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Numerically Probing the Universal Operator Growth Hypothesis

Robin Heveling, Jiaozi Wang, Jochen Gemmer

TL;DR

This study tests the universal operator growth hypothesis by computing Lanczos coefficients from autocorrelations in Ising and Heisenberg spin models. It analyzes a geometry-derived bound on growth via moments and compares actual coefficients to this bound across 1D and 2D systems. The results show near-linear growth of b_n in nonintegrable Ising models and linear-like behavior in 2D Ising, while Heisenberg data are inconclusive within reachable n, and the bound 𝔅_n is not tight but shares the same growth form. Overall, the work supports the plausible universality of operator growth while highlighting limits of current numerics and bounds.

Abstract

Recently, a hypothesis on the complexity growth of unitarily evolving operators was presented. This hypothesis states that in generic, non-integrable 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 the paper at hand, we numerically test this hypothesis for a variety of exemplary systems, including 1d and 2d Ising models as well as 1d 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 eventually stems from geometric arguments involving the locality of the Hamiltonian as well as the lattice configuration. We investigate such a geometric bound and find that it is not sharply achieved for any considered model.

Numerically Probing the Universal Operator Growth Hypothesis

TL;DR

This study tests the universal operator growth hypothesis by computing Lanczos coefficients from autocorrelations in Ising and Heisenberg spin models. It analyzes a geometry-derived bound on growth via moments and compares actual coefficients to this bound across 1D and 2D systems. The results show near-linear growth of b_n in nonintegrable Ising models and linear-like behavior in 2D Ising, while Heisenberg data are inconclusive within reachable n, and the bound 𝔅_n is not tight but shares the same growth form. Overall, the work supports the plausible universality of operator growth while highlighting limits of current numerics and bounds.

Abstract

Recently, a hypothesis on the complexity growth of unitarily evolving operators was presented. This hypothesis states that in generic, non-integrable 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 the paper at hand, we numerically test this hypothesis for a variety of exemplary systems, including 1d and 2d Ising models as well as 1d 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 eventually stems from geometric arguments involving the locality of the Hamiltonian as well as the lattice configuration. We investigate such a geometric bound and find that it is not sharply achieved for any considered model.
Paper Structure (8 sections, 27 equations, 5 figures)

This paper contains 8 sections, 27 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Operator growth hypothesis
  3. Bound on growth via moments
  4. Numerical Analysis
  5. Transverse Ising model
  6. Heisenberg model
  7. Conclusion
  8. Translating between $b_n$ and $\mu_{2n}$

Figures (5)

  • Figure 2: Lanczos coefficients $b_n$ of the transverse Ising model for a local observable $\mathcal{O}^{(3)} = \sigma_0^x$ for various integrability-breaking parameters $B_x$. The distinction between the free and non-nonintegrable curves is not as striking a before. 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 two.
  • Figure 3: Comparison between Lanczos coefficients for the Ising model with $B_x=0.5$ for all four obervables 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 $\mathcal{O} = \sigma_{0,0}^x$ 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 :
  • Figure :