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Physical Review E

Tsallis scaling and the long-range Ising chain: A transfer matrix approach

Roberto F. S. Andrade, Suani T. R. Pinho

Published Feb 25, 2005• 25 citations
Publisher

Abstract

A numerically efficient transfer matrix (TM) approach is introduced to investigate the long-range Ising spin chain. Results obtained within this procedure are primarily used to verify the Tsallis scaling hypothesis for long-range systems with an alpha power-law decay of the interaction constants, both in the extensive (alpha>1) and nonextensive (alpha<1) regimes. Results for finite-size systems, taking into account all interactions between spins up to 24 sites apart, show that the conjecture is satisfied with a very good precision (less than 0.004%) for all temperature intervals. This TM procedure is further used to investigate several other thermodynamic and critical properties of this system, and it may also be extended to similar one-dimensional long-range systems.

DOI
10.1103/physreve.71.026126

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A numerically efficient transfer matrix (TM) approach is introduced to investigate the long-range Ising spin chain. Results obtained within this procedure are primarily used to verify the Tsallis scaling hypothesis for long-range systems with an alpha power-law decay of the interaction constants, both in the extensive (alpha>1) and nonextensive (alpha<1) regimes. Results for finite-size systems, taking into account all interactions between spins up to 24 sites apart, show that the conjecture is satisfied with a very good precision (less than 0.004%) for all temperature intervals. This TM procedure is further used to investigate several other thermodynamic and critical properties of this system, and it may also be extended to similar one-dimensional long-range systems.
A numerically efficient transfer matrix approach is used to investigate the validity of the Tsallis scaling hypothesis in the long-range Ising spin chain with competitive interactions. In this model, the interaction between two spins i and j placed r lattice steps apart is Ji, j = (-1)ζ(i,j)J0/rα, where ζ(i, j) is either 0 or 1. This procedure has succeeded to show the validity of the scaling hypothesis for the well investigated ferromagnetic version of the model, i.e., ζ(i, j)= 0,∀i, j, ∀α > 0. Results are reported for some models of a set, which is defined by requiring ζ(i, j) to be a periodic sequence of 0's and 1's. As expected from symmetry arguments, we find that the hypothesis is not valid when ζ(i, j)= 1,∀i, j and α < 1. however, it is verified, with high degree of numerical accuracy, when α < 1, for sequences in which the occurrence of ζ(i, j)= 0 is more frequent than that of ζ(i, j)= 1.
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.
Small systems consisting of particles interacting with long-range potentials exhibit enormous size effects. The Tsallis conjecture [Tsallis, Fractals 3, 541 (1995)FRACEG0218-348X10.1142/S0218348X95000473], valid for translationally invariant systems with long-range interactions, states a well-known scaling relating different sizes. Here we propose to generalize this conjecture to systems with this symmetry broken, by adjusting one parameter that determines an effective distance to compute the strength of the interaction. We apply this proposal to the one-dimensional Ising model with ferromagnetic interactions that decay as 1/r^{1+σ} in the region where the model has a finite critical temperature. We demonstrate the convenience of using this generalization to study finite-size effects, and we compare this approach with the finite-size scaling theory.
The phase diagram of a two-sited magnetic Ising chain, with a long-range interaction in the form of ${1/r}^{1+\ensuremath{\sigma}},$ is studied. In this investigation, the finite-range scaling technique is employed and a proper transfer matrix is developed. The critical temperature of the chain, as a function of each type of interaction, i.e., the interaction between the similar sites and different sites, in both the classical and nonclassical regions is calculated. The results indicate that the critical temperature is strongly affected by the interaction range parameter \ensuremath{\sigma} for both types of interaction, but its behavior is quite different for each of the interactions. The behavior of the system is explained by calculating the strength of each interaction using the mean-field approximation. The study of the critical exponent of the correlation length indicates that the exponent is, within a good approximation, independent of the interaction constants.
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 (NNN 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=1d=1d=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.