{"id":"afae96f6-96de-42b5-b5d5-3ba2ceb2ef08","arxiv_id":"2502.07079","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Nearest-neighbor electron interactions in the extended Hubbard model can suppress d-wave pairing and instead favor d_xy-wave or p-wave pairing, offering a possible mechanism for the weak superconductivity of electron-doped cuprates.","lead":"Quantum Monte Carlo simulations of the square-lattice t-U-V Hubbard model find that a nearest-neighbor repulsion between electrons suppresses the usual d-wave superconducting pairing and promotes d_xy-wave pairing, while a nearest-neighbor attraction can induce an exotic p-wave spin-triplet pairing.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The p-wave 'phase' rests on real-space correlations that the paper itself describes as exponentially decaying; exponential decay means no long-range order, so the central p-wave claim is not established.","rationale":"The reader's weakest assumption points broadly at CPQMC constrained-path bias and finite-size scaling. I agree those matter, but the sharper, more load-bearing issue is internal: even taking the CPQMC results at face value, the paper's own text says the p-wave correlations decay exponentially. Exponential decay is incompatible with the existence of a genuine zero-temperature pairing phase, so the p-wave region of Fig. 1(a) is not supported by the evidence presented. This directly affects the abstract's headline claim that attractive V 'notably drive[s] an exotic p-wave spin-triplet pairing.' The d-wave suppression and d_xy enhancement on the repulsive-V side may still be plausible, and the qualitative connection to electron-doped cuprates can survive, which is why the appropriate disposition remains conditional rather than rejection. The proposed concrete test uses data the authors already have: fitting the real-space p-wave correlation to power-law versus exponential decay, and checking the scaling of the k=0 momentum distribution with system size. This would settle whether the p-wave 'phase' is a true thermodynamic phase or a finite-size short-range tendency. No independent code or error bars are provided, but that is secondary to the internal inconsistency identified here.","tokens_in":11743,"tokens_out":6272,"duration_ms":65886,"concrete_test":"Take the raw p-wave correlation used for Fig. 1(g) at U=2, V=-0.8, delta=0.153 and fit C_p(r) for each available lattice size (L=8, 10, 12, and 14 if possible) to both C(r) = A e^{-r/xi} + B and C(r) = A r^{-alpha} + B. If the exponential fit wins with B statistically zero and xi remains finite as L grows, then the p-wave state has no ODLRO and cannot be called a thermodynamic pairing phase; the phase diagram should be revised to label this region as a short-range triplet pairing tendency. As a second quantitative handle, check whether N_eff_p(k=0) scales with L^2 (condensate) or saturates (finite correlation length). This reanalysis of already-gathered CPQMC data decides whether the central p-wave claim stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Sec. III and Fig. 1(g) contain an internal tension that the paper does not resolve. For U=2, V=-0.8, delta=0.153 the p-wave effective pair correlation is said to 'show exponential decay in real space and drops to 0 quickly,' i.e., short-range. Yet this is the same point in parameter space used to place the exotic p-wave triplet pairing phase in Fig. 1(a). Exponential decay in r is the standard signature of the absence of off-diagonal long-range order (ODLRO): C_p(r) ~ e^{-r/xi} gives a finite correlation length, not a condensate. The insets of Figs. 1(e)-(g) are described as finite-size extrapolations that 'remain positive,' but positivity of N_eff_p(k=0) = sum_r C_p(r) ~ xi^2 is exactly what an exponentially decaying correlation produces; it does not imply condensation. Thus the CPQMC data shown are compatible with a short-range triplet pairing tendency and do not establish a zero-temperature p-wave pairing phase. Because the abstract and Sec. III lead with this p-wave phase as one of the central results, the load-bearing assumption is not merely generic CPQMC robustness; it is that an exponentially decaying pair correlation has been promoted to a phase without showing either power-law decay or a condensate fraction that grows with L. If instead the intended meaning of 'pairing phase' is only 'dominant short-range pairing susceptibility at the largest computed distance,' then the phase diagram and abstract overstate the result and need to be reworded; as written, the claim is unsupported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript reports constrained-path quantum Monte Carlo (CPQMC) ground-state calculations of the square-lattice t-U-V Hubbard model at electron doping δ=0.153 (with additional doping scans), using t'=0.2 to model electron-doped cuprates. The central claims are: (i) attractive nearest-neighbor interaction V drives an exotic p-wave spin-triplet