Out-of-time-ordered Correlators in Integrable Quantum Ising Model

Bannishikha Banerjee

Abstract Out-of-time-order correlators (OTOCs) have emerged as the preeminent theoretical and experimental probe of quantum information scrambling — the delocalization of quantum information across many-body degrees of freedom. Here we present a comprehensive computational study of OTOCs across three paradigmatic models: the Sachdev-Ye-Kitaev (SYK) model, the non-integrable transverse-field Ising chain, and random unitary circuits. Through exact diagonalization and large-N analytics, we extract Lyapunov exponents, scrambling times, and butterfly velocities, demonstrating that all chaotic systems studied approach, but do not saturate, the Maldacena-Shenker-Stanford (MSS) bound λ_L ≤ 2π/β. Crucially, we introduce the Spectral Out-of-Time-Order Correlator (SOTOC), a novel frequency-domain diagnostic that decomposes information scrambling into its constituent spectral modes. The SOTOC achieves a 32% improvement in Lyapunov exponent extraction precision over conventional log-linear fitting and maintains robust performance at noise levels σ > 0.30, where traditional methods fail. Our phase diagram reveals a sharp transition in scrambling dynamics at h/J ≈ 1.0 in the TFIM, coinciding with the onset of quantum chaos as measured by level-spacing statistics. These results establish SOTOC as a practical and theoretically grounded tool for probing scrambling in near-term quantum devices and holographic systems.

Sergey Syzranov, Victor Galitski

Geoffrey R. Grimmett, Tobias J. Osborne, Petra F. Scudo

Rohit Kumar Shukla

In recent times out-of-time-order correlators (OTOC) have been established as a tool to understand butterfly effects, quantum information scrambling, and many-body localization. They can also be useful in determining different phases of quantum critical systems. OTOCs can identify the quantum chaos within a system undergoing time evolution; and therefore, they can distinguish between chaotic and regular dynamics. This motivates us to study OTOCs in integrable and nonintegrable periodically kicked quantum spin models. A periodically kicked quantum Ising spin system, known as the quantum Ising Floquet system, is a variant of the transverse Ising model. In place of constant transverse magnetic fields in the transverse Ising system, time-periodic fields are applied in the form of delta pulses in the quantum Ising Floquet spin system. It provides very interesting and peculiar dynamics separate from that of the transverse Ising system.

Sei Suzuki, Jun‐ichi Inoue, Bikas K. Chakrabarti

Hua 华 Yan 颜, Jiao-Zi 骄子 Wang 王, Wen-Ge 文阁 Wang 王

Abstract Previous studies show that, in quantum chaotic and integrable systems, the so-called out-of-time-ordered correlator (OTOC) generically behaves differently at long times, while, it may show similar early growth in one-body systems. In this paper, by means of numerical simulations, it is shown that OTOC has similar early growth in two quantum many-body systems, one integrable and one chaotic .