Holographic Entanglement in dS₂ Space

• The paper demonstrates that embedding a dS₂ brane into an AdS₃ bulk resolves conventional QES failures by establishing a doubly holographic framework.
• It reveals non-extremal islands emerging at the brane edge and identifies phase transitions through analytic, SYK, and static patch approaches.
• The study emphasizes how stability criteria and extrinsic curvature influence holographic entanglement, linking it to Gibbons–Hawking entropy and entropy bounds.

Holographic entanglement entropy in two-dimensional de Sitter space (dS$_2$) probes the interplay between quantum information, gravity, and higher-dimensional holographic dualities. The study of EE in dS$_2$ is sharply distinct from standard anti–de Sitter (AdS) holography, exposing the peculiar structure of extremal surfaces, non-trivial phase transitions, and subtle stability phenomena tied to the signature and extrinsic curvature of de Sitter branes. A variety of approaches—analytic continuation from AdS, direct holographic calculations in AdS-dS braneworlds, double-scaled SYK constructions, and covariant QES functionals—collectively demonstrate the nuances of holographic entropy in dS$_2$, the emergence of non-extremal “islands,” and the realization of the Gibbons–Hawking entropy as an entanglement entropy.

1. Doubly Holographic Formulation and Brane Embedding

The central challenge in dS$_2$ arises because the conventional island formula, successful in AdS and quantum black hole studies, fails to yield a satisfactory quantum extremal surface (QES) when naively transplanted to de Sitter space. In (Hao et al., 2024), this issue is resolved by embedding a dS$_2$ “end-of-world” (EOW) brane into a three-dimensional AdS$_3$ bulk, constructing a doubly holographic setup.

The AdS$_3$ geometry in a dS$_2$ slicing,

$$ds^2_3 = L_{\rm AdS}^2 [d\eta^2 + \sinh^2\!\eta\, (-dt^2 + \cosh^2 t\,d\theta^2)],$$

admits a brane at fixed $\eta=\eta_b$, its location and tension $\mathcal{T}$ determined by the Neumann boundary condition: $$K_{ij} - K\,g_{ij} + \mathcal T\,g_{ij} = 0,\qquad \mathcal T = \frac{\coth\eta_b}{L_{\rm AdS}}.$$ The intrinsic brane geometry is

$$ds^2_{\rm brane} = L^2 (-dt^2 + \cosh^2 t\,d\theta^2),\quad L = L_{\rm AdS} \sinh\eta_b,$$

realizing dS$_2$ (up to overall length scale).

2. Generalized Entropy Functional and Non-Extremal Islands

In the AdS/braneworld setup, the generalized entropy functional for a region $\mathcal A$ in the non-gravitating “bath” is

$$S_{\rm gen}\{I\} = \frac{\operatorname{Area}(\partial I)}{4 G_N^{(3)}} + S_{\rm bulk}(\mathcal A \cup I),$$

with $\partial I$ a codimension-2 locus (in dS$_2$-brane: a point or two points) and $S_{\rm bulk}$ the CFT von Neumann entropy of $\mathcal A \cup I$.

The RT “area-term” is set by the length of a geodesic in AdS$_3$ stretching from the AdS boundary to the brane,

$$\cosh D_{PX} = \cosh\eta_\infty\,\cosh\eta_b [\cosh t_A\,\cosh t_I\,\cos(\theta_A-\theta_I) - \sinh t_A\,\sinh t_I] - \sinh\eta_\infty\,\sinh\eta_b,$$

with

$$\frac{\operatorname{Area}(\Gamma)}{4G_N^{(3)}} = \frac{L_{\rm AdS}}{4G_N^{(3)}} D_{PX}.$$

The bulk CFT entropy on intervals $[\theta_A,\theta_I]\cup[-\theta_I,-\theta_A]$ is

$$S_{\rm bulk} = \frac{c}{3}\, \ln \frac{2(1 + \sinh t_A\,\sinh t_I - \cosh t_A\,\cosh t_I\,\cos(\theta_A-\theta_I))}{\epsilon},$$

with UV cutoff $\epsilon$.

Extremizing $S_{\rm gen}$, one finds the only saddle: $\theta_I = \theta_A - \pi,\quad t_I = -t_A,$ is not a true minimum for the spatial direction on the dS brane; it is a local maximum instead. Bulk correlation functions instead pick out a non-extremal saddle at the spatial edge of the dS brane, i.e. $\theta_I = \pm\pi/2$, where the “island” appears. The corresponding time is fixed by

$$\tan t_I^\ast = \frac{\tan t_A}{\lvert \sin\theta_A \rvert}.$$

For a symmetric interval, entanglement entropy is: $$S_{\mathcal A} = \min\left\{ \frac{c}{3}\ln\!\left[\frac{2\cosh t_A\,\sin\theta_A}{\epsilon}\right] ,\; \frac{c}{3}\ln\!\left(\frac{2}{\epsilon}\left[ \cosh\eta_b - \sinh\eta_b \sqrt{\cosh^2 t_A \sin^2\theta_A - \sinh^2 t_A} \right]\right) \right\}.$$

3. Local Instability, Second Variation, and the Distinction to AdS

The physical difference between AdS$_2$ and dS$_2$ branes is rooted in their extrinsic curvature $K$. For a surface to be a local minimum of the generalized entropy functional, the quadratic variation of the geodesic length must be positive: $$\delta^2 L = -K(v,v)|_{\rm brane} + \int_0^1 \{\|\nabla_t v\|^2 - R(v,t,t,v)\} ds.$$ Since $K_{ij} = \mathcal T g_{ij}$ and $\mathcal T > 1/L_{\rm AdS}$ for dS brane, $K > 0$ and consequently surfaces orthogonally ending on the brane are unstable in the spatial direction; they are not quantum extremal in the QES sense. In contrast, AdS branes ($K<0$) permit standard quantum extremal surfaces. Thus, in dS$_2$, the “island” is a non-extremal one—pinned to the edge of the gravitational region, extremal only in time.

