{"id":"586b3095-4d49-49a6-8c19-7babf9c402cd","arxiv_id":"1908.07028","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"Analytic dome and pseudogap temperature formulas fit cuprate data with one shape parameter per material, with a plausible but partly circular pressure extension.","lead":"The authors propose analytic formulas for the superconducting and pseudogap transition temperatures of cuprate superconductors as functions of doping, and claim a single fitted parameter reproduces the doping domes of many materials. A compact theory of the cuprate phase diagram, including layer-number and pressure trends, would be a major organizing result for a long-standing problem, though the pressure evidence is partly built from the model itself.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"As printed, Eq. (72) is not the |M0|->0 limit of Eq. (56): the overdoped branch drops the |mu0|/(2Tc) term, so the central dome derivation needs an independent check.","rationale":"The Reader's weakest assumption, the linear mu0(x) parametrization, is a valid concern: the universal phase diagram and the dome shape depend on it, and it is not derived from the microscopic Hamiltonian. However, the more decisive stress-test finding is internal: even granting that linear ansatz, the overdoped branch printed as Eq. (72) is not the |M0| -> 0 limit of Eq. (56) shown in Eq. (67). The algebra error at Eq. (71) is independent of any physical assumption and affects the exact functional form that is the paper's main result. I credit the paper's genuinely non-perturbative structure, the microscopic gS/gP estimates for LSCO, and the ambient-pressure fits as substantial supporting evidence; those features do not cure this derivation gap. The pressure-section circularity noted by the Reader remains a separate, additional weakness. Because the issue is concrete and checkable, the authors should supply the missing derivation or a corrected formula before the central claim can be accepted. If the corrected formula materially changes the overdoped fits, the headline claim of analytic dome expressions would need to be weakened; if the formula is merely a transcription artifact, the correction is straightforward. The appropriate verdict therefore remains CONDITIONAL, unchanged from the Reader's assessment, but for a more internal reason than the linear-mu0 assumption alone.","tokens_in":30660,"tokens_out":24170,"duration_ms":246699,"concrete_test":"Use a symbolic algebra system to evaluate the |M0| -> 0 limit in Eq. (67) for mu0 = -q < 0, simplifying ln2 + (1/2) ln cosh((|M0| - q)/(2Tc)) + (1/2) ln cosh((|M0| + q)/(2Tc)). If the result is q/(2Tc) + ln(1 + e^{-q/Tc}), then Eq. (72) is missing a term; replace Eq. (72) by the corrected implicit equation, recompute the overdoped Tc curves and x+_SC, and refit the Bi/Hg/LSCO overdoped data. The same check should also be applied to the similar 'putting in evidence' step leading to Eq. (70), since its denominator differs from ln(2 cosh(q/(2Tc))) beyond leading order.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claim requires that the overdoped branch of the dome, Eq. (72), be the exact x>x0 limit of the critical equation Eq. (56). It is not, as printed. Setting mu0 = -q < 0, the denominator in Eq. (56) is ln2 + ln cosh(mu0/(2Tc)) = ln(2 cosh(q/(2Tc))) = q/(2Tc) + ln(1 + e^{-q/Tc}). Eq. (72) instead contains only ln(1 + e^{-mu0/Tc}) = ln(1 + e^{q/Tc}), so the q/(2Tc) term is absent. The origin is Eq. (71): for mu0 = -q and |M0| < q, |mu0 + |M0|| + |mu0 - |M0|| = (q - |M0|) + (q + |M0|) = 2q, so B(x) = q/(2Tc), not 0. The printed evaluation swaps |M0| and |mu0| inside one absolute value. This is not a cosmetic typo: the overdoped dome, the quantum critical point x+_SC in Eq. (73), and the universal phase-diagram variable zeta of Eq. (85) all inherit this error. Even granting the fitted linear mu0(x) that the Reader identified, Eq. (72) does not follow from Eq. (56).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs an effective Hamiltonian for itinerant oxygen holes on a bipartite oxygen lattice, starting from a spin-fermion-Hubbard model. The attractive and repulsive effective couplings gS and gP are derived by tracing out copper spins and by a second-order tp/Up expansion, and the model is used to derive implicit analytic expressions for the superconducting transition temperature Tc(x) and the pseudogap temperature T*(x). After fitting a small number of parameters, the authors compare the resulting domes with data for LSCO, Bi2201/Bi2212/Bi2223, and Hg1201/Hg1212/Hg1223, and they extend the formalism to explain the increase of optimal Tc with the number of CuO2 layers and to describe the pressure dependence of Tc. The abstract claims excellent agreement with experiment