  We propose a scheme for the quantum simulation of sub-Ohmic spin--boson
models by color centers in free-standing hexagonal boron nitride (h-BN)
membranes. The electronic spin of a color center that couples to the membrane
vibrational spectrum constitute the physical model. The spin-motion coupling is
provided by an external magnetic field gradient. In this study, we show that a
class of spectral densities can be attained by engineering geometry and
boundary conditions of the h-BN resonator. We then put our focus on two extreme
cases, i.e. $1/f$- and white-noise spectral densities. Spin coherence and
polarization dynamics are studied. Our calculations show coherence revivals at
periods set by the bath characteristic frequency signaling the non-Markovian
nature of the baths. The nonequilibrium dynamics of the spin polarization
exhibits a coherent localization, a property peculiar to the quantum phase
transition in extremely sub-Ohmic spin-boson models. Our scheme may find
application in understanding sources of decoherence in solid-state quantum
bits.
