Upper and lower critical decay exponents of Ising ferromagnets with long-range interaction

Horita, Toshiki; Suwa, Hidemaro; Todo, Synge

doi:10.1103/physreve.95.012143

Cited by 50 publications

(58 citation statements)

References 36 publications

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“…Since according to scaling relations, γ/ν = 2 − η and η = 2 − σ + δη for a LR system, the scaling of S will be used to evaluate the correction δη to the power law decay of the correlation functions at criticality. In the following it will be shown that our results are in agreement with the widely accepted result δη = 0, which, as already mentioned, has been recently confirmed by extensive numerical simulations and theoretical investigations on the LR Ising model [28,30,34]. We have also introduced and studied the ratio:…”

Section: Observablessupporting

confidence: 88%

“…However up to date numerical simulations on the two-dimensional LR Ising model confirmed the traditional scenario with δη = 0 and σ * = 2 − η SR [28], also indicating an extremely slow convergence as a function of the system size for the critical amplitude of the standard Binder ratio of magnetization. Logarithmic corrections at the boundary value σ * are a possible source of error in numerical approaches.…”

Section: Introductionmentioning

confidence: 93%

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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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Section: Observablessupporting

confidence: 88%

Section: Introductionmentioning

confidence: 93%

One-dimensional long-range percolation: A numerical study

et al. 2017

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“…The relevance of each solution of the flow equations system in Eq. (18) can be deducted by the study of the perturbation spectrum. For σ > σ * the SR fixed point only displays a single relevant direction (connected with the temperature) and no additional solution exists.…”

Section: Sr-and Lr-dimensionsmentioning

confidence: 99%

“…Then, η is a continuous function of σ and there is no correction to the canonical dimension of the field in the case of LR interactions. It is is fair to say that most of the Monte Carlo (MC) results, based on MC algorithms specific for LR interactions [13][14][15], confirmed this picture [15][16][17][18]. Nevertheless, different behaviours compatible with σ * = 2 were discussed in the literature [19][20][21][22][23].…”

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confidence: 99%

Criticality of spin systems with weak long-range interactions

Defenu¹,

Codello²,

Ruffo³

et al. 2020

J. Phys. A: Math. Theor.

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The study of critical properties of systems with long-range interactions has attracted in the last decades a continuing interest and motivated the development of several analytical and numerical techniques, in particular in connection with spin models. From the point of view of the investigation of their criticality, a special role is played by systems in which the interactions are long-range enough that their universality class is different from the short-range case and, nevertheless, they maintain the extensivity of thermodynamical quantities. Such interactions are often called weak long-range. In this paper we focus on the study of the critical behaviour of spin systems with weak-long range couplings using renormalization group, and we review their remarkable properties. For the sake of clarity and self-consistency, we start from the classical O(N ) spin models and we then move to quantum spin systems. I. INTRODUCTIONIn this Special Issue several aspects of the equilibrium and dynamical properties of systems with long-range (LR) interactions are discussed. In the paper we apply concepts and tools developed for LR systems, including the modern renormalization group approach, to the study of spin models with weak LR interactions. Such interactions are able to modify the universal properties of the systems in which they act, but anyway preserve the extensivity of the thermodynamic quantities. Our main goal is to present and apply the formalism of the functional renormalization group (FRG) to weak LR systems and show the capability of the FRG to clarify their critical properties, which are in many cases still unknown and sometimes diffucult to obtain with other approaches.Given the wide framework already presented in the other contributions to this Special Issue, it is not necessary to extensively emphasize that LR interactions play an important and paradigmatic role in the study of many body interacting systems, such as O(N ) symmetric models. Indeed, the importance of LR interactions is motivated by their presence in several systems ranging from plasma physics to astrophysics and cosmology [1,2].In order to understand the typical phenomena occurring in spin models with power-law LR couplings and define the weak LR regime, let us first consider the classical O(N ) symmetric models, whose Hamiltonian reads(1)The spin variables S i are unit vectors with N components, placed at the sites, labeled by the index i, of a d dimensional lattice. The coupling constant J is constant for a decay exponent σ > 0, conversely for σ ≤ 0 the coupling constant J needs to be rescaled by an appropriate power of the system size to absorb the divergence of the interaction energy density the thermodynamic limit [1,2]. When σ > 0 the model may have a second order phase transition. The main result is that three different regimes can occur as a function of the parameter σ [3, 4]:• for σ ≤ d/2 the mean-field approximation correctly describes the universal behavior;• for σ greater than a threshold value, σ * , the model has the same critical...

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“…Importantly, in disagreement with (2), α is found there to be no different than in NNIM for all σ, thus suggesting a universal nonequilibrium behavior. In equilibrium it is well established both theoretically [19][20][21] and in simulations [22][23][24] that critical exponents are not universal. For example, in the d = 2 LRIM, for σ < 1 the critical exponent η takes its mean-field value, followed by an intermediate range 1 < σ < σ × where it is σ-dependent, and for σ > σ × it behaves like in the NNIM.…”

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confidence: 99%

Phase ordering kinetics of the long-range Ising model

2019

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We use an efficient method that eases the daunting task of simulating dynamics in spin systems with long-range interaction. Our Monte Carlo simulations of the long-range Ising model for the nonequilibrium phase ordering dynamics in two spatial dimensions perform significantly faster than the standard Metropolis approach and considerably more efficiently than the kinetic Monte Carlo method. Importantly, this enables us to establish agreement with the theoretical prediction for the time dependence of domain growth, in contrast to previous numerical studies. This method can easily be generalized to applications in other systems.Generic models of statistical physics exhibiting a transition from disordered to ordered states have been proved to be instrumental for understanding the dynamics in diverse fields, from species evolution [1] to traffic flow [1], from economic dynamics [2] to rainfall dynamics [3]. An extensively used paradigm is the Ising model with nearest-neighbor (NNIM) interaction [4,5]. Even the complex neural dynamics of brain depends on similar underlying mechanisms [6]. The maximum entropy models obtained from experimental data upon mapping the spiking activities of the neurons onto spin variables are equivalent to Ising models [7]. However, it is believed that the neuron activities are effectively modelled by longdistance communications [6]. In nature, also many other intermolecular interactions are evidently long-range, e.g., electrostatic forces, polarization forces, etc. Hence, a more complete picture calls for employing models that consider long-range interactions.The simplest generic model system is the long-range Ising model (LRIM), which on a d-dimensional lattice is described by the Hamiltonian

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Upper and lower critical decay exponents of Ising ferromagnets with long-range interaction

Cited by 50 publications

References 36 publications

One-dimensional long-range percolation: A numerical study

One-dimensional long-range percolation: A numerical study

Criticality of spin systems with weak long-range interactions

Phase ordering kinetics of the long-range Ising model

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