Analytical and numerical studies of the one-dimensional sawtooth chain
By using the analytical coupled cluster method, the numerical exact diagonalization method, and the numerical density matrix renormalization group method, we investigated the properties of the one-dimensional sawtooth chain. The results of the coupled cluster method based on Neel state demonstrate that the ground state is in the quasi-Neel-long-range order state when a<ac1. The translational symmetry of the ground state varies and the ground state evolves from the quasi-Neel-long-range order state to the dimerized state at the critical point ac1. The dimerized state is stable in the intermediate parameter regime and vanishes at another critical point ac2. The results drawn from the exact diagonalization show that the precise critical point ac1 and ac2 can be determined by using the spin stiffness fidelity susceptibility and spin gap separately. We compared the results obtained by using the coupled cluster method based on canted state with those obtained based on spiral state, and found that the ground state of the sawtooth chain is in the quasi-canted state if a>ac2. The results of the coupled cluster method and the density matrix renormalization group method both disclose that the type of the quantum phase transition occurring at ac2 belongs to the first-order transition.