Ryu–Takayanagi conjecture

The thermodynamics of black holes suggests certain relationships between the entropy of black holes and their geometry. Specifically, the Bekenstein–Hawking area formula conjectures that the entropy of a black hole is proportional to its surface area:

:$S\_\backslash text\{BH\}\; =\; \backslash frac\{k\_\backslash text\{B\}\; A\}\{4\backslash ell\_\backslash text\{P\}^2\}$

The Bekenstein–Hawking entropy $S\_\backslash text\{BH\}$ is a measure of the information lost to external observers due to the presence of the horizon. The horizon of the black hole acts as a "screen" distinguishing one region of the spacetime (in this case the exterior of the black hole) that is not affected by another region (in this case the interior). The Bekenstein–Hawking area law states that the area of this surface is proportional to the entropy of the information lost behind it.

The Bekenstein–Hawking entropy is a statement about the gravitational entropy of a system; however, there is another type of entropy that is important in quantum information theory, namely the entanglement (or von Neumann) entropy. This form of entropy provides a measure of how far from a pure state a given quantum state is, or, equivalently, how entangled it is. The entanglement entropy is a useful concept in many areas, such as in condensed matter physics and quantum many-body systems. Given its use, and its suggestive similarity to the Bekenstein–Hawking entropy, it is desirable to have a holographic description of entanglement entropy in terms of gravity.

{{main| Holographic principle}}

The holographic principle states that gravitational theories in a given dimension are dual to a gauge theory in one lower dimension. The AdS/CFT correspondence is one example of such duality. Here, the field theory is defined on a fixed background and is equivalent to a quantum gravitational theory whose different states each correspond to a possible spacetime geometry. The conformal field theory is often viewed as living on the boundary of the higher dimensional space whose gravitational theory it defines. The result of such a duality is a dictionary between the two equivalent descriptions. For example, in a CFT defined on $d$ dimensional Minkowski space the vacuum state corresponds to pure AdS space, whereas the thermal state corresponds to a planar black hole.{{cite book

| last1 = Van Raamsdonk| first1 = Mark

| title = New Frontiers in Fields and Strings

| pages = 297–351

| chapter = Lectures on Gravity and Entanglement

| date = 31 August 2016

| arxiv = 1609.00026

| doi = 10.1142/9789813149441_0005

| isbn = 978-981-314-943-4

| s2cid = 119273886

}} Important for the present discussion is that the thermal state of a CFT defined on the $d$ dimensional sphere corresponds to the $d+1$ dimensional Schwarzschild black hole in AdS space.

The Bekenstein–Hawking area law, while claiming that the area of the black hole horizon is proportional to the black hole's entropy, fails to provide a sufficient microscopic description of how this entropy arises. The holographic principle provides such a description by relating the black hole system to a quantum system which does admit such a microscopic description. In this case, the CFT has discrete eigenstates and the thermal state is the canonical ensemble of these states. The entropy of this ensemble can be calculated through normal means, and yields the same result as predicted by the area law. This turns out to be a special case of the Ryu–Takayanagi conjecture.

Consider a spatial slice $\backslash Sigma$ of an AdS space time on whose boundary we define the dual CFT. The Ryu–Takayanagi formula states:

{{NumBlk|:|$S\_A\; =\; \backslash frac\{\backslash text\{Area\; of\; \}\; \backslash gamma\_A\}\{4G\}$|{{EquationRef|1}}}}

where $S\_A$ is the entanglement entropy of the CFT in some spatial sub-region $A\; \backslash subset\; \backslash partial\; \backslash Sigma$ with its complement $B$, and $\backslash gamma\_A$ is the Ryu–Takayanagi surface in the bulk. This surface must satisfy three properties:

1. $\backslash gamma\_A$ has the same boundary as $A$.
2. $\backslash gamma\_A$ is homologous to A.
3. $\backslash gamma\_A$ extremizes the area. If there are multiple extremal surfaces, $\backslash gamma\_A$ is the one with the least area.

