Entanglement entropy in critical phenomena and analog models of quantum gravity

A general geometrical structure of the entanglement entropy for spatial partition of a relativistic QFT system is established by using methods of the effective gravity action and the spectral geometry. A special attention is payed to the subleading terms in the entropy in different dimensions and to behavior in different states. It is conjectured, on the base of relation between the entropy and the action, that in a fundamental theory the ground state entanglement entropy per unit area equals $1/(4G_N)$, where $G_N$ is the Newton constant in the low-energy gravity sector of the theory. The conjecture opens a new avenue in analogue gravity models. For instance, in higher-dimensional condensed matter systems, which near a critical point are described by relativistic QFT’s, the entanglement entropy density defines an effective gravitational coupling. By studying the properties of this constant one can get new insights in quantum gravity phenomena, such as the universality of the low-energy physics, the renormalization group behavior of $G_N$, the statistical meaning of the Bekenstein-Hawking entropy.

Figure 1

The upper picture shows ${\mathcal{M}}_1^{\prime }$ in two dimensions. It is a cylinder with the circumference length $T^{-1}$ and a cut along the axis. The space ${\mathcal{M}}_3$ is schematically drawn on the lower picture. It is obtained by gluing along the cuts of 3 copies of ${\mathcal{M}}_1^{\prime }$. The circumference length of the right boundary of ${\mathcal{M}}_3$ is $3T^{-1}$. The cuts meet at the point $X$ which is a conical singularity.

Figure 2

This figure shows division of a cube by a plane $\mathcal{B}$. The trace is taken over the states of a QFT in one of the halves of the cube.

Figure 3

The manifold ${\mathcal{M}}_3^{(2)}$ is shown for computing the entanglement entropy $S_B$ at a finite temperature when the region $A$ consists of two internal disjoint intervals. ${\mathcal{M}}_3^{(2)}$ is obtained by gluing 4 copies of the manifold shown on Fig. 1. ${\mathcal{M}}_3^{(2)}$ has 4 conical singularities at branch points.

Figure 4

This figure is based on results of 36. It shows the dependence of the entanglement entropy in the Ising spin chain (6) at a fixed $N$ as a function of magnetic field strength $\lambda$. The critical point $\lambda =1$ is ultaviolet fixed point. The arrows show directions of the RG flow from the ultraviolet to infrared regions.