VARIATIONAL AND DENSITY-MATRIX RENORMALIZATION-GROUP STUDIES OF THE FRUSTRATED ANTIFERROMAGNETIC HEISENBERG S=1 QUANTUM SPIN CHAIN

In this paper we study a frustrated antiferromagnetic isotropic Heisen berg S=1 quantum spin chain H=Sigma(i)S(i)S(i+1) +alpha Sigma(i)S(i)S( i+2), using a variational ansatz starting from valence bond states and the density-matrix renormalization group. We find both methods to giv e results in very good qualitative and good quantitative agreement, wh ich clarify the phase diagram as follows: At alpha(D)=0.284(1), there is a disorder point of the first kind, marking the onset of incommensu rate spin-spin correlations in the chain. At alpha(L)=0.3725(25) there is a Lifshitz point, at which the excitation spectrum is found to dev elop a particular doubly degenerate structure. These points are the qu antum remnants of the transition from antiferromagnetic to spiral orde r in the classical frustrated chain. At alpha(T)=0.7444(6) there is a first-order phase transition from an Affleck-Kennedy-Lieb-Tasaki (AKLT ) phase characterized by nonvanishing string order to a phase which ca n be understood as a next-nearest-neighbor generalization of the AKLT model. At the transition, the string order parameter shows (to numeric al precision) a discontinuous jump of 0.085 to zero; the correlation l ength and the gap are both finite at the transition. The problem of ed ge states in open frustrated chains is discussed at length.