Considering a Horndeski scalar-tensor theory of gravity in the bulk (a charged scalar coupled directly to the Einstein tensor), one finds that the AdS/CFT dual is a superconductor with a concentration of impurities (determined by the non-minimal coupling). The impurities hinder the formation of Cooper pairs, the quasiparticles lose energy and hence the critical temperature is lower and the band gap decreases faster with temperature. (This work is carried out in the probe limit, ignoring any backreaction on the metric)

The AdS/CFT correspondence allows one to turn a theory of gravity and a charged scalar (in the bulk) into a theory of a superconductor (on the boundary). Black hole solutions in the bulk have a critical temperature at which the scalar forms a condensate - this condenate then determines the band gap in the superconductor. The case of a scalar minimally coupled to gravity has been well studied, and the dual superconductor conductivity known. Here, we see that the effect of including some non-minimal coupling to gravity is to introduce some impurities into the superconductor which affect its band gap and conduction properties.

Pseudo-linear massive gravity is a ghost-free spin-2 theory with strong coupling at L₃, containing two potentials (which yield the same decoupling limit as the dRGT interactions, with the helicity-0 mode becoming a Galileon) and a non-Einstein two derivative cubic interaction. However, unlike dRGT, there is found to be no region of parameter space in which positivty constraints are obeyed. This suggests that a ghost-free non-linear massive spin-2 theory must contain the Einstein-Hilbert term in order to admit a local, Lorentz invariant UV completeion. Also, a self-interacting massive vector field (whose helicity-0 component also becomes a Galileon in the decoupling limit) can only satisfy posivity constraints if higher derivative terms are present.

The Galileon always marginally breaks the positivity constraints of 2-to-2 scattering (meaning there can be no local Lorentz invariant UV completion). However, it was recently shown that dRGT massive gravity (which contains the Galileon as a decoupling limit) can satisfy these constraints in a small region of parameter space. One might ask if the other possible IR completions of the Galileon can also satisfy these contraints, where the Galileon alone fails. Here we see that pseudo-linear massive gravity cannot (hence no local Lorentz invariant UV completion), but that a (higher derivative) Proca theory can.