A stronger antiferromagnetic pairing vertex need not raise Tc

The claim under pressure

Once antiferromagnetic fluctuations generate a positive $d_{x^2-y^2}$ pairing eigenvalue, making those fluctuations stronger should raise $T_c$.

The first half can be right while the conclusion fails.

Scattering near $\mathbf{Q}=(\pi ,\pi )$ rewards a gap that changes sign under $\mathbf{k}\mapsto \mathbf{k}+\mathbf{Q}$. The same scattering also enters the normal self-energy, reshapes the spectral function, and may slow the magnetic dynamics.

The physical transition can be limited later by phase stiffness or altered by stripe and pseudogap physics. A pairing interaction is therefore only one part of a theory of $T_c$.

The compact statement is

The question

How can antiferromagnetic scattering act as an effective attraction in the $d$-wave Cooper channel while degrading the fermions that must form the pair?

The answer has three layers:

1. Sign algebra: repulsion can produce a positive eigenvalue after projection onto a sign-changing gap.
2. Many-body dynamics: the kernel contains dressed propagators as well as an irreducible vertex.
3. Ordering physics: pair formation and macroscopic phase coherence need not occur at the same temperature.

Keep those layers separate. Much of the confusion comes from proving a statement at one layer and silently promoting it to the next.

1. Start from the full pairing problem

Use the one-band square-lattice Hubbard model as an effective starting point,

Nearest-neighbor hopping is $t$; next-nearest hopping is $t^{\prime }$. This Hamiltonian does not become a controlled weak-coupling spin-fluctuation model merely because its spin susceptibility is large.

For singlet pairing, write the linearized particle-particle Bethe-Salpeter equation as

where $k=(\mathbf{k},i{\omega }_n)$. In this convention, an instability occurs when the leading eigenvalue reaches one:

The kernel is

This is the technical spine of the session. Neither the vertex ${\mathrm{\Gamma }}_{\mathrm{pp}}$ nor the propagator $G$ determines the transition by itself.

Scope note. The eigenvalue condition can define a pair-formation or mean-field instability in an approximation that omits long-wavelength phase fluctuations. Calling that scale the physical $T_c$ requires an additional check, discussed below.

2. Why repulsion at $\mathbf{Q}$ can favor $d$-wave pairing

Ignore frequency dependence for one step and project the kernel onto the Fermi surface:

Suppose the repulsive vertex is largest when ${\mathbf{k}}^{\prime }-\mathbf{k}\simeq \mathbf{Q}$. For

one has

The dominant contribution then has the sign

The real-space interaction has not become attractive. The projected integral operator has acquired a pair-forming eigenfunction.

This is the useful result:

The qualification matters. Momentum geometry, frequency dependence, orbital structure, and competing eigenfunctions decide whether $d_{x^2-y^2}$ is actually leading.

Scalapino’s review gives the standard case for spin-fluctuation pairing across Hubbard-like models.

3. The effective spin-fluctuation kernel has a domain of validity

Near an itinerant antiferromagnetic instability, one often writes

with an overdamped susceptibility

This is an infrared ansatz. It is not an operator identity of the Hubbard model.

The full singlet irreducible vertex can contain an instantaneous term, charge contributions, crossed diagrams, and vertex corrections. Replacing it by $(3/2)g^2\chi$ is a model choice whose quality depends on regime and approximation.

The same proviso applies to the Hertz-Millis form. In two dimensions, gapless fermions and critical bosons feed back on each other. Hot-spot theories can become singular, and simple power counting does not settle every infrared limit.

There are controlled limits, including large-$N$, small-parameter, and special sign-problem-free models. There is no generic cuprate analogue of the phonon Migdal ratio ${\omega }_D/E_F\ll 1$.

4. The same fluctuations dress the fermion

At the schematic one-loop level,

States near Fermi-surface points connected by $\mathbf{Q}$ are the conventional hot spots. The processes that dominate the anomalous kernel can therefore dominate their decay and mass renormalization.

Write

A quasiparticle-like parametrization would be

but this notation must not be mistaken for proof that a sharp pole survives. Near a non-Fermi-liquid regime, $Z$ can be strongly frequency dependent and the pole can be overdamped.

