Finite-range scaling for a one-dimensional system with long-range interactions

Uzelac, Katarina; Glumac, Zvonko

doi:10.1088/0305-4470/21/7/011

Cited by 20 publications

(12 citation statements)

References 9 publications

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“…In order to carry out the calculations we chose σ = 0.75, because for this value of the parameter the critical exponents of the Ising model are expected to be sufficiently different from mean-field values (σ = 0.50) to allow a meaningful comparison with RG results [21][22][23]. Furthermore, one also likes to be as far as possible from σ = 1.00, where strong Kosterlitz-Thouless behaviour is known to occur [24].…”

Section: The Ising Model With Lr Interactions and The Simulation mentioning

confidence: 99%

Study of the nonequilibrium critical quenching and the annealing dynamics for the long-range Ising model in one dimension

2011

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Extensive Monte Carlo simulations are employed in order to study the dynamic critical behaviour of the one-dimensional Ising magnet, with algebraically decaying long-range interactions of the form 1 r d+σ , with σ = 0.75. The critical temperature, as well as the critical exponents, are evaluated from the power-law behaviour of suitable physical observables when the system is quenched from uncorrelated states, corresponding to infinite temperature, to the critical point. These results are compared with those obtained from the dynamic evolution of the system when it is suddenly annealed at the critical point from the ordered state. Also, the critical temperature in the infinite interaction limit is obtained by means of a finite-range scaling analysis of data measured with different truncated-interaction range. All the estimated static critical exponents (γ/ν, β/ν, and 1/ν ) are in good agreement with Renormalization Group (RG) results and previously reported numerical data obtained under equilibrium conditions. On the other hand, the dynamic exponent of the initial increase of the magnetization (θ) was close to RG predictions. However, the dynamic exponent z of the time correlation length is slightly different than the RG results likely due to the fact that either it may depend on the specific dynamics used or because the two-loop expansion used in the RG analysis may be insufficient.

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Section: The Ising Model With Lr Interactions and The Simulation mentioning

confidence: 99%

Study of the nonequilibrium critical quenching and the annealing dynamics for the long-range Ising model in one dimension

2011

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“…It would also be interesting to perform a complementary study of a more general case of the Yang-Lee zeros for the Potts model with infinite range interactions with the power-law decay, where some previous work has been performed [30] for the Ising model by the finite-range scaling approach [31].…”

Section: Discussionmentioning

confidence: 99%

Yang-Lee zeros and the critical behavior of the infinite-range two- and three-state Potts models

Glumac

Uzelac

2013

Phys. Rev. E

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The phase diagram of the two- and three-state Potts model with infinite-range interactions in the external field is analyzed by studying the partition function zeros in the complex field plane. The tricritical point of the three-state model is observed as the approach of the zeros to the real axis at the nonzero field value. Different regimes, involving several first- and second-order transitions of the complicated phase diagram of the three-state model, are identified from the scaling properties of the zeros closest to the real axis. The critical exponents related to the tricritical point and the Yang-Lee edge singularity are well reproduced. Calculations are extended to the negative fields, where the exact implicit expression for the transition line is derived.

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“…To compare, in Fig. 6, we plot our extrapolated critical temperature values together with those extrapolations given by Uzelac [17], Cannas [11], Nagle and Bonner [9], Monroe [6,13], Pires [18], and Luijten [20,21]. Our critical temperatures as a function of p fall within the upper and lower bounds (0.313, 0.658) reported by Monroe [35] and show a better coincidence with many existing approximations for p close to 1 than for the p values near 2.…”

Section: Critical Temperaturementioning

confidence: 99%

“…After some years, in 1982, Fröhlich and Spencer [2] proved the existence of a phase transition for p = 2. Since the original demonstration of phase transition existence by Dyson, who did not provide a critical value, many approximations have been reported to give the critical temperatures for 1 ≤ p ≤ 2 using different methods: variational [5], coherent anomaly [6], ζ-function [7], finite-range [8], series expansion [9], renormalization group [10,11], Monte Carlo [12], Bethe Lattice approximation [13], cluster approach [14], among others [15][16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Critical temperature of one-dimensional Ising model with long-range interaction revisited

Martínez-Herrera,

Rodríguez-López,

Solís

2021

Preprint

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We present a generalized expression for the transfer matrix of finite and infinite one-dimensional spin chains within a magnetic field with spin pair interaction J/r p , where r = 1, 2, . . . , nv is the distance between two spins, nv is the number of nearest neighbors reached by the interaction, and p ∈ [1,2]. With this generalized expression, we calculate the partition function, the Helmholtz free energy, and the specific heat for both finite and infinite ferromagnetic 1D Ising models within a zero external magnetic field. We focus on the temperature Tmax where specific heat reaches its maximum. We calculate J/(kBTmax) numerically for every values of nv ∈ {1, 2, . . . , 25}, which we interpolate and then extrapolate up to the critical temperature as a function of p, using a novel functional approach. Two different procedures are used to reach the infinite spin chain with an infinite interaction range: increasing the chain size and the interaction range by the same amount, and increasing the interaction range for the infinite chain. As we expected, both extrapolations lead to the same critical temperature, although by two different concurrent curves. Our critical temperatures as a function of p fall within the upper and lower bounds reported in the literature and show a better coincidence with many existing approximations for p close to 1 than for the p values near 2. We report an averaged inverse critical temperature J/(kBTc) = 0.532 for the one-dimensional spin chain with p = 2. It is worth mentioning that the well-known cases for near (original Ising model) and next-near neighbor interactions are recovered doing nv = 1 and nv = 2, respectively.

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Finite-range scaling for a one-dimensional system with long-range interactions

Cited by 20 publications

References 9 publications

Study of the nonequilibrium critical quenching and the annealing dynamics for the long-range Ising model in one dimension

Study of the nonequilibrium critical quenching and the annealing dynamics for the long-range Ising model in one dimension

Yang-Lee zeros and the critical behavior of the infinite-range two- and three-state Potts models

Critical temperature of one-dimensional Ising model with long-range interaction revisited

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