Diamond-like Hierarchical Lattice Research Articles

Consensus of Julia Sets of Potts Models on Diamond-Like Hierarchical Lattice

The limit sets of zeros of the partition function for λ-state Potts models on diamond-like hierarchical lattice are the Julia sets of functions in a family of rational functions. In this paper, the consensus problem of Julia sets generated by λ-state Potts models on diamond-like hierarchical lattice is studied. Two types of the consensus problem of Julia sets are considered, one is with a leader and the other is with no leaders. Based on these two types, two different control protocols are proposed respectively to make systems achieve consensus of Julia sets. The simulations confirm the efficacy of control protocols.

The two-layer Ising model on a sequence of diamond-like hierarchical lattices

The two-layer Ising model on diamond-like hierarchical lattices of different structures with equal fractal dimension was investigated. We applied the renormalization group transformations for the considered lattices. The critical values of the shift exponent φ were computed for various intralayer interaction values (J1 and J2). For the simplest diamond-like hierarchical lattice φ≈2.35 at J1=J2. As the structure of the lattice becomes more complex, the shift exponent value increases. At J1=0.5J2 the value φ≈0.5 was obtained, which is consistent with the data for the square lattice. The graphs of the heat capacity, magnetization and magnetic susceptibility were plotted at J1=0.5J2. They showed the absence of the second phase transition at an arbitrarily weak coupling between the layers.

Fractal Dynamics and Control of the Fractional Potts Model on Diamond-Like Hierarchical Lattices

The fractional Potts model on diamond-like hierarchical lattices is introduced in this manuscript, which is a fractional rational system in the complex plane. Then, the fractal dynamics of this model is discussed from the fractal viewpoint. Julia set of the fractional Potts model is given, and control items of this fractional model are designed to control the Julia set. To associate two different Julia sets of the fractional model with different parameters and fractional orders, nonlinear coupling items are taken to make one Julia set change to another. The simulations are provided to illustrate the efficacy of these methods.

Julia sets, Hausdorff dimension and phase transition

The limit set of zeros of partition function of the Potts model on diamond-like hierarchical lattices is studied. It is shown that the limit set is the Julia set of a family rational maps, it is shown in a mathematically exact way that the Julia set tends to a geometrical circle and its Hausdorff dimension tends to 1 when the parameter ∣ λ∣ → +∞, which gives a true answer that Bambi Hu and Bin Lin proposed in 1989, furthermore, in this paper, it give a perfect description about this relations. Also the continuity of level diameter of A λ (1) of this physical model about λ is discussed.

Feigenbaum Julia sets of singularities of free energy

AbstractConsidering the Potts model on diamond-like hierarchical lattices, in this paper, we show that the Julia sets of the singularities of the free energy may be a Feigenbaum Julia set. Furthermore, we prove that these Julia sets are locally connected if they are connected.

Phase diagram of the two-dimensional ferromagnetic three-color Ashkin–Teller model

In the present work we study the critical properties of the ferromagnetic three-color Ashkin–Teller model (3AT) by means of a Migdal–Kadanoff renormalization group approach on a diamond-like hierarchical lattice. The analysis of the fixed points and flux diagram of the recursion relations is used to determine the corresponding phase diagram (including its symmetry properties) and critical exponents. Our numerical results show the presence of four universality classes, three of them are associated to the Potts model with q = 2 , 4 and 6 states. Finally, a connection between our findings and some known results from the literature is presented.

Jordan domain and Fatou set concerning diamond-like hierarchical Potts models**The research was supported by the National Natural Science Foundation of China (Grant No 10625107) and the Program for New Century Excellent Talents in University (Grant No 04-0490).

For the Potts models on the diamond-like hierarchical lattice, the domains of the complex phases are indeed the Fatou components of a family of rational maps. In this paper, we deal with the relationships between this family of Fatou components and the Jordan domains and describe the topological structures of this family of Fatou components completely.

Julia sets and complex singularities in diamond-like hierarchical Potts models

We study the phase transition of the Potts model on diamond-like hierarchical lattices. It is shown that the set of the complex singularities is the Julia set of a rational mapping. An interesting problem is how are these singularities continued to the complex plane. In this paper, by the method of complex dynamics, we give a complete description about the connectivity of the set of the complex singularities.

