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.
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.
We investigate the isotropic-anisotropic phase transition of the twodimensional XY model with six-fold anisotropy, using Monte Carlo renormalization group method. The result indicates difficulty of observing asymptotic critical behavior in Monte Carlo simulations, owing to the marginal flow at the fixed point. PACS numbers: 02.70.Lq, 75.10.Hk Typeset using REVTEX 1 The XY model with Z6 symmetry breaking field represents not only planar spin magnets with Z6 symmetric crystal field [1], but also models with Z2 × Z3 symmetry (which is isomorphic to Z6) such as three-state antiferromagnetic Potts model on the square and the cubic lattice [2] and Ising antiferromagnet on the triangular and the stacked-triangular lattice [3–8]. The critical behavior of the model in two dimension is well understood by scaling argument [1]: The model undergoes two distinct phase transitions, both of which being
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 study the phase diagram and critical behaviour of the two-dimensional square-lattice fully frustrated XY model (FFXY) and of two related models, a lattice discretization of the Landau–Ginzburg–Wilson Hamiltonian for the critical modes of the FFXY model, and a coupled Ising XY model. We present a finite-size-scaling analysis of the results of high-precision Monte Carlo simulations on L × L square lattices, up to L = O (10 3 ). In the FFXY model and in the other models, when the transitions are continuous, there are two very close but separate transitions. There is an Ising chiral transition characterized by the onset of chiral long-range order while spins remain paramagnetic. Then, as temperature decreases, the systems undergo a Kosterlitz–Thouless spin transition to a phase with quasi-long-range order. The FFXY model and the other models, in a rather large parameter region, show a crossover behaviour at the chiral and spin transitions that is universal to some extent. We conjecture that this universal behaviour is due to a multicritical point. The numerical data suggest that the relevant multicritical point is a zero-temperature transition. A possible candidate is the O(4) point that controls the low-temperature behaviour of the 4-vector model.
We present detailed Monte Carlo results for the susceptibility \ensuremath{\chi}, correlation length \ensuremath{\xi}, and specific heat Cv for the XY model. The simulations are done on 642, 1282, 2562, and 5122 lattices over the temperature range 0.98\ensuremath{\le}T\ensuremath{\le}1.43 corresponding to 270. Fits to \ensuremath{\chi} and \ensuremath{\xi} data favor a Kosterlitz-Thouless (KT) singularity over a second-order transition; however, unconstrained four-parameter KT fits do not confirm the predicted values \ensuremath{\nu}=0.5 and \ensuremath{\eta}=0.25. Our best estimate, Tc=0.894(5), is obtained using KT fits with \ensuremath{\nu} fixed at 0.5. The exponent \ensuremath{\eta} is calculated as a function of temperature in the spin-wave phase using Monte Carlo renormalization-group and finite-size-scaling methods. Both methods give consistent results and we find \ensuremath{\eta}\ensuremath{\approxeq}0.235 at T=0.894. We also present results for the behavior of vortex density across the transition and exhibit how the dilute-gas approximation breaks down.
We investigate the dynamical phases and phase transitions arising in a classical two-dimensional anisotropic XY model under the influence of a periodically driven temporal external magnetic field in the form of a symmetric square wave. We use a combination of finite temperature classical Monte Carlo simulation, implemented within a CPU+GPU paradigm, utilizing local dynamics provided by the Glauber algorithm and a phenomenological equation-of-motion approach based on relaxational dynamics governed by the time-dependent free energy within a mean-field approximation to study the model. We investigate several parameter regimes of the variables (magnetic field, anisotropy, and the external drive frequency) that influence the anisotropic XY system. We identify four possible dynamical phases: Ising-SBO, Ising-SRO, XY-SBO, and XY-SRO. Both techniques indicate that only three of them (Ising-SRO, Ising-SBO, and XY-SRO) are stable dynamical phases in the thermodynamic sense. Within the Monte Carlo framework, a finite-size scaling analysis, shows that XY-SBO does not survive in the thermodynamic limit giving way to either an Ising-SBO or a XY-SRO regime. The finite-size scaling analysis further shows that the transitions between the three remaining dynamical phases either belong to the two-dimensional Ising universality class or are first-order in nature. Within the mean-field calculations yield three stable dynamical phases, i.e., Ising-SRO, Ising-SBO and XY-SRO, where the final steady state is independent of the initial condition chosen to evolve the equations of motion, as well as a region of bistability where the system flows to either Ising-SBO or XY-SRO (Ising-SRO) depending on the initial condition. Unlike the stable dynamical phases, the XY-SBO represents a transient feature that is eventually lost to either Ising-SBO or XY-SRO. Our mean-field analysis highlights the importance of the competition between switching of the stationary point(s) of the free energy after each half cycle of the external field and the two-dimensional nature of the phase space for the equations of motion.
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.