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Physical review. E

Binder ratio in the two-dimensional q -state clock model

Luong Minh Tuan, Ta Thanh Long, Duong Xuan Nui, Pham Tuan Minh, Nguyen Duc Trung Kien, Dao Xuan Viet

Published Sep 30, 2022• 16 citations
Publisher

Abstract

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

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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.
We study the phase transition properties of the two-dimentional q-state clock model by an extensive Monte Carlo simulation. Several independent physical quantities, including the specific heat, the magnetization and the helicity modulus had been calculated for q > 5. The specific heat and the magnetization shows the behavior of two distinct phase transitions, including the higher phase transition, T2, between the disordered phase and the quasi-long-range ordered phase and the lower phase transition, T1, between the quasi-long-range ordered phase and the quasi-long-range ordered phase. The helicity modulus not only show that T2 is the Kosterlitz-Thouless phase transition but also has signature of phase transition around T1.
The two-dimensional S=1/2S=1/2S=1/2 XY model is investigated with an extensive quantum Monte Carlo simulation. The helicity modulus is precisely estimated through a continuous-time loop algorithm for systems up to 128×128128 \times 128128×128 near and below the critical temperature. The critical temperature is estimated as TKT=0.3427(2)JT_{\rm KT} = 0.3427(2)JTKT​=0.3427(2)J. The obtained estimates for the helicity modulus are well fitted by a scaling form derived from the Kosterlitz renormalization group equation. The validity of the Kosterlitz-Thouless theory for this model is confirmed.
The critical behavior of the clock model in two-dimensional square lattice is studied numerically using Monte Carlo method with Wolff algorithm. The Kosterlitz-Thouless (KT) transition is observed in the 8-state clock model, where an intermediate phase exists between the low-temperature ordered phase and the high-temperature disordered phase. The bond randomness is introduced to the system by assuming a Gaussian distribution for the coupling coefficients with the mean μ = 1 and different values of variance: from σ2 = 0.1 to σ2 = 3.0. An abrupt jump in the helicity modulus at the transition, which is the key characteristic of the KT transition, is verified with a stability argument. Our results show that, a small amount of disorder (small σ) reduces the critical temperature of the system, without altering the nature of transition. However, a larger amount of disorder changes the transition from the KT-type into that of non-KT-type.
The critical behavior of the clock model in two-dimensional square lattice is studied numerically using Monte Carlo method with Wolff algorithm. The Kosterlitz-Thouless (KT) transition is observed in the six-state clock model, where an intermediate phase exists between the low-temperature ordered phase and the high-temperature disordered phase. The bond randomness is introduced to the system by assuming a Gaussian distribution for the coupling coefficients with the mean μ=1 and different values of variance, from σ2=0.1 to σ2=3.0. An abrupt jump in the helicity modulus at the transition, which is the key characteristic of the KT transition, is verified with a stability argument. The critical temperature Tc for both pure and disordered systems is determined from the critical exponent η(Tc)=1/4. The results showed that a small amount of disorder (small σ) reduces the critical temperature of the system, without altering the nature of transition. However, a larger amount of disorder changes the transition from the KT-type into that of non-KT-type.
The two Berezinskii–Kosterlitz–Thouless phase transitions of the two-dimensional 5-state clock model are studied on infinite strips using the DMRG algorithm. Because of the open boundary conditions, the helicity modulus ϒ 2 is computed by imposing twisted magnetic fields at the two boundaries. Its scaling behavior is in good agreement with the existence of essential singularities with σ = 1/2 at the two transitions. The predicted universal values of ϒ 2 are shown to be reached in the thermodynamic limit. The fourth-order helicity modulus is observed to display a dip at the high-temperature BKT transition, like the XY model, and shown to take a new universal value at the low-temperature BKT transition. Finally, the scaling behavior of magnetization at the low-temperature transition is compatible with η = 1/4.
We performed Monte Carlo simulation for a two-dimensional generalized XY model and calculated the magnetic and nematic Binder parameters. The phase diagram is re-examined based on these Binder parameters, demonstrating their power in studying generalized XY models. The Binder parameter has distinctive behaviors, resulting in different types of phase transition. More importantly, the magnetic Binder parameter gives more insights into the tricritical region where the Kosterlitz-Thouless, Ising, and 1/2 Kosterlitz-Thouless (KT) transition lines meet. It shows signatures for the intermediate region starting from the tricritical point, where the transition line is neither the same physics as the Ising transition below nor the KT transition far above the tricritical point.