Periodic boundary conditions have no unique implementation in magnetic systems where all spins interact with each other through a power law decaying interaction of the form 1/rα, r being the distance between spins. In this work we present a comparative study of the finite size effects oberved in numerical simulations by using first image and infinite image periodic boundary conditions in one- and two-dimensional spin systems with those interactions, including the ferromagnetic, anti-ferromagnetic and competitive interaction cases. Our results show no significative differences between the finite size effects produced by both boundary conditions when the low temperature phase has zero global magnetization, and it depends on the ratio α/d for systems with a low temperature ferromagnetic phase. In the last case the first image convention gives more stronger finite size effects than the other when the system enters into the classical regime α/d≤3/2.
Periodic boundary conditions have no unique implementation in magnetic systems where all spins interact with each other through a power law decaying interaction of the form 1/rα, r being the distance between spins. In this work we present a comparative study of the finite size effects oberved in numerical simulations by using first image and infinite image periodic boundary conditions in one- and two-dimensional spin systems with those interactions, including the ferromagnetic, anti-ferromagnetic and competitive interaction cases. Our results show no significative differences between the finite size effects produced by both boundary conditions when the low temperature phase has zero global magnetization, and it depends on the ratio α/d for systems with a low temperature ferromagnetic phase. In the last case the first image convention gives more stronger finite size effects than the other when the system enters into the classical regime α/d≤3/2.
Periodic boundary conditions have no unique implementation in magnetic systems where all spins interact with each other through a power law decaying interaction of the form 1/rα, r being the distance between spins. In this work we present a comparative study of the finite size effects oberved in numerical simulations by using first image and infinite image periodic boundary conditions in one- and two-dimensional spin systems with those interactions, including the ferromagnetic, anti-ferromagnetic and competitive interaction cases. Our results show no significative differences between the finite size effects produced by both boundary conditions when the low temperature phase has zero global magnetization, and it depends on the ratio α/d for systems with a low temperature ferromagnetic phase. In the last case the first image convention gives more stronger finite size effects than the other when the system enters into the classical regime α/d≤3/2.
We present a possible way to study long-range interacting particles in finite-infinite systems with periodic boundary conditions. A symmetrical lattice and its contributions overall space are being used. In this context, we study the Ising model with ferromagnetic interaction that decays as a 1/r a law. We verify by Monte Carlo heat-bath simulations in the D (dimension) = 1 case that the thermodynamics quantities scale in a way proposed by Tsallis, and we confirm that mean field theory is exact in the last model for all 0 < a < D suggested by Cannas and Tamarit.
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.
Finite-size effects on the first-order phase transition in a three-dimensional classical Heisenberg model with nearest-neighbor ferromagnetic interactions and periodic boundary conditions are studied theoretically and numerically in this paper. We find that the current Fisher-Privman theory fails at temperatures close to criticality and/or for small systems. We rederive explicit scaling functions for the longitudinal magnetization and susceptibility under another phenomenological assumption. The results are in very good agreement with Monte Carlo simulations.
A scaling hypothesis is set up for the magnetization m(T,H;L), susceptibility \ensuremath{\chi}(T,H;L), and correlation length \ensuremath{\xi}(T,H;L) of a finite-sized system, with O(n) symmetry (n\ensuremath{\ge}2) and long-range interactions decaying as 1/${r}^{d+\ensuremath{\sigma}}$ (0<\ensuremath{\sigma}<2), confined to geometry ${L}^{d\mathrm{\ensuremath{-}}d\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\infty}}^{d\mathcal{'}}$ (\ensuremath{\sigma}<d<2\ensuremath{\sigma},d'\ensuremath{\le}\ensuremath{\sigma}) and subjected to periodic boundary conditions. Finite-size effects are predicted, in the region of a first-order phase transition (TTc) as well as in the region of a second-order phase transition (T\ensuremath{\simeq}Tc), for different regimes of the parameters H and L. To test these predictions, a detailed analytical study is carried out in the case of the spherical model of ferromagnetism (n=\ensuremath{\infty}), and all predictions based on the scaling hypothesis are seen to be fully borne out. In situations where L\ensuremath{\gg}\ensuremath{\xi}, finite-size corrections to standard bulk values of the various physical quantities pertaining to these models are found to vary as (\ensuremath{\xi}/L${)}^{d+\ensuremath{\sigma}}$, rather than as ${e}^{\mathrm{\ensuremath{-}}L/\ensuremath{\xi}}$, which is characteristic of models with short-range interactions.
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.