  We present new possibilities for the charge response in the quantum critical
regime in 2+1D using holography, and compare them with field theory and recent
quantum Monte Carlo results. We show that a family of (infinitely many) higher
derivative terms in the gravitational bulk leads to behavior far richer than
what was previously obtained. For example, we prove that the conductivity
becomes unbounded, undermining previously obtained constraints. We further find
a non-trivial and infinite set of theories that have a self-dual conductivity.
Particle-vortex or S duality plays a key role; notably, it maps theories with a
finite number of bulk terms to ones with an infinite number. Many properties
such as sum rules and stability conditions are proved.
