We perform a real-space renormalization-group analysis for the two-dimensional antiferromagnetic Heisenberg model. We first derive an effective model, described by the Hamiltonian: H=J
Jj=1L S(j)\ensuremath{\cdot}S(j+1)+
Jj=1L(-1
)j\ensuremath{\Vert}J(j)\ensuremath{\Vert}S(0)\ensuremath{\cdot}S(j), where S(0) represents a single spin located in the center of the ring. We show that the renormalized coupling constants \ensuremath{\Vert}J(j)\ensuremath{\Vert} tend to nonzero values as one increases the block size by successive renormalization. Using analytic and numerical arguments, we establish the existence of antiferromagnetic long-range order in the effective model and hence, in the original Heisenberg model. We compare our results with those obtained from perturbation theory, spin-wave theory, and exact-diagonalization calculations.
We present a numerical study of single-chain models of doped spin-1 compounds. We use low-energy effective one-dimensional models for both the cases of paramagnetic and spin-1/2 doping. In the case of paramagnetic doping, the effective model is equivalent to the bond disordered spin-1/2 chain model recently analysed by means of the real space renormalization group by Hyman and Yang. By means of exact diagonalizations in the XX limit, we confirm the stability of the Haldane phase for weak disorder. Above a critical amount of disorder, the effective model flows to the so-called random singlet fixed point. In the case of spin-1/2 doping, we argue that the Haldane phase should be destabilized even for weak disorder. This picture is not in contradiction with existing experimental data. We also discuss the possible occurrence of (unobserved) antiferromagnetically ordered phases.
The real-space renormalization group theory using the first order cumulant is applied to the q -state antiferromagnetic Potts model on 2-d (dimensional) and 3-d lattice. The special features of this paper are that the Potts states are represented by q unit vectors in q -1 dimensional space and the two-sublattice structure is preserved after the renormalization transformation. The critical exponents obtained for q =2 in 3-d are in fairly good agreement with the reliable values and those for q =3 and 4 are obtained for the first time.