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REVIEW 3 major objections 4 minor 62 references

Superconductivity, pseudogap and marginal Fermi liquid in a relative-momentum-local theory

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper argues that d-wave superconductivity, the pseudogap with Fermi arcs, and the strange metal are three low-energy organizations of one Cooper-pair channel of the t-J model, unified by a relative-momentum-local theory.

desk verdict Clever and transparent, but the central unification claim rests on postulated relative-momentum locality that the paper itself concedes is underived. read the letter →

arxiv 2608.09785 v1 pith:GRK3B5VX submitted 2026-08-10 cond-mat.supr-con

Jian-Jian Miao This is my paper · ORCID
classification cond-mat.supr-con
keywords cupratesuperconductorspseudogapstrangemetalmarginalFermiliquidCooperchannelt-Jmodelsurface2e-ARPES
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that cuprate superconductivity, the pseudogap, and the strange metal are not separate phenomena with separate mechanisms but three low-energy organizations of the same Cooper-pair channel that arises from antiferromagnetic superexchange in the t-J model. It builds a relative-momentum-local (RML) theory in which low-energy pair correlations factorize into independent relative-momentum blocks, and shows that a static version of this factorization yields an exactly solvable pseudogap Hamiltonian that produces nodal electron poles, antinodal gaps, and resolution-broadened Fermi arcs. A dynamic version of the same factorization yields a one-loop self-energy with marginal-Fermi-liquid scaling, and the underlying pair spectrum contains gapless charge-2e Cooper surfaces whose Lifshitz transition produces logarithmic heat-capacity enhancements. If the argument is right, a single pair-physics thread connects single-particle, pair-sensitive, and thermodynamic measurements, and zero-energy Cooper-surface ridges become a decisive 2e-ARPES signature.

What carries the argument

The workhorse is the exact operator identity that rewrites the superexchange interaction as a sum over Cooper-pair operators: $\hat H_J = -\frac{2J}{N_s}\sum_{Q,\alpha=s,d}\sum_{q,q'}\phi_{q,\alpha}\phi_{q',\alpha}\hat\Delta^\dagger_{Q,q}\hat\Delta_{Q,q'}$, with $\phi_{q,d}=(\cos q_x-\cos q_y)/2$ and $\hat\Delta_{Q,q}$ the spin-singlet pair operator at centre-of-mass momentum $Q$ and relative momentum $q$. On top of this, the paper postulates relative-momentum locality: at low energy the pair kernel factorizes into independent $\{q,-q\}$ blocks. The static prescription sets the intra-block pairing coupling to $J_q(x,T)=\sqrt{2}\Phi_d(x,T)\phi^2_{q,d}$, making $\hat H_{PG}=\sum_q'[\xi_q(\hat n_q+\hat n_{-q})-2J_q\hat\Delta^\dagger_q\hat\Delta_q]$ exactly solvable in a 16-dimensional Fock space per block; the dynamic prescription uses a block-local pair field with inverse propagator $D^{-1}_{\Xi,0}=r_s+i\ell_s\Omega_m+\gamma_s|\Omega_m|$ to generate Ohmic damping and a marginal-Fermi-liquid self-energy. The two prescriptions are the mechanism that translates one Cooper channel into the three observed regimes.

What would settle it

In the pseudogap regime, a momentum- and energy-resolved two-electron photoemission experiment should see zero-energy charge-$2e$ ridges along $S_{2e}:\xi_q=0$ and $S_{2h}:\xi_q=J_q$ near the nodal points, with the two ridges meeting at the pseudogap nodes; observing no such ridges when the pair-removal matrix elements are symmetry-allowed and the resolution is sufficient would falsify the Cooper-surface part of the theory. In the strange-metal regime, measuring the dressed finite-centre-of-mass pair continuum and the single-particle self-energy in the same doping and temperature window should reveal $\omega/T$ collapse of $-\operatorname{Im}\Sigma^R/T$ and a $T$-linear decay rate; their absence would falsify the dynamic RML and marginal-Fermi-liquid claims.

