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arXiv:2401.07403 · Journal of High Energy Physics

Doubled Hilbert space in double-scaled SYK

Kazumi Okuyama

Published Apr 17, 2024• 18 citations
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Abstract

A bstract We consider matter correlators in the double-scaled SYK (DSSYK) model. It turns out that matter correlators have a simple expression in terms of the doubled Hilbert space $$\mathcal{H}\otimes \mathcal{H}$$ , where $$\mathcal{H}$$ is the Fock space of q -deformed oscillator (also known as the chord Hilbert space). In this formalism, we find that the operator which counts the intersection of chords should be conjugated by certain “entangler” and “disentangler”. We explicitly demonstrate this structure for the two- and four-point functions of matter operators in DSSYK.

DOI
10.1007/jhep04(2024)091

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A bstract We consider matter correlators in the double-scaled SYK (DSSYK) model. It turns out that matter correlators have a simple expression in terms of the doubled Hilbert space H⊗H\mathcal{H}\otimes \mathcal{H}H⊗H , where H\mathcal{H}H is the Fock space of q -deformed oscillator (also known as the chord Hilbert space). In this formalism, we find that the operator which counts the intersection of chords should be conjugated by certain “entangler” and “disentangler”. We explicitly demonstrate this structure for the two- and four-point functions of matter operators in DSSYK.
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 holography, gauging symmetries of the boundary theory leads to important modifications in the bulk. In this work, we study constraints to gauge symmetry sectors in the chord Hilbert space of the double-scaled SYK (DSSYK) with matter, and we connect them to different proposals of its bulk dual. These sectors include chord parity symmetry, corresponding to End-Of-The-World (ETW) branes and Euclidean wormholes in sine dilaton gravity; and relative time-translations in a doubled DSSYK model (resulting from a single DSSYK with an infinitely heavy matter chord) used in de Sitter holography. We define and evaluate partition functions and thermal correlation functions of the ETW brane and Euclidean wormhole systems in the boundary theory. We deduce the holographic dictionary by matching geodesic lengths in the bulk with the spread complexity of the parity-gauged DSSYK. The Euclidean wormholes of fixed size are perturbatively stable, and their baby universe Hilbert space is non-trivial only when matter is added. We conclude studying the constraints in the path integral of the doubled DSSYK. We derive the gauge invariant operator algebra of one of the DSSYKs dressed to the other one and discuss its holographic interpretation.
Abstract The matter operator in the double-scaled Sachdev–Ye–Kitaev model exhibits special properties when its dimension is analytically continued to −12-\frac{1}{2}−21​. At this dimension, the operator is in a degenerate representation of the q-deformed oscillator algebra and satisfies a null vector equation. Its peculiar fusion property gives rise to recursion relations among matter correlation functions. We find that these relations allow us to determine the two-point function without having to sum over infinitely many chord diagrams.
A bstract In this work, we formulate the holographic dictionary for the double-scaled SYK (DSSYK) model with matter operators. Based on the two-sided Hartle-Hawking (HH) state, we derive several properties of the DSSYK model, without making assumptions about the specific dual theory, including its semiclassical thermodynamics, correlation functions, and Krylov complexity. We derive these quantities from the saddle points of the DSSYK path integral preparing the HH state. We also construct a Lanczos algorithm that simultaneously evaluates Krylov state and operator complexity in the two-sided Hamiltonian system including finite temperature effects. In the semiclassical limit, both measures are encoded in the saddle points of the path integral. They have a bulk interpretation in terms of minimal geodesic lengths in an effective AdS 2 space with matter backreaction. Different saddle points correspond to geodesic distances with different evolution, and they display different scrambling properties. We also discuss about the quantization of the bulk theory dual to the DSSYK model. At last, we formulate the double-scaled algebra [ JHEP 11 (2022) 060] in bulk terms, and the dual entry of the proper radial momentum of a bulk probe.
In holography, gauging symmetries of the boundary theory leads to important modifications in the bulk. In this work, we study constraints to gauge symmetry sectors in the chord Hilbert space of the double-scaled SYK (DSSYK) with matter, and we connect them to different proposals of its bulk dual. These sectors include chord parity symmetry, corresponding to End-Of-The-World (ETW) branes and Euclidean wormholes in sine dilaton gravity; and relative time-translations in a doubled DSSYK model (resulting from a single DSSYK with an infinitely heavy matter chord) used in de Sitter holography. We define and evaluate partition functions and thermal correlation functions of the ETW brane and Euclidean wormhole systems in the boundary theory. We deduce the holographic dictionary by matching geodesic lengths in the bulk with the spread complexity of the parity-gauged DSSYK. The Euclidean wormholes of fixed size are perturbatively stable, and their baby universe Hilbert space is non-trivial only when matter is added. We conclude studying the constraints in the path integral of the doubled DSSYK. We derive the gauge invariant operator algebra of one of the DSSYKs dressed to the other one and discuss its holographic interpretation.