Recent years have witnessed tremendous progress in developing a fine-grained low-dimensional holographic correspondence, specifically the construction of quantum mechanical boundary theories as holographic duals of two-dimensional gravity. In these developments, quantum chaos played a crucial role, both as source of universality and as a guiding principle for the matching of bulk and boundary signatures of gravity. In this article we review the construction of the chaos-assisted low-dimensional holographic correspondence for non-experts. We open with an introductory discussion of the two main protagonists of the theory, the SYK model and two-dimensional Jackiw-Teitelboim gravity. Within this framework we will discuss two independent 'bridges' between bulk and boundary physics, one pertaining to early time chaotic instabilities, the other to late time quantum chaos up to and including time scales of the order of the gravitational quantum level spacing. We will demonstrate that the resolution of these fine-grained quantum scales requires the extension of semiclassical gravity by elements of string theory. We conclude with an outlook towards higher dimensional generalizations of the chaotic holographic correspondence.
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Chapter for the Quantum Chaos volume in 'Comprehensive Quantum Mechanics', to be published by Elsevier (Main editor: R. B. Mann; volume editors: S. Gnutzmann and K. {Ż}yczkowski)
Recent years have witnessed tremendous progress in developing a fine-grained low-dimensional holographic correspondence, specifically the construction of quantum mechanical boundary theories as holographic duals of two-dimensional gravity. In these developments, quantum chaos played a crucial role, both as source of universality and as a guiding principle for the matching of bulk and boundary signatures of gravity. In this article we review the construction of the chaos-assisted low-dimensional holographic correspondence for non-experts. We open with an introductory discussion of the two main protagonists of the theory, the SYK model and two-dimensional Jackiw-Teitelboim gravity. Within this framework we will discuss two independent 'bridges' between bulk and boundary physics, one pertaining to early time chaotic instabilities, the other to late time quantum chaos up to and including time scales of the order of the gravitational quantum level spacing. We will demonstrate that the resolution of these fine-grained quantum scales requires the extension of semiclassical gravity by elements of string theory. We conclude with an outlook towards higher dimensional generalizations of the chaotic holographic correspondence.
We investigate the bulk-boundary correspondence in the SYK model from a quantum information perspective. The SYK model describes a system of Majorana fermions with random all-to-all interactions, whose disorder average-typically taken over a Gaussian ensemble-admits a dual description in terms of JT gravity in the large-N, low-energy limit. This framework provides a minimal setting for exploring holography and emergent spacetime in nearly AdS2. We probe the holographic principle through diagnostics of quantum chaos and entanglement. In the early-time regime, the SYK model saturates the universal bound on the Lyapunov exponent, signaling maximal chaos consistent with semiclassical black hole dynamics. In the late-time regime, its spectral statistics are governed by random matrix theory, reflecting universal features of strongly chaotic quantum systems. These dynamical properties establish a concrete link between boundary quantum chaos and bulk semiclassical gravity. In parallel, we analyze quantum entanglement and the structure of operator algebras to investigate transitions in the associated von Neumann algebras and their implications for emergent geometry. To explore the robustness of these phenomena, we consider deformations of the SYK model through modified matter couplings and alternative random distributions. Our results clarify how quantum information-theoretic structures encode bulk gravitational dynamics and provide insight into the mechanism of spacetime emergence.
