Second-order intrinsic Wiedemann-Franz law

In recent years, the nonlinear anomalous thermal Hall effect has attracted substantial attention. In this paper, we carry out a theoretical exploration of the intrinsic anomalous thermal Hall and Nernst effect that is induced by the thermal Berry connection polarizability. This effect is independent of the relaxation time and can be present in antiferromagnets possessing $\mathcal{PT}$ symmetry. Additionally, we put forward a second-order intrinsic Wiedemann-Franz law, which represents the ratio of the second-order intrinsic thermal conductivity coefficient to the second-order intrinsic electrical conductivity coefficient. When analyzed within a four-band $\mathcal{PT}$ symmetric Dirac model, we observe that the second-order intrinsic thermal conductivity coefficient is linearly proportional to the second-order intrinsic electrical conductivity coefficient and the second-order intrinsic Wiedemann-Franz law is characterized by the chemical potential $\ensuremath{\mu}$ in the low-temperature regime. These findings provide significant implications for experimental verification.