Although well described by mean-field theory in the thermodynamic limit, scaling has long been puzzling for finite systems in high dimensions. This raised questions about the efficacy of the renormalization group and foundational concepts such as universality, finite-size scaling and hyperscaling, until recently believed not to be applicable above the upper critical dimension. Significant theoretical progress has been made resolving these issues, and tested in numerous simulational studies of spin models. This progress rests upon superlinearity of correlation length, a notion that for a long time encountered resistance but is now broadly accepted. Percolation theory brings added complications such as proliferation of interpenetrating clusters in apparent conflict with suggestions coming from random-graph asymptotics and a dearth of reliable simulational guidance. Here we report on recent theoretical progress in percolation theory in the renormalization group framework in high dimensions that accommodates superlinear correlation and renders most of the above concepts mutually compatible under different boundary conditions. Results from numerical simulations for free and periodic boundary conditions which differentiate between previously competing theories are also presented. Although still fragmentary, these Monte Carlo results support the new framework which restores the renormalization group and foundational concepts on which it rests.
Although well described by mean-field theory in the thermodynamic limit, scaling has long been puzzling for finite systems in high dimensions. This raised questions about the efficacy of the renormalization group and foundational concepts such as universality, finite-size scaling and hyperscaling, until recently believed not to be applicable above the upper critical dimension. Significant theoretical progress has been made resolving these issues, and tested in numerous simulational studies of spin models. This progress rests upon superlinearity of correlation length, a notion that for a long time encountered resistance but is now broadly accepted. Percolation theory brings added complications such as proliferation of interpenetrating clusters in apparent conflict with suggestions coming from random-graph asymptotics and a dearth of reliable simulational guidance. Here we report on recent theoretical progress in percolation theory in the renormalization group framework in high dimensions that accommodates superlinear correlation and renders most of the above concepts mutually compatible under different boundary conditions. Results from numerical simulations for free and periodic boundary conditions which differentiate between previously competing theories are also presented. Although still fragmentary, these Monte Carlo results support the new framework which restores the renormalization group and foundational concepts on which it rests.
Abstract 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 = 2 d /3, and obey random-graph asymptotics. Universality is instead manifested 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.
These lecture notes provide an overview of the renormalization group (RG) as a successful framework to understand critical phenomena above the upper critical dimension duc. After an introduction to the scaling picture of continuous phase transitions, we discuss the apparent failure of the Gaussian fixed point to capture scaling for Landau mean-field theory, which should hold in the thermodynamic limit above duc. We recount how Fisher's dangerous-irrelevant-variable formalism applied to thermodynamic functions partially repairs the situation but at the expense of hyperscaling and finite-size scaling, both of which were, until recently, believed not to apply above duc. We recall limitations of various attempts to match the RG with analytical and numerical results for Ising systems. We explain how the extension of dangerous irrelevancy to the correlation sector is key to marrying the above concepts into a comprehensive RG scaling picture that renders hyperscaling and finite-size scaling valid in all dimensions. We collect what we believe is the current status of the theory, including some new insights and results. This paper is in grateful memory of Michael Fisher who introduced many of the concepts discussed and who, half a century later, contributed to their advancement.
These lecture notes provide an overview of the renormalization group (RG) as a successful framework to understand critical phenomena above the upper critical dimension d_{uc} d u c . After an introduction to the scaling picture of continuous phase transitions, we discuss the apparent failure of the Gaussian fixed point to capture scaling for Landau mean-field theory, which should hold in the thermodynamic limit above d_{uc} d u c . We recount how Fisher’s dangerous-irrelevant-variable formalism applied to thermodynamic functions partially repairs the situation but at the expense of hyperscaling and finite-size scaling, both of which were, until recently, believed not to apply above d_{uc} d u c . We recall limitations of various attempts to match the RG with analytical and numerical results for Ising systems. We explain how the extension of dangerous irrelevancy to the correlation sector is key to marrying the above concepts into a comprehensive RG scaling picture that renders hyperscaling and finite-size scaling valid in all dimensions. We collect what we believe is the current status of the theory, including some new insights and results. This paper is in grateful memory of Michael Fisher who introduced many of the concepts discussed and who, half a century later, contributed to their advancement.
