Out-of-time Correlation Functions Maintain Maldacena Bound Via Instantons in Ring Polymer Molecular Dynamics

The scrambling of information lies at the heart of chaotic systems, and scientists commonly use out-of-time-ordered correlators (OTOCs) to measure the rate at which this scrambling occurs. Andrew C. Hunt from Caius College, along with colleagues, investigates how instantons, quantum mechanical phenomena that govern tunnelling, influence this scrambling rate and whether current computational methods accurately capture this behaviour. This research demonstrates that instantons play a crucial role in upholding a fundamental theoretical bound on scrambling, known as the Maldacena bound, but also reveals limitations in the widely used ring polymer molecular dynamics (RPMD) method for simulating these complex systems. By developing an alternative approach using Matsubara dynamics, the team uncovers distinct dynamical behaviour around instantons, challenging the assumptions of RPMD and offering new insights into the fundamental physics of chaos and information scrambling.

Out-of-time correlation functions in single-body systems Recent studies demonstrate that instantons, localised solutions representing quantum tunnelling, play a crucial role in determining the behaviour of ‘out of time ordered correlators’ (OTOCs), which are commonly used to quantify the rate at which quantum information is scrambled. This work investigates the dynamics of OTOCs in single-body quantum systems, focusing on how initial conditions and complex energy landscapes influence the emergence of chaotic behaviour. The research develops a theoretical framework for analysing OTOCs, providing insights into the mechanisms governing quantum information scrambling.

Specifically, the team found that tunnelling through potential barriers reduces the growth rate of OTOCs. For a symmetric double well potential, this reduction ensures the Maldacena bound, a theoretical limit on scrambling rates, is maintained when using ring polymer molecular dynamics, a method that approximates quantum dynamics with exact quantum statistics. The impact of system confinement on the flattening of OTOCs was also investigated by comparing bounded and scattering systems, revealing that scattering systems exhibit significantly slower growth rates, a result attributed to the Boltzmann operator and interference from the potential energy landscape.

Instantons, Wavepackets, and Quantum Chaos Calculations

This document details the numerical methods and parameters used in a series of calculations related to quantum dynamics, specifically focusing on instantons, wavepacket propagation, and OTOC calculations. Calculations rely on numerical integration using the trapezium rule and the discrete variable representation (DVR) to represent quantum states on a grid. Parameters such as grid length, the number of grid points, and particle mass are carefully chosen to ensure accurate results, and numerical convergence is rigorously checked to validate the reliability of the calculations.

Detailed calculations involving instantons and transition state dynamics are performed to explore potential energy surfaces. Wavepacket propagation simulations are used to model the time evolution of quantum states, and OTOCs are computed to characterise quantum chaos and information scrambling, employing Kubo regularization to ensure convergence. Key concepts underpinning the calculations include instantons, representing quantum tunnelling paths, and transition state theory, a method for calculating reaction rates. Permutational invariance, ensuring calculations remain consistent under variable permutations, is also maintained throughout the process.

Instantons Govern Quantum Information Scrambling Rates

This research has significantly advanced understanding of quantum chaos by investigating the role of instantons in determining the rate of information scrambling. The team demonstrated that instantons contribute to upholding the Maldacena bound in certain quantum systems. Through detailed calculations, they observed that systems allowing for particle scattering exhibit slower scrambling rates and a flattening of growth over time, effects attributed to the influence of the Boltzmann operator and interference from the potential energy landscape.

However, the study also revealed limitations in current methods for modelling these quantum systems. Specifically, the researchers found that the ring polymer molecular dynamics (RPMD) approach does not consistently satisfy the Maldacena bound, suggesting it may not fully capture the complex dynamics governing quantum chaos. To address this, they developed a new theoretical framework based on Matsubara dynamics, which provides a more accurate description of the behaviour around instantons and their fluctuations. This new approach highlights differences in dynamical behaviour compared to predictions from RPMD, suggesting a more nuanced understanding of quantum chaos is required. Future work will focus on further refining this theory and exploring its implications for developing novel quantum rate theories.