A useful caricature of the competition is

so that

Here ${\lambda }_d$ measures the anomalous projection, ${\lambda }_Z$ summarizes normal renormalization, and ${\mathrm{\Omega }}_{\mathrm{pair}}$ is the useful dynamical bandwidth. This formula is a diagnostic model, not a cuprate prediction.

Its lesson is sound: increasing $g^2\chi$ can increase both ${\lambda }_d$ and ${\lambda }_Z$.

The spectral-function version

The dressed pair bubble makes the loss of usable spectral weight explicit:

Using

one obtains, assuming inversion symmetry for compactness,

The formula shows why a large vertex does not settle the problem. Broadening or removing low-energy spectral weight changes the object on which the vertex acts.

It also makes the pseudogap issue sharp. The $d$-wave form factor is largest near the antinodes, where cuprate spectral weight is strongly depleted below $T^{\ast }$.

The depletion can weaken this simple Cooper bubble even if spin correlations grow. It does not determine whether the pseudogap is caused by pairing, a competing order, fractionalization, or a mixture of effects.

5. Criticality can make the interaction stronger and slower

For the overdamped ansatz above, the crossover between the mass term and damping term gives a characteristic relaxation scale

If $\gamma$ remains regular, this is the familiar $z=2$ scaling ${\mathrm{\Omega }}_{\mathrm{sf}}\propto {\xi }^{-2}$.

Thus $\chi (\mathbf{Q},0)$ can rise while the mediator’s characteristic frequency falls. A static neutron peak cannot by itself tell us the pairing scale.

A BCS-like expression would contain both trends:

Near a genuine strong-coupling quantum critical point, that expression may fail. Critical pairing can remain finite even when a naive bosonic scale tends to zero, because the entire frequency-dependent problem reorganizes.

That is not permission to ignore the cutoff. It is a demand to solve the critical integral equation instead of inserting a divergent static susceptibility into BCS.

Sign-problem-free spin-fermion simulations have found regimes in which small hot regions govern pairing and $T_c$ (Wang et al., 2017). This supports the hot-spot mechanism within that model class, not a universal monotonic law for cuprates.

6. Pair formation is not yet the thermodynamic transition

Let

At long wavelengths, a two-dimensional phase-only description contains

For a strictly two-dimensional neutral system in the BKT universality class,

Real cuprates are layered charged superconductors. Interlayer coupling and electromagnetic screening modify the strict two-dimensional story and can produce a three-dimensional crossover.

The conditional lesson survives: a high amplitude scale does not guarantee a high ordering temperature when the superfluid stiffness is small.

It is useful to name the two scales separately:

In a strictly two-dimensional SU(2)-symmetric Hubbard model, finite-temperature antiferromagnetic long-range order is also forbidden by Mermin-Wagner. A large $\xi$ at finite $T$ means strong short-range correlations, not a thermodynamic Néel transition.

7. Stripes are not summarized by the word “competition”

A minimal Landau functional might contain

The sign of ${\gamma }_{\mathrm{\Delta }\rho }$ distinguishes simple competition from cooperation only within this truncated uniform description.

Finite-momentum pairing allows couplings such as

Charge order can then be intertwined with pair-density-wave order rather than acting as an independent scalar competitor.

A 2024 calculation combining two high-accuracy many-body methods found superconductivity on both sides of the doped two-dimensional Hubbard model with $t^{\prime }$. On the hole-doped side, strong superconductivity coexisted with partially filled stripes.

The result matters because it excludes a false dichotomy within that calculation: “stripe” and “superconductor” need not denote mutually exclusive states. It does not prove that every cuprate stripe is pair-enhancing.

See Xu et al., Science 384, eadh7691 (2024).

8. Self-energy, transport, and optics are different projections

A marginal form is often written

This is a phenomenological form associated with marginal-Fermi-liquid physics. It is not the universal output of every antiferromagnetic spin-fermion theory.

Even if a large single-particle decay rate is established, DC resistivity requires current relaxation. In a continuum with exact translation symmetry, electron-electron collisions conserve total momentum and can leave a Drude contribution.

On a lattice, Umklapp, disorder, multiple bands or sectors, and the overlap of current with long-lived quantities determine the result. “The lattice relaxes momentum” is too vague; the available kinematics must permit the relevant process.

