  The two-component cold atom systems with anisotropic hopping amplitudes can
be phenomenologically described by a two-dimensional Ising-XY coupled model
with spatial anisotropy. At low temperatures, theoretical predictions [Phys.
Rev. A 72, 053604 (2005)] and [arXiv:0706.1609] indicate the existence of a
topological ordered phase characterized by Ising and XY disorder but with 2XY
ordering. However, due to ergodic difficulties faced by Monte Carlo methods at
low temperatures, this topological phase has not been numerically explored. We
propose a linear cluster updating Monte Carlo method, which flips spins without
rejection in the anisotropy limit but does not change the energy. Using this
scheme and conventional Monte Carlo methods, we succeed in revealing the nature
of topological phases with half-vortices and domain walls. In the constructed
global phase diagram, Ising and XY type transitions are very close to each
other and differ significantly from the schematic phase diagram reported
earlier. We also propose and explore a wide range of quantities, including
magnetism, superfluidity, specific heat, susceptibility, and even percolation
susceptibility, and obtain consistent results. Furthermore, we observe
first-order transitions characterized by common intersection points in
magnetizations for different system sizes, as opposed to the conventional phase
transition where Binder cumulants of various sizes share common intersections.
The results are useful to help cold atom experiments explore the half-vortex
topological phase.

Quantum Gases;Statistical Mechanics