Unconventional Superconductivity

Unconventional superconductivity is not one mechanism or one material family. It is a disciplined classification problem: declare the normal-state basis and symmetries, impose fermionic exchange, determine what the candidate gap matrix does on the actual Fermi surfaces, and ask which observations constrain its representation, relative phase, nodes, or additional broken symmetries. The result is a bounded pairing-state claim—not an automatic claim about the pairing glue, high transition temperature, or topology.

Required background. BCS Theory provides the controlled superconducting benchmark, while Symmetry of Bloch States provides Bloch fibers, sewing matrices, little groups, and representation labels. This page builds the pair-specific synthesis of those capabilities.

Helpful background. Ginzburg–Landau Theory helps with multicomponent order, Spin–Orbit Coupling in Solids with physical-spin and pseudospin distinctions, and Spectral Functions, Josephson Effect, and the measurement gateway with branch-specific evidence and forward models.

Define the Unconventional Pairing Claim

Begin by stating what unconventional means in the claim under audit. Three questions that are often collapsed are logically different:

• Does the order transform in a nontrivial crystal representation or break an additional symmetry below the superconducting transition?
• Does a symmetry-trivial representation carry a nonuniform relative sign, orbital texture, or multiband phase structure absent from the elementary isotropic benchmark?
• Which microscopic interaction selected that state?

For example, a sign-changing $A_{1g}$ state can preserve every crystal symmetry while remaining structurally unlike a one-band constant-sign gap. Conversely, a nontrivial representation constrains the order parameter but does not identify the boson, fluctuation, or interaction that produced it. Throughout this page, unconventional refers to the first two structural questions. The changing mechanism and material-status record belongs to the frontier layer, which is not yet a live route.

Before calculating or fitting, record all ten fields below. A field may be marked unresolved, but it may not silently disappear.

1. Material, specimen, and geometry. Record composition, batch or device, crystal orientation, dimension, surface or bulk region, contacts, sample shape, and relevant boundaries.
2. Prepared thermodynamic state and tuning variables. State temperature, field, pressure, strain, doping or gate setting, disorder, equilibrium or drive, cooling and field history, and domain preparation.
3. Normal-state low-energy Hilbert space. Declare retained bands, orbitals, sublattices, spin or pseudospin basis, Fermi sheets, spin–orbit coupling, and integrated-out states.
4. Exact and approximate symmetries. State the crystal or magnetic space group, inversion, time reversal, spin rotation, translations, explicit sources, and the symmetry accuracy of the specimen and model.
5. Pair definition and convention. Declare center-of-mass momentum, relative momentum and frequency, the gap-matrix and Nambu convention, fermionic antisymmetry, charge and global phase, Bloch-frame transformation, and candidate irrep or corepresentation.
6. Gap structure on every relevant sheet. Record magnitudes, symmetry-enforced or accidental nodes, deep minima, relative signs and phases, interband or intraband content, multicomponent order, and domains.
7. Observable and canonical forward model. Specify the requested thermodynamic, spectral, magnetic, transport, or phase-sensitive quantity; probe operator; geometry, polarization, field, frequency and temperature window; matrix elements; and surface versus bulk weighting.
8. Nuisance controls and alternatives. Carry disorder, surfaces, interfaces, strain, domains, finite size, nonlocality, heating, resolution, backgrounds, competing order, inhomogeneity, and credible alternative gaps.
9. Numerics and uncertainty. State normalization, fit covariance, parameter and model uncertainty, numerical convergence, reproducibility, sample spread, and cross-probe consistency.
10. Licensed claim and stopping rule. Give the strongest supported conclusion, unresolved alternatives, falsifier or escalation test, and the next canonical owner for any mechanism, finite-$\mathbf Q$, topology, probe, or material-specific question.

Declare the Basis, Symmetries, and Evidence Ledger

Let $c_{\mathbf k}$ be a column of annihilation operators in a declared orbital, sublattice, band, and spin or pseudospin basis. For zero center-of-mass pairing, write the mean-field pairing term as

$$H_{\Delta} = \frac{1}{2} \sum_{\mathbf k} \left[ c_{\mathbf k}^{\dagger} \Delta(\mathbf k) \bigl(c_{-\mathbf k}^{\dagger}\bigr)^{T} + \text{h.c.} \right].$$

The displayed $\Delta$ is a basis-dependent pairing self-energy. It is not the same object as a gauge-invariant measured gap edge, and it is not by itself a pairing mechanism. If the old operators and a new frame obey

$$c_{\mathbf k}=W(\mathbf k)\widetilde c_{\mathbf k},$$

then

$$\widetilde\Delta(\mathbf k) = W^{\dagger}(\mathbf k) \Delta(\mathbf k) W^{*}(-\mathbf k).$$

Individual orbital entries and phases therefore change with the frame. The BdG spectrum, the consistently transformed symmetry content, and properly defined response functions do not.

The normal-state ledger must also say whether the retained subspace is isolated from other bands, whether the crystal is centrosymmetric, whether time reversal is present, and whether a smooth Kramers frame exists over the region used. Near a band degeneracy, strong interband pairing or a poor band separation can invalidate a sheet-by-sheet scalar gap description.

