Recently, it has been found that an effective long-range interaction is realized among local bistable variables (spins) in systems where the elastic interaction causes ordering of the spins. In such systems, we generally expect both long-range and short-range interactions to exist. In the short-range Ising model, the correlation length diverges at the critical point. In contrast, in the long-range interacting model the spin configuration is always uniform and the correlation length is zero. As long as a system has nonzero long-range interactions, it shows criticality in the mean-field universality class, and the spin configuration is uniform beyond a certain scale. Here we study the crossover from the pure short-range interacting model to the long-range interacting model. We investigate the infinite-range model (Husimi-Temperley model) as a prototype of this competition, and we study how the critical temperature changes as a function of the strength of the long-range interaction. This model can also be interpreted as an approximation for the Ising model on a small-world network. We derive a formula for the critical temperature as a function of the strength of the long-range interaction. We also propose a finite-size scaling form for the spin correlation length at the critical point, which is finite as long as the long-range interaction is included, though it diverges in the limit of the pure short-range model. These properties are confirmed by extensive Monte Carlo simulations.Recently, it has been found that an effective long-range interaction is realized among local bistable variables (spins) in systems where the elastic interaction causes ordering of the spins. In such systems, we generally expect both long-range and short-range interactions to exist. In the short-range Ising model, the correlation length diverges at the critical point. In contrast, in the long-range interacting model the spin configuration is always uniform and the correlation length is zero. As long as a system has nonzero long-range interactions, it shows criticality in the mean-field universality class, and the spin configuration is uniform beyond a certain scale. Here we study the crossover from the pure short-range interacting model to the long-range interacting model. We investigate the infinite-range model (Husimi-Temperley model) as a prototype of this competition, and we study how the critical temperature changes as a function of the strength of the long-range interaction. This model can also be interpreted as an approximation for the Ising model on a small-world network. We derive a formula for the critical temperature as a function of the strength of the long-range interaction. We also propose a finite-size scaling form for the spin correlation length at the critical point, which is finite as long as the long-range interaction is included, though it diverges in the limit of the pure short-range model. These properties are confirmed by extensive Monte Carlo simulations.

arXiv:1104.1834Physical Review BAug 10, 201134 citations

Wilson-type expansions of critical exponents for long-range interactions

Masuo Suzuki, Y. Yamazaki, George Igarashi

Physics Letters ADec 1, 197244 citations

Short-Time Critical Behavior Affected by Weakly Long-Range Interactions

Yuan Chen, Guo Shuohong, Zhibing Li

The theoretic renormalization group approach is applied to the study of short-time critical behavior of the Ginzburg–Landau model with weakly long-range interactions . The system initially at a high temperature is firstly quenched to the critical temperature and then released to an evolution with a model A dynamics. A double expansion in and with of order is employed, where is the spatial dimension. The asymptotic scaling laws and the initial slip exponents and for the order parameter and the response function respectively are calculated to the second order in for close to 2.The theoretic renormalization group approach is applied to the study of short-time critical behavior of the Ginzburg–Landau model with weakly long-range interactions . The system initially at a high temperature is firstly quenched to the critical temperature and then released to an evolution with a model A dynamics. A double expansion in and with of order is employed, where is the spatial dimension. The asymptotic scaling laws and the initial slip exponents and for the order parameter and the response function respectively are calculated to the second order in for close to 2.

Communications in Theoretical PhysicsApr 15, 20013 citations

Finite Size Scaling and Long-Range Interactions

Jonathan Shapiro

Jan 1, 19860 citations

Scaling behavior for finite O(n) systems with long-range interaction

Hassan Chamati, N. S. Tonchev

A detailed investigation of the scaling properties of the fully finite $\mathcal{O}(n)$ systems, under periodic boundary conditions, with long-range interaction, decaying algebraically with the interparticle distance r like ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}},$ below their upper critical dimension, is presented. The computation of the scaling functions is done to one loop order in the nonzero modes. The results are obtained in an expansion of powers of $\sqrt{\ensuremath{\varepsilon}},$ where $\ensuremath{\varepsilon}=2\ensuremath{\sigma}\ensuremath{-}d$ up to $\mathcal{O}({\ensuremath{\varepsilon}}^{3/2}).$ The thermodynamic functions are found to depend upon the scaling variable ${z=RU}^{\ensuremath{-}1/2}{L}^{2\ensuremath{-}\ensuremath{\eta}\ensuremath{-}\ensuremath{\varepsilon}/2},$ where R and U are the coupling constants of the constructed effective theory, and L is the linear size of the system. Some simple universal results are obtained.A detailed investigation of the scaling properties of the fully finite $\mathcal{O}(n)$ systems, under periodic boundary conditions, with long-range interaction, decaying algebraically with the interparticle distance r like ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}},$ below their upper critical dimension, is presented. The computation of the scaling functions is done to one loop order in the nonzero modes. The results are obtained in an expansion of powers of $\sqrt{\ensuremath{\varepsilon}},$ where $\ensuremath{\varepsilon}=2\ensuremath{\sigma}\ensuremath{-}d$ up to $\mathcal{O}({\ensuremath{\varepsilon}}^{3/2}).$ The thermodynamic functions are found to depend upon the scaling variable ${z=RU}^{\ensuremath{-}1/2}{L}^{2\ensuremath{-}\ensuremath{\eta}\ensuremath{-}\ensuremath{\varepsilon}/2},$ where R and U are the coupling constants of the constructed effective theory, and L is the linear size of the system. Some simple universal results are obtained.

arXiv:cond-mat/0005335Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJan 18, 20017 citations

A detailed investigation of the scaling properties of the fully finite

\mathcal{O}(n)

systems, under periodic boundary conditions, with long-range interaction, decaying algebraically with the interparticle distance r like ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}},$ below their upper critical dimension, is presented. The computation of the scaling functions is done to one loop order in the nonzero modes. The results are obtained in an expansion of powers of $\sqrt{\ensuremath{\varepsilon}},$ where $\ensuremath{\varepsilon}=2\ensuremath{\sigma}\ensuremath{-}d$ up to $\mathcal{O}({\ensuremath{\varepsilon}}^{3/2}).$ The thermodynamic functions are found to depend upon the scaling variable ${z=RU}^{\ensuremath{-}1/2}{L}^{2\ensuremath{-}\ensuremath{\eta}\ensuremath{-}\ensuremath{\varepsilon}/2},$ where R and U are the coupling constants of the constructed effective theory, and L is the linear size of the system. Some simple universal results are obtained.