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

Xidi Wang, H. Q. Lin

Abstract

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 $ρ_s$ 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) $S(π,π)$ 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 $ρ_s$ forms a dome, mirroring cuprate phenomenology where $T_c$ is limited by the superfluid density (Uemura relation). Vertex corrections remain to be quantified.

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

Abstract

We construct an effective single-particle Hamiltonian 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 sign-problem-free: both sign sectors yield identical dispersions to . captures the exact correlated single-particle spectrum, incorporating all self-energy effects non-perturbatively. Applied to the Hubbard model (, ), reveals a -wave pseudogap with strong nodal-antinodal dichotomy below a computational phase transition at . Three sign-free observables provide evidence consistent with spin-fluctuation pairing: (i) the gap ratio confirms -wave symmetry -- a temperature-independent property of the correlated band structure that provides the medium for pairing; (ii) the superfluid stiffness forms a dome across doping at , , and , exceeding the Berezinskii-Kosterlitz-Thouless threshold by - at the dome peak; (iii) 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 forms a dome, mirroring cuprate phenomenology where is limited by the superfluid density (Uemura relation). Vertex corrections remain to be quantified.

Paper Structure

This paper contains 42 sections, 22 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model and method
  3. The $K_{\mathrm{eff}}$ construction
  4. Sign-sector decomposition
  5. Gap extraction
  6. Superfluid stiffness from $K_{\mathrm{eff}}$
  7. Simulation parameters
  8. Sign transition and computational phase diagram
  9. The $T^*$ boundary
  10. Finite-size dependence of the $T^*$ valley
  11. Correlation with physical observables
  12. Pseudogap spectroscopy
  13. $d$-wave pseudogap
  14. Superfluid stiffness dome
  15. Dome structure and finite-size scaling
  16. ...and 27 more sections

Figures (4)

  • Figure 1: Computational phase diagram ($L = 12$): $\mathrm{Var}(\mathrm{sign})$ across temperature and doping. Important:$\mu = 0$ does not correspond to half-filling when $t'/t = -0.30$; the particle-hole asymmetry from $t'$ shifts half-filling to $\mu/t \approx +0.6$, outside the plotted range. Green ($\mathrm{Var} \approx 0$): sign conserved (strange metal/ Fermi liquid regime above $T^*$). Red ($\mathrm{Var} \approx 1$): sign fluctuates maximally (pseudogap regime below $T^*$). The region below $T^*$ is uniformly red because antiferromagnetic correlations are strong enough at all dopings to produce maximal sign cancellations once the pseudogap opens; $\mathrm{Var}(\mathrm{sign})$ saturates at $1$ and cannot resolve further structure, but the sign-free $K_{\mathrm{eff}}$ observables ($R_g$, $\rho_s$) do vary across this region (Fig. \ref{['fig:dome']}). The white contour marks $T^*(\mu)$ at $\mathrm{Var} = 0.5$; the valley near $\mu/t \approx -0.85$ to $-1.00$ identifies optimal doping where the pseudogap onset is lowest.
  • Figure 2: Dressed quasiparticle band structure $\varepsilon_{K_{\mathrm{eff}}}(\bm{k})$ (red) compared to density-matched free fermions (blue) along $\Gamma \to (\pi,0) \to (\pi,\pi) \to (\pi/2,\pi/2) \to \Gamma$ on the $L = 12$ grid. The overall bandwidth increases by $\sim$20% (from $8.0t$ to $\sim$9.6t), but the key feature is the momentum-dependent self-energy: $\Sigma(\pi,0) = -0.19t$ (antinode pushed toward $E_F$) versus $\Sigma(\pi/2,\pi/2) = +0.09t$ (node pushed away), producing the $d$-wave anisotropy that drives the pseudogap. Parameters: $U/t = 4$, $t'/t = -0.30$, $\mu = -0.60$, $\beta = 24$.
  • Figure 3: Three sign-free $K_{\mathrm{eff}}$ observables versus $\mu/t$ for $L = 8$ (left), $L = 10$ (center), and $L = 12$ (right). Top row (a--c): mean-field superfluid stiffness $\rho_s$; the dome strengthens and narrows with cooling at all system sizes. Middle row (d--f): gap ratio $R_g > 1$ confirms $d$-wave pseudogap symmetry throughout the dome. Bottom row (g--i): $S(\pi,\pi)$ is approximately flat across doping. Open symbols in (c): points with $R_g < 1$, where $\rho_s$ reflects metallic response rather than superconducting stiffness. Parameters: $U/t = 4$, $t'/t = -0.30$.
  • Figure 4: Temperature scan at the dome peak ($\mu/t = -0.90$, $L = 12$). Blue circles (left axis): $R_g$ remains stable at $1.55$--$1.65$, confirming that the $d$-wave pseudogap is temperature-independent below $T^*$. Red squares (right axis): the dressed-particle stiffness $\rho_s$ grows on cooling as the Fermi function sharpens. Gray bars: BKT thresholds $2T/\pi$ at each temperature.