Extreme competition between stripes and superconductivity in the hole-doped Hubbard model

arXiv:2610.30321 · cond-mat.str-el, cond-mat.dis-nn, cond-mat.supr-con, quant-ph · Submitted 2026-10-21 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Extreme competition between stripes and superconductivity in the hole-doped Hubbard model".

Mira: Here's the gist of the paper. The gist The work reveals an extreme competition between stripes and superconductivity in the hole-doped Hubbard model, showing that a low-energy uniform d-wave state exists near competing stripe states at 1/8 doping, regardless of next-nearest-neighbor hopping on page 1.

Kai: Model and Competition The Hubbard model is considered a minimal model for uniform d-wave superconductivity in cuprates, but recent calculations suggest other ground states exist in realistic regimes.

Mira: Numerical studies of the hole-doped Hubbard model find different ordered ground states at the same point in the phase diagram.

Kai: Specifically, at 1/8 doping and U = 8, a near degeneracy is revealed among a low-energy uniform d-wave state, superconducting stripes, and non-superconducting stripe states on the milli-t energy scale.

Title and authors: Kai: So we've just talked about how these different variational methods are used to probe this competition between stripes and superconductivity in the hole-doped Hubbard model on page two <ref:2610.30321#pg1,competition between stripes and superconductivity in the hole-doped Hubbard model>.

Mira: The summary says that at one eighth doping and U equals eight, they find a near degeneracy among the low-energy uniform d-wave state, superconducting stripes, and non-superconducting stripe states on a milli-t energy scale.

Lev: That suggests that this competition isn't just theoretical; it’s present right down to very low energy scales in their numerical simulations.

Kai: They are also looking at how next-nearest-neighbor hopping, t', affects the situation, which is important because it changes the landscape of candidate ground states.

Mira: At t' = zero they see stripe order including SDW and CDW at a period of sixteen for spin and a charge density wave period of eight on page two <ref:2610.30321#pg2>.

Lev: That’s concrete, so it tells us the stripes are definitely present in that limit, but how does that compare to the superconducting state?

Kai: When they look at t' = -zero point two, things shift a bit; they find evidence for weak or vanishing superconductivity alongside strong stripe order on page two <ref:2610.30321#pg2>.

Mira: They also find that for t' = -zero point two, the Pfaffian backflow and the NN-fPEPS both form a half-filled stripe with an SDW of period eight and a CDW of period four coexisting with a finite pair-pair correlation that gets weaker as the system size increases.

Lev: So what does that mean for what we're building? It means if we want to simulate this accurately on real hardware, we need to capture both the stripe order and the pairing correlations simultaneously.

Kai: The paper’s main finding is that at both values of t', an ordered stripe suppresses superconductivity, meaning the ground state is either not superconducting or only weakly superconducting.

Mira: But even with stripes present, there’s still a strongly superconducting uniform d-wave state existing nearby on a milli-t energy scale, which resembles what they saw in a previous paper on page two <ref:2610.30321#pg1>.

Lev: So the point isn't that one wins entirely, but that they all exist in this incredibly close competition.

The paper's summary: Kai: Now let’s talk about how this work itself improves things, moving beyond just finding these states on page two <ref:2610.30321#pg1>.

Mira: They address the uncertainty arising from the different computational methods by using three classes of variational states, and they argue that each relies on an underlying fermionic ansatz that might impose a bias.

Lev: That’s smart because it means they aren't just picking one method and assuming it’s correct; they’re systematically cross-validating results from distinct fermionic references.

Kai: They also mention some methodological advances, like using residual blocks in neural tensor networks to map configuration n to local tensors T(n).

Mira: That residual block idea allows for weight transfer from smaller systems to larger lattices without needing a full retraining from scratch on page three <ref:2610.30321#pg1>.

Lev: That’s efficiency for real hardware; if we want to simulate a bigger system, we don't have to start the training process over again.

Kai: Then there’s the optimization strategy, they use AdamSR with norm-constrained step sizes for training the NN-fPEPS instead of standard stochastic reconfiguration on page three <ref:2610.30321#pg1>.

Mira: AdamSR is used because it’s shown to be superior for training these neural wavefunctions compared to standard methods, which lets them converge faster on those complex parameters.

Lev: Faster convergence is crucial when you’re dealing with the complexity of mapping a configuration n to local tensors Ti(n) in a large system simulation.

Kai: They also use the LSMR solver for large-scale parallel simulations of the NN-fPEPS and the Cholesky solver for all other cases, which is about computational efficiency on page three <ref:2610.30321#pg1>.

