Abstract We systematically develop theories of critical phenomena with a prior formed memory of a power-law decaying long-range temporal interaction parameterized by a constant θ > 0 for a space dimension d both below and above an upper critical dimension d c = 6 − 2/θ. We first provide more evidences to confirm the previous theory that a dimensional constant d t is demanded to rectify a hyperscaling law, to produce correct unique mean-field critical exponents via an effective spatial dimension originating from temporal dimension, and to transform the time and change the dynamic critical exponent. Next, for d < d c , we develop a renormalization-group theory by employing the momentum-shell technique to the leading nontrivial order explicitly and to higher orders formally in ϵ = d c − d but to zero order in ε = 1 − θ and find that more scaling laws besides the hyperscaling law are broken due to the breaking of the fluctuation-dissipation theorem. Moreover, because dynamics and statics are intimately interwoven, even the static critical exponents involve contributions from the dynamics and hence do not restore the short-range exponents even for θ = 1 and the crossover between the short-range and long-range fixed points is discontinuous contrary to the case of long-range spatial interaction. In addition, a new scaling law relating the dynamic critical exponent with the static ones emerges, indicating that the dynamic critical exponent is not independent. However, once d t is displaced by a series of ϵ and ε such that most values of the critical exponents are changed, all scaling laws are saved again, even though the fluctuation-dissipation theorem keeps violating. Then, for d ≥ d c , we develop an effective-dimension theory by carefully discriminating the corrections of both temporal and spatial dimensions and find three different regions. For d ≥ d c0 = 4, the upper critical dimension of the usual short-range theory, the usual Landau mean-field theory with fluctuations confined to the effective-dimension equal to d c0 correctly describe the critical phenomena with memory, while for d c < d ≤ 4, there exist new universality classes whose critical exponents depend only on the space dimension but not at all on θ. Yet another region consists of d = d c only and the previous theory is retrieved. All these results show that the dimensional constant d t is the fundamental ingredient of the theories for critical phenomena with memory. However, its value continuously varies with the space dimension and vanishes exactly at d = 4, reflecting the variation of the amount of the temporal dimension that is transferred to the spatial one with the strength of fluctuations. Moreover, special finite-size scaling appears ubiquitously except for d = d c0.
Abstract We systematically develop theories of critical phenomena with a prior formed memory of a power-law decaying long-range temporal interaction parameterized by a constant θ > 0 for a space dimension d both below and above an upper critical dimension d c = 6 − 2/θ. We first provide more evidences to confirm the previous theory that a dimensional constant d t is demanded to rectify a hyperscaling law, to produce correct unique mean-field critical exponents via an effective spatial dimension originating from temporal dimension, and to transform the time and change the dynamic critical exponent. Next, for d < d c , we develop a renormalization-group theory by employing the momentum-shell technique to the leading nontrivial order explicitly and to higher orders formally in ϵ = d c − d but to zero order in ε = 1 − θ and find that more scaling laws besides the hyperscaling law are broken due to the breaking of the fluctuation-dissipation theorem. Moreover, because dynamics and statics are intimately interwoven, even the static critical exponents involve contributions from the dynamics and hence do not restore the short-range exponents even for θ = 1 and the crossover between the short-range and long-range fixed points is discontinuous contrary to the case of long-range spatial interaction. In addition, a new scaling law relating the dynamic critical exponent with the static ones emerges, indicating that the dynamic critical exponent is not independent. However, once d t is displaced by a series of ϵ and ε such that most values of the critical exponents are changed, all scaling laws are saved again, even though the fluctuation-dissipation theorem keeps violating. Then, for d ≥ d c , we develop an effective-dimension theory by carefully discriminating the corrections of both temporal and spatial dimensions and find three different regions. For d ≥ d c0 = 4, the upper critical dimension of the usual short-range theory, the usual Landau mean-field theory with fluctuations confined to the effective-dimension equal to d c0 correctly describe the critical phenomena with memory, while for d c < d ≤ 4, there exist new universality classes whose critical exponents depend only on the space dimension but not at all on θ. Yet another region consists of d = d c only and the previous theory is retrieved. All these results show that the dimensional constant d t is the fundamental ingredient of the theories for critical phenomena with memory. However, its value continuously varies with the space dimension and vanishes exactly at d = 4, reflecting the variation of the amount of the temporal dimension that is transferred to the spatial one with the strength of fluctuations. Moreover, special finite-size scaling appears ubiquitously except for d = d c0.
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.
Abstract Phase transitions and critical phenomena are among the most intriguing phenomena in nature and the renormalization-group theory for them is one of the greatest achievements of theoretical physics. However, the predictions of the theory above an upper critical dimension d c seriously disagree with reality. In addition to its fundamental significance, the problem is also of practical importance because both complex classical systems with long-range spatial or temporal interactions and quantum phase transitions with long-range interactions can substantially lower d c . The extant scenarios built on a dangerous irrelevant variable (DIV) to resolve the problem introduce two sets of critical exponents and even two sets of scaling laws whose origin is obscure. Here, we consider the DIV from a different perspective and clearly unveil the origin of the two sets of exponents and hence the intrinsic inconsistency in those scenarios. We then develop an effective-dimension theory in which critical fluctuations and system volume are fixed at an effective dimension by the DIV. This enables us to account for all the extant results consistently. A novel asymptotic finite-size scaling behavior for a correlation function together with a new anomalous dimension and its associated scaling law is also predicted. We also apply the theory to quantum phase transitions.
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.
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.
Dimensional correspondences have a long history in critical phenomena. Here, we review the effective dimension approach, which relates the scaling exponents of a critical system in d spatial dimensions with power-law decaying interactions r^{d+σ} to a local system, i.e., with finite-range interactions, in an effective fractal dimension d_{eff}. This method simplifies the study of long-range models by leveraging known results from their local counterparts. While the validity of this approximation beyond the mean-field level has been long debated, we demonstrate that the effective dimension approach, while approximate for non-Gaussian fixed points, accurately estimates the critical exponents of long-range models with an accuracy typically larger than 97%. To do so, we review perturbative renormalization-group (RG) results, extend the approximation's validity using functional RG techniques, and compare our findings with precise numerical data from conformal bootstrap for the two-dimensional Ising model with long-range interactions.
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.