A good approximation for the phase diagram of the quantum anisotropic spin-1/2 Heisenberg ferromagnet on a self-dual hierarchical lattice is calculated within a real-space renormalization-group frame-work. We extend the results to the antiferromagnetic case and show that
Tc vanishes for values of the anisotropy parameter \ensuremath{\Delta}\ensuremath{\le}0.399, i.e., for small anisotropy the order is destroyed by quantum fluctuations at any finite temperature.
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.
A real-space effective-field renormalization-group method (ERFG) recently derived for computing critical properties of Ising spins is extended to treat the quantum spin-1/2 anisotropic Heisenberg model. The formalism is based on a generalized but approximate Callen–Suzuki spin relation and utilizes a convenient differential operator expansion technique. The method is illustrated in several lattice structures by employing its simplest approximation version in which clusters with one (N′=1) and two (N=2) spins are used. The results are compared with those obtained from the standard mean-field (MFRG) and Migdal–Kadanoff (MKRG) renormalization-group treatments and it is shown that this technique leads to rather accurate results. It is shown that, in contrast with the MFRG and MKRG predictions, the EFRG, besides correctly distinguishing the geometries of different lattice structures, also provides a vanishing critical temperature for all two-dimensional lattices in the isotropic Heisenberg limit. For the simple cubic lattice, the dependence of the transition temperature Tc with the exchange anisotropy parameter Δ [i.e., Tc(Δ)], and the resulting value for the critical thermal crossover exponent φ [i.e., Tc≂Tc(0)+AΔ1/φ ] are in quite good agreement with results available in the literature in which more sophisticated treatments are used.
We consider the square-lattice spin-(1/2) anisotropic Heisenberg ferromagnet with interactions whose symmetry can independently (quenched model) and randomly be of two competing types: namely, the isotropic Heisenberg type and the Ising type. Within a real-space renormalization-group framework, we perform a quite precise numerical calculation of the critical frontier, and establish its main asymptotic behaviors. We also characterize the relevant universality classes through the analysis of the correlation-length critical exponent.
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
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.