Hubbard model on the square lattice at 1/8 doping

In brief. At $U/t=8$ and filling $n=7/8$ the square-lattice Hubbard model is a doped Mott insulator in the regime most relevant to the cuprates, and a particularly difficult point of the 2015 community benchmark. Its ground state is a filled stripe with wavelength 8 and no superconductivity; all controlled methods agree on $e_\infty=-0.767\pm0.004\,t$ , but no exact energy exists for any torus beyond $4\times4$ . Determinant and Pfaffian wavefunctions dressed by neural networks now give the lowest variational energies on tori, where DMRG cannot go, and the spread between ansätze on a fixed cluster is a hundred times the statistical error bars.

Notation follows the conventions page. This page fixes the conventions of the Ketboard instances hubbard-square-8x8-pbc-u8-n56-sz0, hubbard-rectangular-Lx4-pbc-u8-nNe-sz0 with $L=8,12,16$ , and hubbard-square-16x16-pbc-u8-n224-sz0, explains why this point of the phase diagram is hard, and summarizes what is known about its energy. The same clusters with a next-nearest-neighbour hopping have their own page. Reviewed entries live in Ketboard, not here.

Hamiltonian and instances

H=-t\sum_{\langle i,j\rangle,\sigma}\bigl(c^\dagger_{i\sigma}c_{j\sigma}+\mathrm{h.c.}\bigr)+U\sum_i n_{i\uparrow}n_{i\downarrow}, \qquad t=1,\quad U=8,\quad t'=0 .

Lattices: $8\times8$ (64 sites), the four-leg tori $8\times4$ , $12\times4$ and $16\times4$ ( $L$ along $x$ and 4 along $y$ ), and $16\times16$ (256 sites), periodic in both directions with no twist and no antiperiodic direction. Nearest-neighbour hopping only; each undirected bond counted once per spin.
Filling: $N_\uparrow=N_\downarrow$ and $N_e=\tfrac78N$ , i.e. 14+14 on $8\times4$ , 21+21 on $12\times4$ , 28+28 on the 64-site clusters and 112+112 on $16\times16$ , a hole doping of $\delta=1/8$ . On all five tori the noninteracting problem at this filling is open-shell (the Fermi level falls inside a degenerate set of $\mathbf k$ points), so the mean-field determinant from which a fermionic ansatz starts is itself a choice.
Sector: $S^z=0$ and fixed particle numbers. No momentum or point-group sector is imposed by the instance; a projected or symmetry-restricted calculation must say which sector it targets, since the lowest state of a finite torus need not have zero momentum.
Energy: reported per site, $e=E/N$ , in units of $t$ . Energies per electron, which differ by the factor $8/7$ , must be converted. The variational energy is a strict upper bound to $e_0$ .

Cylinders (periodic along $y$ , open along $x$ ), used by DMRG, are different instances; a $16\times4$ cylinder and a $16\times4$ torus are not comparable (Moreno, Carleo, Georges, Stokes, arXiv:2111.10420, Table IV, lists both). Energies per site differ between the tori by up to $0.02\,t$ because of shell effects and because each cluster accommodates different stripe patterns, so results are compared only within one instance.

Why $U/t=8$ and $n=7/8$

No unbiased method on tori. Away from half filling the auxiliary-field quantum Monte Carlo sign problem is severe and the constrained-path approximation is needed; it is accurate but not variational, and its bias is estimated by cross-checks with DMRG on cylinders (Zheng et al., arXiv:1701.00054; Qin et al., arXiv:1910.08931). DMRG itself is limited to cylinders of width 4 to 8. Exact diagonalization stops at $4\times4$ . On a torus of 64 sites or more there is no exact energy, and a lower variational energy is the only strict statement available.

