Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9607210 · Physical review. B, Condensed matter

Long-range interactions and nonextensivity in ferromagnetic spin models

Sergio Alejandro Cannas, Francisco A. Tamarit

Published Nov 1, 1996• 122 citations
arXivHTMLPDF

Abstract

The Ising model with ferromagnetic interactions that decay as $\frac{1}{{r}^{\ensuremath{\alpha}}}$ is analyzed in the nonextensive regime $0<~\ensuremath{\alpha}<~d$, where the thermodynamic limit is not defined. In order to study the asymptotic properties of the model in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit ($N$ being the number of spins) we propose a generalization of the Curie-Weiss model, for which the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit is well defined for all $\ensuremath{\alpha}>~0$. We conjecture that mean-field theory is exact in the last model for all $0<~\ensuremath{\alpha}<~d$. This conjecture is supported by Monte Carlo heat-bath simulations in the $d=1$ case. Moreover, we confirm a recently conjectured scaling by Tsallis that allows for a unification of extensive ($\ensuremath{\alpha}>d$) and nonextensive ($0<~\ensuremath{\alpha}<~d$) regimes.

DOI
10.1103/physrevb.54.r12661

Related papers

Long-range interactions and periodic boundary conditions in ferromagnetic spin models

Sergio Curilef

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.

•Jan 1, 2002•1 citation

Evidence of exactness of the mean-field theory in the nonextensive regime of long-range classical spin models

Sergio Alejandro Cannas, A C N de Magalhães, Francisco A. Tamarit

The q-state Potts model with long-range interactions that decay as ${1/r}^{\ensuremath{\alpha}}$ subjected to a uniform magnetic field on d-dimensional lattices is analyzed for different values of q in the nonextensive regime $0<~\ensuremath{\alpha}<~d.$ We calculate the mean-field solution of the model for all q and performed, for some values of q, Monte Carlo simulations for the spontaneous magnetization in the one-dimensional case. We show that, using a derived scaling which properly describes the nonextensive thermodynamic behavior, both types of calculations present an excellent agreement for $0<~\ensuremath{\alpha}<d.$ We also consider the two-dimensional antiferromagnetic Ising model with competing antiferromagnetic long-range interactions and ferromagnetic first neighbor ones in the presence of a uniform magnetic field. We calculate the mean-field magnetization for this case and compare it with Monte Carlo numerical data from Sampaio et al. They also show a very good agreement for $\ensuremath{\alpha}<d.$ These results, together with some previous ones, led us to conjecture that the mean-field theory is exact for nonextensive classical spin models with $0<~\ensuremath{\alpha}<d.$

arXiv:cond-mat/9906340•Physical review. B, Condensed matter•May 1, 2000•46 citations

Thermodynamics from a scaling Hamiltonian

Luis Alberto del Pino, P. Troncoso, Sergio Curilef

There are problems with defining the thermodynamic limit of systems with long-range interactions; as a result, the thermodynamic behavior of these types of systems is anomalous. In the present work, we review some concepts from both extensive and nonextensive thermodynamic perspectives. We use a model, whose Hamiltonian takes into account spins ferromagnetically coupled in a chain via a power law that decays at large interparticle distance $r$ as $1∕{r}^{\ensuremath{\alpha}}$ for $\ensuremath{\alpha}\ensuremath{\geqslant}0$. Here, we review old nonextensive scaling. In addition, we propose a Hamiltonian scaled by $2\frac{{(N∕2)}^{1\ensuremath{-}\ensuremath{\alpha}}\ensuremath{-}1}{1\ensuremath{-}\ensuremath{\alpha}}$ that explicitly includes symmetry of the lattice and dependence on the size $N$ of the system. The approach enabled us to improve upon previous results. A numerical test is conducted through Monte Carlo simulations. In the model, periodic boundary conditions are adopted to eliminate surface effects.

arXiv:0710.5644•Physical Review B•Nov 8, 2007•10 citations

Finite-size analysis of a two-dimensional Ising model within a nonextensive approach

