Electrons at the monkey saddle: A multicritical Lifshitz point

We consider two-dimensional interacting electrons at a monkey saddle with dispersion ∝pₓ³-3pₓpy². Such a dispersion naturally arises at the multicritical Lifshitz point when three Van Hove saddles merge in an elliptical umbilic elementary catastrophe, which we show can be realized in biased bilayer graphene. A multicritical Lifshitz point of this kind can be identified by its signature Landau level behavior Eₘ∝(Bm)3/2 and related oscillations in thermodynamic and transport properties, such as de Haas--Van Alphen and 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 noninteracting 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 waves.