Role of Fourier Modes in Finite-Size Scaling above the Upper Critical Dimension

Flores-Sola, Emilio; Berche, Bertrand; Kenna, Ralph; Weigel, Martin

doi:10.1103/physrevlett.116.115701

Cited by 42 publications

(94 citation statements)

References 38 publications

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“…(14), occurring in LR systems. Similar deviations were already noticed in LR classical systems [55,56], but their consequences appear to be much more striking in the dynamics of quantum systems.…”

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confidence: 81%

Universal dynamical scaling of long-range topological superconductors

et al. 2019

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We study the out-of-equilibrium dynamics of p-wave superconducting quantum wires with longrange interactions, when the chemical potential is linearly ramped across the topological phase transition. We show that the heat produced after the quench scales with the quench rate δ according to the scaling law δ θ , where the exponent θ depends on the power law exponent of the longrange interactions. We identify the parameter regimes where this scaling can be cast in terms of the universal equilibrium critical exponents and can thus be understood within the Kibble-Zurek framework. When the electron hopping decays more slowly in space than pairing, it dominates the equilibrium scaling. Surprisingly, in this regime the dynamical critical behaviour arises only from paring and, thus, exhibits anomalous dynamical universality unrelated to equilibrium scaling. The discrepancy from the expected Kibble-Zurek scenario can be traced back to the presence of multiple universal terms in the equilibrium scaling functions of long-range interacting systems close to a second order critical point.

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confidence: 81%

Universal dynamical scaling of long-range topological superconductors

et al. 2019

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“…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%

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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“…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.…”

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confidence: 99%

“…In this Letter, we apply a geometric approach to reexamine a long-standing debate concerning the FSS of the n-vector model with d > d c [5][6][7][8][9][10][11][12]. The majority of this debate has focused on the boundary-dependent FSS of the ferromagnetic Ising model when d > d c ; particularly on the case d = 5.…”

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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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Role of Fourier Modes in Finite-Size Scaling above the Upper Critical Dimension

Cited by 42 publications

References 38 publications

Universal dynamical scaling of long-range topological superconductors

Universal dynamical scaling of long-range topological superconductors

Universal finite-size scaling for percolation theory in high dimensions

Geometric Explanation of Anomalous Finite-Size Scaling in High Dimensions

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