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 paper 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., 31 Jul 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., 31 Jul 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, 21 Mar 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.

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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., 31 Jul 2024, Aguilar-Gutierrez, 5 Nov 2025, Giataganas et al., 2019, Jiang et al., 12 Feb 2025, Franken, 21 Mar 2024). This web of results reflects the broader ambiguity, non-uniqueness, and non-unitarity challenges at the heart of de Sitter holography.