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SCIENTIFIC JOURNAL OF TAN TRAO UNIVERSITY

Mô đun xoắn trong mô hình q-state clock

Tuấn Lương Minh, Long Ta Thanh, Nui Duong xuan, Việt Đào Xuân, Duc Nguyen Ba, Kiên Nguyễn Đức Trung

Published Apr 16, 2021• 0 citations
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Abstract

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.

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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.
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 have examined the critical properties of the q=6 clock (vector Potts) model in two dimensions through Monte Carlo simulations. The model was investigated on L\ifmmode\times\else\texttimes\fi{}L square lattices of size L=4 to L=72 with periodic boundary conditions. We found an intermediate XY-like phase between a low-temperature ordered phase and a high-temperature disordered phase. The phase transitions occur at kB{k}_{B}kB​T1{T}_{1}T1​/J=0.68\ifmmode\pm\else\textpm\fi{}0.02 and kB{k}_{B}kB​T2{T}_{2}T2​/J=0.92\ifmmode\pm\else\textpm\fi{}0.01 and are of the Kosterlitz-Thouless type. The susceptibility diverges in the intermediate phase and the exponent \ensuremath{\eta} varies between 0.100 at T1{T}_{1}T1​ and 0.275 at T2{T}_{2}T2​.
Using Monte Carlo simulation, the one dimensional N-state clock model with long range inverse square interaction has been studied. In the case of N = 6, the temperature dependence of the specific heat shows two peaks at T1 and T2 (< T1). The critical exponent η calculated from the system size dependence of magnetization changes with temperature in the region between T1 and T2. The behaviors are very similar to those of the two-dimensional 6-state clock model with the nearest neighbor interaction which shows two steps of phase transition, Berezinskii-Kosterlitz-Thouless transition and ordinary long range order. We have also confirmed that the temperature T2 shift toward zero with the increase of N.
From consideration of the order-parameter distribution, we propose an observable which makes a clear distinction between true and quasi-long-range orders in the two-dimensional generalized q-state clock model. Measuring this quantity by Monte Carlo simulations for q=8, we construct a phase diagram and identify critical properties across the phase-separation lines among the true long-range order, quasi-long-range order, and disorder. Our result supports the theoretical prediction that there appears a discontinuous order-disorder transition as soon as the two phase-separation lines merge.
We study the phases and phase transitions of a disordered one-dimensional quantum qqq-state clock Hamiltonian using large-scale Monte Carlo simulations. Making contact with earlier results, we confirm that the clean, translational invariant version of the model, for q=6q=6q=6, hosts an intermediate emergent quasi-long-range ordered (QLRO) phase between the symmetry-broken true long-range ordered (TLRO) phase and the disordered (paramagnetic) phase. With increasing disorder strength, the quasi-long-range ordered phase shrinks and finally vanishes at a multicritical point, beyond which there is a direct transition from the TLRO phase to the paramagnetic phase. After establishing the phase diagram, we characterize the critical behaviors of the various quantum phase transitions in the model. We find that weak disorder is an irrelevant perturbation of the Berezinskii-Kosterlitz-Thouless transitions that separate the QLRO phase from the TLRO and paramagnetic phases. For stronger disorder, some of the critical exponents become disorder-dependent already before the system reaches the multicritical point. We also show that beyond the multicritical point, the direct transition from the TLRO phase to the paramagnetic phase is governed by an infinite-randomness critical point in line with strong-disorder renormalization group predictions. While our numerical results are for q=6q=6q=6, we expect the qualitative features of the behavior to hold for all q>4q>4q>4.
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.