Integrability as an attractor of adiabatic flows

TL;DR

This paper explores how integrable regions in quantum spin chains act as attractors for adiabatic flows, revealing universal transition behaviors from integrability to chaos with implications for quantum dynamics.

Contribution

It introduces a geometric framework using the quantum geometric tensor to analyze adiabatic flows and demonstrates the universality of the integrability-chaos transition in different spin chain models.

Findings

Integrable regions act as attractors in coupling space for adiabatic flows.

Transition from integrability to chaos exhibits universal, phase transition-like behavior.

Local integrability breaking leads to rapid chaos but avoids ergodicity.

Abstract

The interplay between quantum chaos and integrability has been extensively studied in the past decades. We approach this topic from the point of view of geometry encoded in the quantum geometric tensor, which describes the complexity of adiabatic transformations. In particular, we consider two generic models of spin chains that are parameterized by two independent couplings. In one, the integrability breaking perturbation is global while, in the other, integrability is broken only at the boundary. In both cases, the shortest paths in the coupling space lead towards integrable regions and we argue that this behavior is generic. These regions thus act as attractors of adiabatic flows similar to river basins in nature. Physically, the directions towards integrable regions are characterized by faster relaxation dynamics than those parallel to integrability, and the anisotropy between them diverges in the thermodynamic limit as the system approaches the integrable point. We also provide evidence that the transition from integrable to chaotic behavior is universal for both models, similar to continuous phase transitions, and that the model with local integrability breaking quickly becomes chaotic but avoids ergodicity.