Enforce Fermionic Antisymmetry

With combined internal indices $a,b$ and fermionic Matsubara energy $\varepsilon_n=(2n+1)\pi k_{\mathrm B}T$, the complete exchange constraint is

$$\Delta_{ab}(\mathbf k,i\varepsilon_n) = -\Delta_{ba}(-\mathbf k,-i\varepsilon_n).$$

For even-frequency pairing this reduces to

$$\Delta(\mathbf k)=-\Delta^{T}(-\mathbf k).$$

This matrix equation—not a verbal singlet/triplet label—is the reliable starting point. Assign exchange signatures $S$, $O$, $P$, and $T$ to spin, orbital, momentum parity, and relative-frequency parity. Each is $+1$ for a symmetric factor and $-1$ for an antisymmetric factor. Fermionic exchange requires

$$SOPT=-1.$$

Thus an even-frequency, even-parity spin-triplet pair is allowed when its orbital factor is antisymmetric. The one-orbital shortcut would incorrectly discard it.

Only for an isolated centrosymmetric Kramers doublet, with an intraband and even-frequency approximation in a declared pseudospin gauge, may one write

$$\Delta(\mathbf k) = \left[ \psi(\mathbf k)\sigma_0 + \mathbf d(\mathbf k)\boldsymbol\cdot\boldsymbol\sigma \right] i\sigma_y,$$

where

$$\psi(-\mathbf k)=\psi(\mathbf k), \qquad \mathbf d(-\mathbf k)=-\mathbf d(\mathbf k).$$

The Pauli matrices here act on Kramers pseudospin. With strong spin–orbit coupling, pseudospin expectation values need not equal physical spin, so a Knight-shift or susceptibility statement requires the actual magnetic matrix elements and any Van Vleck background. Without inversion, even- and odd-parity components can mix. With magnetic order or a nonsymmorphic space group, corepresentations, fractional translations, and momentum-dependent sewing must remain explicit.

Classify Zero-Momentum Pairs by Crystal Symmetry

Choose the creation-operator sewing convention

$$g c_{\mathbf k}^{\dagger} g^{-1} = c_{g\mathbf k}^{\dagger}U_g(\mathbf k).$$

The pairing coefficient transformed by $g$ then satisfies

$$\Delta^{g}(g\mathbf k) = U_g(\mathbf k) \Delta(\mathbf k) U_g^{T}(-\mathbf k).$$

Classify the span of gap matrices under this action. For an irreducible channel $\Gamma$ of dimension $d_{\Gamma}$, expand

$$\Delta(\mathbf k) = \sum_{\mu=1}^{d_{\Gamma}} \eta_{\mu} \Phi_{\Gamma\mu}(\mathbf k).$$

The basis matrices $\Phi_{\Gamma\mu}$ transform among themselves, while the complex amplitudes $\eta_{\mu}$ select the superconducting state within that representation. A one-dimensional irrep fixes the symmetry character but not the detailed momentum dependence. A multidimensional irrep permits distinct real, complex, nematic, or chiral combinations whose energetic selection is a separate question.

For a square lattice, familiar even-parity examples near the Brillouin-zone center include

$$\Phi_{A_{1g}}\sim 1, \qquad \Phi_{B_{1g}}\sim \cos k_xa-\cos k_ya, \qquad \Phi_{B_{2g}}\sim \sin k_xa\sin k_ya.$$

These functions are illustrations, not universal gap fits. Orbital matrices, three-dimensional dispersion, nonsymmorphic sewing, or a Fermi surface far from the zone center can change which zeros are enforced and which are merely features of a chosen basis function.

Add Spin–Orbit Coupling, Pseudospin, and Multiorbital Structure

Spin–orbit coupling reorganizes both the normal-state states and the pair classification. In a spinful centrosymmetric, time-reversal-symmetric crystal, the usual commuting inversion and time-reversal operations give $(\mathcal P\Theta)^2=-1$ and hence a same-$\mathbf k$ Kramers doublet at generic momentum. A pseudospin classification is useful only after its sewing convention has been declared. Physical spin response follows from matrix elements within and outside that doublet, not from the word singlet or triplet alone.

Multiorbital pairing adds three further distinctions:

• orbital-symmetric and orbital-antisymmetric tensors obey different parity constraints under the full exchange relation;
• a gap diagonal in an orbital basis need not remain diagonal in the band basis, especially near hybridization gaps or degeneracies; and
• relative phases between band-resolved amplitudes become physical only through a declared interband coupling or probe process. A raw sign attached to an independently rephased Bloch state is not an observable.

The scalar sheet gap $\Delta_n(\mathbf k)$ and dispersion

$$E_n(\mathbf k) = \sqrt{ \xi_n(\mathbf k)^2 + \lvert\Delta_n(\mathbf k)\rvert^2 }$$

apply only in a weak-pairing, intraband, unitary limit with well-separated bands. Here $\xi_n$ and the normal-state matrix $h$ below are measured relative to the chemical potential. Otherwise the full BdG matrix must be diagonalized:

$$\mathcal H_{\mathrm{BdG}}(\mathbf k) = \begin{pmatrix} h(\mathbf k) & \Delta(\mathbf k) \\ \Delta^{\dagger}(\mathbf k) & -h^{T}(-\mathbf k) \end{pmatrix}.$$

BCS Mean-Field Theory owns the reusable saddle construction. This page uses its output only after the normal-state subspace, basis, and symmetry action have been made auditable.

