Scaling and Finite-Size Scaling above the Upper Critical Dimension

Kenna, Ralph; Berche, Bertrand

doi:10.1142/9789814632683_0001

Cited by 9 publications

(28 citation statements)

References 44 publications

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“…Here we have seen, however, that the pseudocritical point is key to linking the various sectors, for it is at p L and not at p ∞ that universality resides [35][36][37][38][39][40]. The existence of the non-trivial, universal, physical, pseudocritical exponent ϙ, which enters a corrected or extended hyperscaling relation, ensures that the largest clusters at p L are universally described by X = 0 and D = 2d/3 when measured on the scale L of the underlying lattice.…”

Section: Discussionmentioning

confidence: 72%

“…Under the new Q-scheme [35][36][37][38][39][40], the thermodynamic length is obviated and hyperscaling (in a new form), finite-size scaling as well as universality all hold above the upper critical dimension. A crucial ingredient that differentiates Q-theory from the earlier set-up is that dangerous irrelevant variables are also allowed to affect the correlation sector of the model.…”

Section: Universal and Non-universal Fss For φ 4 -Theory In High Dimementioning

confidence: 99%

“…Recently, we introduced a new formulation for scaling and finite-size scaling above the upper critical dimension [35][36][37][38][39][40]. The new framework is variously termed Q-scaling or Q-FSS or simply Q-theory to distinguish it from the older scaling scenario and its modifications [41].…”

Section: Introductionmentioning

confidence: 99%

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Universal finite-size scaling for percolation theory in high dimensions

Kenna¹,

Berche²

2017

J. Phys. A: Math. Theor.

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We present a unifying, consistent, finite-size-scaling picture for percolation theory bringing it into the framework of a general, renormalization-group-based, scaling scheme for systems above their upper critical dimensions d c . Behaviour at the critical point is non-universal in d > d c = 6 dimensions. Proliferation of the largest clusters, with fractal dimension 4, is associated with the breakdown of hyperscaling there when free boundary conditions are used. But when the boundary conditions are periodic, the maximal clusters have dimension D = 2d/3, and obey random-graph asymptotics. Universality is instead manifest at the pseudocritical point, where the failure of hyperscaling in its traditional form is universally associated with random-graph-type asymptotics for critical cluster sizes, independent of boundary conditions.

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Section: Discussionmentioning

confidence: 72%

Section: Universal and Non-universal Fss For φ 4 -Theory In High Dimementioning

confidence: 99%

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Universal finite-size scaling for percolation theory in high dimensions

Kenna¹,

Berche²

2017

J. Phys. A: Math. Theor.

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“…Moreover, with periodic boundaries, it was numerically observed [10] that the correlation between spins at distance L 2 scales like L −d 2 , in contrast to the standard mean-field prediction L −(d−2) expected for free boundaries. Recent renormalizationgroup arguments attempt to explain this anomalous FSS by postulating a modified scaling of the correlation length ξ ∼ L ϙ when d > d c , where ϙ ∶= d d c [17]. Moreover, an additional exponent η Q , related to ϙ, was introduced to explain the anomalous large-distance behaviour of the spin-spin correlation function.…”

mentioning

confidence: 99%

Geometric Explanation of Anomalous Finite-Size Scaling in High Dimensions

Grimm¹,

Elçi²,

Zhou³

et al. 2017

Phys. Rev. Lett.

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We give an intuitive geometric explanation for the apparent breakdown of standard finite-size scaling in systems with periodic boundaries above the upper critical dimension. The Ising model and self-avoiding walk are simulated on five-dimensional hypercubic lattices with free and periodic boundary conditions, by using geometric representations and recently introduced Markov-chain Monte Carlo algorithms. We show that previously observed anomalous behaviour for correlation functions, measured on the standard Euclidean scale, can be removed by defining correlation functions on a scale which correctly accounts for windings.Finite-size Scaling (FSS) is a fundamental physical theory within statistical mechanics, describing the asymptotic approach to the thermodynamic limit of finite systems in the neighbourhood of a critical phase transi-It is well-known [3] that models of critical phenomena typically possess an upper critical dimension, d c , such that in dimensions d ≥ d c , their thermodynamic behaviour is governed by critical exponents taking simple mean-field values [4]. In contrast to the simplicity of the thermodynamic behaviour, however, the theory of FSS in dimensions above d c is surprisingly subtle, and remains the subject of ongoing debate [5][6][7][8][9][10][11][12]. We will show here that such subtleties can be explained in a simple way, by taking an appropriate geometric perspective.Perhaps the most important class of models in equilibrium statistical mechanics are the n-vector models [13], describing systems of pairwise-interacting unit-vector spins in R n [14]. The cases n = 1, 2, 3 respectively correspond to the Ising, XY and Heisenberg models of ferromagnetism, while the limiting case n = 0 corresponds to the Self-avoiding Walk (SAW) model of polymers [3].The n-vector model has wide-ranging applications in condensed matter physics, particularly in the theory of superfluidity/superconductivity and quantum magnetism. In addition, the case n = 2 is related to the Bose-Hubbard model [15] which is actively studied in the field of ultra-cold atom physics. In such quantum applications, the quantum system in d spatial dimensions is related to the classical model in d + 1 dimensions. Since [3] d c = 4 for the nearest-neighbour n-vector model, this shows that understanding its FSS when d ≥ d c is of importance not only to the theory of FSS itself, but also in the field of condensed matter physics more generally. We also note that the value of d c can be reduced by the introduction of long-range interactions.

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“…Above the upper critical dimension the finite-size laws have been recently studied in Refs. [33][34][35]. This regime, in the case of the one-dimensional Ising model with long-range interactions, is given by the condition σ < 1/2.…”

Section: B Finite-size Scaling Theory In the Mean-field Regionmentioning

confidence: 99%

Size effects in finite systems with long-range interactions

Loscar

Horowitz

2018

Phys. Rev. E

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Small systems consisting of particles interacting with long-range potentials exhibit enormous size effects. The Tsallis conjecture [Tsallis, Fractals 3, 541 (1995)FRACEG0218-348X10.1142/S0218348X95000473], valid for translationally invariant systems with long-range interactions, states a well-known scaling relating different sizes. Here we propose to generalize this conjecture to systems with this symmetry broken, by adjusting one parameter that determines an effective distance to compute the strength of the interaction. We apply this proposal to the one-dimensional Ising model with ferromagnetic interactions that decay as 1/r^{1+σ} in the region where the model has a finite critical temperature. We demonstrate the convenience of using this generalization to study finite-size effects, and we compare this approach with the finite-size scaling theory.

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Scaling and Finite-Size Scaling above the Upper Critical Dimension

Cited by 9 publications

References 44 publications

Universal finite-size scaling for percolation theory in high dimensions

Universal finite-size scaling for percolation theory in high dimensions

Geometric Explanation of Anomalous Finite-Size Scaling in High Dimensions

Size effects in finite systems with long-range interactions

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