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arxiv: 2509.24439 · v1 · submitted 2025-09-29 · ❄️ cond-mat.str-el

Spin-stripes in the Hubbard model: a combined DMFT and Bethe-Salpeter analysis

Ruslan Mushkaev , Francesco Petocchi , Shintaro Hoshino , Philipp Werner This is my paper

Pith reviewed 2026-05-18 12:42 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords Hubbard modelspin stripesDMFTBethe-Salpeter equationdoping dependencecupratesantiferromagnetic phasenext-nearest neighbor hopping
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The pith

The square lattice Hubbard model supports broad regions of horizontal and vertical spin stripes whose wavelength varies nonlinearly with hole doping.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines spin-stripe order in the Hubbard model on the square lattice by solving the Bethe-Salpeter equation with a local vertex taken from dynamical mean-field theory. The method is applied both inside and outside the antiferromagnetic phase for models with and without next-nearest-neighbor hopping. Calculations show stable stripe phases over wide ranges of doping and low temperatures. The stripe wavelength changes nonlinearly with hole doping and depends strongly on the ratio of nearest- and next-nearest-neighbor hopping amplitudes. These findings bear on the electronic orders thought to underlie the physics of cuprate superconductors.

Core claim

Solving the Bethe-Salpeter equation with the local vertex from DMFT yields broad regions of horizontal and vertical spin stripes at low temperatures. Their wavelength depends on hole doping in a nonlinear fashion and is highly sensitive to the ratio of nearest and next-nearest neighbor hoppings, both with and without next-nearest neighbor hopping.

What carries the argument

Bethe-Salpeter equation solved with the local vertex extracted from DMFT, used to obtain the momentum-dependent spin susceptibility.

If this is right

  • Spin-stripe order occupies broad regions of doping and low temperature in the model with and without next-nearest-neighbor hopping.
  • Stripe wavelength varies nonlinearly with hole doping.
  • Stripe wavelength is highly sensitive to the ratio of nearest- and next-nearest-neighbor hoppings.
  • Stripe order exists both inside and outside the antiferromagnetic phase.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same local-vertex approach might be applied to other lattice geometries or interaction strengths to predict stripe periods without additional parameters.
  • Material-specific band structures could be used to tune the observed stripe wavelength through the hopping-ratio dependence.
  • If the approximation holds, similar stripe patterns should appear in related models that include longer-range Coulomb terms.

Load-bearing premise

The local vertex obtained from DMFT continues to approximate the momentum-dependent spin response accurately even when long-wavelength stripe fluctuations develop.

What would settle it

Neutron scattering measurements on a cuprate compound at fixed doping that show a stripe wavelength differing by more than 10 percent from the nonlinear doping curve predicted for the corresponding hopping ratio would falsify the central claim.

Figures

Figures reproduced from arXiv: 2509.24439 by Francesco Petocchi, Philipp Werner, Ruslan Mushkaev, Shintaro Hoshino.

Figure 1
Figure 1. Figure 1: FIG. 1: Spin-stripe and AFM order in the Hubbard [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Temperature dependence of the spin [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Spin incommensurability parameter for the [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Full momentum dependence of the static spin [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
read the original abstract

The Hubbard model is known to accommodate various electronic orders, including stripes, which are important for understanding the physics of cuprates. We study spin-stripe order in the square lattice Hubbard model as a function of doping and temperature, by solving the Bethe-Salpeter equation with the local vertex from dynamical mean field theory (DMFT), both inside and outside the antiferromagnetic phase. We find broad regions of horizontal/vertical spin stripes at low temperatures for the model with and without next-nearest neighbor hopping. Their wavelength depends on hole doping in a nonlinear fashion, and is highly sensitive to the ratio of nearest and next-nearest neighbor hoppings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript investigates spin-stripe order in the square-lattice Hubbard model by extracting the local two-particle vertex from DMFT and solving the Bethe-Salpeter equation for the momentum-dependent spin susceptibility, both inside and outside the antiferromagnetic phase. The central findings are the existence of broad low-temperature regions of horizontal and vertical spin stripes for models with and without next-nearest-neighbor hopping, with stripe wavelengths that vary nonlinearly with hole doping and depend sensitively on the ratio t'/t.