Watch

Extended reading notes

Core claim

The central claim is that the exact spin-singlet Cooper representation of antiferromagnetic superexchange in the t-J model, in which the interaction is written as attractive scattering of Cooper pairs conserving centre-of-mass momentum, is the common thread behind superconductivity, the pseudogap, and the strange metal. The paper derives, from a static relative-momentum-local prescription with coupling $J_q(x,T)=\sqrt{2}\Phi_d(x,T)\phi^2_{q,d}$, an exactly solvable block Hamiltonian whose ground states are empty, singly paired, or doubly paired blocks; this produces nodal poles, antinodal gaps, Fermi arcs controlled by the energy resolution, and two gapless Cooper surfaces $S_{2e}:\xi_q=0$ and $S_{2h}:\xi_q=J_q$. It then shows that a dynamic RML prescription with pair-field inverse propagator $D^{-1}=r_s+i\ell_s\Omega_m+\gamma_s|\Omega_m|$ gives a one-loop fermion self-energy with marginal-Fermi-liquid scaling $-\operatorname{Im}\Sigma^R_s(q_F,\omega)\simeq\lambda_s\,\omega\coth(\omega/2T)$. The same pair spectrum supports a Cooper-surface Lifshitz transition at the doping $x_{L,2e}=0.1934$ where a Cooper surface passes through the band-structure saddle point, yielding logarithmic enhancements of $c_V/T$ and the charge response with no spin singularity. The paper concludes that superconductivity is marked by phase stiffness and global coherence, the pseudogap by phase-incoherent quasi-static pair weight and gapless Cooper surfaces, and the strange metal by phase-incoherent dynamic pair fluctuations, with zero-energy Cooper-surface ridges as the decisive experimental signature.

Load-bearing premise

The framework rests on the assumption that, at low energies, pairing interactions in the t-J model act independently in each momentum pair $\{q,-q\}$, so that the simple block-local coupling $J_q=\sqrt{2}\Phi_d\phi^2_{q,d}$ and the dynamic inverse propagator $D^{-1}=r_s+i\ell_s\Omega_m+\gamma_s|\Omega_m|$ capture the full pair physics—an assumption the paper itself says is not yet derived from BKT disordering, the Cooper logarithm, or the $Q\to 0$ limit.

Editorial extensions

If this is right

  • In the pseudogap regime, the visible Fermi arc is a thermally redistributed and resolution-broadened spectral feature, not a continuous locus of gapless quasiparticles; its width is set by the energy resolution $\Gamma$ and its length scales as $\sqrt{\Gamma/\kappa_n}$.
  • The pseudogap hosts two gapless charge-$2e$ Cooper surfaces, $S_{2e}:\xi_q=0$ and $S_{2h}:\xi_q=J_q$; when a Cooper surface crosses the band-structure saddle point at $x_{L,2e}=0.1934$, a neck-reconnection Lifshitz transition produces logarithmic enhancements in $c_V/T$ and the charge response but not in the spin susceptibility.
  • In the strange-metal regime the single-particle decay rate is $-\operatorname{Im}\Sigma^R\simeq 2\lambda_s T$ for $|\omega|\ll T$ and $\lambda_s|\omega|$ for $T\ll|\omega|$, so the quasiparticle weight vanishes logarithmically and the spectral function is incoherent; whether this produces $T$-linear dc resistivity requires a separate treatment of momentum relaxation and current vertices.
  • The pseudogap-to-strange-metal crossover can be tracked by a spectral-weight-transfer fraction $f_{SWT}(x,T)$ in the physical pair spectrum, while the superconducting transition is identified by nonzero phase stiffness and global phase coherence rather than by pair correlations alone.
  • Zero-energy Cooper-surface ridges in two-electron ARPES would be a decisive signature of the theory; their absence at the predicted nodal geometry, with allowed matrix elements and sufficient resolution, would falsify the Cooper-surface prediction.

