  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.
