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We construct a class of solvable models for 2+1D quantum critical points byattaching 1+1D conformal field theories (CFTs) to fluctuating domain wallsforming a ``loop soup''. Specifically, our local Hamiltonian attaches gaplessspin chains to the domain walls of a triangular lattice Ising antiferromagnet.The macroscopic degeneracy between antiferromagnetic configurations is split bythe Casimir energy of each decorating CFT, which is usually negative and thusfavors a short loop phase with a finite gap. However, we found a set of 1D CFTHamiltonians for which the Casimir energy is effectively positive, making itfavorable for domain walls to coalesce into a single ``snake'' which ismacroscopically long and thus hosts a CFT with a vanishing gap. The snakeconfigurations are geometrical objects also known as fully-packed self-avoidingwalks or Hamiltonian walks which are described by an $\mathrm{O}(n=0)$ loopensemble with a non-unitary 2+0D CFT description. Combining this descriptionwith the 1+1D decoration CFT, we obtain a 2+1D theory with unusual criticalexponents and entanglement properties. Regarding the latter, we show that the$\log$ contributions from the decoration CFTs conspire with the spatialdistribution of loops crossing the entanglement cut to generate a ``non-localarea law''. Our predictions are verified by Monte Carlo simulations.

Solvable models for 2+1D quantum critical points: Loop soups of 1+1D conformal field theories