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A general geometrical structure of the entanglement entropy for spatialpartition of a relativistic QFT system is established by using methods of theeffective gravity action and the spectral geometry. A special attention ispayed to the subleading terms in the entropy in different dimensions and tobehaviour in different states. It is conjectured, on the base of relationbetween the entropy and the action, that in a fundamental theory the groundstate entanglement entropy per unit area equals $1/(4G_N)$, where $G_N$ is theNewton constant in the low-energy gravity sector of the theory. The conjectureopens a new avenue in analogue gravity models. For instance, in higherdimensional condensed matter systems, which near a critical point are describedby relativistic QFT's, the entanglement entropy density defines an effectivegravitational coupling. By studying the properties of this constant one can getnew insights in quantum gravity phenomena, such as the universality of thelow-energy physics, the renormalization group behavior of $G_N$, thestatistical meaning of the Bekenstein-Hawking entropy.Comment: 13 pages, published version, minor changes in the abstract, new  reference

Entanglement entropy in critical phenomena and analog models of quantum gravity