Valence bond mapping of antiferromagnetic spin chains

Antiferromagnetic spin chains have been the subject of intense interest in recent years, largely due to the conjecture by Haldane that chains built from integer spins should exhibit a gap in their energy spectrum. The existence of this gap, originally predicted on the basis of simple fieldtheoretic considerations, was subsequently confirmed experimentally. Much insight into the properties of antiferromagnetic spin chains has been provided by simple models. The fieldtheoretic nonlinear sigma model ~NLSM!, for example, provided the original motivation for Haldane’s conjecture. Equally illuminating insight into the diverse properties of spin chain systems has been provided by the introduction of the valence bonds solid ~VBS! state. Detailed descriptions of these systems, however, have depended on very complex numerical analyses, using either quantum Monte Carlo ~QMC! simulations or density-matrix renormalization-group ~DMRG! methods. These ‘‘exact’’ treatments provide striking confirmation of the features conjectured by Haldane. Ideally, it would be nice to have a simpler method for a reliable quantitative treatment of quantum spin chains. Valence bonds provide a physically motivated starting point for such a description. The VBS is the exact ground state of specific spin-chain Hamiltonians involving quadratic and quartic terms, suggesting that wave functions constructed in terms of valence bonds might be good trial states more generally. Unfortunately, such wave functions are still too complex, for reasons to be discussed later, to be of general use in a variational analysis. In the present work, we propose a method that permits valence bonds to be used efficiently in variational calculations. The method makes use of boson mapping techniques, whereby a mapping is carried out to a space in which a valence bond is represented exactly by a boson. We also report here an application of this method to study the phase transitions in spin chains governed by the alternating bond Hamiltonian. In this application, we consider an especially simple variational ansatz for the different phases of the system. Nevertheless, we find that our analysis reproduces perfectly the ‘‘exact’’ results for the critical points obtained in DMRG and QMC calculations.