Nuno Crokidakis, Diogo O. Soares-Pinto, Mario S. Reis +3 more

In this work we present a thorough analysis of the phase transitions that occur in a ferromagnetic two-dimensional Ising model, with only nearest-neighbors interactions, in the framework of the Tsallis nonextensive statistics. We performed Monte Carlo simulations on square lattices with linear sizes L ranging from 32 up to 512. The statistical weight of the Metropolis algorithm was changed according to the nonextensive statistics. Discontinuities in the m(T) curve are observed for $q\ensuremath{\le}0.5$. However, we have verified only one peak on the energy histograms at the critical temperatures, indicating the occurrence of continuous phase transitions. For the $0.5<q\ensuremath{\le}1.0$ regime, we have found continuous phase transitions between the ordered and the disordered phases, and determined the critical exponents via finite-size scaling. We verified that the critical exponents $\ensuremath{\alpha}$, $\ensuremath{\beta}$, and $\ensuremath{\gamma}$ depend on the entropic index $q$ in the range $0.5<q\ensuremath{\le}1.0$ in the form $\ensuremath{\alpha}(q)=(10{q}^{2}\ensuremath{-}33q+23)/20$, $\ensuremath{\beta}(q)=(2q\ensuremath{-}1)/8$, and $\ensuremath{\gamma}(q)=({q}^{2}\ensuremath{-}q+7)/4$. On the other hand, the critical exponent $\ensuremath{\nu}$ does not depend on $q$. This suggests a violation of the scaling relations $2\ensuremath{\beta}+\ensuremath{\gamma}=d\ensuremath{\nu}$ and $\ensuremath{\alpha}+2\ensuremath{\beta}+\ensuremath{\gamma}=2$ and a nonuniversality of the critical exponents along the ferro-paramagnetic frontier.

arXiv:0910.2218•Physical Review E•Nov 2, 2009•14 citations

Effect of long-range interactions on the critical behavior of the continuous Ising model

E. Bayong, H. T. Diep

Critical behavior of the one- and two-dimensional general continuous Ising models with long-range ferromagnetic interactions decaying as ${1/r}^{d+\ensuremath{\sigma}}$ is studied using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between $\ensuremath{-}1$ and 1. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents which depend on $\ensuremath{\sigma}.$ Results for various values of $\ensuremath{\sigma}$ will be shown and compared to predictions from renormalization-group theory. Though there is an agreement with the overall tendency predicted, there are several fundamental differences. Discussion is given.

Physical review. B, Condensed matter•May 1, 1999•28 citations

Infinite-ranged models of spin-glasses

Scott Kirkpatrick, David Sherrington

A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and $\mathrm{XY}$ spins are evaluated using a many-replica procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for $T$ greater than ${T}_{c}$, the spin-glass transition temperature. At and below ${T}_{c}$, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.