Separate Nodes, Minima, Signs, and Multiband Gaps

A zero of a convenient basis function is not automatically a quasiparticle node. In the scalar intraband limit, a node requires both

$$\xi_n(\mathbf k)=0 \qquad\text{and}\qquad \Delta_n(\mathbf k)=0.$$

This gives four distinct possibilities:

• symmetry-enforced node: every allowed order parameter in the declared representation vanishes on a locus intersected by the Fermi surface;
• accidental node: coefficients within a representation happen to cancel, and a symmetry-preserving perturbation can lift the zero;
• deep minimum: the gap is nonzero but smaller than a probe’s temperature, disorder, lifetime, or energy resolution;
• matrix node: a gap singular value vanishes in a compatible normal-state Fermi subspace; in general the definitive criterion is a zero eigenvalue of the full BdG problem, even when no single orbital entry tells the story.

Power laws are therefore evidence about low-energy phase space, not unique representation labels. Disorder can change a linear penetration-depth law to a quadratic one, lift accidental nodes, create subgap states, or average a strongly anisotropic gap. Nonlocal electrodynamics, multiple sheets, surface reconstruction, and finite resolution can mimic the same crossover.

Gap magnitude and relative phase also require different probes. ARPES and tunneling can constrain $\lvert\Delta_n(\mathbf k)\rvert$ after their matrix elements and surface sensitivity are modeled. A sign reversal requires a phase-sensitive process, an interband coherence factor, or a controlled impurity response with a declared gauge convention. A neutron resonance or a QPI contrast can be consistent with a sign change, but neither is a universal binary sign detector.

Distinguish Multicomponent, Nematic, and Time-Reversal-Breaking Order

For a two-component order parameter $\boldsymbol\eta=(\eta_1,\eta_2)$, a useful minimal homogeneous illustration is

$$f = \alpha R^2 + \beta_1 R^4 + \beta_2 \left\lvert \eta_1^{*}\eta_2-\eta_2^{*}\eta_1 \right\rvert^2, \qquad R^2=\lvert\eta_1\rvert^2+\lvert\eta_2\rvert^2.$$

For fixed $R$, a real relative phase makes the last term vanish. The chiral combinations $\boldsymbol\eta=R(1,\pm i)/\sqrt2$ maximize it. Hence $\beta_2<0$ favors the chiral manifold when $\beta_1+\beta_2>0$; $\beta_2>0$ favors a real-relative-phase manifold, while crystal and higher-order invariants select its orientation. Boundedness of this illustrative quartic form requires $\beta_1>0$ along the real manifold and $\beta_1+\beta_2>0$ along the chiral manifold when $\beta_2<0$.

In a real irrep basis, the gauge-invariant bilinear

$$\mathcal L = i\left( \eta_1\eta_2^{*} - \eta_2\eta_1^{*} \right)$$

changes sign under time reversal. It vanishes for a real nematic state and is $\pm R^2$ for the two chiral combinations. This is a symmetry diagnostic, not a Chern number. Whether a chiral state is topological depends on the full BdG gap, dimension, symmetry class, and invariant, all owned by Topological Superconductors.

Strain can couple to bilinears such as

$$f_{\mathrm{strain}} = g\varepsilon \left( \lvert\eta_1\rvert^2 - \lvert\eta_2\rvert^2 \right),$$

lifting a degeneracy or moving transition temperatures. A split transition is not unique evidence for a multidimensional irrep: strain gradients, two bands, two competing orders, and sample inhomogeneity must be tested. Likewise, a Kerr angle or zero-field local magnetic signal requires controls for trapped flux, magnetic inclusions, domains, training, and instrumental offsets before it supports bulk time-reversal breaking.

Build a Multi-Probe Evidence Ladder

No single probe determines a general gap matrix. Build the claim in stages.

1. Establish bulk superconductivity. Heat capacity, magnetic screening, and transport constrain transition temperatures, volume fraction, and homogeneity. Heat Capacity and Thermodynamics owns calorimetric subtraction and entropy balance; London Theory owns penetration-depth electrodynamics.
2. Constrain low-energy gap structure. Low-temperature thermodynamics, penetration depth, thermal response, ARPES, STM/STS, and Raman and Optical Spectroscopy weight different momenta, bands, and surfaces. Agreement matters more than a shared fit label.
3. Test relative phase. Controlled corner junctions, tricrystal loops, and other Josephson geometries can compare crystal orientations. Intersheet neutron scattering and QPI can add conditional coherence-factor evidence after their orbital, interaction, and impurity models are fixed.
4. Test spin and pseudospin. Susceptibility or NMR must separate Pauli, orbital, Van Vleck, diamagnetic, heating, and field-orientation effects. A physical-spin response cannot be read directly from a pseudospin label.
5. Test additional broken symmetries. Strain, ultrasound, Kerr rotation, and zero-field local magnetism can constrain multicomponent, nematic, or time-reversal-breaking order only with domain, training, and background controls.
6. Demand same-sample consistency. Sample quality, disorder, surface termination, strain, and transition width must be carried across the evidence package. Combining incompatible sample regimes can manufacture a false consensus.

The measurement gateway owns calibration and inference discipline. Spectral Functions owns poles, continua, and matrix-element-weighted spectra. This page owns only the cross-probe pairing-state synthesis.

Disorder, Surfaces, Domains, Strain, and Resolution

Nonmagnetic disorder leaves an ideal isotropic single-band constant-sign $s$-wave transition unusually robust under the assumptions of Anderson’s theorem. The same statement does not extend without qualification to an anisotropic, multiband, sign-changing, magnetic, strongly inelastic, or nonuniform system. Rapid $T_c$ suppression can support pair breaking, but it does not by itself identify the representation or the pairing glue.

