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Violations of Hyperscaling in Phase Transitions and Critical Phenomena

Published Sep 24, 2025• 0 citations
Publisher

Abstract

This Special Issue was initiated by Prof. Ralph Kenna (August 27, 1964October 26, 2023), who, at the time, was a member of the Entropy Editorial Board and it is dedicated to his memory. Universality is a striking feature of critical phenomena, and its origin is largely illuminated by the renormalization group. Universality allows for the essential aspects of critical behaviour in complex physical systems to be captured through highly simplified models, where only key components such as spatial dimensionality, symmetry, and interaction range are retained. Both theoretical models and real systems are thus grouped into universality classes, which are characterised by sets of critical exponents connected through scaling relations. When these relations explicitly involve dimensionality, they are referred to as hyperscaling.In many systems with a high degree of connectivity, mean-field theory provides accurate predictions independently of spatial dimension, leading to the common assertion that hyperscaling breaks down in such cases. However, this conventional view has been reconsidered in light of recent renormalization group developments, which suggest mechanisms by which hyperscaling might be preserved in an extended version even above the upper critical dimension. The contributions collected in this reprint address these issues, focusing on scaling properties beyond the upper critical dimensionality as well as related challenges in the broader theory of phase transitions and critical phenomena.

DOI
10.3390/books978-3-7258-5148-5

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This Special Issue was initiated by Prof. Ralph Kenna (August 27, 1964October 26, 2023), who, at the time, was a member of the Entropy Editorial Board and it is dedicated to his memory. Universality is a striking feature of critical phenomena, and its origin is largely illuminated by the renormalization group. Universality allows for the essential aspects of critical behaviour in complex physical systems to be captured through highly simplified models, where only key components such as spatial dimensionality, symmetry, and interaction range are retained. Both theoretical models and real systems are thus grouped into universality classes, which are characterised by sets of critical exponents connected through scaling relations. When these relations explicitly involve dimensionality, they are referred to as hyperscaling.In many systems with a high degree of connectivity, mean-field theory provides accurate predictions independently of spatial dimension, leading to the common assertion that hyperscaling breaks down in such cases. However, this conventional view has been reconsidered in light of recent renormalization group developments, which suggest mechanisms by which hyperscaling might be preserved in an extended version even above the upper critical dimension. The contributions collected in this reprint address these issues, focusing on scaling properties beyond the upper critical dimensionality as well as related challenges in the broader theory of phase transitions and critical phenomena.
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.
These lecture notes provide an overview of the renormalization group (RG) as a successful framework to understand critical phenomena above the upper critical dimension ducd_{\rm uc}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 ducd_{\rm uc}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 ducd_{\rm uc}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.
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.
Memory is a ubiquitous characteristic of complex systems, and critical phenomena are one of the most intriguing phenomena in nature. Here, we propose an Ising model with memory, develop a corresponding theory of critical phenomena with memory for complex systems, and discover a series of surprising novel results. We show that a naive theory of a usual Hamiltonian with a direct inclusion of a power-law decaying long-range temporal interaction violates radically a hyperscaling law for all spatial dimensions even at and below the upper critical dimension. This entails both indispensable consideration of the Hamiltonian for dynamics, rather than the usual practice of just focusing on the corresponding dynamic Lagrangian alone, and transformations that result in a correct theory in which space and time are inextricably interwoven, leading to an effective spatial dimension that repairs the hyperscaling law. The theory gives rise to a set of novel mean-field critical exponents, which are different from the usual Landau ones, as well as new universality classes. These exponents are verified by numerical simulations of the Ising model with memory in two and three spatial dimensions.
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.