4. Connections to Alternative dS$_2$ Holography Approaches

Several independent constructions provide insight and confirm this picture:

• Analytic continuation from AdS$_2$: Under double Wick-rotation, the RT geodesic length in EAdS$_2$ becomes complex, rendering the dS$_2$ entropy formula $$S_A^{(\mathrm{dS}_2)} = -i \frac{c}{3}\ln(\Delta x / \epsilon)$$. The negative imaginary factor and lack of a known unitary dual signals peculiarities in dS/CFT frameworks (Sato, 2015).
• Double-scaled SYK and edge modes: In the DSSYK model, subsystem definition ties to quantum reference frames and edge modes. Here, the holographic EE recovers a real-valued, unitary result even in dS$_2$, and the area term matches a generalized RT formula with entangling surfaces at $\mathcal{I}^\pm$. The entropy depends on Krylov complexity and at $t=0$ reproduces Gibbons–Hawking entropy (Aguilar-Gutierrez, 5 Nov 2025).
• Static patch and horizon calculations: Using Fefferman–Graham slicing and regulating with a UV cutoff, the RT prescription gives $S_{\rm EE} \to \frac{1}{2G_2}$ at the dS$_2$ horizon, exactly matching Wald entropy (Giataganas et al., 2019, Tetradis, 2021).

Approach	RT Surface Location	Stability	Entropy Formula
AdS$_3$ brane (Hao et al., 2024)	Edge of dS brane	Non-extremal (maximal bend)	$S_{\mathcal A}$ as above
EAdS$_2$ Wick rotation (Sato, 2015)	Usual RT geodesic (complexified)	Not unitary	$$S_A^{(\mathrm{dS}_2)} = -i \frac{c}{3}\ln(\Delta x/\epsilon)$$
SYK/dS JT (Aguilar-Gutierrez, 5 Nov 2025)	dS$_2$ horizon / $\mathcal{I}^\pm$	Unitary (Krylov)	$S_{EE}(t) = (1/4G_N) e^{i\ell_{dS}(t)/2}$
Static patch(Giataganas et al., 2019, Tetradis, 2021)	dS horizon	N/A	$S_{\rm EE} = \frac{1}{2G_2}$

5. Phase Transitions and Multi-Saddle Structures

In closed dS$_2$ spacetimes, doubly holographic models reveal the existence of several distinct extremal and non-extremal surfaces. Three entropic saddles control EE:

• Connected–Hartman–Maldacena: Geodesics connecting bath endpoints.
• New doubly-connected saddle: Includes segments ending on dS brane, which may have negative length inside bulk horizons.
• Island saddle: Geodesics stretching between bath and brane endpoints, saturating late-time entropy.

As a function of temperature and time, there is a tri-critical phase structure. At low temperatures, the new saddle dominates early times, with transitions to connected and finally island saddles as time increases. The negative-entropy regions introduced by behind-horizon geodesics are regulated using a “multiplicity one” rule: only a single branch is counted per endpoint (Jiang et al., 12 Feb 2025).

6. Quantum Extremal Surface (QES) Prescriptions and the Role of Matter

Covariant bilayer QES functionals in dS$_2$ clarify the emergent structure of entanglement wedges and their reconstructibility. Since all points on the dS$_2$ horizon are classically degenerate (yielding the same RT length), quantum corrections from bulk matter, notably the CFT contribution,

$$S_{\rm semicl}(\Delta \theta) = \frac{c}{3} \ln\left(\frac{1}{\epsilon} \sin(\Delta \theta/2)\right),$$

select between candidate surfaces. At the critical slice, there is a first-order phase transition in the entanglement wedge, corresponding to changing reconstructibility between antipodal boundary screens (Franken, 2024). The inclusion of higher-order corrections such as conformal anomaly terms leads to subleading log corrections in higher dimensions, but dS$_2$ remains dominated by the leading order.

7. Entropy Bounds, Swampland Criteria, and Ambiguities

In the DS$_3$/dS$_2$ correspondence, a one-parameter family of minimal geodesics (both "horizon-hugger" and "slice" geodesics) yield the same dS entropy, $S_{dS} = {\pi L}/{2G_3}$. When matter is introduced, only the "horizon-hugger" surface maintains this equality; others can exceed it, potentially violating entropy bounds. This suggests that dS entropy serves as a "swampland" bound constraining semiclassical matter configurations (Geng et al., 2019). The redundancy of minimal surfaces persists due to the degeneracy arising from the dS$_2$/CFT$_1$ or DS$_3$/dS$_2$ geometries.

---

Collectively, these results establish that holographic entanglement entropy in dS$_2$ cannot be formulated solely in terms of conventional quantum extremal surfaces. The correct recipe picks out non-extremal, edge-pinned islands, or relies on careful inclusion of quantum and matter contributions, with the Gibbons–Hawking entropy emerging as an entanglement entropy in several distinct constructions (Hao et al., 2024, Aguilar-Gutierrez, 5 Nov 2025, Giataganas et al., 2019, Jiang et al., 12 Feb 2025, Franken, 2024). This web of results reflects the broader ambiguity, non-uniqueness, and non-unitarity challenges at the heart of de Sitter holography.