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Physical review. B, Condensed matter

Effect of small interchain coupling on one-dimensional antiferromagnetic quantum Heisenberg spin systems: The integer-spin case

Mohamed Moussadek Azzouz, Benoît Douçot

Published Apr 1, 1993• 23 citations
Publisher

Abstract

A two-dimensional array of weakly coupled antiferromagnetic spin chains is investigated using the Schwinger-boson technique. This method is known to give a good description of integer spins already at the mean-field level. We find that a finite coupling is necessary to introduce long-range antiferromagnetic order, consistently with the presence of a gap for a single chain. The energy gap and spin-spin correlations are calculated. These calculations correspond to the behavior of a nonlinear \ensuremath{\sigma} model without the one-dimensional topological term.

DOI
10.1103/physrevb.47.8660

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A two-dimensional array of weakly coupled antiferromagnetic spin chains is investigated using the Schwinger-boson technique. This method is known to give a good description of integer spins already at the mean-field level. We find that a finite coupling is necessary to introduce long-range antiferromagnetic order, consistently with the presence of a gap for a single chain. The energy gap and spin-spin correlations are calculated. These calculations correspond to the behavior of a nonlinear \ensuremath{\sigma} model without the one-dimensional topological term.
Within a mean-field theory, the disordered spin-1/2 single-chain state is shown to be unstable towards an antiferromagnetically, long-range ordered one, when any finite transverse coupling, ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$, is introduced between chains. A mean-field Hamiltonian is derived from the Heisenberg model, using an extension of the Wigner-Jordan transformation in two dimensions. The staggered magnetization and the ground-state energy are calculated for values of transverse couplings ranging from 0 to 1. The main consequence of these calculations is that the value of ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$, required to have antiferromagnetic long-range order, must only be nonzero. This result is consistent with that of Sakai and Takahashi indicating that the critical transverse coupling should be of order of the inverse of the one-dimensional staggered magnetic susceptibility. Following a qualitative argument, it can also suggest that the Luttinger-liquid state is unstable at least at nearly half-filling for the Hubbard model in the limit of large Coulomb repulsion.
The double chain Heisenberg antiferromagnet with anisotropy is studied using the mapping onto the coupled anisotropic nonlinear σ model. This model consists of two anisotropic antiferromagnetic quantum Heisenberg chains with half-integer spin coupled with each other via the ferromagnetic interchain coupling J '. For small J ', our analysis suggests that J ' is an irrelevant parameter and consequently the two chains behave independently and no gap is generated. On the other hand, for large enough J ', the two chains behave as a single Heisenberg chain with integer spin which has a gap in the excitation spectrum. The expected behavior in the isotropic limit is also discussed.
Quasi-one-dimensional quantum antiferromagnets formed by a ddd-dimensional hypercubic lattice of weakly coupled spin- 12\frac{1}{2}21​ antiferromagnetic Heisenberg chains are studied by combining exact results in one-dimension and renormalization group analyses of the interchain correlations. It is shown that ddd-dimensional magnetic long-range order develops at zero temperature for infinitesimal antiferromagnetic or ferromagnetic interchain couplings. In the presence of weak bond alternations, the order-disorder transition occurs at a finite interchain coupling. Relevances to the lightly doped quantum antiferromagnets and multilayer quantum Hall systems are discussed.
The effect of antiferromagnetic interchain coupling in alternating spin-(1,1/2) chains is studied by means of spin-wave theory and density-matrix renormalization group (DMRG). In particular, two limiting cases are investigated, the two-leg ladder and its two-dimensional (2D) generalization. Results of the ground-state properties like energy, spin gap, magnetizations, and correlation functions are reported for the whole range of the interchain coupling ${J}_{\ensuremath{\perp}}.$ For the 2D case the spin-wave results predict a smooth dimensional crossover from 1D to 2D keeping the ground state always ordered. For the ladder system, the DMRG results show that any ${J}_{\ensuremath{\perp}}>0$ drives the system to a gapped ground state. Furthermore the behavior of the correlation functions closely resemble the uniform spin-1/2 ladder. For ${J}_{\ensuremath{\perp}}$ lower than 0.3, however, the gap behaves quadratically as $\ensuremath{\Delta}\ensuremath{\sim}{0.6J}_{\ensuremath{\perp}}^{2}.$ Finally, it is argued that the behavior of the spin gap for an arbitrary number of mixed coupled spin chains is analogous to that of the uniform spin-1/2 chains.
We study the quenching of the Haldane gap in quasi-one-dimensional systems of weakly coupled spin-1 antiferromagnetic Heisenberg chains. The critical interchain coupling Jc required to stabilize long range magnetic order can be accurately determined from large scale quantum Monte Carlo calculations. Several different geometries of coupled chains are studied, illustrating the dependence of Jc on the coordination of chains. For bipartite geometries, ferromagnetically coupled chains yield similar magnitudes for Jc.
The Schwinger-boson mean-field theory is applied to the quantum ferrimagnetic Heisenberg chain. There is a ferrimagnetic long-range order in the ground state. We observe two branches of the low-lying excitation and calculate the spin reduction, the gap of the antiferromagnetic branch, and the spin fluctuation at T=0K.T=0 \mathrm{K}.T=0K. These results agree with the established numerical results quite well. At finite temperatures, the long-range order is destroyed because of the disappearance of the Bose condensation. The thermodynamic observables, such as the free energy, magnetic susceptibility, specific heat, and the spin correlation at T>0K,T>0 \mathrm{K},T>0K, are calculated. The $T{\ensuremath{\chi}}_{\mathrm{uni}}$ has a minimum at intermediate temperatures and the spin-correlation length behaves as ${T}^{\ensuremath{-}1}$ at low temperatures. These qualitatively agree with the numerical results and the difference is small at low temperatures.