Hokkaido University, Center of Education & Research for Topological Science & Technology

Topology in relation to critical phenomena

Various systems in nature have their own characteristic lengths that describe their sizes. In general, the number of lengths characterizing a system depends on the type of system. However, sometimes the number of such characteristic lengths can be zero, i.e., systems lack the characteristic lengths. This type of system is called a "critical system". Critical systems are known to exist at the critical point of thermal or quantum phase transitions. It is also known that a system can spontaneously shift to a critical state in many of non-equilibrium and nonlinear processes in various physical, biological, and social phenomena. Many such critical systems can be observed in nature, as in surface irregularities of ground, living tissue, and the universe. Metric geometry for critical structures is known as fractal geometry, and has long been studied. However, there have been few cases in which critical structures were studied from the topological perspective. This project tries to understand and elucidate properties of critical systems and mechanisms of self-organization of systems developing into critical states, from the viewpoint of network topology. We will also establish methods for quantitative analyses of topological structures observed in a variety of interesting systems obtained in the "Topological Materials Project" and the "Novel Topology-Related Technologies Project". In practice, we have several avenues of investigation:

A mode pattern of fractons, which are localized vibrational excitations on a critical percolation network.