We calculate entanglement entropy in a non-relativistic field theory described by the Schrödinger operator. We demonstrate that the entropy is characterized by i) the area law and ii) UV divergences that are identical to those in the relativistic field theory. These observations are further supported by a holographic consideration. We use the non-relativistic symmetry and completely specify entanglement entropy in large class of non-relativistic theories described by the field operators polynomial in derivatives. The entropy of interacting fields is analyzed in some detail. We suggest that the area law of the entropy can be tested in experiments with condensed matter systems such as liquid helium.
Here, we introduce the notion of low-energy interactions in non-relativistic quantum mechanics. The goal is to reproduce low-energy phenomenology only, which generally generates infinities, just as in relativistic quantum field theories. The resulting low-energy theory is called effective field theory of interactions, in analogy with the non-relativistic ‘pionless’ effective field theory widely used in nuclear physics. The predictive power of the effective theory depends upon the renormalizability of the theory, which removes the infinities while obtaining sound low-energy physics. Most of the chapter deals with perturbative and non-perturbative renormalization, which will be subsequently needed throughout the rest of the book.
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/(4GN), where
GN 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
GN, the statistical meaning of the Bekenstein-Hawking entropy.