Invaded cluster algorithm for critical properties of periodic and aperiodic planar Ising models

Redner, Oliver; Baake, Michael

doi:10.1088/0305-4470/33/16/304

Cited by 8 publications

(10 citation statements)

References 29 publications

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“…It is the Fourier transform of the connected pair correlation function, defined by what is inside the square brackets. Furthermore,χ(q) is 3 Universality of the critical exponents of ferromagnetic Ising models on quasiperiodic lattices has been confirmed for non-Z-invariant cases also using real-space renormalization group techniques, (22) Monte Carlo simulations, (23,24,25,26,27,28,29) series expansion methods, (30,31,32) and the study of Yang-Lee zeros. (33,34,35) 4 Unlike the q-dependent susceptibility, thermodynamic quantities like the free energy, the specific heat, and the bulk susceptibility do not probe the lattice structure, although subtle lattice effects do show up in corrections to scaling.…”

Section: Introductionmentioning

confidence: 86%

Wavevector-Dependent Susceptibility in Z-Invariant Pentagrid Ising Model

Au-Yang

Perk

2007

J Stat Phys

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We study the q-dependent susceptibility χ(q) of a Z-invariant ferromagnetic Ising model on a Penrose tiling, as first introduced by Korepin using de Bruijn's pentagrid for the rapidity lines. The pair-correlation function for this model can be calculated exactly using the quadratic difference equations from our previous papers. Its Fourier transform χ(q) is studied using a novel way to calculate the joint probability for the pentagrid neighborhoods of the two spins, reducing this calculation to linear programming. Since the lattice is quasiperiodic, we find that χ(q) is aperiodic and has everywhere dense peaks, which are not all visible at very low or high temperatures. More and more peaks become visible as the correlation length increases-that is, as the temperature approaches the critical temperature.

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Section: Introductionmentioning

confidence: 86%

Wavevector-Dependent Susceptibility in Z-Invariant Pentagrid Ising Model

Au-Yang

Perk

2007

J Stat Phys

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“…It is obtained from the average over p {σ} defined in the Eq. (7). Data for p c (L) with a three-parameter fit are illustrated in Figure 5.…”

Section: B the Thermal Critical Exponent Yt And Critical Temperaturementioning

confidence: 99%

“…Our EIC algorithm approach follows the same IC procedure but restrains the excessive fluctuations in p {σ} in order to regain the equilibrium fluctuations. The main idea is to constrain the width of distribution of p {σ} obtained from the simulations by the relation (7) to the width which is compatible with the distribution of b {σ} in (6) and which has the same scaling in L. The width of the binomial distribution in Eq. (6) gives…”

Section: Algorithmmentioning

confidence: 99%

“…An interesting direction undertaken by Machta et al [5] was to construct an algorithm which would, in a self organized way, drive the system to critical temperature without prior knowledge of it. It was applied to various classical models from discrete to continuous ones including some more complicated effects such as frustration [6], quasi periodic ordering [7], or tricritical points [8]. Their method, based on the invasion percolation [9] and the cluster algorithm, appeared to converge much faster than in the Swendsen-Wang (SW) algorithm, with a dynamical exponent z close to zero.…”

Section: Introductionmentioning

confidence: 99%

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Equilibriumlike invaded cluster algorithm: Critical exponents and dynamical properties

Balog

Uzelac

2010

Phys. Rev. E

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We present a detailed study of the Equilibriumlike invaded cluster algorithm (EIC), recently proposed as an extension of the invaded cluster (IC) algorithm, designed to drive the system to criticality while still preserving the equilibrium ensemble. We perform extensive simulations on two special cases of the Potts model and examine the precision of critical exponents by including the leading corrections. We show that both thermal and magnetic critical exponents can be obtained with high accuracy compared to the best available results. The choice of the auxiliary parameters of the algorithm is discussed in context of dynamical properties. We also discuss the relation to the Li-Sokal bound for the dynamical exponent z.

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“…[1][2][3][4][5] This algorithm, and others of the same general approach, 6,7 have the property that without prior knowledge of the critical temperature they evolve a spin system to the vicinity of the critical temperature. Let us briefly review the ICA algorithm.…”

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confidence: 99%

Locally converging algorithms for determining the critical temperature in Ising systems

Faraggi

Robb

2008

Phys. Rev. B

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We introduce a class of algorithms that converge to criticality automatically, in a way similar to the invaded cluster algorithm. Unlike the invaded cluster algorithm which uses global percolation as a test for criticality, these local algorithms use an average over local observables, specifically the number of satisfied bonds, in a feedback loop which drives the system toward criticality. Two specific algorithms are introduced, the average algorithm and the locally converging Wolff algorithm. We apply these algorithms to study the Ising square lattice and the Ising Bethe lattice. We find reasonable convergence to the critical temperature for both systems under the locally converging Wolff algorithm. We also re-examine the phase diagram of the dilute twodimensional ͑2D͒ Ising model and find results supporting our previously reported conclusions regarding the existence of a local regime of magnetization below the percolations threshold. In addition, the presented algorithms are computationally more efficient than the invaded cluster algorithm, requiring less CPU time and memory.A useful technique in the computational study of phase transitions in spin systems is the invaded cluster algorithm ͑ICA͒. 1-5 This algorithm, and others of the same general approach, 6,7 have the property that without prior knowledge of the critical temperature they evolve a spin system to the vicinity of the critical temperature. Let us briefly review the ICA algorithm. Starting in an arbitrary spin state, for example the positive aligned state, the ICA forms one by one a collection of satisfied bonds from the set of all satisfied bonds associated with the spin state. After each bond is added to a cluster, the ICA tests whether this cluster spans the lattice, i.e., has an extent that is of the order of the size of the system. When such a spanning cluster is found, the ICA performs the final step of the Swendsen-Wang ͑SW͒ algorithm 8 using this set of clusters. This consists of assigning a random spin value to each of the clusters formed-for the two-state Ising model considered in this paper, this is equivalent to flipping each cluster's spin with probability 1/2.The SW algorithm relies on the relationship p͑T c ͒ =1 − e −2J/k b T c , established by Fortuin and Kasteleyn, 9 between the critical temperature T c of the Potts spin lattice with exchange coupling J and the critical bond probability p of the associated ͑multispecies͒ bond percolation problem. When the ICA algorithm acts on a spin state typical of T Ͻ T c , because the state is relatively ordered, the fraction of satisfied bonds needed to achieve percolation is lower than the fraction needed at a temperature of T c . Through the FortuinKastelyn relationship, this small fraction of satisfied bonds results in the execution of a SW Monte Carlo step at a high temperature. In a similar way, the ICA generates a large fraction of satisfied bonds when applied to a spin state at T Ͼ T c , and so executes a SW step at a temperature T Ͻ T c . As the ICA algorithm is repeated, the temperatur...

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Invaded cluster algorithm for critical properties of periodic and aperiodic planar Ising models

Cited by 8 publications

References 29 publications

Wavevector-Dependent Susceptibility in Z-Invariant Pentagrid Ising Model

Wavevector-Dependent Susceptibility in Z-Invariant Pentagrid Ising Model

Equilibriumlike invaded cluster algorithm: Critical exponents and dynamical properties

Locally converging algorithms for determining the critical temperature in Ising systems

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