In this paper, we study the entanglement entropy of a single interval on a cylinder in two-dimensional
TT-deformed conformal field theory (CFT). For such case, the (R\'enyi) entanglement entropy takes a universal form in a CFT. We compute the correction due to the deformation up to the leading order of the deformation parameter in the framework of the conformal perturbation theory. We find that the correction to the entanglement entropy is nonvanishing in the finite temperature case, while it is vanishing in the finite size case. For the deformed holographic large
c CFT, which is proposed to be dual to a
AdS3 gravity in a finite region, we find the agreement with the holographic entanglement entropy via the Ryu-Takayanagi formula. Moreover, we compute the leading order correction to the R\'enyi entropy, and discuss its holographic picture as well.
In this Dissertation, different aspects of holographic entanglement entropy are explored. In quantum information theory, entanglement is a useful resource needed to perform quantum operations. Entanglement entropy quantifies this resource and even if computable in quantum field theories, it is quite hard to calculate explicitly. However, in the context of gauge/gravity duality, (holographic) entanglement entropy was proposed by Ryu and Takayanagi (RT) to be captured by the area of a minimal surface in Anti DeSitter (AdS) space, which is extremely simple to compute. We begin by using the holographic dictionary to derive the RT formula. This is done by first considering smooth bulk geometries which are dual to n copies of the boundary geometry glued in a particular way. The action of these geometries computes the Renyi entropies S_n for any integer n. We give a prescription to analytically continue the action to non-integer n and argue that, as n goes to 1,S_1, the entanglement entropy, is given by the area of a minimal surface, reproducing the RT conjecture. That is, we reduced the RT proposal to the equality between bulk and boundary partition functions, a standard entry in the dictionary. We then generalize the previous formalism to account for corrections due to bulk quantum fields (1/N corrections in the boundary). This allows us to derive a new formula: boundary entanglement entropy is given by the area of the minimal surface plus bulk entanglement entropy. This extends RT beyond the planar limit and we present several predictions of the proposal. Our exploration of holographic entanglement entropy continues by considering different states: by comparing the quantum corrected entropy for nearby states we obtain the modular hamiltonian operator in bulk perturbation theory. From this expression of the modular hamiltonian, we derive that relative entropy is bulk relative entropy and that the modular flow is the bulk modular flow.
Entanglement entropy is a useful probe of compressible quantum matter because it can detect the existence of Fermi surfaces, both of microscopic fermionic degrees of freedom and of ``hidden'' gauge-charged fermions. Much recent attention has focused on holographic efforts to model strongly interacting compressible matter of interest for condensed matter physics. We complete the entanglement analysis initiated by Huijse et al. [Phys. Rev. B 85, 035121 (2012)] and Ogawa et al. [JHEP 1 (2012) 125] using the recent proposal of Faulkner et al. (arXiv:1307.2892) to analyze the entanglement entropy of the visible fermions which arises from bulk loop corrections. We find perfect agreement between holographic and field-theoretic calculations.