PHYSICAL REVIEW E 106, 034138 (2022) Binder ratio in the two-dimensional q-state clock model Luong Minh Tuan , 1, 2 Ta Thanh Long , 1 Duong Xuan Nui , 1, 3 Pham Tuan Minh, 4 Nguyen Duc Trung Kien, 1 and Dao Xuan Viet 1 1 Advanced Institute for Science and Technology, Hanoi University of Science and Technology, Hanoi 10000, Vietnam 2 Faculty of Mechanical Engineering, National University of Civil Engineering, Hanoi 10000, Vietnam 3 Faculty of Electromechanical and Civil Engineering, Vietnam National University of Forestry, Hanoi 10000, Vietnam 4 Institute of Physics, Vietnam Academy of Science and Technology, Hanoi 10000, Vietnam (Received 1 May 2021; revised 24 July 2022; accepted 12 September 2022; published 30 September 2022) We study phase transition properties of the two-dimensional q-state clock model by an extensive Monte Carlo simulation. By analyzing the Binder ratio and its temperature derivative, we confirm that the two-dimensional q-state clock model exhibits two distinct Kosterlitz-Thouless phase transitions for q = 5, 6 but it has one second- order phase transition for q = 4. The critical temperatures are estimated quite accurately from the crossing behavior of the Binder ratio (for q < 5) and from negative divergent dips of the derivative of the Binder ratio (for q  5) around these critical points. We also calculate the correlation length, the helicity modulus, and the derivative of the helicity modulus, and analyze their behaviors in different phases in detail. DOI: 10.1103/PhysRevE.106.034138 I. INTRODUCTION The two-dimensional (2D) q-state clock model has been studied extensively in phase transition phenomena for years [1–9]. It is a generalization of the 2D Ising model for q = 2, but it approaches the 2D XY model in the limit q →∞. The 2D Ising model exhibits a second-order phase transition be- tween the long-range ordered phase and the disordered phase. For the 2D XY model, in early days it was indicated that there is no sign of a phase transition [10]. But then it was proved to have the Kosterlitz-Thouless (KT) phase transition between the quasi-long-range ordered phase and the disordered phase [11]. The KT phase transition is one of the most important concepts in statistical physics. The KT phase transition is observed in many 2D systems, such as a planar array of coupled Josephson junctions in a transverse magnetic field [12], a liquid crystal [13], and a 2D Coulomb crystal [14]. Interestingly, this type of phase transition also appears in the 2D q-state clock model, depending on the number of single spin states q. It was theoretically predicted [1] and later confirmed by several numerical works [2–4] that the 2D q-state clock model has only one second-order phase transition at T c for q < 5 and two distinct KT phase transitions at finite critical tempera- tures T 1 and T 2 (T 2 > T 1 ) for q  5. The intermediate phase between these critical temperatures is a quasi-long-range or- dered phase like that of the 2D XY model. The phase above T 2 is a disordered phase, and the phase below T 1 is a long-range ordered phase [2]. Recently, Monte Carlo simulation studies of phase tran- sitions in the 2D q-state clock model have mainly tried to clarify q c , the critical value of the boundary between the KT transition type (for q  q c ) and the non-KT transition type (for q < q c ). From the behavior of the helicity modulus and the fourth-order helicity modulus, Lapilli et al. [15] claimed that the KT transitions occur only for q  8. Based on the helicity modulus and its temperature derivative, Baek et al. [6] concluded that the transitions are KT type for q  6. However, more recent studies [8,9,16] are in favor of the previous sce- nario where q c = 5. Kumano et al. [8] and Chatelain et al. [16] demonstrated that the 2D five-state clock model exhibits two KT phase transitions via the behavior of the discrete helicity modulus. Surungan et al. [9] approached in a different way: they calculated correlation lengths for both the 2D q-state clock model and the Villain model. By comparing the be- havior of correlation lengths of these two models, they also argued about having two KT transitions for q = 5. The phase transition type of the model in case of q = 5 is controversial because its different physical quantities lead to different re- sults of q c . For q = 5, while the helicity modulus shows the behavior of non-KT phase transitions [6], the discrete helicity modulus [8] and the correlation length [9] show the behavior of KT phase transitions, but it seem to be unclear because of the narrow intermediate phase. In order to clarify further the type of phase transitions in the 2D q-state clock model, we perform an extensive Monte Carlo simulation but focus on q = 4, 5, and 6. We calculate several independent physical quantities including the Binder ratio, the derivative of the Binder ratio, the correlation length ratio, the helicity modulus, and the derivative of the helicity modulus. II. MODEL AND METHODS The q-state clock model in a square lattice is defined by the Hamiltonian H =−J  〈ij 〉 cos(θ i − θ j ), (1) 2470-0045/2022/106(3)/034138(8) 034138-1 ©2022 American Physical Society