  Hexagonal ferrites do not only have enormous commercial impact ({\pounds}2
billion/year in sales) due to applications that include ultra-high density
memories, credit card stripes, magnetic bar codes, small motors and low-loss
microwave devices, they also have fascinating magnetic and ferroelectric
quantum properties at low temperatures. Here we report the results of tuning
the magnetic ordering temperature in PbFe$_{12-x}$Ga$_x$O$_{19}$ to zero by
chemical substitution $x$. The phase transition boundary is found to vary as
$T_N \sim (1-x/x_c)^{2/3}$ with $x_c$ very close to the calculated spin
percolation threshold which we determine by Monte Carlo simulations, indicating
that the zero-temperature phase transition is geometrically driven. We find
that this produces a form of compositionally-tuned, insulating, ferrimagnetic
quantum criticality. Close to the zero temperature phase transition we observe
the emergence of an electric-dipole glass induced by magneto-electric coupling.
The strong frequency behaviour of the glass freezing temperature $T_m$ has a
Vogel-Fulcher dependence with $T_m$ finite, or suppressed below zero in the
zero frequency limit, depending on composition $x$. These quantum-mechanical
properties, along with the multiplicity of low-lying modes near to the
zero-temperature phase transition, are likely to greatly extend applications of
hexaferrites into the realm of quantum and cryogenic technologies.
