Within a real-space renormalization-group framework, the spin-\textonehalf{} Heisenberg ferromagnet in the presence of an Ising-like anisotropy on a self-dual hierarchical lattice is discussed. The results are exact for this lattice but can also be considered as a quite satisfactory approximation for the simple square one. The controversial point on how
Tc vanishes in the isotropic Heisenberg limit is analyzed: Quite strong evidence is presented favoring a continuous function of anisotropy. The crossover from the isotropic Heisenberg model to the pure Ising one is exhibited.
With a real-space renormalization-group method, the anisotropic quantum Heisenberg model on three-dimensional diamond-type hierarchical lattice is studied, and the phase diagram and critical properties are obtained. For the ferromagnetic system, it is shown that there is a finite-temperature phase transition for Δ=0 where Δ is the anisotropy parameter. The order parameter and critical exponents are also calculated. For the antiferromagnetic model, we find that the critical temperature is not equal to zero for Δ=0 and there is not reentrant behavior on the critical line.
In this paper, the phase diagrams and the critical behavior of the spin-1/2 anisotropic XXZ ferromagnetic model (the anisotropic parameter Δ\in(-\infty,1]) on two kinds of diamond-type hierarchical (DH) lattices with fractal dimensions d_{f}=2.58 and 3, respectively, are studied via the real-space renormalization group method. It is found that in the isotropic Heisenberg limit (Δ=0), there exist finite temperature phase transitions for the two kinds of DH lattices above. The systems are also investigated in the range of -\infty
Using a real-space renormalization-group approximation, we study the anisotropic quantum Heisenberg model on hierarchical lattices, with interactions following aperiodic sequences. Three different sequences are considered, with relevant and irrelevant fluctuations, according to the Luck-Harris criterion. The phase diagram is discussed as a function of the anisotropy parameter Delta (such that Delta=0 and 1 correspond to the isotropic Heisenberg and Ising models, respectively). We find three different types of phase diagrams, with general characteristics: the isotropic Heisenberg plane is always an invariant one (as expected by symmetry arguments) and the critical behavior of the anisotropic Heisenberg model is governed by fixed points on the Ising-model plane. Our results for the isotropic Heisenberg model show that the relevance or irrelevance of aperiodic models, when compared to their uniform counterpart, is as predicted by the Harris-Luck criterion. A low-temperature renormalization-group procedure was applied to the classical isotropic Heisenberg model in two-dimensional hierarchical lattices: the relevance criterion is obtained, again in accordance with the Harris-Luck criterion.
We study the anisotropic Heisenberg spin-glass model on a three-dimensional hierarchical lattice (designed to approximate the cubic lattice), within a real-space renormalization-group approach. Two different initial probability distributions for the exchange interaction
(Jij), Gaussian and uniform, are used, with zero mean and width
J. The $(kT∕\overline{J})\ifmmode\times\else\texttimes\fi{}{\ensuremath{\Delta}}_{0}$ phase diagram is obtained, where
T is the temperature, ${\ensuremath{\Delta}}_{0}$ is the first moment of the probability distribution for the uniaxial anisotropy, and
k is the Boltzmann constant. For the Ising model $({\ensuremath{\Delta}}_{0}=1)$, there is a spin-glass phase at low temperatures (high
J) and a paramagnetic phase at high temperatures (low
J). For the isotropic Heisenberg model $({\ensuremath{\Delta}}_{0}=0)$, our results indicate no spin-glass phase at finite temperatures. The transition temperature between the spin-glass and paramagnetic phase decreases with ${\ensuremath{\Delta}}_{0}$, as expected, but goes to zero at a finite value of the anisotropy parameter, namely ${\ensuremath{\Delta}}_{0}={\ensuremath{\Delta}}_{c}\ensuremath{\sim}0.59$. Our results indicate that the whole transition line, between the paramagnetic and the spin-glass phases, for ${\ensuremath{\Delta}}_{c}<{\ensuremath{\Delta}}_{0}<1$, belongs to the same universality class as the transition for the Ising spin glass.
In this work we apply two different real-space renormalization-group (RSRG) approaches to the anisotropic antiferromagnetic spin-1/2 Heisenberg model on the square lattice. Our calculations allow for an approximate evaluation of the T vs \ensuremath{\Delta} phase diagram: the results suggest the existence of a critical value of $\ensuremath{\Delta}>0,$ at which the critical temperature goes to zero, and the presence of reentrant behavior on the critical line between the ordered and disordered phases. This whole critical line is found to belong to the same universality class as the Ising model. Our results are in accordance with previous RSRG approaches but not with numerical simulations and spin-wave calculations.