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 translationally invariant pair potentials, isotropic in spin space, and of the long-range form [Formula: see text] Here ∊ is a positive constant setting energy and temperature scales (i.e. T*=k B T /∊), P2 denotes the second Legendre polynomial, and xj are dimensionless coordinates of the lattice sites. Available theorems entail the existence of an ordering transition at finite temperature when 0 < σ < 2, and its absence when σ ≥ 2. We have studied the border case σ=2, by means of computer simulation. Similarly to the nearest-neighbour counterpart of the present model, and to other long-range models, we found evidence suggesting a transition to a low-temperature phase with slow decay of correlations and infinite susceptibility, i.e. a Berezhinski[Formula: see text]–Kosterlitz–Thouless-like transition; the transition temperature was estimated to be Θ=1.112 ± 0.005.
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 translationally invariant pair potentials, isotropic in spin space, and of the long-range form [Formula: see text] Here ∊ is a positive constant setting energy and temperature scales (i.e. T*=k B T /∊), P2 denotes the second Legendre polynomial, and xj are dimensionless coordinates of the lattice sites. Available theorems entail the existence of an ordering transition at finite temperature when 0 < σ < 2, and its absence when σ ≥ 2. We have studied the border case σ=2, by means of computer simulation. Similarly to the nearest-neighbour counterpart of the present model, and to other long-range models, we found evidence suggesting a transition to a low-temperature phase with slow decay of correlations and infinite susceptibility, i.e. a Berezhinski[Formula: see text]–Kosterlitz–Thouless-like transition; the transition temperature was estimated to be Θ=1.112 ± 0.005.
We have considered classical spin systems, consisting of n-component unit vectors (n=2,3), associated with a one-dimensional lattice {uk, k ∈ Z}, and interacting via translationally and rotationally invariant pair potentials, of the long-range and ferromagnetic form [Formula: see text] here ε is a positive constant setting energy and temperature scales (i.e. T*=k B T/ε). Available theorems entail the existence of an ordering transition at finite temperature when 0<σ<1, and its absence for σ≥1. When σ=1, spin-wave arguments and previous simulation results suggest the existence of a transition to a low-temperature phase with slow decay of correlations and infinite susceptibility, i.e. a Berezhinskiǐ–Kosterlitz–Thouless-like transition. We have reexamined this case by a more detailed simulation study, and especially addressed the sample-size dependence of calculated observables: simulation results suggest the existence of such a transition, and the proposed estimates for the transition temperatures are Θ2=0.775±0.025 and Θ3=0.500±0.025, when n=2 and n=3, respectively.
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 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.
We have considered a classical lattice-gas model, consisting of a two-dimensional lattice Z2, each site of which hosts at most one two-component unit vector; particles occupying pairs of nearest-neighbouring sites interact via the ferromagnetic potential [Formula: see text] where νj=0,1 denotes occupation numbers, uj are the unit vectors (classical spins) and ∊ is a positive constant setting energy and temperature scales; the total Hamiltonian is given by [Formula: see text] where ∑{j<k} denotes sum over all distinct nearest-neighbouring pairs of lattice sites. The saturated-lattice version of this model, where all sites are occupied, supports the well-known Berezhinskiĭ–Kosterlitz–Thouless transition; we report here a simulation study, carried out for both μ= 0.1 and μ=-0.2, showing evidence of a transition of this kind, in broad qualitative agreement with previous Renormalization-Group studies.
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 present detailed Monte Carlo results for a two-dimensional grain boundary model on a lattice. The effective Hamiltonian of the system results from the microscopic interaction of grains with orientations described by spins of unit length, and leads to a nearest-neighbour interaction proportional to the absolute value of the angle between the grains. Our analysis of the correlation length ξ and susceptibility χ in the high-temperature phase favour a Kosterlitz-Thouless-like (KT) singularity over a second-order phase transition. Unconstrained KT fits of χ and ξ confirm the predicted value for the critical exponent ν, while the values of η deviate from the theoretical prediction. Additionally we apply finite-size scaling theory and investigate the question of multiplicative logarithmic corrections to a KT transition. As for the critical exponents our results are similar to data obtained from the XY model, so that both models probably lie in the same universality class. 1