High magnetic field phases of the $J_1$-$J_2$-$J_3$ triangular antiferromagnet

We present a thorough study of the $J_1$-$J_2$-$J_3$ triangular lattice antiferromagnet close to the saturation field, where the magnetic structure is determined by the condensation of magnons. We focus on the case of ferromagnetic $J_1$, that is particularly rich because frustration effects can allow for magnons of different (commensurate or incommensurate) wave-vectors to condense simultaneously. Our calculation includes an interlayer coupling $J_0$, that can be taken as small as $10^{-4}$ (in units of $J_1$), in which case the system is nearly two-dimensional. Besides the well-known spiral and fan phases, we find a new double-$q$ phase (superposition of two modes), dubbed ``01'' phase, whose features (including a new type of multiferroic behavior) can be seen as intermediate between the two. Furthermore, in some regions of the parameter space, we show that a dilute gas of magnon can not be stable and phase separation (corresponding to a magnetization jump) is expected. Related to this, we discuss the presence of quantum tricritical points. In the $J_1$-$J_2$ model ($J_3=0$) bound states of two and three magnons may also appear, but it is an open issue whether or not they form a stable condensate and then give rise to nematic or octupolar order.