Electrons at the monkey saddle: A multicritical Lifshitz point

Abstract

We consider 2D interacting electrons at a monkey saddle with dispersion ∝ p 3x − 3pxp 2 y . 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 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 manybody instabilities, namely s-wave superconductivity, ferromagnetism, and spin-and charge-density wave. PACS numbers:arXiv:1606.04950v1 [cond-mat.str-el]

Classification of critical points in energy bands based on topology, scaling, and symmetry

“…the dispersion becomes binomial y ,Ẽ = sgn(α)E (9) we arrive at the parameter-free binomial form of the canonical dispersion…”

“…The high-order critical points have been proposed occasionally in specific materials under different names in the context of DOS singularity, including the so-called extended VHS [2][3][4][5][6][7][8], multicritical points [9][10][11][12] and highorder VHS [13]. At the energy of a high-order critical point, the DOS can be power-law divergent, stronger than ordinary VHS, and hence we call it a high-order VHS of DOS.…”

“…This work is about the single-particle physics at zero temperature. Interaction effects between ordinary saddle points have been studied intensively [23][24][25][26][27][28][29][30][31][32], and some theoretical treatments have been carried out for highorder critical points [9,33]. The high-order critical points can also affect transport properties at finite temperature such as thermoelectric effect [34].…”

Catastrophe theory classification of Fermi surface topological transitions in two dimensions

“…Singularities and the associated divergence of DOS are a signature of Fermi surface topological transitions [2,3] in which the Fermi surface undergoes a change in topology from electron type to hole type across the critical energy, with two or more branches of the Fermi surface touching at the critical point in a singular way. Historically, when Lifshitz [4] first studied the change of Fermi surface topology, he dealt with two cases: the appearance or collapsing of a neck and the appearance or collapsing of a pocket in the Fermi surface.…”

“…Higher order singularities display more exotic Fermi surface topological transitions. They have recently been associated with phenomena such as the unusual Landau level structure and tripling of de Haas−van Alphen and Shubnikov−de Haas oscillation periods in biased bilayer graphene [2], the non-trivial thermodynamic and transport properties in Sr 3 Ru 2 O 7 [3], correlated electron phenomena in twisted bilayer graphene near half filling [14] and the so called supermetal with diverging susceptibilities in the absence of long range order [15]. In the present work we develop a classification scheme for Fermi surface topological transitions and their associated DOS divergence using catastrophe theory.…”