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arXiv:2604.01737

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

Xidi Wang, H. Q. Lin

Published Apr 2, 2026• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2604.01737

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We compute high-resolution angle-resolved photoemission spectroscopy of the Hubbard model using the unbiased determinant quantum Monte Carlo algorithm, revealing an asymmetry between electron and hole doping. Electron doping exhibits more coherent quasiparticles and stronger antiferromagnetic correlations compared to hole doping. At low doping, a nodal-antinodal dichotomy on the Fermi surface is observed, similar to cuprate experiments. The dichotomy reflects the momentum dependence of the Mott gap, as manifested in both the spectral function and the self-energy. For hole doping, we observe a transition towards the pseudogap, without signature of pocket formation. The simulated nuclear magnetic resonance pseudogap temperatures do not necessarily agree with the temperature determined by spectroscopy. These findings collectively suggest the pseudogap is a smooth crossover driven by strong correlations.
While it is over 30 years since the high-T c superconductor was discovered, the cuprates have attracted great interests in many scientific fields.Understanding high-T c is one of the most challenging issues in the condensed matter theory.The mechanism of the superconductivity in these materials is still not fully understood, and continues to be discussed intensely.Specifically, the two-dimensional repulsive Hubbard model, which is a representative model for the cuprates, still has many questions, and many analytical and numerical methods have been proposed for studying the superconductivity, magnetism and other instabilities.In the present thesis, we propose to combine the non-perturbative dynamical mean field theory (DMFT) with the perturbative diagrammatic expansion techniques to treat the local correlation effects ("Mott physics") and momentum dependent pairing interaction (that produces the d-wave pairing superconductivity) to explore the superconductivity and various instabilities in the Hubbard model.First, we formulate the combination of the fluctuation exchange (FLEX) approximation and DMFT in a self-consistent manner in terms of the Luttinger-Ward functional for superconducting phases.The resultant phase diagram exhibits a clear T c dome structure against the band filling as observed experimentally, while in the ordinary FLEX, T c almost monotonically increases toward the half-filling.We trace the origin of the T c -dome back to a combined effect of the filling dependence of the local vertex correction and the filling dependence of the pairing interaction.As another effect inherent in the strongly correlated systems, we study a phase transition for the Pomeranchuk instability, where the four-fold rotational symmetry in the electronic structure on the tetragonal lattice is spontaneously broken into two-fold due to an interaction effect.We find that the Pomeranchuk instability is remarkably more sensitive to the Fermi surface warping than the superconducting T c dome.An interesting question is the effect of the distorted Fermi surface due to the Pomeranchuk instability on the superconductivity, and we have revealed that the gap function is slightly changed from ordinary d-wave pairing to (d+s)-wave.Furthermore, we find that the Fermi surface distortion can enhance the superconducting T c in frustrated systems.i Finally, we consider the extension of the DMFT vertex to study the vertex correction effect on nonlocal self-energy and the pairing interaction.For this purpose, we employ the DΓA (dynamical vertex approximation) formalism, which is extended here to accommodate superconducting phases.For the Hubbard model we obtain the T c -dome due to a combined effect as in the FLEX+DMFT case, but the transition temperature is reduced by the vertex correction.We also find that the pairing interaction is strongly screened by the vertex correction effect.ii
The ``dynamic'' Hubbard Hamiltonian describes interacting fermions on a lattice whose on-site repulsion is modulated by a coupling to a fluctuating bosonic field. We investigate one such model, introduced by Hirsch, using the determinant quantum Monte Carlo method. Our key result is that the extended sss-wave pairing vertex, repulsive in the usual static Hubbard model, becomes attractive as the coupling to the fluctuating Bose field increases. The sign problem prevents us from exploring a low enough temperature to see if a superconducting transition occurs. We also observe a stabilization of antiferromagnetic correlations and the Mott gap near half-filling, and a near linear behavior of the energy as a function of particle density which indicates a tendency toward phase separation.
Abstract The spectral properties of the doped t – t ’ Hubbard model, using parameters typical for high-temperature cuprate superconductors, and the mechanism of d-wave pairing remain among the longstanding problems of many-body fermionic materials. We used a strong-coupling Green’s function expansion around a correlated reference system, namely a particle-hole-symmetric undoped Hubbard lattice with t′=0{t}^{{\prime} }=0t′=0 t ′ = 0 , which can be treated numerically exactly using sign-problem-free lattice Quantum Monte Carlo calculations. This reference system exhibits a large antiferromagnetic Mott-Hubbard-Slater gap in the electronic spectrum. We investigate how the Mott-like spectrum is reconstructed under finite doping and nonzero t′{t}^{{\prime} }t′ t ′ using a dual-fermion-inspired perturbation expansion. For a large next-nearest-neighbor hopping t′=−0.3t{t}^{{\prime} }=-0.3tt′=−0.3t t ′ = − 0.3 t , characteristic of cuprate families with T c around 100 K, the electronic spectral function reveals a strongly renormalized flat-band feature with a pseudogap near the antinodal point. The superconducting response of this system to a small dx2−y2{d}_{{x}^{2}-{y}^{2}}dx2−y2​ d x 2 − y 2 -like external field shows a pseudogap at the antinodal point in the normal part of the Nambu Green’s function, associated with “bad-fermion” behavior in the normal phase. At the same time, the anomalous Green’s function exhibits a d-wave-like structure with zero response at the nodal point of the Brillouin zone.
Previous theoretical studies [Nat. Phys. {\bf 16}, 1175 (2020)] based on the Hatsugai-Kohmoto model have examined the stability of sss-wave superconductivity in strongly correlated systems, demonstrating that correlations alone can substantially modify superconducting behavior. Motivated by this perspective, but going beyond these studies, we perform self-consistent microscopic calculations of ddd-wave superconductivity in strongly correlated systems by employing an exactly solvable correlated model that hosts a pseudogap phase and a partially flat band [Phys. Rev. Lett. {\bf 133}, 166501 (2024)]. We show that pseudogap correlations and superconducting order affect the low-energy spectrum in qualitatively different ways: the former leads to a momentum-localized suppression of spectral weight, whereas the latter induces a coherent reorganization of quasiparticle excitations. Moreover, we demonstrate that the interplay between superconducting order and pseudogap correlations naturally generates a superconducting dome in the temperature-doping phase diagram, with optimal doping located near the quantum critical point separating the pseudogap and metallic phases. Furthermore, dx2−y2d_{x^2-y^2}dx2−y2​-wave superconductivity is found to be remarkably robust, remaining energetically dominant over both dxyd_{xy}dxy​-wave and sss-wave pairing channels across a wide doping range. Our results offer a potential route for a direct and controlled connection between pseudogap correlations and the emergence of the superconducting dome in cuprates.