Surface scattering can average momentum directions, suppress one component, or create Andreev bound states. A zero-bias feature may arise from sign-changing surface trajectories, ordinary Andreev physics, disorder, Kondo scattering, heating, or instrumental convolution. Induced odd-frequency or converted pair amplitudes at an interface are not automatically a new bulk phase; the Superconducting Proximity Effect owns that spatial-correlation problem.

Domains can cancel a macroscopic time-reversal-odd or nematic signal while leaving local signatures. Strain can train domains or split transitions, but strain inhomogeneity can broaden them. A probe that averages over many domains and one that sees a single surface may legitimately report different apparent symmetries. The forward model must include those weights before the results are called inconsistent.

Finally, a fitted zero is only as strong as the energy and temperature floor. If a reported minimum $\Delta_{\min}$ is comparable to a lifetime broadening $\Gamma$, thermal scale $k_{\mathrm B}T$, or resolution $\delta E$, the licensed statement is a resolution-limited minimum—not a node.

Worked Audit: A Tetragonal Nodal Candidate

This first record is synthetic. Its numbers are chosen to make the inference reproducible; they are not a fit to a named material.

1. Material, specimen, and geometry. The synthetic crystal is centrosymmetric and tetragonal. Three platelets from one batch supply bulk thermodynamics and penetration depth; a cleaved $(001)$ surface supplies spectroscopy; registered $(100)$ and $(010)$ faces form the corner device. One quasi-two-dimensional Fermi cylinder surrounds $\Gamma$.
2. Prepared thermodynamic state and tuning variables. The clean specimens have $T_c=14.0\pm0.1\ \mathrm K$ at zero applied field and strain after zero-field cooling. A matched irradiated specimen raises the residual resistivity by $8\pm1\ \mu\Omega\,\mathrm{cm}$ and lowers $T_c$ to $12.2\pm0.2\ \mathrm K$. Field-cool, reverse-field, and orientation-reversed junction histories are retained.
3. Normal-state low-energy Hilbert space. The retained sheet is one isolated Kramers doublet. The next band remains $75\pm10\ \mathrm{meV}$ away, while $E_F=0.42\pm0.04\ \mathrm{eV}$. Orbital weights and the Bloch-to-pseudospin frame are registered; remote bands enter only through fitted normal-state parameters.
4. Exact and approximate symmetries. The normal state has $D_{4h}$, inversion, and time reversal. X-ray bounds orthorhombic strain below $2\times10^{-4}$, and no magnetic order is resolved. The pairing model keeps translations and a spin–orbit-entangled Kramers basis but not spin rotation.
5. Pair definition and convention. Pair momentum is zero and the state is even frequency. Around the cylinder, $\phi=0$ points along $+k_x$. The candidates are $\Delta_{A_{1g}}=\Delta_0(1-r\cos4\phi)$ and $\Delta_{B_{1g}}=\Delta_0\cos2\phi$ in one sewn Kramers gauge. Only relative phases sampled by the registered junction are treated as observable.
6. Gap structure on every relevant sheet. The fitted maximum is $\Delta_{\max}=2.40\pm0.10\ \mathrm{meV}$. Four diagonal locations are zero or below the $0.15\ \mathrm{meV}$ energy and $2^{\circ}$ angular resolution. The $B_{1g}$ model enforces diagonal nodes; the competing $A_{1g}$ fit has $r=-1.10\pm0.08$ and therefore permits movable accidental zeros, but its central zeros lie about $6.2^{\circ}$ from the diagonals and worsen the fit at the declared angular resolution.
7. Observable and canonical forward model. Between $0.08T_c$ and $0.20T_c$, $C_{\mathrm{el}}/T=(1.2\pm0.2)\ \mathrm{mJ\,mol^{-1}K^{-2}} +(2.8\pm0.2)T\ \mathrm{mJ\,mol^{-1}K^{-3}}$, while $\Delta\lambda=(4.8\pm0.3)T\ \mathrm{nm\,K^{-1}}$. Calorimetry includes phonon subtraction and entropy balance; penetration depth includes the nonlocal correction. ARPES carries polarization, matrix elements, and energy convolution. The corner-SQUID forward model includes junction asymmetry, self-inductance, and the measured facet distribution.
8. Nuisance controls and alternatives. The registered junction phase is $0.98\pi\pm0.08\pi$, equivalent to a near-half-flux offset after self-inductance correction. Facets lie within $\pm2^{\circ}$ of their target orientations. Trapped flux, current crowding, surface reconstruction, nonlocal electrodynamics, domains, a second unresolved sheet, and accidental $A_{1g}$ nodes remain explicit alternatives. Irradiation changes the penetration law from $T$ toward $T^2$ rather than supplying a unique irrep.
9. Numerics and uncertainty. Joint fits share $\Delta_{\max}$ and the measured elastic width. Doubling the angular mesh from $720$ to $1440$ points changes fitted minima by less than $0.01\ \mathrm{meV}$; varying the spectral kernel within calibration changes $r$ by $0.04$. The $r$–broadening correlation coefficient is $0.62$, and the three bulk specimens agree within their stated one-standard-deviation spread.
10. Licensed claim and stopping rule. Reproducing the phase shift on devices rotated by $90^{\circ}$, preserving diagonal minima in a bulk-sensitive probe, and matching the disorder evolution license a $B_{1g}$-compatible sign-changing nodal state. Loss of the registered phase shift or a resolved nodeless bulk sheet reopens $A_{1g}$ and multiband alternatives. Even a successful audit stops before pairing glue or a BdG invariant.

