy at the Z-point but still overestimated it by a factor of three compared to experimental results. QSGW significantly improved the agreement of the calculated SW spectra with INS data, particularly reducing the acoustic SW energies. The application of QSGW on top of DFT + U (QSGW + U) yielded further improvement in agreement with the experimental data. Calculations revealed a correlation-enhanced interlayer super-superexchange (J₂), absent in DFT calculations but present in DFT + U and QSGW calculations. This interaction, corresponding to a Cr-I-I-Cr super-superexchange path, induces a magnon gap along the Γ-K-M path. This gap opening is unexpected without relativistic exchanges, such as DMI or Kitaev interactions, previously thought to be responsible for the gap. The study also demonstrates that the explicit treatment of electron correlations correctly predicts the stacking-dependent magnetic ordering (A-type AFM ordering) in M-CrI₃, while DFT predicts the wrong FM ground state. The calculated gap size (~1.8 meV in QSGW + U) is smaller than the experimental value (~4 meV). The gap is caused by the combination of vanishing J₂₀ and correlation-enhanced J₂, not the equivalence of J₂₀ and J₂ as previously assumed. Calculations revealed helical Dirac nodal lines in the SW spectra, winding around the edges of the hexagonal Brillouin zone. Calculations showed that in M-CrI₃, the acoustic SW calculated with the FM configuration is negative along the Γ-Z path, indicating the instability of the FM interlayer configuration. In contrast, the SW spectra calculated with the AFM configuration are positive throughout the Brillouin zone.

The findings address the long-standing challenge of accurately describing spin excitations and exchange interactions in 2D magnetic vdW materials. The significant improvement in SW spectra obtained using QSGW demonstrates the crucial role of nonlocal electron correlations in determining magnetic interactions, particularly interlayer couplings. The discovery of the correlation-enhanced interlayer super-superexchange mechanism for gap opening in CrI₃ provides a new perspective and challenges previous explanations based on relativistic interactions. The ability to correctly predict the stacking-dependent magnetic ordering in CrI₃ further validates the importance of explicitly including electron correlations. The smaller calculated gap size compared to experiments suggests that additional factors, or more comprehensive theoretical frameworks (e.g., dynamical mean-field theory (DMFT) + QSGW), should be considered in future studies.

This study demonstrates that ab initio calculations including explicit electron correlations accurately describe SW spectra and stacking-dependent magnetism in CrI₃. A correlation-enhanced interlayer super-superexchange interaction is identified as a key mechanism for gap opening, contrasting with previous explanations. Future research should focus on experimental verification through high-resolution INS or other techniques to confirm the predicted gap and nodal lines, ideally in monolayer CrI₃ to isolate the interlayer exchange effects. The work emphasizes the need for explicit treatment of electron correlations in understanding layered magnetic materials, including those with topological properties.

The calculated gap size in QSGW + U is smaller than the experimental value, suggesting that additional subtle interactions or effects not included in the model might contribute. The rigid spin approximation used in the calculations may also introduce some limitations. More comprehensive theoretical approaches, such as DMFT + QSGW, could be explored to improve accuracy. Experimental verification of the predicted gap in monolayer CrI₃ is challenging due to the difficulty in performing INS measurements on such a small system.