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We study defect energy levels in hexagonal boron-nitride with varying numberof layers using a fragment many-body $GW$ formalism, taking as examples theparadigmatic carbon-dimer and $C_BV_N$ defects. We show that a single layer canbe fragmented in polarizable finite-size areas reproducing faithfully theeffect of the dielectric environment, dramatically facilitating the study atthe many-body level of point defects in the dilute limit. The evolution ofdefect energy levels from the monolayer to a $n$-layer system due to increasedscreening, labeled polarization energies, follow a simple $({\Delta P}/n +P_{\infty})$ behavior. The coefficients $\Delta P$ and $P_{\infty}$ are foundto be close-to-universal, with opposite signs for holes and electrons,characterizing mainly the host and the position of the defect (surface orbulk), but hardly the defect type. Our results rationalize the evolution ofdefect energy levels with layers number, allowing to safely extrapolate resultsobtained for the monolayer to few-layers, surface or bulk \textit{h}-BN. Thepresent many-body fragment approach further opens the door to studyingdisordered 2D layers.

Universal polarization energies for defects in monolayer, surface and bulk hexagonal boron nitride : A finite-size fragments GW approach