We show that including pairing and repulsion into the description of one-dimensional spinless fermions, as in the domain wall theory of commensurate melting or the interacting Kitaev chain, leads, for strong enough repulsion, to a line of critical points in the eight-vertex universality class terminating floating phases with emergent $U\left(1\right)$ symmetry. For nearest-neighbor repulsion and pairing, the variation of the critical exponents along the line that can be extracted from Baxter's exact solution of the $XYZ$ chain at $J_x=-J_z$ is fully confirmed by extensive density matrix renormalization group (DMRG) simulations of the entire phase diagram, and the qualitative features of the phase diagram are shown to be independent of the precise form of the interactions.

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• Quantum phase transitions
• 1-dimensional spin chains
• Lattice models in statistical physics
• Superconducting qubits
• Critical exponents
• Density matrix renormalization group
• Kitaev model
• XYZ model