  We study the famous example of weakly first order phase transitions in the
1+1D quantum Q-state Potts model at Q>4. We numerically show that these weakly
first order transitions have approximately conformal invariance. Specifically,
we find entanglement entropy on considerably large system sizes fits perfectly
with the universal scaling law of this quantity in the conformal field theories
(CFTs). This supports that the weakly first order transitions is proximate to
complex fixed points, which are described by recent conjectured complex CFTs.
Moreover, the central charge extracted from this fitting is close to the real
part of the complex central charge of these complex CFTs. We also study the
conformal towers and the drifting behaviors of these conformal data (e.g.,
central charge and scaling dimensions).
