Power law decay of local density of states oscillations near a line defect in a system with semi-Dirac points

We theoretically study the power-law decay behavior of the local density of states (LDOS) oscillations near a line defect in a system with semi-Dirac points by using a low-energy $k\ifmmode\cdot\else\textperiodcentered\fi{}p$ Hamiltonian. We find that the LDOS oscillations are strongly anisotropic and sensitively depend on the orientation of the line defect. We analytically obtain the decay indexes of the LDOS oscillations near a line defect running along different directions by using the stationary phase approximation. Specifically, when the line defect is perpendicular to the linear dispersion direction, the decay index is $\ensuremath{-}5/4$ whereas it becomes $\ensuremath{-}1/4$ if the system is gapped, both of which are different from the decay index $\ensuremath{-}3/2$ in isotropic Dirac systems. In contrast, when the line defect is perpendicular to the parabolic dispersion direction, the decay index is always $\ensuremath{-}1/2$ regardless of whether the system is gapped or not, which is the same as that in a conventional semimetal. In general, when the defect runs along an arbitrary direction, the decay index sensitively depends on the incident energy for a certain orientation of the line defect. It varies from $\ensuremath{-}5/4$ to $\ensuremath{-}1/2$ due to the absence of strict stationary phase point. Our results indicate that the decay index $\ensuremath{-}5/4$ provides a fingerprint to identify semi-Dirac points in 2D electron systems.