and a unified explanation of the d-wave symmetry of both order parameters and of the ARPES Fermi pockets.","tokens_in":30987,"tokens_out":25232,"duration_ms":236958,"significance":"If the derivation were correct, the paper would provide a universal analytic form for the cuprate phase diagram, a microscopic rationale for d-wave symmetry in both channels, a quantitative account of the layer-number dependence of Tc, and a falsifiable prediction that T* is pressure-independent. The manuscript is rich in explicit analytic expressions and direct comparisons with published data, and its prediction that pressure does not affect the pseudogap temperature is a clear testable statement. However, the central algebraic reduction from Eq. (56) to Eqs. (70)-(72) is incorrect on both branches of the dome, and the pressure analysis in Sec. 7.3 fits the model to data that were themselves generated with the model's equations. As printed, the paper does not establish its main quantitative claims, although the framework may be salvageable after a substantial reworking.","major_comments":[{"comment":"The overdoped branch is not the |M0|->0 limit of Eq. (56). For mu0 = -q < 0, Eq. (56) gives a denominator ln2 + ln cosh(q/(2Tc)) = q/(2Tc) + ln(1 + e^{-q/Tc}), whereas Eq. (72) contains only ln(1 + e^{-mu0/Tc}) = ln(1 + e^{q/Tc}); the linear term q/(2Tc) is missing. The error originates in Eq. (71): with mu0 = -q and |M0| < q, one has |mu0 + |M0|| + |mu0 - |M0|| = 2q, so B(x) = q/(2Tc), not 0. Because x_SC^+ in Eq. (73) and the universal variable zeta in Eq. (85) are built on Eq. (72), this algebraic error propagates into the central quantitative claims of the paper.","section":"Section 4.4, Eqs. (67)-(73)"},{"comment":"The underdoped expression has the same type of problem. From Eq. (56) with mu0 > 0, the exact denominator is mu0/(2Tc) + ln(1 + e^{-mu0/Tc}). Equation (70) instead contains mu0/(2Tc) + ln2 + (1/2)(e^{-mu0/Tc} - 1), which is not an identity and is numerically poor away from optimal doping. For example, at mu0/Tmax = 1.12, which is still inside the dome whose exact edge from Eq. (56) is mu0/Tmax = 2 ln2, Eq. (56) gives Tc approximately 0.89 Tmax, whereas Eq. (70) gives approximately 0.53 Tmax. Consequently the quantum critical points in Eq. (73) are also inconsistent with Eq. (56), which yields x0 +/- Tmax ln2/gamma rather than the printed values. The statement that Eqs. (70) and (72) are derived from Eq. (56) therefore fails on both branches.","section":"Section 4.4, Eqs. (68)-(70)"},{"comment":"The pressure comparison is circular. The text states that the experimental pressure data were obtained from Ref. [61] using the paper's equations (70) and (72), rather than a parabola, to convert each measured Tc into a doping value. Those converted points are then fit with the same equations, with gamma and x0 refitted at every pressure (Figs. 29 and 30), and kappa_g is fitted in Sec. 7.2. The agreement in Figs. 27 and 28 is therefore not an independent test of the model. The authors should compare against the original Tc(P) data without converting doping through the model, or at least clearly separate the converted data from the fitted quantities.","section":"Section 7.3, Figs. 27-30"},{"comment":"The claim that only one parameter is fitted is overstated. For each material, Tmax and x0 are experimental inputs (Table I), gamma is fitted for Tc(x), and a separate tilde gamma is fitted for T*(x) (Table II). The linear relation mu0(x) = 2 gamma (x0 - x) is introduced as the simplest parametrization and is not derived from the microscopic Hamiltonian; the dome shape and the universal zeta depend on this assumed linearity. In the pressure section, kappa_g and the linear pressure dependences of gamma and x0 are additional fitted inputs. The paper should present these as phenomenological assumptions and adjust the wording of the one-parameter claim accordingly.","section":"Section 4.2, Eqs. (57)-(58); Tables I-II; Section 7.3"}],"minor_comments":[{"comment":"There are several typographical errors, including 'Altough' and 'approachs' in the introduction, and a dangling fragment 'The independent curve will be [h]' in the caption of Fig. 6; these should be corrected.","section":"Throughout"},{"comment":"The assumption that the moduli of compressibility kappa_J for J_AF and J_K are approximately equal is introduced without justification; it should be labeled as an additional assumption rather than a consequence of the model. The reference in the sentence 'Considering (19)' should point to Eq. (16), where gP is defined, rather than to the numerical value in Eq. (19).","section":"Section 7.1"},{"comment":"The caption states 'Only gamma has been adjusted,' but Tmax and x0 are taken from experiment; the caption should specify that gamma is the only fitted parameter, with Tmax and x0 treated as experimental inputs.","section":"Table I caption"},{"comment":"The statement that the chemical potential is proportional to x, 'mu(x) proportional to x', is inconsistent with the later affine relation mu0 = 2 gamma (x0 - x); the wording should be adjusted to avoid the contradiction.","section":"Section 3.2"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: the model framework is worth knowing about, but the dome equations at the center of the paper do not survive a check of the algebra. The pressure section is circular. The ambient-pressure N-scaling is the most interesting part.