Extensively degenerate superradiant phase in a giant-atom quantum Rabi ring

We show that an $N$-site quantum Rabi ring with matched giant-atom couplings has $2^N$ degenerate superradiant configurations as the atomic-to-cavity frequency ratio $\eta =\mathrm{\Delta }\mathrm{/}\omega$ tends to infinity. Each atom couples to two cavities two sites apart, and the photon hopping carries a phase $\pi \mathrm{/}2$. Tuning the ratio of contact strengths to the photon hopping reduces the mean-field energy to a sum of single-site Rabi energies, with a common critical coupling for all momentum-pair sectors. The degeneracy survives quadratic quantum fluctuations despite a dispersive excitation spectrum. At finite $\eta$, second-order processes involving the local cubic anharmonicity generate a same-sublattice Ising coupling of order $\omega \mathrm{/}\eta$, selecting ferro- and antiferro-superradiant configurations. This quantum order-by-disorder effect has no classical counterpart. Four-site calculations retaining all spin states quantitatively support the predicted energy coefficients. The separation between the induced coupling and the excitation gap allows a broad temperature interval with macroscopic photon occupation and short-range correlations between displacement signs.