Entanglement entropy: holography and renormalization group

Abstract

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 C-theorem that constrains renormalization group flows of quantum field theories in diverse dimensions.

Figures (23)

• Figure 1: The decomposition of the system into a subsystem $A$ shown in blue 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) and $B$ (red).
• Figure 3: A system of $N$-coupled harmonic oscillators. Here the total size of the system is $N=10$ and the subsystem $A$ (blue) 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.

Theorems & Definitions (3)

• Theorem : Zamolodchikov's $c$-theorem
• Theorem : $F$-theorem
• Theorem : $a$-theorem