This appendix records the compact derivations behind the loop-supercurrent results chapter. The purpose is not to reproduce the research notebooks, but to keep the analytic steps that make the thesis claims checkable.

Phase-Only Threshold

With four equal-amplitude components (\Delta_j=d e^{i\phi_j}) and a uniform phase step (\phi_{j+1}-\phi_j=\theta), the frustrated Josephson part of the free energy is

Stationarity gives

The time-reversal-symmetric branch has (\theta=\pi). The winding branch has

and exists only for (J_d>J/2). The corresponding minimum energy is

Thus the analytic backbone has a continuous phase-only boundary at (J_d=J/2). Microscopic BdG calculations can shift the practical boundary because the amplitudes, quasiparticle spectrum, Hartree/Fock fields, and staggered competitors are no longer frozen.

Relative-Phase Curvature

Expanding about a stationary solution (\theta_0),

with

For the TRS branch,

so the relative-phase mode softens at the same (J_d=J/2) line. For the TRSB branch,

In the full four-phase problem the Hessian has one zero global-(U(1)) mode and three relative-phase modes. The Leggett-response diagnostics in the chapter are the microscopic version of this curvature analysis.

Microscopic Current

The phase-only model defines a coarse Josephson current by differentiating an assumed free energy:

The BdG calculation instead uses the hopping current associated with the normal-state bond:

These currents need not match unless the phase-only coupling has been derived from the same microscopic Hamiltonian and the phases are self-consistently minimized in that Hamiltonian. The Results chapter therefore uses phase-only currents as design intuition and BdG hopping currents for microscopic magnetic response.

Four-Site Block Limit

In the analytically solvable four-site ring with only intracell hopping (t_{\rm intra}), define orbital Fourier modes

The normal-state energies are

For a common pairing magnitude (d), each sector gives a (2\times2) BdG block

with eigenvalues

This block limit is not the full microscopic model. It is the reference that explains why the four-component problem naturally organizes into relative phase sectors before hopping, diagonal frustration, and self-consistency are restored.

Selection-Rule Algebra

For the C3 molecule, the winding phases ((1,\omega,\omega^2)) satisfy

Any central anomalous amplitude proportional to the phase sum is therefore suppressed in the winding branches and finite in the uniform branch. This is the cleanest microscopic selection rule.

For the baseline C4 molecule the central sum is less selective. Both winding ((1,i,-1,-i)) and staggered ((1,-1,1,-1)) patterns can cancel a single central site. Split-centre or screw geometries replace the central penalty by diagonal-channel sums,

so winding can regain a selective energetic advantage over the staggered competitor. This is the algebraic reason the thesis treats C4 as a geometry problem rather than as a trivial extension of C3.