Heisenberg spin chains with additional isotropic three-site exchange interactions

The $J_1-J_3$ Heisenberg spin models with nearest-neighbor ($J_1$) and additional isotropic three-site ($J_3$) spin interactions remain relatively less explored, although such types of competing exchange terms can naturally emerge from different sources, including the strong-coupling expansion of the multiorbital Hubbard model. Below we present a short survey of the recently published research in this field, the emphasis being on the characteristics of the variety of quantum phases supported by a few generic uniform- and alternating-spin $J_1-J_3$ Heisenberg chains. For the reason that the positive ($J_3>0$) three-site couplings tend towards the formation of local quantum dimers, the $J_1-J_3$ spin models typically experience some spontaneous dimerization upon increasing $J_3$. Actually, it occurred that the established dimer phases in spin-$S$ $J_1-J_3$ Heisenberg chains ($S>{1}/{2}$) serve as complete analogues of the famous gapped Majumdar-Ghosh dimer phase in the spin-${1}/{2}$ Heisenberg chain with next-nearest-neighbor couplings. The same dimerizations have been observed in the alternating-spin ($S,\sigma$) $J_1-J_3$ chains ($S>\sigma$), provided that the cell spin $S+\sigma=\rm{integer}$, whereas for half-integer cell spin, the local dimer formation produces gapless spin-liquid ground states. The alternating-spin $J_1-J_3$ chains also provide some typical examples of spin models supporting the so-called non-Lieb-Mattis magnetic phases.