Self-avoiding walk on fractal complex networks: Exactly solvable cases

Self-avoiding walks and connective constants in clustered scale-free networks

Various types of walks on complex networks have been used in recent years to model search and navigation in several kinds of systems, with particular emphasis on random walks. This gives valuable information on network properties, but self-avoiding walks (SAWs) may be more suitable than unrestricted random walks to study long-distance characteristics of complex systems. Here we study SAWs in clustered scale-free networks, characterized by a degree distribution of the form P(k)∼k^{-γ} for large k. Clustering is introduced in these networks by inserting three-node loops (triangles). The long-distance behavior of SAWs gives us information on asymptotic characteristics of such networks. The number of self-avoiding walks, a_{n}, has been obtained by direct enumeration, allowing us to determine the connective constant μ of these networks as the large-n limit of the ratio a_{n}/a_{n-1}. An analytical approach is presented to account for the results derived from walk enumeration, and both methods give results agreeing with each other. In general, the average number of SAWs a_{n} is larger for clustered networks than for unclustered ones with the same degree distribution. The asymptotic limit of the connective constant for large system size N depends on the exponent γ of the degree distribution: For γ>3, μ converges to a finite value as N→∞; for γ=3, the size-dependent μ_{N} diverges as lnN, and for γ<3 we have μ_{N}∼N^{(3-γ)/2}.

Devil's staircase continuum in the chiral clock spin glass with competing ferromagnetic-antiferromagnetic and left-right chiral interactions

The chiral clock spin-glass model with q=5 states, with both competing ferromagnetic-antiferromagnetic and left-right chiral frustrations, is studied in d=3 spatial dimensions by renormalization-group theory. The global phase diagram is calculated in temperature, antiferromagnetic bond concentration p, random chirality strength, and right-chirality concentration c. The system has a ferromagnetic phase, a multitude of different chiral phases, a chiral spin-glass phase, and a critical (algebraically) ordered phase. The ferromagnetic and chiral phases accumulate at the disordered phase boundary and form a spectrum of devil's staircases, where different ordered phases characteristically intercede at all scales of phase-diagram space. Shallow and deep reentrances of the disordered phase, bordered by fragments of regular and temperature-inverted devil's staircases, are seen. The extremely rich phase diagrams are presented as continuously and qualitatively changing videos.

Chiral Potts spin glass in d=2 and 3 dimensions

The chiral spin-glass Potts system with q=3 states is studied in d=2 and 3 spatial dimensions by renormalization-group theory and the global phase diagrams are calculated in temperature, chirality concentration p, and chirality-breaking concentration c, with determination of phase chaos and phase-boundary chaos. In d=3, the system has ferromagnetic, left-chiral, right-chiral, chiral spin-glass, and disordered phases. The phase boundaries to the ferromagnetic, left- and right-chiral phases show, differently, an unusual, fibrous patchwork (microreentrances) of all four (ferromagnetic, left-chiral, right-chiral, chiral spin-glass) ordered phases, especially in the multicritical region. The chaotic behavior of the interactions, under scale change, are determined in the chiral spin-glass phase and on the boundary between the chiral spin-glass and disordered phases, showing Lyapunov exponents in magnitudes reversed from the usual ferromagnetic-antiferromagnetic spin-glass systems. At low temperatures, the boundaries of the left- and right-chiral phases become thresholded in p and c. In d=2, the chiral spin-glass Potts system does not have a spin-glass phase, consistently with the lower-critical dimension of ferromagnetic-antiferromagnetic spin glasses. The left- and right-chirally ordered phases show reentrance in chirality concentration p.

Stepwise positional-orientational order and the multicritical-multistructural global phase diagram of the s=3/2 Ising model from renormalization-group theory

The spin-3/2 Ising model, with nearest-neighbor interactions only, is the prototypical system with two different ordering species, with concentrations regulated by a chemical potential. Its global phase diagram, obtained in d=3 by renormalization-group theory in the Migdal-Kadanoff approximation or equivalently as an exact solution of a d=3 hierarchical lattice, with flows subtended by 40 different fixed points, presents a very rich structure containing eight different ordered and disordered phases, with more than 14 different types of phase diagrams in temperature and chemical potential. It exhibits phases with orientational and/or positional order. It also exhibits quintuple phase transition reentrances. Universality of critical exponents is conserved across different renormalization-group flow basins via redundant fixed points. One of the phase diagrams contains a plastic crystal sequence, with positional and orientational ordering encountered consecutively as temperature is lowered. The global phase diagram also contains double critical points, first-order and critical lines between two ordered phases, critical end points, usual and unusual (inverted) bicritical points, tricritical points, multiple tetracritical points, and zero-temperature criticality and bicriticality. The four-state Potts permutation-symmetric subspace is contained in this model.

Lower-critical spin-glass dimension from 23 sequenced hierarchical models

The lower-critical dimension for the existence of the Ising spin-glass phase is calculated, numerically exactly, as dL=2.520 for a family of hierarchical lattices, from an essentially exact (correlation coefficent R2=0.999999) near-linear fit to 23 different diminishing fractional dimensions. To obtain this result, the phase transition temperature between the disordered and spin-glass phases, the corresponding critical exponent yT, and the runaway exponent yR of the spin-glass phase are calculated for consecutive hierarchical lattices as dimension is lowered.