Topological charge-entropy scaling in kagome Chern magnet TbMn6Sn6

Abstract

In ordinary materials, electrons conduct both electricity and heat, where their charge-entropy relations observe the Mott formula and the Wiedemann-Franz law. In topological quantum materials, the transverse motion of relativistic electrons can be strongly affected by the quantum field arising around the topological fermions, where a simple model description of their charge-entropy relations remains elusive. Here we report the topological charge-entropy scaling in the kagome Chern magnet TbMn₆Sn₆, featuring pristine Mn kagome lattices with strong out-of-plane magnetization. Through both electric and thermoelectric transports, we observe quantum oscillations with a nontrivial Berry phase, a large Fermi velocity and two-dimensionality, supporting the existence of Dirac fermions in the magnetic kagome lattice. This quantum magnet further exhibits large anomalous Hall, anomalous Nernst, and anomalous thermal Hall effects, all of which persist to above room temperature. Remarkably, we show that the charge-entropy scaling relations of these anomalous transverse transports can be ubiquitously described by the Berry curvature field effects in a Chern-gapped Dirac model. Our work points to a model kagome Chern magnet for the proof-of-principle elaboration of the topological charge-entropy scaling.

Fig. 1. Topological quantum oscillations.

a Magnetic phase diagram of TbMn₆Sn₆ when the field B is applied along the crystallographic c axis. Three main regions can be resolved, including the out-plane ferrimagnetic state where Chern-gapped Dirac states are supported, in-plane ferrimagnetic state, and paramagnetic state above 420 K. Inset shows the magnetic Mn kagome lattice and corresponding Chern-gapped Dirac cone at the Brillouin Zone corner of the momentum space. b Quantum oscillations revealed in magneto-Seebeck signal S_xx and magnetoresistance ρ_xx. There is a π/2 phase shift between oscillatory parts of S_xx and ρ_xx. c Landau fan diagram for the oscillations, suggesting a non-trivial Berry phase. d Fast Fourier transform spectrum of the oscillatory component in S_xx and ρ_xx, showing dominant contribution of the α orbit. e Cyclotron mass fitting of oscillatory amplitude to the Lifshitz-Kosevich formulas. f Oscillatory parts in Seebeck signals when the field is tilted away from the c direction. g Angle dependent oscillatory frequencies and corresponding cyclotron mass for orbit α, both showing an inverse cosine behavior.

Fig. 2. Topological transverse transport.

a–c The electric Hall conductivity σ_xy, the Nernst thermopower divided by temperature − S_xy/T and thermal Hall conductivity divided by temperature κ_xy/T at representative temperatures, showing dominant contribution of anomalous terms. Curves are shifted vertically for clarity. The resemblance of σ_xy, S_xy and κ_xy profiles indicates a shared origin from Berry curvature contributions. Insets show the sketches for anomalous Hall, Nernst and thermal Hall effect. d–f The temperature dependence of ${\sigma }_{xy}^A$, $S_{xy}^A$ and ${\kappa }_{xy}^A$, respectively.

Fig. 3. Topological charge-entropy scaling.

a Scaling of the anomalous Hall conductivity. The longitudinal conductivity σ_xx for TbMn₆Sn₆ lies within the good metal region, suggesting a dominant intrinsic contribution. Inset shows a polynomial fitting of the intrinsic Hall conductivity, amounting to 0.13 e²/h per kagome layer. b The pondering function for anomalous Hall conductivity ${\sigma }_{xy}^A$, anomalous thermoelectric Hall conductivity ${\alpha }_{xy}^A$, and the anomalous thermal Hall conductivity ${\kappa }_{xy}^A$ at 100 K and 300 K, respectively, together with the $2\pi {\tilde{\sigma }}_{xy}\left(\epsilon \right)$ for the Chern gapped Dirac fermion with a gap size of Δ. For clarity, the pondering functions at 300 K are multiplied by 3 times. Top-right sketch shows the Chern-gapped Dirac cone with gap size Δ ~ 34 meV and Dirac cone energy E_D ~ 130 meV. c The ratio ${\alpha }_{xy}^A/{\sigma }_{xy}^A$ at different temperatures, approaching k_B/e at around 330 K. This ratio scales with the linear function of k_BT/E_D, which is obtained from the Chern-gapped Dirac model. The error bars reflect the uncertainty in determining the sample size and low temperature anomalous Nernst signals. d The ratio ${\kappa }_{xy}^A/{\sigma }_{xy}^A$ at different temperatures. Above 100 K, this ratio significantly enhances over the T-linear function expected by the Wiedemann-Franz law at elevated temperatures, which matches the $L_{0}T\left(1+\eta k_B^2{T}^2/E_D^2\right)$ behavior for the Chern-gapped Dirac model. The error bars come from the uncertainty in the sample’s geometric factor.