The discovery of Dirac and Weyl fermions as low-energy quasiparticles in topological materials has revolutionized contemporary physics. These quasiparticles exhibit quantum-mechanical geometric phases, leading to various transport phenomena amplified by Berry phase theory, often exhibiting quantized effects. Beyond DC transport, the low-energy optical responses, reflecting optical transitions between topological bands, are also of significant interest. Nonlinear optical and magneto-optical (MO) effects are enhanced in these materials; spectroscopic studies offer insights into the energy-dependent electronic structure. Two-dimensional Dirac systems are prime examples. A mass gap at the Dirac point, induced by breaking time-reversal symmetry, creates massive Dirac fermions. Controlling the Fermi level within this gap leads to a quantized anomalous Hall response (e²/h), establishing the quantum anomalous Hall (QAH) state. Analogously, MO effects show quantized optical rotation at photon energies below the mass gap. While quantized MO Faraday and Kerr rotations have been observed in the terahertz region in magnetically doped topological insulators, the resonant enhancement of MO responses expected at higher photon energies exceeding the mass gap remains elusive. Recent research has revealed massive Dirac bands in various systems beyond doped materials, potentially exhibiting high-temperature QAH effects. Kagome-lattice magnets, with their unique lattice geometry, frequently host massive Dirac bands strongly coupled to long-range magnetic orders. This allows for intrinsic QAH states without magnetic dopants, ideal for high-temperature realization and investigation of electromagnetic responses. This paper focuses on the low-energy MO resonance stemming from massive Dirac fermions in the kagome-lattice magnet TbMn6Sn6, utilizing broadband MO spectroscopy and first-principles calculations to elucidate the contribution of massive Dirac bands to the observed optical Hall conductivity resonance. The analysis quantifies the overall response function of these fermions and their contribution to the DC anomalous Hall response, relating the findings to universal electrodynamics in the QAH state.

Extensive research has explored Dirac and Weyl fermions in topological materials, focusing on transport phenomena governed by Berry phase theory and the resulting anomalous Hall effect (AHE). Studies have demonstrated the enhancement and quantization of the AHE in various materials, including magnetically doped topological insulators and intrinsic magnetic topological insulators. However, the optical analogues of these effects, particularly the resonant behavior in magneto-optical (MO) response, remain less explored. While theoretical work predicted the giant magneto-optical Kerr effect and universal Faraday effect in topological insulators, experimental verification of resonant enhancement in the optical Hall conductivity remained limited. Recent breakthroughs have unveiled massive Dirac bands in numerous systems, including kagome-lattice magnets. These magnets, possessing unique geometric properties, offer a promising platform for studying the interplay between Dirac fermions and magnetism, leading to the potential for high-temperature QAH states. Previous studies have examined similar systems, highlighting the presence of Dirac fermions and their connection to various transport properties. This work builds upon this foundation by investigating the optical response of these Dirac fermions and relating it to the anomalous Hall effect.

Single crystals of TbMn6Sn6 were grown using the Sn-flux method. Plate-like crystals with (0001) kagome planes as the flat surface were obtained and confirmed by Laue x-ray diffraction. Transport measurements of magneto-resistivity and Hall resistivity were performed using a Physical Property Measurement System (Quantum Design). Optical conductivity spectra were deduced from reflectivity measurements (0.01–4 eV) using Kramers-Kronig analysis. Reflectivity spectra were measured using Fourier-transform and monochromator-type spectrometers. For high-energy extrapolation, reflectivity was assumed proportional to ω⁻². Magneto-optical Kerr rotation spectra were measured using Fourier-transform and monochromator-type spectrometers with a photoelastic modulator (PEM), enabling measurement of Kerr ellipticity and rotation. Single-domain states were prepared by applying magnetic fields (±1 T below 150 K, ±0.5 T above 150 K). The optical Hall conductivity was calculated from optical conductivity and magneto-optical spectra. Density functional theory (DFT) calculations were performed using Quantum ESPRESSO, employing full relativistic projector augmented-wave (PAW) pseudopotentials, the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation, and a k-point mesh of 7 × 7 × 5. A penalty term was used to adjust the exchange splitting. Wannier representation and Kubo-Greenwood formula were used to calculate optical conductivities, both total and orbital-resolved (focusing on Mn 3dxy orbitals). An analytical model, based on the Hamiltonian for a two-dimensional massive Dirac fermion, was used to calculate the optical Hall conductivity using the Kubo formula. Temperature effects were incorporated via the Fermi distribution function.

