Valence-bond crystal phase of the crossed-chain quantum spin model

Using a series expansion based on the flow-equation method we study the ground-state energy and the elementary triplet excitations of a generalized model of crossed spin-$1/2$ chains starting from the limit of decoupled quadrumers. The triplet dispersion is shown to be very sensitive to the interquadrumer frustration, exhibiting a line of almost complete localization as well as lines of quantum phase transitions limiting the stability of the valence-bond crystal phase. In the vicinity of the checkerboard point, a finite window of exchange couplings is found with a nonzero spin-gap, consistent with known results from exact diagonalization. The ground-state energy is lower than that of the bare quadrumer case for all exchange couplings investigated. In the limiting situation of the fully frustrated checkerboard magnet our results agree with earlier series expansion studies.