The basic symmetry calculation is short. Under a fourfold rotation,

$$\cos\bigl[2(\phi+\pi/2)\bigr] = -\cos2\phi,$$

and the nodes are

$$\phi = \frac{\pi}{4} + n\frac{\pi}{2}.$$

The two orthogonal crystal faces sampled by the corner device therefore see a relative phase $\pi$ in the ideal $B_{1g}$ model. Also,

$$k_{\mathrm B}T_c = 1.206\ \mathrm{meV}, \qquad \frac{2\Delta_0}{k_{\mathrm B}T_c} = 3.98\pm0.17.$$

For the $B_{1g}$ candidate, $\Delta_0=\Delta_{\max}$. The resulting ratio is above the elementary isotropic weak-coupling value within this synthetic record, but it does not identify a bosonic glue. The combined record licenses a $B_{1g}$-compatible sign-changing nodal state under the declared one-sheet and junction controls. The phase-sensitive result carries the representation claim; the power laws and ARPES constrain its low-energy realization.

Worked Audit: A Multiband A1g Sign-Structure Candidate

This second record is also synthetic. It is designed to keep gap magnitude, crystal representation, and intersheet relative phase separate.

1. Material, specimen, and geometry. A synthetic centrosymmetric $D_{4h}$ metal has one hole sheet around $\Gamma$ and one electron sheet around the zone corner. One batch supplies bulk calorimetry and neutron crystals; a registered $(001)$ surface supplies ARPES, STS, and impurity-QPI records.
2. Prepared thermodynamic state and tuning variables. The batch has a sharp zero-field transition at $T_c=24.0\pm0.2\ \mathrm K$ after zero-field cooling, with no second anomaly. Clean and controlled-irradiation states are compared at the same temperature and carrier density.
3. Normal-state low-energy Hilbert space. Two resolved Kramers-degenerate sheets with measured orbital weights are retained. Interband pairing is neglected in the scalar spectra, while a finite interband Josephson coupling locks their relative phase. The registered orbital-to-band matrices enter every coherence-factor calculation.
4. Exact and approximate symmetries. The normal state preserves $D_{4h}$, inversion, time reversal, and translations within experimental accuracy. Spin rotation is not assumed. No static reconstruction at the intersheet wavevector is detected above $T_c$ within the neutron and diffraction limits.
5. Pair definition and convention. Pair momentum is zero and the state is even frequency and $A_{1g}$. The momentum $\mathbf Q$ connects the dominant hole- and electron-sheet regions. In the common sewn Bloch frame, $s_{++}$ means relative phase zero and $s_{\pm}$ means relative phase $\pi$; neither is an independently observable raw sign on one rephased band.
6. Gap structure on every relevant sheet. Magnitude probes give $\lvert\Delta_h\rvert=5.0\pm0.3\ \mathrm{meV}$ and $\lvert\Delta_e\rvert=7.2\pm0.4\ \mathrm{meV}$, with no node resolved below $0.25\ \mathrm{meV}$. Both $s_{++}$ and $s_{\pm}$ are nodeless two-gap $A_{1g}$ states; their distinction is the intersheet relative phase.
7. Observable and canonical forward model. Heat capacity carries band weights and entropy balance; ARPES carries polarization and orbital matrix elements; STS carries surface tunneling weights and lifetime convolution. The $9.5\pm0.4\ \mathrm{meV}$ neutron feature onsets below $T_c$ with a compensating superconductivity-induced redistribution of spectral weight. It is compared with a registered orbital RPA susceptibility including the normal continuum and resolution. QPI uses a calibrated impurity $T$-matrix, tunneling matrix elements, and a phase-referenced field control.
8. Nuisance controls and alternatives. Controlled irradiation suppresses $T_c$ faster than the constant-sign reference calculation, and the QPI contrast agrees with the sign-changing model. Retained alternatives are an $s_{++}$ state with a paramagnon or another collective mode, orbital-selective damping, an incipient band, magnetic defects, a surface-selective gap, normal-state reconstruction, and the wrong impurity potential.
9. Numerics and uncertainty. Doubling the momentum mesh from $256\times256$ to $512\times512$ moves the fitted neutron pole by $0.06\ \mathrm{meV}$; varying the resolution kernel moves it by $0.12\ \mathrm{meV}$. Joint heat-capacity and spectral fits retain the stated gap uncertainties, a gap-error correlation coefficient $\rho_{he}=0.05$, and a $-0.48$ correlation between band weight and the smaller gap. Normal and superconducting neutron data use one interaction parameter set; three samples reproduce the feature within $0.3\ \mathrm{meV}$.
10. Licensed claim and stopping rule. Magnitude agreement licenses nodeless multigap $A_{1g}$ superconductivity. The neutron, disorder, and QPI package favors a relative sign reversal only at the level licensed by those forward models. A calibrated phase-sensitive contradiction or a normal-state collective-mode fit of equal quality reopens $s_{++}$. The audit stops before a unique interaction or topology claim.

