  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.
