  Pseudogap phenomena and the formation of Fermi arcs in underdoped cuprates
are numerically studied in the presence of phase fluctuations that are
simulated by an XY model. Most importantly the spectral function for each Monte
Carlo sample is calculated directly and efficiently by the Chebyshev
polynomials without having to diagonalize the fermion Hamiltonian, which
enables us to handle a system large enough to achieve sufficient
momentum/energy resolution. We find that the momentum dependence of the energy
gap is identical to that of a pure d-wave superconductor well below the
KT-transition temperature ($T_{KT}$), while displays an upturn deviation from
$\cos k_x - \cos k_y$ with increasing temperature. An abrupt onset of the Fermi
arcs is observed above $T_{KT}$ and the arc length exhibits a similar
temperature dependence to the thermally activated vortex excitations.
