| Ryu–Takayanagi formula | |
|---|---|
| Name | Ryu–Takayanagi formula |
| Field | Theoretical physics; Quantum field theory; General relativity |
| Introduced | 2006 |
| Authors | Shinsei Ryu; Tadashi Takayanagi |
| Related | AdS/CFT correspondence; Entanglement entropy; Holography (physics) |
Ryu–Takayanagi formula
The Ryu–Takayanagi formula is a conjectured geometric prescription that computes the entanglement entropy of a spatial region in a conformal field theory (CFT) via the area of a minimal surface in a dual anti-de Sitter space (AdS). It plays a central role in connecting quantum information theory measures to geometric quantities in quantum gravity, providing a quantitative instantiation of the holographic principle and the AdS/CFT correspondence.
The Ryu–Takayanagi proposal states that for a static asymptotically AdS bulk geometry dual to a boundary CFT state, the entanglement entropy S(A) of a boundary spatial region A equals the area of the minimal codimension-two bulk surface γ_A homologous to A divided by 4G_N: S(A)=Area(γ_A)/(4G_Nħ). This parallels the Bekenstein–Hawking entropy formula for black hole horizons and uses concepts from general relativity such as minimal (extremal) surfaces and the Einstein field equations. The original statement targeted static spacetimes and was later generalized to time-dependent and quantum-corrected regimes.
The formula was motivated within the framework of the AdS/CFT correspondence formulated by Juan Maldacena and further developed by researchers such as Edward Witten and Steven Gubser. Ryu and Takayanagi provided the holographic dictionary entry linking boundary entanglement entropy in a CFT_{d} to a geometric object in AdS_{d+1}. Subsequent derivations and justifications used the replica trick in the bulk, Euclidean gravitational path integrals, and the modular Hamiltonian; notable works include those by Lewkowycz–Maldacena which used the replica method to derive the RT formula under certain assumptions. Connections were established to the gravitational entropy proposals of Wald entropy and to Jacobson-type derivations linking entanglement variations to linearized Einstein equations.
Practically, computing S(A) holographically reduces to finding minimal (or extremal) surfaces in explicit bulk metrics, using techniques from differential geometry and numerical relativity. Classic examples include the computation of entanglement for intervals in the 2D CFT dual to global or Poincaré AdS3 where the RT surface reduces to geodesics; results match independent CFT calculations by methods of Cardy and Calabrese–Cardy. For higher dimensions, computations in AdS5/N=4 SYM backgrounds illustrate area scalings, phase transitions between competing surfaces (entanglement plateaux), and connections to confinement/deconfinement via the Hawking–Page transition. Numerical techniques include the use of minimal surface solvers, the shooting method, and time evolution for extremal surfaces.
The Hubeny–Rangamani–Takayanagi (HRT) proposal generalized RT to time-dependent (Lorentzian) settings by replacing minimal with extremal surfaces anchored on the boundary. Quantum corrections lead to the Quantum Extremal Surface (QES) prescription: the generalized entropy S_gen = Area(γ)/(4G_Nħ) + S_bulk, where S_bulk is the bulk entanglement contribution of quantum fields across the surface; extremizing S_gen yields the QES. These ideas underpin recent calculations of Page curves for evaporating black holes and the role of islands in reconciling information paradox computations. Perturbative corrections involve semiclassical expansions around classical saddles and inputs from effective field theory and the gravitational path integral.
The RT formula provides a geometrization of quantum information quantities: entanglement entropy, mutual information, and relative entropy have holographic duals computed via areas and bulk modular flows. It has motivated the view that spacetime geometry emerges from entanglement structure, encapsulated in slogans like "entanglement builds geometry" and developed in tensor network models such as MERA and random tensor networks. The formula relates to quantum error correction descriptions of holography by groups including Almheiri, Dong, Harlow and clarifies subsystem reconstruction, entanglement wedge reconstruction, and the role of modular Hamiltonians in encoding bulk operators.
In quantum gravity research, RT and its generalizations serve as probes of the microstructure of spacetime, tests of semiclassical gravity, and tools in the study of black hole evaporation and entropy bounds. In condensed matter physics, holographic entanglement entropy has been applied to characterize phases, critical points, and topological order in strongly correlated systems using gauge/gravity duality models; examples include holographic superconductors and strange metals studied via the gauge/gravity toolkit. Cross-disciplinary applications involve quantum quenches, thermalization, and transport phenomena where entanglement dynamics provide insight beyond conventional correlators.
Key open problems include a rigorous derivation of the RT and QES prescriptions from first principles in quantum gravity, a microscopic accounting of entanglement contributions for general bulk quantum fields, and the extension to non-AdS holography (e.g., de Sitter space and flat space holography). Understanding the limitations of entanglement-based reconstruction, the role of complexity measures (such as "complexity = volume/action"), and connections to emergent spacetime programs remain active. Computational challenges persist in numerical extremal-surface finding in realistic string-theory compactifications and in integrating RT-based methods with lattice and tensor network approaches for concrete condensed matter systems.
Category:Quantum gravity Category:Entanglement entropy Category:AdS/CFT correspondence