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arXiv:2604.14387 · Journal of High Energy Physics

Emergent states and algebras from the double-scaling limit of pure states in SYK

Harshit Rajgadia, Jiuci Xu

Published Sep 25, 2026• 0 citations
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Abstract

A bstract Recent work has emphasized a subtlety of large- N limits in AdS/CFT: a sequence of pure states in the microscopic theory need not remain pure with respect to the emergent algebra of observables. We study this phenomenon for Kourkoulou-Maldacena (KM) states in the double-scaling limit of the SYK model, and show that their ensemble-averaged algebraic description depends crucially on which observables are retained in the limit. For fermionic operators of size p ∼ N 1/2 , generic operators converge to the usual chord operators of double-scaled SYK. The resulting von Neumann algebra is the standard Type II 1 factor, and the KM pure states at infinite temperature converge to the tracial state, so generic probes lose access to microscopic purity. We then identify a class of operators adapted to the KM state that also survive the double-scaling limit. Since the KM state may be viewed as a projection inside the tracial state, these operators become dressed chord creation and annihilation operators, with the dressing encoding the distance to the KM state through the bulk. Once included, the limiting algebra becomes Type I ∞ and the sequence of KM states converges to a pure state. This provides a solvable example in which adding a sufficiently state-adapted operator to the emergent algebra restores access to the purity of the underlying state. We derive exact modified chord-diagram rules for correlators of the dressed operators, including general uncrossed 2 n -point functions and crossed four-point functions. In the semi-classical limit these correlators exhibit a novel factorization pattern, while in the Schwarzian limit they reduce to correlators of boundary primaries dressed to Schwarzian modes. We also study a deformation of the chord Hamiltonian, the emergence of bound states above a critical coupling, and its relation to JT gravity coupled to an EOW brane of general tension. We identify an emergent U (1) global symmetry in the Type I ∞ algebra as well as its violation by 1 /N corrections from double-trace wormholes. Finally, we discuss analogies with boundary algebras proposed for black hole interiors and closed universes, and suggest lessons from our construction for both.