Reading between the lines

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

  • Extension: the same static and dynamic RML split could be tested in controlled toy models, such as attractive Hubbard or BCS-BEC crossover models, by computing the block-resolved pair correlator $P_{qq'}$ and checking whether it is nearly diagonal in relative momentum before relying on exact-solvability shortcuts.
  • Extension: the Cooper-surface Lifshitz transition predicts a rounded logarithmic peak in $c_V/T$ at the reconnection doping, but superconductivity preempts the zero-temperature limit; a field-tuned normal-state measurement in the underdoped-to-optimally-doped range could look for the charge-spin asymmetry the paper predicts.
  • Extension: because the paper leaves the derivation of both RML prescriptions from the dressed Cooper kernel open, direct numerical diagonalization of small t-J clusters could test whether block-diagonal pair autocorrelation dominates and whether the effective $J_q$ tracks $\sqrt{2}\Phi_d\phi^2_{q,d}$.
  • Extension: if the marginal-Fermi-liquid self-energy is real, the same pair continuum should contribute to optical conductivity; computing the current vertex consistent with the self-energy via the Ward identity would connect the single-particle $\omega/T$ scaling to the measured $A_1 T$ resistivity, which the paper flags as an open problem.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a relative-momentum-local (RML) theory of cuprates in which the antiferromagnetic superexchange of the t-J model, rewritten as an exact spin-singlet Cooper-pair interaction (Eq. 1), organizes low-energy physics into three regimes: phase-coherent static d-wave superconductivity, a phase-incoherent quasi-static pseudogap, and a phase-incoherent dynamic strange metal. A static RML prescription (Eq. 4), coupling independent relative-momentum blocks with J_q(x,T) = √2 Φ_d φ²_{q,d}, yields an exactly solvable block Hamiltonian with nodal electron poles, antinodal gaps, resolution-controlled Fermi arcs, and gapless charge-2e Cooper surfaces; a Cooper-surface Lifshitz transition is predicted to produce logarithmic enhancements in c_V/T and the charge response without a spin singularity. A dynamic RML prescription, with a block-local pair-field inverse propagator D^{-1}_Ξ = r_s + iℓ_s Ω_m + γ_s |Ω_m|, yields a one-loop marginal-Fermi-liquid self-energy. The paper predicts zero-energy Cooper-surface ridges in two-electron ARPES as a decisive signature. Both RML prescriptions are labeled as postulates, and Section 8.1 states that deriving them from a common dressed Cooper kernel of the t-J model is an open problem.

Significance. The unification claim would be a substantial advance if the RML prescriptions could indeed be derived from the t-J model, since it would connect single-particle, pair-sensitive, and thermodynamic probes through a single Cooper channel. The manuscript has genuine strengths: the exact Cooper representation identity (Eq. 1) is a correct starting point; the 16-dimensional block solution of H_PG is genuinely exact and internally consistent (Eqs. 5-6); the Lifshitz-transition mechanism rests on a parameter-free symmetry argument at the van Hove points; the nodal scaling forms for arc width and length are explicit and falsifiable; and the paper names concrete experimental signatures (2e-ARPES ridges, charge-spin asymmetry at the Lifshitz doping, and ω/T collapse of −Im Σ/T). The paper is also unusually candid about its open problems. However, the central claim is not established: the PG and marginal-Fermi-liquid results follow from two assumed models with free parameters, not from the t-J model, and Section 8.1 identifies the derivation of the RML prescriptions as the 'central microscopic task' that remains open.

