{"id":"5a5bbb32-66ee-444c-827a-ab2ebed6f099","arxiv_id":"1908.10000","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"At 1/8 hole doping in the t-J model, fermionic projected entangled pair states yield a ground state with stable diagonal stripes of hole filling 0.5 and suppressed long-range superconductivity.","lead":"Using a high-accuracy tensor network method, the authors find that the ground state of the t-J model at 1/8 hole doping has diagonal stripes, not vertical ones. The result bears on the long-running debate about stripe order and superconductivity in models of copper-oxide superconductors.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Same-method vertical-stripe comparison is missing: the paper demonstrates robustness of a diagonal-stripe fPEPS solution, not that it is the global ground state.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing gap: the computation assumes the diagonal-stripe fPEPS solution is the global ground state, but no same-method comparison with a vertical-stripe state is reported. The paper's energy comparison with prior DMRG and iPEPS results is cross-method and therefore cannot decide orientation, since different methods differ in ansatz, bond dimension, boundary conditions, and extrapolation procedures. My review of the full text finds no internal contradiction that would invalidate the calculation itself; the 4x4 exact-diagonalization check and the reported convergence at Dc=48 support the numerical quality of the fPEPS solution. The concern is therefore a missing control experiment rather than a demonstrated error, which is precisely why a conditional verdict is appropriate. I would not strengthen or weaken the reader's verdict; the paper should be accepted only if the orientation comparison is supplied or the claim is explicitly softened to a variational prediction. The proposed concrete test is the minimal check that would settle the issue, and it is well within the reach of the authors' existing code.","tokens_in":7928,"tokens_out":3494,"duration_ms":41324,"concrete_test":"Using the same fPEPS method, D=12, Dc=48, and identical open boundary conditions, prepare two initial states on identical lattices (e.g., 8x8 and 12x12 at n_h=1/8): one diagonal-stripe pattern as in the paper, and one vertical-stripe pattern with period-4 charge modulation along x and uniform along y (e.g., initialized from the DMRG/Ref. 17 charge pattern). Optimize both with the same gradient descent protocol and compare converged energies and hole densities. If the vertical-initialized state converges to a lower energy than the diagonal state, the headline claim is refuted; if it converges to the same diagonal state, that is direct evidence of orientation stability. Report the energy difference with Monte Carlo error bars and repeat for at least three lattice sizes to check whether the orientation energy difference survives the thermodynamic extrapolation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the t-J ground state at n_h=1/8 has stable diagonal stripes rather than vertical stripes. The evidence for this orientation is a single fPEPS optimization at D=12 on open-boundary lattices, with no reported same-method energy comparison against a vertical-stripe state. The paper compares its hole energy with DMRG and iPEPS results, but those are different methods with different boundary conditions and ansatz restrictions, so the comparison cannot isolate orientation stability. The text explicitly acknowledges that iPEPS found diagonal stripes to have higher energy than vertical stripes, and proposes a DMRG calculation with diagonal periodic boundary conditions as a possible resolution, but does not perform it. The statement that 'all L1 x L2 tensors are independent and free to change' and that the same stripe is found from random initial states shows reproducibility of one local minimum, not that a vertical-stripe initial state would converge to a higher energy. Open boundary conditions can pin domain-wall orientation through corner and boundary effects, so the orientation claim is exactly the part that needs a direct variational comparison.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":8149,"tokens_out":4557,"duration_ms":44785,"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":[{"comment":"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.","section":"Fig. 1, Table I, and the paragraphs comparing with iPEPS"},{"comment":"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.'","section":"Fig. 1 and E∞=-0.6701 extrapolation"},{"comment":"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.","section":"Fig. 4 and Eqs. (2)-(3)"}],"minor_comments":[{"comment":"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.","section":"Eq. (1)"},{"comment":"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.","section":"Abstract and Fig. 3"},{"comment":"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.","section":"Table I"},{"comment":"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.","section":"Fig. 3"},{"comment":"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.","section":"Paragraph on robustness against random initial states"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a serious variational study with a credible method, but the central claim of diagonal stripes requires a same-method comparison with vertical stripes; the current evidence establishes only that a good diagonal-stripe state exists. The lack of error estimates for the energy extrapolation and correlation-function fits is a second concern. I would encourage the editor to request the missing comparison and convergence data as part of a major revision, since the question is empirically and theoretically important."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's real contribution is a new numerical scenario: fPEPS with open boundary conditions finds diagonal, site-centered stripes with hole filling rho_l=0.5 at J/t=0.4 and n_h=1/8, and a hole energy (-1.619) that beats published DMRG and iPEPS values. The 4x4 check against exact diagonalization is clean, and the authors are explicit about how their diagonal stripes differ from the iPEPS ones (filling, tensor constraints). They also name the obvious test—DMRG with diagonal periodic boundary conditions—without dodging it. That is honest.