Sign-Free Evidence for a d-Wave Superfluid Stiffness Dome in the Doped Hubbard Model

We construct an effective single-particle Hamiltonian $K_{\mathrm{eff}}$ from Monte Carlo--averaged matrix logarithms of the imaginary-time propagator in determinant quantum Monte Carlo (DQMC). The logarithm maps the multiplicative sign problem into an additive framework where the central limit theorem guarantees convergence, rendering $K_{\mathrm{eff}}$ sign-problem-free: both sign sectors yield identical dispersions to $<1\%$. $K_{\mathrm{eff}}$ captures the exact correlated single-particle spectrum, incorporating all self-energy effects non-perturbatively. Applied to the Hubbard model ($t'/t = -0.30$, $U/t = 4$), $K_{\mathrm{eff}}$ reveals a $d$-wave pseudogap with strong nodal-antinodal dichotomy below a computational phase transition at $T^*$. Three sign-free observables provide evidence consistent with spin-fluctuation pairing: (i) the gap ratio $R_g > 1$ confirms $d$-wave symmetry -- a temperature-independent property of the correlated band structure that provides the medium for pairing; (ii) the superfluid stiffness $\rho_s$ forms a dome across doping at $L = 8$, $10$, and $12$, exceeding the Berezinskii-Kosterlitz-Thouless threshold by $5$-$7\times$ at the dome peak; (iii) $S(\pi,\pi)$ is approximately flat across doping, establishing that the dome originates from Fermi-surface geometry responding to uniform spin-fluctuation glue. The pseudogap grows monotonically toward half-filling while $\rho_s$ forms a dome, mirroring cuprate phenomenology where $T_c$ is limited by the superfluid density (Uemura relation). Vertex corrections remain to be quantified.