Finite-range scaling study of the 1D long-range Ising model

Glumac, Zvonko; Uzelac, Katarina

doi:10.1088/0305-4470/22/20/020

Cited by 41 publications

(54 citation statements)

References 19 publications

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“…The numerical results apply to both inversesquare interactions [12][13][14][15] and general algebraically decaying interactions. [16][17][18][19][20][21][22][23][24][25][26][27] The work by Anderson, Yuval, and Hamann, [28][29][30][31] which greatly stimulated the interest in spin chains with long-range interactions, deserves special mention. They also developed a renormalizationlike approach to the one-dimensional ͑1D͒ inverse-square model.…”

Section: Introductionmentioning

confidence: 99%

Classical critical behavior of spin models with long-range interactions

1997

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We present the results of extensive Monte Carlo simulations of Ising models with algebraically decaying ferromagnetic interactions in the regime where classical critical behavior is expected for these systems. We corroborate the values for the exponents predicted by renormalization theory for systems in one, two, and three dimensions and accurately observe the predicted logarithmic corrections at the upper critical dimension. We give both theoretical and numerical evidence that above the upper critical dimension the decay of the critical spin-spin correlation function in finite systems consists of two different regimes. For one-dimensional systems our estimates for the critical couplings are more than two orders of magnitude more accurate than existing estimates. In two and three dimensions we are unaware of any other results for the critical couplings.

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Section: Introductionmentioning

confidence: 99%

Classical critical behavior of spin models with long-range interactions

1997

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“…This way, it is possible to obtain a partition function based on 2 × 2 TM M g,c , instead of 2 g × 2 g TM M g introduced in reference [7]. Besides working with smaller TM's, this approach has the advantage of avoiding the use of numerical diagonalization procedures to evaluate the largest eigenvalues.…”

Section: Transfer Matrix Frameworkmentioning

confidence: 99%

“…To obtain the scaling properties of ξ we will make use of the FRS framework [7]. This scheme proposes a scaling hypothesis which is formally similar to the well known finite size scaling, which compares the behaviors of finite size systems with different number of components.…”

Section: Transfer Matrix Frameworkmentioning

confidence: 99%

“…The most important features of these rigorous results are: for α > 2, the system shows only a disordered phase, ∀T [1,2]; for 1 < α ≤ 2, there is a phase transition at finite temperature T c [3,4]; critical mean field behavior occurs for 1 < α ≤ 1.5 [1,2]; for α < 1, only one ordered phase exists, ∀T . Regarding the evaluation of approximate results, both renormalization group schemes [5]- [8] and numerical calculations of finite size systems [7]- [10] have been used to estimate the thermodynamical properties, the critical temperature and the critical exponents when 1 < α ≤ 2.…”

Section: Introductionmentioning

confidence: 99%

“…For the evaluation of ν, this framework is quite useful, as it leads to analytical expressions for the derivatives of the Boltzmann weights that can be similarly stored in very compact matrices. So, the derivatives of thermodynamical properties can be directly computed, avoiding the use of numerical differentiation [7]. This work is organized as follows: in Section II we discuss the essential aspects of transfer matrix method (TM) used to evaluate the critical exponents.…”

Section: Introductionmentioning

confidence: 99%

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Critical exponents for the long-range Ising chain using a transfer matrix approach

Andrade

Pinho

2006

Eur. Phys. J. B

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Presentation: OralTopic: long range Ising model, critical behaviorThe critical behavior of the Ising chain with long-range ferromagnetic interactions decaying with distance r α , 1 < α < 2, is investigated using a numerically efficient transfer matrix (TM) method. Finite size approximations to the infinite chain are considered, in which both the number of spins and the number of interaction constants can be independently increased. Systems with interactions between spins up to 18 sites apart and up to 2500 spins in the chain are considered. We obtain data for the critical exponents ν associated with the correlation length based on the Finite Range Scaling (FRS) hypothesis. FRS expressions require the evaluation of derivatives of the thermodynamical properties, which are obtained with the help of analytical recurrence expressions obtained within the TM framework. The Van den Broeck extrapolation procedure is applied in order to estimate the convergence of the exponents. The TM procedure reduces the dimension of the matrices and circumvents several numerical matrix operations.

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Infinite-range mean-field percolation: Transfer matrix study of longitudinal correlation length

Privman

Schulman

1991

J Stat Phys

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Finite-range scaling study of the 1D long-range Ising model

Cited by 41 publications

References 19 publications

Classical critical behavior of spin models with long-range interactions

Classical critical behavior of spin models with long-range interactions

Critical exponents for the long-range Ising chain using a transfer matrix approach

Infinite-range mean-field percolation: Transfer matrix study of longitudinal correlation length

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