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arXiv:2003.04405 · Journal of High Energy Physics

The double scaled limit of super-symmetric SYK models

Micha Berkooz A, Nadav Brukner, Vladimir Narovlansky, Amir Raz

Published Dec 1, 2020• 48 citations
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Abstract

Abstract We compute the exact density of states and 2-point function of the $$ \mathcal{N} $$ 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 interaction length limit. They imply that the conformal ansatz of the 2-point function is inconsistent due to the large number of ground states, and we show how to add this contribution. We also discuss the relation of the model to SLq(2|1). For completeness we present an overview of the $$ \mathcal{N} $$ N = 1 super-symmetric SYK model in the large N double-scaled limit.

DOI
10.1007/jhep12(2020)110

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Abstract We compute the exact density of states and 2-point function of the N\mathcal{N}N 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 interaction length limit. They imply that the conformal ansatz of the 2-point function is inconsistent due to the large number of ground states, and we show how to add this contribution. We also discuss the relation of the model to SLq(2|1). For completeness we present an overview of the N\mathcal{N}N N = 1 super-symmetric SYK model in the large N double-scaled limit.
Abstract We compute the exact, all energy scale, 4-point function of the large N doublescaled 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.
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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 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 a model of fermions with random couplings similar to conventional SYK with N number of flavours of fermions, at large N . Unlike the conventional SYK model, which has all-to-all couplings, the model we study, which we call local SYK, has a much less number of random couplings, just N in number and with only local interactions. It is shown that there exists a limit in which the local SYK model can be solved using the chord diagram techniques, analogous to the double-scaled limit of conventional SYK. This limit corresponds to taking the size of the fermion coupling terms, q , to scale linearly with N . A further triple-scaling limit is taken to analyze the low energy limit and it is shown that the OTOCs saturate the chaos bound, paralleling the analysis in the conventional SYK.