Phase control in multiply looped rings

A material-designing principle in which a hierarchical structure is achieved by a series of steps of folding a low-order structure into a high-order structure is introduced in comparison with periodic arrays of unit cells. Multiply looped rings are a typical example, in which a ring consists of L components, which are themselves rings consisting of L components, and so on. A lattice of multiply looped rings when the number of hierarchies is greater than two is three dimensional, as suggested by the existence of an Anderson transition, in which a quantum particle runs through a random potential. The Curie temperature of ferromagnetic transition and the critical strength of electron–electron interaction in a Mott transition are demonstrated to be well modulated by changing L in the hierarchical structure, suggesting that material design can be achieved using the hierarchical structure to yield physical properties as desired.