Proximity-induced sequence of field transitions in Kitaev candidate BaCo$_2$(AsO$_4$)$_2$

We study field-induced phase transitions of the minimal exchange model proposed earlier for BaCo$_2$(AsO$_4$)$_2$, a candidate for Kitaev honeycomb model, using numerical minimization of classical spin clusters. We show that experimentally observed sequence of step-like transitions in magnetic field is realized in the phase diagram of the minimal model. Surprisingly, intermediate up-up-down plateau phase is stabilized only in the proximity of a double-zigzag$-$zigzag phase boundary. We systematically map out the region of stability of experimentally observed cascade of transitions and argue that BaCo$_2$(AsO$_4$)$_2$ exchange parameters are close to a region of strong phase competition, which can explain suppressed saturation field.