First Order Transition in the Frustrated Antiferromagnetic HeisenbergS=1Quantum Spin Chain

We study a frustrated Heisenberg $S=1\mathrm{}$ quantum spin chain with next-nearest neighbor (NNN) coupling $\alpha$ using a variational ansatz and the density matrix renormalization group. We find as quantum remnants of the phase transition in the classical chain a disorder point at ${\alpha }_D=0.284\mathrm{}\left(1\right)$ and a Lifshitz point at ${\alpha }_L=0.3725\mathrm{}\left(25\right)$. Our main finding is a first-order transition from an Affleck-Kennedy-Lieb-Tasaki (AKLT) phase to a NNN generalization of the AKLT phase at ${\alpha }_T=0.7444\mathrm{}\left(6\right)$. At the transition, string order jumps discontinuously by $\approx 0.085$ to 0; correlation length and gap are finite.