  We show that the strongly coupled field theory holographically dual to the
Gubser-Rocha anti-de-Sitter Einstein-Maxwell-Dilaton theory describes not a
single non-trivial AdS$_2$ IR fixed point, but a one-parameter family. It is
dual to a local quantum critical phase instead of a quantum critical point.
This result follows from a detailed analysis of the possible quantizations of
the gravitational theory that is consistent with the thermodynamics of the
analytical Gubser-Rocha black hole solution. The analytic Gubser-Rocha black
hole is only a 2-parameter subset of all possible solutions, and we construct
other members numerically. These new numerical solutions correspond to turning
on an additional scalar charge. Moreover, each solution has multiple
holographic interpretations depending on the quantization chosen. In one
particular quantization involving a multitrace deformation the scalar charge is
a marginal operator. In other quantizations where the marginal multitrace
operator is turned off, the analytic Gubser-Rocha black hole does not describe
a finite temperature conformal fluid.
