Level spectroscopy of the square-lattice three-state Potts model with a   ferromagnetic next-nearest-neighbor coupling
  We study the square-lattice three-state Potts model with the ferromagnetic next-nearest-neighbor coupling at finite temperature. Using the level-spectroscopy method, we numerically analyze the excitation spectrum of the transfer matrices and precisely determine the global phase diagram. Then we find that, contrary to a previous result based on the finite-size scaling, the massless region continues up to the decoupling point with ${\bf Z}_3\times{\bf Z}_3$ criticality in the antiferromagnetic region. We also check the universal relations among excitation levels to provide the reliability of our result. 
Hiromi Otsuka, Koutaro Mori, Yutaka Okabe, and Kiyohide Nomura