Theory of Critical Phenomena with Memory

TL;DR

This paper develops a new theoretical framework for critical phenomena in systems with memory, revealing violations of hyperscaling laws and introducing novel critical exponents and universality classes, validated by numerical simulations.

Contribution

It introduces a theory of critical phenomena with memory that accounts for dynamic Hamiltonians and space-time interwoven transformations, leading to new universality classes.

Findings

Violation of hyperscaling law in naive models

Introduction of effective spatial dimension to restore hyperscaling

Discovery of new mean-field critical exponents and universality classes

Abstract

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 and develop a corresponding theory of critical phenomena with memory for complex systems and discovered 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.