\n\nThe soft spot is exactly what the stress-test note says: the orientation claim is not tested by a same-method energy comparison. All L1 x L2 tensors being free is a good property, and random initial states reproducing the same diagonal-stripe state shows that this is a stable variational minimum. It does not show it is the global minimum. A vertical-stripe initialization that converges to a higher energy is the missing control. Open boundary conditions can pin stripe orientation through corners, so the paper's central claim is conditional until that comparison is done. The extrapolation also has no error bars, and D is fixed at 12 without a systematic D-scaling study; only Dc convergence is shown. The superconductivity suppression argument relies on power-law fits with exponents ~4.4-4.9 but no error bars, and fitting a power law across a stripe-ordered state is a bit hand-wavy. These are addressable gaps, not internal contradictions.\n\nThe citation pattern is fine; the authors engage the competing DMRG, iPEPS, and vQMC results directly and do not hide the discrepancies. The paper is written clearly and the method description is sufficient for a specialist to see what was done.\n\nWho gets value: people working on stripe order in the t-J model and on tensor-network methods for doped Mott insulators. It is a serious challenge to the vertical-stripe consensus, not a settled result. I would send it to peer review and ask for the same-method diagonal-versus-vertical comparison and explicit error estimates. That request is the whole ballgame: if the comparison comes out in favor of diagonal stripes, this is an important result; if not, it is a cautionary tale about boundary conditions in fPEPS. Either way, referees should see it.","headline":"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.","tokens_in":8668,"tokens_out":1351,"would_cite":false,"duration_ms":16024,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["t-J model","1/8 hole doping","diagonal stripes","fermionic projected entangled pair states","fPEPS","stripe order","superconductivity suppression","pair correlations"],"falsifier":"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.","tokens_in":7750,"feed_emoji":"🧲","tokens_out":7149,"duration_ms":62314,"temperature":0.7,"pith_summary":"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.","feed_headline":"t-J model at 1/8 doping forms diagonal stripes, not vertical","feed_subtitle":"A high-accuracy tensor-network state beats earlier methods and finds stable stripes with no long-range pairing.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Reports a uniform ground state from VQMC plus Lanczos steps at the same parameters; this is the contrasting result whose hole energy (-1.546) the paper compares and must beat.","marker":"[16]"},{"why":"Supplies the iPEPS simple-update and DMRG hole energies (-1.593 and -1.612) and the vertical-stripe ground-state picture that this paper challenges.","marker":"[17]"},{"why":"Provides the earlier iPEPS full-update study with vertical stripes and a diagonal-stripe comparison, as well as the anisotropic t-J parameters used for the tx/ty=0.85 check.","marker":"[18]"},{"why":"Establishes the site-centered vertical stripe state with width 4 and rho_l=0.5 in DMRG; the structural reference for stripe filling and the contrast case for orientation.","marker":"[14]"},{"why":"The fPEPS gradient-optimization method with Monte Carlo sampling that the paper uses, including the convergence criterion Dc=3D for the t-J model at 1/8 doping.","marker":"[36]"},{"why":"Provides exact-diagonalization results on the 4x4 lattice used to validate the fPEPS energy and to show that the small lattice is uniform.","marker":"[13]"},{"why":"Supplies the zero-doping energy E0=-0.467775 used to convert ground-state energies into hole energies.","marker":"[41]"},{"why":"The finite-size scaling formalism used to extrapolate energies from finite open-boundary lattices to the thermodynamic limit.","marker":"[43]"}],"fun_headline_variants":["t-J model at 1/8 doping: diagonal stripes, not vertical","Diagonal stripes stable in t-J model at 1/8 doping","fPEPS finds diagonal stripes in t-J model, no long-range pairing","Ground state of t-J model at 1/8 doping is diagonal stripes","t-J model: diagonal stripes win over vertical at 1/8 doping"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["t-J model at 1/8 doping: diagonal stripes, not vertical","Diagonal stripes stable in t-J model at 1/8 doping","fPEPS finds diagonal stripes in t-J model, no long-range pairing","Ground state of t-J model at 1/8 doping is diagonal stripes","t-J model: diagonal stripes win over vertical at 1/8 doping"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000453,"raw_usage":{"total_tokens":2249,"prompt_tokens":884,"completion_tokens":1365,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":500,"completion_tokens_details":{"reasoning_tokens":1266}},"tokens_in":500,"tokens_out":1365,"duration_ms":9952,"temperature":1.0,"reasoning_tokens":1266,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:55:43.571885+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports a uniform ground state from VQMC plus Lanczos steps at the same parameters; this is the contrasting result whose hole energy (-1.546) the paper compares and must beat."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the iPEPS simple-update and DMRG hole energies (-1.593 and -1.612) and the vertical-stripe ground-state picture that this paper challenges."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the site-centered vertical stripe state with width 4 and rho_l=0.5 in DMRG; the structural reference for stripe filling and the contrast case for orientation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The fPEPS gradient-optimization method with Monte Carlo sampling that the paper uses, including the convergence criterion Dc=3D for the t-J model at 1/8 doping."},{"cited_title":"Capello, M","cited_arxiv_id":null,"evidence_quote":"Provides exact-diagonalization results on the 4x4 lattice used to validate the fPEPS energy and to show that the small lattice is uniform."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the zero-doping energy E0=-0.467775 used to convert ground-state energies into hole energies."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The finite-size scaling formalism used to extrapolate energies from finite open-boundary lattices to the thermodynamic limit."}],"review_version":1}