Anisotropic deformation of the 6-state clock model: Tricritical-point classification

Abstract

The two-dimensional $q$-state clock models exhibit the Berezinskii-Kosterlitz-Thouless (BKT) transition for $q\geq5$ since they are a subset of the isotropic XY model. We examine the $6$-state clock model with an anisotropic deformation. Selecting the $6$-state Potts model as a source of the deformation, the model naturally violates the discrete rotational symmetry of the clock model. We introduce the anisotropic deformation parameter $\alpha$ in the clock model interpolating the clock ($\alpha = 1$) and the Potts ($\alpha = 0$) models. We employ the corner transfer matrix renormalization group method to analyze the phase transitions on the square lattice in the thermodynamic limit. Three different phases and phase transitions are identified. The phase diagram is constructed, and we determine a tricritical point at $\alpha_{\rm c} = 0.21405(4)$ and $T_{\rm c} = 0.834017(5)$. Analyzing the latent heat and the entanglement entropy in the vicinity of the $T_{\rm c}(\alpha_{\rm c})$, we observe a single discontinuous phase transition and two BKT phase transitions meeting in the tricritical point. The tricritical point exhibits a phase transition of the second order with the critical exponents $\beta \approx 1/10$ and $\delta \approx 14$. We conjecture that an infinitesimal surrounding of the tricritical point consists of the three fundamental phase transitions, in which the first and the BKT orders gradually weaken into the second-order tricritical point.