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arXiv:2403.09021

Von Neumann Algebras in Double-Scaled SYK

Jiuci Xu

Published Mar 14, 2024• 1 citation
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Abstract

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 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 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.

DOI
10.48550/arxiv.2403.09021

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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 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 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.
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.
Entanglement entropy in type II1_11​ von Neumann algebra: examples in Double-Scaled SYK
An intriguing feature of type II1_11​ von Neumann algebra is that the entropy of the mixed states is negative. Although the type classification of von Neumann algebra and its consequence in holography have been extensively explored recently, there has not been an explicit calculation of entropy in some physically interesting models with type II1_11​ algebra. In this paper, we study the entanglement entropy SnS_nSn​ of the fixed length state {∣n⟩}\{|n\rangle\}{∣n⟩} in Double-Scaled Sachdev-Ye-Kitaev model, which has been recently shown to exhibit type II1_11​ von Neumann algebra. These states furnish an orthogonal basis for 0-particle chord Hilbert space. We systematically study SnS_nSn​ and its Rényi generalizations Sn(m)S_n^{(m)}Sn(m)​ in various limit of DSSYK model, ranging q∈[0,1]q\in[0,1]q∈[0,1]. We obtain exotic analytical expressions for the scaling behavior of Sn(m)S_n^{(m)}Sn(m)​ at large nnn for random matrix theory limit (q=0q=0q=0) and SYK2_22​ limit (q=1q=1q=1), for the former we observe highly non-flat entanglement spectrum. We then dive into triple scaling limits where the fixed chord number states become the geodesic wormholes with definite length connecting left/right AdS2_22​ boundary in Jackiw-Teitelboim gravity. In semi-classical regime, we match the boundary calculation of entanglement entropy with the dilaton value at the center of geodesic, as a nontrivial check of the Ryu-Takayanagi formula.
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)\text{T}\bar{\text{T}}(+Λ_2)TTˉ(+Λ2​) deformations, denoted T2(+Λ1)T^2(+Λ_1)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 nnn-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 T2T^2T2 and T2+Λ1T^2+Λ_1T2+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.