  Local quantum criticality in itinerant fermion systems has been extensively
investigated through the soft-gap Anderson impurity model, wherein a localized,
correlated impurity, hybridizes with a broad conduction band with a singular,
$|\omega|^r$, density of states. However, lattice models hosting quantum
critical points (QCPs), do not appear to have such a spectrum emerging at the
QCP. In this work, we report the emergence of such a singular form of the
density of states in a three-orbital lattice model, within dynamical mean field
theory, precisely at a quantum critical point, separating a gapless, Fermi
liquid, metallic phase from a gapped, Mott insulating phase. A
temperature-dependent exponent, $\alpha$, defined using the corresponding
Matsubara self-energy, is found to vary from $+1$ deep in the FL regime, to
$-1$ in the Mott insulator regime. Interestingly, we find that $\alpha$ becomes
temperature independent, and hence isosbestic, precisely at the QCP. The
isosbestic exponent is shown to lead to an emergent soft-gap spectrum,
$|\omega|^r$ at the QCP, where $r = |\alpha_{\rm iso}|$. We discuss the
implications of our findings for non-Fermi liquid behaviour in the quantum
critical region of the phase diagram.
