Gap magnitude and gap sign require different experimental tests

For an illustrative circular two-dimensional Fermi surface, take a real d-wave gap proportional to cos2φ, where φ is measured from a crystal axis. It has four nodal directions and changes sign under a 90-degree rotation. Quasiparticle energies contain the squared gap magnitude, so a negative gap does not mean a negative excitation cost. The sign refers to a relative pairing phase between momentum directions in a common convention, which can affect pair tunneling when an appropriate reference and junction geometry connect those directions.

Fermionic antisymmetry constrains the combined spin, orbital, and any additional internal wavefunction. With inversion symmetry and a simple spin-rotation-invariant single-band description, spin-singlet pairing has even orbital parity and spin-triplet pairing has odd parity. Strong spin–orbit coupling, absent inversion, multiorbital structure, or frequency-dependent pairing require a more careful classification. A nodal gap is also not automatically symmetry-enforced: an anisotropic even s-wave state can have accidental nodes. Determining the gap structure therefore requires both symmetry reasoning and evidence that distinguishes plausible alternatives.

Near a simple line node, the quasiparticle dispersion is linear in the two transverse directions. Counting the available low-energy states gives a density of states proportional to energy. The thermal energy then scales as T cubed and the electronic heat capacity as T squared in the clean low-temperature regime. The same power arises for isolated nodal points on a quasi-two-dimensional cylindrical Fermi surface with the appropriate linear dispersion. A full minimum gap instead suppresses electronic thermal excitations exponentially at temperatures well below that gap.

Experiments must separate electronic and lattice terms. Three-dimensional acoustic phonons commonly contribute T cubed heat capacity, while disorder can generate residual low-energy states and change apparent powers. Point nodes, deep gap minima, multiple bands, magnetic impurities, and a limited fitting window can complicate inference. A T-squared electronic term supports a nodal interpretation under its assumptions but does not uniquely identify d-wave symmetry. Combining heat capacity, penetration depth, thermal transport, spectroscopy, and controlled disorder studies constrains more alternatives than any one fitted exponent.

A Josephson coupling can become a π junction when its orientation and pairing structure favor opposite signs. A superconducting loop with an odd number of such intrinsic phase shifts can be frustrated and favor spontaneous flux near half a flux quantum in an appropriate large-inductance regime. Tsuei and Kirtley’s tricrystal experiments compared deliberately chosen crystal orientations and control geometries. The diagnostic information is the reproducible relationship between orientation, phase frustration, and flux, not a measurement of an absolute local order-parameter phase.

Magnetic junction mechanisms can also produce π shifts, and trapped ordinary flux or geometric asymmetry can complicate interpretation. Controls and complementary measurements are therefore essential. The waves lab illustrates the sign-reference idea only: Reset and set both controls to phase difference zero and amplitude ratio one, predicting amplitude two. Reset and set phase difference 180 degrees and ratio one, predicting cancellation. Changing a relative sign changes interference while individual amplitudes stay fixed. These are equal-frequency classical waves, not a model of tricrystal tunneling, magnetic screening, or a proof of gap symmetry.