The magnitude ledger gives

$$\frac{2\lvert\Delta_h\rvert}{k_{\mathrm B}T_c} = 4.84\pm0.29, \qquad \frac{2\lvert\Delta_e\rvert}{k_{\mathrm B}T_c} = 6.96\pm0.39.$$

The two-quasiparticle continuum edge at the connecting momentum is generally

$$\Omega_c(\mathbf Q) = \min_{\mathbf k} \left[ E_h(\mathbf k) + E_e(\mathbf k+\mathbf Q) \right].$$

For the declared locally constant gaps and $\mathbf Q$-connected Fermi points, this reduces to

$$\lvert\Delta_h\rvert+\lvert\Delta_e\rvert = 12.2\pm0.5\ \mathrm{meV},$$

so the feature lies below it by $2.7\pm0.6\ \mathrm{meV}$ within the declared model, where the uncertainty includes the declared gap-error covariance. The subthreshold location together with superconductivity-induced onset and weight redistribution supports an intersheet spin-exciton interpretation when the coherence factor and interaction model are valid, but it does not prove $s_{\pm}$ by kinematics alone. The strongest unconditional conclusion is nodeless multigap $A_{1g}$ superconductivity. The sign-reversing conclusion is consistent and model-supported, but not yet uniquely phase resolved.

Common Failure Modes and Canonical Handoffs

• Defining unconventional by a presumed glue. A phonon-mediated state can be anisotropic or symmetry nontrivial, and a nonphononic interaction does not determine a unique gap. Heavy Fermions and Moiré Superconductivity own their platform evidence; changing mechanism status remains a frontier coverage gap.
• Equating nodal with unconventional. Accidental $A_{1g}$ nodes and deep minima exist; some nontrivial or sign-changing states are fully gapped.
• Calling every $A_{1g}$ state conventional. The irrep label does not fix intersheet relative phase, orbital tensor, or microscopic structure.
• Using singlet/triplet language outside its license. Inversion, Kramers structure, frequency parity, and orbital exchange must be declared first.
• Reading physical spin from pseudospin. Magnetic response needs material matrix elements, orbital terms, and field-dependent backgrounds.
• Treating one power law as a representation measurement. Disorder, nonlocality, multiple bands, dimensional crossover, and resolution can change exponents.
• Treating a zero-bias peak, neutron resonance, or QPI contrast as a binary sign detector. Each requires its full forward model and alternatives.
• Treating Kerr or zero-field magnetism as automatic chirality. Establish superconducting onset, bulk origin, training, domain behavior, and magnetic controls. Time-reversal breaking is not itself a topological invariant.
• Using a finite-$\mathbf Q$ or composite state as a zero-momentum irrep. Pair-Density Waves and Exotic Orders owns finite-center-of-mass and composite pairing.
• Promoting a competing order into pairing evidence. Competing Orders owns coexistence and reconstruction; this page requires a pairing-specific operator and onset.
• Promoting a gap sign into topology. Odd parity, chirality, a surface bound state, or a phase shift does not replace a bulk BdG invariant and protecting gap. Route that question to Topological Superconductors.
• Confusing anomalous amplitude, pair potential, and spectral edge. The general Order Parameters page owns reusable distinctions, while bulk Proximity owns spatially induced correlations.

When a fast-moving mechanism or material-family claim outruns that stable classification, Quantum Matter Frontiers and Open Problems supplies the dated claim-and-evidence audit and routes future living status without changing this page’s canonical scope.

This page is deliberately stable. It ends with symmetry, gap structure, and a bounded evidence claim. It does not rank unsettled mechanisms, declare a material family solved, reproduce instrument inversion, or run production BdG/Eliashberg calculations.

Exercises

Exercise 1: Complete the exchange ledger

For even-frequency pairing, classify the required momentum parity for four factorized internal tensors: (a) spin singlet and orbital symmetric; (b) spin triplet and orbital symmetric; (c) spin singlet and orbital antisymmetric; and (d) spin triplet and orbital antisymmetric. Which case defeats the slogan “triplet means odd parity”?

Solution

Use $SOPT=-1$ with $T=+1$. A singlet has $S=-1$, a triplet has $S=+1$, an orbital-symmetric tensor has $O=+1$, and an orbital-antisymmetric tensor has $O=-1$.

• (a) requires $P=+1$;
• (b) requires $P=-1$;
• (c) requires $P=-1$;
• (d) requires $P=+1$.

Case (d) is an even-parity spin-triplet state made possible by the orbital-antisymmetric factor. For odd-frequency pairing, $T=-1$ flips each required momentum parity. The matrix exchange equation remains valid in every case and is safer than the slogan.

Exercise 2: Verify frame covariance

Starting from $c_{\mathbf k}=W(\mathbf k)\widetilde c_{\mathbf k}$, derive the transformation of $\Delta$. Show that if $\Delta(\mathbf k)=-\Delta^T(-\mathbf k)$, then the transformed matrix obeys the same exchange relation. What remains invariant?

Solution

Substituting the operator change into $H_{\Delta}$ gives

$$\widetilde\Delta(\mathbf k) = W^{\dagger}(\mathbf k) \Delta(\mathbf k) W^{*}(-\mathbf k).$$

Then

$$\widetilde\Delta^{T}(-\mathbf k) = W^{\dagger}(\mathbf k) \Delta^{T}(-\mathbf k) W^{*}(-\mathbf k) = -\widetilde\Delta(\mathbf k).$$

The individual entries, orbital weights, and raw sheet phases are frame dependent. The normal matrix transforms as

$$\widetilde h(\mathbf k) = W^{\dagger}(\mathbf k)h(\mathbf k)W(\mathbf k).$$

In the Nambu basis the complete transformation is block diagonal with $W(\mathbf k)$ in the particle block and $W^{*}(-\mathbf k)$ in the hole block. It is unitary, so the BdG eigenvalues are invariant. The representation carried by the full gap-matrix span is likewise invariant when the symmetry sewing matrices are transformed with the same frame.