pairing; (ii) repulsive V suppresses d_{x^2-y^2}-wave pairing and promotes d_xy-wave pairing; (iii) repulsive V enhances CDW and suppresses SDW; and (iv) a doping-dependent V can reproduce the narrow, low-T_c superconducting dome of electron-doped cuprates, thus identifying the t-U-V model as a minimal model for these materials. The paper presents schematic U-V phase diagrams, momentum-space pairing distributions, real-space pairing correlations, density-wave structure factors, and doping scans of d-wave pairing strength.","tokens_in":12043,"tokens_out":5555,"duration_ms":51006,"significance":"If the central claims were established, the paper would offer a concrete microscopic mechanism for the electron-hole asymmetry in cuprates: nearest-neighbor Coulomb repulsion suppresses d-wave pairing in electron-doped systems, while attractive V gives an unusual triplet channel. The systematic scans over U, V, and doping, and the simultaneous treatment of pairing and density-wave channels, are strengths of the study. However, the p-wave 'phase' is based on exponentially decaying real-space correlations, which do not by themselves establish long-range order; the doping-dependent V in Fig. 5 is fitted to the experimental T_c dome and then used as evidence for the model's explanatory power; and no statistical uncertainties are reported. With careful rephrasing of the p-wave claim, addition of error bars, and an honest reframing of the V(doping) extraction, the qualitative trends could still be a useful contribution, but the current presentation overstates several conclusions.","major_comments":[{"comment":"The p-wave pairing phase is a central result, but the evidence is inconsistent with a long-range ordered phase. The text states that 'the p-wave pairing shows exponential decay in real space and drops to 0 quickly in the distance, indicating short-range correlations.' Exponential decay of the effective real-space correlation C_eff_{p-pair}(r) implies a finite correlation length and therefore no off-diagonal long-range order. The peak in N_eff_{p-pair}(k=0) defined in Eq. (2) is the momentum-space sum of this correlation and remains positive for any exponentially decaying function, so it does not by itself diagnose condensation. The finite-size extrapolation in the inset of Fig. 1(g), described only as 'remains positive,' also cannot distinguish a short-range tendency from a condensate; what is needed is either power-law decay of C_eff(r) or growth of a condensate fraction with system size. As written, the abstract and the phase diagram in Fig. 1(a) overstate the p-wave claim; the authors should either provide thermodynamic-limit evidence for long-range p-wave pairing or explicitly rephrase the p-wave region as a dominant short-range pairing tendency.","section":"Section III, last paragraph (Fig. 5)"},{"comment":"The d-wave pairing strength as a function of doping for different V is fitted to a black dotted dome-like curve 'resembling T_c domes in the typical phase diagram of cuprates.' Because V is adjusted doping-by-doping to reproduce the dome, the subsequent claim that this demonstrates 'the critical role of V in capturing the superconducting behavior of cuprates' is partly circular. The inference that V/U ~ 1/4 to 1/3, made 'assuming that the suppression of the electron-doped SC region arises solely from the NN repulsion V,' is presented as an output even though V is a free parameter chosen to match the target. The authors should reframe Fig. 5 as a consistency check under a stated assumption rather than as an independent derivation, and ideally compare the extracted V(δ) with a microscopic estimate or with independent constraints.","section":"Section II and III, throughout (Figs. 1-5)"},{"comment":"No statistical uncertainties are reported for the CPQMC data. CPQMC is a stochastic method, and the data points in Figs. 1-5 are shown without error bars; the schematic phase boundaries in Figs. 1(a) and 4(a)-(d) therefore carry unquantified uncertainty. Adding error bars is essential to assess whether, for instance, the rapid suppression of d-wave pairing for V ≳ 1.3 in Fig. 2 is statistically significant and whether the 'negative' d-wave pairing strengths are genuine or noise. A statement about the statistical error and the constrained-path bias would be needed to support the quantitative comparisons drawn in the text.","section":"Section III, paragraph after Fig. 1(d)"}],"minor_comments":[{"comment":"The sentence 'From another perspective, the d-wave pairing suggests a d-wave PDW (π,0) state' is a strong claim that is not supported by a real-space pair-density-wave analysis. A peak in the effective momentum-space pairing distribution at (π,0) is not by itself sufficient to identify pair-density-wave order; the authors should either provide supporting real-space evidence or mark this as speculation.","section":"Section II, Eq. (2) and following definitions"},{"comment":"There are notation