When momentum is the dominant slow mode, memory-matrix notation exposes the distinction:

$M_{PP}$ measures momentum relaxation and ${\chi }_{JP}$ is the current-momentum susceptibility. Neither is identical to $-\mathrm{Im}\mathrm{\Sigma }$.

Likewise, optical conductivity contains current vertices and a frequency-dependent memory function. ARPES probes a one-particle spectral function. Numerical agreement among their fitted rates is a result to establish, not an identity to assume.

An LSCO sample near the pseudogap endpoint showed optical $\hslash \omega /k_{B}T$ scaling and a logarithmic low-frequency effective mass (Michon et al., 2023). This is evidence for scaling in that material and regime.

It does not identify a unique critical operator or microscopic scattering mechanism.

What “Planckian” safely means here

Many strange metals exhibit a fitted relaxation scale

with $C$ of order unity.

That empirical regularity is important. It is not a general theorem imposing one universal lower bound on every microscopic lifetime. The extracted $C$ also depends on which rate is measured and how effective mass and carrier density are assigned.

9. What the recent frontier changes

The clean picture of a narrow strange-metal fan above one antiferromagnetic quantum critical point is no longer the only serious organization.

A disordered magnetic route

Neutron scattering across overdoped LSCO has found low-energy, temperature-dependent spin fluctuations with scaling across part of the superconducting dome.

The 2026 analysis reports consistency with a disordered spin-density-wave transition and a quantum-Griffiths description (Nature Communications 17, 2026).

“Consistent with” is the correct phrase. The data do not uniquely derive the microscopic theory of the strange metal.

Numerically exact simulations of a two-dimensional model with spatially random antiferromagnetic interactions found an extended gapless regime with approximately Planckian $T$-linear transport, rather than strange metallicity confined to a point.

That result is currently available as a preprint: Patel, Lunts, and Albergo, arXiv:2410.05365. Its controlled statement concerns the specified random-interaction model.

Correlations across transport and superconductivity

High-field measurements in Bi2201, LSCO, and Tl2201 found a robust correlation between the $H$-linear magnetoresistance slope and the $T$-linear resistivity coefficient.

The authors proposed a real-space inhomogeneity model in which doping changes the fraction of strange-metal carriers (Nature Communications 15, 2024).

This is a useful counterexample to a common inference. A correlation between a transport slope and $T_c$ need not mean that one bosonic coupling controls both quantities.

Evidence map

Result	What it supports	What it does not establish
$\mathbf{Q}$-peaked repulsion favors a sign-changing eigenfunction	a viable $d$-wave pairing channel	the magnitude or doping dependence of physical $T_c$
Hot-spot spin-fermion calculations produce superconductivity	magnetic critical modes can pair fermions in controlled models	that all cuprates reduce to the same low-energy model
Hubbard numerics find superconductivity with partially filled stripes	coexistence and intertwined energetics are possible	that stripe order universally raises $T_c$
LSCO neutron data show scaling spin fluctuations	magnetic dynamics remain active in the strange-metal regime	a unique clean QCP or unique pairing glue
$T$-linear and $H$-linear transport coefficients correlate	shared phenomenology across three cuprate families	equality of transport and pairing kernels
Random-interaction models yield extended $T$-linear transport	disorder can organize a gapless strange-metal phase	that disorder is the sole cuprate mechanism

The modern frontier is therefore comparative. One must distinguish a clean hot-spot critical point, an extended disordered critical regime, and an intertwined stripe-pseudogap metal using observables that differ among them.

The deliberately false inference

If $A_1$ in $\rho (T)={\rho }_0+A_{1}T$ tracks $T_c$ across doping, the fluctuations responsible for $T$-linear resistivity must also cause pairing.

The inference is invalid.

Transport weights current relaxation. Pairing weights a symmetry-resolved two-particle eigenfunction multiplied by dressed propagators. The two kernels may share microscopic interactions without being proportional.

The correlation is evidence that the sectors are connected. It is not operator identification.

Collision: four statements that must not be merged

1. Algebra

A repulsive vertex concentrated near $(\pi ,\pi )$ can favor a gap that changes sign under translation by $\mathbf{Q}$.