The paper's improvements: Mira: Wrapping up this paper, they conclude that stripes, striped superconductivity, and uniform superconductivity all exist in extreme competition in the hole-doped Hubbard model at this doping level.

Kai: And resolving this competition requires an extreme level of accuracy because the states are so close together on a milli-t energy scale.

Lev: From an engineering standpoint, capturing that precision means we need to get way up to a lattice size where finite-size errors stop dominating the physics, which they say needs N greater than four hundred sites for a state with SC around zero point zero two on page five.

Mira: So the overall picture is that stripes, striped superconductivity, and uniform superconductivity all exist in extreme competition in the hole-doped Hubbard model, and resolving this competition requires an extreme level of accuracy as they point out at the end of page two <ref:2610.30321#pg1,in the hole-doped Hubbard model>.

Kai: This paper presents a concrete step toward understanding how to approach this problem with state-of-the-art neural tensor network and neural backflow state simulations enabled by large-scale GPU computation on up to one thousand twenty-four sites on page one <ref:2610.30321#pg1,state-of-the-art neural tensor network and neural backflow state simulations>.

Lev: So, for us, it means the path forward is getting those methods accurate enough so we can actually simulate systems big enough where they aren't just guessing at the physics.

Mira: We’ll keep an eye out for how this competition plays out in other regimes of the Hubbard model next time on air.

Kai: Until then, that's what we have for now on this paper.

Conclusion: Kai: So we just finished looking at "Extreme competition between stripes and superconductivity in the hole-doped Hubbard model." The main thing is that this paper shows that at one eighth doping and U equals eight, you get a near degeneracy between the low-energy uniform d-wave state, superconducting stripes, and non-superconducting stripe states on a milli-t energy scale.

Mira: That suggests the competition between those stripe orders and the uniform superconductivity isn't just theoretical; it’s happening right at very low energy scales in their simulations.

Lev: So they found these three distinct ground states existing at the same point in the phase diagram, which means we’re looking at a really complex landscape there.

Kai: They tested this by varying next-nearest-neighbor hopping, t', and when t' is zero, they see stripe order with a spin density wave period of sixteen and a charge density wave period of eight on page two.

Mira: And when they look at negative t', like-zero point two, the evidence for superconductivity gets weaker, but strong stripe order is still there on page two.

Lev: That tells us that whatever t' you pick, the stripes tend to suppress the superconductivity, or make it really weak.

Kai: But even when stripes are present, they found a strongly superconducting uniform d-wave state existing nearby on a milli-t energy scale which looks like something we saw before.

Mira: So the overall picture is that stripes, striped superconductivity, and uniform superconductivity all exist in extreme competition in the hole-doped Hubbard model.

Lev: It means resolving this competition requires an extreme level of accuracy because you need to see those subtle energy differences on page two.

Kai: The study presents a concrete step toward understanding this by using neural tensor networks and backflow states to probe these complex states, which is a big deal for how we do simulations.

Mira: The authors show how combining different variational forms helps them overcome the intrinsic bias that comes from picking one specific fermionic ansatz.

Lev: From an engineering standpoint, they’re using AdamSR for training and a Cholesky solver for all those large-scale parallel simulations, which shows how to make it run efficiently on real hardware.

Kai: So we’ve seen that the competition is fierce, but it's not settled; there are these three competing candidates right at the edge of what we can measure.

Mira: This work on "Extreme competition between stripes and superconductivity in the hole-doped Hubbard model" gives us a clearer map of where to look for these subtle effects.

Lev: It sets a benchmark for how accurately we need to simulate these correlated systems if we want to find those states.

California Institute of Technology

cond-mat.str-el, cond-mat.dis-nn, cond-mat.supr-con, quant-ph

Submitted: 2026-10-21

Updated: 2026-10-21

Comments: 33 pages, 25 figures

Code: https://github.com/ChenAo-Phys/fermix

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 81/100

The gist: The gist The work reveals an extreme competition between stripes and superconductivity in the hole-doped Hubbard model, showing that a low-energy uniform d-wave state exists near competing stripe

Key concepts

Hubbard Model: This is a simplified mathematical model used to describe the physics of strongly correlated electrons in materials like cuprate superconductors. It includes terms for electron hopping (movement) and on-site repulsion (electrons avoiding being on the same site), which are crucial for understanding superconductivity.
d-wave Superconductivity ($ ext{dSC}$): This describes a specific type of superconducting state where electrons pair up in a way that has an angular dependence similar to the d-orbit of an atom. It is the primary candidate for high-temperature superconductivity in many cuprates.
Stripe Order ($ ext{M2str}$): This refers to a pattern where charge and spin order form stripes—alternating lines or domains of high and low density/spin. The paper explores how this stripe order competes directly with the formation of the d-wave superconducting state.
Variational States: These are different mathematical approximations (like BCS backflow or NN-fPEPS) used to estimate the true ground state energy. Using multiple methods helps check if a result is reliable, even when each method has its own inherent assumptions or biases.

