Schwinger-boson mean-field theory for S

in the study of one-dimensional spin chains with nearestneighbor antiferromagnetic exchange interaction J, following the conjecture by Haldane' that integer spin chains have a singlet ground state with a finite gap EH in the excitation spectrum, whereas the excitation spectrum for half-integer spins is gapless. More recently, the physics of S =1 open spin chains has received new attention stimulated by the discovery of fractional S= 2 spin states localized at the ends of the chain. ' It is generally believed that the qualitative behavior of the S=1 spin chain can be understood in terms of the valence-bond-solid (VBS) model where S = I spin on each site is represented by symmetrization of two S= 2 spins and a manybody singlet ground-state wave function is formed by making two valence bonds from each site to adjacent neighbors. It has been shown rigorously that the correlation function decays exponentially in the VBS state and that there exists an energy gap in the excitation spectrum. For an open chain, unpaired bonds are left at each end of the chain, corresponding to effective localized S= 2 objects. A similar picture for the ground state of the Heisenberg model is believed to be valid except that the valence bonds are not restricted to form only between nearest-neighbor sites and a residual effective interaction -Je i~ (L is the length of chain, g is the correlation length) between the two end states is believed to exist. The residue interaction is believed to be antiferromagnetic for chains with an even number of sites and ferromagnetic for chains with an odd number of sites, resulting in a S=O singlet ground state for even chains and S=l triplet ground state for odd chains. This physical picture has been confirmed by exact diagonalization on open chains up to 14 sites. " In this paper we shall study open S =1 Heisenberg spin chains using the Schwinger-boson mean-field theory (SBMFT) proposed by Arovas and Auerbach for the Heisenberg model. The theory has been applied successfully to infinite Heisenberg spin systems in both oneand two-dimensions. ' (In the one-dimensional case, the theory works well only with integer spin chains, presumably due to the neglect of topogical excitations in the SBMFT. ) The theory is the leading term in a systematic I/N expansion and has the advantage that computations can be made easily on long chains. It also provides a simple physical picture for understanding the properties of the S= & end states. We shall study chains with various lengths and compare the mean-field results with theoretical expectations and experimental results. Effects of fluctuations beyond mean-field theory and various asymmetric terms in the Hamiltonian will also be discussed. To begin with, we first briefly review properties of infinite Heisenberg spin chains in the SBMFT. The SBMFT can be interpreted as a variational approach with VBS-type trial ground-state wave functions, except that (I ) valence bonds joining arbitrary sites on opposite sublattices are allowed "and (2) the constraint that there are two spin—, objects on each site is relaxed and is satisfied only on average. " The probability distribution of the valence bonds is determined variationally in the SBMFT. The major source of error in the theory is believed to be the relaxation of constraint ' which results