Exploring the properties of quantum scars in a toy model

TL;DR

This paper introduces a toy model to study quantum scars, demonstrating how they deviate from ergodic behavior, and explores their properties and controllability in many-body quantum systems.

Contribution

The paper presents a simple, pedagogical toy model that captures key features of quantum scars and introduces a projector method to control their mixing with ergodic states.

Findings

Quantum scars can be generated and controlled using projectors.

Entanglement spectrum reveals classicality of scarred states.

Out-of-time-ordered correlators show scar dynamics.

Abstract

We introduce the concept of ergodicity and explore its deviation caused by quantum scars in an isolated quantum system, employing a pedagogical approach based on a toy model. Quantum scars, originally identified as traces of classically unstable orbits in certain wavefunctions of chaotic systems, have recently regained interest for their role in non-ergodic dynamics, as they retain memory of their initial states. We elucidate these features of quantum scars within the same framework of this toy model. The integrable part of the model consists of two large spins, with a classical counterpart, which we combine with a random matrix to induce ergodic behavior. Scarred states can be selectively generated from the integrable spin Hamiltonian by protecting them from the ergodic states using a projector method. Deformed projectors mimic the `quantum leakage' of scarred states, enabling tunable mixing with ergodic states and thereby controlling the degree of scarring. In this simple model, we investigate various properties of quantum scarring and shed light on different aspects of many-body quantum scars observed in more complex quantum systems. Notably, the underlying classicality can be revealed through the entanglement spectrum and the dynamics of `out-of-time-ordered correlators'.