Abstract
We study a precise and computationally tractable notion of operator complexity in holographic quantum theories, including the ensemble dual of Jackiw-Teitelboim gravity and two-dimensional holographic conformal field theories. This is a refined, “microcanonical” version of K-complexity that applies to theories with infinite or continuous spectra (including quantum field theories), and in the holographic theories we study exhibits exponential growth for a scrambling time, followed by linear growth until saturation at a time exponential in the entropy — a behavior that is characteristic of chaos. We show that the linear growth regime implies a universal random matrix description of the operator dynamics after scrambling. Our main tool for establishing this connection is a “complexity renormalization group” framework we develop that allows us to study the effective operator dynamics for different timescales by “integrating out” large K-complexities. In the dual gravity setting, we comment on the empirical match between our version of K-complexity and the maximal volume proposal, and speculate on a connection between the universal random matrix theory dynamics of operator growth after scrambling and the spatial translation symmetry of smooth black hole interiors.
Highlights
The horizon of an Anti-de Sitter (AdS) black hole is a reflection of the maximally chaotic dynamics of the dual conformal field theory (CFT)
We show that the linear growth regime implies a universal random matrix description of the operator dynamics after scrambling
The growing momentum results in gravitational backreaction which, in turn, imprints this exponential growth on the time-dependence of the volume of the maximal volume slices, and by extension on the particle’s distance from the boundary along these slices. This characteristic behavior of momentum has been compellingly linked to the dual operator “size” which increases at the same exponential rate due to scrambling [1,2,3,4,5]; the corresponding maximal volume growth is expected to reflect the evolution of operator complexity [1, 6,7,8,9] — assuming an appropriate definition of the latter
Summary
We start by reviewing the definitions required for the discussion of Krylov complexity, and adapt them to systems with a non-compact spectrum. The key in constructing a meaningful complexity ladder is to introduce an operator inner product, completing the definition of HGNS, that will allow us to orthonormalize the basis (2.1) This is the point where the second ingredient of the Krylov formalism comes in: the reference state. In the presence of other conserved charges of the Liouvillian, a similar approach ought to be followed: a K-complexity basis should be constructed for each fixed charge sector separately This separates the quantum chaotic aspects of the theory from those which are determined by symmetry.. The recurrence coefficients bEn in this context are called Lanczos coefficients This allows us to express the time evolution of an operator O(t) in terms of a Schrödinger equation for its wavefunction on the one-dimensional Krylov chain (2.15): φE,n(t) = bEn+1φE,n+1(t) − bEn φE,n−1(t). The moment method, while somewhat cumbersome to implement numerically due to stability issues, is a very useful tool for computing Lanczos coefficients and we will use it in section 3.2 to compute them for the ensemble dual to JT gravity
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