Quantum phase transitions in Heisenberg $J_1-J_2$ triangular antiferromagnet in a magnetic field

We present the zero temperature phase diagram of a Heisenberg antiferromagnet on a frustrated triangular lattice with nearest neighbor ($J_1$) and next nearest neighbor ($J_2$) interactions, in a magnetic field. We show that the classical model has an accidental degeneracy for all $J_2\mathrm{/}{J}_1$ and all fields, but the degeneracy is lifted by quantum fluctuations. We show that at large $S$, for $J_2\mathrm{/}{J}_1<1\mathrm{/}8$, quantum fluctuations select the same sequence of three sublattice co-planar states in a field as for $J_2=0$, and for $1/8<J_2/J_1 <1$ they select the canted stripe state for all non-zero fields. The transition between the two states is first order in all fields, with the hysteresis width set by quantum fluctuations. We study the model with arbitrary $S$, including $S=1\mathrm{/}2$, near the saturation field by exploring the fact that near saturation the density of bosons is small for all $S$. We show that for $S>1$, the transition remains first order, with a finite hysteresis width, but for $S=1\mathrm{/}2$ and, possibly, $S=1$, there appears a new intermediate phase, likely without a spontaneous long-range order.