Effects of inhomogeneities and thermal fluctuations on the spectral function of a model d-wave superconductor

We compute the spectral function of a model for high-temperature superconductors, at both zero and finite temperatures $T$. The model consists of a two-dimensional BCS Hamiltonian with $d$-wave symmetry, which has a spatially varying, thermally fluctuating, complex gap $\Delta$. Thermal fluctuations are governed by a Ginzburg-Landau free energy functional. We assume that a fraction $c_{\beta}$ of the superconductor area has a large $\Delta$ ($\beta$ regions), while the rest has a smaller $\Delta$ ($\alpha$ regions). $\alpha$ and $\beta$ regions are randomly distributed in space. We find that the inhomogeneous gap distribution of $\Delta$ affects the spectral function primarily near $\mathbf k = (\pi,0)$. For $c_{\beta}\simeq 0.5$, a split band appears if the difference between the gap magnitudes in the $\alpha$ and $\beta$ regions is sufficiently large; otherwise, the band is only broadened. Thermal fluctuations also affect the spectral function most strongly near $\mathbf k = (\pi,0)$, where the peaks that are sharp and high at zero temperature become reduced, widened, and shifted toward smaller energies as $T$ increases through the Kosterlitz-Thouless transition temperature.