Superconducting quantum criticality in three-dimensional Luttinger semimetals

Abstract

We study a simple model of three-dimensional fermions close to a quadratic band touching point, built from the celebrated Luttinger single-particle Hamiltonian and an attractive contact interaction between the particles. Such a system displays a quantum critical point between the semimetallic and an $s$-wave superconducting phase at which the low-energy ``Luttinger fermions'' are inextricably coupled to the order parameter fluctuations. The quantum critical point is perturbatively accessible near four spatial dimensions, where it features nontrivial scaling with dynamical exponent $1<z<2$ and emergent rotational and particle-hole symmetries. Some features of the criticality, such as oscillatory corrections to scaling and its enhanced symmetry, are discussed.