RT Surfaces in Holographic Entanglement

• Ryu-Takayanagi surfaces are codimension-two extremal surfaces that compute leading contributions to entanglement entropy in holographic theories, bridging geometry and quantum information.
• They are constructed via minimal area prescriptions on static slices or through covariant extremization in dynamic spacetimes, often requiring analytic continuation for complex cases.
• Research shows that complex extremal surfaces can dominate in specific geometries, challenging standard interpretations and impacting holographic entanglement and black hole thermodynamics.

Ryu-Takayanagi surfaces are codimension-two extremal surfaces in a gravitational bulk geometry whose area—properly regulated—computes the leading contribution to the entanglement entropy of subregions in the boundary theory, as formalized by the Ryu-Takayanagi (RT) and Hubeny-Rangamani-Takayanagi (HRT) prescriptions. These surfaces are central objects in the geometric realization of holographic entanglement entropy, with deep implications for AdS/CFT, black hole thermodynamics, quantum information, and emergent spacetime. Key technical developments over the past decade have refined, generalized, and challenged the foundational picture of RT surfaces in a broad landscape of gravitational backgrounds and quantum field theories.

1. Definition and General Construction

The RT prescription: for a spatial subregion $A$ in a holographic boundary theory, the entanglement entropy $S_A$ is computed as

$S_A = \frac{\mathrm{Area}(\Sigma_A)}{4 G_N}$

where $\Sigma_A$ is the codimension-two surface in the bulk spacetime homologous to $A$ that extremizes the area functional, subject to given boundary conditions.

For static asymptotically AdS geometries, these are minimal surfaces on a constant-time slice. In dynamical or covariant cases, HRT surfaces extremize area in the full spacetime. For bulk metrics of the form

$$ds^2 = \frac{1}{z^2} \left(-f(z) dt^2 + \frac{dz^2}{g(z)} + dx^2 + d\vec{y}^2 \right),$$

the surface is parameterized by functions (e.g., $x(z)$, $t(z)$) subject to extremality equations, frequently reformulated via conserved charges and effective potentials.

The regulated area typically diverges near the AdS boundary, necessitating the subtraction of divergent counterterms.

2. Analytic Structure: Complex Extremal Surfaces

A rigorous semiclassical path integral analysis reveals that the area functional, after analytic continuation (e.g., from Euclidean to Lorentzian signature), generally defines a multi-valued function across a Riemann surface with additional "complex" extremal surfaces living on secondary sheets (Fischetti et al., 2014).

• BTZ (AdS$_3$): The complex surfaces are trivial copies (differing by quantized imaginary offsets) of real geodesic minimal surfaces. The physical entropy is always determined by the real area.
• Schwarzschild-AdS and Lifshitz Black Holes: Analytic continuation generates complex-valued critical energies and families of extremal surfaces on secondary Riemann sheets. In these cases, the real part of the area $\mathrm{Re}~A_{\text{ren}}$ for some complex surfaces on secondary sheets can be strictly less than that of any real surface:

$$\mathrm{Re}\,A_{\text{ren}}(\text{complex}) < \mathrm{Re}\,A_{\text{ren}}(\text{real}).$$

If the CFT entropy is determined by $$S_{\text{CFT}} \approx \mathrm{Re}\,A_{\text{ren}}/(4 G_N)$$, then it may be dominated by genuinely complex saddles—a generalization of the standard picture.

• Saddle Dominance and Physical Consistency: The dominance of complex extremal surfaces is plausible in sectors where their $\mathrm{Re}\,A_{\text{ren}}$ is minimal and their area functional exhibits expected physical monotonicity (e.g., minimal at $t_b = 0$, symmetric and rising with $|t_b|$), but theoretical questions about strong subadditivity and path integral contour deformation remain unresolved.

3. Explicit Examples and Families of Solutions

BTZ Geometry

The area and time separation are given by: $\Delta t = \beta \left[ -\frac{1}{\pi} \arctanh(E) + \frac{i}{2} \right], \qquad A_{\text{ren}} = \ell \ln \left( \frac{4}{1-E^2} \right),$ with the principal sheet yielding real geodesics for $E \in (-1,1)$.

