Superconducting and Pseudogap Transition Temperatures in High-Tc Cuprates and the $T_{c}$ Dependence on Pressure

E. C. Marino; Lizardo H. C. M. Nunes; R. Arouca; Reginaldo O. Corr\^ea Jr; Van S\'ergio Alves

REVIEW 4 major objections 4 minor 88 references

Superconducting and Pseudogap Transition Temperatures in High-Tc Cuprates and the $T_{c}$ Dependence on Pressure

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Two competing oxygen-hole forces yield the cuprate Tc dome

desk verdict 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. read the letter →

arxiv 1908.07028 v2 pith:MYUUYPDE submitted 2019-08-19 cond-mat.supr-con cond-mat.str-el

E. C. Marino , Reginaldo O. Corrêa Jr , R. Arouca , Lizardo H. C. M. Nunes , Van Sérgio Alves This is my paper · ORCID

classification cond-mat.supr-concond-mat.str-el

keywords cupratesuperconductivitypseudogapTcdomed-wavepairingbipartiteoxygenlatticespin-fermionHubbardmodelpressuredependenceuniversalphasediagram

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

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.

What carries the argument

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 $\gamma$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

What'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.

The 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.

The 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).

Who 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.

Referee Report

4 major / 4 minor

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.

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 (4)

[Section 4.4, Eqs. (67)-(73)] 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 4.4, Eqs. (68)-(70)] 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 7.3, Figs. 27-30] 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 4.2, Eqs. (57)-(58); Tables I-II; Section 7.3] 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.

minor comments (4)

[Throughout] 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 7.1] 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).
[Table I caption] 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 3.2] 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.

Circularity Check

1 steps flagged · score 6.0 of 10

Pressure-dome comparison is circular: the plotted data are mapped by the model's own Eqs. (70)-(72), then fitted and replotted as 'prediction'; the ambient dome is a one-parameter fit.

fitted input called prediction [Section 7.3, around Eqs. (70) and (72) and Figs. 27-30]
"The experimental data of the two previous figures were obtained from [61], using our equations (70) and (72), instead of a parabola for obtaining the different doping values corresponding to each value of Tc following the same procedure as in [61]."

The plotted pressure-dome points are assigned their x-coordinates by inverting the very Tc(x) equations that are then drawn as the theoretical prediction, so each point lies on the model curve by construction. The parameters x0(P) and gamma(P) shown in Figs. 29 and 30 are fitted to these model-converted points, and the phase diagrams in Figs. 27 and 28 compare the model with data that have been re-expressed through the model. This makes the pressure phase-diagram agreement a self-consistency check rather than an independent confirmation of the derived Tc(x,P).

full rationale

The main ambient-pressure Tc(x) derivation is not circular in the strict sense: Eq. (56) follows from the fermion functional integral, and the self-citations (Refs. 28, 29, 38, 42) are methodological, with the traced-spin and perturbative steps exhibited in the appendices, so they are not load-bearing self-citation. The chemical potential relation mu0(x)=2gamma(x0-x) is, however, explicitly introduced as 'the simplest parametrization' and gamma is adjusted to the experimental Tc(x) curves, so the excellent ambient-pressure dome agreement is a one-parameter fit rather than an independent prediction. The genuine circular step is in the pressure analysis: Section 7.3 obtains the plotted doping values from the model's own Eqs. (70) and (72), then fits x0(P) and gamma(P) to those converted points, so the claimed agreement is by construction. Separately, as printed, Eq. (71) is not the |M0|->0 limit of Eq. (56) for mu0<0: with mu0=-q<0, |mu0+|M0||+|mu0-|M0|| = 2q, so B(x)=q/(2Tc), not 0, and Eq. (72) omits the q/(2Tc) term. That is an internal consistency/correctness defect, not a circularity; it would need correction before the overdoped branch, x+_SC, and zeta in Eq. (85) could be regarded as derived. Overall the pressure-dome prediction reduces in part to the model's own equations, giving a partial circularity score of 6.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central results rest on several inputs: fitted shape parameters gamma and tilde gamma per material, experimental values of Tmax and x0, a linear chemical-potential ansatz, and a family-constancy assumption for the couplings. No fundamentally new microscopic entity is introduced; the order parameters Delta and M are standard. The 'one parameter' claim in the abstract omits the tilde gamma fit, the kappa_g fit, and the per-pressure refits of gamma and x0.

