Information scrambling, characterized by the out-of-time-ordered correlator (OTOC), has attracted much attention, as it sheds new light on chaotic dynamics in quantum many-body systems. The scale invariance, which appears near the quantum critical region in condensed matter physics, is considered to be important for the fast decay of the OTOC. In this paper, we focus on the one-dimensional spin-1/2 XXZ model, which exhibits quantum criticality in a certain parameter region, and investigate the relationship between scrambling and the scale invariance. We quantify scrambling by the averaged OTOC over the Pauli operator basis, which is related to the operator space entanglement entropy (OSEE). Using the infinite time-evolving block decimation (iTEBD) method, we numerically calculate time dependence of the OSEE in the early time region in the thermodynamic limit. We show that the averaged OTOC decays faster in the gapless region than in the gapped region. In the gapless region, the averaged OTOC behaves in the same manner regardless of the anisotropy parameter. This result is consistent with the fact that the low energy excitations of the gapless region belong to the same universality class as the Tomonaga-Luttinger liquid with the central charge c = 1. Furthermore, we estimate c by fitting the numerical data of the OSEE with an analytical result of the two-dimensional conformal field theory, and confirmed that c is close to unity. Thus, our numerical results suggest that the scale invariance is crucial for the universal behavior of the OTOC.
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