World Scientific lecture notes in physics•Nov 1, 1986•1 citation
The Ising model with ferromagnetic interactions that decay as 1r\ensuremathα\frac{1}{{r}^{\ensuremath{\alpha}}}r\ensuremathα1​ is analyzed in the nonextensive regime 0< \ensuremathα< d0<~\ensuremath{\alpha}<~d0< \ensuremathα< d, where the thermodynamic limit is not defined. In order to study the asymptotic properties of the model in the N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞ limit (NNN being the number of spins) we propose a generalization of the Curie-Weiss model, for which the N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞ limit is well defined for all \ensuremathα> 0\ensuremath{\alpha}>~0\ensuremathα> 0. We conjecture that mean-field theory is exact in the last model for all 0< \ensuremathα< d0<~\ensuremath{\alpha}<~d0< \ensuremathα< d. This conjecture is supported by Monte Carlo heat-bath simulations in the d=1d=1d=1 case. Moreover, we confirm a recently conjectured scaling by Tsallis that allows for a unification of extensive (\ensuremathα>d\ensuremath{\alpha}>d\ensuremathα>d) and nonextensive (0< \ensuremathα< d0<~\ensuremath{\alpha}<~d0< \ensuremathα< d) regimes.
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.
The q-state Potts model with long-range interactions that decay as ${1/r}^{\ensuremath{\alpha}}$ subjected to a uniform magnetic field on d-dimensional lattices is analyzed for different values of q in the nonextensive regime $0<~\ensuremath{\alpha}<~d.$ We calculate the mean-field solution of the model for all q and performed, for some values of q, Monte Carlo simulations for the spontaneous magnetization in the one-dimensional case. We show that, using a derived scaling which properly describes the nonextensive thermodynamic behavior, both types of calculations present an excellent agreement for $0<~\ensuremath{\alpha}<d.$ We also consider the two-dimensional antiferromagnetic Ising model with competing antiferromagnetic long-range interactions and ferromagnetic first neighbor ones in the presence of a uniform magnetic field. We calculate the mean-field magnetization for this case and compare it with Monte Carlo numerical data from Sampaio et al. They also show a very good agreement for $\ensuremath{\alpha}<d.$ These results, together with some previous ones, led us to conjecture that the mean-field theory is exact for nonextensive classical spin models with $0<~\ensuremath{\alpha}<d.$
There are problems with defining the thermodynamic limit of systems with long-range interactions; as a result, the thermodynamic behavior of these types of systems is anomalous. In the present work, we review some concepts from both extensive and nonextensive thermodynamic perspectives. We use a model, whose Hamiltonian takes into account spins ferromagnetically coupled in a chain via a power law that decays at large interparticle distance rrr as $1∕{r}^{\ensuremath{\alpha}}$ for $\ensuremath{\alpha}\ensuremath{\geqslant}0$. Here, we review old nonextensive scaling. In addition, we propose a Hamiltonian scaled by $2\frac{{(N∕2)}^{1\ensuremath{-}\ensuremath{\alpha}}\ensuremath{-}1}{1\ensuremath{-}\ensuremath{\alpha}}$ that explicitly includes symmetry of the lattice and dependence on the size NNN of the system. The approach enabled us to improve upon previous results. A numerical test is conducted through Monte Carlo simulations. In the model, periodic boundary conditions are adopted to eliminate surface effects.
In this work we present a thorough analysis of the phase transitions that occur in a ferromagnetic two-dimensional Ising model, with only nearest-neighbors interactions, in the framework of the Tsallis nonextensive statistics. We performed Monte Carlo simulations on square lattices with linear sizes L ranging from 32 up to 512. The statistical weight of the Metropolis algorithm was changed according to the nonextensive statistics. Discontinuities in the m(T) curve are observed for $q\ensuremath{\le}0.5$. However, we have verified only one peak on the energy histograms at the critical temperatures, indicating the occurrence of continuous phase transitions. For the $0.5<q\ensuremath{\le}1.0$ regime, we have found continuous phase transitions between the ordered and the disordered phases, and determined the critical exponents via finite-size scaling. We verified that the critical exponents $\ensuremath{\alpha}$, $\ensuremath{\beta}$, and $\ensuremath{\gamma}$ depend on the entropic index qqq in the range $0.5<q\ensuremath{\le}1.0$ in the form $\ensuremath{\alpha}(q)=(10{q}^{2}\ensuremath{-}33q+23)/20$, $\ensuremath{\beta}(q)=(2q\ensuremath{-}1)/8$, and $\ensuremath{\gamma}(q)=({q}^{2}\ensuremath{-}q+7)/4$. On the other hand, the critical exponent $\ensuremath{\nu}$ does not depend on qqq. This suggests a violation of the scaling relations $2\ensuremath{\beta}+\ensuremath{\gamma}=d\ensuremath{\nu}$ and $\ensuremath{\alpha}+2\ensuremath{\beta}+\ensuremath{\gamma}=2$ and a nonuniversality of the critical exponents along the ferro-paramagnetic frontier.
Critical behavior of the one- and two-dimensional general continuous Ising models with long-range ferromagnetic interactions decaying as ${1/r}^{d+\ensuremath{\sigma}}$ is studied using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between $\ensuremath{-}1$ and 1. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents which depend on $\ensuremath{\sigma}.$ Results for various values of $\ensuremath{\sigma}$ will be shown and compared to predictions from renormalization-group theory. Though there is an agreement with the overall tendency predicted, there are several fundamental differences. Discussion is given.
A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and XY\mathrm{XY}XY spins are evaluated using a many-replica procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for TTT greater than Tc{T}_{c}Tc​, the spin-glass transition temperature. At and below Tc{T}_{c}Tc​, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.