free parameters (4)

gamma (shape parameter for Tc dome) = 0.012 to 0.054 eV depending on material (Table I)
Adjusted per material so that the solution of Eqs. (70)/(72)/(74) fits the experimental Tc(x) dome. The paper's claim of a single fitted parameter applies only here.
tilde gamma (shape parameter for T* line) = 0.089 to 0.2708 eV depending on material (Table II)
Adjusted per material for the pseudogap transition line T*(x), Eq. (76). This is a separate fitted parameter not mentioned in the 'only one parameter' claim.
kappa_g (pressure compressibility of gS) = 1/17 GPa^-1
Adjusted once to match the Tmax(P) curves for both Hg1212 and Hg1223 in Section 7.2.
gamma(P) and x0(P) linear slopes = gamma: 1% and 1.79% per GPa; x0: -0.003 and -0.001 per GPa (Figs. 29-30)
Linear fits of gamma and x0 versus pressure for Hg1212 and Hg1223, used to produce the Tc(x,P) phase diagrams. These are additional fitted parameters in the pressure section.

assumptions (5)

domain assumption The Spin-Fermion-Hubbard model with parameters from Hansmann et al. (2014) is the correct starting point, and the same family coupling gS applies to all members of a cuprate family.
Used in Sections 2 and 5.1 to express gS and gP in terms of 3BHM parameters and to derive the layer scaling relations. No independent verification is provided for the family-wide constancy.
domain assumption The oxygen lattice is bipartite with only one of the px or py orbitals hybridizing with copper d-orbitals, leading to the d-wave form of both order parameters.
Structural assumption used in Section 2.1 to define the bipartite sublattices and in Section 2.5 to derive the d-wave gap symmetry. It is motivated by crystal structure and is plausible.
ad hoc to paper The linear chemical potential-doping relation μ0(x) = 2γ(x0 - x) is the correct connection between stoichiometric doping and hole chemical potential.
Introduced in Section 4.2, Eq. (57), as 'the simplest parametrization' and determined by fitting. All dome-shape results depend on this linear ansatz.
domain assumption Interplane interactions are negligible; the only effect of multiple CuO2 planes is the enhancement gS -> NgS.
Stated in Section 2.5 before Eq. (28). Used to derive the layer-number scaling of Tmax. For N=4,5 the assumption fails and is patched with a charge-reservoir argument.
ad hoc to paper The pressure dependence of magnetic couplings is exponential, J(P) = J(0) exp(kappa P), with the same kappa for JAF and JK, and gP is pressure-independent because Up scales as tp^2.
Assumed in Section 7.1, Eqs. (95)-(96). The exponential form is motivated by overlap integrals but the common kappa is not demonstrated.

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Pith. "Pith review of Superconducting and Pseudogap Transition Temperatures in High-Tc Cuprates and the $T_{c}$ Dependence on Pressure." pith.science (2026). https://pith.science/paper/MYUUYPDE

@misc{pith2026190807028,
  author       = {Pith},
  title        = {Pith review of: Superconducting and Pseudogap Transition Temperatures in High-Tc Cuprates and the $T_c$ Dependence on Pressure},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MYUUYPDE}},
  note         = {Machine review of arXiv:1908.07028}
}

abstract

We derive analytic expressions for the critical temperatures of the superconducting (SC) and pseudogap (PG) transitions of the high-Tc cuprates as a function of doping. These are in excellent agreement with the experimental data both for single-layered materials such as LSCO, Bi2201 and Hg1201 and multi-layered ones, such as Bi2212, Bi2223, Hg1212 and Hg1223. Optimal doping occurs when the chemical potential vanishes, thus leading to an universal expression for the optimal SC transition temperatures. This allows for the obtainment of a quantitative description of the growth of such temperatures with the number of layers, N, which accurately applies to the $Bi$, $Hg$ and $Tl$ families of cuprates. We study the pressure dependence of the SC transition temperatures, obtaining excellent agreement with the experimental data for different materials and dopings. These results are obtained from an effective Hamiltonian for the itinerant oxygen holes, which includes both the electric repulsion between them and their magnetic interactions with the localized copper ions. We show that the former interaction is responsible for the SC and the latter, for the PG phases, the phase diagram of cuprates resulting from the competition of both. The Hamiltonian is defined on a bipartite oxygen lattice, which results from the fact that only the $p_x$ and $p_y$ oxygen orbitals alternatively hybridize with the $3d$ copper orbitals. From this, we can provide an unified explanation for the $d_{x^2-y^2}$ symmetry of both the SC and PG order parameters and obtain the Fermi pockets observed in ARPES experiments.