major comments (3)
  1. [§4, Eq. (4); §8.1] The static RML prescription is postulated, not derived. Equation (4) restricts the exact interaction (Eq. 1) to a block-diagonal Hamiltonian with J_q(x,T) = √2 Φ_d φ²_{q,d}, and the PG phenomenology that follows (gap ∝ φ²_{q,d} along the PG contour, Fermi arcs, Cooper surfaces) is a property of this assumed block structure. Section 8.1 concedes the point: 'Neither BKT phase disordering, the Cooper logarithm nor the Q→0 limit establishes relative-momentum locality,' and 'the reduction to J_q δ_{qq′} remains to be derived.' The three candidate mechanisms are indeed insufficient: BKT disordering acts in real space, the Cooper logarithm organizes the Q=0 channel, and Q→0 does not restrict the relative momentum q. A concrete test would be to evaluate the inter-block correlation ratio P_{qq′}(0,0)/[P_{qq}(0,0)P_{q′q′}(0,0)]^{1/2} in the Gaussian theory of Section 6; if this ratio is not parametrically small in the relevant window, the block-diagonal model (4) is not the infrared limit of the t-J Cooper channel. Until such a check is supplied, the exact solvability and the arc and Cooper-surface results are results for a model with locality imposed, not consequences of the t-J model.
  2. [§6, Eq. (7); §8.1] The dynamic RML prescription assumes the inverse propagator D^{-1}_{Ξ,0}(iΩ_m) = r_s + iℓ_s Ω_m + γ_s |Ω_m| with free coefficients r_s, ℓ_s, γ_s and a further free coupling constant J_s, and it is labeled 'postulated to have the same infrared form as D^{-1}_{Ξ̄}'. The marginal-Fermi-liquid decay rate (Eq. 7), the Kramers–Kronig logarithm in Re Σ^R_s, and the logarithmic suppression of the quasiparticle weight Z_s(ω) all follow mechanically from this assumed form, so the MFL scaling is built into the model by construction rather than obtained from the t-J model. Section 8.1 acknowledges that 'reducing the finite-Q sector to a channel-independent Ξ_q requires microscopic justification' and that the MFL benchmark additionally needs r_s > 0, a causal ultraviolet completion, and a non-collapsing scaling window. The Gaussian particle-particle bubble of Section 6 provides a kinematic origin for Ohmic damping, but the step from γ_{αα′}(Q,T) to a channel-independent, approximately isotropic γ_s|Ω_m| form is not shown. I would regard the unification claim as supported only if the dressed inverse propagator obtained from the fermion determinant is shown to reduce to, or be bounded by, the assumed form in the transverse-intersection annulus T ≪ v̄_F |Q| ≪ E_F.
  3. [§2, §4, §7, §8.1] There is a matching gap between the operational definition of the pseudogap and the model used to describe it. Section 2 defines the PG by a finite-width collective peak with ΔQ ∼ ξ^{-1}_PG and ΔΩ ∼ τ^{-1}_PG, whereas H_PG retains strictly zero-COM, static pair operators. Section 8.1 states that a normalized weighted average over |Q| ≲ ΔQ and |Ω| ≲ ΔΩ is only a 'possible matching prescription,' and that H_PG 'does not determine finite-Q pair propagation and lifetime' because the reduction to J_q δ_{qq′} 'remains to be derived.' Consequently, the abstract's central claim that the RML theory 'relates single-particle, pair-sensitive and thermodynamic probes to three infrared organizations of one Cooper channel' is not demonstrated: the three regimes are organizations of two separately postulated models, and the common dressed Cooper kernel that would unite them is exactly the missing step, as Section 7 also states ('deriving both RML prescriptions from a common dressed Cooper kernel of the t-J model remains an open problem').
minor comments (4)
  1. [Abstract, throughout] The manuscript contains numerous typesetting artifacts, including 'hostd-wave' in the abstract, 'thet-J model,' and inconsistent subscript rendering ('TΔd', 'T F', 'T (0) BKT'); a careful production pass is needed.
  2. [Figure 1] The caption of Figure 1 reproduces the adjacent main-text paragraph almost verbatim and should be shortened to describe the panels.
  3. [§3] The claim that the d-wave saddle has a lower Helmholtz free energy than the extended-s-wave saddle 'for the parameters used here' is asserted without a shown calculation; please display the free-energy comparison or provide a reference.
  4. [§6] The symbols D^{-1}_{Ξ,0} and D^{-1}_{Ξ̄} differ only by subscripts and are easy to confuse; distinct notation would help.

Circularity Check

3 steps flagged · score 7.0 of 10

Central PG and MFL 'predictions' are restatements of the two postulated RML prescriptions; the paper concedes neither is derived from the t-J Cooper channel.

  1. self definitional [Section 4, Eq. (4)]
    "A postulated static RML prescription implements this essential inter-block dephasing and block factorization, yielding the exactly solvable PG Hamiltonian: ... The static d-wave coupling satisfies Jq(x,T) = √2 Φd(x,T)ϕ2 q,d ≥ 0"

    The PG Hamiltonian is not obtained from the exact Cooper-channel identity of Eq. (1); instead, block factorization and the d-wave form Jq ∝ ϕ2 q,d are imposed. All central PG predictions then follow by construction: the single-particle gap is ∆sp ∝ ϕ2 q,d with nodes at the d-wave nodal lines, the Fermi arcs are bounded by the resolution-broadened contours of this imposed coupling, and the Cooper surfaces S2e (ξq = 0) and S2h (ξq = Jq) are simply the level sets of the assumed HPG. The paper even notes the reduction to Jqδqq′ 'remains to be derived' (Sec. 8.1), so the PG phenomenology is a property of the ansatz, not a consequence of the t-J model.

  2. fitted input called prediction [Section 6, Eq. (7)]
    "A complementary dynamic RML prescription provides the block-local pair field Ξq with inverse propagator D−1 Ξ,0(iΩm) = rs + iℓsΩm + γs|Ωm|, which is postulated to have the same infrared form as D−1 ¯Ξ for |Ωm|≪EF."

    The marginal-Fermi-liquid scaling is encoded in the assumed inverse propagator before any calculation. The dissipative term γs|Ωm| plus the free parameters rs, ℓs, γs produce, after integrating out Ξq, the one-loop self-energy −Im ΣR s ≃ λs ω coth(ω/2T) with λs = J2 s γs/r2 s. This is the MFL line shape that was put in, so the 'prediction' is equivalent to the input. The paper does not derive rs, ℓs, γs, or the required finite-Q transverse-intersection sector from the t-J model; it states only that the form is 'postulated' and that reducing the finite-Q sector 'requires microscopic justification' (Sec. 8.1).

