  We investigate the role of disorder in a two-dimensional semi-Dirac material
characterized by a linear dispersion in one, and a parabolic dispersion in the
orthogonal, direction. Using the self-consistent Born approximation, we show
that disorder can drive a topological Lifshitz transition from an insulator to
a semi-metal, as it generates a momentum independent off-diagonal contribution
to the self-energy. Breaking time-reversal symmetry enriches the topological
phase diagram with three distinct regimes-- single-node trivial, two-node
trivial and two-node Chern. We find that disorder can drive topological
transitions from both the single- and two-node trivial to the two-node Chern
regime. We further analyze these transitions in an appropriate tight-binding
Hamiltonian of an anisotropic hexagonal lattice, by calculating the real-space
Chern number. Additionally we compute the disorder-averaged entanglement
entropy which signals both the topological Lifshitz and Chern transition as a
function of the anisotropy of the hexagonal lattice. Finally, we discuss
experimental aspects of our results.
