In this paper, we study the phase diagram of the one-dimensional deformed t–J1–J2 model with frustrated exchange interaction. The introduction of the deformed parameter ζ softens the strength of the induced four- and six-electron interactions caused by the no-double occupancy constraint. For the small ζ and in the weak-coupling regime, we use the bosonization approach and the renormalization group techniques to obtain the phase diagram, which is characterized by the singlet-(SS) and triplet-superconducting (TS), spin-density-wave (SDW) as well as bond-charge-density wave (BCDW) instabilities. The TS and BCDW phases are realized in the whole regime while the SDW and SS correlations depend on J2/J1 < (J2/J1)c or J2/J1 > (J2/J1)c, where (J2/J1)c = 1/3. Furthermore, the deformed model with ζ ≪ 1 is expected to adiabatically connect to the conventional t–J1–J2 model with ζ = 1.
We analyze the stability of the spontaneously dimerized phase of the frustrated Heisenberg antiferromagnet---the ${J}_{1}{\ensuremath{-}J}_{2}$ model---in two dimensions. The lowest triplet excitation, corresponding to breaking of a singlet bond, is found to be stable in the region $0.38\ensuremath{\lesssim}{J}_{2}{/J}_{1}\ensuremath{\lesssim}0.62.$ In addition we find a stable low-energy collective singlet mode, which reflects the spontaneous violation of the discrete symmetry. The spontaneous dimerization vanishes at the point of second-order quantum transition into the N\'eel ordered phase $[{(J}_{2}{/J}_{1}{)}_{c1}\ensuremath{\approx}0.38].$ We argue that the disappearance of dimer order is related to the vanishing of the singlet energy gap at the transition point.
We study the competition between antiferromagnetic order and valence-bond-crystal formation in a two-dimensional frustrated spin-
1∕2 model. The ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model on the square lattice is further frustrated by introducing products of three-spin projectors which stabilize four dimer-product states as degenerate ground state. These four states are reminiscent of the dimerized singlet ground state of the Shastry-Sutherland model. Using exact diagonalization, we study the evolution of the ground state by varying the ratio of interactions. For a large range of parameters $({J}_{2}\ensuremath{\gtrsim}0.25{J}_{1})$, our model shows a direct transition between the valence-bond-crystal phase and the collinear antiferromagnetic phase. For small values of
J2, several intermediate phases appear which are also analyzed.