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 Kosterlitz-Thouless (KT) transition temperature $\left(T_{\text{KT}}\right)$ while displays an upturn deviation from $\text{cos}{k}_x-\text{cos}{k}_y$ with increasing temperature. An abrupt onset of the Fermi arcs is observed above $T_{\text{KT}}$ and the arc length exhibits a similar temperature dependence to the thermally activated vortex excitations.

• Received 6 July 2010

DOI:https://doi.org/10.1103/PhysRevB.82.052503