We summarize our theoretical findings on the ground-state phase diagram of the spin-1/2 XXZ chain having competing nearest-neighbor (J1) and antiferromagnetic next-nearest-neighbor (J2) couplings. Our study is mainly concerned with the case of ferromagnetic J1, and the case of antiferromagnetic J1 is briefly reviewed for comparison. The phase diagram contains a rich variety of phases in the plane of J1/J 2 versus the XXZ anisotropy Δ: vector-chiral phases, Néel phases, several dimer phases, and Tomonaga-Luttinger liquid phases. We discuss the vector-chiral order that appears for a remarkably wide parameter space, successive Néel-dimer phase transitions, and an emergent nonlocal string order in a narrow region of ferromagnetic J1 side.
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