Fermi arcs and pseudogap phase in a minimal microscopic model of $d$-wave superconductivity

We show conclusively that a pseudogap state can arise at $T > T_c$, for reasonable pairing interaction strength, from order parameter fluctuations in a two dimensional minimal model of $d$-wave superconductivity. The occurrence of the pseudogap requires neither strong correlation nor the presence of competing order. We study a model with attractive nearest neighbor interaction and establish our result using a combination of cluster based Monte Carlo for the order parameter field and a twisted-boundary scheme to compute the momentum-resolved spectral function. Apart from a dip in the density of states that characterizes the pseudogap, the momentum and frequency resolution on our effective lattice size $\sim 160 \times 160$ allows two major conclusions: (i)~at $T < T_c$, despite the presence of thermal phase fluctuations the superconductor has only nodal Fermi points while all non nodal points on the normal state Fermi surface show a two peak spectral function with a dip at $\omega =0$, and (ii)~for $T > T_c$ the Fermi points develops into arcs, characterized by a single quasiparticle peak, and the arcs connect up to recover the normal state Fermi surface at a temperature $T^* > T_c$. We show the variation of $T_c$ and $T^*$ with coupling strength and provide detailed spectral results at a coupling where $T^* \sim 1.5T_c$.