We present the phase diagram of a 2D isotropic triangular Heisenberg antiferromagnet in a magnetic field. We consider spin-S model with nearest-neighbor (J1) and next-nearest-neighbor (J2) interactions. We focus on the range of 1/8<J2/J1<1, where the ordered states are different from those in the model with only nearest-neighbor exchange. A classical ground state in this range is infinitely degenerate in any field. The actual order is then determined by quantum fluctuations via "order from disorder" phenomenon. We argue that the phase diagram is rich due to competition between competing quantum states which break either orientational or sublattice symmetry. At small and high fields, the ground state is a canted stripe state, which breaks orientational symmetry, but at intermediate fields the ordered states break sublattice symmetry. The most noticeable of such states is "three up, one down" state in which spins in three sublattices are directed along the field and in one sublattice opposite to the field. In such a state, magnetization is quantized at exactly one half of the saturation value. We identify gapless states, which border the "three up, one down" state and discuss the transitions between these states and the canted stripe state.
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Acknowledgments . We acknowledge with thanks useful conversations with C. Batista, A. Coldea, R. Coldea, S.-W. Cheong, J. Kang, N. Perkins, and O. Starykh. We are thankful to R. Coldea and C. Sun for careful reading of the manuscript and useful comments. The work was supported by the NSF DMR-1523036.