Geometric Explanation of Anomalous Finite-Size Scaling in High Dimensions

Abstract

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.

Random-Length Random Walks and Finite-Size Scaling in High Dimensions

“…No such definition has previously been studied however. We introduce such a definition here (see also [27]), and verify numerically the universality of the prediction (4); see Section 4. Combining (4) with (3) leads to the expectation that the Ising and SAW two-point functions satisfy…”

“…In this article, we focus on PBCs; we defer the discussion of the FBC case to [17]. In recent years, several studies [13,14,18] have investigated the FSS behaviour of the critical Ising two-point function g Ising,crit (x) above d c . It was conjectured in [18], based on random-current and random-path representations [4,5,6] of the Ising model, that the Ising two-point function g Ising,crit (x) satisfies the piecewise asymptotic behaviour g Ising,crit (x) ≈ c 1…”

The length of self-avoiding walks on the complete graph

“…Cases (i) and (ii) then follow by combining with (16) and (20), respectively. In Cases (iii) and (iv), Eq.…”

Finite-size scaling of O(n) systems at the upper critical dimensionality

“…Since the length ξ 1 is vanishingly small compared to the linear size, ξ 1 / L → 0, the plateau effectively dominates the scaling behavior of g ( r , L ) and the FSS of χ. The two-length scaling form ( 8 ) has been numerically confirmed for the 5D Ising model and self-avoiding random walk, with a geometric explanation based on the introduction of an unwrapped length on the torus [ 18 ]. It is also consistent with the rigorous calculations for the so-called random-length random-walk model [ 20 ].…”

“…and the FSS of the free-energy density f ( t , h ) becomes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}\begin{equation*} f(t,h) = L^{-d} \tilde{f} (t L^{y_t^{*}}, hL^{y_h^{*}}). \end{equation*}\end{document} In this scenario of the dangerously irrelevant field, the FSS of the critical susceptibility becomes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\chi \asymp L^{2y_h^*-d} = L^{d/2}$\end{document} , consistent with the numerical results [ 13–15 , 17 , 18 ]. It was further assumed that the scaling behavior of g ( r , L ) is modified as [ 16 ] \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}\begin{equation*} g(r,L) \asymp r^{-(d-2+\eta _Q)} \tilde{g}(r/L) \end{equation*}\end{document} with η Q = 2 − d /2, such that the decay of g ( r , L ) is no longer Gaussian-like.…”