1 more flagged steps
  1. other [Section 8.1]
    "Deriving both RML prescriptions from a common dressed Cooper kernel of the t-J model remains an open problem. Neither BKT phase disordering, the Cooper logarithm nor the Q→0 limit establishes relative-momentum locality."

    This is an explicit admission that the load-bearing premise of the paper is unproven. The central claim that SC, PG and SM are three infrared organizations of one Cooper channel of the t-J model requires block-diagonal relative-momentum locality, but the paper states that no standard mechanism (BKT disordering, Cooper logarithm, or Q→0 limit) produces that locality. Consequently, the exactly solvable PG model and the MFL self-energy are results for a model with RML locality imposed, not derivations from the t-J model; the advertised unification is therefore supported only by the assumed prescriptions.

full rationale

The paper contains an exact operator identity, Eq. (1), rewriting the t-J superexchange as an attractive Cooper-channel interaction, and the d-wave mean-field SC analysis of Section 3 is a legitimate use of that identity. The circularity lies in the two RML prescriptions that carry the PG and SM claims. The static prescription fixes Jq = √2 Φd ϕ2 q,d, so the PG gap anisotropy, nodal structure, Fermi arcs and Cooper surfaces are mathematically forced by this input; the dynamic prescription fixes D−1 = rs + iℓsΩm + γs|Ωm|, so the one-loop MFL self-energy is a direct algebraic consequence of the assumed propagator. Section 8.1 explicitly concedes that deriving these prescriptions from the t-J Cooper kernel remains open and that BKT disordering, the Cooper logarithm and Q→0 do not establish relative-momentum locality. Thus the central unification claim is not a derivation from the microscopic model; the headline phenomenological outputs reduce, by construction, to the postulated low-energy forms. There are no relevant self-citations, and the mean-field SC part is not circular, but the PG and SM 'predictions' are the main deliverables and they are built into the inputs. This is partial but structural circularity, warranting a score of 7 rather than a lower score.

Assumptions & free parameters 5 free parameters · 6 assumptions · 3 invented entities

The central claim rests on the exact Cooper representation of superexchange, which is free of free parameters, plus two postulated RML prescriptions with several free coefficients. The t-J band parameters and resolution broadening are chosen inputs; the dynamic MFL scale and the static d-wave coupling are not derived. The paper itself identifies the derivation of both prescriptions as the main open problem.

free parameters (5)
  • Band parameters t, t', t'', J = t = 0.4 eV, t'/t = -0.22, t''/t = 0.045, J/t = 0.16
    Chosen as standard cuprate parameter values; all numerical results including the Cooper-surface Lifshitz doping x_L depend on them.
  • Static RML coupling J_q(x,T) = J_q = sqrt(2) Phi_d(x,T) phi^2_{q,d}
    The coupling is postulated to be proportional to the mean-field d-wave pairing amplitude; this fixes the d-wave gap shape and Fermi-arc structure by hand.
  • Dynamic RML parameters r_s, l_s, gamma_s, J_s = not fixed; lambda_s = J_s^2 gamma_s / r_s^2
    The inverse propagator D^{-1} = r_s + i l_s Omega_m + gamma_s |Omega_m| is postulated, and the MFL scale lambda_s is a free parameter in the effective action.
  • Resolution broadening Gamma = Gamma/t = 0.01 (4.0 meV)
    The Fermi-arc width and length are resolution controlled; the paper states the absolute arc length is non-universal and depends on Gamma and the chosen threshold.
  • SWT windows Q0, Q1, Q2, Omega_c = not specified
    The crossover diagnostic fSWT is an operational, window-dependent quantity and is not a sum rule.
assumptions (6)
  • domain assumption The t-J model with Gutzwiller factors g_t = 2x/(1+x) and g_J = 4/(1+x)^2 captures cuprate low-energy physics.
    Used throughout; a standard but uncontrolled approximation, introduced in Section 3.
  • ad hoc to paper Static RML prescription: relative-momentum-local block factorization with J_q = sqrt(2) Phi_d phi^2_{q,d}.
    Postulated in Section 4; Section 8.1 states the derivation from a common dressed Cooper kernel remains open.
  • ad hoc to paper Dynamic RML prescription: block-local pair field with D^{-1} = r_s + i l_s Omega_m + gamma_s |Omega_m|.
    Postulated in Section 6; the paper says reducing the finite-Q sector to a channel-independent Xi_q requires microscopic justification.
  • domain assumption BKT phase disordering explains the pseudogap above Tc.
    Assumed in Section 4; it is one of several competing pseudogap mechanisms in the literature.
  • domain assumption Gaussian approximation and one-loop self-energy are adequate for the strange metal sector.
    Section 6 neglects self-energy feedback and vertex corrections; Section 8.1 says this is not a closed many-body theory.
  • standard math Exact operator identity for AFM superexchange in the Cooper representation, Eq. (1).
    Algebraic identity for the superexchange interaction rewritten in terms of spin-singlet Cooper-pair operators; no physical assumption beyond the t-J model.
invented entities (3)
  • Cooper surfaces S2e and S2h independent evidence
    purpose: Zero-frequency pole loci of the retarded Cooper-pair Green's function; proposed as the pair analog of Fermi surfaces and the locus of a Lifshitz transition.
    The paper predicts zero-energy Cooper-surface ridges observable by 2e-ARPES, which is a falsifiable experimental handle.
  • Block-local pair field Xi_q
    purpose: Effective low-energy bosonic pair field in the dynamic RML prescription; integrating it out produces the marginal-Fermi-liquid self-energy.
    Introduced via Hubbard-Stratonovich and then given a postulated infrared form; no independent evidence is provided.
  • Quasi-pairs
    purpose: Conjectured elementary excitations of an interacting Cooper liquid, adiabatically connected to the exact block-pair states of H_PG.
    Explicitly conjectured in Section 8.1; no experimental handle or independent evidence is provided.