Significance. If the results are robust, the work supplies a systematic numerical map of incommensurate stripe wavelengths across doping and t'/t, which is directly relevant to cuprate phenomenology. The DMFT+BSE construction extends single-site DMFT to capture finite-wavelength magnetic instabilities in a computationally accessible manner and generates concrete, falsifiable predictions for how stripe periods change with parameters.

major comments (1)
  1. [Bethe-Salpeter analysis / method section] The central claim rests on solving the Bethe-Salpeter equation with the strictly local DMFT vertex (abstract and method description). This construction assumes that all relevant momentum dependence resides in the bubble while the irreducible vertex remains local. In the 2D Hubbard model, however, non-local correlations are expected to produce q-dependent vertex corrections precisely near stripe instabilities; if sizable, such corrections can shift the susceptibility maximum away from the wave-vector obtained with the local vertex and modify its doping dependence. Because the reported nonlinear wavelength-versus-doping relation and t'/t sensitivity are extracted from the location of these maxima, this approximation is load-bearing and requires either a quantitative estimate of the neglected corrections or a direct comparison with a method that retains momentum-dependent vertices (e.g., a
minor comments (2)
  1. [Abstract] The abstract states the findings clearly but does not indicate the precise doping and temperature windows explored; adding this information would help readers assess the breadth of the reported stripe regions.
  2. [Figures] In figures showing the susceptibility, the extracted stripe wavelengths should be marked explicitly together with any uncertainty arising from finite-size or frequency-grid effects.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. We are pleased that the referee recognizes the potential relevance of our results to cuprate phenomenology. Below we provide a point-by-point response to the major comment.

read point-by-point responses

  1. Referee: [Bethe-Salpeter analysis / method section] The central claim rests on solving the Bethe-Salpeter equation with the strictly local DMFT vertex (abstract and method description). This construction assumes that all relevant momentum dependence resides in the bubble while the irreducible vertex remains local. In the 2D Hubbard model, however, non-local correlations are expected to produce q-dependent vertex corrections precisely near stripe instabilities; if sizable, such corrections can shift the susceptibility maximum away from the wave-vector obtained with the local vertex and modify its doping dependence. Because the reported nonlinear wavelength-versus-doping relation and t'/t sensitivity are extracted from the location of these maxima, this approximation is load-bearing and requires either a quantitative estimate of the neglected corrections or a direct comparison with a method that

    Authors: We agree that the use of a strictly local vertex from DMFT in the Bethe-Salpeter equation is an approximation whose validity near stripe instabilities merits careful consideration. This method has been widely used to study magnetic and pairing instabilities, and the results provide a systematic map within this framework. However, we acknowledge that non-local vertex corrections could quantitatively affect the doping dependence of the stripe wavelength. In the revised manuscript, we will include an expanded discussion of this approximation in the methods section, highlighting its limitations and referencing works that explore vertex corrections in the Hubbard model. We note that performing a full quantitative estimate or comparison with cluster methods would require significant additional computational resources and is left for future work. revision: partial

standing simulated objections not resolved
  • Providing a quantitative estimate of the neglected q-dependent vertex corrections or a direct comparison with methods that include momentum-dependent vertices.

Circularity Check

0 steps flagged

No significant circularity in DMFT-BSE derivation of spin-stripe properties

full rationale

The paper computes spin-stripe regions and their doping-dependent wavelengths by first obtaining the local irreducible vertex from DMFT on the Hubbard Hamiltonian and then solving the Bethe-Salpeter equation for the spin susceptibility. This is a standard numerical workflow whose outputs (stripe wavevectors, temperature ranges, t'/t sensitivity) are generated by the equation solver rather than being equivalent to any input parameter or self-citation by construction. No load-bearing step reduces to a fitted quantity renamed as a prediction or to an ansatz imported from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The study relies on the standard DMFT approximation of a local self-energy and the assumption that the Bethe-Salpeter equation with this vertex captures stripe instabilities. No new particles or forces are introduced. The interaction strength U, hoppings t and t', doping, and temperature are model parameters rather than fitted quantities.

axioms (1)
  • domain assumption DMFT provides a sufficiently accurate local vertex for the subsequent Bethe-Salpeter calculation of momentum-dependent instabilities.
    Invoked when the Bethe-Salpeter equation is solved with the DMFT vertex inside and outside the antiferromagnetic phase.

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