Unveiling chiral phases: Finite-size scaling as a probe of quantum phase transition in symmetry-enriched $c=1$ conformal field theories

Abstract

We study the low-energy properties of the chiral Heisenberg chain, namely, a one-dimensional spin-1/2 isotropic Heisenberg chain with time-reversal symmetry-breaking pseudo-scalar chiral interaction. We employ the thermodynamic Bethe ansatz to find "chiralization", the response of the ground state versus the strength of the chiral interaction of a chiral Heisenberg chain. Unlike the magnetization case, the chirality of the ground state remains zero until the transition point corresponding to critical coupling $\alpha_c=2J/\pi$ with $J$ being the antiferromagnetic spin-exchange interaction. The central-charge $c=1$ conformal field theories (CFTs) describe the two phases with zero and finite chirality. We suggest that the difference lies in the symmetry of their ground state (lightest weight) primary fields, i.e., the two phases are symmetry-enriched CFTs. At finite but small temperatures, the non-chiral Heisenberg phase acquires a finite chirality that scales with the temperature quadratically. We show that the finite-size effect around the transition point probes the transition.