Multiband and multichannel pairing

Scope

Multiband pairing describes coupled gaps on different electronic bands. Multichannel pairing describes different interaction contributions to a Cooper-pair kernel. These are distinct from parallel normal-state conductivities and from spatially separate superconducting phases. Observing two gaps does not identify two mediating bosons; observing a resistance drop in two phases does not establish one homogeneous condensate. The band basis, sample and measurement conditions must accompany the claim. R1 R3 R5

Mathematical assumptions

In a real, reciprocal two-band weak-coupling illustration with a shared cutoff energy $E_c$, write $\Delta_i=L(T_c)\sum_j\lambda_{ij}\Delta_j$, where $L(T)=\ln[2e^{\gamma_E}E_c/(\pi k_BT)]$ and $\lambda_{ij}=-V_{ij}N_j$. Here $V_{ij}=V_{ji}$ is the pair interaction, attraction has negative $V$, and $N_j>0$ is the one-spin density of states in band $j$. The largest positive eigenvalue gives the leading instability, $k_BT_c=(2e^{\gamma_E}/\pi)E_c\exp(-1/\lambda_+)$. The approximation assumes $k_BT_c\ll E_c$; a nonpositive leading eigenvalue gives no instability in this restricted model. This is a stated modern illustration, not a claim that the historical Suhl paper has been fully audited. R1 R3

Although $\lambda$ need not be symmetric, $K_{ij}=-\sqrt{N_i}V_{ij}\sqrt{N_j}$ is a symmetric similar matrix with eigenvector $\eta_i=\sqrt{N_i}\Delta_i$. For two bands, $\lambda_+=[\lambda_{11}+\lambda_{22}+\sqrt{(\lambda_{11}-\lambda_{22})^2+4\lambda_{12}\lambda_{21}}]/2$. An independent check of 64 reciprocal kernels verifies this formula, the density-of-states transformation and the gap-vector residuals. All parameters are synthetic; no MgB2 or FeSe coupling is fitted. The eigenvalue is not the sum of every matrix entry. R1 R3

Consider $K=\begin{pmatrix}0.3&-0.1\\-0.1&0.3\end{pmatrix}$, whose leading eigenvalue is $0.4$ with opposite gap signs. Adding only positive off-diagonal entries of $0.05$ lowers it to $0.35$. That added matrix is not positive semidefinite: entrywise attraction and a nonnegative quadratic form are different conditions. In contrast, 31 positive-semidefinite additions do not lower the largest eigenvalue of the fixed kernel. This bounded example shows why cooperation must be assessed after projecting the interaction, with the other assumptions held fixed. R1 R3

With nonzero reciprocal interband coupling, changing both off-diagonal signs leaves the two eigenvalues unchanged but reverses the relative gap sign of the nondegenerate leading mode. A transition temperature alone therefore cannot determine the relative phase. With zero interband coupling, the two bands have independent instabilities, and the weaker band need not acquire a gap at the higher transition. If the two uncoupled eigenvalues coincide, this quadratic model selects neither a unique gap vector nor a relative phase; additional terms are needed. A full calculation must also retain frequency dependence, different cutoffs, Coulomb terms and normal self-energy. Even an illustrative scalar increase from $0.3$ to $0.4$ loses to a mass factor $Z=1.5$, because $0.4/1.5<0.3$; that arithmetic is not a complete Eliashberg calculation. R1 R3

Experimental criteria

Nagamatsu and colleagues reported a resistive onset at 39 K and endpoint at 38 K in a polycrystalline MgB2 pellet, alongside separate 10 Oe magnetic measurements. The original paper's 196 MPa pressure concerns synthesis, not the superconducting measurement. This page now records the onset criterion explicitly instead of treating every reported 39 K value as an unspecified transition. Geometry, sample quality and criterion remain part of the record. R2

Choi and colleagues calculated anisotropic MgB2 gaps using an electron–phonon Eliashberg model. Their band-dependent gap distributions close at a common calculated transition. These are theoretical results, not direct measurements of two independent mediators, and numerical agreement does not independently identify every interaction in a material. The corrected DOI is the MgB2 paper, not the unrelated biology article inherited by the pinned draft. R3

Szabo and colleagues used Cu point contacts and two weighted BTK contributions to describe MgB2 spectra. Their selected author methods keep interface barrier and weighting parameters fixed across temperature, with applied fields helping distinguish the gap contributions. This is spectroscopic evidence interpreted through an interface model; it does not determine the relative gap sign or assign separate bosons to each gap. Cross-check band assignment and bulk thermodynamics before equating a fitted component with a unique condensate. R5

Material interpretation

Lee and colleagues observed replica bands in one-unit-cell FeSe/SrTiO3 and interpreted them through forward-focused interface phonons. In the identified author v6, a mean-field fit gives a gap-closing scale of $58\pm7$ K, with Figure 2 showing closure between 50 and 64 K. This is an ARPES-derived scale, not a measured zero-resistance temperature. Their predicted enhancement assumes an electronic magnetic-pairing baseline and a specified model; neither it nor the fitted coupling is an experimentally separated budget of all pairing channels. R4

Replica-band interpretation requires controls. Li and Sawatzky proposed predominantly extrinsic energy loss by the escaping photoelectron, using a dielectric model. Faeth and colleagues subsequently tested photon-energy dependence at 15 K over 21–75 eV and found a nearly constant replica-to-main intensity ratio; higher-order separations and line shapes also supported intrinsic coupling in their samples. Their forward-scattering comparison inferred $\lambda=0.19\pm0.02$. This selected-method check is stronger than a bare replica observation, but does not by itself uniquely partition the superconducting pairing kernel. R6 R7

The Faeth analysis also notes that the estimated gap enhancement under its chosen model cannot fully explain the difference between the reported high gap-closing scale and electron-doped bulk comparators. Distinguish that conditional inference from a universal phonon percentage, and distinguish measured spectra from a thermodynamic or zero-resistance transition. Differences between papers require matched coupling definitions, background subtraction, sample preparation and model assumptions; a single scalar cannot settle the mechanism. R7

Limits and disputes

A shared transition and multiple gap features are compatible with interband coupling, but do not independently establish beneficial cooperation between different interactions. Carrier transfer, density of states, scattering and phase coherence can change together. Claims of a unique mechanism require discriminating controls; phase-sensitive probes, spectroscopy and thermodynamics constrain different parts of the problem. The original Suhl text, full FeSe supplementary controls and complete raw-data analyses remain outside the completed checks. Complete scientific acceptance remains open. R1 R3 R4 R5 R7

Discovery implications

Specify what changes when an interaction is added and what is held fixed. Report the band and momentum basis, cutoff, Coulomb and self-energy treatment, and the eigenfunction as well as the leading eigenvalue. Avoid double-counting one screened interaction under two names. A useful proposal predicts a response to a controlled perturbation and explains which competing interpretation that measurement would exclude. Do not substitute a list of plausible bosons or an unsourced sum of coupling constants for a tested kernel. R1 R3 R4 R7

Related topics

See s± pairing, MgB2, iron-based materials, two-dimensional interfaces and multichannel transport.

Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.