Terminology

Hubbard Model — 1× in this paper · explained in 6 episodes

Summary

The gist The work reveals an extreme competition between stripes and superconductivity in the hole-doped Hubbard model, showing that a low-energy uniform d-wave state exists near competing stripe states at 1/8 doping, regardless of next-nearest-neighbor hopping on page 1.

Model and Competition

The Hubbard model is considered a minimal model for uniform d-wave superconductivity in cuprates, but recent calculations suggest other ground states exist in realistic regimes. Numerical studies of the hole-doped Hubbard model find different ordered ground states at the same point in the phase diagram. Specifically, at 1/8 doping and U = 8, a near degeneracy is revealed among a low-energy uniform d-wave state, superconducting stripes, and non-superconducting stripe states on the milli-t energy scale. This finding rebalances the narrative that the Hubbard model supports a single dominant order towards a picture of extreme competition between stripes and superconductivity. The Hamiltonian is defined by terms including nearest-neighbor hopping t, next-nearest-neighbor hopping t', and on-site repulsion U on page 3. The ground state is characterized by two order parameters, the d-wave pairing ∆2SC and the stripe order M2str. These two parameters are used to distinguish three candidate ground states: the stripe state with M2str > 0 and ∆2SC = 0, the uniform d-wave superconductor with ∆2SC > 0 and M2str = 0, and a striped d-wave superconductor where both orders are nonzero.

Variational Methods

The study employs three classes of variational states to address the uncertainty arising from different computational methods. These include the BCS backflow state, the Pfaffian backflow state, and the neural-network fermionic PEPS (NN-fPEPS). The rationale for using multiple variational forms is that each relies on an underlying fermionic ansatz that may impose a bias in the calculation. The NN-fPEPS is built on the fPEPS, whose amplitude is given by Eq. (8) and which favors locally entangled (area-law) ground states. In contrast, backflow wavefunctions capture pairing through mean-field wavefunctions and have been argued to favor paired states. The NN-fPEPS architecture uses a residual convolutional neural network (CNN) to map the configuration n to local tensors Ti, which are then contracted by the higher-order tensor renormalization group (HOTRG).

Results at t' = 0

At t' = 0 and 1/8 doping, the work reaches the lowest variational energies reported so far on both the 16 × 16 and 32 × 32 PBC lattices. On the pure Hubbard model at t' = 0, extrapolating a pure fPEPS simple-update calculation with PBC gives an estimated exact energy of E/N = −0.7602(14) (SM Sec. VI [34]). The energies of all variational wavefunctions lie within 0.002 t per site (0.3%) of this estimate, and the lowest one (BCS backflow) lies 0.0014 t above it. In the pure Hubbard model at t' = 0, the NN-fPEPS forms a filled stripe with a spin-density wave (SDW) of period 16 and a charge-density wave (CDW) of period 8, whose spin and charge correlations persist to the largest distance while its pair-pair correlation decays.

Results at t' = -0.2

At t' = −0.2, evidence for weak or vanishing superconductivity with strong stripe order is found in the ground state. For this model, the Pfaffian backflow state achieves the lowest energies on all lattice sizes from 16×16 to 32×32. The Pfaffian backflow and the NN-fPEPS both form a half-filled stripe with an SDW of period 8 and a CDW of period 4, coexisting with a finite pair-pair correlation that becomes weaker when the system size increases. The BCS backflow result at t' = −0.2 is from Ref. [26] instead of this work.

Conclusion and Outlook

Both with and without t', the research finds at this doping that an ordered stripe suppresses superconductivity, so that the ground state is either not superconducting or only weakly superconducting. However, at both values of t', a strongly superconducting uniform d-wave state exists nearby on a milli-t energy scale, resembling the proposal in Ref. [13]. The overall picture is that stripes, striped superconductivity, and uniform superconductivity all exist in extreme competition in the hole-doped Hubbard model, and resolving this competition requires an extreme level of accuracy. This work presents a concrete step in this direction.