errors in the definitions of the effective correlations: 'G^σ_{i,j}G^σ_{i,j}' should presumably be 'G^σ_{i,j}G^σ_{i+δ,j+δ'}' (and similarly for the p-wave case). Please also define δ_ζ and δ'_ζ consistently for each pairing symmetry and state which spin indices are used for the triplet p-wave operator.","section":"Section III, Fig. 1 caption"},{"comment":"The insets of Figs. 1(e)-(g) are described as 'fitted using exponential function in 1/L.' Please specify which quantity is fitted, how many system sizes are used, and report the fitted parameters or at least the goodness of fit, since the extrapolation is used to argue for the robustness of the pairing states.","section":"Section III, text near Fig. 3"},{"comment":"The phrase 'consistent with its have incommensurate condensation points' is grammatically unclear; it should be 'consistent with its incommensurate condensation points.' Also, the connection between the slightly staggered real-space behavior of d_xy-wave pairing and the incommensurate peaks near (π,π) is not demonstrated quantitatively.","section":"Section IV"},{"comment":"The word 'validness' in the summary ('proves the validness of') should be 'validity.' In addition, the summary repeats the overstatement about the p-wave pairing phase without the caveat of short-range correlations; the summary should be made consistent with the corrected interpretation.","section":"Introduction, Ref. [38]"},{"comment":"The authors' prior work (Ref. [38], Phys. Rev. B 111, 024509 (2025)) studied the same t-U-V model for hole doping and is cited only briefly. Since the present paper uses the same methodology and definitions, please state explicitly what is new here compared with that work (electron doping, density-wave channels, and the doping dependence of V).","section":"Section III, Fig. 5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is closely related to the authors' earlier publication (Ref. [38]) on the hole-doped side; the electron-doped focus is a reasonable extension, but the novelty should be stated more explicitly. The main concerns are the unsupported p-wave 'phase' label based on short-range correlations and the post hoc fitting of V to the cuprate T_c dome in Fig. 5. If the authors can either supply genuine long-range-order evidence for p-wave pairing or rephrase the results as pairing tendencies, and if they add error bars, the paper could become acceptable. The fit in Fig. 5 should be presented as an illustrative consistency check rather than as a parameter-free success."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the repulsive-V story is a real addition to the electron-doped cuprate puzzle, but the p-wave phase claim doesn't survive contact with the paper's own correlation functions.\n\nWhat's actually new: a systematic CPQMC scan of nearest-neighbor V in the t-U-V model at electron doping, showing that repulsive V suppresses d-wave pairing, promotes d_xy, and shifts the CDW/SDW balance. The qualitative trends are consistent across U and doping, and the idea that a nonlocal Coulomb term shrinks the electron-doped SC dome is physically reasonable. The d-wave/d_xy competition and the CDW enhancement at larger V are the useful parts.\n\nThe soft spots: the p-wave 'phase' is built from real-space correlations that the text itself calls exponentially decaying and short-ranged. Exponential decay means no ODLRO; a positive zero-momentum N_eff is just the correlation-length squared. The finite-size insets don't fix this—they extrapolate a finite number. So the attractive-V p-wave region in Fig. 1(a) is at best a short-range pairing tendency, not a thermodynamic phase. That needs rewording or stronger evidence (power-law decay, condensate fraction scaling). Also, Fig. 5 fits V to reproduce the cuprate Tc dome and then uses that V to explain the same suppression; that's partly circular, though the authors do note V could vary with doping. Minor: no error bars anywhere, no code/data release, and CDW/SDW are only shown at (pi,pi) without size scaling.\n\nOverall: the repulsive-V mechanism is a credible candidate worth serious refereeing, but the paper oversells the p-wave result. I'd send it to review with the expectation of major revision: either drop 'phase' for the p-wave channel or provide evidence of long-range order; and separate the dome fit from the prediction.","headline":"Repulsive-V suppression of d-wave looks plausible and worth a look; the p-wave 'phase' is an overreach built from exponentially decaying correlations.","tokens_in":12611,"tokens_out":2539,"would_cite":false,"duration_ms":24733,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["74.20.