Superconductivity in the Twisted Sector: Pairing Channels, Phase Solitons and Defects

The Möbius Klein Transformation (MKT) splits fields on a space with a twist into the untwisted and the twisted sectors of a double cover. It does not set the coupling, the energy scale or T_c, and it does not raise T_c. Under the antiferromagnetic momentum shift, an s-wave gap is untwisted and a sign-changing gap is twisted. The linearized gap operator commutes with this shift exactly when the pair susceptibility does, and it then splits into 2 blocks. A repulsion that is non-negative on each sublattice leaves no attraction in the untwisted block. With 2 shifts, fourfold rotation and a band of next-nearest-neighbour hopping only at half filling, an extended s and a d channel form an exactly degenerate even-parity doublet. A π twist of the pair phase costs the stiffness at zero temperature; in 2 dimensions 2 half-integer windings lower it by T ln 2. For bipolarons the momentum shift by π does not select the light pair. On a Möbius strip only the transverse-odd subband is antiperiodic, and the T_c difference to an annulus is its odd-winding part in a Poisson sum. It reverses sign at 1 superconducting flux quantum, whatever its sign at zero flux, and in a ballistic aluminium strip it decays over about 4 micrometres. Born pair breaking by disorder splits exactly into an anisotropy term and a term of the shifted disorder correlator. In stripe order a product rule cancels the 1st-order Josephson coupling of adjacent and 3rd-neighbour layers. A ring driven with a time screw fixes the pair momentum of each harmonic of the pair amplitude. A Möbius Josephson ring returns a fluxon as an antifluxon, forces a fluxon above a critical length only for an odd fluxoid number, and carries no dc flux flow. Deterministic scripts check the computed statements.