An anisotropic Heisenberg spin S = 1 model on a two-dimensional hexagonal lattice (formed from interacting chains) with antiferromagnetic nearest-neighbor interactions of two types is analyzed by the real space renormalization group method. The problem of the influence of interchain pairing on the critical properties of the model is studied, and the phase diagram of the model is constructed. The two-dimensional density matrix renormalization group algorithm is used to calculate the ground state energy for the isotropic case as a function of the ratio between interchain and intrachain interactions.
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.
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.
The
S=1/2 and
S=1 two-dimensional quantum Heisenberg antiferromagnets on the anisotropic dimerized square lattice are investigated by the quantum Monte Carlo method. By finite-size-scaling analyses on the correlation lengths, the ground-state phase diagram parametrized by strengths of the dimerization and of the spatial anisotropy is determined much more accurately than the previous works. It is confirmed that the quantum critical phenomena on the phase boundaries belong to the same universality class as that of the classical three-dimensional Heisenberg model. Furthermore, for
S=1, we show that all the spin-gapped phases, such as the Haldane and dimer phases, are adiabatically connected in the extended-parameter space, though they are classified into different classes in terms of the string order parameter in the one-dimensional, i.e., the zero-interchain-coupling, case.
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 study the anisotropic Heisenberg spin-glass model in 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
Jˉ. The
(kT/Jˉ)×Δ0 phase diagram is obtained, where
T is the temperature,
Δ0 is the first moment of the probability distribution for the uniaxial anisotropy, and
k is the Boltzmann constant. For the Ising model (
Δ0=1), there is a spin-glass phase at low temperatures (high
Jˉ) and a paramagnetic phase at high temperatures (low
Jˉ). For the isotropic Heisenberg model (
Δ0=0), our results indicate no spin-glass phase at finite temperatures. The transition temperature between the spin-glass and paramagnetic phase decreases with
Δ0, as expected, but goes to zero at a finite value of the anisotropy parameter, namely
Δ0=Δc∼0.59. Our results indicate that the whole transition line, between the paramagnetic and the spin-glass phases, for
Δc≤Δ0<1, belongs to the same universality class as the transition for the Ising spin glass.