Comprehensive study of Entanglement Entropy in Quantum Field Theory: Analysis of Conformal Field Theory to Massive Field Extensions and Holographic Entanglement
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
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
Quantum systems display counterintuitive features, escaping our everyday-life intuition, whose origin is related to entanglement. This phenomenon arises when different parts of the same systems display correlations, which cannot be understood via classical mechanics, and they are dubbed as quantum. To characterize the correlations above, it is necessary to find good entanglement measures which are able to capture quantitatively the quantum features of a given system. This motivates the investigation and the computation of quantities such as the von Neumann entropy of regions, which provides information about how a system is entangled. Our focus is quantum one-dimensional systems close to their critical point, where quantum correlations are enhanced, and a characterization via Quantum Field Theory (QFT) can be provided. In particular, we aim to understand the role of symmetries and localized defects on the entanglement properties. Those aspects display an intimate connection. First, via replica trick, the entropy is expressed via a replica permutation symmetry. Then, single defects break symmetries that are present in the corresponding homogeneous systems. Due to these reasons, the characterization of symmetry actions on subsystems can be viewed as a unifying framework for quantifying entanglement. We split this thesis into two parts. The first part is devoted to the symmetry-resolved entanglement. We first compute it in massive integrable field theories, giving analytical predictions for free fermions/bosons and the 3-state Potts model. For instance, we characterize the universal corrections to area law via form factor bootstrap. We also study excited states of field theories, and we characterize both low-energy states of Conformal Field Theory (CFT) and quasi-particle states. The second part deals with quantum correlations across defects. We provide a CFT framework to compute Rényi entropy and negativity in the ground state of conformal junctions, giving analytical predictions for free fermions and bosons. We find logarithmic growth of the entanglement measures above, and we relate the universal prefactor to the boundary conditions, encoded in a boundary scattering matrix. Then, we focus on the entanglement dynamics, and we study the time evolution of correlations for two paradigmatic classes of quench protocols. For a global quench, we find a linear growth of entropy across the defect whose slope is computed both in CFT, for thermal states, and for free fermions, via a modification of the quasi-particle picture. For a domain wall melting protocol, also known as inhomogeneous quench, we find that, while local observables are easily understood via hydrodynamics and local generalized Gibbs ensembles, the same is not true for the entropy. In contrast, by employing the quasi-particle picture, we provide precise predictions for the dynamics of entanglement, revealing the influence of long-range correlations between points that are specular with respect to the defect.
Entanglement entropy (EE) is a pivotal quantity in quantum field theory (QFT) that captures the nonlocal correlations between subsystems, offering deep insights into the quantum structure of spacetime, field dynamics, and phase transitions. This work derives EE in QFT using the covariance matrix formalism, establishing its foundations through the von Neumann entropy of reduced density matrices. We analytically compute EE for coupled harmonic oscillators as a prototypical model, extending the results to quantum fields via perturbative expansions. In the context of Maxwell QFT, we analyze entropy corrections induced by external perturbations, revealing their fractal dynamics through the emergence of Julia sets. The study further incorporates noncommutative geometry, where the deformation of spacetime modifies the covariance matrix, field strength tensors, and modular indices. Numerical simulations validate the derived scaling laws, entropy corrections, and noncommutative effects. These findings provide a mathematically rigorous framework to explore the interplay between EE, field theory, and geometric structures in high-energy physics.
A bstract We consider linear superpositions of single particle excitations in a scalar field theory on AdS 3 and evaluate their contribution to the bulk entanglement entropy across the Ryu-Takayanagi surface. We compare the entanglement entropy of these excitations obtained using the Faulkner-Lewkowycz-Maldacena formula to the entanglement entropy of linear superposition of global descendants of a conformal primary in a large c CFT obtained using the replica trick. We show that the closed form expressions for the entanglement entropy in the small interval expansion both in gravity and the CFT precisely agree. The agreement serves as a non-trivial check of the FLM formula for the quantum corrections to holographic entanglement entropy as well as the methods developed in the CFT to evaluate entanglement entropy of descendants. Our checks includes an example in which the state is time dependent and spatially in-homogenous as well another example involving a coherent state with a Bañados geometry as its holographic dual.
In these notes we give a pedagogical account of the replica trick derivation of CFT entanglement and its holographic counterpart, i.e. the Lewkowycz Maldacena derivation of the Ryu-Takayanagi formula. The application to an ‘island set-up’ for the calculation of black hole radiation entropy is briefly discussed. Further topics focused on are the relation to thermal entropy, thermofield double constructions and statements about the emergence of gravity from entanglement through reinterpretations of gravitational first laws.
This note presents a purely geometric construction of the so-called twist-field correlation functions in Conformal Field Theory (CFT), derived from conical singularities. This approach provides a purely mathematical interpretation of the seminal results in physics by Cardy and Calabrese on the entanglement entropy of quantum systems. Specifically, we begin by defining CFT partition functions on surfaces with conical singularities, using a ``Cauchy-Hadamard renormalization'' of the Polyakov anomaly integral. Next, we demonstrate that for a branched cover f:Σd→Σ with d sheets, where the cover inherits the pullback of a smooth metric from the base, a specific ratio of partition functions on the cover to the base transforms under conformal changes of the base metric in the same way as a correlation function of CFT primary fields with specific conformal weights. We also provide a discussion of the physical background and motivation for entanglement entropy, focusing on path integrals and the replica trick, which serves as an introduction to these ideas for a mathematical audience.
Abstract We compute the entanglement entropy in a two dimensional conformal field theory at finite size and finite temperature in the large central charge limit via the replica trick. We first generalize the known monodromy method for the calculation of conformal blocks on the plane to the torus. Then, we derive a monodromy method for the zero-point conformal blocks of the replica partition function. We explain the differences between the two monodromy methods before applying them to the calculation of the entanglement entropy. We find that the contribution of the vacuum exchange dominates the entanglement entropy for a large class of CFTs, leading to universal results in agreement with holographic predictions from the RT formula. Moreover, we determine in which regime the replica partition function agrees with a correlation function of local twist operators on the torus.