  We investigate a system of frustrated hardcore bosons, modeled by an XY
antiferromagnet on the spatially anisotropic triangular lattice, using
Takahashi's modified spin-wave (MSW) theory. In particular we implement
ordering vector optimization on the ordered reference state of MSW theory,
which leads to significant improvement of the theory and accounts for quantum
corrections to the classically ordered state. The MSW results at zero
temperature compare favorably to exact diagonalization (ED) and projected
entangled-pair state (PEPS) calculations. The resulting zero-temperature phase
diagram includes a 1D quasi-ordered phase, a 2D Neel ordered phase, and a 2D
spiraling ordered phase. We have strong indications that the various ordered or
quasi-ordered phases are separated by spin-liquid phases with short-range
correlations, in analogy to what has been predicted for the Heisenberg model on
the same lattice. Within MSW theory we also explore the finite-temperature
phase diagram. We find that the zero-temperature long-range-ordered phases turn
into quasi-ordered phases (up to a Berezinskii-Kosterlitz-Thouless
temperature), while zero-temperature quasi-ordered phases become short-range
correlated at finite temperature. These results show that modified spin-wave
theory is very well suited for describing ordered and quasi-ordered phases of
frustrated XY spins (or, equivalently, of frustrated lattice bosons) both at
zero and finite temperatures. While MSW theory, just as other theoretical
methods, cannot describe spin-liquid phases, its breakdown provides a fast
method for singling out Hamiltonians which may feature these intriguing quantum
phases. We thus suggest a tool for guiding our search for interesting systems
whose properties are necessarily studied with a physical quantum simulator.