Ising spin glass on a D = 3 hierarchical lattice: Effects of tailed coupling distributions

Within a real-space renormalization-group framework, we approach the cubic lattice through a D = 3 diamond-like hierarchical lattice. The model is a standard, nearest-neighbor, Ising spin glass with coupling constants { J ij } distributed according to the family of continuous probability distributions P q(J ij) ∝ 1 [1 + (q − 1)J ij 2 2J 2 ] 1 (q − 1) (if 1 + (q − 1) J ij 2 2J 2 > 0 , and zero otherwise; q ∈ R ). Such distributions, which arise naturally in the treatment, within the recently proposed nonextensive thermostatistics, of anomalous diffusion, reproduce the usual, Gaussian case, for q → 1. Moreover, they present a second moment 〈 J ij 2〉 proportional to (5 − 3 q) −1 for q < 5 3 , diverging for q ≥ 5 3 , but keeping a finite width at midheight. In the limit q → 3, P q ( J ij ) collapses with the abscissa, and so the width at midheight diverges. We compute the q-dependence of the spin-glass critical temperature T c . We show numerically that T c does not scale with 〈J ij 2〉 1 2 (contrary to the usual belief), but rather with the width at midheight of P q ( J ij ). Our results suggest that T c vanishes as −1 q when q → −∞; furthermore, we verified that T c diverges exponentially when q approaches 3 from below.

Aperiodic interactions on hierarchical lattices: an exact criterion for the Potts ferromagnet criticality

We discuss the critical behaviour of the q-state Potts model on a diamond-like hierarchical lattice with ferromagnetic interactions according to an aperiodic two-letter substitutional sequence. We show that the geometric (deterministic) fluctuations become relevant for , where is the wandering exponent of the sequence, D is the fractal dimension of the lattice, and is the critical exponent associated with the specific heat of the uniform model. We also point out that the criteria for analysing the relevance of deterministic and random fluctuations are generically different.

Ising models, julia sets, and similarity of the maximal entropy measures

We study the phase transition of Ising models on diamondlike hierarchical lattices. Following an idea of Lee and Yang, one can make an analytic continuation of free energy of this model to the complex temperature plane. It is known that the Migdal-Kadanoff renormalization group of this model is a rational endomorphism (denoted byf) of Ĉ and that the singularities of the free energy lie on the Julia setJ(f). The aim of this paper is to prove that the free energy can be represented as the logarithmic potential of the maximal entropy measure onJ(f). Moreover, using this representation, we can show a close relationship between the critical exponent and local similarity of this measure.

A generalised Hubbard Hamiltonian : Influence of temperature and fractality

We consider a generalised Hubbard Hamiltonian which is invariant under a Real Space Renormalisation Group. Within the present approximation, the Bravais lattice is replaced by a diamond-like hierarchical lattice. We calculate the phase diagram associated with the half-filled band Hubbard Hamiltonian in d = 1, 2 and 3, in the full range of the dimensionless intra-atomic interaction U and hopping t parameters. The influence of the hierarchical lattice fractality is analysed as well.

Julia sets and complex singularities in hierarchical Ising models

We study the analytical continuation in the complex plane of free energy of the Ising model on diamond-like hierarchical lattices. It is known that the singularities of free energy of this model lie on the Julia set of some rational endomorphism $f$ related to the action of the Migdal-Kadanoff renorm-group. We study the asymptotics of free energy when temperature goes along hyperbolic geodesics to the boundary of an attractive basin of $f$. We prove that for almost all (with respect to the harmonic measure) geodesics the complex critical exponent is common, and compute it.

Surface effects in the Potts ferromagnet

Within a real-space renormalisation-group framework, the authors study the phase diagram of a semi-infinite cubic lattice q-state Potts ferromagnet, in which the free-surface coupling constant Js=(1+ Delta )JB might be different from the bulk one JB. The authors calculate the starting value Delta c(q) above which surface order is possible even if bulk order is absent. Their results can be alternatively seen as approximate for the simple cubic lattice (as a matter of fact, the Ising value Delta c(2) the authors obtain approaches reasonably well the series and Monte Carlo results; consequently Delta c(q) is expected to be quite satisfactory even for q not=2) or as exact for a well defined diamond-like hierarchical lattice. In the q to 0 limit, Delta c diverges as q-12/.