Operator growth in the transverse-field Ising spin chain with integrability-breaking longitudinal field

We investigate the operator growth dynamics of the transverse field Ising spin chain in one dimension as varying the strength of the longitudinal field. An operator in the Heisenberg picture spreads in the extended Hilbert space. Recently, it has been proposed that the spreading dynamics has a universal feature signaling chaoticity of underlying quantum dynamics. We demonstrate numerically that the operator growth dynamics in the presence of the longitudinal field follows the universal scaling law for one-dimensional chaotic systems. We also find that the operator growth dynamics satisfies a crossover scaling law when the longitudinal field is weak. The crossover scaling confirms that the uniform longitudinal field makes the system chaotic at any nonzero value. We also discuss the implication of the crossover scaling on the thermalization dynamics and the effect of a nonuniform local longitudinal field.

Figure 1

Plots of $b_n$ for the t-Ising model ($h=1$ and $g_l=0$) with respect to $n$ in (a) and $\sqrt{n}$ in (b). $b_n$ of $O^z$ is in perfect agreement with the straight line in (b).

Figure 2

Lanczos coefficients for the operator $O^x={\sigma }_0^x$ in the tl-Ising model with $h=1$. (a) The Lanczos coefficients at several values of the uniform longitudinal fields $g$ are compared. (b) Plots of $n/b_n$ against $W\left(n\right)$. Data for $n>n_c\left(g\right)$ are in agreement with the straight lines, which indicates that $b_n\propto n/W\left(n\right)$ for $n>n_c\left(g\right)$. (c) Scaling plot of $g^{-2}\mathrm{\Delta }{b}_n$ vs $n$.

Figure 3

The same plots as in Fig. 2 for the operator $O^z={\sigma }_0^z$.

Figure 4

Lanczos coefficients for the operator ${\sigma }_0^x$ in the tl-Ising model with $h=1$ and $g_l=g{\delta }_{l0}$ with $g\le 0.1$. (a) The Lanczos coefficients at several values of the local longitudinal fields $g$ are compared. There is a crossover from the type-I behavior. (b) Plots of $b_n$ against $\sqrt{n}$. The straight line represents a linear fit of the data with $g=0.1$ for $n\ge 13$. (c) Plots of $n/b_n$ against $W\left(n\right)$. The straight line also represents a linear fit of the data with $g=0.1$.

Figure 5

Lanczos coefficients for the operator ${\sigma }_0^x$ in the tl-Ising model with $h=1$ and $g_l=g{\delta }_{l0}$ with $g\ge 0.1$. The data are plotted in the same way as in Fig. 4.