-z","74.25.Dw","71.10.Fd"],"model":"deepseek-v4-flash","headline":"In the t-U-V Hubbard model at electron doping δ = 0.153, nearest-neighbor attraction drives p-wave spin-triplet pairing while nearest-neighbor repulsion suppresses d-wave pairing and induces d_xy-wave pairing.","keywords":["t-U-V Hubbard model","electron-doped cuprates","nearest-neighbor interaction","d-wave pairing","d_xy-wave pairing","p-wave triplet pairing","constrained-path quantum Monte Carlo","charge and spin density waves"],"falsifier":"A calculation with a bias-free or sign-problem-free method at $U=2$, $V=-0.8$, and $\\delta=0.153$ - for instance, density-matrix renormalization group on $8\\times L$ cylinders - that finds no surviving $p$-wave pairing, or a constrained-path run on $16\\times16$ and $24\\times24$ lattices whose $p$-wave extrapolation trends to zero, would settle that the claimed $p$-wave phase is not a thermodynamic ground state.","tokens_in":11542,"feed_emoji":"⚛️","tokens_out":14723,"duration_ms":111571,"temperature":0.7,"pith_summary":"The paper sets out to explain why electron-doped cuprates show a much smaller superconducting region than standard Hubbard-model simulations predict. It argues that the missing ingredient is the nearest-neighbor interaction $V$ between electrons, treated alongside the on-site repulsion $U$ in the $t$-$U$-$V$ Hubbard model on a square lattice. Using constrained-path quantum Monte Carlo at doping $\\delta = 0.153$, the authors find that attractive $V$ ($V<0$) drives a spin-triplet $p$-wave pairing, whereas repulsive $V$ ($V>0$) suppresses the conventional $d_{x^2-y^2}$-wave pairing and instead strengthens $d_{xy}$-wave pairing. They further report that repulsive $V$ enhances charge-density-wave correlations and suppresses spin-density-wave correlations. If correct, the $t$-$U$-$V$ model would serve as a minimal model for the electron-doped cuprate phase diagram, with repulsive $V$ acting as the factor that shrinks the $d$-wave superconducting dome.","feed_headline":"NN repulsion shrinks d-wave pairing in electron-doped cuprates","feed_subtitle":"A repulsive nearest-neighbor interaction suppresses d-wave pairing and favors d_xy-wave pairing instead.","key_machinery":"The carrying object is the $t$-$U$-$V$ Hubbard Hamiltonian: nearest-neighbor hopping $t$ (with next-nearest-neighbor hopping $t'=0.2t$ for electron doping), on-site Coulomb repulsion $U$, and nearest-neighbor density-density interaction $V$ (negative $V$ attractive, positive $V$ repulsive). The diagnostic that carries the phase assignment is the effective pair momentum distribution and its real-space counterpart, which separate the $d$-wave, $d_{xy}$-wave, and spin-triplet $p$-wave pairing channels. The constrained-path quantum Monte Carlo method supplies the ground-state correlations, using a path constraint to partially control the fermion sign problem. The mechanism is a competition between $U$ and $V$: on-site repulsion favors $d$-wave pairing through spin fluctuations, while nearest-neighbor interaction shifts the balance, with attraction favoring triplet $p$-wave pairing and repulsion favoring $d_{xy}$-wave pairing and suppressing $d$-wave pairing.","core_discovery":"On the paper's own terms, the central discovery is a zero-temperature pairing phase diagram for the $t$-$U$-$V$ Hubbard model at electron doping $\\delta=0.153$, computed with constrained-path quantum Monte Carlo on $12\\times12$ lattices with finite-size extrapolation. At $V=0$, $d$-wave pairing dominates and its strength grows with $U$. Once $V$ becomes attractive, even a small $|V|$ drives the system into a spin-triplet $p$-wave phase, especially at weak coupling $U=1$-$2$, where the $p$-wave channel condenses at zero center-of-mass momentum. Once $V$ becomes repulsive, $d$-wave pairing is progressively suppressed, and in the intermediate coupling regime $U=3$-$4$ the system enters a $d_{xy}$-wave pairing phase whose strength increases with $V$. The paper interprets repulsive $V$ as the agent that suppresses $d$-wave pairing in electron-doped cuprates, and estimates that $V/U$ around $1/4$ to $1/3$ would reproduce the observed suppression of superconductivity. The authors also note that the $p$-wave real-space correlations decay exponentially, yet the finite-size extrapolations remain positive and they assign a $p$-wave phase to that region; separately, repulsive $V$ enhances charge-density-wave order and suppresses spin-density-wave order.","pith_inferences":["The paper leaves it implicit, but its $V/U \\approx 1/4$ to $1/3$ estimate is a quantitative target that could be checked against ab initio estimates of the effective nearest-neighbor Coulomb repulsion in specific electron-doped cuprate compounds.","A natural next test, not performed in the paper, is whether the $p$-wave phase survives in methods without the constrained-path bias, such as density-matrix renormalization group on long cylinders.","The phase diagram implies a material-level prediction the paper does not draw: altering the dielectric environment or applying strain to reduce nonlocal Coulomb screening should suppress $d$-wave superconductivity and strengthen $d_{xy}$-wave or charge-order correlations in electron-doped cuprates."],"forward_implications":["If