Schwarzschild–AdS and Lifshitz Black Holes

For higher-dimensional AdS black holes: $$f(r) = g(r) = \frac{r^2}{\ell^2} \left( 1 - \frac{r_h^d}{r^d} \right),$$ the extremal surface equations reduce to an effective single-variable "mechanical" problem,

$|{\dot R}|^2 + V_{\text{eff}}(R) = 0.$

Branch points and critical energies arise where two roots of the denominator coalesce. For complex surfaces, analytic continuation across these branch cuts accesses new families with lower $\mathrm{Re}~A_{\text{ren}}$.

For Lifshitz black holes: $$f(r) = \left( \frac{r}{\ell} \right)^{2z}\left[1 - \left( \frac{r_h}{r} \right)^{d+z-1} \right], \quad g(r) = \left( \frac{r}{\ell} \right)^2 \left[1 - \left( \frac{r_h}{r} \right)^{d+z-1} \right],$$ rotations in the radial coordinate map real contours to complex sheets which may further lower $\mathrm{Re}~A_{\text{ren}}$.

4. Real vs. Imaginary Area Contributions

In all cases, the renormalized area is complex: $$A_{\text{ren}} = \mathrm{Re}\,A_{\text{ren}} + i\,\mathrm{Im}\,A_{\text{ren}}.$$ The entropy formula is taken to be

$$S_{\text{CFT}} \approx \frac{\mathrm{Re}\,A_{\text{ren}}}{4 G_N},$$

with the imaginary part discarded (either because it is quantized, as in BTZ, or subdominant in the semiclassical limit). The choice of which extremal surface (real or complex) dominates the CFT entropy reduces to minimizing $\mathrm{Re}~A_{\text{ren}}$ over all allowed extremal surfaces.

5. Physical and Mathematical Consequences

• Multivaluedness and analytic continuation: The presence of complex extremal surfaces is intrinsic to the multivalued analytic structure of the RT functional in backgrounds with nontrivial causal structure or multiple horizons.
• Correlation functions analogy: The necessity for complex geodesic contours is reminiscent of the need to deform integration contours for bulk propagators and two-point functions (e.g., to regulate physically unacceptable divergences at finite time).
• Dominance and phase transitions: For Schwarzschild–AdS and Lifshitz cases, secondary sheet solutions may better match expected CFT entropy, even dominating real ones. This motivates a generalized prescription sensitive to the full analytic structure of the area functional.
• Consistency and open problems: While for real surfaces strong subadditivity and other key properties of entanglement entropy are proven, their extension to complex saddles is not guaranteed. The gravitational path integral's integration cycle may need to be specified carefully if complex saddles are to contribute, a subtlety with implications for the quantum-corrected geometry and the emergence of semiclassical gravity.

6. Summary Table: Complex Extremal Surfaces Across Geometries

Geometry	Complex Surfaces	Dominance Criteria	Area Behavior
BTZ (AdS₃)	Trivial copies	No effect; real dominates	Quantized imaginary part
Schwarzschild–AdS	New families	May dominate if lower Re	Re area < real for some t_b
Schwarzschild–Lifshitz	New families	May dominate if lower Re	Re area < real for same Δt

7. Implications for Holographic Entanglement

These findings establish that the geometry of RT/HRT surfaces is more intricate than previously assumed, particularly in time-dependent or nonsimply connected backgrounds. The analytic continuation leading to complex extremal surfaces opens the door to richer behavior in holographic entanglement entropy, suggesting that the holographic dictionary in quantum gravity must incorporate the possibility of complex saddles and their physical interpretation in the dual CFT. The situation in higher dimensions and Lifshitz/HV backgrounds is more subtle, indicating that semiclassical gravity and its quantum corrections probe a higher level of analytic complexity than strictly real geometries admit.

Overall, the inclusion of complex codimension-two extremal surfaces into the RT/HRT prescriptions constitutes a significant step toward a more complete and physically accurate holographic entanglement theory, while also posing challenging questions about the physical meaning, dominance, and mathematical consistency of such generalized saddles (Fischetti et al., 2014).