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Results in Physics

Critical properties in the two-dimensional XY model with four-fold and eight-fold crystal fields

Truong Thi Bach Yen, Duong Xuan Nui, Lê Thị Ngọc Tú, Nguyễn Trí Tuấn, Nguyen Duc Dung, Dao Xuan Viet

Published Oct 7, 2025• 1 citation
Publisher

Abstract

We numerically investigate the critical properties of the two-dimensional XY model with four-fold and eight-fold crystal fields. Using extensive Monte Carlo simulations, we compute the specific heat, magnetization, orientational magnetization, and correlation length ratio to examine how these anisotropies influence the nature of phase transitions. The critical temperature and the critical exponent are systematically mapped across both compatible and competing field regimes, spanning from weak to strong field strengths. The numerical results reveal that the critical exponent exhibits a linear dependence on the inverse of the four-fold field strength in the strong field regime, while remaining finite and non-divergent in the weak field regime. Furthermore, the orientational magnetization effectively identifies and characterizes Ising-type phase transitions in the competing field regime.

DOI
10.1016/j.rinp.2025.108474

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We numerically investigate the critical properties of the two-dimensional XY model with four-fold and eight-fold crystal fields. Using extensive Monte Carlo simulations, we compute the specific heat, magnetization, orientational magnetization, and correlation length ratio to examine how these anisotropies influence the nature of phase transitions. The critical temperature and the critical exponent are systematically mapped across both compatible and competing field regimes, spanning from weak to strong field strengths. The numerical results reveal that the critical exponent exhibits a linear dependence on the inverse of the four-fold field strength in the strong field regime, while remaining finite and non-divergent in the weak field regime. Furthermore, the orientational magnetization effectively identifies and characterizes Ising-type phase transitions in the competing field regime.
Early theoretical studies of the two-dimensional XY model with four-fold anisotropy (XY h 4 ) predicted that any finite anisotropy should completely suppress Kosterlitz–Thouless (KT) transition of the isotropic system and replace it with a second-order phase transition. This prediction is inconsistent with Monte Carlo (MC) simulations, which suggested KT-like behavior at weak anisotropy. To resolve this discrepancy, we performed large-scale MC simulations to systematically examine the XY h 4 model across a wide range of anisotropic strengths. Our results indicate that while weak anisotropy can mimic KT behavior in small systems, a detailed finite-size scaling analysis of larger systems reveals that the temperature derivative of Binder ratio and the correlation length ratio displays scaling behaviors characteristic of a second-order transition even at the weak h 4 values. The critical exponent ν is non-universal and varies continuously with h 4 from weak to strong anisotropy. These findings confirm that XY h 4 model exhibits a second-order phase transition for weak finite anisotropy, effectively reconciling theoretical predictions with numerical observations.
We compare the critical properties of the two-dimensional (2D) XY\mathrm{XY}XY model in a transverse magnetic field with filling factors f=1/3f\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/3f=1/3 and 2/52/52/5. To obtain a comparison with recent experiments, we investigate the effect of weak quenched bond disorder for f=2/5f\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2/5f=2/5. A finite-size scaling analysis of extensive Monte Carlo simulations strongly suggests that the critical exponents of the phase transition for f=1/3f\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/3f=1/3, and for f=2/5f\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2/5f=2/5 with disorder, are those of the pure 2D Ising model. Studying the possible domain walls in the system provides some explanations for our results.
We study the critical properties of the two-dimensional (2D) XY\mathrm{XY}XY model in a transverse magnetic field with filling factors f=1/3f=1/3f=1/3 and 2/5.2/5.2/5. To obtain a comparison with recent experiments, we investigate the effect of weak quenched bond disorder for f=2/5.f=2/5.f=2/5. A finite-size scaling analysis of extensive Monte Carlo simulations strongly suggests that the critical exponents of the phase transition for f=1/3f=1/3f=1/3 and for f=2/5f=2/5f=2/5 with disorder are those of the pure 2D Ising model. The relevant low-energy excitations are domain walls, and we show that their properties determine the nature of the phase transition.
This Thesis addresses the thermodynamic behaviour of two-dimensional mag nets. The objective is to evaluate the role played by external perturbations in these systems by use of analytical and computational techniques. To this end, extensive Monte Carlo simulations have been performed on Ising, XY and Heisenberg spin models in the presence of symmetry breaking crystal fields, external magnetic fields and long range dipolar interactions. The effects of four-fold symmetry are examined in detail, since this is common in nature. In the 2dXY model, such a crystal field acts as a marginal perturbation which drives the system to non-universal critical behaviour only upon reseating to exponentially large length scales. This has the implication that real, finite systems are unexpectedly robust to their presence. On the other hand, the four-state clock model is recovered in the limit of extremely strong fields, which exhibits Ising critical behaviour. The nature of this crossover from XY to Ising behaviour is studied in detail, using numerical and analytical methods. The picture that emerges from this analysis is that there are two principal categories of two-dimensional magnets: those belonging to the so-called Ising uni versality class and those belonging to the XY class. A detailed survey of the experimental literature confirms this classification. However, a considerable num ber of experimental systems also exhibit behaviour wrhich is intermediate between these two regimes. It is argued these materials are susceptible to the crossover mechanism induced by the presence of four-fold fields.
We present results of numerical studies of the two-dimensional XY model with four- and eightfold symmetry-breaking fields. This model has recently been shown to describe hydrogen-induced reconstruction on the W(100) surface. Based on mean-field and renormalization-group arguments, we first show how the interplay between the anisotropy fields can give rise to different phase transitions in the model. When the fields are compatible with each other there is a continuous phase transition when the fourth-order field is varied from negative to positive values. This transition becomes discontinuous at low temperatures. These two regimes are separated by a multicritical point. In the case of competing four- and eightfold fields, the first-order transition at low temperatures opens up into two Ising transitions. We then use numerical methods to accurately locate the position of the multicritical point, and to verify the nature of the transitions. The different techniques used include Monte Carlo histogram methods combined with finite-size scaling analysis, the real-space Monte Carlo renormalization-group method, and the Monte Carlo transfer-matrix method. Our numerical results are in good agreement with the theoretical arguments.
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