Related Experiment Video
Updated: May 18, 2026

Modeling the Functional Network for Spatial Navigation in the Human Brain
Published on: October 13, 2023
Criticality governed by the stable renormalization fixed point of the Ising model in the hierarchical small-world
Tomoaki Nogawa1, Takehisa Hasegawa, Koji Nemoto
1Department of Mathematics, Tohoku University, 6-3-09, Aramaki-Aza-Aoba, Sendai, Miyagi 980-8579, Japan. nogawa@serow.t.u-tokyo.ac.jp
Abstract:
We study the Ising model in a hierarchical small-world network by renormalization group analysis and find a phase transition between an ordered phase and a critical phase, which is driven by the coupling strength of the shortcut edges. Unlike ordinary phase transitions, which are related to unstable renormalization fixed points (FPs), the singularity in the ordered phase of the present model is governed by the FP that coincides with the stable FP of the ordered phase. The weak stability of the FP yields peculiar criticalities, including logarithmic behavior. On the other hand, the critical phase is related to a nontrivial FP, which depends on the coupling strength and is continuously connected to the ordered FP at the transition point. We show that this continuity indicates the existence of a finite correlation-length-like quantity inside the critical phase, which diverges upon approaching the transition point.
Related Concept Videos
Routh-Hurwitz Criterion I
To apply the Routh-Hurwitz criterion, a Routh table is constructed. The table's rows are labeled with powers of the complex frequency variable s, starting from the...
Woodward–Hoffmann Selection Rules and Microscopic Reversibility
Protein Networks
These interactions can be represented through maps depicting protein-protein interaction networks, represented as nodes and edges. Nodes are circles that are representative of a protein,...
Stability of structures
Complexation Equilibria: Factors Influencing Stability of Complexes
Imperfections in Crystal Structure: Stoichiometric Point Defects