Competing orders within $10^{-2}t$ . The doped Mott insulator has stripe states of different wavelengths, uniform antiferromagnetic states and paired states within about $0.01\,t$ per site of each other (arXiv:1701.00054, Fig. 3). Which state wins on a given cluster depends on whether its wavelength fits: a wavelength-8 filled stripe fits the $8\times8$ torus once, the $16\times4$ torus twice along $x$ , and the $16\times16$ torus twice. Pinning fields, mean-field initializations with a chosen unit cell and boundary conditions all select a pattern, and a variational calculation can converge to a metastable stripe (arXiv:2111.10420, Sec. 3B; Sharma et al., arXiv:2510.11710). On the $16\times4$ torus the stripe ground state is well established; on the $8\times8$ torus Sharma et al. find features of a doped Mott insulator with antiferromagnetic correlations and suppressed density fluctuations rather than a clear stripe.

A fermionic sign structure. The amplitudes in the occupation-number basis are antisymmetric under particle exchange and have no simple sign rule at finite doping. Every competitive ansatz carries a determinant or a Pfaffian for this reason, and the network corrects the orbitals (backflow, hidden fermions) or multiplies the determinant by a Jastrow factor; see the fermionic states page. The accuracy of an entry is therefore the accuracy of its orbital correction, and the open-shell starting point makes the optimization landscape rugged.

What is known

Filled stripes, no superconductivity. Combining DMRG, AFQMC, DMET and iPEPS, Zheng et al. found a vertical stripe of wavelength 8 (one hole per unit length of the stripe, "filled") as the ground state at $U/t=8$ , $\delta=1/8$ , with the wavelength-5 to wavelength-10 stripes within $0.01\,t$ per site and all methods agreeing on an energy in the range $-0.767\pm0.004\,t$ (arXiv:1701.00054). Qin et al. established with AFQMC and DMRG that the pure model has no superconducting long-range order at intermediate to strong coupling near optimal doping (arXiv:1910.08931).
The 2015 benchmark. LeBlanc et al. tabulate thermodynamic-limit energies at $U/t=8$ , $n=0.875$ : AFQMC $-0.766(1)$ , DMRG $-0.759(4)$ , fixed-node $-0.749(2)$ , and DMET $-0.749(7)$ with the caveat that different clusters show different orders (arXiv:1505.02290, Table V). These are the reference numbers behind every "relative error" quoted on this instance; none of them is a finite-torus energy.
Large tori with neural networks. Convolutional backflow states reach $32\times32$ at this doping and find horizontal and vertical filled stripes degenerate within their resolution (Gu et al., arXiv:2604.25775); hidden-fermion Pfaffian states on $16\times16$ find the filled stripe $0.011\,t$ per site below the half-filled stripe with pairing at $t'=0$ (Roth, Chen, Sengupta, Georges, arXiv:2511.07566, Table V); tensor backflow on $16\times16$ finds the linear stripe from an unrestricted Hartree–Fock start (Liang, arXiv:2507.01856).

Known energies

Energies per site in units of $t$ , with the source of each number. Ketboard entries carry the arXiv version and the exact printed token; no uncertainty means none was printed.

$16\times4$ torus, 28+28 electrons. The DMRG reference for the $\infty\times4$ cylinder, $-0.7659(4\times10^{-5})$ , is quoted by Luo and Clark (arXiv:1807.10770, Fig. 2) and is not a torus energy.

Method	$e$	Source
Hidden-fermion determinant state, 16 hidden fermions	$-0.753(2)$	Moreno et al., arXiv:2111.10420, Table IV
Tensor backflow, one Lanczos step	$-0.7597$ , $-0.7618$	Zhou, Zhou, Liang, arXiv:2308.11823, Table III, two backflow forms
Deep neural backflow with low-rank multi-determinant updates	$-0.7598$ ; $-0.7622$ with one Lanczos step	Huang, Ou, Ozolins, arXiv:2609.26544, p. 6
MLP neural backflow, single determinant, 8192 hidden units	$-0.76258(1)$	Loehr and Clark, arXiv:2510.26906, p. 3
Transformer neural backflow	$-0.76298$	Gu et al., arXiv:2507.02644, p. 31
MLP neural backflow, four determinants from optimized symmetries	$-0.76340(2)$	Loehr and Clark, arXiv:2510.26906, p. 5
Hidden-fermion Pfaffian state, group-convolutional backbone	$-0.76413(3)$	Chen, Wan, Sengupta, Georges, Roth, arXiv:2507.10705, Table IV
MLP neural backflow, 32 determinants from optimized symmetries	$-0.76486(2)$	Loehr and Clark, arXiv:2510.26906, p. 5
Same state projected onto the fully symmetric sector	$-0.76560(1)$	Loehr and Clark, arXiv:2510.26906, p. 5

$8\times4$ torus, 14+14 electrons.

