d-wave superconductivity in the hole-doped two-dimensional Hubbard model: establishing the low-temperature, sign-problem-blocked regime by controlled computation and quantum simulation

The repulsive Hubbard model on the square lattice is the minimal model for doped Mott insulators and the standard theoretical setting for cuprate high-temperature superconductivity. Simple BCS-type mean-field and Gutzwiller-projected variational treatments readily produce $d_{x^2-y^2}$ superconducting states in it, but whether the model itself, in the thermodynamic limit, actually hosts such order at finite hole doping — and where in the (doping, interaction, hopping) phase diagram — has never been established: at finite doping and low temperature the fermion sign problem blocks unbiased finite-temperature quantum Monte Carlo exactly in the regime of interest. The problem, posed from the cold-atom Fermi–Hubbard program of Mazurenko et al. (2017), is to establish the presence (or absence, and location) of d-wave superconducting order in the low-temperature doped two-dimensional Hubbard model — the motivating window is $T/t\approx 0.05$, $\delta\approx 0.15$, $U/t\approx 7$–$8$ — by controlled, unbiased means: either many-body computation with controlled or certifiable error on the doped model, or quantum simulation (ultracold fermions in optical lattices, or other quantum platforms) reaching this regime at finite doping and measuring superconducting order. A genuine partial resolution exists — at $t'=0$, $U/t=8$, $\delta=1/8$ the ground state is consensus-stripe without coexisting d-wave order — but the surviving core is wide open and actively contested in 2025–2026: neural-network variational studies claim robust d-wave order for $t'\ne 0$ and at weaker coupling, density-matrix-renormalization-group cylinder studies find charge order with only short-ranged pairing, and cold-atom simulators have reached $T/t=0.05$ only at half filling and observed the pseudogap, not superconductivity.

Background

The Hubbard model is the minimal model of interacting electrons on a lattice: on the square lattice its Hamiltonian is $H=-t\sum_{\langle i,j\rangle,\sigma}(c^{\dagger}_{i\sigma}c_{j\sigma}+\mathrm{h.c.})+U\sum_{i}n_{i\uparrow}n_{i\downarrow}$, with hopping $t$ and on-site repulsion $U$. At half filling (one electron per site) and $U/t\gg 1$ it is a Mott insulator — an insulator driven by charge blocking rather than band filling — with antiferromagnetic order generated by the superexchange scale $J=4t^{2}/U$. Removing electrons (hole doping $\delta$) turns it into the minimal description of the CuO$_2$ planes of cuprate high-temperature superconductors (Lee, Nagaosa and Wen 2006).

The central open question of this problem is superconductivity. Superconducting order of $d_{x^2-y^2}$ symmetry means Cooper pairs whose gap changes sign between the $x$ and $y$ bond directions, as observed in the cuprates; in a two-dimensional system at nonzero temperature such order is quasi-long-ranged (Berezinskii–Kosterlitz–Thouless type: algebraically decaying correlations below a stiffness-driven transition), while at zero temperature true long-range off-diagonal order is possible. Simple BCS-type mean-field and Gutzwiller-projected variational treatments (trial wavefunctions in which doubly occupied sites are suppressed by projection) of the doped model readily produce $d_{x^2-y^2}$ states, so the question is not whether one can write down such a wavefunction but whether the model itself, in the thermodynamic limit, actually supports such a phase, and in which region of its phase diagram (doping $\delta$, coupling $U/t$, and next-nearest-neighbor hopping $t'/t$, the standard cuprate-motivated extension).

Exact unbiased computation is blocked precisely in the regime of interest. Determinant quantum Monte Carlo (auxiliary-field sampling in imaginary time) is numerically exact, but away from half filling the Monte Carlo weight — a product of two determinants — is not non-negative. This is the fermion sign problem: the signal-to-noise ratio decays exponentially with system size and inverse temperature, so the low-temperature doped regime, exactly where superconductivity is expected, has no controlled finite-temperature method. At half filling, particle-hole symmetry restores a sign-free formulation, which is why the antiferromagnet is quantitatively benchmarked while the doped regime is not.

