Entanglement entropy: Holography and renormalization group

Entanglement entropy plays a variety of roles in quantum field theory, including the connections between quantum states and gravitation through the holographic principle. This article provides a review of entanglement entropy from a mixed viewpoint of field theory and holography. A set of basic methods for the computation is developed and illustrated with simple examples such as free theories and conformal field theories. The structures of the ultraviolet divergences and the universal parts are determined and compared with the holographic descriptions of entanglement entropy. The utility of quantum inequalities of entanglement are discussed and shown to derive the $\mathcal{C}$ theorem that constrains renormalization group flows of quantum field theories in diverse dimensions.

Figure 1

The decomposition of the system into a subsystem $A$ shown in blue (light gray) and its complement $B$. (a) Two spin systems. The subsystems $A$ and $B$ are left and right spins, respectively. (b) In $d$-dimensional quantum field theory, a spatial region at a given time slice is split into the subsystems $A$ and $B$ whose common boundary $\partial A=\partial B$ is always a codimension-two hypersurface in $d$ dimensions.

Figure 2

The mutual information between two disjoint regions $A$ (blue, light gray) and $B$ red (gray).

Figure 3

A system of $N$-coupled harmonic oscillators. Here the total size of the system is $N=10$ and the subsystem $A$ in blue (light gray) consists of four oscillators from the left.

Figure 4

Path integral representations of a wave function. The integral is performed over the shaded regions.

Figure 5

The $n$-fold cover of ${\mathbb{R}}^d$ with a cut along the subregion $A$ ($x_1>0$) at $t=0$ in the polar coordinates.

Figure 6

The Rindler spacetime (shaded region) embedded in the Minkowski spacetime.

Figure 7

A cone that has a singularity at $r=0$ is smoothened by replacing the tip with a disk.

Figure 8

The $n$-fold cover ${\mathcal{M}}_n$ of $\mathbb{C}$ with a cut between $z=-l/2$ and $l/2$ is conformally mapped to a cylinder. The intervals are mapped to lines at $\phi =k\ell$ ($k=0,1,\dots ,n-1$) where the $n$ copies are glued together. This shows the conformal transformation for $n=3$.

Figure 9

The coordinate transformation Eq. (242) induces the conformal map from a flat space (left) to a hyperbolic space times a circle (middle). The latter is further conformally mapped to a sphere (right) by the coordinate transformation $\sinh u=\cot \theta$. The blue dashed circles represent the entangling surface ${\mathbb{S}}^{d-2}$, and the orange circles with arrows stand for the modular time, respectively.

Figure 10

The $\mathrm{AdS}/\mathrm{CFT}$ correspondence in the Poincaré coordinates. The (shaded) boundary of ${\mathrm{AdS}}_{d+1}$ at $z=\epsilon$ is the flat space ${\mathbb{R}}^d$ on which ${\mathrm{CFT}}_d$ lives.

Figure 11

The minimal surface ${\gamma }_A$ in orange (light gray) anchored on the boundary of the region $A$ in blue (gray) located at the AdS boundary $z=\epsilon$. The holographic entanglement entropy of the region $A$ is given by the area of the surface ${\gamma }_A$ divided by 4 times the Newton constant $G_N$ as in Eq. (278).

Figure 12

The smooth bulk manifold ${\mathcal{B}}_n$ whose boundary is the $n$-fold cover $\partial {\mathcal{B}}_n={\mathcal{M}}_n$ used for the boundary QFT in the replica trick by gluing $n$ copies of a manifold along the entangling region $A$ (the blue dotted lines). The entangling surface $\mathrm{\Sigma }=\partial A$ is the fixed point of the ${\mathbb{Z}}_n$ symmetry acting as the shift $\tau \to \tau +2\pi$, which extends to the bulk as a codimension-two hypersurface ${\gamma }_A^{(n)}$ (the orange dashed curves).

Figure 13

The holographic proof of the strong subadditivity. (a) The minimal surfaces ${\gamma }_{A\cup B}$ and ${\gamma }_{B\cup C}$ are light gray and black on the leftmost side, which can be reconnected to surfaces ${\gamma }_B^{\prime }$ (light gray) and ${\gamma }_{A\cup B\cup C}^{\prime }$ (black) bounding the regions $B$ and $A\cup B\cup C$ in the middle. The total area is minimized by the minimal surfaces ${\gamma }_B$ (light gray) and ${\gamma }_{A\cup B\cup C}$ (black) on the rightmost side, proving the strong subadditivity $S_{A\cup B\cup C}+S_B\le S_{A\cup B}+S_{B\cup C}$. (b) A similar setup for proving the inequality $S_A+S_C\le S_{A\cup B}+S_{B\cup C}$.

Figure 14

A schematic proof of the monogamy of mutual information.

Figure 15

The minimal surface of a spherical entangling surface in the Poincaré coordinates of ${\mathrm{AdS}}_{d+1}$.

Figure 16

The global AdS space obtained by the conformal map Eq. (300) from the Poincaré coordinates. The outside of the AdS topological black hole horizon covers half of the global coordinates. The minimal surface (the thick dotted orange curve) at $r=R$ in Fig. 15 is on the horizon of the AdS topological black hole.

Figure 17

(Left) An example of RG flows in the theory space $\mathcal{T}$. The black dots stand for the fixed points of RG flows. (Right) A $\mathcal{C}$ function monotonically decreases as the RG time $t$ is increased.

Figure 18

An interval of width $R$ as an entangling region.

Figure 19

A proof of the entropic $c$ theorem. Two intervals $A$ and $C$ are on the light rays $t=\pm x$ while an interval $B$ is on a time slice $t=0$.

Figure 20

$N=3$ boosted disks of radii $\sqrt{rR}$ (blue circles) touching on the null cone (gray). The dashed and dotted curves are circles of radii $R$ and $r$ at constant time slices $t=R$ and $r$, respectively.

Figure 21

The dimensional reduction of a ($3+1$)-dimensional field theory to the ($2+1$)-dimensional field theory.

Figure 22

The numerical plot of the $\mathcal{F}$ function of a free massive scalar field (the black solid curve). The dotted red line is the UV fixed point value ${\mathcal{F}}_{\mathrm{UV}}\approx 0.064$ of the $\mathcal{F}$ function.

Figure 23

Two types of a minimal surface. The dashed blue and solid orange curves are disk and cylinder type surfaces, respectively.