A bstract We revisit the bulk Hilbert space interpretation of chords in the double-scaled SYK (DSSYK) model and introduce a notion of intertwiner that constructs bulk states from states with fixed boundary conditions. This leads to an isometric map that factorizes the one-particle bulk Hilbert space into a tensor product of two boundary Hilbert spaces without particle insertion. The map enables a systematic derivation of a family of correlation functions with arbitrary finite amount of matter insertions, relevant for capturing the switchback effect — a feature of holographic complexity. We further develop a path integral framework that describes multiple shockwave configurations in the semiclassical limit. For the two-body scattering processes in semi-classical regime, we show it exhibits sub-maximal chaos at finite temperature, consistent with the scramblon dynamics associated with the “fake disk” geometry. The effective “fake temperature” governing this behavior emerges from the semiclassical limit of the quantum 6 j -symbol associated with the out-of-time-order correlator. We further analyze multi-shockwave configurations and derive precise conditions under which the switchback effect is realized, both in terms of the total chord number and the Krylov complexity of precursor operators. Our results clarify the structure of correlation functions with multiple operator insertions, their bulk interpretation in terms of shockwave geometries in the semi-classical regime, and provide a microscopic derivation of the switchback effect in the DSSYK model.
A bstract We revisit the bulk Hilbert space interpretation of chords in the double-scaled SYK (DSSYK) model and introduce a notion of intertwiner that constructs bulk states from states with fixed boundary conditions. This leads to an isometric map that factorizes the one-particle bulk Hilbert space into a tensor product of two boundary Hilbert spaces without particle insertion. The map enables a systematic derivation of a family of correlation functions with arbitrary finite amount of matter insertions, relevant for capturing the switchback effect — a feature of holographic complexity. We further develop a path integral framework that describes multiple shockwave configurations in the semiclassical limit. For the two-body scattering processes in semi-classical regime, we show it exhibits sub-maximal chaos at finite temperature, consistent with the scramblon dynamics associated with the “fake disk” geometry. The effective “fake temperature” governing this behavior emerges from the semiclassical limit of the quantum 6 j -symbol associated with the out-of-time-order correlator. We further analyze multi-shockwave configurations and derive precise conditions under which the switchback effect is realized, both in terms of the total chord number and the Krylov complexity of precursor operators. Our results clarify the structure of correlation functions with multiple operator insertions, their bulk interpretation in terms of shockwave geometries in the semi-classical regime, and provide a microscopic derivation of the switchback effect in the DSSYK model.
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 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 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.
A bstract In this paper we study the notion of complexity under time evolution in chaotic quantum systems with holographic duals. Continuing on from our previous work, we turn our attention to the issue of Krylov complexity upon the insertion of a class of single-particle operators in the double-scaled SYK model. Such an operator is described by a matter-chord insertion, which splits the theory into left/right sectors, allowing us, via chord-diagram technology, to compute two different notions of complexity associated to the operator insertion: first a Krylov operator complexity, and second the Krylov complexity of a state obtained by an operator acting on the thermofield double state. We will provide both an analytic proof and detailed numerical evidence, that both Krylov complexities arise from a recursively defined basis of states characterized by a constant total chord number. As a consequence, in all cases we are able to establish that Krylov complexity is given by the expectation value of a length operator acting on the Hilbert space of the theory, expressed in terms of basis states, organized by left and right chord number. We find analytic expressions for the semiclassical limit of K-complexity, and study how the size of the operator encodes the scrambling dynamics upon the matter insertion in Krylov language. We furthermore determine the effective Hamiltonian governing the evolution of K-complexity, showing that evolution on the Krylov chain can equivalently be understood as a particle moving in a Morse potential. A particular type of triple scaling limit allows to access the gravitational sector of the theory, in which the geometrical nature of K-complexity is assured by virtue of being a total chord length, in an analogous fashion to what was found in [1] for the K-complexity of the thermofield double state.
A bstract We extend the notion of chord number in the strict large N double-scaled Sachdev-Ye-Kitaev (DSSYK) model to the corresponding finite N ETH matrix model. The chord number in the strict large N DSSYK model is known to correspond to the discrete length of the Einstein-Rosen bridge in the gravity dual, which reduces to the renormalized geodesic length in JT gravity at weak coupling. At finite N , these chord number states form an over-complete basis of the non-perturbative Hilbert space, as the structure of the inner product gets significantly modified due to the Cayley-Hamilton theorem: There are infinitely many null states. In this paper, by considering “EFT for gravitational observables” or a version of “non-isometric code”, we construct a family of chord number operators at finite N . While the constructed chord number operator depends on the reference chord number state, it realizes approximate q -deformed oscillator algebra and reproduces semiclassical bulk geometry around the reference state. As a special case, we will show that when the reference is chosen to be the chord number zero state, the chord number operator precisely matches with the Krylov state complexity, leading to the “ramp-slope-plateau” behavior at late times, implying the formation of “gray hole”.
We study the properties of the double-scaled SYK (DSSYK) model under chord Hamiltonian deformations based on finite cutoff holography for general dilaton gravity theories with Dirichlet boundaries. The formalism immediately incorporates a lower-dimensional analog of TTˉ(+Λ2) deformations, denoted T2(+Λ1), as special cases. In general, the deformation mixes the chord basis of the Hilbert space in the seed theory, which we order through the Lanczos algorithm. The resulting Krylov complexity for the Hartle-Hawking state represents a wormhole length at a finite cutoff in the bulk. We study the thermodynamic properties of the deformed theory; the growth of Krylov complexity; the evolution of n-point correlation functions with matter chords; and the entanglement entropy between the double-scaled algebras of the DSSYK model for a given chord state. The latter, in the triple-scaling limit, manifests as the minimal codimension-two area in the bulk following the Ryu-Takayanagi formula. By performing a sequence of T2 and T2+Λ1 deformations in the upper tail of the energy spectrum in the deformed DSSYK, we concretely realize the cosmological stretched horizon proposal in de Sitter holography by Susskind. We discuss other extensions with sine dilaton gravity, end-of-the-world branes, and the Almheiri-Goel-Hu model.
We derive a large-N wormhole saddle in the Sachdev-Ye-Kitaev (SYK) model for the variance of the thermal one-point function of a heavy operator across realizations of the microscopic couplings. We propose that this saddle describes a double trumpet geometry. In the double-scaling limit, we express the wormhole as a transition amplitude in an emergent bulk Hilbert space of chords connecting the two boundaries. In the Schwarzian limit, the chords proliferate and the wormhole defines a quantum transition amplitude in the open-universe Hilbert space of Jackiw-Teitelboim (JT) gravity. Comparing the SYK wormhole amplitude with its counterpart in JT gravity coupled to matter provides a direct test of the heavy-operator randomness hypothesis in SYK. The microscopic SYK couplings specify the α-parameters of the wormhole. For a fixed realization of the couplings, we compute the observable in the corresponding α-state. We provide the bulk wavefunction of the associated half-wormhole, which exhibits erratic correlations in the length basis that do not self-average. We extend the bulk description to the Hilbert space of a closed universe. We point out that the correlation chords from which the wormhole geometry emerges structurally resemble local observer worldlines.