Ground-State of the Dimerized 1\emph{D} Heisenberg Model with Next Nearest Neighbor Interaction

A well-known variant of the one-dimensional antiferromagnetic spin $1/2$ Heisenberg model includes explicit dimerization and was first studied by Cross and Fisher many years ago. The Hamiltonian is given by H=J_{1}\sum_{l=1}^{2N-1}\left( 1-\left( -1\right) ^{l}\delta\right) \vec{S}_{l}\cdot\vec{S}_{l+1}+J_{2}\sum_{l=1}^{2N-2}\vec{S}_{l}\cdot\vec {S}_{l+2}% where $J_{1}$ is the nearest neighbor interaction (here we take $J_{1}=1$), $\delta$ ($0\leq\delta\leq1$) is the dimerization and $J_{2}$ ($0\leq J_{2}\leq2$) is the next-nearest neighbor interaction. Here we shall apply both a Lanczos matrix truncation as well as a Connected Moments approach to study both the ground-state energy as well as the energy gap.