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Matrix product state approach to a frustrated spin chain with long-range interactions

We make extensive simulations over a spin chain model that combines the frustrated $J_1\textrm{-}J_2$ spin chain and the long-range nonfrustrated $(-1)^{(r-1)}r^{-α}$ decay interactions through the variational matrix product state method for both finite and infinite lengths. We study both the ground state entanglement and phase diagram. We find that it is most entangled in the rotation invariant long-range ordered antiferromagnetic phase, where the entanglement scales approximately logarithmically. We determine the development of the Majudar-Ghosh point to a disorder line from entanglement. And we determine approximately the transition from the dimerized and incommensurate phase of the $J_1\textrm{-}J_2$ model to a decoupled phase by studying spin correlation and the dimerization order parameter. Some implications for entanglement in systems with long-range interactions are stated.