# 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:

- To discover relationships between critical exponents characterizing the critical system and topological invariants, and to understand the universality of systems from the viewpoint of topological invariance,
- To find a universal law in time-development of various topological invariants by regarding self-organized criticality as a dynamics of topology change of the system,
- To carry out quantitative analysis of results obtained in the "Topological Materials Project" and the "Novel Topology-Related Technologies Project", and to establish a method for topological diagnosis of living tissue anomalies,
- To clarify the process by which various network structures in economical and social systems are self-organized into critical states due to variations in scales, technological developments, and environmental changes by theoretical and numerical studies as well as field surveys, and to provide a new vision for policies from a topological perspective,
- To realize stable control of superposed quantum states, and to study new quantum information technologies utilizing topological quantity of states.

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