Critical phenomena in 3.99 dimensions

Robert J. Myerson

Eytan Domany

Shaolong Zeng, Sue ping Szeto, Fan Zhong

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.

Ekaterina S. Pisanova, N. S. Tonchev, Hristo Kisov +2 more

Usually, the critical phenomena properties at their upper critical dimensions are characterized by leading logarithmic terms in the corresponding equations and the Lambert W‐function can be used to obtain exact results. A class of models with long‐range interactions and developed classical and quantum fluctuations is considered at the upper quantum critical dimension The exact results, obtained here in the Lambert W‐function terms, and the asymptotic ones for the inverse susceptibility of a lattice model of structural phase transitions in an anharmonic crystal are compared on the basis of their graphic presentations.