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

TL;DR

This study investigates the phase transitions and tricritical point in an anisotropically deformed 6-state clock model, revealing a complex phase diagram with BKT and second-order transitions using advanced numerical methods.

Contribution

It introduces an anisotropic deformation parameter interpolating between the clock and Potts models and characterizes the tricritical point with detailed critical exponents.

Findings

Identified three phases and phase transitions in the model.

Located the tricritical point at specific deformation and temperature values.

Observed a combination of first-order, BKT, and second-order transitions near the tricritical point.

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.