Method	$e$	Source
Tensor backflow, one Lanczos step	$-0.7591$ , $-0.7633$	Zhou, Zhou, Liang, arXiv:2308.11823, Table III, two backflow forms
Hidden-fermion determinant state	$-0.7633(7)$	Moreno et al., arXiv:2111.10420, Table IV
Deep neural backflow	$-0.7659$ ; $-0.7671$ with one Lanczos step	Huang, Ou, Ozolins, arXiv:2609.26544, p. 6
Hidden-fermion Pfaffian state	$-0.76690(1)$	Chen et al., arXiv:2507.10705, Table IV
MLP neural backflow, fully symmetrized, 128 determinants	$-0.768337(2)$	Loehr and Clark, arXiv:2510.26906, p. 6

$12\times4$ torus, 21+21 electrons.

Method	$e$	Source
Tensor backflow, one Lanczos step	$-0.7608$ , $-0.7636$	Zhou, Zhou, Liang, arXiv:2308.11823, Table III, two backflow forms
Deep neural backflow	$-0.7640$ ; $-0.7662$ with one Lanczos step	Huang, Ou, Ozolins, arXiv:2609.26544, p. 6
Hidden-fermion Pfaffian state	$-0.76608(1)$	Chen et al., arXiv:2507.10705, Table IV

$8\times8$ torus, 28+28 electrons. All from Sharma et al., arXiv:2510.11710, Appendix E, Table I.

Method	$e$
Hidden-fermion determinant state, ViT backbone	$-0.736(1)$
Jastrow–backflow determinant, ViT backbone	$-0.7422(3)$
Hidden-fermion determinant state	$-0.7454(9)$
Jastrow–backflow determinant	$-0.7458(6)$

$16\times16$ torus, 112+112 electrons.

Method	$e$	Source
Hidden-fermion Pfaffian, half-filled stripe with pairing	$-0.7401(1)$	Roth et al., arXiv:2511.07566, Table V
Tensor backflow, $p=0$	$-0.7509$	Liang, arXiv:2507.01856, Table I
Hidden-fermion Pfaffian, filled stripe	$-0.7515(1)$	Roth et al., arXiv:2511.07566, Table V
Sparse convolutional backflow (SCALE)	$-0.7529$	Gu et al., arXiv:2604.25775, Table I
Tensor backflow, one Lanczos step	$-0.7552$	Liang, arXiv:2507.01856, Table I
Deep convolutional backflow (ACE)	$-0.7573$	Gu et al., arXiv:2604.25775, Table I

On a fixed cluster the ansätze differ by up to $10^{-2}t$ per site while the printed error bars are $10^{-4}t$ or smaller, so the ranking is informative here in a way it is not for the J₁–J₂ instance.

For the hidden-fermion Pfaffian and the deep backflow state the energy rises from $8\times4$ to $12\times4$ to $16\times4$ . Tensor backflow does not follow that order ( $-0.7633$ , $-0.7636$ , $-0.7618$ for BW2), so the trend with $L$ reflects how well each ansatz converged on each cluster as well as the clusters themselves. The Loehr–Clark $8\times4$ state lies $8\times10^{-5}$ above their own variance-extrapolated estimate.

The best $16\times16$ energy is $0.01\,t$ above the thermodynamic-limit range. The finite-size correction and the variational error are of the same order, and a size series at fixed ansatz is needed to separate them.

