  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.
