The presence of nearby complex conformal field theories (CCFTs) 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. Recently, we have numerically confirmed the presence of such a CCFT in a loop model which derives from a high-temperature formulation of the O(n) model. Surprisingly, we found that the CCFT only survives until n=12.34, beyond which the transfer matrix acquires a gap. In this talk, I will discuss ongoing work in trying to explain this loss of complex conformality at large n, using a mapping to a hard hexagon model for n going to infinity. I will also discuss the connection between the original O(n) model and its loop version and the consequences for CCFTs in these models.
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