Variety of order-by-disorder phases in the asymmetric J1-J2 zigzag ladder: From the delta chain to the J1-J2 chain

We study an asymmetric J₁-J₂ zigzag ladder consisting of two different spin-1/2 antiferromagnetic (AFM; J₂, γ J₂>0 ) Heisenberg legs coupled by zigzag-shaped ferromagnetic (FM; J₁<0 ) interleg interaction. On the basis of density-matrix renormalization group based calculations, the ground-state phase diagram is obtained as functions of γ and J₂/|J₁| . It contains four kinds of frustration-induced ordered phases except a trivial FM phase. Two of the ordered phases are valence bond solid (VBS) with spin-singlet dimerization, which is a rather conventional order by disorder. Still, it is interesting to note that the VBS states possesses an Affleck-Kennedy-Lieb-Tasaki-type topological hidden order. The remaining two phases are ferrimagnetic orders, each of which is distinguished by commensurate or incommensurate spin-spin correlation. It is striking that the ferrimagnetic orders are not associated with geometrical symmetry breaking; instead, the global spin-rotation symmetry is broken. In other words, the system lowers its energy via the FM interleg interaction by polarizing both of the AFM Heisenberg legs. This is a rare type of order by disorder. Besides, the incommensurate ferrimagnetic state appears as a consequence of the competition between a polarization and a critical Tomonaga-Luttinger-liquid behavior in the AFM Heisenberg legs.