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arXiv:2008.12274 · Journal of High Energy Physics

Complexity growth of operators in the SYK model and in JT gravity

Shao-Kai Jian, Brian G. Swingle, Zhuo-Yu Xian

Published Mar 1, 2021• 116 citations
arXivHTMLPDF

Abstract

Abstract The concepts of operator size and computational complexity play important roles in the study of quantum chaos and holographic duality because they help characterize the structure of time-evolving Heisenberg operators. It is particularly important to understand how these microscopically defined measures of complexity are related to notions of complexity defined in terms of a dual holographic geometry, such as complexity-volume (CV) duality. Here we study partially entangled thermal states in the Sachdev-Ye-Kitaev (SYK) model and their dual description in terms of operators inserted in the interior of a black hole in Jackiw-Teitelboim (JT) gravity. We compare a microscopic definition of complexity in the SYK model known as K-complexity to calculations using CV duality in JT gravity and find that both quantities show an exponential-to-linear growth behavior. We also calculate the growth of operator size under time evolution and find connections between size and complexity. While the notion of operator size saturates at the scrambling time, our study suggests that complexity, which is well defined in both quantum systems and gravity theories, can serve as a useful measure of operator evolution at both early and late times.

DOI
10.1007/jhep03(2021)014

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Abstract The concepts of operator size and computational complexity play important roles in the study of quantum chaos and holographic duality because they help characterize the structure of time-evolving Heisenberg operators. It is particularly important to understand how these microscopically defined measures of complexity are related to notions of complexity defined in terms of a dual holographic geometry, such as complexity-volume (CV) duality. Here we study partially entangled thermal states in the Sachdev-Ye-Kitaev (SYK) model and their dual description in terms of operators inserted in the interior of a black hole in Jackiw-Teitelboim (JT) gravity. We compare a microscopic definition of complexity in the SYK model known as K-complexity to calculations using CV duality in JT gravity and find that both quantities show an exponential-to-linear growth behavior. We also calculate the growth of operator size under time evolution and find connections between size and complexity. While the notion of operator size saturates at the scrambling time, our study suggests that complexity, which is well defined in both quantum systems and gravity theories, can serve as a useful measure of operator evolution at both early and late times.
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.
Abstract The study of quantum gravity in the form of the holographic duality has uncovered and motivated the detailed investigation of various diagnostics of quantum chaos. One such measure is the operator size distribution, which characterizes the size of the support region of an operator and its evolution under Heisenberg evolution. In this work, we examine the role of the operator size distribution in holographic duality for the Sachdev-Ye-Kitaev (SYK) model. Using an explicit construction of AdS2 bulk fermion operators in a putative dual of the low temperature SYK model, we study the operator size distribution of the boundary and bulk fermions. Our result provides a direct derivation of the relationship between (effective) operator size of both the boundary and bulk fermions and bulk SL(2; ℝ) generators.
We present a collection of recent results concerning quantum information theory applied to quantum gravity. We first study the entanglement structure of Euclidean path integral states in SU(2) Chern-Simons theory, where we elucidate a connection between topological entanglement and quantum mechanical entanglement. We prove that the topology of certain three-manifolds controls the entanglement structure of the resulting quantum state, and conjecture a more general relationship for arbitrary three-manifolds. We then analyze the quantum circuit complexity of the time evolution operator in the Sachdev-Ye-Kitaev model, a theory of near-extremal black hole microstates. We find that this complexity grows linearly for a time exponential in the entropy, modulo a caveat concerning global obstructions to the growth of the distance function along geodesics, which we do not rule out. This constitutes a partial proof of Susskind’s conjecture about the complexity growth of black holes. Finally, we address the black hole information paradox in three dimensions by considering a toy model of black hole microstates in the form of an end-of-the-world brane. This brane carries a quantum theory which is itself holographic, and we compute entanglement entropies in the glued dual geometry by using a Ryu-Takayanagi formula where the minimal entropy surface can pass through the gluing surface.
A bstract Previous work has explored the connections between three concepts — operator size, complexity, and the bulk radial momentum of an infalling object — in the context of JT gravity and the SYK model. In this paper we investigate the higher dimensional generalizations of these connections. We use a toy model to study the growth of an operator when perturbing the vacuum of a CFT. From circuit analysis we relate the operator growth to the rate of increase of complexity and check it by complexity-volume duality. We further give an empirical formula relating complexity and the bulk radial momentum that works from the time that the perturbation just comes in from the cutoff boundary, to after the scrambling time.
We investigate the growth of the black hole interior in three-dimensional gravity from the boundary theory. For the two-sided BTZ black hole, we propose a boundary reconstruction of the time dependence of a codimension-one surface in terms of correlation functions of smeared operators in the thermofield double state, reproducing the characteristic late-time linear growth predicted by the complexity-volume proposal. Using the Chern--Simons formulation of three-dimensional gravity, these nonlocal correlators are represented by bulk Wilson lines with smeared endpoints, extending the familiar connection between Wilson lines and codimension-two observables underlying holographic entanglement entropy to codimension-one observables. We then ask whether the same geometric growth is captured by Krylov complexity. For the smeared operators, we extract the Lanczos data from their correlation functions and find that operator Krylov complexity reproduces the late-time linear growth of the black hole interior, extending previous connections between operator growth and bulk geometry to AdS3_33​. By contrast, the Krylov spread complexity of the thermofield double state, obtained from the semiclassical gravitational partition function, does not exhibit the linear growth of the bulk volume within the regime accessible to our analysis. Our results therefore point to a distinguished role for operator Krylov complexity in encoding black hole interior growth beyond two-dimensional gravity, while highlighting a qualitative distinction between operator and state notions of Krylov complexity.
The exact role of the internal degrees of freedom (a.k.a. d.o.f.) in holography is not well-understood. Thus, in this thesis, we study a toy model of holography without space: the Sachdev-Ye-Kitaev (SYK) Model. This 0+1 theory of all possible 4-body interactions of N fermion "flavors"/"colors" features a low energy limit reproducing aspects of 1+1 Jackiw-Teitelboim gravity. First, we show that the inherent discreteness of the quantum spectrum results in universal late-time behavior due to eigenvalue repulsion. We then note that the theory's four-point functions probe the phenomenon of operator growth, where an internal d.o.f. goes on to epidemically evolve into larger products of internal d.o.f.s. In this manner where small operators smoothly grow into superpositions of increasing products of operators, we observe a sort of "size" locality, which is intimately tied with the notion of a conformal primary "descending" along its descendants. In fact, we find that the underlying structure of the SYK epidemic limits to that of a probe particle falling into a AdS2AdS_2AdS2​ black hole. In other words, similar to how nearest neighbor interactions lead to dynamics on a flat space background, we demonstrate that many internal interactions lead to dynamics on a higher dimensional geometry.