Scaling, hyperscaling and finite-size scaling were long considered problematic in theories of critical phenomena in high dimensions. The scaling relations themselves form a model-independent structure that any model-specific theory must adhere to, and they are accounted for by the simple principle of homogeneity. Finite-size scaling is similarly founded on the fundamental idea that only two length scales enter the game — namely system length and correlation length. While all scaling relations are quite satisfactory for multitudes of physical systems in low dimensions, one fails in high dimensions. The aberrant scaling relation is called hyperscaling and involves dimensionality itself. Finite-size scaling also appears to fail in high dimensions. Developed in the 1930s, Landau mean-field theory is valid in such high-dimensional systems. However, it too does not accord with hyperscaling and finite-size scaling there. The advent of renormalization-group theory in the 1970s brought deeper fundamental insights into critical phenomena, allowing systems to be viewed at different scales. Above a critical dimensionality, higher-order Renormalization Group (RG) eigenvalues become irrelevant and scaling is governed by the Gaussian fixed point. Although obeying all scaling relations including hyperscaling, and although it appears to successfully explain scaling in the correlation sector, the Gaussian fixed point fails to capture the free energy and derivatives, even in infinite volume. In the 1980s, to fix this for the magnetisation, specific heat and susceptibility, Fisher introduced the notion of dangerous irrelevant variables. Since the correlation sector did not appear to be broken, no attempt was made to repair it and Fisher’s modified RG formalism worked quite well in the thermodynamic limit of infinite volume. However, finite-size scaling fails. Also in the 1980’s, Binder, Nauenberg, Privman and Young extended Fisher’s concept to the free energy itself and to finite-size systems. While putting Fisher’s ideas on a more fundamental footing, the failure of finite-size scaling there still presented a problem. This appeared to be resolved by the introduction of “thermodynamic length” to replace correlation length as the length scale that controls Finite-Size Scaling (FSS). Thus hyperscaling and FSS were both sacrificed in favour of the RG patched together by ad hoc solutions. In the 1990’s, Luijten and Blöte went a long way to resolving the dilemma by adding corrections to scaling to the above considerations. It was clear that both of these played a role. However, as with previous authors, and adhering to the principle of not fixing that appears not to be broken, they did not address correlation length directly. Here we report on developments over the past decade which went a long way to addressing these long-standing problems The key to unlocking these, and extending their validity to the high-dimensional regime, was to relax assumptions that the correlation length has to be bounded by the physical length of bounded systems. This allowed and necessitated the extension of Fisher’s concept of dangerous irrelevant variables to the correlation sector.
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. 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 also lead to the introduction of a new scaling relation analogous to one introduced by Fisher 50 years ago.
For statistical mechanical systems with continuous phase transitions, there are two closely related but subtly different mean-field treatments, the Gaussian fixed point (GFP) in the renormalization group framework and the Landau mean-field theory or the complete-graph (CG) asymptotics. By large-scale Monte Carlo simulations, we systematically study the interplay of the GFP and CG effects to the finite-size scaling of percolation above the upper critical dimension d_{c}=6 with periodic and cylindrical boundary conditions. Our results suggest that, with periodic boundaries, the unwrapped correlation length scales as L^{d/6} at the critical point, diverging faster than L above d_{c}. As a consequence, the scaling behaviors of macroscopic quantities with respect to the linear system size L follow the CG asymptotics. The distance-dependent properties, such as the short-distance behavior of the two-point correlation function and the Fourier transformed quantities with nonzero modes, are still controlled by the GFP. With cylindrical boundaries, due to the interplay of the GFP and CG effects, the correlation length along the axial direction of the cylinder scales as ξ_{L}∼L^{(d-1)/5} within the critical window of size O(L^{-2(d-1)/5}), distinct from periodic boundary. A field-theoretical calculation for deriving the scaling of ξ_{L} is also presented. Moreover, the one-point surface correlation function along the axial direction of the cylinder is observed to scale as τ^{(1-d)/2} when the distance τ is short, but then enter a plateau of order L^{-3(d-1)/5} before it decays significantly fast.