Numerically probing the universal operator growth hypothesis

Recently, a hypothesis on the complexity growth of unitarily evolving operators was presented. This hypothesis states that in generic, nonintegrable many-body systems, the so-called Lanczos coefficients associated with an autocorrelation function grow asymptotically linear, with a logarithmic correction in one-dimensional systems. In contrast, the growth is expected to be slower in integrable or free models. In this paper, we numerically test this hypothesis for a variety of exemplary systems, including one-dimensional and two-dimensional Ising models as well as one-dimensional Heisenberg models. While we find the hypothesis to be practically fulfilled for all considered Ising models, the onset of the hypothesized universal behavior could not be observed in the attainable numerical data for the Heisenberg model. The proposed linear bound on operator growth associated with the hypothesis eventually stems from geometric arguments involving the locality of the Hamiltonian as well as the lattice configuration. We derive and investigate a related geometric bound, and we find that while the bound itself is not sharply achieved for any considered model, the hypothesis is nonetheless fulfilled in most cases.

Figure 1

Lanczos coefficients $b_n$ of the transverse Ising model for the 2-local observables (a) $O^{\left(1\right)}$, (b) slow mode of $O_q^{\left(2\right)}$ with $q=\pi /13$, and (c) fast mode of $O_q^{\left(2\right)}$ with $q=\pi$. The integrability-breaking magnetic field attains values from $B_x=0.0$ to 0.5. For all observables, the transition from a free model to a nonintegrable model is evident. The coefficients ${\mathcal{B}}_n$ are explicitly depicted in (a) for the case $B_x=0.5$ as yellow triangles. Dashed lines indicate the “lower branches” of the corresponding ${\mathcal{B}}_n$. To avoid clutter, only the dashed lines are depicted as a guide to the eye for other values of $B_x$ and in all subsequent figures. The coefficients ${\mathcal{B}}_n$ are larger than the physical $b_n$ by a factor of about 2 for all observables.

Figure 2

Lanczos coefficients $b_n$ of the transverse Ising model for a local observable $O^{\left(3\right)}={\sigma }_0^x$ for various integrability-breaking parameters $B_x$. The distinction between the free and nonintegrable curves is not as striking as before. The green dash-dotted line indicates a fit $\propto \sqrt{n}$ to the data of the integrable model. Dashed lines serve as a guide to the eye for the coefficients ${\mathcal{B}}_n$, which are larger by a factor of about 2.

Figure 3

Comparison between Lanczos coefficients for the Ising model with $B_x=0.5$ for all four observables considered thus far. The growth is quite similar for larger $n$ hinting at a universality of operator growth.

Figure 4

Lanczos coefficients $b_n$ of the two-dimensional transverse Ising model for an observable $O={\sigma }_{0,0}^x$ for various $B_x$. For all values of $B_x$ the growth is nicely linear. Dashed lines serve as a guide to eye for the coefficients ${\mathcal{B}}_n$, which are much larger (note the additional vertical axis on the right).

Figure 5

Lanczos coefficients $b_n$ of the Heisenberg model for spin density waves. Depicted are various combinations of anisotropies and momenta: $\left(\mathrm{a}\right) \mathrm{\Delta }=0.5, q=\pi /13$; $\left(\mathrm{b}\right) \mathrm{\Delta }=0.5, q=\pi$; $\left(\mathrm{c}\right) \mathrm{\Delta }=1.5, q=\pi /13$; $\left(\mathrm{d}\right) \mathrm{\Delta }=1.5, q=\pi$. The integrability-breaking parameter ${\mathrm{\Delta }}^{\prime }$ attains values from 0.0 to 0.5. Dashed lines serve as a guide to the eye for the coefficients ${\mathcal{B}}_n$, which are much larger (note the additional vertical axes on the right).