Exercise 3: Distinguish square-lattice irreps

Let

$$f_0(\mathbf k)=1, \qquad f_s(\mathbf k)=\cos k_xa+\cos k_ya,$$

and

$$f_1(\mathbf k)=\cos k_xa-\cos k_ya, \qquad f_2(\mathbf k)=\sin k_xa\sin k_ya.$$

Find all four functions’ signs under a $90^{\circ}$ rotation and the reflection $k_x\mapsto-k_x$. Identify the $D_{4h}$ irreps and the nontrivial functions’ near-$\Gamma$ nodal directions.

Solution

Both $f_0$ and $f_s$ are unchanged by $C_4$ and the reflection, so they are $A_{1g}$ examples. Under $C_4:(k_x,k_y)\mapsto(-k_y,k_x)$, $f_1$ and $f_2$ both change sign. Under the $k_x$ reflection, $f_1$ is even while $f_2$ is odd. Hence $f_1$ transforms as $B_{1g}$ and $f_2$ as $B_{2g}$. Near $\Gamma$, $f_1\propto k_y^2-k_x^2$, so its nodes lie along $k_x=\pm k_y$. Meanwhile $f_2\propto k_xk_y$, so its nodes lie along the crystal axes. The actual quasiparticle nodes exist only where those loci intersect a Fermi surface. These scalar harmonics classify only the momentum factor. With strong spin–orbit coupling or multiple orbitals, the internal gap matrix and the momentum-dependent Bloch sewing matrices transform too; the full matrix span, not the scalar harmonic alone, determines the pair representation.

Exercise 4: Enforced node, accidental node, or deep minimum?

On a connected, fourfold-symmetric cylindrical Fermi surface, compare

$$\Delta_B(\phi)=\Delta_0\cos2\phi,$$ $$\Delta_A(\phi)=\Delta_0(r+\cos4\phi),$$

and

$$\Delta_M(\phi)=\Delta_0(0.20+0.15\cos4\phi).$$

Take $r=0.15$ and an experimental resolution $\delta E=0.08\Delta_0$. Classify the zeros or minima and state a falsifier.

Solution

$\Delta_B$ changes sign under $C_4$. Continuity on the connected symmetric sheet forces four diagonal intersections. More directly, a diagonal mirror leaves each diagonal momentum fixed but acts with character $-1$ on $B_{1g}$, so $\Delta_B=-\Delta_B=0$ there. These are symmetry-enforced nodes within the stated one-component model.

$\Delta_A$ is $A_{1g}$. At $r=0.15$, the equation $\cos4\phi=-0.15$ has solutions. In general, $|r|<1$ gives accidental sign-changing zeros, $|r|=1$ gives fine-tuned touching zeros, and $|r|>1$ is nodeless. Changing $r$ across that range is symmetry allowed, so the zeros are accidental.

$\Delta_M$ has minimum $0.05\Delta_0$, so it is nodeless. Because the minimum is smaller than $\delta E$, the supplied experiment cannot distinguish it from a node. Lowering temperature and broadening below $0.05\Delta_0$, then checking bulk scaling and disorder evolution, is the falsifier.

Exercise 5: Test the scalar-sheet approximation

Case A has two bands separated by $40\ \mathrm{meV}$ wherever pairing is important and a $2\ \mathrm{meV}$ gap. Case B has an avoided crossing of $0.4\ \mathrm{meV}$ and a $1.2\ \mathrm{meV}$ gap with appreciable interband matrix elements. Write the minimal two-band BdG matrix and decide which case licenses independent scalar sheet gaps. Give a quantitative failure test.

Solution

Work in the instantaneous normal-band basis, where the avoided-crossing hybridization is encoded in $\xi_i$ and in the momentum-dependent band frame. Suppressing the Kramers block, the relevant matrix is

$$\mathcal H_{\mathrm{BdG}} = \begin{pmatrix} \xi_1 & 0 & \Delta_1 & \Delta_{12} \\ 0 & \xi_2 & \Delta_{21} & \Delta_2 \\ \Delta_1^{*} & \Delta_{21}^{*} & -\xi_1(-\mathbf k) & 0 \\ \Delta_{12}^{*} & \Delta_2^{*} & 0 & -\xi_2(-\mathbf k) \end{pmatrix}.$$

Independent scalar sheets discard $\Delta_{12}$ and $\Delta_{21}$. Case A has a separation-to-gap ratio of $20$, so that projection is a controlled starting point if the frame is smooth and other bands remain remote. Case B has ratio $0.4/1.2=0.33$; interband pairing and the rapid variation of the normal-band frame cannot be treated perturbatively. Its scalar signs or nodes can be basis artifacts. In an orbital basis, the avoided-crossing hybridization would instead appear explicitly in the normal block.

A quantitative acceptance test is to enlarge the retained subspace, keep the interband block, refine the momentum mesh, and require every claimed minimum and low BdG eigenvalue to change by less than the target tolerance—say $5\%$. Case B must use the full matrix unless it passes that test.

Exercise 6: Minimize a two-component functional

For

$$f_4 = \beta_1R^4 + \beta_2 \left\lvert \eta_1^{*}\eta_2-\eta_2^{*}\eta_1 \right\rvert^2,$$

compare $R(1,0)$, $R(1,1)/\sqrt2$, and $R(1,i)/\sqrt2$. Which state is favored for each sign of $\beta_2$, and what does the time-reversal-odd bilinear say?