Reporting a result for this instance

State the boundary conditions in both directions (periodic, antiperiodic, or twisted), the orientation of a rectangular cluster, and the sign of the hopping. A $16\times4$ cylinder, a $4\times16$ torus and a $16\times4$ torus are three different things; only the last two are the same instance.
State $N_\uparrow$ and $N_\downarrow$ and the energy normalization (per site, not per electron).
State the mean-field starting point, any pinning field, and the stripe pattern the state converged to (wavelength and orientation). The reported energy must be that of $H$ alone, with every pinning field switched off in the final evaluation.
State the symmetry sector and whether a projection was applied after optimization. A Lanczos step is a variational improvement and qualifies; diffusion Monte Carlo, variance extrapolation and energies quoted from other papers do not.
Evaluate the final state with a dedicated run and the binning analysis; report $e$ , one standard error, the number of samples and $\operatorname{Var}(H)/N$ .
Give the printed digits exactly as in the paper, with the arXiv version, table and row, through the Ketboard form.

References

J. P. F. LeBlanc et al., Solutions of the two-dimensional Hubbard model: benchmarks and results from a wide range of numerical algorithms, arXiv:1505.02290 (2015). The community benchmark; thermodynamic-limit energies at $U/t=8$ , $n=0.875$ .
B.-X. Zheng et al., Stripe order in the underdoped region of the two-dimensional Hubbard model, arXiv:1701.00054 (2016). Filled stripes of wavelength 8; the $-0.767\pm0.004\,t$ energy range.
M. Qin et al., Absence of superconductivity in the pure two-dimensional Hubbard model, arXiv:1910.08931 (2019).
D. Luo, B. K. Clark, Backflow transformations via neural networks for quantum many-body wave functions, arXiv:1807.10770 (2018). Neural backflow; the first NQS results on $L\times4$ Hubbard clusters at this doping.
J. R. Moreno, G. Carleo, A. Georges, J. Stokes, Fermionic wave functions from neural-network constrained hidden states, arXiv:2111.10420 (2021). Hidden-fermion determinant states; $4\times L$ tori and cylinders.
Y.-T. Zhou, Z.-W. Zhou, X. Liang, Solving Fermi-Hubbard-type models by tensor representations of backflow corrections, arXiv:2308.11823 (2023).
X. Liang, Investigating the Fermi-Hubbard model by the tensor-backflow method, arXiv:2507.01856 (2025). $16\times16$ tori at $t'=0$ and $t'=-0.2$ .
Y. Gu et al., Solving the Hubbard model with neural quantum states, arXiv:2507.02644 (2025). Transformer neural backflow.
A. Chen, Z.-Q. Wan, A. Sengupta, A. Georges, C. Roth, Neural network-augmented Pfaffian wave-functions for scalable simulations of interacting fermions, arXiv:2507.10705 (2025). Hidden-fermion Pfaffian states; the $L\times4$ series.
L. Sharma, A. Shokry, R. Nutakki, O. Simard, M. Ferrero, F. Vicentini, Comparing symmetrized determinant neural quantum states for the Hubbard model, arXiv:2510.11710 (2025). Hidden-fermion versus Jastrow–backflow determinants with transformer backbones; $8\times8$ torus.
K. Loehr, B. K. Clark, Enhancing neural network backflow, arXiv:2510.26906 (2025). Multi-determinant backflow from symmetry projection and Lanczos steps.
C. Roth, A. Chen, A. Sengupta, A. Georges, Superconductivity in the two-dimensional Hubbard model revealed by neural quantum states, arXiv:2511.07566 (2025). Hidden-fermion Pfaffians on $16\times16$ ; filled versus half-filled stripes as a function of $t'$ .
Y. Gu et al., Pareto frontier of neural quantum states: scalable, affordable, and accurate convolutional backflow for strongly correlated lattice fermions, arXiv:2604.25775 (2026). SCALE and ACE convolutional backflow; tori up to $32\times32$ .
T. Huang, X. Ou, V. Ozolins, Neural network backflow with low-rank multi-determinant updates, arXiv:2609.26544 (2026).