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Perturbative Structure in Entanglement, Gauge Theory, and Holography

Allic Sivaramakrishnan

Published Nov 6, 2017• 0 citations
OpenAlex

Abstract

We explore aspects of perturbative quantum field theory towards the goal of illuminating features of quantum gravity. In Chapter 1, we begin with the color-kinematic duality, a surprising relationship between gauge theoretic and kinematic properties of scattering amplitudes that, via the double-copy property, leads to a deep connection between gauge theory and gravity. We investigate the relationship between the color-kinematic duality and amplitude relations in ABJM theory and find that the duality is satisfied eight points without associated amplitude relations present in all other known instances of the duality. In Chapter 2, we derive a basic consistency check for AdS3/CFT2, a tractable example of the well-known AdS/CFT duality. We show that, under certain mild assumptions on the light spectrum of the CFT, CFT correlators of light operators match those computed in AdS perturbatively in 1/N: in a black hole background for high temperatures, and in thermal AdS for low temperatures. Next we turn to entanglement entropy, an information-theoretic quantity that in the CFT may encode dual AdS geometry. Time-dependent entanglement entropy has been studied for generic excited states. However, localized unitary operators in particular are in correspondence with Hamiltonian perturbations, and are basic building blocks of local excitations. In Chapter 3, we detail these operators' locality properties as well as the differences between excited states created by Hermitian operators and those created with localized unitary operators. In Chapter 4, we initiate the perturbative exploration of entanglement entropy with a time-dependent Hamiltonian, computing past first order in perturbation theory. We find a universal structure of entanglement propagation: interactions entangle unentangled excitations according to entanglement diagrams that track the flow of entanglement. We provide diagrammatic tools to simplify computations of loop entanglement diagrams.

