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REVIEW 3 major objections 5 minor 46 references

Stable diagonal stripes in the t-J model at $\bar{n}_h$=1/8 doping from fPEPS calculations

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Using fermionic projected entangled pair states, this paper finds that the ground state of the t-J model at 1/8 hole doping and J/t=0.4 has stable diagonal stripes, and that long-range superconductivity is suppressed.

desk verdict A credible fPEPS claim of stable diagonal stripes at 1/8 doping with the lowest energy so far, but the orientation claim needs a same-method comparison that isn't in the paper. read the letter →

arxiv 1908.10000 v1 pith:PC27CSK2 submitted 2019-08-27 cond-mat.str-el

Shao-Jun Dong , Chao Wang , Yong-Jian Han , Chao Yang , Lixin He This is my paper · ORCID
classification cond-mat.str-el
keywords t-Jmodel1/8holedopingdiagonalstripesfermionicprojectedentangledpairstatesfPEPSstripeordersuperconductivitysuppressioncorrelations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper addresses a long-standing dispute over the ground state of the two-dimensional t-J model at 1/8 hole doping, a parameter region relevant to stripe order in copper-oxide superconductors. Using fermionic projected entangled pair states (fPEPS) with gradient optimization and Monte Carlo sampling, it reports a thermodynamic-limit hole energy of -1.6186 at J/t=0.4, lower than previous variational, DMRG, and iPEPS estimates. The central finding is that the ground state has stable stripes running along the lattice diagonal, with hole-density period 4, staggered-magnetization period 8, and a hole filling of rho_l=0.5 per stripe; earlier vertical-stripe results are attributed to boundary conditions that favor the vertical orientation. The same calculation shows that both s-wave and d-wave superconducting pair correlations decay quickly, with power-law exponents of about 4.9 and 4.4, so long-range superconductivity is suppressed in this stripe state. If correct, the paper establishes that the true ground state at this doping is diagonal stripes rather than uniform or vertical-stripe order.

What carries the argument

The load-bearing machinery is the fermionic projected entangled pair state (fPEPS) tensor network, a systematically improvable variational ansatz for interacting fermions on a lattice. The wavefunction is first prepared by imaginary-time evolution with a simple update and then refined by stochastic gradient descent, with energies and gradients estimated by Monte Carlo sampling; U(1) particle-number symmetry is enforced and open boundary conditions are used. With bond dimension D=12 and environment truncation Dc=48, energies on lattices from 4x4 to 12x12 are extrapolated to the thermodynamic limit by a second-order polynomial in 1/sqrt(L1 L2). Because every tensor is optimized independently with no imposed periodicity, the method claims to be unbiased with respect to stripe orientation; the 4x4 result matches exact diagonalization to about 1e-4, validating the larger D=12 runs.

What would settle it

A direct same-method test: on an 8x8 or 12x12 lattice with boundary conditions that do not favor either orientation, optimize fPEPS wavefunctions initialized respectively with diagonal stripes, vertical stripes, and a uniform state, all at the same D=12 and Dc=48; if the vertical-stripe or uniform state has lower energy, or if increasing D to 16 changes the ordering, the central claim fails. Equivalently, a DMRG calculation with periodic boundary conditions along the diagonal direction that finds vertical stripes at equal or lower energy would contradict the orientation claim.

Watch

Extended reading notes

Core claim

The discovery is that the ground state of the t-J model at n_h=1/8, t=1, J/t=0.4, is a diagonal stripe phase rather than the vertical stripe phase found in earlier DMRG and iPEPS calculations. In 12x12 and other open-boundary lattices, the hole density forms site-centered stripes with period 4 along the diagonal direction, while the staggered magnetization has period 8 with a pi phase shift across the domain wall; the stripe hole filling is rho_l=0.5. The authors emphasize that their simulation is unbiased: all L1 x L2 tensors are independent, random initial states converge to the same pattern, and the thermodynamic-limit hole energy (-1.6186) is lower than previously reported values. They also find that s-wave and d-wave pair correlations decay as $r^{{-alpha}}$ with $\alpha$ about 4.9 and 4.4 respectively, so superconductivity does not survive to long range in this phase. The difference in stripe orientation from earlier work is attributed to the different boundary conditions used in those calculations.