Solution

For the first two real states, the second invariant is zero and $\mathcal L=0$. For $R(1,i)/\sqrt2$, the invariant equals $R^4$ and $\mathcal L=+R^2$; its complex-conjugate partner has $\mathcal L=-R^2$. Therefore $\beta_2<0$ favors the chiral pair when $\beta_1+\beta_2>0$ and the real direction is stable only when $\beta_1>0$. For $\beta_2>0$, the real-relative-phase manifold is favored, but this displayed term alone does not choose between its orientations; other same-order crystal-allowed invariants, strain, or higher-order terms can select a nematic state.

In the declared real irrep basis, time reversal sends $\boldsymbol\eta$ to $\boldsymbol\eta^*$. A state is invariant only if $\boldsymbol\eta^*=e^{i\chi}\boldsymbol\eta$ for one global phase. Real states pass. For $(1,i)$, the first component requires $e^{i\chi}=1$ while the second requires $e^{i\chi}=-1$, so no such phase exists. The nonzero bilinear therefore marks time-reversal breaking, not a Chern number or Majorana boundary state.

Exercise 7: Rank an evidence package

A sample shows $C_{\mathrm{el}}/T\propto T$ with a $7\%$ slope error, a penetration-depth crossover from $T$ to $T^2$ after weak disorder, diagonal ARPES minima below $0.15\ \mathrm{meV}$, nodeless surface STS with $0.25\ \mathrm{meV}$ resolution, a $9.5\pm0.4\ \mathrm{meV}$ neutron feature, an in-plane susceptibility drop of $45\%\pm5\%$ but a $c$-axis drop of $5\%\pm7\%$, and a corner-junction phase $0.98\pi\pm0.08\pi$. State what each record constrains, rank the evidence for a $B_{1g}$ claim, and identify which spin, time-reversal, and mechanism claims remain unlicensed.

Solution

Heat capacity and penetration depth constrain bulk low-energy density of states and its disorder crossover, not a unique irrep. ARPES localizes candidate minima; nodeless STS can remain compatible because it weights a reconstructed surface and has poorer resolution. The neutron feature is conditional evidence for a coherence-factor sign change only after the normal continuum, orbital vertices, interactions, and resolution are fit. The anisotropic susceptibility drop constrains the physical magnetic response after orbital, Van Vleck, diamagnetic, and heating subtraction; it is not a direct pseudospin label. The registered corner-junction phase shift is the most direct representation-level evidence when its Josephson forward model and controls are valid.

Required alternatives include accidental $A_{1g}$ nodes or deep minima; an unresolved second band; nonlocal electrodynamics; facet averaging and current crowding; trapped flux; surface reconstruction; and a normal collective mode. The combined record can support $B_{1g}$ only after the phase shift is repeated on rotated devices and the diagonal low-energy structure is confirmed in bulk. No time-reversal-breaking measurement is supplied, the susceptibility does not uniquely fix singlet or triplet structure, and none of the records identifies the microscopic interaction.

Exercise 8: Route a hybrid claim

A moiré sample has a zero-bias edge feature, a modulation at nonzero wavevector, strong sample-to-sample $T_c$ variation, a fitted odd-frequency interface amplitude, and a spontaneous-field signal below the resistive transition. Fill all ten ledger fields at the level possible from those facts, give a falsifier for each retained descriptive claim, and route the unresolved questions. What is the strongest immediate claim?

Solution

1. Material, specimen, and geometry: only “moiré device” is known; contacts, edges, dimensions, and compared specimens are unresolved.
2. Prepared thermodynamic state and tuning variables: the resistive transition is known, but temperature, field, filling, disorder, cooling history, and drive are not.
3. Normal-state low-energy Hilbert space: bands, valleys, orbitals, spin/pseudospin, and retained Fermi surfaces are missing.
4. Exact and approximate symmetries: translations, point group, time reversal, strain, and their specimen accuracy are missing.
5. Pair definition and convention: the modulation suggests a finite wavevector and the fit labels an interface frequency structure, but neither a bulk gap matrix nor its convention is supplied.
6. Gap structure on every relevant sheet: magnitudes, nodes, phases, sheet dependence, and domains are unresolved.
7. Observable and canonical forward model: the spectral, imaging, interface-fit, and spontaneous-field operators, contacts, resolution, and backgrounds are missing.
8. Nuisance controls and alternatives: disorder, heating, edge confinement, trapped flux, magnetic inclusions, strain, and inhomogeneity remain viable.
9. Numerics and uncertainty: fit covariance, model comparison, convergence, sample spread, and cross-probe reproducibility are absent.
10. Licensed claim and stopping rule: only the coexistence of several superconductivity-adjacent anomalies is presently licensed.

Route the finite-$\mathbf Q$ order to Pair-Density Waves and Exotic Orders; the edge/topology question to Topological Superconductors; the odd-frequency interface amplitude to Superconducting Proximity Effect; spectra and acquisition to their probe owners; platform variability to Moiré Superconductivity; the spontaneous-field acquisition through magnetic probe controls; any independent magnetic or coexisting phase to Competing Orders; and any mechanism ranking to the future frontier owner.

The modulation claim is falsified if it persists unchanged above the bulk transition or lacks phase coherence; the zero-bias claim is falsified by a calibrated background or contact resonance; the odd-frequency fit is falsified by an equally good conventional interface model; and the spontaneous-field claim is falsified by trapped-flux, magnetic-inclusion, or off-sample controls. A bulk, zero-momentum unconventional representation is not yet established. A same-device symmetry-resolved bulk transition plus calibrated phase-sensitive or nodal evidence would refine the claim.

References

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