Fixed Point Properties of the Ising Ferromagnet on the Hanoi Networks

Abstract

The Ising model with ferromagnetic couplings on the Hanoi networks is analyzed with an exact renormalization group. In particular, the fixed-points are determined and the renormalization-group flow for certain initial conditions is analyzed. Hanoi networks combine a one-dimensional lattice structure with a hierarchy of small-world bonds to create a mix of geometric and mean-field properties. Generically, the small-world bonds result in non-universal behavior, i.e. fixed points and scaling exponents that depend on temperature and the initial choice of coupling strengths. It is shown that a diversity of different behaviors can be observed with seemingly small changes in the structure of the networks. Defining interpolating families of such networks, we find tunable transitions between regimes with power-law and certain essential singularities in the critical scaling of the correlation length, similar to the so-called inverted Berezinskii-Kosterlitz-Thouless transition previously observed only in scale-free or dense networks.

13 pages, revtex, 12 fig. incl.; fixed confusing labels, published version. For related publications, see http://www.physics.emory.edu/faculty/boettcher/

References in corpus (13)

• Spatial Networks
• Critical phenomena in complex networks
• Monte Carlo studies of the one-dimensional Ising spin glass with power-law interactions
• BKT-like transition in the Potts model on an inhomogeneous annealed network
• Small-World Bonds and Patchy Percolation on the Hanoi Network
• Hierarchical, Regular Small-World Networks
• Griffiths singularities and algebraic order in the exact solution of an Ising model on a fractal modular network
• Ising model on the Apollonian network with node dependent interactions
• Monte-Carlo simulation study of the two-stage percolation transition in enhanced binary trees
• Anomalous Diffusion on the Hanoi Networks
• Generating-function approach for bond percolations in hierarchical networks
• Geometry and Dynamics for Hierarchical Regular Networks
• Totally Asymmetric Exclusion Process with Hierarchical Long-Range Connections

Cited by in corpus (20)

• Ordinary Percolation with Discontinuous Transitions
• Topological Percolation on Hyperbolic Simplicial Complexes
• Transition-type change between an inverted Berezinskii-Kosterlitz-Thouless transition and an abrupt transition in the bond percolation on a random hierarchical small-world network
• Renormalization group for link percolation on planar hyperbolic manifolds
• Scaling of Clusters near Discontinuous Percolation Transitions in Hyperbolic Networks
• Classification of critical phenomena in hierarchical small-world networks
• Generalized scaling theory for critical phenomena including essential singularity and infinite dimensionality
• Criticality governed by the stable renormalization fixed point of the Ising model in the hierarchical small-world network
• Renormalization group theory of percolation on pseudo-fractal simplicial and cell complexes
• Profile and scaling of the fractal exponent of percolations in complex networks
• From explosive to infinite-order transitions on a hyperbolic network
• Real-Space Renormalization Group for Spectral Properties of Hierarchical Networks
• Absence of the non-percolating phase for percolation on the non-planar Hanoi network
• The Griffiths Phase on Hierarchical Modular Networks with Small-world Edges
• Upper transition point for percolation on the enhanced binary tree: A sharpened lower bound
• Evaluation of Spectral Zeta-Functions with the Renormalization Group
• Renormalization-group theory of the abnormal singularities at the critical-order transition in bond percolation on pointed hierarchical graphs
• Optimal Vertex Cover for the Small-World Hanoi Networks
• Jamming in Hierarchical Networks
• Renormalization Group Solution of the Chutes&Ladder Model