Figures

Figures reproduced from arXiv: 1908.07028 by the authors.

**Figure 3.** FIG. 3: The bipartite oxygen crystal structure, showing two [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** FIG. 4: The hybridization of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (25 more)

**Figure 5.** Figure 5: FIG. 5: Fermi surfaces for different levels of doping, at [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: FIG. 6: Fermi surfaces for different levels of doping at [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: FIG. 7: Solution of Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: FIG. 8: Solution of Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

**Figure 9.** Figure 9: FIG. 9: Solution of Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

**Figure 10.** Figure 10: FIG. 10: Solution of Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

**Figure 11.** Figure 11: FIG. 11: Solution of Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]

**Figure 12.** Figure 12: FIG. 12: Solution of Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]

**Figure 14.** Figure 14: FIG. 14: Solution of Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]

**Figure 15.** Figure 15: FIG. 15: Solution of Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p016_15.png]

**Figure 18.** Figure 18: FIG. 18: Solution of Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p017_18.png]

**Figure 19.** Figure 19: FIG. 19: Solution of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p017_19.png]

**Figure 22.** Figure 22: FIG. 22: Universal Phase Diagram for all compounds studied [PITH_FULL_IMAGE:figures/full_fig_p018_22.png]

**Figure 23.** Figure 23: FIG. 23: The [PITH_FULL_IMAGE:figures/full_fig_p019_23.png]

**Figure 25.** Figure 25: FIG. 25: Optimal temperature of Hg1223 as a function of [PITH_FULL_IMAGE:figures/full_fig_p020_25.png]

**Figure 26.** Figure 26: FIG. 26: Optimal Temperature calculated from our model [PITH_FULL_IMAGE:figures/full_fig_p020_26.png]

**Figure 24.** Figure 24: FIG. 24: Optimal temperature of Hg1212 as a function of [PITH_FULL_IMAGE:figures/full_fig_p020_24.png]

**Figure 27.** Figure 27: FIG. 27: Phase diagram of Hg1212 as a function of pressure. [PITH_FULL_IMAGE:figures/full_fig_p021_27.png]

**Figure 29.** Figure 29: FIG. 29: Dependence of [PITH_FULL_IMAGE:figures/full_fig_p021_29.png]

**Figure 30.** Figure 30: FIG. 30: Dependence of [PITH_FULL_IMAGE:figures/full_fig_p021_30.png]

**Figure 31.** Figure 31: FIG. 31 [PITH_FULL_IMAGE:figures/full_fig_p022_31.png]

**Figure 32.** Figure 32: FIG. 32: Temperature [PITH_FULL_IMAGE:figures/full_fig_p022_32.png]

**Figure 33.** Figure 33: FIG. 33 [PITH_FULL_IMAGE:figures/full_fig_p022_33.png]

**Figure 34.** Figure 34: FIG. 34: Unperturbed ground state [PITH_FULL_IMAGE:figures/full_fig_p025_34.png]

**Figure 35.** Figure 35: FIG. 35: Unperturbed excited state [PITH_FULL_IMAGE:figures/full_fig_p025_35.png]

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Reference graph

Works this paper leans on

88 extracted references · 76 canonical work pages

[61]

B. H. Bransden and C. J. Joachain, Introduction to Quantum Mechanics, Longman, UK (1989), p. 360

work page 1989
[1]

Tmax(n = 2) Tmax(n = 1) = η(n = 2) η(n = 1)

work page
[2]

The CuO 2 planes have a lattice structure in which Cu ++ ions occupy the sites and oxygen ions the links of a square lattice, with a lattice parameter a = 3.8˚ A

