Asymptotic behavior of generic Tomonaga-Luttinger liquid in the vicinity of phase-separated regions is known to produce an instability where well-known relativistic conformal field theory (CFT) techniques fail. In this paper, we introduce an analytic paradigm that provides a continuum description of this important issue. We show that there is an emergent Carrollian symmetry when phase separation is reached, and techniques of Carroll CFT, as opposed to its relativistic relative, are central to the understanding of the physics. We work with the analogous spinless fermionic system in this region and capture the transition across this phase separation. Our numerical results corroborate the density-density correlations intrinsically computed using Carroll CFT. We further test the framework in a number of lattice systems, namely the spinless and spinfull fermionic models with distinct microscopic content, and find the same scaling at the transition. We discuss the scope of the framework and broader perspective.

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• Conformal field theory
• Critical phenomena
• Lower-dimensional field theories
• Quantum criticality
• Symmetries
• Density matrix methods
• Luttinger liquid model