Out-of-Time-Ordered Density Correlators in Luttinger Liquids

Information scrambling and the butterfly effect in chaotic quantum systems can be diagnosed by out-of-time-ordered (OTO) commutators through an exponential growth and large late time value. We show that the latter feature shows up in a strongly correlated many-body system, a Luttinger liquid, whose density fluctuations we study at long and short wavelengths, both in equilibrium and after a quantum quench. We find rich behavior combining robustly universal and nonuniversal features. The OTO commutators display temperature- and initial-state-independent behavior and grow as $t^2$ for short times. For the short-wavelength density operator, they reach a sizable value after the light cone only in an interacting Luttinger liquid, where the bare excitations break up into collective modes. This challenges the common interpretation of the OTO commutator in chaotic systems. We benchmark our findings numerically on an interacting spinless fermion model in 1D and find persistence of central features even in the nonintegrable case. As a nonuniversal feature, the short-time growth exhibits a distance-dependent power.

Figure 1

ED result for $N=22$ for the OTO commutator for the local charge density in the interacting spinless fermion model. Red squares and blue circles denote the late time limit in equilibrium and after a sudden quench from the noninteracting limit, respectively. The solid line is $f\sin (K\pi {)}^2$ from Eq. (9) with only the parameter $f=0.15$ adjustable. Top inset: Numerical time evolution of the OTO commutator with the representative value $J_z/J=0.6$ in equilibrium (red dashed line) and after a sudden quench (blue solid line) from the noninteracting case, $J_z=0$. For a short time ($t<1/J$), it follows the predicted $t^2$ (bottom inset).

Figure 2

ED result for $N=10$ for the OTO commutator between the local charge densities $n_1$ and $n_{1+x}$ in the interacting spinless fermion model with $x=2$ (blue line), 3 (red line), and 4 (green line) in equilibrium for $J_z/J=0.6$; the quench data are indistinguishable in this time window. The thin black dashed lines denote $t^{2x}/(2x)!$.