The Eﬀective Hamiltonian 2.1) The Oxygen Sublattices An outstanding feature of all High-Tc cuprates is the presence of one or more CuO 2 planes, intercepting the primitive unit cell of such compounds. The CuO 2 planes have a lattice structure in which Cu ++ ions occupy the sites and oxygen ions the links of a square lattice, with a lattice parameter a = 3...
[3]

In this work, we will consider only hole doping, in which electrons are progressively removed from the oxygenpx andpy orbitals, thereby creating holes in such orbitals

The Doping Process 3.1) The Mechanism of Doping The process of doping plays a central role in the physics of High-Tc cuprates. In this work, we will consider only hole doping, in which electrons are progressively removed from the oxygenpx andpy orbitals, thereby creating holes in such orbitals. The oxygen ions are themselves, placed on the sites of a squa...
[4]

The SC and PG T ransition T emperatures: Derivation 4.1) F ermion Integration We shall now integrate over the fermions, taking into account the doping constraint term, in order to obtain an eﬀective potential in terms of the SC and PG order parameters and the chemical potential. The eﬀective po- tential, despite being a function of the ground-state ex- pe...
[5]

The SC and PG T ransition T emperatures: Applications 5.1) Determination of Parameters We present a summary of the relevant parameters for each of the compounds considered in this study in Table I. For determining the values of the relevant parameters for each family group (Bismuth family, Mercury family, LSCO) we proceed through the following steps: a) F...

work page
[6]

We now consider the following regimes: a) T >T ∗ In this case M = ∆ = 0 and µ2(x) = ǫ2(k) = v2 eff (cosk+a +cosk−a)2 (41) FIG

This leads to a second degree equation whose solution is µ(x) = ∓ √ ǫ2(k) + |M (k)|2 ±i|∆( k)| (40) Notice that the above expression becomes complex wherever a nonzero SC gap exists, reﬂecting the fact that no Fermi surface exists in the presence of a SC gap. We now consider the following regimes: a) T >T ∗ In this case M = ∆ = 0 and µ2(x) = ǫ2(k) = v2 ef...
[7]

For an inﬁnitesimal change of pressure, dP , L(P ) would satisfy the linear diﬀerential equation, 1 L dL dP = −κL, (89) which is solved by L(P ) = L0e−κ LP

Eﬀects of an Applied External Pressure 7.1) Preliminary Considerations Under the eﬀect of a change in pressure, given by ∆ P = P −P0 a linear segment with original length L0 would shrink to L, such that L −L0 L0 = −κL∆ P, (88) whereκL is the linear modulus of compressibility. For an inﬁnitesimal change of pressure, dP , L(P ) would satisfy the linear diﬀe...

work page

Show all 88 references

[8]

The ﬁrst is the SC phase, while the second is the PG phase

(53) We see that since gS ⁄= gP it is impossible to satisfy (47) and (48) simultaneously with both ∆ 0 ⁄= 0 and M0 ⁄= 0, so we must have either ∆ 0 ⁄= 0 and M0 = 0 or ∆ 0 = 0 and M0 ⁄= 0. The ﬁrst is the SC phase, while the second is the PG phase. For a ﬁxed value of the dopin...
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Honma and P

T. Honma and P. H. Hor, Uniﬁed electronic phase dia- gram for hole-doped high-T c cuprates, Phys. Rev. B 77, 184520 (2008)

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2.2, we have: JK = 1.17eV and JAF = 0.43eV

and ( 12), we have gS = J 2 K 8JAF gP = 2t2 p Up (16) Using the values of the magnetic coupling parameters valid for LSCO, provided in Sect. 2.2, we have: JK = 1.17eV and JAF = 0.43eV . From this, we obtain J 2 K 8JAF = 0.39793 eV (17) This corresponds, with excellent accuracy...
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as Σ = nA +−nB −+ +nA −+nB +− Π = nA +−nB +− +nA −+nB −+ Ξ 1 =nA ++nB −− +nA −−nB ++ Ξ 2 =nA ++nB ++ +nA −−nB −− (111) and using ( 110), we establish ( 15), up to a constant. 25 Appendix C In this Appendix, we perform a perturbation expansion in tp/Up, in H0 +HU , given by H0 ...
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η(n = 3) = η(n = 1) 3 + 2 3
[16]