2. Eigenvalue

The superconducting instability depends on $-{\mathrm{\Gamma }}_{\mathrm{pp}}GG$, not on ${\mathrm{\Gamma }}_{\mathrm{pp}}$ alone.

3. Dynamics

The relevant frequency range, self-energy, vertex corrections, spectral redistribution, competing orders, and phase stiffness all enter the scale problem.

4. Infrared identification

$T$-linear transport, $\omega /T$ scaling, Griffiths physics, and clean quantum criticality are distinct claims. One observation need not select one theory.

The defensible conclusion is

Do this now: a two-hot-patch model

Patch 1 and patch 2 are connected by $\mathbf{Q}=(\pi ,\pi )$. Assume a repulsive interpatch interaction $V>0$, negligible intrapatch interaction, patch density of states $N_h$, and cutoff $\mathrm{\Omega }$.

Define

Let the same fluctuations produce the toy frequency renormalization

After the energy integral, assume the reduced linearized equations are

where

The use of one factor of $Z$ is part of this reduced model. In a microscopic derivation, residues, dispersion renormalization, and vertex corrections must be treated consistently.

Target A

Find both eigenvectors of the patch-exchange matrix. Which one has a positive pairing-kernel eigenvalue?

Target B

Derive $T_c$ in the sign-changing channel. Identify exactly where the normal self-energy enters.

Target C

Model the approach to a $z=2$ magnetic critical point by

Derive $T_c(\xi )$ and determine whether it is monotonic.

Target D: judge the model

List three reasons why the maximum you find is not a universal derivation of the cuprate superconducting dome.

Oral check 1

Why does $V>0$ favor the antisymmetric patch combination?

Oral check 2

If $-\mathrm{Im}{\mathrm{\Sigma }}^R(0,T)\propto T$, what else must be known before concluding that $\rho \propto T$?

Hints

• Use ${\mathrm{\Phi }}_{\pm }=({\mathrm{\Phi }}_1\pm {\mathrm{\Phi }}_2)/\sqrt{2}$.
• The overall minus sign in the gap equation reverses the matrix eigenvalue.
• Maximize $\log T_c$, not $T_c$.
• For Target D, inspect the assumed cutoff, momentum structure, and distinction between pair formation and phase coherence.

Solution outline

The patch-exchange matrix has eigenvectors

$${\mathrm{\Phi }}_{+}=\frac{1}{\sqrt{2}}(1,1),{\mathrm{\Phi }}_{-}=\frac{1}{\sqrt{2}}(1,-1),$$Copy LaTeX

with matrix eigenvalues $+1$ and $-1$. Because the gap equation contains an overall minus sign, the pairing-kernel eigenvalues are

$${\kappa }_{+}=-\frac{g_{p}L}{Z},{\kappa }_{-}=+\frac{g_{p}L}{Z}.$$Copy LaTeX

Only the sign-changing state can reach the instability condition ${\kappa }_{-}=1$ for $L>0$:

$$1=\frac{g_{p}L(T_c)}{Z}.$$Copy LaTeX

Therefore

$$L(T_c)=\frac{1+b{g}_p}{g_p}=\frac{1}{g_p}+b,$$Copy LaTeX

and

$$\boxed{{\displaystyle T_c=1.13\mathrm{\Omega }\exp (-\frac{1}{g_p}-b).}}$$Copy LaTeX

The sign change converts repulsive interpatch scattering into a positive kernel eigenvalue. Normal-state renormalization supplies the extra suppression $e^{-b}$ in this chosen parametrization.

Now insert $g_p=a{\xi }^2$ and $\mathrm{\Omega }={\mathrm{\Omega }}_0{\xi }^{-2}$:

$$T_c(\xi )=1.13{\mathrm{\Omega }}_0{\xi }^{-2}\exp [-\frac{1}{a{\xi }^2}-b].$$Copy LaTeX

Thus

$$\log T_c=\text{const}-2\log \xi -\frac{1}{a{\xi }^2},$$Copy LaTeX

and

$$\frac{d\log T_c}{d\xi }=-\frac{2}{\xi }+\frac{2}{a{\xi }^3}.$$Copy LaTeX

The stationary point is

$${\xi }^2=\frac{1}{a},g_p=1.$$Copy LaTeX

The second derivative there is $-4a<0$, so this toy $T_c$ has a maximum. It rises while the Cooper exponential wins, then falls because the imposed cutoff collapses.

