We present a theoretical investigation of the electronic band structure and optical properties of a two-dimensional anisotropic semimetal described by a tilted semi-Dirac type spectrum with a pair of Weyl nodes. We observe that a tilt along the quadratic direction can give rise to an energy imbalance between these nodes, contrary to the effect of tilt along the linear direction. We investigate the optical response of such a system subjected to an external ac bias, aiming to probe the energy imbalance between the nodes. We show that the anisotropic interband optical conductivity gives a clear signature of imbalanced nodes by exciting electrons at two different chemical potentials at near zero frequency, where the difference between these two chemical potentials is a direct measure of the energy imbalance. We also investigate the intraband dc conductivity, which reveals that, contrary to the tilted Dirac materials, tilt can convert a semi-Dirac material from a semimetallic phase to a metallic phase. Furthermore, we periodically drive the system with an external time-periodic perturbation to open up a topological gap at those nodes. We also show that the presence of imbalanced Weyl nodes prevents the semi-Dirac material from switching to a Chern topological phase even after opening topological gaps at the nodes, as the bulk remains gapless. Such state cannot be probed by the usual anomalous Hall response, as it will be overshadowed by the bulk contribution. Here, we show that those gaps at different chemical potentials can be probed by optical excitation. Finally, we extend our study to the nonlinear regime, where we particularly focus on second-harmonic generation in an inversion-symmetry-broken tilted semi-Dirac system. A clear signature of energy-imbalanced Weyl nodes can also be detected here.

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• Quantum transport
• Dirac semimetal
• Green's function methods
• Linear response theory