From the three equations above, using the experimen- tal values of Tmax(n = 2) and Tmax(n = 1), we deter- mine η(n = 1) = 0 .23077, η(n = 2) = 0 .61538, η(n =

η(n = 2) = η(n = 1) 2 + 1 2 (80) where the last two equations derive from ( 79), assum- ing that the coupling parameters gS are the same for all compounds of the same family. From the three equations above, using the experimen- tal values of Tmax(n = 2) and Tmax(n = 1), we det...
[17]

We ﬁnd for all the materials of the Bi and Hg families: Λ = 0 .018eV

= 0.74358 for the Bi family and η(n = 1) = 0 .61577, η(n = 2) = 0 .80788, η(n = 3) = 0 .87192 for the Hg family; c) Now use Tmax(n) = η(n) 2 ln 2Λ (81) to determine Λ. We ﬁnd for all the materials of the Bi and Hg families: Λ = 0 .018eV . d) for LSCO, use the value of Λ found ...
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charge reservoir

Increasing of Tmax with the Number of Layers It is an evident experimental fact that the optimal transition temperature becomes higher as one increases the number of CuO 2 planes per primitive unit cell. Bi2201 and Hg1201, for instance, are single-layered ma- terials, which ha...
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Concluding Remarks We report here the obtainment of analytical expres- sions for Tc(x), T ∗(x) and Tc(x,P ), for several High-Tc cuprate compounds, which show an excellent agreement with the experimental data. Our starting point is a Spin- Fermion-Hubbard (SFH) Model, describi...
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In that case, dimerization pro- duces a nonzero ground-state expectation value of that ﬁeld, which generates a gap for the electrons

that ∆( k) = ∑ i=1,.., 4 ∆( di) exp [ ik · di ] M (k) = ∑ i=1,.., 4 M (di) exp [ ik · di ] (30) Then, notice that ∆ and M in ( 28), eﬀectively act as hopping parameters for the fermion ﬁeld, similarly to the dimerization ﬁeld in the Su-Schriefer-Heeger model for polyacetylene ...
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T. Honma, P. H. Hor, H. H. Hsieh and M. Tanimoto Universal intrinsic scale of the hole concentration in high - Tc cuprates, Phys. Rev. B 70, 214517 (2004)

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and ( 49) imply µ0(x) = d(x) gc 2η(NgS) (54) where gc =α/Λ. In order to ﬁnd the critical temperature Tc(x), we im- pose on ( 47) the condition ∆ 0 = 0 and M0 = 0, which expresses the fact that the system is in one of the points belonging to the critical curve which separates t...
[48]

(64) Now (49) yields the following expression for the chem- ical potential ˜µ0 = 2˜γ(NgP )(˜x0 −x)

in the limit M0 → 0, which leads to T ∗(x) = lim M0→0 αη (Ng P ) 2gc F (∆ 0 = 0,M 0, ˜µ0(x)). (64) Now (49) yields the following expression for the chem- ical potential ˜µ0 = 2˜γ(NgP )(˜x0 −x). (65) Observe that, because M0 ⁄= 0 in the PG phase, the chemical potential ˜ µ0(x),...
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considered at the values: m = 2,...,N , we have N − 1 linear equations relating the η(mg)’s, namely, η(mg) = m − 1 m + 1 mη(1g). (78) Notice that, in order to obtain the above equation, we must assume the coupling parameters, either g =gS (or g =gP , below) in η(Ng ) are the s...
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with ( 55) we can express the optimal temperature as Tmax = Λ 2 ln 2η(NgS) (59) This should be compared with the corresponding BCS result, namely (see [ 38], for instance) Tc;BCS = 2γ π ℏωDe− 1 gBCS N (EF ) (60) where γ is the exponential of the Euler’s constant C ≃ 0.577 and ...
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for determining the SC critical temperature It is instructive to compare our result with the empir- ical curve, known since a long time [ 8, 9, 43], obtained by ﬁtting the data for the LSCO dome, by the following parabola Tc(x) = Tmax [ 1 − 82.616(x0 −x)2] . In Fig. 21 we supe...
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A summary of the results found by following the steps described above is shown in Table I

