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

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.