Universal scrambling in gapless quantum spin chains

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.

Figure 1

Schematics of the iTEBD. (a) After the Trotter decomposition of $e^{-\frac{\beta }{4}\widehat{H}}$, we obtain the two-site translation-invariant iMPO composed of two tensors $\left\{A_1,A_2\right\}$ by performing the singular value decompositions. (b) Transformation of the iMPO into the canonical form composed of four tensors.

Figure 2

Imaginary-time evolution of the OSEE $S_{\mathrm{A}}^{\mathrm{OSEE}\left(2\right)}$ for the operator $e^{-\frac{\beta }{4}\widehat{H}}$ obtained by the iTEBD. The bond dimension $\chi$ is 512 and the Trotter decomposition width is $\mathrm{\Delta }\beta =0.01$. Small anisotropy parameters $\mathrm{\Delta }=0.25,0.5,0.75,1$ correspond to the gapless region and large ones $\mathrm{\Delta }=1.6,1.8,2,3$ correspond to the gapped region. As the temperature decreases, the difference between these regions becomes more visible.

Figure 3

OSEE $S_{\mathrm{A}}^{\mathrm{OSEE}\left(2\right)}$ for $e^{-\frac{\beta }{4}\widehat{H}}$ with $\mathrm{\Delta }=3$ in Fig. 2 is extracted. The open black circles represent $S_{\mathrm{A}}^{\mathrm{OSEE}\left(2\right)}+ln2 (7.2\le \beta )$ and the black dashed line represents the average of them. If there were no artificial symmetry breaking, the OSEE at low temperature would saturate near the black dashed line.

Figure 4

(a) Real-time dependence of the OSEE $S_{\mathrm{A}}^{\mathrm{OSEE}\left(2\right)}$ for the complex-time evolution operator $e^{-\left(\frac{\beta }{4}+it\right)\widehat{H}}$ for the XXZ model. We subtracted $S_0$, which is the value of the OSEE at $t=0$. (b) Real-time dependence of the averaged OTOC ${\overline{F}}_{\mathrm{AB}}$ corresponding to the OSEE by Eq. (9). The vertical axis is normalized by dividing by ${\overline{F}}_{\mathrm{AB}}\left(0\right)$. The bond dimension $\chi$ is 512 and the Trotter decomposition widths on the imaginary-time axis and the real-time axis are $\mathrm{\Delta }\beta =\mathrm{\Delta }t=0.005$. Small anisotropy parameters $\mathrm{\Delta }=0.25,0.5,0.75,1$ correspond to the gapless region and large ones $\mathrm{\Delta }=1.6,1.8,2$ correspond to the gapped region. The averaged OTOC ${\overline{F}}_{\mathrm{AB}}$ decays faster in the gapless region. The averaged OTOCs in the gapless region decay in the same manner.

Figure 5

The effective-length dependence of the Rényi-2 OSEE $S_{\mathrm{A}}^{\mathrm{OSEE}\left(2\right)}$ for the complex-time evolution operator $e^{-\left(\frac{\beta }{4}+it\right)\widehat{H}}$ in the gapless region ($\mathrm{\Delta }=0.5$). The horizontal axis is $ln{L}_{\mathrm{eff}}=ln[\frac{\beta v}{\pi }cosh\left(\frac{2\pi }{\beta }t\right)]$. For each of the inverse temperatures $\beta =2,3,4,5,6$, we numerically calculate the real-time evolution by fixing $\beta$. The solid line is obtained by the fitting with the fitting range [3,4].

Figure 6

Imaginary-time evolution of the OSEE with $\mathrm{\Delta }\beta =0.04,0.02,0.01,0.005$. The bond dimension is taken as $\chi =512$. The upper figure is for the gapless region $\mathrm{\Delta }=0.5$ and the lower one is for the gapped region $\mathrm{\Delta }=3$.

Figure 7

Imaginary-time evolution of the OSEE with $\chi =256,384,512$. The width in the Trotter decomposition is taken as $\mathrm{\Delta }\beta =0.01$. The upper figure is for the gapless region $\mathrm{\Delta }=0.5$ and the lower one is for the gapped region $\mathrm{\Delta }=3$.

Figure 8

Real-time evolution of OSEE with $\mathrm{\Delta }t=0.02,0.01,0.005,0.0025$. The inverse temperature is $\beta =4$ and the bond dimension is $\chi =512$. The upper panel is for the gapless region $\mathrm{\Delta }=0.5$ and the lower one is for the gapped region $\mathrm{\Delta }=2$.

Figure 9

Real-time evolution of the OSEE with $\chi =256,384,512$. The inverse temperature is $\beta =4$ and the width in the Trotter decomposition is $\mathrm{\Delta }t=0.005$. The upper panel is for the gapless region $\mathrm{\Delta }=0.5$ and the lower one is for the gapped region $\mathrm{\Delta }=2$.

Figure 10

The $ln{L}_{\mathrm{eff}}$ dependence of the OSEE of the complex-time evolution operator $e^{-\left(\frac{\beta }{4}+it\right)\widehat{H}}$ in the gapless region $\mathrm{\Delta }=0.5$ with $\chi =256,384,512$.

Figure 11

The $ln{L}_{\mathrm{eff}}$ dependence of the OSEE of the complex-time evolution operator $e^{-\left(\frac{\beta }{4}+it\right)\widehat{H}}$ in the gapless region $\mathrm{\Delta }=0.5$ with $\chi =256$. The solid line represents the fitting result corresponding to the fitting range [3,4] in Table 2.

Figure 12

The $ln{L}_{\mathrm{eff}}$ dependence of the OSEE of the complex-time evolution operator $e^{-\left(\frac{\beta }{4}+it\right)\widehat{H}}$ in the gapless region $\mathrm{\Delta }=0.5$ with $\chi =384$. The solid line represents the fitting result corresponding to the fitting range [3,4] in Table 3.