Universal scrambling in gapless quantum spin chains

Abstract

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 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 confirm that $c$ is close to unity. Thus, our numerical results suggest that the scale invariance leads to a universal behavior of the OTOC that is independent of the anisotropic parameter, which reflects the universality of the two-dimensional conformal field theory at low temperatures. Although the one-dimensional XXZ model is integrable, our results suggest such a universal behavior of generic nonintegrable systems, because the Tomonaga-Luttinger liquid serves as a low-energy effective theory for many nonintegrable and integrable systems. On the other hand, the OTOC in our numerical result does not exhibit the exponential decay, as our parameter regime is far from the semiclassical limit.