Band warping, band non-parabolicity and Dirac points in fundamental lattice and electronic structures

We demonstrate from a fundamental perspective the physical and mathematical origins of band warping and band non-parabolicity in electronic and vibrational structures. Remarkably, we find a robust presence and connection with pairs of topologically induced Dirac points in a primitive-rectangular lattice using a $p$-type tight-binding approximation. We provide a transparent analysis of two-dimensional primitive-rectangular and square Bravais lattices whose basic implications generalize to more complex structures. Band warping typically arises at the onset of a singular transition to a crystal lattice with a larger symmetry group, suddenly allowing the possibility of irreducible representations of higher dimensions at special symmetry points in reciprocal space. Band non-parabolicities are altogether different higher-order features, although they may merge into band warping at the onset of a larger symmetry group. Quite separately, although still maintaining a clear connection with that merging, band non-parabolicities may produce pairs of conical intersections at relatively low-symmetry points. Apparently, such conical intersections are robustly maintained by global topology requirements, rather than any local symmetry protection. For two $p$-type tight-binding bands, we find such pairs of conical intersections drifting along the edges of restricted Brillouin zones of primitive-rectangular Bravais lattices as lattice constants vary relatively, until they merge into degenerate warped bands at high-symmetry points at the onset of a square lattice. The conical intersections that we found appear to have similar topological characteristics as Dirac points extensively studied in graphene and other topological insulators, although our conical intersections have none of the symmetry complexity and protection afforded by the latter more complex structures.