This is not a derivation of the cuprate dome. The model assumes $g_p\propto {\xi }^2$, fixes $b$, imposes $\mathrm{\Omega }\propto {\xi }^{-2}$, discards momentum and frequency structure within each patch, and identifies the amplitude instability with $T_c$.

Its legitimate conclusion is narrower:

$$\boxed{{\displaystyle \frac{d{g}_p}{d\xi }>0\nRightarrow \frac{d{T}_c}{d\xi }>0.}}$$Copy LaTeX

Exit ticket

A calculation finds that the leading $d$-wave Bethe-Salpeter eigenvalue rises sharply near a putative magnetic critical doping.

What additional calculation or measurement would let you decide whether the physical $T_c$ must rise as well?

A strong answer should name at least two of the following: the dressed spectral weight in $GG$, the fluctuation spectrum in frequency, the consistency of vertex corrections, the superfluid stiffness, and the leading competing or intertwined order.

Research checks worth doing next

1. Kernel decomposition. Track separately the temperature and doping evolution of the irreducible vertex and the dressed pair bubble without treating either as directly observable.
2. Momentum resolution. Compare hot-spot, antinodal, and nodal contributions to the leading eigenvalue.
3. Optics versus ARPES. Test whether one self-energy and one vertex model can fit both data sets while satisfying conductivity sum rules.
4. Momentum relaxation. Compute Umklapp phase space or the relevant memory matrix instead of inferring resistivity from a lifetime.
5. Stiffness audit. Compare the pair-formation scale with ${\rho }_s(T)$ and the dimensional crossover scale.
6. Disorder discrimination. Seek observables that distinguish a clean critical fan from a Griffiths-like extended regime, such as spatial distributions of local scales and sample-quality dependence.
7. Stripe energetics. Compare uniform $d$-wave, stripe-coexisting, and pair-density-wave states at matched microscopic parameters and controlled finite-size error.

Further reading

• A common thread: the pairing interaction for unconventional superconductors — the standard spin-fluctuation pairing review and the symmetry-resolved vertex viewpoint.
• From quantum matter to high-temperature superconductivity in copper oxides — the 2015 assessment separating the comparatively clear superconducting symmetry from the unresolved normal state.
• Superconductivity mediated by quantum critical antiferromagnetic fluctuations: The rise and fall of hot spots — sign-problem-free spin-fermion numerics and hot-spot control of pairing.
• Coexistence of superconductivity with partially filled stripes in the Hubbard model — high-accuracy 2024 evidence for stripe-superconductivity coexistence in the $t^{\prime }$ Hubbard model.
• Reconciling optical-conductivity scaling with Planckian resistivity and specific heat — optical scaling near the LSCO pseudogap endpoint and the assumptions behind rate extraction.
• Universal correlation between $H$-linear magnetoresistance and $T$-linear resistivity — the three-family transport correlation and an inhomogeneous-carrier interpretation.
• Strange metals and Planckian transport in a gapless phase from spatially random interactions — numerically exact results for a specified random antiferromagnetic-interaction model; currently a preprint.
• Critical spin fluctuations across the superconducting dome in LSCO — 2026 neutron evidence and the disordered-SDW/Griffiths interpretation.

What to retain

• Repulsion can favor pairing after projection onto a sign-changing eigenfunction.
• The transition kernel is $-{\mathrm{\Gamma }}_{\mathrm{pp}}GG$.
• Static magnetic strength and dynamical pairing bandwidth are different quantities.
• The pseudogap can remove spectral weight where the $d$-wave form factor is largest.
• A single-particle lifetime is not a transport relaxation rate.
• “Planckian” is a phenomenological scale unless a particular theory derives more.
• Stripes can coexist or intertwine with superconductivity; “competition” is not a complete model.
• Pair formation and phase coherence can set different temperatures.
• A larger pairing eigenvalue does not by itself imply a larger physical $T_c$.

Next: derive a frequency-dependent two-hot-patch Eliashberg system and test whether the Cooper logarithm survives a non-Fermi-liquid self-energy.