Krylov Winding and Emergent Coherence in Operator Growth Dynamics

Area of Science:

• Quantum Information Theory
• Condensed Matter Physics
• Quantum Chaos

Background:

• Operator wave functions describe quantum chaos and operator complexity growth.
• Size winding, a phase increasing with operator size at finite temperatures, is observed but not fully understood in thermalizing systems.

Purpose of the Study:

• To elucidate the phenomenon of size winding in quantum chaotic systems.
• To introduce and explain the concept of Krylov winding.

Main Methods:

• Introducing Krylov winding as a phase acquired by the operator wave function proportional to the Krylov index.
• Demonstrating Krylov winding is a generic feature of quantum chaotic systems, linked to the operator growth bound hypothesis.
• Analyzing conditions for Krylov winding to induce size winding: low-rank basis mapping and saturation of the chaos-operator growth bound.

Main Results:

• Krylov winding is shown to be a universal feature of quantum chaotic systems.
• Size winding emerges from Krylov winding under specific conditions related to basis mapping and bound saturation.
• Deviations from bound saturation lead to superlinear winding (ℓ^{1/h}) for systems with h < 1.

Conclusions:

• Krylov winding provides a fundamental explanation for size winding in quantum chaotic systems.
• The study connects operator growth, quantum chaos, and thermalization through the operator growth bound.
• Results are illustrated using the Sachdev-Ye-Kitaev (SYK) model and disordered k-local spin models.