\n\nWhat's new: closed-form Tc(x) and T*(x) expressions, the zeta universality variable, and the Tmax(N) layer scaling. These go beyond the authors' earlier SFHM work. The bipartite-oxygen derivation of d-wave symmetry is clean, and the competing gS/gP mechanism is a plausible organizing idea. For N=2,3 the Tmax scaling works well; the charge-reservoir explanation for the N>3 downturn is sensible.\n\nThe soft spots are serious. Eq. (72) is not the |M0|->0 limit of Eq. (56). For mu0=-q, the B(x) term is q/(2Tc), not 0; the printed evaluation drops it. Eq. (70) also does not match Eq. (56) -- the denominator has ln2 + (1/2)(e^{-mu/Tc}-1) where the correct limit gives ln(1+e^{-mu/Tc}). So the asymmetric dome, x+_SC, and zeta all inherit an error. The correct Eq. (56) would produce a dome symmetric in mu0, so the paper's signature asymmetry is an artifact of the algebra. Even granting the linear mu0(x) parametrization, this is a load-bearing flaw, not a typo.\n\nThe pressure section has a circular step: Section 7.3 says the experimental pressure data were converted from Ref. [61] using Eqs. (70) and (72), and the model is then fit to those converted points. That cannot confirm the model. The claim of \"only one fitted parameter\" is also overstated: Tmax and x0 are inputs, tilde-gamma is fit separately, kappa_g is fit, and gamma and x0 are refit at every pressure. There are dimensional inconsistencies around Eqs. (51)-(53).\n\nWho should read it: people working on phenomenological phase diagrams and effective Hamiltonians for cuprates. The model framework and the N-scaling idea deserve a serious referee. But the paper as printed needs major revision: fix the limit, redo the pressure comparison with raw doping data, and disclose all fitted parameters. I would not cite it in its current form.","headline":"The paper's effective-Hamiltonian picture and layer scaling are worth a look, but the central dome equations have an algebraic error and the pressure comparison is circular; it needs major revision before it can be cited.","tokens_in":31564,"tokens_out":9683,"would_cite":false,"duration_ms":95121,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Two competing oxygen-hole forces yield the cuprate Tc dome","keywords":["cuprate superconductivity","pseudogap","Tc dome","d-wave pairing","bipartite oxygen lattice","spin-fermion Hubbard model","pressure dependence","universal phase diagram"],"falsifier":"Measure $T^*(x)$ under applied hydrostatic pressure in a cuprate such as Hg1201 or Hg1212: the model predicts $g_P$ is pressure-independent, so the pseudogap line should remain fixed while the superconducting dome moves; a measurable pressure shift of $T^*$ would falsify the central claim. A second check is the universal ratio $|\\Delta_0(0,x_0)|/T_{\\max} = 2\\ln 2$, which can be tested by scanning tunneling or optical measurements of the zero-temperature gap at optimal doping.","tokens_in":30394,"feed_emoji":"⚡","tokens_out":7537,"duration_ms":71417,"temperature":0.7,"pith_summary":"This paper claims that the doping-temperature phase diagram of the high-Tc cuprates follows from one effective Hamiltonian for holes on the oxygen sublattices of the CuO2 planes. Two interaction terms compete: an attractive coupling $g_S = J_K^2/(8J_{AF})$ that forms Cooper pairs and a repulsive coupling $g_P = 2t_p^2/U_p$ that forms exciton (d-density-wave) order; superconducting order wins at low temperature, while the pseudogap appears first on cooling from high temperature. The authors integrate out the fermions and minimize the resulting effective potential, obtaining analytic expressions for $T_c(x)$ and $T^*(x)$ that reproduce the measured superconducting domes and pseudogap lines for single-layer and multilayer cuprates. They also derive how $T_c$ grows with the number of CuO2 layers and how pressure shifts the superconducting dome, while predicting that the pseudogap temperature is pressure-independent. If correct, the paper supplies a universal functional form for the cuprate phase diagram and an account of the d-wave symmetry of both order