Load-bearing premise

The load-bearing premise is that the diagonal-stripe pattern found at bond dimension D=12 on finite open-boundary lattices is the variational ground state, so that no lower-energy vertical-stripe, uniform, or other state was missed; the paper does not report a same-method energy comparison of diagonal versus vertical stripes.

Editorial extensions

If this is right

  • At J/t=0.4 and 1/8 doping, the t-J model ground state is charge-ordered rather than uniform: the hole density has period 4 and the staggered spin period 8, robust across lattice sizes and aspect ratios.
  • The stripe hole filling rho_l=0.5 matches the earlier stripe picture, but the orientation is diagonal; vertical stripes found by DMRG and iPEPS may be artifacts of the anisotropic boundary conditions those methods use.
  • Long-range superconductivity does not coexist with this stripe state: s- and d-wave pair correlations decay with exponents near 4.9 and 4.4, well above the threshold for true long-range order.
  • The reported thermodynamic-limit hole energy of -1.6186 is the lowest among the methods compared, so if the variational energies are converged, the diagonal-stripe state is the best available estimate of the true ground state at this doping.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An implication the authors leave implicit: if diagonal stripes are the true ground state, the widely used vertical-stripe results from DMRG and iPEPS at the same parameters are probably influenced by the boundary conditions, and a direct same-method comparison of both orientations on equivalent lattices would be a decisive check.
  • A testable extension would be to increase the bond dimension D beyond 12 and confirm that the diagonal-stripe hole energy continues to decrease and remains below a constrained vertical-stripe state; if the gap closes, the orientation claim weakens.
  • The paper's suppression of pairing suggests a sharp experimental signature: in a clean two-dimensional t-J-like system at 1/8 doping, one should see diagonal charge modulations and no bulk superconducting phase, which could distinguish this strong-coupling stripe scenario from a uniform resonating-valence-bond picture.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This manuscript applies a fermionic projected entangled pair states (fPEPS) method with stochastic gradient descent and Monte Carlo sampling to the two-dimensional t-J model at J/t=0.4 and hole doping n_h=1/8 on finite open-boundary lattices up to 12x12. The authors report a hole energy E_h=-1.6186 after extrapolation to the thermodynamic limit, and they find stripe order along diagonal directions with a hole-density period of 4 and a staggered-magnetization period of 8. They further report that s-wave and d-wave superconducting pair correlations decay with exponents fit to alpha_s≈4.9 and alpha_d≈4.4, from which they conclude that long-range superconductivity is suppressed. The paper compares its hole energy with previous VQMC, iPEPS, and DMRG results and argues that the diagonal orientation is the true ground-state stripe orientation.

Significance. If the diagonal-stripe result is correct, it would be a significant revision of the prevailing vertical-stripe picture for the t-J model at 1/8 doping and would sharpen the debate about stripe order and superconductivity. The paper has genuine strengths: the 4x4 fPEPS energy agrees with exact diagonalization to about 1e-4, the decision to use a larger environment truncation (Dc=6D) for correlation functions is a thoughtful accuracy measure, and the fPEPS optimizations use independent tensors on each site, which is less constrained than the supercell ansatze used in earlier iPEPS calculations. However, the central orientation claim and the superconducting-suppression claim are not yet supported by the evidence actually reported, because no same-method energy comparison against vertical stripes is given and no error estimates accompany the extrapolations and fits.

