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We study the interplay between disorder and interactions for emergent bosonicdegrees of freedom induced by an external magnetic field in the Br-dopedspin-gapped antiferromagnetic material Ni(Cl$_{1-x}$Br$_x$)$_2$-4SC(NH$_2$)$_2$(DTNX). Building on nuclear magnetic resonance experiments at high magneticfield [A. Orlova et al., Phys. Rev. Lett. 118, 067203 (2017)], we describe thelocalization of isolated impurity states, providing a realistic theoreticalmodeling for DTNX. Going beyond single impurity localization we use quantumMonte Carlo simulations to explore many-body effects from which pairwiseeffective interactions lead to a (impurity-induced) BEC revival [M. Dupont, S.Capponi, and N. Laflorencie, Phys. Rev. Lett. 118, 067204 (2017)]. We furtheraddress the question of the existence of a many-body localized Bose-glass (BG)phase in DTNX, which is found to compete with a series of a new kind of BECregimes made out of the multi-impurity states. The global magneticfield-temperature phase diagram of DTNX reveals a very rich structure for lowimpurity concentration, with consecutive disorder-induced BEC mini-domesseparated by intervening many-body localized BG regimes. Upon increasing theimpurity level, multiple mini-BEC phases start to overlap, while intermediateBG regions vanish.

physical review. b./physical review. b

Competing Bose-Glass physics with disorder-induced Bose-Einstein  condensation in the doped $S=1$ antiferromagnet  Ni(Cl$_{1-x}$Br$_x$)$_2$-4SC(NH$_2$)$_2$ at high magnetic fields