The study revealed a prominent resonance peak around 0.4 eV in the imaginary part of the optical Hall conductivity (Im σxy(ω)) at 8 K, comparable to the DC AHC (~200 Ω⁻¹cm⁻¹). This resonance persists at higher temperatures. DFT calculations qualitatively reproduced the experimental σxy(ω) spectra, confirming the significant contribution of interband transitions within massive Dirac bands to the 0.4 eV resonance. Restricting the optical transitions to Mn 3dxy orbitals in the DFT calculations still yielded a marked resonance peak at 0.4 eV, further supporting this conclusion. The experimental and calculated spectral features of σxy(ω) were consistent, although some discrepancies (e.g., the sign change in Re σxy(ω), low-energy increase) may be due to contributions from trivial bands. By modifying exchange splitting in the DFT calculations, a clear shift in the resonance peak was observed, confirming its relationship to the Dirac node position. An analytical model for a two-dimensional massive Dirac fermion quantitatively reproduced the experimental optical Hall conductivity at 8 K, yielding band parameters (mass gap, chemical potential) consistent with previous STM studies. The model also accurately predicted the temperature dependence of the optical Hall conductivity due to the Fermi edge smearing at higher temperatures. The AHC arising from the massive Dirac bands (σMDxy(ω=0,T)) was evaluated, accounting for a dominant part of the total DC AHC. The massive Dirac bands contributed almost 15% of the quantized Hall conductance (e²/h) per kagome layer, even at room temperature, suggesting robustness against thermal agitation.

The observed anomalous Hall resonance in TbMn6Sn6 directly demonstrates the significant contribution of massive Dirac fermions to the material's magneto-optical response. The quantitative agreement between experimental results and both DFT calculations and the analytical model firmly establishes the link between the resonance and the underlying band topology. The robustness of the anomalous Hall response at elevated temperatures highlights the inherent stability of the Dirac fermions in this kagome-lattice magnet. The close relationship between the observed magneto-optical resonance and the large DC AHE underscores the significant role of Berry curvature arising from the massive Dirac fermions in the material's electrodynamic properties. This study further connects the findings to the universal electrodynamics of the QAH state, even though the present system deviates from the quantized limit due to the finite energy distance between Fermi level and Dirac point and the presence of trivial bands. The results offer valuable insights into the fundamental properties of massive Dirac fermions and their potential applications in terahertz and infrared magneto-optical devices.

This research demonstrates the anomalous Hall resonance originating from two-dimensional massive Dirac fermions in the kagome-lattice ferrimagnet TbMn6Sn6, using experimental and theoretical spectroscopies. The spectral characteristics, accurately modeled by a simple analytical approach, establish the general optical response directly linked to the robust AHE. Both the MO resonance and the large DC AHE are consequences of the intense Berry curvature from the massive Dirac fermions. The findings relate to the universal electrodynamics of the QAH state, although the system deviates from quantized limits due to finite energy separation between Fermi level and Dirac point and the contribution of other bands. These findings are applicable to other massive Dirac fermion systems and hold promise for future terahertz and infrared magneto-optical applications.

The DFT calculations, while providing a good qualitative agreement, do not perfectly reproduce all aspects of the experimental data. Discrepancies, such as the low-energy increase and sign change in the real part of the optical Hall conductivity, highlight the influence of other bands beyond the massive Dirac bands. Furthermore, the analytical model is a simplification, neglecting factors such as the finite kz dispersion and interactions between the Dirac bands and other energy bands. Finally, the study is limited to TbMn6Sn6. Further investigations of different kagome magnets and the systematic analysis of the relationship between the material parameters and the resulting optical Hall conductivity are important to broaden the findings.