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

We show that a magnetic field in the high-symmetry direction lifts the macroscopic classical ground-state degeneracy of the honeycomb $\mathrm{\Gamma }$ model and induces a long-range magnetic order. While a simple spin-polarized state is stabilized for the ferromagnetic $\mathrm{\Gamma }$-exchange, a periodic $\sqrt{3}\times \sqrt{3}$ magnetic order is selected by magnetic field for the antiferromagnetic interaction. We show that the complex spin structure of the tripled unit cell can be described by the magnetization vector and a N'eel order parameter, similar to those for the spin-flop state of a bipartite antiferromagnet. Indeed, the transition from the low-field plaquette-ordered spin liquid to the field-induced magnetic order can be viewed as a generalized spin-flop transition. An accidental O(2) degeneracy associated with rotation symmetry of the N'eel vector is broken by either quantum or thermal fluctuations, leaving a six-fold degenerate ground state. At high fields, the breaking of the ground-state $Z_6$ symmetry is through two Berezinskii-Kosterlitz-Thouless transitions that enclose a critical XY phase.