Disorder-induced Revival of the Bose-Einstein Condensation at High Magnetic Fields in Ni(Cl$_{1-x}$Br$_x$)$_2$-4SC(NH$_2$)$_2$

Building on recent NMR experiments [arXiv:1607.02360], we theoretically investigate the high magnetic field regime of the disordered quasi-one-dimensional $S=1$ antiferromagnetic material Ni(Cl$_{1-x}$Br$_x$)$_2$-4SC(NH$_2$)$_2$. The interplay between disorder, chemically controlled by Br-doping, interactions, and the external magnetic field, leads to a very rich phase diagram. Beyond the well-known antiferromagnetically ordered regime, analog of a Bose condensate of magnons, which disappears when $H\ge 12.3$ T, we unveil a resurgence of phase coherence at higher field $H\sim 13.6$ T, induced by the doping. Interchain couplings stabilize finite temperature long-range order whose extension in the field -- temperature space is governed by the concentration of impurities $x$. Such a ``mini-condensation'' contrasts with previously reported Bose-glass physics in the same regime by Yu {\it{et al.}} [Nature 489, 379 (2012)], and should be accessible to future experiments.