Edge states in Schwinger-boson mean-field theory of low-dimensional quantum antiferromagnets.

Recently there has been much interest in the study of finite one-dimensional quantum spin chains following the discovery of fractional S = —, ' spin states localized at the ends of S = 1 spin chains. ' In a recent paper, we showed numerically that the S = —, ' spin states can be understood within the framework of the Schwinger-boson mean-field theory (SBMFT) for the Heisenberg model. ' In this paper, we shall derive an analytic explanation within the SBMFT framework for the end states. We shall show that corresponding to the generation of end states, a weak (spin) dimerization effect occurs near the ends of spin chains. The situation for general (integer) values of spin S will be discussed. Furthermore, we shall show using a similar analysis in two dimensions (2D) that corresponding edge states also exist in 2D quantum antiferromagnets (on a square lattice) with structures depending periodically on spin value S. It is helpful to first review some basic physics of the edge states in S =1 quantum spin chains. It is generally believed that the qualitative behavior of the S=1 spin chain can be understood in terms of the valence-bondsolid (VBS) model where S =1 spin on each site is represented by symmetrization of two S = —, ' spins and a many-body singlet ground-state wave function is formed by making two valence bonds from each site to adjacent neighbors. For an open chain, unpaired bonds are left at each end of the chain, corresponding to effective localized S = —, ' objects. The model is believed to be a good representation of the ground state of the Heisenberg spin chain, except that the valence bonds are not restricted to form only between nearest-neighbor sites in the "real" wave function. The existence of the edge states can also be understood from Berry phase consideration in the corresponding continuum nonlinear-sigma-model (NLo. M). For a finite spin chain with length L, the Berry phase contribution to the effective action is ' S/2, located at the end points x=O and L. The extra Berry phases modify the rule of angular momentum quantization at the end points, leading to effective S =—, '