parameters.","feed_headline":"Two competing oxygen-hole forces yield the cuprate Tc dome","feed_subtitle":"One fitted parameter reproduces the cuprate superconducting and pseudogap curves across families, layers, pressure.","key_machinery":"The load-bearing object is an effective Hamiltonian defined on the bipartite oxygen lattice of the CuO2 planes, formed because only one of the two oxygen p orbitals ($p_x$ or $p_y$) hybridizes with each copper $3d$ orbital. In this Hamiltonian, the attractive channel, written with a Hubbard-Stratonovich pairing field $\\Phi$, and the repulsive channel, written with a field $\\chi$, both inherit the sign alternation $\\Delta(d_{1,3}) = -\\Delta(d_{2,4})$, $M(d_{1,3}) = -M(d_{2,4})$; in momentum space this gives the d-wave factor $\\cos(k_+ a) - \\cos(k_- a)$ for both the superconducting gap and the pseudogap. The mechanism that produces the dome formulas is the effective potential $V_{\\rm eff}[\\Delta, M, \\mu]$, obtained from a Nambu four-component fermion integration; minimizing it with respect to $\\Delta$, $M$, and the chemical potential $\\mu$ produces the gap equations whose $\\Delta, M \\to 0$ limits are the transition-temperature equations. The chemical potential is coupled to doping by $\\mu_0(x) = 2\\gamma(g_S)(x_0 - x)$, a linear relation containing the single fitted parameter $\\gamma$.","core_discovery":"The central claim is that the cuprate phase diagram emerges from a duality between two condensates with the same d-wave symmetry, with no phonons needed. Starting from a spin-fermion-Hubbard model on the CuO2 planes, the authors trace out the localized copper spins and make a second-order expansion in $t_p/U_p$; the resulting effective Hamiltonian for oxygen holes contains an attractive term of coupling $g_S$ and a repulsive term of coupling $g_P$. The two couplings come out numerically close to values obtained from published LSCO parameters, and the two order parameters cannot be nonzero at the same time except at $g_S = g_P$, so the system chooses either a superconducting phase or a pseudogap (d-density-wave) phase. Functional integration over fermions and minimization of the effective potential yields implicit equations for $T_c(x)$ and $T^*(x)$, with optimal doping occurring when the chemical potential vanishes. The same machinery yields a universal rescaled phase diagram, an enhancement $g_S \\to N g_S$ with the number of CuO2 layers, and a pressure dependence $g_S(P) = g_S e^{\\kappa_g P}$ that accounts for measured $T_c$ shifts without changing the pseudogap line.","pith_inferences":["A testable consequence of the competition picture that goes beyond the reported fits: if the linear $\\mu_0(x)$ assumption is replaced by a nonlinear material-specific relation, the dome formulas should be re-derived; the same effective-potential machinery would still predict superconducting/pseudogap mutual exclusion as long as $g_S \\neq g_P$.","The prediction that pressure does not move the pseudogap line is a distinguishing signature: $g_P = 2t_p^2/U_p$ is argued to be pressure-invariant, so a pressure scan of $T^*$ by Nernst or ARPES measurements would either support or sharply conflict with the model.","If the universal ratio $|\\Delta_0(0,x_0)|/T_{\\max} = 2\\ln 2$ is checked across multiple cuprate families with different optimal dopings, it becomes a sharp test of the whole derivation, since it follows directly from the logarithmic structure of the gap equation rather than from any fitted parameter.","The same Hamiltonian could be pushed toward transport: the strange-metal and charge-ordering regimes above $T^*$ are mentioned by the authors as natural next targets, and the present formalism appears to have the ingredients (two competing order parameters and a doping-dependent chemical potential) to model them."],"forward_implications":["The same equations (70) and (72) generate the $T_c(x)$ domes for LSCO, Bi2201, Bi2212, Bi2223, Hg1201, Hg1212, and Hg1223 after adjusting one parameter $\\gamma$ per material, with reported close agreement to measured data.","Optimal doping is fixed by the vanishing chemical potential, and the zero-temperature gap satisfies $|\\Delta_0(0,x_0)|/T_{\\max} = 2\\ln 2$ for all cuprates covered by the theory.","Multi-layer enhancement follows from $g_S \\to Ng_S$, predicting $T_{\\max}(N)$ for Bi and Hg families up to three layers; beyond three layers, inner planes are far from charge reservoirs and are poorly doped, which explains the observed saturation and decline.","Under pressure, $g_S$ grows exponentially with an effective compressibility $\\kappa_g$, giving $T_{\\max}(P)$ and $T_c(x,P)$ curves fitted to mercury-family data, while the predicted pseudogap temperature $T^*(x)$ is independent of pressure.","Rescaling by $p = x/x_0$ and $\\tau_c = T_c/T_{\\max}$ collapses the calculated domes onto a universal curve governed by a single dimensionless parameter $\\zeta = \\gamma x_0/T_{\\max}$, in line with the reported universal phase diagram."],"supporting_citations":[{"why":"Reports the first high-Tc cuprate superconductor and defines the experimental phenomenon the paper aims to explain.","marker":"[1]"},{"why":"Three-band Hubbard model with copper d and oxygen p orbitals, supplying the orbital structure behind the oxygen sublattices.","marker":"[17]"},{"why":"Spin-fermion model for oxygen holes and localized copper spins, a starting point for the effective Hamiltonian.","marker":"[24]"},{"why":"Spin-fermion Hubbard model whose Hamiltonian is the paper's starting point and which provides the magnetic pairing mechanism.","marker":"[28]"},{"why":"Shows competing effective interactions in the spin-fermion system, the duality used to obtain the SC and PG terms.","marker":"[29]"},{"why":"Provides LSCO three-band parameters used to compute $g_S$ and $g_P$ from the model couplings.","marker":"[30]"},{"why":"Quantum field theory methods for tracing out spins, functional integration, and constructing the effective potential.","marker":"[38]"},{"why":"Gives the pressure dependence of copper-oxide magnetic exchange couplings used to derive $g_S(P)$.","marker":"[60]"},{"why":"Supplies high-pressure $T_c$ data for Hg1212 and Hg1223 used to test the pressure predictions.","marker":"[61]"}],"fun_headline_variants":["Cuprate Tc dome from competing oxygen-hole forces","One model reproduces cuprate Tc and pseudogap curves","Pressure tunes cuprate Tc but not pseudogap","Oxygen-hole duality explains cuprate phase diagram","Same d-wave symmetry two phases one model"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the chemical potential is a linear function of doping, $\\mu_0(x) = 2\\gamma(g_S)(x_0 - x)$, with $\\gamma$ fixed by fitting; if the true relation is nonlinear and material-dependent, the specific analytic forms of the $T_c(x)$ dome and the universal phase diagram do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Cuprate Tc dome from competing oxygen-hole forces","One model reproduces cuprate Tc and pseudogap curves","Pressure tunes cuprate Tc but not pseudogap","Oxygen-hole duality explains cuprate phase diagram","Same d-wave symmetry two phases one model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000836,"raw_usage":{"total_tokens":3734,"prompt_tokens":1119,"completion_tokens":2615,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":735,"completion_tokens_details":{"reasoning_tokens":2537}},"tokens_in":735,"tokens_out":2615,"duration_ms":18640,"temperature":1.0,"reasoning_tokens":2537,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:29:01.319469+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure $T^*(x)$ under applied hydrostatic pressure in a cuprate such as Hg1201 or Hg1212: the model predicts $g_P$ is pressure-independent, so the pseudogap line should remain fixed while the superconducting dome moves; a measurable pressure shift of $T^*$ would falsify the central claim. A second check is the universal ratio $|\\Delta_0(0,x_0)|/T_{\\max} = 2\\ln 2$, which can be tested by scanning tunneling or optical measurements of the zero-temperature gap at optimal doping.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports the first high-Tc cuprate superconductor and defines the experimental phenomenon the paper aims to explain."},{"cited_title":"We find for all the materials of the Bi and Hg families: Λ = 0 .018eV","cited_arxiv_id":null,"evidence_quote":"Three-band Hubbard model with copper d and oxygen p orbitals, supplying the orbital structure behind the oxygen sublattices."},{"cited_title":"Damascelli, Z.-X","cited_arxiv_id":null,"evidence_quote":"Spin-fermion Hubbard model whose Hamiltonian is the paper's starting point and which provides the magnetic pairing mechanism."},{"cited_title":"H¨ ufner, M","cited_arxiv_id":null,"evidence_quote":"Shows competing effective interactions in the spin-fermion system, the duality used to obtain the SC and PG terms."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides LSCO three-band parameters used to compute $g_S$ and $g_P$ from the model couplings."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Quantum field theory methods for tracing out spins, functional integration, and constructing the effective potential."},{"cited_title":"Chakravarty, C","cited_arxiv_id":null,"evidence_quote":"Gives the pressure dependence of copper-oxide magnetic exchange couplings used to derive $g_S(P)$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies high-pressure $T_c$ data for Hg1212 and Hg1223 used to test the pressure predictions."}],"review_version":1}