major comments (3)
  1. [Fig. 1, Table I, and the paragraphs comparing with iPEPS] The central claim that the ground state has diagonal stripes rather than vertical stripes is not supported by a same-method variational comparison. The paper reports fPEPS optimizations that converge to diagonal stripes from random initial states, which demonstrates reproducibility of a single local minimum, not that the diagonal orientation is globally lower in energy. The energy comparisons in Table I are against DMRG and iPEPS results that use different boundary conditions and different ansatz restrictions, so they cannot isolate the stripe orientation. The text explicitly acknowledges that iPEPS calculations found diagonal stripes to have somewhat higher energy than vertical stripes, and it proposes a DMRG calculation with diagonal periodic boundary conditions as a possible resolution without performing it. A direct fPEPS calculation initialized with vertical-stripe states, or a constrained vertical-stripe ansatz with the same D and Dc, is needed to establish that diagonal stripes are lower in energy.
  2. [Fig. 1 and E∞=-0.6701 extrapolation] The thermodynamic limit extrapolation is presented without error bars, fit coefficients, or a documented fit procedure. The reported hole energy E_h=-1.6186 depends on a second-order polynomial fit in sqrt(L1L2) over lattices from 4x4 to 12x12, combined with the zero-doping energy E0 from Ref. 41, but the paper does not state the statistical errors on the individual energies, the sensitivity of E∞ to the fitting form, or the bond-dimension convergence data for the largest systems beyond the assertion that Dc=48 is converged at D=12. Because the competition between stripe and uniform phases is known to be very close in energy, an estimate of the uncertainty in E∞ is essential for claiming the 'most competitive ground state hole energy.'
  3. [Fig. 4 and Eqs. (2)-(3)] The conclusion that long-range superconductivity is suppressed is based on power-law fits with exponents alpha_s≈4.9 and alpha_d≈4.4 extracted from pair correlation functions over distances up to r~10 on a 12x12 lattice with a single reference point r_i=(6,2). No fit residuals, error bars, or alternative decay forms are reported, and the available distance range is short relative to the system size and likely affected by the stripe structure and boundaries. To support the claim, the authors should show log-log fits for multiple reference points and system sizes, or otherwise provide a more controlled finite-size analysis of the correlation decay.
minor comments (5)
  1. [Eq. (1)] In Eq. (1), the phrase 'where ⟨i,j⟩ are the nearest-neighbor sites' should be 'where ⟨i,j⟩ runs over nearest-neighbor pairs of sites' to avoid the impression that each site itself is a pair.
  2. [Abstract and Fig. 3] The abstract describes 'width of 4 unit cells' for the stripes, while the text and Fig. 3 report a hole-density period of 4 and a staggered-magnetization period of 8; 'width' is ambiguous and should be replaced with a precise statement about the charge and spin periodicities.
  3. [Table I] The full-update iPEPS row in Table I corresponds to n_h=0.120, not exactly 1/8; this caveat should appear in the table caption so that the reader does not compare the energies as if they were at identical doping.
  4. [Fig. 3] Figure 3 does not fully label its vertical axes; the green curve should be labeled as the average hole density per site and the red curve as the staggered magnetization (-1)^{i-1}⟨S_i^z⟩, with units indicated.
  5. [Paragraph on robustness against random initial states] The statement that 'we always obtain the same stripe ground states for randomly chosen initial states' would be more informative if the number of random initializations, the range of final energies, and the criterion for identifying the same state were reported.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the diagonal-stripe ground-state claim is an unbiased variational output benchmarked against exact and external results.

