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Journal of Applied Physics

Perturbative approximation scheme for isolated impurity bonds in the two-dimensional spin-1/2 Heisenberg antiferromagnet

P. Schlottmann

Published May 15, 1994• 3 citations
Publisher

Abstract

An isolated impurity bond of coupling−K replacing a link J in the spin-1/2 Heisenberg antiferromagnet on a square lattice is considered. A simple perturbative approximation scheme is deviced, in which the two-site cluster formed by the spins linked by the impurity bond is embedded into the linearized spin wave approximation for the antiferromagnet. The resolvent for the local states of the cluster is calculated in three successive levels of approximation. The ground state wave function, the magnetization at the impurity link, and the transverse correlation across the impurity link as a function of the coupling K are discussed.

DOI
10.1063/1.355679

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An isolated impurity bond of coupling−K replacing a link J in the spin-1/2 Heisenberg antiferromagnet on a square lattice is considered. A simple perturbative approximation scheme is deviced, in which the two-site cluster formed by the spins linked by the impurity bond is embedded into the linearized spin wave approximation for the antiferromagnet. The resolvent for the local states of the cluster is calculated in three successive levels of approximation. The ground state wave function, the magnetization at the impurity link, and the transverse correlation across the impurity link as a function of the coupling K are discussed.
The two-dimensional quantum Heisenberg antiferromagnet at zero temperature, with locally frustrating magnetic defects, is studied. We consider two types of defects---an isolated ferromagnetic bond (FMB) and a quantum impurity spin, coupled symmetrically to the two sublattices. A technique is developed to study strong defect-environment interaction. In the case of a FMB we find that, contrary to the previous linear spin-wave result, the local magnetization stays finite even for strong frustration. For an antiferromagnetic coupling of a quantum impurity spin with its environment, we find a local change in the ground state. All our calculations are compared with numerical results from exact diagonalization.
The two-dimensional Heisenberg model at zero temperature with a quantum S=1/2 impurity spin, coupled to one site, is studied. The ground state properties of the model are calculated for large coupling, in order to study the impurity-host spin local singlet formation and related local suppression of the magnetization in the host lattice. Analytic results are obtained by developing perturbation theory around the exactly solvable two-body (impurity plus neighbor) problem and compared with results obtained by exact diagonalization of small clusters. We find that perfect screening of the impurity is only achieved for infinitely large interaction, while at intermediate coupling the local magnetization is suppressed, but nonzero.
Magnetic impurities coupled antiferromagnetically to a one-dimensional Heisenberg model are studied by numerical diagonalization of chains of finite clusters. By calculating the binding energy and the correlation function, it is shown that a local singlet develops around each impurity. This holds true for systems with a single impurity, with two impurities, and for impurities forming a lattice. The local character of the singlet is found to be little affected by the presence of other impurity spins. A small effective interaction is found between a pair of impurity spins, which oscillates depending on impurity distances. For impurity lattices, the energy spectrum shows a gap which is found to be much smaller than the binding energy per impurity if the coupling constants are small. For larger coupling constants, it increases to the same order of magnitude as the binding energy, indicating that a local singlet is broken to create excited states. Impurity lattices with ferromagnetic couplings are also studied and their connection to the Haldane problem is discussed.
Two-dimensional spin 1/2 antiferromagnetic Heisenberg models are numerically studied using Entanglement Perturbation Theory, where the ground state wave function is described by a product of local matrices defined at every site, and each matrix is optimized variationally to minimize the energy. We first apply this method for the spin 1/2 antiferromagnetic Heisenberg model on the square lattices to reproduce correctly the known ground state energy and the spin structure factors. Then, we study the spin 1/2 antiferromagnetic Heisenberg model on the triangular lattice with spatially anisotropic nearest neighbor couplings, J and J ', to establish the ground state phase diagram as a function of J'/ J.
The s=1 Heisenberg antiferromagnet is studied in the presence of two kinds of local impurities. First, a perturbed antiferromagnetic bond with J'\ensuremath{\ne}J at the center of an even-length open chain is considered. Using the density-matrix renormalization-group method we find that, for sufficiently strong or weak J', a bound state is localized at the impurity site, giving rise to an energy level in the Haldane gap. The energy of the bound state is in agreement with perturbative results, based on s=1/2 chain-end excitations, both in the weak- and strong-coupling limit. In a region around the uniform limit, J'=J, no states are found with energy below the Haldane gap. Second, a s=1/2 impurity at the center of an otherwise even-length open chain is considered. The coupling to the s=1/2 impurity is varied. Bound states in the Haldane gap are found only for sufficiently weak (antiferromagnetic) coupling. For a s=1/2 impurity coupled with a strong (antiferromagnetic) bond, no states are found in the Haldane gap. Our results are in good qualitative agreement with recent experiments on doped NENP and Y2{\mathrm{Y}}_{2}Y2​BaNiO5{\mathrm{BaNiO}}_{5}BaNiO5​.
In this work we use the linear spin-wave approximation to study the quantum two-dimensional Heisenberg model on triangular lattice in the presence of nonmagnetic impurities. In particular we discuss how site disorder changes the ground-state properties of this important frustrated antiferromagnetic spin system. Our main focus is on the role of the nonmagnetic impurities which generates an influence of the quantum order-disorder transition and the effect of the ground-state spin structure in many spin systems.