In this note we study the spectral form factor in the SYK model in large $q$ limit at infinite temperature. We construct analytic solutions for the saddle point equations that describe the slope and the ramp regions of the spectral form factor time dependence. These saddle points are obtained by taking different approaches to the large $q$ limit: the slope region is described by a replica-diagonal solution and the ramp region is described by a replica-nondiagonal solution. We find that the onset of the ramp behavior happens at the Thouless time of order $q \log q$. We also evaluate the one-loop corrections to the slope and ramp solutions for late times, and study the transition from the slope to the ramp. We show this transition is accompanied by the breakdown of the perturbative $1/q$ expansion, and that the Thouless time is defined by the consistency of extrapolation of this expansion to late times.
I want to call attention to a simple previously noted fact about the double-scaled version of the SYK model which suggests that it may be holographically dual to de Sitter space.
We argue that at finite energies, double-scaled SYK has a semiclassical approximation controlled by a coupling $λ$ in which all observables are governed by a non-trivial saddle point. The Liouville description of double-scaled SYK suggests that the correlation functions define a geometry in a two-dimensional bulk, with the 2-point function describing the metric. For small coupling, the fluctuations are highly suppressed, and the bulk describes a rigid (A)dS spacetime. As the coupling increases, the fluctuations become stronger. We study the correction to the curvature of the bulk geometry indu…
In the semiclassical limit of de Sitter gravity a separation of scales takes place that divides the theory into a "cosmic" sector and a "microscopic" sector. A similar separation takes place in the double-scaled limit of SYK theory. We examine the scaling behaviors that accompany these limits and find parallels that support the previously conjectured duality between Jackiw-Teitelboim gravity (with positive cosmological constant), and double-scaled SYK. This paper is a companion to "dS JT Gravity and Double-Scaled SYK" by Adel Rahman, to appear simultaneously with this paper.
Probing the Chaos to Integrability Transition in Double-Scaled SYK
Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；Julius-Maximilians-Universität Würzburg（维尔茨堡大学）；Weizmann Institute of Science（魏茨曼科学研究所）；Massachusetts Institute of Technology（麻省理工学院）；Freie Universität Berlin（柏林自由大学）
We compute the exact, all energy scale, 4-point function of the large $N$ double-scaled SYK model, by using only combinatorial tools and relating the correlation functions to sums over chord diagrams. We apply the result to obtain corrections to the maximal Lyapunov exponent at low temperatures. We present the rules for the non-perturbative diagrammatic description of correlation functions of the entire model. The latter indicate that the model can be solved by a reduction of a quantum deformation of SL$(2)$, that generalizes the Schwarzian to the complete range of energies.
We compute the exact density of states and 2-point function of the $\mathcal{N} =2$ super-symmetric SYK model in the large $N$ double-scaled limit, by using combinatorial tools that relate the moments of the distribution to sums over oriented chord diagrams. In particular we show how SUSY is realized on the (highly degenerate) Hilbert space of chords. We further calculate analytically the number of ground states of the model in each charge sector at finite $N$, and compare it to the results from the double-scaled limit. Our results reduce to the super-Schwarzian action in the low energy short …
We study the double-scaling limit of SYK (DS-SYK) model and elucidate the underlying quantum group symmetry. The DS-SYK model is characterized by a parameter $q$, and in the $q\rightarrow 1$ and low-energy limit it goes over to the familiar Schwarzian theory. We relate the chord and transfer-matrix picture to the motion of a ``boundary particle" on the Euclidean Poincar{é} disk, which underlies the single-sided Schwarzian model. $AdS_2$ carries an action of $\mathfrak{s}\mathfrak{l}(2,{\mathbb R}) \simeq \mathfrak{s}\mathfrak{u}(1,1)$, and we argue that the symmetry of the full DS-SYK model is…
Toward Krylov-based holography in double-scaled SYK
Gwangju Institute of Science and Technology（光州科学技术院）；Instituto de Física Teórica UAM/CSIC（马德里自治大学/西班牙国家科研中心理论物理研究所）；Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；Pohang University of Science and Technology（浦项科技大学）；Research Center for Photon Science Technology, Gwangju Institute of Science and Technology（光州科学技术院光子科学与技术研究中心）
We show that the quantum critical point (QCP) between a diffusive metal and ferromagnetic (or antiferromagnetic) phases in the SYK chain has a gravitational description corresponding to the double-trace deformation in an AdS$_2$ chain. Specifically, by studying a double-trace deformation of a $Z_2$ scalar in an AdS$_2$ chain where the $Z_2$ scalar is dual to the order parameter in the SYK chain, we find that the susceptibility and renormalization group equation describing the QCP in the SYK chain can be exactly reproduced in the holographic model. Our results suggest that the infrared geometry…
Geometry of Chord Intertwiner, Multiple Shocks and Switchback in Double-Scaled SYK
Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；Department of Physics, University of California, Santa Barbara（加州大学圣塔芭芭拉分校物理系）
According to holography, entanglement is the building block of spacetime; therefore, drastic changes of entanglement will lead to interesting transitions in the dual spacetime. In this paper, we study the effect of projective measurements on the Sachdev-Ye-Kitaev (SYK) model's thermofield double state, dual to an eternal black hole in Jackiw-Teitelboim (JT) gravity. We calculate the (Renyi-2) mutual information between the two copies of the SYK model upon projective measurement of a subset of fermions in one copy. We propose a dual JT gravity model that can account for the change of entangleme…
