  The active driving of the electron spin of a color center is known as a
method for the hyperpolarization of the surrounding nuclear spin bath and to
initialize a system with large number of spins. Here, we investigate the
efficiency of this approach for various spin coupling schemes in a
one-dimensional Heisenberg chain coupled to a central spin. To extend our study
to the realistic systems with a large number of interacting spins, we employ an
approximate method based on Holstein-Primakoff transformation. The validity of
the method for describing spin polarization dynamics is benchmarked by the
exact numerics for a small lattice, where the accuracy of the bosonic
Holstein-Primakoff approximation approach is confirmed. We, thus, extend our
analysis to larger spin systems where the exact numerics are out of reach. The
results prove the efficiency of the active driving method when the central spin
interaction with the spin bath is long range and the inter-spin interactions in
the bath spins is large enough. The method is then applied to the realistic
case of optically active negatively charged boron vacancy centers ($V_B$) in
hexagonal boron nitride. Our results suggest that a high degree of
hyperpolarization in the boron and nitrogen nuclear spin lattices is achievable
even starting from a fully thermal bath. As an initialization, our work
provides the first step toward the realization of a two-dimensional quantum
simulator based on natural nuclear spins and it can prove useful for extending
the coherence time of the $V_B$ centers.
