The lattice spin models considered in the present paper consist of three-component unit vectors, associated with a D-dimensional lattice (say d), parameterized by usual spherical angles (θk,k), and interacting via a ferromagnetic potential restricted to nearest neighbours, of the form Wjk = − (sin θj sin θk)p cos (j − k), p , p ≥ 1; here is a positive quantity setting energy and temperature scales. The models were recently introduced, and rigorous comparison inequalities holding for them were investigated, and used to prove the existence of an ordering transition when D = 3 (Romano S. and Zagrebnov V. A., Phys. Lett. A, 301 (2002) 402), investigated by other approximate techniques as well (Mol L. A. S., Pereira A. R. and Moura-Melo W. A., Phys. Lett. A, 319 (2003) 114). We report here an extensive Monte Carlo study of the critical behaviour for D = 3 and p ≤ 4; our results are consistent with the same universality class as the xy model.
The lattice spin models considered in the present paper consist of three-component unit vectors, associated with a D-dimensional lattice (say d), parameterized by usual spherical angles (θk,k), and interacting via a ferromagnetic potential restricted to nearest neighbours, of the form Wjk = − (sin θj sin θk)p cos (j − k), p , p ≥ 1; here is a positive quantity setting energy and temperature scales. The models were recently introduced, and rigorous comparison inequalities holding for them were investigated, and used to prove the existence of an ordering transition when D = 3 (Romano S. and Zagrebnov V. A., Phys. Lett. A, 301 (2002) 402), investigated by other approximate techniques as well (Mol L. A. S., Pereira A. R. and Moura-Melo W. A., Phys. Lett. A, 319 (2003) 114). We report here an extensive Monte Carlo study of the critical behaviour for D = 3 and p ≤ 4; our results are consistent with the same universality class as the xy model.
We have considered a classical spin system, consisting of n-component unit vectors (n = 2, 3), associated with a semi-infinite lattice in one dimension {uk, k ∈ N+}, and interacting via inhomogeneous pair potentials, in general anisotropic in spin space, and of the long-range ferromagnetic form [Formula: see text] here ∊ is a positive constant setting energy and temperature scales (i.e. T* = k B T/∊), a ≥ 0, b ≥ 0, and the symbols uj, α denote cartesian components of the spins. For some specific values of the two parameters a and b, and on the basis of available theoretical results, one can prove the existence of an ordering transition taking place at finite temperature, and obtain rigorous upper and lower bounds on the transition temperatures. This holds, for example, when n = 2, 3, a > b = 0 (the models studied in our previous paper), as well as for n = 2, a = b > 0 and n = 3, b > a = 0, where a continuous O(2) symmetry of the interaction is involved. We have studied these two latter cases by computer simulation, and made comparison with mean-field treatment; simulation results show a broad qualitative similarity between the four models, and a closer, quantitative one, between pairs of models with the same number of spin components, especially for n = 2.
We have considered a classical spin system, consisting of n-component unit vectors (n = 2, 3), associated with a semi-infinite lattice in one dimension {uk, k ∈ N+}, and interacting via inhomogeneous pair potentials, anisotropic in spin space, and of the long-range ferromagnetic form [Formula: see text] here ∊ is a positive constant setting energy and temperature scales (i.e. T* = k B T/∊), a > 0, b ≥ 0, and the symbols uj,α denote Cartesian components of the spins. On the basis of available theoretical results, one can prove the existence of an ordering transition at a finite temperature for n ≥ 2, b = 0, as well as upper and lower bounds on the transition temperatures. We have studied the two cases n = 2, 3, b = 0, by computer simulation, and made a comparison with the Mean Field treatment.
A generalisation of the two-dimensional XY model is studied using Monte Carlo simulation. In this model, two-component classical spins interact both ferromagnetically and nematically. Three phases occur as temperature and the two interaction strengths are varied: a high-temperature, disordered phase, and two low-temperature phases with algebraic correlations in, respectively, the ferromagnetic and nematic order parameters. The critical behaviour at the phase boundaries is examined: the high-temperature phase is entered via Kosterlitz-Thouless transitions, whilst the low-temperature phases are separated by an Ising transition.
