Designing singularities in electronic dispersions

Abstract

This dissertation is devoted to the study of singularities in electronic dispersions and their implications for electronic systems. First, we consider two-dimensional interacting electrons at a monkey saddle with dispersion ~ $p_x^3 − 3*p_xp_y2$. Such dispersion naturally arises as a singularity when three van Hove saddles merge in an elliptical umbilic elementary catastrophe and can be interpreted as a multicritical Lifshitz point. It can be realized in biased bilayer graphene and can be identified by its signature Landau level behavior $E_m ~ (B_m)^{3/2} and related oscillations in thermodynamic and transport properties, such as Shubnikov-de Haas oscillations, whose period triples as the system crosses the singularity. We show, in the case of a single monkey saddle, that the non-interacting electron fixed point is unstable to interactions under the renormalization group flow, developing either a superconducting instability or non-Fermi liquid features. Biased bilayer graphene, where there are two non-nested monkey saddles at the K and K` points, exhibits an interplay of competing many-body instabilities, namely s−wave superconductivity, ferromagnetism, and spin- and charge-density wave. Next, we show that electronic bands in silicon have nontrivial topological structures that are captured by a network of Berry flux lines. These flux lines link at points of high symmetry in the Brillouin zone, forming singular ice-nodal points where fluxes satisfy ice rules, making silicon a "nodal-chain insulator". This complex Berry-flux network implies a topologically stable two-fold degeneracy along the X-W direction in all of silicon bands. Similarly to nodal-chain semimetals, we find drumhead-like states in the regions that are delimited by the projections of the bulk Berry flux network on the surface Brillouin zone.Finally, we discuss Meissner effect in a nonequilibrium superconducting state. By carefully tuning the system so as to match electron and hole velocities, a singularity in the effective density of states can be achieved leading to a strong non-BCS pairing. We calculate the superfluid density for such a nonequilibrium paired state, and find it to be positive for repulsive interactions and interband pairing. The positivity of the superfluid density implies the stability of the photo-induced superconducting state as well as the existence of the Meissner effect.