A Bond weighted tensor renormalization group study of the q-state ferromagnetic Potts models on the square lattice

It is known rigorously that the phase transition of the $q$-state ferromagnetic Potts model on the square lattice is second order for $q=4$. Despite this fact, some observables of the $q=4$ model show features of a first-order phase transition. For example, negative peak appears for the quantity of Binder ratio $Q_2$ of this model. Such a non-monotonic behavior of $Q_2$ is typically a consequence of phase coexistence, hence is served as a signal of a first-order phase transition. In particular, the negative peak should diverge with linear system size $L$ squared. Since the mentioned divergence phenomenon is not observed for the 4-state Potts model, the scenario of a first-order phase transition for this model is ruled out. Interestingly, a recent large scale Monte Carlo investigation of the 4-state Potts model observes that the two-peak structure of the energy density distribution becomes more noticeable when $L$ increases. This finding indicates the signal of coexistence of phases is getting stronger with $L$. Due to these unusual critical behaviors, here we study the energy density $E$ and the specific heat $C_v$ of the 4-state Potts model on the square lattice using the technique of bond weighted tensor renormalization group (BWTRG). For a comparison purpose, $q=2$ and $q=5$ ferromagnetic Potts models on the square lattice are investigated using the same method as well. Remarkably, our results do imply there may be a small energy gap for $q=4$ model. While the appearance of the mentioned small energy gap can be explained plausibly and it will disappear with a more sophisticated investigation, our finding suggests that whether a message of a first-order phase transition is genuine or is an artificial effect requires further and detailed investigations.