Frustration and Multicriticality in the Antiferromagnetic Spin-1 Chain

We study the spin $S=1$ Heisenberg chain, with nearest neighbor, next nearest neighbor ($\alpha$) and biquadratic ($\beta$) interactions using a combination of the density matrix renormalization group (DMRG), an analytic variational matrix product state wavefunction, and non-Abelian bosonization. We study the effect of frustration ($\alpha>0$) on the Haldane phase with $-1\leq \beta < 1$ which reveals a rich phase diagram. For $-1<\beta<\beta^\ast$, we establish the existence of a spontaneously dimerized phase for large $\alpha>\alpha_c$, separated from the Haldane phase by the critical line $\alpha_c(\beta)$ of second-order phase transitions connected to the Takhtajan--Babudjian integrable point $\alpha_c(\beta=-1)=0$. In the opposite regime, $\beta>\beta^\ast$, the transition from the Haldane phase becomes first-order into the next nearest neighbor (NNN) AKLT phase. Based on field theoretical arguments and DMRG calculations, we conjecture that these two regimes are separated by a multicritical point ($\beta^\ast, \alpha^\ast$) of a different universality class, described by the $SU(2)_4$ Wess--Zumino--Witten critical theory. From the DMRG calculations we estimate this multicritical point to lie in the range $-0.2<\beta^\ast<-0.15$ and $0.47<\alpha^\ast < 0.53$. We find that the dimerized and NNN-AKLT phases are separated by a line of first-order phase transitions that terminates at the multicritical point. Inside the Haldane phase, we show the existence of two incommensurate crossovers: the Lifshitz transition and the disorder transition of the first kind, marking incommensurate correlations in momentum and real space, respectively. We show these crossover lines stretch across the entire $(\beta,\alpha)$ phase diagram, merging into a single incommensurate-to-commensurate transition line for negative $\beta\lesssim \beta^\ast$ outside the Haldane phase.