This paper provides an in-depth analysis of entanglement entropy (EE) in quantum field theory (QFT), with a particular focus on its computation using the replica trick and its applications to both conformal and non-conformal systems. Beginning with an introduction to the basics of QFT, the study explains how entanglement entropy quantifies the quantum correlations between subsystems in a pure state, represented by the von Neumann entropy of the reduced density matrix. The replica trick is employed to derive the entanglement entropy, involving path integrals over n-sheeted Riemann surfaces. The paper demonstrates the method’s utility in 1+1-dimensional conformal field theory (CFT), where the central charge governs universal properties of entropy. It further extends the analysis to massive field theories, finite systems, and topological phases, exploring how deviations from conformal symmetry affect entanglement. The work also delves into the AdS/CFT correspondence, showcasing how holographic techniques facilitate entanglement entropy calculations in higher-dimensional systems via the Ryu-Takayanagi formula. In addition, the study investigates entanglement entropy in quantum lattice systems, considering the effects of spatial and thermal fluctuations. Numerical methods are used to compute the scaling of entanglement entropy with interval length for varying system masses and temperatures, highlighting the influence of these parameters on entropy behavior. Fractal lattice structures are explored to uncover unique entropy scaling laws and self-similar entanglement patterns. This study advances the under-standing of EE in diverse quantum systems and establishes a foundation for exploring its role in quantum criticality and fractal geometries. The paper also explores multiscale entanglement entropy (MSE), offering a comprehensive framework for future research in quantum critical phenomena and non-equilibrium systems, shedding light on the complex relationship between quantum entanglement and thermal effects. This work serves as a foundational reference for future theoretical and computational studies in quantum entanglement
Entanglement entropy is a proposal to quantify quantum entanglement of two disjoint regions in a pure system. It is a relatively new topic and is developing rapidly. The current main motives to study it are its applications to studying quantum gravity, thermalization and phase transitions in condensed matter systems. Mutual information is a related quantity and it can be used to measure the amount of information two disjoint regions share. The purpose of this thesis is to give an introduction to the general results of the field without considering specific systems. The two prominent approaches used are two-dimensional conformal field theory and the holographic entanglement entropy conjecture. The first approach is to calculate entanglement entropy using conformal field theory, the results of which are known to be exact although technically more difficult to derive and only available for 1+1 dimensional systems. Both static and dynamic systems will be discussed. The results are reproduced and generalized to higher dimensions using holography. As a more recent topic, the holographic approach is used to rederive the entanglement entropy of systems with a Fermi surface using AdS/Vaidya metric with Lifshitz scaling and hyperscaling violation. The final chapter of the thesis discusses mutual information in various static and dynamic cases considered in the previous chapters. For the first time, the evolution of mutual information is calculated in AdS/Vaidya metric with Lifshitz scaling and hyperscaling violation in the critical theta=d-1 case.
This thesis collects three works about holographic computations of entanglement entropy. In the first one it is shown how to compute numerically, following the Ryu-Takayanagi prescription, the entanglement entropy for arbitrarily shaped entangling regions in three dimensional conformal field theories. The other two focus on holographic theories with hyperscaling violating exponents: the time dependence after a holographic quench is analyzed and the arising of new universal terms due to the presence of non smooth boundaries is shown.
Entanglement entropy is crucial for understanding the link between quantum mechanics and information theory. This thesis investigates how energy fluctuations and acceleration affect entanglement entropy through three key scenarios. First, we analyze entanglement between disjoint Rindler observers in de Sitter spacetime. We introduce 1-loop corrections to the partition function of the conformal field theory, disrupting the previously flat entanglement spectrum and resulting in an area law for entanglement entropy, while preserving the finiteness of the Hilbert space. Next, we extend the holographic entanglement entropy framework to field theories with global
U(1) symmetry, considering non-linearly charged theories and their thermal and quantum fluctuations. We find that the dual theory experiences spontaneous symmetry breaking and a reduction in the leading-order coefficient of the
⟨TT⟩ correlator due to quantum corrections. Using conformal electrodynamics coupled with three-dimensional gravity as an example, we explore a black hole solution and its connection to two-dimensional free bosons. Finally, we investigate three-dimensional accelerating black holes, revealing that increased acceleration reduces the accessible boundary region and, consequently, the entanglement entropy. This suggests a loss of information in the dual theory due to acceleration.