This is why quantum simulation is pursued as a complementary route. Ultracold fermionic atoms in optical lattices — periodic trapping potentials for neutral atoms created by interfering laser beams — realize the Hubbard Hamiltonian with tunable $t$, $U$ and $\delta$, and quantum gas microscopy — site-resolved fluorescence imaging of single atoms — gives direct access to spin and density correlations. Mazurenko et al. (2017) realized the two-dimensional Hubbard antiferromagnet at $T/t=0.25(2)$ with $U/t=7.2$, using entropy redistribution (shifting entropy into a low-density reservoir engineered by a programmable optical potential) as the cooling strategy, and found antiferromagnetic order persisting to about 15% hole doping. Their stated outlook, which is the source of this problem: at lower temperatures $T/t\approx 0.05$ and dopings $\delta\approx 0.15$, theoretical work on cuprate phenomenology indicates a transition to a d-wave superconducting state; the low-temperature doped-Hubbard phase diagram is numerically intractable because of the sign problem; reaching such temperatures via entropy redistribution schemes is the proposed route.

The computational state of the art shapes what a solution must overcome. The main methods are the density-matrix renormalization group (DMRG; matrix-product-state wavefunctions, essentially exact on quasi-one-dimensional cylinders but geometry-restricted), constrained-path auxiliary-field quantum Monte Carlo (an approximate QMC whose constraint removes the sign problem at the price of uncontrolled bias), infinite projected entangled-pair states (iPEPS; tensor networks defined directly in the thermodynamic limit), density-matrix embedding theory (DMET; a method that solves each local fragment of the lattice quantum-exactly inside a self-consistently determined bath), and variational Monte Carlo including modern neural-network quantum states (NQS). A multi-method benchmark (LeBlanc et al. 2015) already showed the methods disagreeing on pairing in the doped regime. By now a consensus exists at one parameter point — $t'=0$, $U/t=8$, $\delta=1/8$ — where the ground state is a charge-stripe state — a state with unidirectional modulation of charge and spin density, here of period 8 lattice spacings — without coexisting d-wave order (Zheng et al. 2017; Qin et al. 2022), with the uniform d-wave state higher in energy by only about $0.01t$ per site. Whether superconductivity appears elsewhere — other dopings, weaker coupling, nonzero $t'$, in partially filled stripes, or at finite temperature — remains unsettled and intensely contested: 2025–2026 neural-network variational studies report robust d-wave order for $t'\ne 0$ and at $U/t=4$, while the largest DMRG cylinder studies find charge order with extremely short-ranged pairing correlations. On the experimental side, cold-atom simulators have reached $T/t=0.05$ at half filling (Xu et al. 2025) and observed the pseudogap metal — the pseudogap being a partial suppression of low-energy spectral weight that in the cuprates appears above the superconducting transition — at finite doping (Kendrick et al. 2026), but not superconductivity.

Problem Statement

Does the repulsive Hubbard model on the square lattice, doped away from half filling, develop $d_{x^2-y^2}$ superconducting order at low temperature in the thermodynamic limit — and if so, in which region of the phase diagram (hole doping $\delta$, coupling $U/t$, and next-nearest-neighbor hopping $t'/t$)? Establish this — presence and location, or absence throughout the low-temperature doped regime — by controlled and unbiased means, in the regime where exact finite-temperature quantum Monte Carlo is blocked by the fermion sign problem (the motivating window from the source is $T/t\approx 0.05$, $\delta\approx 0.15$, $U/t\approx 7$–$8$). Two routes are admissible and address the same objective: (i) a many-body computation whose errors are controlled or certifiable on the doped model, delivering the superconducting order parameter (off-diagonal pair correlations — the pair-pair correlation function at large separation — or the superfluid stiffness, the energy cost of twisting the superconducting phase) with a thermodynamic-limit extrapolation; or (ii) a quantum-simulation experiment — e.g. ultracold fermions in an optical lattice — that reaches the low-temperature doped regime and measures superconducting order with a controlled finite-size or homogeneity extrapolation. The single overarching objective is a settled answer to the existence and location of d-wave superconducting order in the doped two-dimensional Hubbard model; an answer establishing that no such order occurs (for example because charge-ordered or stripe states dominate throughout this regime) resolves the problem equally. The numerical values above are the source's motivating window, not a restriction: an answer that establishes the presence of the order at other dopings, couplings or hoppings within the low-temperature doped regime — or that establishes its absence throughout that regime — answers the stated question.

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