Violations of Hyperscaling in Phase Transitions and Critical Phenomena

This Special Issue was initiated by Prof. Ralph Kenna (August 27, 1964–October 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.