Entanglement entropy (EE) is a concept of paramount importance in modern theoretical physics. Nevertheless, calculating EE in generic interacting theories with non-trivial time dependence is notoriously difficult. In this thesis, we address the real-time dynamics of EE in certain strongly-coupled quantum field theories (QFT) using the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence. We first study the time evolution of EE in an exactly solvable model of two-dimensional CFT with an inhomogeneous sine-square deformation (SSD). We argue that the non-trivial time evolutions driven by the SSD Hamiltonian can be regarded as changing the foliations of spacetime. Remarkably, this leads to the presence of new horizons in the dual three-dimensional bulk. We then compute the EE in SSD CFT using holography and found agreement with CFT results in all comparable cases. Afterwards, we proceed to entanglement dynamics in generic chaotic quantum many-body systems. In the limit of large subsystems and late times, entanglement dynamics is believed to be described by an entanglement membrane effective theory. For 2d CFTs, a degenerate tension function reproduces several results regarding the dynamics of the entropy; this stands in contrast to higher dimensions where the tension is non-degenerate. We use holography to show that, in order to correctly capture the reflected entropy in 2d CFT, one needs to add an additional degree of freedom to the membrane description. Furthermore, we show that the conventional non-degenerate membrane tension function emerges upon introducing a relevant deformation to the CFT, dual to a planar Banados-Teitelboim-Zanelli (BTZ) black hole with scalar hair and with an interior Kasner universe. Subsequently, we establish a membrane effective theory for the dynamics of reflected entropy in higher dimensions. Lastly, we apply the membrane theories for reflected entropy to study the time evolution of multiparty entanglement signals. We found that scrambling dynamics generate multiparty entanglement with rich structures.
Entanglement and symmetries are two pillars of our understanding of modern Physics, but surprisingly the study of the interplay between these two fundamental concepts has become the subject of an intense research area only in the last few years. In particular, the resolution of entanglement in the various symmetry sectors of a quantum many-body system with a global conserved charge gained a lot of interest within the entanglement community. As shown in some experimental papers, the study of this symmetry resolved entanglement is necessary for a better understanding of the non-equilibrium dynamics of quantum many-body systems. This Thesis is devoted to the study of the symmetry resolution of the entanglement measures in free fermionic systems and Conformal Field Theories. It is organised as follows. In Chap. 1 we give an overview of the main definitions and concepts related to the entanglement and its measures in quantum mechanics and quantum field theory. Then, we specialize to the case of conformal field theories, to derive expressions of the entanglement entropy and negativity in all the cases that will be examined in the second part of the Thesis. We finally illustrate the main free-fermionic techniques that will be used to perform the analytical as well as numerical lattice calculations. In Chap. 2, we consider a
1-dimensional system of free fermions hopping on a lattice, in which the conserved
U(1) charge is the total particle number. We evaluate the symmetry resolved R'enyi and entanglement entropies determining exactly all the non-universal constants and logarithmic corrections to the scaling, thanks to the so-called Fisher-Hartwig conjecture. In Chap. 3, we study the symmetry resolved entanglement entropies in
1-dimensional systems in presence of boundaries, considering the cases of an interval starting from the boundary and disconnected from it. We evaluate the charged and symmetry resolved entropies for conformal invariant theories and a semi-infinite chain of free fermions, finding entanglement equipartition. Using a generalization of the lattice calculations performed in Chap. 2, we also evaluate the first term breaking equipartition for the free fermionic chain. In Chap. 4, we study the symmetry resolution of the entanglement negativity in free fermionic systems, which admits a decomposition in terms of the charge imbalance between the two subsystems. We use conformal field theory to derive the universal behavior of the charge imbalance resolved negativity for a free Dirac field at finite temperature and size. For the lattice version of the Dirac filed, we also determine exactly the model-dependent terms using the Fisher-Hartwig conjecture. We find that at leading order the charge imbalance resolved negativity verifies equipartition. For the lattice model, we also evaluate the first corrections to this equipartition. In Chap. 5, we study the time evolution of the symmetry resolved entanglement entropy and a suitably defined symmetry resolved mutual information in a system of free fermions after a quench. In the context of free fermions and conformal field theories, we are able to provide analytical results for the time-dependent charged and resolved entropies, and shed light on two new physical phenomena. First, the symmetry resolved R'enyi entropies start evolving with a calculable time delay that depends on the charge sector. Second, there is an effective equipartition of the entanglement in the limit of large subsystem size. For free fermions, we also find that effective equipartition is verified by the mutual information. We expect that these phenomena occur in more general quantum systems, as they can be understood from the quasiparticle picture for the spreading of entanglement. In the Appendix, we give an overview of conformal field theory, reporting the main concepts and formulas that are needed for a more complete understanding of the previous Chapters.