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Physical review. B, Condensed matter

Magnetic phase diagram of an alternating Ising chain with long-range interactions

M. Barati, Abdolali Ramazani

Published Jun 18, 2001• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.64.024407

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The phase diagram of a two-sited magnetic Ising chain, with a long-range interaction in the form of 1/r1+\ensuremathσ,{1/r}^{1+\ensuremath{\sigma}},1/r1+\ensuremathσ, 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 phase diagram and critical exponent of the correlation length for an Ising chain with a long-range interaction in the form of ${1/r}^{1+\ensuremath{\sigma}}$ for S=1/2,1,3/2,2S=1/2,1,3/2,2S=1/2,1,3/2,2 is calculated by the finite-range scaling technique. The spin dependence of the critical temperature and critical exponent $\ensuremath{\nu}$ in the classical and nonclassical regions is also studied. It is observed that, with a good approximation, the critical exponent of the correlation length is independent of the magnitude of spin in both the classical and nonclassical regions.
The classical phase diagram of an Ising chain for S> 12S>~\frac{1}{2}S> 21​ with long-range interaction in the form of ${1/r}^{1+\ensuremath{\sigma}}$ is determined. The calculation is performed by using the exact value of the critical exponent of the correlation length, $\ensuremath{\nu}=1/\ensuremath{\sigma},$ given by the renormalzation-group technique in the classical region, and applying a searching program based on the finite-range-scaling method. The results for the critical temperature in this approach are in excellent agreement with those calculated by the extensive Monte Carlo technique. This is strong support for the range-scaling method in the classical region.
The coherent anomaly method and a cluster mean-field approach combined with finite-sizescaling extrapolation were used to determine the critical temperature and the critical exponents of aS =1 Ising chain with a long-range interaction in the form of 1/ r1+σ . The results of criticaltemperature are in good agreement with the recent results of the finite-range scaling method and aremore accurate in the classical region. The critical exponents are in agreement with the results of spin-21Ising chains and indicate that the critical exponents are independent of the magnitude of spin in Isingmodels with long-range interactions.
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 long-range Ising chain, which has ferromagnetic 1/r 3 type interactions, has been studied by exact numerical calculation of thermodynamic properties of the sequence of systems of N spins (N<or=30). By expressing the sequence in terms of a series and using extrapolation methods such as the Pade approximant, the authors have found a consistent way to obtain critical temperatures which is accurate to 0.4% for the very long-range equivalent neighbour model. The authors have estimated the high-temperature susceptibility and the low- temperature long-range correlations of the infinite chain and thence have made estimates for the critical exponents gamma and beta . Although the nature of the phase transition appears to be classical for 1<s<or approximately=1.3, it is certainly not classical for 1.6<s<2; in particular, as s approaches 2 from below, gamma increases regularly to about 2 or greater and beta decreases regularly to near 0.
A next-neighbor Ising model with disordered but long range correlated coupling constants is investigated. The model is built on a hierarchical lattice and the correlation strength depends on a tuning parameter $\ensuremath{\alpha}$. The results are obtained within a transfer-matrix framework, which allows for the evaluation of the properties of individual samples. Collective behavior is computed by averaging over a large number of independent realizations. The dependence of the thermodynamic and magnetic functions with respect to the temperature is investigated for each value of $\ensuremath{\alpha}$. Phase diagrams in the $(\ensuremath{\alpha},T)$ plane are constructed for two distinct versions of the model, indicating the existence of regions of paramagnetic and ordered phases. Critical values ${\ensuremath{\alpha}}_{c}$, below which the system always assumes the paramagnetic phase, are found for both versions.