Methodological Advances

The study highlights two aspects of current simulation methodology. First, despite being highly parametrized, modern neural wavefunctions retain biases that are not overcome in training, necessitating cross-validation using different techniques to identify the qualitative nature of the ground state. Second, neural tensor networks now offer state-of-the-art performance in large-scale simulations without using a mean-field reference. The NN-fPEPS and backflow wavefunctions retain different inductive biases arising from how the fermionic sign structure is enforced. The HOTRG contraction cost is O(Nχ7) for N = LxLy sites, or O(N log N) for the translation-restored fPEPS (trans). The AdamSR optimizer was introduced to stabilize first-order optimizers by combining momentum with the natural gradient. The LSMR solver is used for large-scale parallel simulations of the NN-fPEPS and the Cholesky solver is used in all other cases.

Future Directions

A finite lattice can resolve pairing only once the gap ∆ exceeds roughly 8t divided by N, requiring N ≳ 400 sites for a state with ∆SC ∼ 0.02 comparable to the uniform d-wave superconductors. Running on up to 1024 GPUs, the researchers simulated N ≈ 1000 sites, reaching a regime where finite-size errors arguably no longer dominate the physics and TDL extrapolation is reliable. The overall picture is that stripes, striped superconductivity, and uniform superconductivity all exist in extreme competition in the hole-doped Hubbard model, and resolving this competition requires an extreme level of accuracy.

Data Availability

The trained wavefunctions, the code to evaluate and measure them, and the data and scripts behind all figures and tables are available at Ref. [70]. All results are obtained with 32-bit floating-point (FP32) precision and TensorFloat-32 (TF32) tensor cores for convolutions and matrix multiplications. The computational resources used include the facilities of the National Energy Research Scientific Computing Center (NERSC). Some computations were also performed using the facilities of the Caltech Resnick High Performance Computing Center. All code, computations, and text were reviewed and verified by the authors who take full responsibility for the content. The paper was supported by the U.S. Department of Energy, Office of Science, through Award No. DE-SC0018140. Most computations were performed using the facilities of NERSC. Some computations were also performed using the facilities of Caltech Resnick High Performance Computing Center. We acknowledge Yuntian Gu, Christopher Roth, Riccardo Rende, and Antoine Georges for useful discussions about their earlier results. We thank Yuntian Gu and Yufei Chen for sharing the determinant backflow data of Ref. [21] and Riccardo Rende for sharing the BCS backflow data of Ref. [26]. We would like to thank the NVIDIA Corporation for technical support. Claude Code, an AI coding assistant developed by Anthropic, was used to write parts of the simulation code, to submit and monitor part of the computations on the high-performance computing clusters, and to draft parts of this manuscript.

References

[1] D. P. Arovas, E. Berg, S. A. Kivelson, and S. Raghu, The Hubbard model, Annual Review of Condensed Matter Physics 13, 239 (2022)

[4] P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a Mott insulator: Physics of high-temperature superconductivity, Rev. Mod. Phys. 78, 17 (2006)

[5] T.

Improvements for AI systems

Neural network architecture improvement by using residual blocks in neural tensor networks to map configuration n to local tensors T(n), which allows for weight transfer from smaller systems to larger lattices without retraining from scratch. This enables the AI system to perform state-of-the-art simulations on large system sizes efficiently.
Variational method refinement by combining different variational forms, such as BCS backflow states and NN-fPEPS, to resolve the intrinsic bias of different computational methods. This allows the AI system to reach an unprecedented variational accuracy by systematically cross-validating results from distinct fermionic references.
Optimization strategy enhancement using AdamSR with norm-constrained step sizes, which is shown to be superior for training NN-fPEPS compared to standard stochastic reconfiguration (SR) or other optimizers. This allows the AI system to converge faster on complex neural wavefunction parameters by effectively managing the update direction and magnitude during training.
Linear solver selection based on network size and parallelism, utilizing the Cholesky solver for networks with up to a few million parameters and LSMR for larger distributed simulations. This improves computational efficiency by selecting the most appropriate linear algebra tool based on hardware constraints like device memory versus communication cost.
Time cost reduction through fused GPU kernels for HOTRG steps, which are implemented using custom kernels instead of direct evaluation of Eqs (S3)–(S5). This allows the AI system to execute complex tensor network contractions significantly faster, leading to a lower time per amplitude evaluation.