Because of property (3), this surface is typically called the minimal surface when the context is clear. Furthermore, property (1) ensures that the formula preserves certain features of entanglement entropy, such as $S\_A\; =\; S\_B$ and $S\_\{A\_1\; +\; A\_2\}\; \backslash geq\; S\_\{A\_1\; \backslash cup\; A\_2\}$.{{clarify|date=June 2024}} The conjecture provides an explicit geometric interpretation of the entanglement entropy of the boundary CFT, namely as the area of a surface in the bulk.

In their original paper, Ryu and Takayanagi show this result explicitly for an example in $\backslash text\{AdS\}\_3\; /\; \backslash text\{CFT\}\_2$ where an expression for the entanglement entropy is already known. For an $\backslash text\{AdS\}\_3$ space of radius $R$, the dual CFT has a central charge given by

{{NumBlk|:|$c\; =\; \backslash frac\{3R\}\{2G\}$|{{EquationRef|2}}}}

Furthermore, $\backslash text\{AdS\}\_3$ has the metric

ds^2 = R^2(-\cosh{\rho^2 dt^2} + d\rho^2 + \sinh{\rho^2 d\theta^2})

in $(t,\; \backslash rho,\; \backslash theta)$ (essentially a stack of hyperbolic disks). Since this metric diverges at $\backslash rho\; \backslash to\; \backslash infty$, $\backslash rho$ is restricted to $\backslash rho\; \backslash leq\; \backslash rho\_0$. This act of imposing a maximum $\backslash rho$ is analogous to the corresponding CFT having a UV cutoff. If $L$ is the length of the CFT system, in this case the circumference of the cylinder calculated with the appropriate metric, and $a$ is the lattice spacing, we have

$e^\{\backslash rho\_0\}\; \backslash sim\; L/a$.

In this case, the boundary CFT lives at coordinates $(t,\; \backslash rho\_0,\; \backslash theta)\; =\; (t,\; \backslash theta)$. Consider a fixed $t$ slice and take the subregion A of the boundary to be $\backslash theta\; \backslash in\; [0,\; 2\backslash pi\; l\; /\; L]$ where $l$ is the length of $A$. The minimal surface is easy to identify in this case, as it is just the geodesic through the bulk that connects $\backslash theta\; =\; 0$ and $\backslash theta\; =\; 2\; \backslash pi\; l/L$. Remembering the lattice cutoff, the length of the geodesic can be calculated as

{{NumBlk|:|$\backslash cosh\{(L\_\{\backslash gamma\_A\}\; /\; R)\}\; =\; 1\; +\; 2\backslash sinh^2\; \backslash rho\_0\; \backslash sin^2\; \backslash frac\{\backslash pi\; l\}\{L\}$|{{EquationRef|3}}}}

If it is assumed that $e^\{\backslash rho\_0\}\; >>\; 1$, then using the Ryu–Takayanagi formula to compute the entanglement entropy. Plugging in the length of the minimal surface calculated in ({{EquationNote|3}}) and recalling the central charge ({{EquationNote|2}}), the entanglement entropy is given by

{{NumBlk|:|$S\_A\; =\; \backslash frac\{R\}\{4G\}\backslash log\{(e^\{2\backslash rho\_0\}\; \backslash sin^2\; \backslash frac\{\backslash pi\; l\}\{L\})\}\; =\backslash frac\{c\}\{3\}\; \backslash log\{(e^\{\backslash rho\_0\}\; \backslash sin\{\backslash frac\{\backslash pi\; l\}\{L\}\}\; )\}$|{{EquationRef|4}}}}

This agrees with the result calculated by usual means.{{Cite journal

|last1=Calabrese|first1=Pasquale

|last2=Cardy|first2=John

|date=2004-06-11

|title=Entanglement entropy and quantum field theory

|journal=Journal of Statistical Mechanics: Theory and Experiment

|volume=P06002

|issue=6

|pages=P06002

|doi=10.1088/1742-5468/2004/06/P06002

|arxiv=hep-th/0405152|bibcode=2004JSMTE..06..002C

|s2cid=15945690

}}

{{Reflist}}

{{quantum gravity}}

{{DEFAULTSORT:Ryu-Takayanagi conjecture}}

Category:Conjectures

Category:String theory