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We explore aspects of perturbative quantum field theory towards the goal of illuminating features of quantum gravity. In Chapter 1, we begin with the color-kinematic duality, a surprising relationship between gauge theoretic and kinematic properties of scattering amplitudes that, via the double-copy property, leads to a deep connection between gauge theory and gravity. We investigate the relationship between the color-kinematic duality and amplitude relations in ABJM theory and find that the duality is satisfied eight points without associated amplitude relations present in all other known instances of the duality. In Chapter 2, we derive a basic consistency check for AdS3/CFT2, a tractable example of the well-known AdS/CFT duality. We show that, under certain mild assumptions on the light spectrum of the CFT, CFT correlators of light operators match those computed in AdS perturbatively in 1/N: in a black hole background for high temperatures, and in thermal AdS for low temperatures. Next we turn to entanglement entropy, an information-theoretic quantity that in the CFT may encode dual AdS geometry. Time-dependent entanglement entropy has been studied for generic excited states. However, localized unitary operators in particular are in correspondence with Hamiltonian perturbations, and are basic building blocks of local excitations. In Chapter 3, we detail these operators' locality properties as well as the differences between excited states created by Hermitian operators and those created with localized unitary operators. In Chapter 4, we initiate the perturbative exploration of entanglement entropy with a time-dependent Hamiltonian, computing past first order in perturbation theory. We find a universal structure of entanglement propagation: interactions entangle unentangled excitations according to entanglement diagrams that track the flow of entanglement. We provide diagrammatic tools to simplify computations of loop entanglement diagrams.
The time evolution of entanglement tracks how information propagates in interacting quantum systems. We study entanglement entropy in CFT2{\mathrm{CFT}}_{2}CFT2​ with a time-dependent Hamiltonian. We perturb by operators with time-dependent source functions and use the replica trick to calculate higher-order corrections to entanglement entropy. At first order, we compute the correction due to a metric perturbation in AdS3/CFT2{\mathrm{AdS}}_{3}/{\mathrm{CFT}}_{2}AdS3​/CFT2​ and find agreement on both sides of the duality. Past first order, we find evidence of a universal structure of entanglement propagation to all orders. The central feature is that interactions entangle unentangled excitations. Entanglement propagates according to ``entanglement diagrams,'' proposed structures that are motivated by accessory spacetime diagrams for real-time perturbation theory. To illustrate the mechanisms involved, we compute higher-order corrections to free fermion entanglement entropy. We identify an unentangled operator, one which does not change the entanglement entropy to any order. Then, we introduce an interaction and find it changes entanglement entropy by entangling the unentangled excitations. The entanglement propagates in line with our conjecture. We compute several entanglement diagrams. We provide tools to simplify the computation of loop entanglement diagrams, which probe UV effects in entanglement propagation in CFT and holography.
In this thesis we apply techniques from quantum information theory to study quantum gravity within the framework of the anti-de Sitter / conformal field theory correspondence (AdS/CFT). Through AdS/CFT, progress has been made in understanding the structure of entanglement in quantum field theories, and in how gravitational physics can emerge from these structures. However, this understanding is far from complete and will require the development of new tools to quantify correlations in CFT. This thesis presents refinements of a duality between operator product expansion (OPE) blocks in the CFT, giving the contribution of a conformal family to the OPE, and geodesic integrated fields in AdS which are diffeomorphism invariant quantities. This duality was originally discovered in the maximally symmetric setting of pure AdS dual to the CFT ground state. In less symmetric states the duality must be modified. Working with excited states within AdS3_33​/CFT2_22​, this thesis shows how the OPE block decomposes into more fine-grained CFT observables that are dual to AdS fields integrated over non-minimal geodesics. Additionally, this thesis contains results on the dynamics of entanglement measures for general quantum systems. Results are presented for the family of quantum Rényi entropies and entanglement negativity. Rényi entropies are studied for general dynamics by imposing special initial conditions. Around pure, separable initial states, all Rényi entropies grow with the same timescale at leading, and next-to-leading order. Mathematical tools are developed for the differentiation of non-analytic matrix functions with respect to constrained arguments and are used to construct analytic expressions for derivatives of negativity. We establish bounds on the rate of change of state distinguishability and the rate of entanglement growth for closed systems. Note: Abstract shortened.
It was found, by studying black holes, that the compatibility of the theory of gravity and the laws of quantum mechanics demands that the universe must act like a hologram. That is, all the information inside a region of the universe should be encoded on a so-called holographic screen of one-lesser dimension. This is termed the holographic principle. In a generic spacetime, such as an expanding universe, holographic screens depend on the choice of an observer, which is consistent with the notion that the observer is part of the system in cosmology. In the first part of this dissertation, we study the observer-dependence of holographic screens. This will help us understand how the fundamental description of the universe depends on the choice of the observer. Furthermore, we study the dynamics of the holographic screens and their geometry in the first part of this dissertation.An example where the holographic principle is manifest is the AdS-CFT correspondence. This correspondence connects quantum information quantities, like entanglement and complexity, to geometric quantities, like area and volume, respectively. In the second and the third part of this dissertation, we use the AdS-CFT correspondence as a tool to calculate the entanglement entropy and the computational complexity of a quenched CFT state. Studying the time evolution of the entanglement entropy following a quantum quench teaches us how a CFT state thermalizes. On the other hand, studying the time evolution of the computational complexity allows us to check the validity of various recent conjectures involving black holes and complexity.
Entanglement entropy of holographic CFTs is expected to play a crucial role in the reconstruction of semiclassical bulk gravity. We consider the entanglement entropy of spherical regions of vacuum, which is known to contain universal contributions. After perturbing the CFT with a relevant scalar operator, also the first order change of this quantity gives a universal term which only depends on a discrete set of basic CFT parameters. We show that in gravity this statement corresponds to the uniqueness of the ghost-free graviton propagator on an AdS background geometry. While the gravitational dynamics in this context contains little information about the structure of the bulk theory, there is a discrete set of dimensionless parameters of the theory which determines the entanglement entropy. We argue that for every (not necessarily holographic) CFT, any reasonable gravity model can be used to compute this particular entanglement entropy. We elucidate how this statement is consistent with AdS/CFT and also give various generalizations. On the one hand this illustrates the remarkable usefulness of geometric concepts for understanding entanglement in general CFTs. On the other hand, it provides hints as to what entanglement data can be expected to provide enough information to distinguish, e.g., bulk theories with different higher curvature couplings.
The AdS/CFT correspondence, a remarkable duality between certain gravitational theories in anti-de Sitter (AdS) spacetime and quantum field theories with conformal symmetry (CFT), has had a profound effect on the development of theoretical physics in the past two decades. Recently, many connections of AdS/CFT to quantum information theory have been found, in particular by providing gravitationally dual descriptions of various entanglement measures. Understanding these manifestions of AdS/CFT --- or more generally, the conjectured holographic principle encompassing it --- requires the combination of tools from both high-energy theory and quantum information physics. In this cumulative thesis, the convergence between these two fields is approached from two fronts: First, by calculations within the dual gravitational theory, and second, using a tensor network ansatz to describe the quantum states suspected to possess such a gravitational description. In the first approach, using the gravitational side of AdS/CFT, entanglement entropies of complicated 2+1-dimensional excited CFTs are computed, thus showing how the holographic approach provides access to systems previously out of reach of practical methods, while introducing new numerical methods that this approach necessitates. The second approach is given by tensor networks, a highly successful ansatz for computing properties of one- and two-dimensional quantum systems. Efficiently computable classes of tensor networks are tested in their ability to represent simple holographic systems, successfully reproducing both hyperbolic geometrical features as well as critical boundary states. In addition, the general properties of tensor networks on regular hyperbolic tesselations are considered, leading to new connections to models not previously considered in the context of holography. This interplay of different approaches to quantum information holography showcases the richness of this new field and suggests that a wide range of physical phenomena is accessible via the new tools now at our disposal.
In the first part of this thesis we explore the entanglement structure of relativistic field theories in momentum space. We discuss a Wilsonian path integral formulation and a perturbative approach. Using perturbation theory we obtain results for specific quantum field theories. These are understood through scaling and decoupling properties of field theories. Convergence of the perturbation theory taking loop diagrams into account is also discussed. We then discuss the entanglement structure in systems where Lorentz invariance is broken by a Fermi surface. The Fermi surface helps the convergence of perturbation theory and entanglement of modes near the Fermi surface is shown to be amplified, even in the presence of a large momentum cutoff. In the second part of this thesis we explore the connection between entanglement and gravity in the context of the AdS/CFT correspondence. We show that there are certain thermodynamic-like relations common to all conformal field theories, which when mapped via the AdS/CFT correspondence to the bulk are tantamount to Einstein's equations, to lowest order in the metric.