Recently, the physics of many-body quantum chaotic systems close to their ground states has come under intensified scrutiny. Such studies are motivated by the emergence of model systems exhibiting chaotic fluctuations throughout the entire spectrum [the Sachdev-Ye-Kitaev (SYK) model being a renowned representative] as well as by the physics of holographic principles, which likewise unfold close to ground states. Interpreting the edge of the spectrum as a quantum critical point, here we combine a wide range of analytical and numerical methods to the identification and comprehensive description of two different universality classes: the near edge physics of “sparse” and the near edge of “dense” chaotic systems. The distinction lies in the ratio between the number of a system's random parameters and its Hilbert space dimension, which is exponentially small or algebraically small in the sparse and dense case, respectively. Notable representatives of the two classes are generic chaotic many-body models (sparse) and invariant random matrix ensembles or chaotic gravitational systems (dense). While the two families share identical spectral correlations at energy scales comparable to the level spacing, the density of states and its fluctuations near the edge are different. Considering the SYK model as a representative of the sparse class, we apply a combination of field theory and exact diagonalization to a detailed discussion of its edge spectrum. Conversely, Jackiw-Teitelboim gravity is our reference model for the dense class, where an analysis of the gravitational path integral and random matrix theory reveal universal differences to the sparse class, whose implications for the construction of holographic principles we discuss. Published by the American Physical Society 2024
A bstract The Sachdev-Ye-Kitaev (SYK) model is a system of N Majorana fermions with random interactions and strongly chaotic dynamics, which at low energy admits a holographically dual description as two-dimensional Jackiw-Teitelboim gravity. Hence the SYK model provides a toy model of quantum gravity that might be feasible to simulate with near-term quantum hardware. Motivated by the goal of reducing the resources needed for such a simulation, we study a sparsified version of the SYK model, in which interaction terms are deleted with probability 1 −p . Specifically, we compute numerically the spectral form factor (SFF, the Fourier transform of the Hamiltonian’s eigenvalue pair correlation function) and the nearest-neighbor eigenvalue gap ratio r (characterizing the distribution of gaps between consecutive eigenvalues). We find that when p is greater than a transition value p 1 , which scales as 1/ N 3 , both the SFF and r match the values attained by the full unsparsified model and with expectations from random matrix theory (RMT). But for p < p 1 , deviations from unsparsified SYK and RMT occur, indicating a breakdown of holography in the highly sparsified regime. Below an even smaller value p 2 , which also scales as 1/ N 3 , even the spacing of consecutive eigenvalues differs from RMT values, signaling a complete breakdown of spectral rigidity. Our results cast doubt on the holographic interpretation of very highly sparsified SYK models obtained via machine learning using teleportation infidelity as a loss function.
This Thesis explores the notion of Krylov complexity as a probe of quantum chaos and as a candidate for holographic complexity. The first Part is devoted to presenting the fundamental notions required to conduct research in this area. Namely, an extensive introduction to the Lanczos algorithm, its properties and associated algebraic structures, as well as technical details related to its practical implementation, is given. Subsequently, an overview of the seminal references and the main debates regarding Krylov complexity and its relation to chaos and holography is provided. The text throughout this first Part combines review material with original analyses which either intend to contextualize, compare and criticize results in the literature, or are the fruit of the investigations leading to the publications on which this Thesis is based. These research projects are the subject of the second Part of the manuscript. In them, methods for the efficient implementation of the Lanczos algorithm in finite many-body systems were developed, allowing to compute numerically the Krylov complexity of models like SYK or the XXZ spin chain up to time scales exponentially large in system size. It was observed that the operator Krylov complexity profile in SYK, a paradigmatic low-dimensional chaotic system with a holographic dual, agrees with holographic expectations, while in the case of integrable models like XXZ complexity is affected by a novel localization effect in the so-called Krylov space which hinders its growth. Finally, an exact, analytical, correspondence between the Krylov complexity of the infinite-temperature thermofield double state in the low-energy regime of the double-scaled SYK model and bulk length in the theory of JT gravity is established.
There has been renewed interest in understanding the details and origins of chaos in quantum systems with many degrees of freedom. Chaos plays a significant role in holographic theories, hydrodynamic transport, and even the strange metal phase of condensed matter systems. With this importance, discovering a unified origin that yields universal results for chaotic systems is clearly desirable. In this thesis, we investigate the conjectured hydrodynamic origin of quantum many-body chaos, first posited in [1], by testing it with the next-to-leading order SYK model. We provide a review of how hydrodynamic theories are constructed, and how hydrodynamic theories with a certain symmetry possess all the standard features of chaos. We then review the leading order SYK model, demonstrate its chaotic behaviour, and compare it with the predictions of the hydrodynamic theory. We finally perform an in-depth investigation of the next-to-leading order SYK model, demonstrating that, while one sector of the theory satisfies the conjecture, another sector does not admit a hydrodynamic description. This shows that the conjecture must be modified to account for near-maximally chaotic theories.
We suggest that the holographic principle, combined with recent technological advances in atomic, molecular, and optical physics, can lead to experimental studies of quantum gravity. As a specific example, we consider the Sachdev–Ye–Kitaev (SYK) model, which consists of spin-polarized fermions with an all-to-all complex random two-body hopping and has been conjectured to be dual to a certain quantum-gravitational system. Achieving low-temperature states of the SYK model is interpreted as a realization of a stringy black hole, provided that the holographic duality is true. We introduce a variant of the SYK model, in which the random two-body hopping is real. This model is equivalent to the original SYK model in the large-|N| limit. We show that this model can be created in principle by confining ultracold fermionic atoms into optical lattices and coupling two atoms with molecular states via photo-association lasers. This development serves as an important first step towards an experimental realization of such systems dual to quantum black holes. We also show how to measure out-of-time-order correlation functions of the SYK model, which allow for identifying the maximally chaotic property of the black hole.