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arXiv:2303.03474 · Journal of High Energy Physics

Large p SYK from chord diagrams

Baur Mukhametzhanov

Published Sep 22, 2023• 21 citations
arXivHTMLPDF

Abstract

A bstract The p-body SYK model at finite temperature exhibits submaximal chaos and contains stringy-like corrections to the dual JT gravity. It can be solved exactly in two different limits: “large p” SYK 1 ≪ p ≪ N and “double-scaled” SYK N , p → ∞ with λ = 2 p 2 / N fixed. We clarify the relation between the two. Starting from the exact results in the double-scaled limit, we derive several observables in the large p limit. We compute euclidean 2 n -point correlators and out-of-time-order four-point function at long lorentzian times. To compute the correlators we find the relevant asymptototics of the $$ {\mathcal{U}}_q\left( su\left(1,1\right)\right) $$ U q su 1 1 6j-symbol.

DOI
10.1007/jhep09(2023)154

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A bstract The p-body SYK model at finite temperature exhibits submaximal chaos and contains stringy-like corrections to the dual JT gravity. It can be solved exactly in two different limits: “large p” SYK 1 ≪ p ≪ N and “double-scaled” SYK N , p → ∞ with λ = 2 p 2 / N fixed. We clarify the relation between the two. Starting from the exact results in the double-scaled limit, we derive several observables in the large p limit. We compute euclidean 2 n -point correlators and out-of-time-order four-point function at long lorentzian times. To compute the correlators we find the relevant asymptototics of the Uq(su(1,1)){\mathcal{U}}_q\left( su\left(1,1\right)\right)Uq​(su(1,1)) U q su 1 1 6j-symbol.
A bstract We study the time evolution governed by the two-sided chord Hamiltonian in the double-scaled SYK model, which induces a probability distribution over operators in the double-scaled algebra. Through the bulk-to-boundary map, this distribution translates into dynamical profiles of bulk states within the chord Hilbert space. We derive analytic expressions for such profiles, valid across a broad parameter range and all time scales. Additionally, we demonstrate how distinct semi-classical behaviors emerge by localizing within specific energy regions in the semi-classical limit. We revisit the doubled Hilbert space formalism as an isometric map between the one-particle sector of the chord Hilbert space and the doubled zero-particle sector. Utilizing this map, we obtain analytic results for correlation functions and investigate the dynamical evolution for chord operators. Specifically, we establish an equivalence between the chord number generating function in presence of matter chords and the crossed four-point correlation function, the latter is closely related to the 6 j -symbol of Uq(su(1,1)) {U}_{\sqrt{q}}\left(\mathfrak{su}\left(1,1\right)\right) Uq​​(su(1,1)) U q su 1 1 . We also explore finite-temperature effects, showing that operator spreading slows as temperature decreases. In the semi-classical limit, we perform a saddle point analysis and incorporate the one-loop determinant to derive the normalized time-ordered four-point correlation function at infinite temperature. The leading correction reproduces the 1/ N connected contribution observed in the large- p SYK model. Finally, we examine the time evolution of total chord number in presence of matter in the triple-scaled regime, linking it to the renormalized two-sided length in JT gravity with matter.
A bstract In this paper, we study one-loop contributions in the double-scaling limit of the SYK model from the chord diagrams and Liouville type effective action. We compute and clarify the meaning of each component consisting of the one-loop corrections for the two- and time-ordered four-point functions of light operators. We also reproduce the exact expression of the out-of-time-ordered four-point function at arbitrary temperatures within the one-loop level, which were previously computed from different methods.
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