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Cite this review

Pith. "Pith review of Superconductivity, pseudogap and marginal Fermi liquid in a relative-momentum-local theory." pith.science (2026). https://pith.science/paper/GRK3B5VX

@misc{pith2026260809785,
  author       = {Pith},
  title        = {Pith review of: Superconductivity, pseudogap and marginal Fermi liquid in a relative-momentum-local theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GRK3B5VX}},
  note         = {Machine review of arXiv:2608.09785}
}
abstract

Cuprate superconductors host \(d\)-wave superconductivity (SC), a pseudogap (PG) with Fermi arcs and a strange metal (SM) with incoherent excitations and temperature-linear resistivity. These regimes emerge from the same doped Mott system but are usually described separately. Here a relative-momentum-local (RML) theory is constructed, with the exact spin-singlet Cooper representation of antiferromagnetic superexchange in the \(t\)-\(J\) model serving as a common thread. Pair correlations organize the low-energy behaviour of this Cooper channel into phase-coherent static SC, phase-incoherent quasi-static PG and phase-incoherent dynamic SM. A static RML prescription yields an exactly solvable PG Hamiltonian with nodal electron poles, antinodal gaps, resolution-broadened Fermi arcs and gapless charge-\(2e\) Cooper surfaces. A Cooper-surface Lifshitz transition generates logarithmic enhancements in the specific-heat coefficient and charge response without a spin singularity. The particle-particle continuum at finite centre-of-mass momentum produces Ohmic damping, while a dynamic RML prescription yields a one-loop self-energy with marginal-Fermi-liquid scaling. Spectral-weight transfer in the physical pair spectrum diagnoses the PG--SM crossover, whereas nonzero phase stiffness and global phase coherence identify SC. The RML theory relates single-particle, pair-sensitive and thermodynamic probes to three infrared organizations of one Cooper channel. A decisive experimental signature would be zero-energy Cooper-surface ridges in two-electron angle-resolved photoemission spectroscopy (2e-ARPES).

Figures

Figures reproduced from arXiv: 2608.09785 by the authors.

Figure 1
Figure 1. Three infrared organizations of pair correlation functions. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. BKT, pseudogap, and Fermi arc. a, Mean-field T∆d , defined as the highest temperature at which Hˆ G,MF t-J has a nonzero d-wave saddle, and bare T (0) BKT obtained from the phase-only action with bare phase stiffness, which neglects long￾distance vortex renormalization. b, Ground-state occupations of HˆPG at x = 0.10. The occupation boundaries are S2e : ξq = 0 and S2h : ξq = Jq, while the punctured Luttinger surface… view at source ↗
Figure 3
Figure 3. Cooper surfaces and Lifshitz transition. a, [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

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