Phases of spatially anisotropic triangular antiferromagnet in high magnetic field

We investigate phases of the Heisenberg spin model on a spatially anisotropic triangular lattice as a function of $J'/J <1$ and a magnetic field $H$ ($J$ is the exchange along the horizontal bonds, and $J'$ is the exchange along the diagonal bonds). The anisotropy of $J$'s competes with quantum fluctuations and this competition leads to a rich phase diagram. Immediately below the saturation field $H_s$ we find three phases: three-sublattice commensurate phase, incommensurate co-planar ``fan'' phase, and incommensurate non-coplanar ``cone'' phase. The former two are supersolids while the latter is a superfluid in the terminology of strongly interacting bosons. At a finite boson density ($H < H_s$) and on approach to the fan-cone phase boundary from within the cone phase with ordering momentum $Q$, we observe softening of the ``roton'' minima at momentum $Q'$ different from $-Q$, which one would expect for a direct cone-fan transition. This points on the existence of the intermediate double-spiral state in which boson density exhibits incommensurate modulations with momenta $Q$ and $Q'$. The extrapolation of our results to $H \sim H_s/3$ predicts that $Q'=Q$, and the intermediate state becomes similar to the ``distorted umbrella'' state that emerges out of up-up-down phase. We discuss the implications of our findings for the global phase diagram of the anisotropic triangular Heisenberg antiferromagnet.