Entanglement entropy of a compact region in a 2D-dilaton gravity theory

Abstract

The two-point function and the entanglement entropy can be seen as quan-tities to describe properties of the quark-gluon plasma. For this an anisotropicc 5-dimensional system is considered. A homogeneous anisotropic but O (2) symmetric solution to the 5-dimensional vacuum Einstein equations is equivalent to a homogeneous and isotropic solution to the Einstein equations including a scalar field. The corresponding 5-dimensional Einstein-Hilbert action without scalar field can be dimensionally reduced to a 2-dimensional dilaton gravity action with two scalar fields. The same 2-dimensional dilaton gravity action arises from reducing isotropic Ein- stein gravity with a minimally coupled massless scalar field. Therefore, the 2-dimensional dilaton gravity is equivalent to the 5-dimensional anisotropic system with matter and the 5-dimensional isotropic system with a scalar field. In this thesis the two-point function of operators of large conformal weight and the holographic entanglement entropy for a spherical region in a 2-dimensional dilaton gravity theory are calculated numerically using the Anti-de Sitter/conformal field theory correspondence. The two-point function, which is computed in the geodesic approximation, amounts to a geodesic-s length in the gravity theory. Similarly the calculation of the entanglement entropy reduces to finding geodesics in an auxiliary spacetime. To obtain the geometry the Einstein equations need to be solved numerically. The Einstein equations of the 2-dimensional dilaton gravity theory are mapped to the vacuum Einstein equations for an anisotropic geometry. The numerical calculation of geodesics is performed using a Mathematica package, which calculates the geodesics with a relaxation method. The results for the entanglement entropy and two-point function vary strongly, depending on the chosen boundary region.