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Integrability-to-chaos transition through the Krylov approach for state evolution.

Gastón F Scialchi^1,2, Augusto J Roncaglia^1,2, Diego A Wisniacki^1,2

¹<a href="https://ror.org/0081fs513">Universidad de Buenos Aires</a>, Facultad de Ciencias Exactas y Naturales, Departamento de Física, Ciudad Universitaria, 1428 Buenos Aires, Argentina.

Physical Review. E

|June 22, 2024

Summary

The Krylov basis effectively measures quantum chaos by analyzing the spread of quantum evolutions. Initial conditions significantly influence Krylov complexity saturation and Lanczos coefficient spread, crucial for gauging dynamical quantum chaos.

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Area of Science:

Quantum mechanics
Chaos theory
Statistical physics

Background:

Quantum evolution complexity is understood by basis spread.
Krylov basis minimizes spread, aiding quantum chaos investigation.
Transition from integrability to chaos is a key research area.

Purpose of the Study:

Investigate the transition from integrability to chaos using the Krylov approach.
Analyze the role of initial conditions in Krylov complexity and Lanczos coefficients.
Determine the efficacy of Krylov-based measures for dynamical quantum chaos.

Main Methods:

Employed the Krylov approach for analyzing quantum evolutions.
Utilized an Ising spin chain and a banded random matrix model as test systems.
Examined the spread of quantum evolutions in the Krylov basis.

Main Results:

Krylov complexity saturation shows dependence on initial conditions.
Spread of Lanczos coefficients is also sensitive to the initial state.
Both quantities can effectively gauge dynamical quantum chaos.

Conclusions:

The Krylov approach provides valuable insights into the transition from integrability to chaos.
Initial state selection is critical for accurately assessing quantum chaos using Krylov complexity and Lanczos coefficients.
This work highlights the importance of the Krylov basis in understanding complex quantum dynamics.