Anomalous Lorenz number in massive and tilted Dirac systems

We analytically calculate the anomalous transverse electric and thermal currents in massive and tilted Dirac systems, using $\beta$-borophene as a representative material, and report on conditions under which the corresponding Lorenz number $\left(\mathcal{L}_{an}\right)$ deviates from its classically accepted value $\left(\mathcal{L}_{0}\right)$. The deviations in the high-temperature regime are shown to be an outcome of the quantitative difference in the respective kinetic transport expressions for electric $\left(\sigma\right)$ and thermal $\left(\kappa\right)$ conductivity, and are further weighted through a convolution integral with a non-linearly energy-dependent Berry curvature that naturally arises in a Dirac material. In addition, the tilt and anisotropy of the Dirac system that are amenable to change via external stimulus are found to quantitatively influence $\mathcal{L}_{an}$. The reported deviations from $\mathcal{L}_{0}$ hold practical utility inasmuch as they allow an independent tuning of $\sigma$ and $\kappa$, useful in optimizing the output of thermoelectric devices.