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.
An intriguing feature of type II
1 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 II
1 algebra. In this paper, we study the entanglement entropy
Sn of the fixed length state
{∣n⟩} in Double-Scaled Sachdev-Ye-Kitaev model, which has been recently shown to exhibit type II
1 von Neumann algebra. These states furnish an orthogonal basis for 0-particle chord Hilbert space. We systematically study
Sn and its Rényi generalizations
Sn(m) in various limit of DSSYK model, ranging
q∈[0,1]. We obtain exotic analytical expressions for the scaling behavior of
Sn(m) at large
n for random matrix theory limit (
q=0) and SYK
2 limit (
q=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 AdS
2 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 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 SU(2) gauge theory. We study the operator algebras of the deformed theories. The algebras can be type II
1 or type I
∞ factors, depending on the operators that are included. The entanglement entropy between the type II
1 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.
ABSTRACT 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 deformations, denoted , 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 ‐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 and 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.
The definition of entanglement entropy in quantum systems relies on the factorization of the total Hilbert space into subsystems, which allows the construction of reduced density matrices through the partial trace operation. However, there exist important physical systems for which such a factorization is not available. In these cases, reduced density matrices and entanglement entropy are not well defined in the conventional sense. In this thesis, we develop an operator-algebraic framework for describing entanglement in such systems using the theory of von Neumann algebras. As a primary example, we consider eternal Schwarzschild black holes in Anti-de Sitter (AdS) spacetime. Starting from the dual boundary theory, we show how a suitable notion of trace can be defined, leading to a well-defined von Neumann entropy. We then reproduce the same entropy from the bulk gravitational description and demonstrate the agreement between the two calculations. Finally, we show that this quantity can be naturally interpreted as the generalized entropy.
The holographic Ryu-Takayanagi formula for entanglement entropy connects the entanglement of a field theory to the geometry of a dual gravitational theory in a straightforward and universal way. The first part of this thesis applies this formula to study the entanglement entropy in strongly coupled noncommutative field theories. It is found that the ground state of these theories have substantial entanglement at the length scale of the noncommutativity. The entanglement entropy in a different perturbative regime is also computed, where in contrast it is found that noncommutative interactions do not induce long range entanglement in the ground state to leading order in perturbations theory. The second part of this thesis explores some general consequences of this holographic formula for the entanglement entropy. Identities involving entanglement entropies are related to nontrivial geometric constraints on gravitational duals. In particular, the strong subadditivity of entanglement entropy is used to show that dual three dimensional asymptotically anti-de Sitter gravitational states must obey an averaged null energy condition. Finally, this holographic formula allows us at least in principle to express the entanglement entropy of a region in a holographic field theory in terms of the one-point functions in that theory. This is explored in the context of a two dimensional conformal field theory where explicit calculations are possible. Our results in this case allow us to extend a recent proposal that the entanglement entropy of states near the vacuum of conformal theories can be understood by propagation in an auxiliary de Sitter space.