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We show that a magnetic field in the high-symmetry direction lifts themacroscopic classical ground-state degeneracy of the honeycomb $\Gamma$ modeland induces a long-range magnetic order. While a simple spin-polarized state isstabilized for the ferromagnetic $\Gamma$-exchange, a periodic $\sqrt{3}\times\sqrt{3}$ magnetic order is selected by magnetic field for theantiferromagnetic interaction. We show that the complex spin structure of thetripled unit cell can be described by the magnetization vector and a N\'eelorder parameter, similar to those for the spin-flop state of a bipartiteantiferromagnet. Indeed, the transition from the low-field plaquette-orderedspin liquid to the field-induced magnetic order can be viewed as a generalizedspin-flop transition. An accidental O(2) degeneracy associated with rotationsymmetry of the N\'eel vector is broken by either quantum or thermalfluctuations, leaving a six-fold degenerate ground state. At high fields, thebreaking of the ground-state $Z_6$ symmetry is through twoBerezinskii-Kosterlitz-Thouless transitions that enclose a critical XY phase.

Honeycomb-lattice Gamma model in a magnetic field: hidden N\'eel order and spin-flop transition