Frustrated antiferromagnets at high fields: the Bose-Einstein condensation in degenerate spectra

Quantum phase transition at the saturation field is studied for a class of frustrated quantum antiferromagnets. The considered models include (i) the $J_1$-$J_2$ frustrated square-lattice antiferromagnet with $J_2={1/2}J_1$ and (ii) the nearest-neighbor Heisenberg antiferromagnet on a face centered cubic lattice. In the fully saturated phase the magnon spectra for the two models have lines of degenerate minima. Transition into partially magnetized state is treated via a mapping to a dilute gas of hard core bosons and by complementary spin-wave calculations. Momentum dependence of the exact four-point boson vertex removes the degeneracy of the single-particle excitation spectra and selects the ordering wave-vectors at $(\pi,\pi)$ and $(\pi,0,0)$ for the two models. The asymptotic behavior of the magnetization curve differs significantly from that of conventional antiferromagnet in $d$-spatial dimensions. We predict a unique form for the magnetization curve $\Delta M=S-M\simeq \mu^{(d-1)/2}(\log\mu)^{(d-1)}$, where $\mu$ is a distance from the quantum critical point.