electron-doped cuprates carry a nearest-neighbor repulsion $V$ with $V/U \\approx 1/4$ to $1/3$, the model predicts a strongly suppressed $d$-wave pairing strength, yielding a small superconducting dome and low $T_c$.","As electron doping increases beyond the optimal value, the $p$-wave and $d_{xy}$-wave regions expand and further suppress $d$-wave pairing, so the superconducting region stays narrow in the overdoped direction.","With an attractive nearest-neighbor interaction of magnitude comparable to the hopping, the square-lattice $t$-$U$-$V$ model becomes a candidate for spin-triplet $p$-wave superconductivity at zero center-of-mass momentum.","Repulsive $V$ shifts the density-wave balance: charge-density-wave correlations strengthen and spin-density-wave correlations weaken, giving an experimental handle on $V$ through CDW/SDW competition."],"supporting_citations":[{"why":"Supplies the constrained-path quantum Monte Carlo method used for all ground-state pairing and density-wave results.","marker":"[51]"},{"why":"Reports the anomalously strong nearest-neighbor attraction in 1D cuprate chains, motivating the attractive-V region of the phase diagram.","marker":"[34]"},{"why":"Estimates a nearest-neighbor Coulomb repulsion of about 400 meV in cuprates, used to justify the repulsive-V region.","marker":"[47]"},{"why":"Finds a dome-like d-wave superconducting region in the t-t'-U Hubbard model, the numerical baseline this paper argues is incomplete without V.","marker":"[20]"},{"why":"Earlier study showing nearest-neighbor attraction drives p-wave pairing in the hole-doped extended Hubbard model, extended here to electron doping.","marker":"[38]"},{"why":"Phase diagram of the square-lattice t-J-V model for electron-doped cuprates, the nearest-neighbor-interaction extension this work builds on.","marker":"[23]"}],"fun_headline_variants":["Repulsive NN V suppresses d-wave, boosts dxy pairing in cuprates","Attractive NN V flips pairing to p-wave in electron-doped Hubbard","dxy-wave pairing emerges as repulsive V grows in electron-doped cuprates","t-U-V Hubbard model reveals pairing phase diagram for electron-doped cuprates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central assumption is that constrained-path quantum Monte Carlo on a $12\\times12$ lattice, extrapolated in system size, identifies the true thermodynamic pairing phases; in particular, the short-range $p$-wave correlations are taken as evidence of a genuine $p$-wave phase.","fun_headline_variants_meta":{"raw":{"variants":["Repulsive NN V suppresses d-wave, boosts dxy pairing in cuprates","Attractive NN V flips pairing to p-wave in electron-doped Hubbard","dxy-wave pairing emerges as repulsive V grows in electron-doped cuprates","t-U-V Hubbard model reveals pairing phase diagram for electron-doped cuprates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000853,"raw_usage":{"total_tokens":3785,"prompt_tokens":1102,"completion_tokens":2683,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":718,"completion_tokens_details":{"reasoning_tokens":2599}},"tokens_in":718,"tokens_out":2683,"duration_ms":20613,"temperature":1.0,"reasoning_tokens":2599,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:33:44.544284+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A calculation with a bias-free or sign-problem-free method at $U=2$, $V=-0.8$, and $\\delta=0.153$ - for instance, density-matrix renormalization group on $8\\times L$ cylinders - that finds no surviving $p$-wave pairing, or a constrained-path run on $16\\times16$ and $24\\times24$ lattices whose $p$-wave extrapolation trends to zero, would settle that the claimed $p$-wave phase is not a thermodynamic ground state.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the constrained-path quantum Monte Carlo method used for all ground-state pairing and density-wave results."},{"cited_title":"Pairing phase diagram for electron-doped cuprates in the square-lattice $t-U-V$ Hubbard model","cited_arxiv_id":"2502.07079","evidence_quote":"Reports the anomalously strong nearest-neighbor attraction in 1D cuprate chains, motivating the attractive-V region of the phase diagram."},{"cited_title":"Boschini, M","cited_arxiv_id":null,"evidence_quote":"Estimates a nearest-neighbor Coulomb repulsion of about 400 meV in cuprates, used to justify the repulsive-V region."},{"cited_title":"Xu, C.-M","cited_arxiv_id":null,"evidence_quote":"Finds a dome-like d-wave superconducting region in the t-t'-U Hubbard model, the numerical baseline this paper argues is incomplete without V."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier study showing nearest-neighbor attraction drives p-wave pairing in the hole-doped extended Hubbard model, extended here to electron doping."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Phase diagram of the square-lattice t-J-V model for electron-doped cuprates, the nearest-neighbor-interaction extension this work builds on."}],"review_version":1}