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Krylov Winding and Emergent Coherence in Operator Growth Dynamics.

Rishik Perugu1, Bryce Kobrin2, Michael O Flynn3

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We introduce Krylov winding, a new concept explaining operator growth in quantum chaotic systems. This phenomenon clarifies the mysterious size winding observed at finite temperatures, linking it to universal operator growth bounds.

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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.