Universality in the critical exponents δ and η

Abstract

The critical exponents δ and η are obtained for the three-dimensional quantum spin- 1 2 anisotropic Heisenberg (QAH) model by the mean field renormalization group (MFRG) approach. The method is illustrated using its simplest approximation version in which clusters with N = 1,2,4 and 8 spins are used. The exponents δ and η are numerically estimated as a function of the anisotropy parameter Δ ( Δ = 0 and Δ = 1 correspond to the isotropic Heisenberg and Ising models, respectively) for all renormalization between clusters of sizes N ≥ 2. In all types of renormalizations the exponents analysed are independent of Δ. I also have studied the classical D-vector model by MFRG approach with clusters of sizes N′ = 1 and N = 2 spins. It was observed that δ and η do not depend on D and the numerical results are equivalent to the quantum case with the same clusters in MFRG approach. The results, qualitative and quantitative, of the present work are in excellent agreement with more accurate methods (Monte Carlo and series expansion).