DOI
10.1007/jhep09(2026)263

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A bstract Recent work has emphasized a subtlety of large- N limits in AdS/CFT: a sequence of pure states in the microscopic theory need not remain pure with respect to the emergent algebra of observables. We study this phenomenon for Kourkoulou-Maldacena (KM) states in the double-scaling limit of the SYK model, and show that their ensemble-averaged algebraic description depends crucially on which observables are retained in the limit. For fermionic operators of size p ∼ N 1/2 , generic operators converge to the usual chord operators of double-scaled SYK. The resulting von Neumann algebra is the standard Type II 1 factor, and the KM pure states at infinite temperature converge to the tracial state, so generic probes lose access to microscopic purity. We then identify a class of operators adapted to the KM state that also survive the double-scaling limit. Since the KM state may be viewed as a projection inside the tracial state, these operators become dressed chord creation and annihilation operators, with the dressing encoding the distance to the KM state through the bulk. Once included, the limiting algebra becomes Type I ∞ and the sequence of KM states converges to a pure state. This provides a solvable example in which adding a sufficiently state-adapted operator to the emergent algebra restores access to the purity of the underlying state. We derive exact modified chord-diagram rules for correlators of the dressed operators, including general uncrossed 2 n -point functions and crossed four-point functions. In the semi-classical limit these correlators exhibit a novel factorization pattern, while in the Schwarzian limit they reduce to correlators of boundary primaries dressed to Schwarzian modes. We also study a deformation of the chord Hamiltonian, the emergence of bound states above a critical coupling, and its relation to JT gravity coupled to an EOW brane of general tension. We identify an emergent U (1) global symmetry in the Type I ∞ algebra as well as its violation by 1 /N corrections from double-trace wormholes. Finally, we discuss analogies with boundary algebras proposed for black hole interiors and closed universes, and suggest lessons from our construction for both.
It has been argued that a finite effective temperature emerges and characterizes the thermal property of double-scaled SYK model in the infinite temperature limit. Meanwhile, in the static patch of de Sitter, the maximally entangled state satisfies a KMS condition at infinite temperature, suggesting the Type II1_11​ nature of the observable algebra gravitationally dressed to the observer. In this work, we analyze the double-scaled algebra generated by chord operators in the double-scaled SYK model and demonstrate that it exhibits features reflecting both perspectives. Specifically, we prove that the algebra is a Type II1_11​ factor, and that the empty state with no chord satisfies the tracial property, in agreement with expectations from earlier work. We further show that this state is cyclic and separating for the double-scaled algebra, based on which we explore its modular structure. We then explore various physical limits of the theory, drawing connections to JT gravity, the Hilbert space of baby universes, and Brownian double-scaled SYK. We also present analytic solutions to the energy spectrum in both the zero- and one-particle sectors of the left/right chord Hamiltonian.
A bstract It has been argued that a finite effective temperature emerges and characterizes the thermal property of double-scaled SYK model in the infinite temperature limit [1]. Meanwhile, in the static patch of de Sitter, the maximally entangled state satisfies a KMS condition at infinite temperature [2], suggesting the Type II 1 nature of the observable algebra gravitationally dressed to the observer. In this work, we analyze the double-scaled algebra generated by chord operators in the double-scaled SYK model and demonstrate that it exhibits features reflecting both perspectives. Specifically, we prove that the algebra is a Type II 1 factor, and that the empty state with no chord satisfies the tracial property, in agreement with expectations from [3]. We further show that this state is cyclic and separating for the double-scaled algebra, based on which we explore its modular structure. We then explore various physical limits of the theory, drawing connections to JT gravity, the Hilbert space of baby universes, and Brownian double-scaled SYK. We also present analytic solutions to the energy spectrum in both the zero- and one-particle sectors of the left/right chord Hamiltonian.
A bstract We study a single-sided black hole with an end-of-the-world (EoW) brane behind the horizon in the double-scaled SYK (DSSYK). The new Hamiltonian is a deformation of the original DSSYK Hamiltonian with an extra exponential wormhole length operator, which leads to a new chord diagram rule. The boundary algebra is defined as generated by the new Hamiltonian and boundary matter. There is an alternative but equivalent definition with a q -coherent state due to a nontrivial isomorphism of the vN algebra of DSSYK. This isomorphism induces a unitary equivalence, which yields a surprising result that the boundary algebra of a single-sided black hole in DSSYK has a non-trivial commutant and is a type II 1 vN factor. It follows that the full bulk reconstruction from the boundary is impossible, and there is a “no man’s island” behind the horizon in the semiclassical JT limit. Inspired by the EoW brane, we construct a family of matter-brane states with an arbitrary number of matter chords and behaving like an EoW brane. They exactly solve the full spectrum of DSSYK. We take different ways to understand the nontrivial commutant. We argue that the commutant is complex on chord number basis and thus non-geometric. In the semiclassical JT limit, the commutant becomes the canonical purification of the boundary algebra and claims the no man’s island. In the context of Hawking radiation after Page time, the unitary equivalence is interpreted as encoding the canonical purification into the old Hawking radiation, and the no man’s island has the same essence as the island. Including the exponential wormhole length operator independently, the boundary algebra is extended to all bounded operators and reconstructs the no man’s island. This can be regarded as a different choice for the definition of boundary algebra. This type I ∞ algebra is closely related to the EoW brane in Kourkoulou-Maldacena.
We construct families of deformations of the double-scaled SYK (DSSYK) model and investigate their bulk interpretation. We introduce microscopic deformations of the SYK model which, after ensemble averaging and in the double-scaling limit, are described by a transfer matrix encoding the recurrence relations of basic orthogonal polynomials in the q-Askey scheme. For certain families of deformations in the semiclassical limit at finite temperature, the chord number (encoding Krylov complexity) corresponds to the length of an Einstein-Rosen bridge connecting an End-Of-The-World brane to an anti-de Sitter asymptotic boundary. By increasing one of the deformation parameters, the models eventually exhibit discrete energy levels, signaling a new geometric transition in sine dilaton gravity. Via the SYK-Schur duality, Krylov complexity also admits a representation-theoretic interpretation as the spread of the SU(2) spin in the index of an N=2\mathcal{N}=2N=2 SU(2) gauge theory. We study the operator algebras of the deformed theories. The algebras can be type II1_11​ or type I∞_\infty∞​ factors, depending on the operators that are included. The entanglement entropy between the type II1_11​ algebras for a pure state manifests as an extremal surface through the Ryu-Takayanagi formula. We discuss connections between our results and the emergence of baby universes in the bulk.
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.
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.