We have considered a classical spin system, consisting of 3-component unit vectors, associated with a two-dimensional lattice {uk, k ∈ Z2}, and interacting via a translationally and rotationally invariant pair potential, of the long-range ferromagnetic form [Formula: see text] Here ∊ is a positive constant setting energy and temperature scales (i.e. T* = k B T/∊), and xj denotes dimensionless coordinates of the lattice sites. This potential model is known rigorously to possess an ordering transition at finite temperature, and has been characterized quantitatively by Monte Carlo simulation, whose results suggest a continuous transition taking place at [Formula: see text]; comparison is also reported with other theoretical treatments, such as Spherical Model, Molecular Field and Two Site Cluster approximations.
We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin-1/2 Ising model with nearest-neighbour interactions, a spin-1/2 model with nearest-neighbour and third-neighbour interactions, and a spin-1 model with nearest-neighbour interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbour interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as y t =1.587 (2), y h =2.4815 (15) and y i =-0.82 (6). The universal ratio Q=(m 2 ) 2 /(m 4 ) is equal to 0.6233 (4) for periodic systems with cubic symmetry. The critical point of the nearest-neighbour spin-1/2 model is K c =0.2216546 (10).
In this master thesis the criticality of the 3D XY model with cubic anisotropy has been examined by the means of Monte Carlo simulation. The 3D XY model is a special case of the O(n) spin-vector model which describe numerous phenomena's. Here, n is the number of vector dimensions and for the XY model n=2. The O(n)-model is invariant for global rotations. This means that it is arbitrary which direction in the symmetry breaking occurs. In real system one should always expect perturbations from a perfect system, here it is done by adding a cubic anisotropy (order p=4) then the symmetry of the Hamiltonian is lowered and it will be advantageous for the spins to be ordered in directions 90o apart. This symmetry is called C4, which is a part of the cyclic group, and it restricts the symmetry breaking to one of these four directions. \n\nThe criticality of this model is when the spins suddenly align as the temperature is lowered to a critical temperature. Sufficiently close to the critical temperature quantities like the magnetization, which is the sum of spins, will show a power law growth characterised by critical exponents specified by the class of universality of the transition. The critical exponents can be obtained by finite-size scaling. By applying renormalization theory on the critical region one can show that some of parameters in the Hamiltonian are: relevant and effect the scaling, irrelevant and does not effect the scaling and dangerously irrelevant meaning that they only affect some of the critical exponents. Previous Monte Carlo and renormalization calculus show that anisotropy which is relevant for p<4 becomes dangerously irrelevant for pgeq4. The question of interest is what kind of effect the dangerously irrelevant parameter in the Hamiltonian has on the scaling. This is interesting because the effect of dangerously irrelevant parameters is hardly studied and anisotropy arise in many crystal systems because of the lattice geometry. \n\nThe results are obtained by Monte Carlo simulations and the critical slowing down close to the phase transition is solved by the method of parallel tempering. This method was chosen because it is simple to implement in a parallel code.\n\nThe Binder cumulant for different quantities and the peak of the magnetic susceptibility was used support a critical temperature Tc=2.2019textnormalk for no field applied and no anisotropy. The known critical exponents for the no field and no anisotropy was used. The scaling made a good collapse, meaning that the right Tc and the correct critical exponents where used. The scaling did not take into account a magnetic field and therefore a good collapse of curves where not obtained in this case. When the cubic anisotropy was added the critical exponent for the magnetic susceptibility and the cubic order parameter changed, while the other exponents where unchanged. This supports the view that cubic anisotropy is dangerously irrelevant for the 3D XY model. The scaling of the magnetic susceptibility showed a possible new kind of scaling, a scaling which is different on the two sides of the transition. Although a more accurate determination of the critical exponent in the ordered phase require more extensive simulations. It appears that this exponent is larger than the exponent in the disordered phase, where one recovered the isotropic exponent. The difference in the critical exponents has been predicted to increase with p and an investigation of the hexagonal anisotropy p=6 has been started.