I use an improved version of the two-step density-matrix renormalization group method to study ground-state properties of the two-dimensional (2D) Heisenberg model on the checkerboard lattice. In this version, the Hamiltonian is projected on a tensor product of two-leg ladders instead of chains. This allows investigations of 2D isotropic models. I show that this method can describe both the magnetically disordered and ordered phases. The ground-state phases of the checkerboard model as $J_2$ increases are (i) Néel with $Q=(\pi ,\pi )$, (ii) a valence-bond crystal (VBC) of plaquettes, (iii) Néel with $Q=(\pi ∕2,\pi )$, and (iv) a VBC of crossed dimers. In agreement with previous results, I find that at the isotropic point $J_2=J_1$, the ground state is made of weakly interacting plaquettes with a large gap $\Delta \approx 0.67J_1$ to triplet excitations. The same approach is also applied to the $J_1\text{−}{J}_2$ model. There is no evidence of a columnar dimer phase in the highly frustrated regime.

• Received 17 January 2008

DOI:https://doi.org/10.1103/PhysRevB.77.052408