Multifractal properties of aperiodic Ising model on hierarchical lattices: role of the geometric fluctuations

Abstract

The role of the geometric fluctuations on the multifractal properties of the local magnetization of aperiodic ferromagnetic Ising models on hierachical lattices is investigated. The geometric fluctuations are introduced by generalized Fibonacci sequences. The local magnetization is evaluated via an exact recurrent procedure encompassing a real space renormalization group decimation. The symmetries of the local magnetization patterns induced by the aperiodic couplings is found to be strongly (weakly) different, with respect to the ones of the corresponding homogeneous systems, when the geometric fluctuations are relevant (irrelevant) to change the critical properties of the system. At the criticality, the measure defined by the local magnetization is found to exhibit a non-trivial F (α) spectra being shifted to higher values of α when relevant geometric fluctuations are considered. The critical exponents are found to be related with some special points of the F (α) function and agree with previous results obtained by the quite distinct transfer matrix approach.

Random-field Ising model on hierarchical lattices: thermodynamics and ground-state critical properties

“…Finally, the unknown e ective ÿelds and couplings can be eliminated ending up with coupled recursive relations giving the local magnetization and the pair-correlation functions involving the internal sites in terms of the corresponding values involving the root sites. This approach has been applied to investigate the local order parameters of several magnetic models comprising the ferromagnetic Ising model with uniform [37,38] and aperiodic interactions [39], the SGIM [40,41] and the ferromagnetic Potts [42] model deÿned on DHLs, as well as the Ising model (both the pure and the spin-glass cases) deÿned on the Wheatstone bridge hierarchical lattice [43] and on the m-sheet Sierpinskii Gasket fractal lattice [44].…”

“…This approach has also been extensively applied to investigate the local order parameter of several magnetic models defined on hierarchical lattices. Those studies com-prise the ferromagnetic Ising model with uniform 10,11 and aperiodic interactions 12 , the spin-glass Ising model 13,14 and the ferromagnetic Potts model defined on the diamond hierarchical lattices, as well as the Ising model (both the pure and the spin-glass cases) defined on the Wheatstone bridge hierarchical lattice 15 and on the msheet Sierpinskii Gasket fractal lattice 16 .…”

The random field Ising model on hierarchical lattices

“…This approach has been extensively applied to investigate the local order parameter of several magnetic models defined on hierarchical lattices. Those studies comprise the ferromagnetic Ising model with uniform 18,19 and aperiodic interactions 20 , the spin-glass Ising model 12,13 and the ferromagnetic Potts 21 model defined on diamond hierarchical lattices, as well as the Ising model (both the pure and the spin-glass cases) defined on the Wheatstone bridge hierarchical lattice 22 and on the m-sheet Sierpinskii Gasket fractal lattice 23 .…”