ﬁnd ˜η(N = 1); i) Use ( 78) to determine ˜η(n) from ˜η(n = 1); j) Determine gP from NgP gc = 1 1 − ˜η(N ) ; (83) and from gc, found in item d. A summary of the results found by following the steps described above is shown in Table I. 5.2) The SC Crit- ical T emperatures We now...
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into ( 51), we obtain the following curves for Tmax(P ), respectively, for Hg 1212 andHg 1223, after adjusting the parameter κg to the sin- gle value κg = 1 17GPa −1 for both compounds. FIG. 24: Optimal temperature of Hg1212 as a function of pressure, according to our theoreti...
[98]

(100) Here H[sN] = HAF [sN] +HK[sN,ψ ]

as a functional integral over N (see, for in- stance [ 38]): TrSIe −β [ HAF [SI ]+HK [SI,ψ ] ] = ∫ DN exp { ∫ β 0 dτ [ ⟨N(τ )| d dτ |N(τ )⟩ −H[sN] ] } . (100) Here H[sN] = HAF [sN] +HK[sN,ψ ]. (101) 24 We now separate N in antiferromagnetic and ferro- magnetic components, deno...

Pith tools

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

[1] [61]

B. H. Bransden and C. J. Joachain, Introduction to Quantum Mechanics, Longman, UK (1989), p. 360

work page 1989

[2] [1]

Tmax(n = 2) Tmax(n = 1) = η(n = 2) η(n = 1)

work page

[3] [2]

The CuO 2 planes have a lattice structure in which Cu ++ ions occupy the sites and oxygen ions the links of a square lattice, with a lattice parameter a = 3.8˚ A

The Eﬀective Hamiltonian 2.1) The Oxygen Sublattices An outstanding feature of all High-Tc cuprates is the presence of one or more CuO 2 planes, intercepting the primitive unit cell of such compounds. The CuO 2 planes have a lattice structure in which Cu ++ ions occupy the sites and oxygen ions the links of a square lattice, with a lattice parameter a = 3...

[4] [3]

In this work, we will consider only hole doping, in which electrons are progressively removed from the oxygenpx andpy orbitals, thereby creating holes in such orbitals

The Doping Process 3.1) The Mechanism of Doping The process of doping plays a central role in the physics of High-Tc cuprates. In this work, we will consider only hole doping, in which electrons are progressively removed from the oxygenpx andpy orbitals, thereby creating holes in such orbitals. The oxygen ions are themselves, placed on the sites of a squa...

[5] [4]

The SC and PG T ransition T emperatures: Derivation 4.1) F ermion Integration We shall now integrate over the fermions, taking into account the doping constraint term, in order to obtain an eﬀective potential in terms of the SC and PG order parameters and the chemical potential. The eﬀective po- tential, despite being a function of the ground-state ex- pe...

[6] [5]

The SC and PG T ransition T emperatures: Applications 5.1) Determination of Parameters We present a summary of the relevant parameters for each of the compounds considered in this study in Table I. For determining the values of the relevant parameters for each family group (Bismuth family, Mercury family, LSCO) we proceed through the following steps: a) F...

work page

[7] [6]

We now consider the following regimes: a) T >T ∗ In this case M = ∆ = 0 and µ2(x) = ǫ2(k) = v2 eff (cosk+a +cosk−a)2 (41) FIG

This leads to a second degree equation whose solution is µ(x) = ∓ √ ǫ2(k) + |M (k)|2 ±i|∆( k)| (40) Notice that the above expression becomes complex wherever a nonzero SC gap exists, reﬂecting the fact that no Fermi surface exists in the presence of a SC gap. We now consider the following regimes: a) T >T ∗ In this case M = ∆ = 0 and µ2(x) = ǫ2(k) = v2 ef...

[8] [7]

For an inﬁnitesimal change of pressure, dP , L(P ) would satisfy the linear diﬀerential equation, 1 L dL dP = −κL, (89) which is solved by L(P ) = L0e−κ LP

Eﬀects of an Applied External Pressure 7.1) Preliminary Considerations Under the eﬀect of a change in pressure, given by ∆ P = P −P0 a linear segment with original length L0 would shrink to L, such that L −L0 L0 = −κL∆ P, (88) whereκL is the linear modulus of compressibility. For an inﬁnitesimal change of pressure, dP , L(P ) would satisfy the linear diﬀe...

work page