  Iridates provide a fertile ground to investigate correlated electrons in the
presence of strong spin-orbit coupling. Bringing these systems to the proximity
of a metal-insulator quantum phase transition is a challenge that must be met
to access quantum critical fluctuations with charge and spin-orbital degrees of
freedom. Here, electrical transport and Raman scattering measurements provide
evidence that a metal-insulator quantum critical point is effectively reached
in 5 % Co-doped Sr$_2$IrO$_4$ with high structural quality. The dc-electrical
conductivity shows a linear temperature dependence that is successfully
captured by a model involving a Co acceptor level at the Fermi energy that
becomes gradually populated at finite temperatures, creating
thermally-activated holes in the $J_{\text {eff}}=1/2$ lower Hubbard band. The
so-formed quantum critical fluctuations are exceptionally heavy and the
resulting electronic continuum couples with an optical phonon at all
temperatures. The magnetic order and pseudospin-phonon coupling are preserved
under the Co doping. This work brings quantum phase transitions, iridates and
heavy-fermion physics to the same arena.
