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We study the phase diagram of spin-1 antiferromagnetic chain with isotropicantiferromagnetic interactions decaying with a power-law $\propto r^{-\alpha}$($\alpha\ge 1$) accompanied by modulated single-ion anisotropy. Employing thetechniques of the density-matrix renormalization group, effects of long-rangeinteractions and single-ion anisotropy on a variety of correlations areinvestigated. In order to check the consistency, the fidelity susceptibilitiesare evaluated across quantum phase transitions. The quantum critical points arefaithfully detected and orders of phase transitions are determined. Thecorrelation-length critical exponent is extracted from scaling functions of thefidelity susceptibility. The presence of long-range interactions leads toquantitative change of the phase boundaries and reduces the order of phasetransition under certain conditions. A direct first-order transition betweenthe periodic N\'eel phase and the large-$D$ phase occurs for slowly decayingantiferromagnetic interactions.

Quantum phase transitions in a spin-1 antiferromagnetic chain with long-range interactions and modulated single-ion anisotropy