Hidden Critical Points in the Two-Dimensional $O(n>2)$ model: Exact Numerical Study of a Complex Conformal Field Theory

TL;DR

This paper investigates a complex conformal field theory in the 2D $O(n)$ model, revealing hidden fixed points in the complex plane that explain the absence of symmetry-breaking for $n>2$ and mapping the phase diagram.

Contribution

It provides the first exact numerical study of a complex CFT in the 2D $O(n)$ model, confirming the existence of hidden fixed points and analyzing their properties.

Findings

Confirmed the presence of a complex CFT in the $O(n)$ model

Extracted real and imaginary parts of central charge and scaling dimensions

Determined the validity range of analytic continuation up to $n_g \\approx 12.34$

Abstract

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 "weakly first-order" transitions in condensed matter and high-energy systems. In this work, we perform an exact microscopic study of such a complex CFT (CCFT) in the two-dimensional $O(n)$ loop model. The well-known absence of symmetry-breaking of the $O(n>2)$ model is understood as arising from the displacement of the non-trivial fixed points into the complex temperature plane. Thanks to a numerical finite-size study of the transfer matrix, we confirm the presence of a CCFT in the complex plane and extract the real and imaginary parts 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 $n_g \approx 12.34$, beyond which the CCFT gives way to a gapped state. Finally, we propose a beta function which reproduces the main features of the phase diagram and which suggests an interpretation of the CCFT as a liquid-gas critical point at the end of a first-order transition line.