Lattice realization of complex CFTs: Two-dimensional Potts model with $Q>4$ states

The two-dimensional $Q$-state Potts model with real couplings has a first-order transition for $Q>4$. We study a loop-model realization in which $Q$ is a continuous parameter. This model allows for the collision of a critical and a tricritical fixed point at $Q=4$, which then emerge as complex conformally invariant theories at $Q>4$, or even complex $Q$, for suitable complex coupling constants. All critical exponents can be obtained as analytic continuation of known exact results for $Q \le 4$. We verify this scenario in detail for $Q=5$ using transfer-matrix computations.