Investigation of quantum chaos in local and non-local Ising models

TL;DR

This work addresses identifying chaotic versus integrable dynamics in quantum many-body spin chains, focusing on local and non-local Ising models under transverse and longitudinal fields. It combines spectral diagnostics (energy-level statistics via the level-spacing ratio $ar{r}$ and its GOE/Poisson benchmarks) with dynamical probes (Krylov complexity $C(t)$ computed through Lanczos Krylov space) to map the integrable-chaotic crossover. The key findings show that non-local all-to-all interactions strongly promote chaos, with GOE-like statistics emerging at relatively small couplings, and that chaotic dynamics are accompanied by a rapid growth and a pronounced peak in Krylov complexity, saturating at higher values than integrable regimes; these patterns correlate with $ar{r}$ across both local and non-local models. Overall, the study provides a robust, multifaceted framework for diagnosing quantum chaos in many-body systems and highlights non-locality as a powerful accelerator of scrambling and complexity.

Abstract

We investigate signatures of quantum chaos within Ising spin chains subjected to transverse and longitudinal fields, incorporating both local (nearest-neighbor) and non-local (long-range) couplings. While local Ising models may exhibit integrable or chaotic dynamics contingent on interaction strengths and field parameters, systems with non-local interactions generally display a stronger propensity toward chaos, even when the non-local couplings are weak. By examining the distribution of energy level spacings through the level spacing ratio, we delineate the transition from integrable to chaotic regimes and characterize the emergence of quantum chaos in these systems. Our analysis demonstrates that non-local couplings facilitate faster operator spreading and more intricate dynamical behavior, enabling these systems to approach maximal chaos more readily than their local counterparts. Additionally, we analyze Krylov complexity as a dynamical probe of chaos, observing a characteristic peak followed by a plateau at late times in chaotic regimes. This behavior provides a quantitative means to distinguish between integrable and chaotic phases, with the growth rate and saturation level of the complexity serving as effective indicators. Our findings underscore the role of non-local interactions in accelerating the onset of chaos and modifying dynamical complexity in quantum spin chains.

Figures (5)

• Figure 1: The Lanczos coefficients for the local version of the mixed-field Ising model (top plots) and the non-local version (bottom plot), in both integrable and chaotic forms are shown.
• Figure 2: We examine the time dependence of the Krylov complexity for the initial state defined in Eq.\ref{['eq.26']}, evaluated at $\beta=0$, which corresponds to the infinite-temperature limit. In the regime where the system exhibits chaotic dynamics, we observe that the Krylov complexity increases rapidly, reaches a pronounced maximum, and subsequently saturates due to the finite dimension of the accessible Hilbert space. The black dashed line denotes the infinite-time average of the Krylov complexity.
• Figure 3: Energy level spacing distributions are presented for the positive parity sector of both the local and non-local Ising models at $h_z= 0$ and $0.5$, Also, in these plots, $L=13$ is considered. The blue curve corresponds to the Poisson distribution, illustrating the behavior expected in integrable regimes, while the red curve represents the Wigner-Dyson distribution, characteristic of chaotic dynamics.
• Figure 4: Energy level spacing distributions in the positive parity sector of the local and non-local Ising models at $h_{z}=1,1.5$, Also, in these plots, $L=13$ is considered. For the local Ising model, increasing $h_{z}$ suppresses chaotic signatures, shifting the level spacing statistics from Wigner-Dyson (red) toward Poisson (blue) distributions. In contrast, the non-local Ising model exhibits an enhanced tendency toward chaos under increasing $h_{z}$, reflected in a transition from Poisson-like to Wigner-Dyson statistics across specific parameter regimes.
• Figure 5: The panels display the ratio of adjacent energy level spacings in the mixed-field Ising model with non-local interactions. In these plots, we have considered $L=13$. The top-left panel shows the distribution of level spacing ratios for the transverse-field Ising model $(g = 0)$ under various values of the longitudinal field $h_{z}$. The top-right panel presents the results for the mixed-field Ising model with fixed $g = -1$ and varying $h_{z}$. The bottom panel shows the level spacing ratio distributions for fixed $h_{z}=0.5$ while varying the non-local interaction strength $g$.