Deforming the Double-Scaled SYK & Reaching the Stretched Horizon From Finite Cutoff Holography

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 $\text{T}\bar{\text{T}}(+Λ_2)$ deformations, denoted $T^2(+Λ_1)$, as special cases. In general, the deformation mixes the chord basis of the Hilbert space in the seed theory, which we order through a modification of the Lanczos algorithm. The resulting chord number in the ordered basis represents a wormhole length at a finite cutoff in the bulk. We study the thermodynamic properties of the deformed theory; the evolution of $n$-point correlation functions with matter chords; the growth of complexity of the Hartle-Hawking state; and the entanglement entropy between the double-scaled algebras 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 $T^2$ and $T^2+Λ_1$ 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.

Figures (16)

• Figure 1: Bulk interpretation of the $T^2$ and $T^2+\Lambda_1$ flow in the boundary theory (concretely explained in Sec. \ref{['ssec:IR UV']}) describing (a,b) an AdS$_2$ black hole, and (c, d) dS$_2$ space. The gray regions represent the parts of the bulk that are cut out. (a) By implementing the $T^2$ deformation, with spectrum \ref{['eq:E(theta) eta+1']}, there is a finite boundary cutoff (red solid curve) in the Rindler-AdS$_2$ patch til it reaches the horizon. (b) We implement the $T^2+\Lambda_1$ deformation \ref{['eq:E(theta) eta-1']} which moves the boundary inside the black hole horizon. A similar procedure applies for dS$_2$ space for (c) $T^2$ and (d) $T^2+\Lambda_1$ deformations. In the latter, the cosmological stretched horizon Susskind:2021esx is realized when the Dirichlet boundaries are very close to the dS$_2$ horizon.
• Figure 2: (a) The energy spectrum (\ref{['eq:energy spectrum']}), (b) thermodynamic entropy (\ref{['eq:thermodynamic entropy']}), (c) density of states (\ref{['eq:DOS TTbar']}), and (d) heat capacity (\ref{['eq:heat capacity']}) for generic values of the deformation parameter $y$ (\ref{['eq:def parameter']}) (shown in the legends) and fixing $\eta=+1$. We use $\lambda=10^{-4}$ in all subfigures, except in (c) where $\lambda=0.1$ for visualization purposes.
• Figure 3: Top: Expectation value of the chord number $\hat{n}_y$\ref{['eq:new chord number op']} in the IR limit \ref{['eq:LIR']} of the deformed HH state \ref{['eq:length dS exp ']} after (a) a $T^2$ ($\eta=+1$) and (b) $T^2+\Lambda_1$ ($\eta=-1$) deformation. Bottom: Corresponding bulk interpretation in terms of (c) spacelike and (d) timelike extremal Einstein-Rosen bridges (blue solid curves) connecting the finite Dirichlet boundaries (red solid curve), located at a constant $r=r_B$ in the Rindler-AdS$_2$ coordinates \ref{['eq:netric more gen']}). We marked the minimal extremal codimension-two area (measured by the dilaton) surfaces $\gamma$ (orange dots) subject to the homology constraint to the entangling surface (blue dots at the Dirichlet boundaries). Increasing the deformation parameter $y$\ref{['eq:def parameter']} enhances the rate of growth of the wormhole length for the $\eta=+1$ case. In finite cutoff holography, this is moves the location of the boundaries towards the corresponding black hole horizon, which increases the Tolman temperature, which determines the growth of the length with respect to the boundary time. Meanwhile for the $\eta=-1$ case, increasing $y$ moves the boundary cutoff away from the horizon in the bulk, leading to a decrease in the Tolman temperature. However, there is no eternal growth unlike the $\eta=+1$ case, since the timelike geodesic eventually reaches $r\rightarrow\infty$ in Rindler-AdS$_2$ space \ref{['eq:AdS blackening']} as one increases the spacelike coordinate $t$.
• Figure 4: $(2m+2)$-correlation function in \ref{['eq:G m operators']} with $\hat{\mathcal{O}}_{\Delta_0}$ (blue) being light operators and $\hat{\mathcal{O}}_{\Delta_{1\leq i\leq m}}$ (red) at different locations within the thermal circle.
• Figure 5: Evolution of the two-point correlation function \ref{['eq:correlator Euclidean']} for (a) $T^2$, and (b) $T^2+\Lambda_1$ deformations. As the deformation parameter reaches the critical value $y_0$\ref{['eq:y0']} the fake temperature \ref{['eq:fake temperature']} increases resulting in faster decay of the correlation function. The parameters in this evaluation are: $\lambda=10^{-4}$, $J=1$, $\Delta=1$ and $\theta=3\pi/4$.