Microscopic study of a complex conformal theory in the O(n) loop model in 2D

The presence of nearby conformal field theories (CFTs) hidden in the complex plane of the tuning parameter was recently proposed as an elegant explanation for the ubiquity of "weak" first order transitions in condensed matter and high-energy systems. However, studying such complex CFTs is even more challenging than for real CFTs, and few results exist about them. In this work, we present an exact microscopic study of a complex CFT (CCFT) for a loop formulation of the O(n) model in 2D. Thanks to a numerical finite-size study of the transfer matrix, we locate the CCFT in the complex plane and extract the real and imaginary part of the central charge and scaling dimensions. By comparing those with the analytic continuation of predictions from Coulomb gas techniques, we determine the range of validity of the analytic continuation to extend up to n∼13, beyond which the CCFT disappears. Finally, we propose an approximate real-space renormalization group model which reproduces the main features of the spiral RG flow around the CCFTs.