Non-vanishing boundary effects and quasi-first-order phase transitions in high dimensional Ising models

Lundow, Per-Håkan; Markström, Klas

doi:10.1016/j.nuclphysb.2010.12.002

Cited by 42 publications

(87 citation statements)

References 30 publications

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“…However, it agrees with the finding in [16] that for the Ising model with cyclic boundary in 5d the energy distribution at the critical temperature becomes bimodal. The second plot in Figure 4 shows the distribution at κ = 0.44 for L = 6, 8, 10, 12, 16 and it clearly shows the distribution retaining its bimodal form with increasing L. …”

Section: A the Largest Clustersupporting

confidence: 92%

Complete graph asymptotics for the Ising and random-cluster models on five-dimensional grids with a cyclic boundary

Lundow

Markström

2015

Phys. Rev. E

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The finite size scaling behaviour for the Ising model in five dimensions, with either free or cyclic boundary, has been the subject for a long running debate. The older papers have been based on ideas from e.g. field theory or renormalization. In this paper we propose a detailed and exact scaling picture for critical region of the model with cyclic boundary. Unlike the previous papers our approach is based on a comparison with the existing exact and rigorous results for the FKrandom-cluster model on a complete graph. Based on those results we identify several distinct scaling regions in an L-dependent window around the critical point. We test these predictions by comparing with data from Monte Carlo simulations and find a good agreement. The main feature which differs between the complete graph and the five dimensional model with free boundary is the existence of a bimodal energy distribution near the critical point in the latter. This feature was found by the same authors in an earlier paper in the form of a quasi-first order phase transition for the same Ising model. * Electronic address: per.hakan.lundow@math.umu.se † Electronic address: klas.markstrom@math.umu.se 1 arXiv:1408.2155v1 [cond-mat.stat-mech]

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Section: A the Largest Clustersupporting

confidence: 92%

Complete graph asymptotics for the Ising and random-cluster models on five-dimensional grids with a cyclic boundary

Lundow

Markström

2015

Phys. Rev. E

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“…53,54,60,61 The standard picture is that the FBC case is governed by Gaussian FSS at the critical point, in which χ L ∼ L 2 due to a belief that the second expression in Eq. (6.45) "cannot hold for FBC's because it lies above a strict upper bound" 60 (namely L γ/ν = L 2 ) established in Ref.…”

Section: An Unsatisfactory Paradigmmentioning

confidence: 99%

“…A recent numerical study of the five-dimensional Ising model with FBC's supports Gaussian FSS χ L ∼ L 2 at the critical point T c . 61 However, a number of unsettling issues with the standard picture arise. Firstly, the mechanism outlined in Sec.…”

Section: An Unsatisfactory Paradigmmentioning

confidence: 99%

Scaling and Finite-Size Scaling above the Upper Critical Dimension

Kenna¹,

Berche²

2015

Order, Disorder and Criticality

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In the 1960's, four famous scaling relations were developed which relate the six standard critical exponents describing continuous phase transitions in the thermodynamic limit of statistical physics models. They are well understood at a fundamental level through the renormalization group. They have been verified in multitudes of theoretical, computational and experimental studies and are firmly established and profoundly important for our understanding of critical phenomena. One of the scaling relations, hyperscaling, fails above the upper critical dimension. There, critical phenomena are governed by Gaussian fixed points in the renormalization-group formalism. Dangerous irrelevant variables are required to deliver the mean-field and Landau values of the critical exponents, which are deemed valid by the Ginzburg criterion. Also above the upper critical dimension, the standard picture is that, unlike for low-dimensional systems, finite-size scaling is non-universal, at least at the critical point. Here we report on new developments which indicate that the current paradigm is flawed and incomplete. In particular, the introduction of a new exponent characterising the finite-size correlation length allows one to extend hyperscaling beyond the upper critical dimension. Moreover, finite-size scaling is shown to be universal provided the correct scaling window is chosen. These recent developments above the upper critical dimension also lead to the introduction of a new scaling relation analogous to one introduced by Fisher 50 years ago and * r.kenna@coventry.ac.uk † bertrand.berche@univ-lorraine.fr 1 Order, Disorder and Criticality Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 08/22/15. For personal use only. 2 R. Kenna and B. Berche deliver a statistical physics explanation for the emergence of effective four dimensionality as characteristic of generic field theories.

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“…where γ = 1, β = 1/2 and δ = 3 for d = 5 [29,30,41,42]. The log-log plots of M (L) at T = T c (L, h) versus h for h in the interval 0 ≤ h ≤ 0.0025 yield to 1/δ(L) (Fig.…”

Section: Resultsmentioning

confidence: 97%

“…In addition, the four-dimensional ferromagnetic Ising model solution is approximated by using Creutz cellular automaton algorithm with nearest neighbor interactions and near the critical region [14][15][16][17][18][19][20][21][22][23]. The algorithm of approximating finite size behavior of ferromagnetic Ising model is extended to higher dimensions [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. It is established that the algorithm has been powerful in terms of providing the values of static critical exponents near the critical region in four and higher dimensions with nearest neighbor interactions [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]…”

Section: Introductionmentioning

confidence: 99%

The Finite-Size Scaling Study of Five-Dimensional Ising Model

2016

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The five-dimensional ferromagnetic Ising model is simulated on the Creutz cellular automaton algorithm using finite-size lattices with linear dimension 4 ≤ L ≤ 8. The critical temperature value of infinite lattice is found to be T χ (∞) = 8.7811 (1) using 4 ≤ L ≤ 8 which is also in very good agreement with the precise result. The value of the field critical exponent (δ = 3.0067 (2)) is good agreement with δ = 3 which is obtained from scaling law of Widom. The exponents in the finite-size scaling relations for the magnetic susceptibility and the order parameter at the infinite-lattice critical temperature are computed to be 2.5080 (1), 2.5005 (3) and 1.2501 (1) using 4 ≤ L ≤ 8, respectively, which are in very good agreement with the theoretical predictions of . The finite-size scaling plots of magnetic susceptibility and the order parameter verify the finite-size scaling relations about the infinitelattice temperature.

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Non-vanishing boundary effects and quasi-first-order phase transitions in high dimensional Ising models

Cited by 42 publications

References 30 publications

Complete graph asymptotics for the Ising and random-cluster models on five-dimensional grids with a cyclic boundary

Complete graph asymptotics for the Ising and random-cluster models on five-dimensional grids with a cyclic boundary

Scaling and Finite-Size Scaling above the Upper Critical Dimension

The Finite-Size Scaling Study of Five-Dimensional Ising Model

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