full rationale

No circularity is present in the derivation chain. The central claim—that the fPEPS ground state of the t-J model at n_h=1/8 and J/t=0.4 has stable diagonal stripes with stripe filling rho_l=0.5 and suppressed long-range superconductivity—is the output of an unbiased variational optimization, not an input. The paper states that no constraints are imposed on the tensors: 'All L1 x L2 tensors are independent and free to change during the optimization. We always obtain the same stripe ground states for randomly chosen initial states.' The 4x4 fPEPS energy is checked against exact diagonalization (difference about 1e-4), and the thermodynamic-limit hole energy is compared with independent DMRG, iPEPS, and vQMC values. The zero-doping energy E0 used to define E_hole is taken from Sandvik (Ref. 41), an external source, so the hole-energy definition is not self-referential. The only self-citations are to the authors' own fPEPS method papers for optimization details and convergence criteria, but these are not load-bearing in a circular manner: the exact 4x4 agreement and external benchmarks independently support the method. The absence of a same-method vertical-stripe energy comparison is a real limitation, and the paper explicitly acknowledges it by proposing future DMRG calculations with diagonal periodic boundary conditions; however, that is a completeness or correctness risk, not circularity, because the diagonal-stripe result is not derived from an assumption of vertical stripes or from any parameter fitted to the target energy. The stripe filling rho_l = W * n_h = 0.5 is an arithmetic consequence of the observed period-4 hole stripes, not a self-definitional prediction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The central inputs are the t-J model, the fPEPS method, and an external zero-doping energy. The main free parameters are finite-size and correlation-function fits; the main unproved load-bearing assumptions are D-convergence and boundary-condition neutrality.

free parameters (2)
  • Finite-size scaling coefficients (a and b)
    A second-order polynomial in (L1L2)^-1/2 is fit to five finite-size energies to obtain the thermodynamic-limit energy; the fitted coefficients are not reported.
  • Superconducting correlation decay exponents alpha_s and alpha_d = alpha_s ~ 4.9, alpha_d ~ 4.4
    Power-law fits P_s,d(r) ~ r^{-alpha} to pair correlations on a 12x12 lattice are used to argue that long-range superconductivity is suppressed.
assumptions (5)
  • domain assumption The fPEPS variational ansatz with bond dimension D=12 spans the relevant ground-state manifold at 1/8 doping.
    D is fixed at 12; convergence is checked via Dc and a 4x4 exact comparison, but no systematic D sweep is shown at large lattice sizes.
  • domain assumption The thermodynamic-limit energy is obtained by a second-order polynomial in (L1L2)^-1/2.
    The functional form is a standard finite-size scaling choice, but its validity for these stripe states is not demonstrated in detail.
  • domain assumption Open boundary conditions do not qualitatively change the stripe orientation.
    The paper attributes DMRG vertical stripes to its boundary conditions, but it does not test its own open-boundary geometry against other boundary conditions.
  • domain assumption E0=-0.467775 at zero doping from Sandvik (Ref 41) is accurate.
    This external value is used to define the hole energy and is taken without re-evaluation in this paper.
  • standard math Variational principle: a lower variational energy implies a state closer to the true ground state.
    This is the basis for comparing hole energies across different numerical methods.

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Cite this review

Pith. "Pith review of Stable diagonal stripes in the t-J model at $\bar{n}_h$=1/8 doping from fPEPS calculations." pith.science (2026). https://pith.science/paper/PC27CSK2

@misc{pith2026190810000,
  author       = {Pith},
  title        = {Pith review of: Stable diagonal stripes in the t-J model at $\barn_h$=1/8 doping from fPEPS calculations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PC27CSK2}},
  note         = {Machine review of arXiv:1908.10000}
}
abstract

We investigate the 2D t-J model at a hole doping of $\bar{n}_h$=1/8 using recently developed high accuracy fermionic projected entangled pair states(fPEPS) method. By applying stochastic gradient descent method combined with Monte Carlo sampling technique, we obtain the ground state hole energy $E_{\rm hole}$=-1.6186 for $J/t$=0.4. We show that the ground state has stable diagonal stripes instead of vertical stripes with width of 4 unit cells, and stripe filling $\rho_l$=0.5. We further show that the long range superconductivity order is suppressed at this point.

Figures

Figures reproduced from arXiv: 1908.10000 by the authors.

Figure 1
Figure 1. FIG. 1: The ground state energies of t-J model with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The ground state hole density and spin moment on [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The average hole density [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The pair correlation function for the d-wave and the s [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

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Reviewed August 14, 2026 · model on record in the stance chip above.