Unconventional Pairing Symmetries and Diagnostics

An unconventional pairing state is classified by the antisymmetric gap matrix on its actual spin–orbital–band Hilbert space and by how that matrix transforms under the crystal and antiunitary symmetries. Nodes, phase winding, spin response, disorder sensitivity, and phase-sensitive interference can falsify candidate representations. None of those symmetry diagnostics, alone or together, identifies the interaction that caused pairing; mechanism and topological BdG classification are separate questions.

Required background. Nambu–Gor’kov notation supplies the gap matrix. Discrete and antiunitary symmetries supplies the transformation rules. Invertible background responses supplies the distinction between symmetry labels and topology.

Helpful background. Band topology and symmetry protection provides the normal-state geometric setting.

Gap matrix and Fermi-statistics constraint

For internal indices $a,b$, define

$$\Delta_{ab}(\mathbf k) =\sum_{\mathbf k'}V_{ab,cd}(\mathbf k,\mathbf k') \langle c_{-\mathbf k'd}c_{\mathbf k'c}\rangle.$$

Fermi statistics requires

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

With inversion and one spin-$1/2$ orbital, one often writes

$$\Delta(\mathbf k)= \bigl[\psi(\mathbf k)\sigma_0+\mathbf d(\mathbf k)\mathbin{\cdot}\boldsymbol\sigma\bigr]i\sigma_y,$$

where $\psi$ is even and $\mathbf d$ is odd. Without inversion, singlet and triplet components may mix. With several orbitals, orbital exchange contributes to antisymmetry, so parity cannot be inferred from a spin label alone.

For a point-group operation $g$ represented on Bloch states by $U_g(\mathbf k)$,

$$\Delta(\mathbf k)\mapsto U_g(\mathbf k)\Delta(g^{-1}\mathbf k)U_g^T(-\mathbf k).$$

Linearized pairing kernels decompose into irreducible representations, but a multidimensional representation leaves a degeneracy. Quartic free-energy coefficients determine whether, for example, two real components combine nematically or as a time-reversal-breaking complex pair. Sigrist and Ueda 1991, §§II–III develops this classification.

Nodes are intersections, not representation names

The quasiparticle gap on band $n$ is the relevant projected singular value of $\Delta$ on that Fermi surface. A symmetry operation can force it to vanish on a plane or line, but only if the Fermi surface intersects that locus and interband pairing does not lift the zero. Accidental nodes can occur within a representation and move under symmetry-preserving parameter changes. Blount 1985, §§II–III states the classic spin–orbit-coupled odd-parity node constraints.

Near a line node in three dimensions, the quasiparticle density of states behaves as $N(E)\propto E$ in the clean limit, leading to power laws such as $C\propto T^2$. Point nodes give a different power. Disorder, nonlocal electrodynamics, multiband gaps, and finite measurement windows can mimic or obscure these asymptotes, so a fitted power over one decade is not a representation measurement.

Phase-sensitive interference is stronger. A junction or corner geometry can compare the sign of the projected pair amplitude on different crystal faces. The forward model must include surface orientation, faceting, tunnelling matrix elements, and possible subdominant surface order. A half-flux pattern can test a sign change; it does not by itself determine whether phonons, spin fluctuations, or another interaction generated it.

A falsification program

For every candidate representation, predict at least:

• symmetry-enforced versus accidental nodes on the measured Fermi surfaces;
• the transformation of the gap under the actual crystal group;
• spin susceptibility for controlled field directions, including spin–orbit coupling;
• impurity and boundary response for specified scattering channels;
• phase-sensitive junction signs or quasiparticle-interference coherence factors; and
• whether time reversal or another symmetry is broken, with a domain-sensitive test.

The correct conclusion is the intersection of surviving tests. A null result can be decisive only when its coupling to the candidate order is quantitatively nonzero and the experimental resolution is sufficient. Conversely, a positive local-field signal may have magnetic, trapped-flux, or inhomogeneous alternatives.

The validity map keeps symmetry, mechanism, and topology on different branches.

Symmetry classification predicts a linked set of falsifiable observables. Agreement licenses a representation within the tested model; it does not identify the pairing interaction or a topological boundary mode. Original schematic, not to scale.

The paired-matter claim test matrix compares these claim types directly.

Exercise

Apply antisymmetry. In a one-orbital centrosymmetric system, show that $\Delta=\psi i\sigma_y$ requires even $\psi(\mathbf k)$, while $\Delta=(\mathbf d\cdot\boldsymbol\sigma)i\sigma_y$ requires odd $\mathbf d(\mathbf k)$.

Solution

$i\sigma_y$ is antisymmetric, so $-[\psi(-\mathbf k)i\sigma_y]^T=\psi(-\mathbf k)i\sigma_y$. Equating this with $\psi(\mathbf k)i\sigma_y$ gives even $\psi$. The three matrices $\sigma_i i\sigma_y$ are symmetric, so the same constraint gives $\mathbf d(\mathbf k)=-\mathbf d(-\mathbf k)$. Additional orbital indices can reverse these assignments through their own exchange parity.

References

• Blount, E. I. (1985). “Symmetry properties of triplet superconductors.” Physical Review B 32, 2935–2944. doi:10.1103/PhysRevB.32.2935.
• Sigrist, M., and Ueda, K. (1991). “Phenomenological theory of unconventional superconductivity.” Reviews of Modern Physics 63, 239–311. doi:10.1103/RevModPhys.63.239.