Recursion Relations and Fixed Points for Ferromagnets with Long-Range Interactions

Sak, J.

doi:10.1103/physrevb.8.281

Cited by 285 publications

(380 citation statements)

References 12 publications

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“…For Ͼ2 they take their short-range values. Sak, 36 however, found that already for Ͼ2Ϫ sr the critical behavior is Isinglike, where sr denotes the exponent in the corresponding model with short-range interactions. In this article we concentrate on the classical range, for which we have performed extensive Monte Carlo simulations of spin models in dϭ1,2,3.…”

Section: Finite-size Analysis Of the Critical Behaviormentioning

confidence: 99%

“…The j 2 k 2 term normally representing the short-range interactions is included because it will appear anyway in the renormalization process and will compete with the long-range term. 36 Under a renormalization transformation with a rescaling factor bϭe l , the term j k is transformed into j kЈ , with kЈϭkb. To keep the coefficient of the k term fixed, we rescale the field k to…”

Section: Finite-size Analysis Of the Critical Behaviormentioning

confidence: 99%

“…12 and 32-34. A major contribution was made by Fisher, Ma, and Nickel 35 and Sak, 36 who obtained renormalization predictions for the critical exponents of models of general dimensionality dϽ4 with algebraically decaying interactions ͑obtained independently by Suzuki et al 37 ͒. Other works concerning dϾ1 are two conjectures on, respectively, the boundary between long-range and shortrange behavior and the boundary between classical ͑mean-field͒ and nonclassical behavior, both by Stell, 38 a ͑refuted͒ conjecture by Griffiths, 39 a rigorous confirmation of the upper critical dimension by Aizenman and Fernández, 10 and a variational approach to the Ising model with long-range interactions.…”

Section: Introductionmentioning

confidence: 99%

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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: Finite-size Analysis Of the Critical Behaviormentioning

confidence: 99%

Section: Finite-size Analysis Of the Critical Behaviormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Classical critical behavior of spin models with long-range interactions

1997

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“…For σ < 2 the upper critical dimension is reduced to d c (σ) = 2σ [24,25] and the values of the critical exponents depend on σ for σ < d < d c (σ) [24,26,27]. The crossover from short-ranged to long-ranged critical behavior occurs for σ = 2 − η sr (d), where η sr (d) is the critical exponent for the short-ranged system (for a given fixed spatial dimension d) [28,29,30,31,32]. This crossover has recently been reexamined numerically in d = 2 in Ref.…”

Section: A Bulk Systemsmentioning

confidence: 99%

Universality of the thermodynamic Casimir effect

2003

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Recently a nonuniversal character of the leading spatial behavior of the thermodynamic Casimir force has been reported [X. S. Chen and V. Dohm, Phys. Rev. E 66, 016102 (2002)]. We reconsider the arguments leading to this observation and show that there is no such leading nonuniversal term in systems with short-ranged interactions if one treats properly the effects generated by a sharp momentum cutoff in the Fourier transform of the interaction potential. We also conclude that lattice and continuum models then produce results in mutual agreement independent of the cutoff scheme, contrary to the aforementioned report. All results are consistent with the universal character of the Casimir force in systems with short-ranged interactions. The effects due to dispersion forces are discussed for systems with periodic or realistic boundary conditions. In contrast to systems with short-ranged interactions, for L/ξ ≫ 1 one observes leading finite-size contributions governed by power laws in L due to the subleading long-ranged character of the interaction, where L is the finite system size and ξ is the correlation length.

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“…Successive investigations complemented this result introducing the threshold value σ * = 2 − η SR , where η SR is the anomalous dimension of a SR model in the same dimension of the LR one under study [23]. At σ = σ * LR interactions become irrelevant with respect to SR ones [23]. These results, in agreement with Monte Carlo findings [24], have been questioned by numerical simulations on the two-dimensional LR Ising model indicating δη = 0 and σ * = 2 [25,26].…”

Section: Introductionmentioning

confidence: 99%

One-dimensional long-range percolation: A numerical study

et al. 2017

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In this paper we study bond percolation on a one-dimensional chain with power-law bond probability C/r 1+σ , where r is the distance length between distinct sites. We introduce and test an order N Monte Carlo algorithm and we determine as a function of σ the critical value Cc at which percolation occurs. The critical exponents in the range 0 < σ < 1 are reported and compared with mean-field and ε-expansion results. Our analysis is in agreement, up to a numerical precision ≈ 10 −3 , with the mean field result for the anomalous dimension η = 2 − σ, showing that there is no correction to η due to correlation effects.

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Recursion Relations and Fixed Points for Ferromagnets with Long-Range Interactions

Cited by 285 publications

References 12 publications

Classical critical behavior of spin models with long-range interactions

Classical critical behavior of spin models with long-range interactions

Universality of the thermodynamic Casimir effect

One-dimensional long-range percolation: A numerical study

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