Critical behaviour of the 1D q-state Potts model with long-range interactions

Glumac, Zvonko; Uzelac, Katarina

doi:10.1088/0305-4470/26/20/013

Cited by 34 publications

(63 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular the results for from Ref. 42 are, for small , in better agreement with the theoretically predicted values than our estimates. However, all previous results, both for and for ␤, deviate seriously from the predicted values when approaches 1/2, which is not the case for our values.…”

Section: ͑27͒supporting

confidence: 85%

“…41 However, to our knowledge, neither any further verifications of the renormalization predictions nor any other results are available for higher-dimensional (dϾ1) models. To conclude this summary, we mention that the one-dimensional q-state Potts model with long-range interactions has been studied analytically, 9,11 numerically, 42,43 and in a mean-field approximation on the Bethe lattice. 44 Why are these models interesting?…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Classical critical behavior of spin models with long-range interactions

1997

View full text Add to dashboard Cite

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.

show abstract

Section: ͑27͒supporting

confidence: 85%

Section: Introductionmentioning

confidence: 99%

Classical critical behavior of spin models with long-range interactions

1997

View full text Add to dashboard Cite

show abstract

“…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%

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

Andrade

Pinho

2006

Eur. Phys. J. B

View full text Add to dashboard Cite

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.

show abstract

“…In Fig. 1 one can observe that the maximum of R L (T ) shrinks and shifts towards T c with increasing L. The rough fit of the difference T max −T c to the power-law [8], FRS [9] and Cannas and Magalaes [10]. For the meaning of labels a and b, see the text.…”

mentioning

confidence: 98%

“…Exact analytical expressions for the critical exponents may be derived for σ ≤ 0.5 [3], corresponding to the classical regime, while for σ > 0.5 only the approximate results exist. Various analytical and numerical approaches have been applied, from direct numerical calculations on finite chains [4], to several approaches based on the renormalization group (RG) and scaling [5][6][7][8][9][10]. The nonlocal character of interactions reduces the efficiency of most of the standard approaches so that the values of critical exponents obtained by all these methods differ considerably.…”

mentioning

confidence: 99%

Critical behavior of the long-range Ising chain from the largest-cluster probability distribution

Uzelac

Glumac²,

Aničić

2001

Phys. Rev. E

View full text Add to dashboard Cite

Monte Carlo simulations of the 1D Ising model with ferromagnetic interactions decaying with distance r as 1/r 1+σ are performed by applying the Swendsen-Wang cluster algorithm with cumulative probabilities. The critical behavior in the non-classical critical regime corresponding to 0.5 < σ < 1 is derived from the finite-size scaling analysis of the largest cluster.PACS numbers: 05.50.+q, 64.60.CnThe calculation of the critical behavior of the Ising model with long-range (LR) interactions decaying with distance with a power law 1/r d+σ is not an easy task, even in one dimension, where phase transition at finite temperature occurs for 0 < σ ≤ 1 [1,2]. Exact analytical expressions for the critical exponents may be derived for σ ≤ 0.5 [3], corresponding to the classical regime, while for σ > 0.5 only the approximate results exist. Various analytical and numerical approaches have been applied, from direct numerical calculations on finite chains [4], to several approaches based on the renormalization group (RG) and scaling [5][6][7][8][9][10]. The nonlocal character of interactions reduces the efficiency of most of the standard approaches so that the values of critical exponents obtained by all these methods differ considerably.In numerous cases of phase transitions in systems with short-range interactions, a useful complementary tool for obtaining both qualitative and quantitative results is provided by Monte Carlo (MC) simulations in combination with finite-size scaling.Such systematic studies were lacking for the LR models until recently. Namely, when applied to models with LR interactions, the standard MC approaches based either on Metropolis or on various cluster algorithms are particularly time consuming, since the number of operations per spin-flip is proportional there to the size of the system. Recently, this problem was successfully resolved by Luijten and Blöte [11] who used the cumulative probabilities within the Wolff cluster algorithm [12], which they applied to the Ising and similar models [13][14][15] reducing the computing time by several orders of magnitude.Their very exhaustive studies concentrate on questions related to the mean-field (MF) regime, while very little or no interest has been dedicated yet to the regime of the non-classical critical behavior corresponding to σ > 0.5.The purpose of this work is to extend the MC studies of the critical behavior of the LR Ising model to the non-classical regime. At the same time this is a suitable example to examine the efficiency of using only the cluster statistics in deriving the critical properties of the LR model. The 1D Ising model with LR interactions written in form of a special case (q = 2) of the Potts model is described by the Hamiltonianwhere J > 0, s i is a two-state Potts variable at the site i, δ is the Kronecker symbol and the summation is over all pairs of the system. By the substitution δ si,sj = (S i · S j + 1)/2, and J I = J/2, where S i = ±1 and J I denote the Ising spins and the interaction constant, respectively, one recovers the st...

show abstract

Critical behaviour of the 1D q-state Potts model with long-range interactions

Cited by 34 publications

References 24 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

Critical behavior of the long-range Ising chain from the largest-cluster probability distribution

Contact Info

Product

Resources

About