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Momentum-Krylov complexity correspondence

Zhong-Ying Fan

TL;DR

The paper proposes a momentum-Krylov complexity correspondence, linking the growth rate of Krylov complexity in the boundary to the radial momentum of an infalling bulk particle in AdS/CFT. By formulating dK/dt = P and computing K for various AdS black hole backgrounds, the authors demonstrate universal early quadratic and late-time exponential growth, with an exact BTZ/CFT2 match at finite temperature. Their testing across AdS vacua and higher-dimensional black holes reveals dimensional nuances but preserves key late-time universality, while also discussing connections to circuit complexity. The work provides a compelling approximate dual for Krylov complexity and motivates further exploration of refined momentum concepts and global charge extensions.

Abstract

In this work, we relate the growth rate of Krylov complexity in the boundary to the radial momentum of an infalling particle in AdS geometry. We show that in general AdS black hole background, our proposal captures the universal behaviors of Krylov complexity at both initial and late times. Hence it can be generally considered as an approximate dual of the Krylov complexity at least in diverse dimensions. Remarkably, for BTZ black holes, our holographic Krylov complexity perfectly matches with that of CFT$_2$ at finite temperatures.

Momentum-Krylov complexity correspondence

TL;DR

The paper proposes a momentum-Krylov complexity correspondence, linking the growth rate of Krylov complexity in the boundary to the radial momentum of an infalling bulk particle in AdS/CFT. By formulating dK/dt = P and computing K for various AdS black hole backgrounds, the authors demonstrate universal early quadratic and late-time exponential growth, with an exact BTZ/CFT2 match at finite temperature. Their testing across AdS vacua and higher-dimensional black holes reveals dimensional nuances but preserves key late-time universality, while also discussing connections to circuit complexity. The work provides a compelling approximate dual for Krylov complexity and motivates further exploration of refined momentum concepts and global charge extensions.

Abstract

In this work, we relate the growth rate of Krylov complexity in the boundary to the radial momentum of an infalling particle in AdS geometry. We show that in general AdS black hole background, our proposal captures the universal behaviors of Krylov complexity at both initial and late times. Hence it can be generally considered as an approximate dual of the Krylov complexity at least in diverse dimensions. Remarkably, for BTZ black holes, our holographic Krylov complexity perfectly matches with that of CFT at finite temperatures.

Paper Structure

This paper contains 14 sections, 69 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Universal behaviors of Krylov complexity
  3. The proposal for holographic Krylov complexity
  4. Testing the proposal
  5. AdS vacuum
  6. BTZ black hole
  7. Higher dimensional black holes
  8. $D=5$ dimension
  9. $D=4$ dimension
  10. Universal results at late times
  11. The relation to circuit complexity
  12. Conclusion
  13. Continuum limit analysis at long times
  14. Free falling particles in AdS spacetimes

Figures (1)

  • Figure 1: The Krylov complexity for five (green) and four (blue) dimensional Schwarzschild black holes. The dashed line is the refined complexity in the four dimension, which equals to half of the K-complexity at late times approximately. We have set $T=1/\pi$ and $E=2\ell_{AdS}$.