The J1-J3 Heisenberg spin models with nearest-neighbor (J1) and additional isotropic three-site (J3) 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 J1-J3 Heisenberg chains. For the reason that the positive (J3>0) three-site couplings tend towards the formation of local quantum dimers, the J1-J3 spin models typically experience some spontaneous dimerization upon increasing J3. Actually, it occurred that the established dimer phases in spin-S J1-J3 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,σ) J1-J3 chains (S>σ), provided that the cell spin S+σ= integer, whereas for half-integer cell spin, the local dimer formation produces gapless spin-liquid ground states. The alternating-spin J1-J3 chains also provide some typical examples of spin models supporting the so-called non-Lieb-Mattis magnetic phases.
The J1-J3 Heisenberg spin models with nearest-neighbor (J1) and additional isotropic three-site (J3) 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 J1-J3 Heisenberg chains. For the reason that the positive (J3>0) three-site couplings tend towards the formation of local quantum dimers, the J1-J3 spin models typically experience some spontaneous dimerization upon increasing J3. Actually, it occurred that the established dimer phases in spin-S J1-J3 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,σ) J1-J3 chains (S>σ), provided that the cell spin S+σ= integer, whereas for half-integer cell spin, the local dimer formation produces gapless spin-liquid ground states. The alternating-spin J1-J3 chains also provide some typical examples of spin models supporting the so-called non-Lieb-Mattis magnetic phases.
Additional three-site exchange interactions, naturally arising, e.g., in the strong-coupling expansion of the two-band Hubbard model and in various effective spin Hamiltonians, may stabilize various exotic spin phases which are not typical for the pure Heisenberg models on frustrated lattices and/or with extra biquadratic exchange terms. The reason is hidden in some specific features of the isotropic three-site exchange interactions, such as the promotion of collinear spin configurations as well as the reinforcing tendency towards clustering of the quantum spins on the shortest exchange bonds. As a result, systems with integer and halfinteger composite spins in the unit cell might be expected to support different quantum phase diagrams. In this work, we study this phenomenon by comparing the quantum phase diagrams of two Heisenberg chains with extra three-site exchange terms and two different alternating pairs of site spins S and σ [( S , σ) = and ( ], resulting in half-integer and integer cell spins S + σ , respectively.
We perform a systematic DMRG investigation of the two natural spin-1 generalizations of the spin-1/2 Majumdar-Ghosh chain, the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg chain, where J2 is a next-nearest-neighbor Heisenberg coupling, and the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ model, where J3 refers to a three-site interaction defined by ${J}_{3}\left[({\mathbf{S}}_{i\ensuremath{-}1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i})({\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i+1})+\mathrm{H}.\mathrm{c}.\right]$. Although both models are rigorously equivalent to the Majumdar-Ghosh chain for spin 1/2, their physics appears to be quite different for spin 1. Indeed, when all couplings are antiferromagnetic, the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model undergoes an effective decoupling into two next-nearest-neighbor (NNN) Haldane chains upon increasing J2, while the ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ chain undergoes a spontaneous dimerization similar to the spin-1/2 Majumdar-Ghosh chain upon increasing J3. By extending the phase diagram to all signs of the couplings, we show that both the dimerized and the NNN-Haldane phase are actually present in the ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ model, the former one adjacent to the Haldane one, the latter to the ferromagneric one, with an Ising transition between them. By contrast, the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chain only has a NNN-Haldane phase between the Haldane phase and the ferromagnetic phase for positive J2. In both cases, our DMRG data are consistent with a continuous Kosterlitz-Thouless transition between the NNN-Haldane and the ferromagnetic phases.
We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (J1F) and antiferromagnetic (J1A) nearest-neighbor interactions and ferromagnetic next-nearest-neighbor interaction (J2F). In this model frustration is induced by the non-zero J2F and the effect of dimerization is realized by changing the strength of J1A for a fixed value of J1F. The model with site spin s behaves like a Haldane spin chain with site spin 2s in the limit of vanishing J2F and large J1F/J1A. We show that the exact ground state of the model can be found along a line in the parameter space. The phase diagram in the space of J1A−J2F is determined using numerical techniques complemented by some analytical calculations. A number of quantities, including the ground state spin, energy gap, structure factor, entanglement entropy, and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and application-wise important feature of this model is that it exhibits a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.
We show that spin S Heisenberg spin chains with an additional three-body interaction of the form $({\mathbf{S}}_{\mathbf{i}\mathbf{\ensuremath{-}}\mathbf{1}}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{\mathbf{i}})({\mathbf{S}}_{\mathbf{i}}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{\mathbf{i}+\mathbf{1}})+\mathrm{H}.\mathrm{c}.$ possess fully dimerized ground states if the ratio of the three-body interaction to the bilinear one is equal to $1/[4S(S+1)\ensuremath{-}2]$. This result generalizes the Majumdar-Ghosh point of the ${J}_{1}\ensuremath{-}{J}_{2}$ chain, to which the present model reduces for S=1/2. For S=1, we use the density matrix renormalization group method to show that the transition between the Haldane and the dimerized phases is continuous with a central charge c=3/2. Finally, we show that such a three-body interaction appears naturally in a strong-coupling expansion of the Hubbard model, and we discuss the consequences for the dimerization of actual antiferromagnetic chains.
We study the ground-state properties of the quantum spin liquid (QSL) phases of the spin-1/2 antiferromagnetic Heisenberg model on the triangular lattice with nearest- (J1), next-nearest- (J2), and third-neighbor (J3) interactions by using the density-matrix renormalization group (DMRG) method. By combining parallel DMRG with SU(2) spin rotational symmetry, we are able to obtain accurate results on large cylinders with length up to Lx=48 and circumference Ly=6−−12. Our results suggest that the QSL phase of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model is gapped, which is characterized by the absence of a gapless mode and by short-range spin-spin and dimer-dimer correlations. In the presence of J3 interaction, we find that a critical QSL with a single gapless mode emerges. While both spin-spin and scalar chiral-chiral correlations are short ranged, dimer-dimer correlations are quasi long ranged and decay as a power law at long distances.
Starting from the exact result of Majumdar and Ghosh for the dimerized ground state of a Heisenberg linear-chain antiferromagnet with competing next-nearest-neighbor interaction J2=1/2J1, it is shown that this state is an exact eigenstate of the same Hamiltonian with couplings extended up to fourth-nearest neighbors, if the relations J1-2J2+J3=0 and J3=2J4 hold. Furthermore, it is proved to all orders in perturbation theory that this state is the ground state of this Hamiltonian in a finite interval of values of the parameter \ensuremath{\lambda}==1/2-j2 around \ensuremath{\lambda}=0. The actual extension of this interval is computed numerically for finite chains and is shown to be significant.