  We analyze the dynamical response functions of strongly interacting quantum
critical states described by conformal field theories (CFTs). We construct a
self-consistent holographic model that incorporates the relevant scalar
operator driving the quantum critical phase transition. Focusing on the finite
temperature dynamical conductivity $\sigma(\omega,T)$, we study its dependence
on our model parameters, notably the scaling dimension of the relevant
operator. It is found that the conductivity is well-approximated by a simple
ansatz proposed by Katz et al [1] for a wide range of parameters. We further
dissect the conductivity at large frequencies $\omega >> T$ using the operator
product expansion, and show how it reveals the spectrum of our model CFT. Our
results provide a physically-constrained framework to study the analytic
continuation of quantum Monte Carlo data, as we illustrate using the O(2)
Wilson-Fisher CFT. Finally, we comment on the variation of the conductivity as
we tune away from the quantum critical point, setting the stage for a
comprehensive analysis of the phase diagram near the transition.
