Studies in: entanglement entropy of two dimensional quasi-topological quantum field theory and geometry of the exact renormalization group and higher spin holography
Part I: Entanglement Entropy of 2d Quasi-Topological Quantum Field Theory We compute the entanglement entropy of two-dimensional quasi-topological quantum field the- ories (QTFTs). These are theories in which the correlation functions depend on the topology and on the total area of the underlying space-time, but are blind to all local details of the geometry, and include topological quantum field theory (TFT) as their limiting case. We use two complimentary methods to compute the entanglement entropy; the first method is the replica trick and the other is to devise a novel tensor network representation, or more precisely, matrix product state (MPS) representation, of the quantum states of QTFT. We demonstrate that the two calculations are in agreement. Part II: Geometry of the Exact Renormalization Group and Higher Spin Holography We consider the Wilson-Polchinski exact renormalization group (RG) applied to the generating functional of single-trace operators at a free-fixed point in d = 2 + 1 dimensions. By exploiting the rich symmetry structure of free-field theory, we study the geometric nature of the RG equations and the associated Ward identities. The geometry, as expected, is holographic, with anti-de Sitter spacetime emerging correspondent with RG fixed points. In particular, we are able to cast the renormalization group equations as Hamilton equations of radial evolution in AdSd+1. We solve these bulk equations of motion in terms of a boundary source and derive an on-shell bulk action. We demonstrate that it correctly encodes all of the correlation functions of the field theory, written as “Witten diagrams.” Going further, we show that the field theory construction gives us a par- ticular vector bundle over the (d + 1)-dimensional RG mapping space, called a jet bundle, whose structure group arises from the bilocal transformations of the bare fields in the path integral. The sources for quadratic operators constitute a connection on this bundle and a section of its endomor- phism bundle. We make comparisons to Vasiliev-type higher spin theories. Detailed calculations are carried out for the case of Majorana fermions. Results and comments are presented for complex scalars. Additional details can be found in [1, 2].
Part I: Entanglement Entropy of 2d Quasi-Topological Quantum Field Theory We compute the entanglement entropy of two-dimensional quasi-topological quantum field the- ories (QTFTs). These are theories in which the correlation functions depend on the topology and on the total area of the underlying space-time, but are blind to all local details of the geometry, and include topological quantum field theory (TFT) as their limiting case. We use two complimentary methods to compute the entanglement entropy; the first method is the replica trick and the other is to devise a novel tensor network representation, or more precisely, matrix product state (MPS) representation, of the quantum states of QTFT. We demonstrate that the two calculations are in agreement. Part II: Geometry of the Exact Renormalization Group and Higher Spin Holography We consider the Wilson-Polchinski exact renormalization group (RG) applied to the generating functional of single-trace operators at a free-fixed point in d = 2 + 1 dimensions. By exploiting the rich symmetry structure of free-field theory, we study the geometric nature of the RG equations and the associated Ward identities. The geometry, as expected, is holographic, with anti-de Sitter spacetime emerging correspondent with RG fixed points. In particular, we are able to cast the renormalization group equations as Hamilton equations of radial evolution in AdSd+1. We solve these bulk equations of motion in terms of a boundary source and derive an on-shell bulk action. We demonstrate that it correctly encodes all of the correlation functions of the field theory, written as “Witten diagrams.” Going further, we show that the field theory construction gives us a par- ticular vector bundle over the (d + 1)-dimensional RG mapping space, called a jet bundle, whose structure group arises from the bilocal transformations of the bare fields in the path integral. The sources for quadratic operators constitute a connection on this bundle and a section of its endomor- phism bundle. We make comparisons to Vasiliev-type higher spin theories. Detailed calculations are carried out for the case of Majorana fermions. Results and comments are presented for complex scalars. Additional details can be found in [1, 2].
A holographic correspondence has been recently developed between higher spin theories in three-dimensional anti-de Sitter space (AdS3) and two-dimensional Conformal Field Theories (CFTs) with extended symmetries. A class of such dualities involves SL(N, R) × SL(N, R) Chern-Simons gauge theories in the (2 + 1)-dimensional bulk space-time, and CFTs with WN symmetry algebras on the (1 + 1)-dimensional boundary. The topological character of the bulk theory forces one to reconsider standard geometric notions such as black hole horizons and entropy, as well as the usual holographic dictionary. Motivated by this challenge, in this note we present a proposal to compute entanglement entropy in the WN CFTs via holographic methods. In particular, we introduce a functional constructed from Wilson lines in the bulk Chern-Simons theory that captures the entanglement entropy in the CFTs dual to standard AdS3 gravity, corresponding to SL(2, R) × SL(2, R) gauge group, and admits an immediate generalization to the higher spin case. We explicitly evaluate this functional for several known solutions of the bulk theory, including charged black holes dual to thermal CFT states carrying higher spin charge, and show that it reproduces expected features of entanglement entropy, study whether it obeys strong subadditivity, and moreover show that it reduces to the thermal entropy in the appropriate limit.
We further develop perturbative methods used to calculate entanglement entropy (EE) away from an interacting CFT fixed point. At second order we find certain universal terms in the renormalized EE which were predicted previously from holography and which we find hold universally for relevant deformations of any CFT in any dimension. We use both replica methods and direct methods to calculate the EE and in both cases find a non-local integral expression involving the CFT two point function. We show that this integral expression can be written as a local integral over a higher dimensional bulk modular hamiltonian in an emergent AdS space-time. This bulk modular hamiltonian is associated to an emergent scalar field dual to the deforming operator. We generalize to arbitrary spatially dependent couplings where a linearized metric emerges naturally as a way of efficiently encoding the field theory entanglement: by demanding that Einstein’s equations coupled to the bulk scalar field are satisfied, we show that EE can be calculated as the area of this metric. Not only does this show a direct emergence of a higher dimensional gravitational theory from any CFT, it allows for effective evaluation of the the integrals required to calculate EE perturbativly. Our results can also be interpreted as relating the non-locality of the modular hamiltonian for a spherical region in non-CFTs and the non-locality of the holographic bulk to boundary map.
We show that the path integral of conformal field theories in D dimensions ( CFT D ) can be constructed by solving for eigenstates of a renormalization group (RG) operator following from the Turaev-Viro formulation of a topological field theory (topological quantum field theory) (TQFT) in D + 1 dimensions ( TQFT D + 1 ), explicitly realizing the holographic sandwich relation between a symmetric theory and a TQFT. Generically, exact eigenstates corresponding to symmetric TQFT D follow from Frobenius algebra in TQFT D + 1 . For D = 2 , we construct eigenstates that produce 2D rational CFT path integrals exactly, which curiously connect a continuous-field theoretic path integral with the Turaev-Viro state sum. We also devise and illustrate numerical methods for D = 2 , 3 to search for CFT D as phase transition points between symmetric TQFT D . Finally, since the RG operator is in fact an exact analytic holographic tensor network, we compute “bulk-boundary” correlators and compare them with the AdS/CFT dictionary at D = 2 . Promisingly, they are numerically compatible given our accuracy, although further works will be needed to explore the precise connection to the AdS/CFT correspondence. Published by the American Physical Society 2024
In this thesis we will explore entanglement of various subsystems in quantum field theory, and its uses for describing underlying structure of states. We begin this journey by focusing on topological quantum field theory in chapters 1 and 2. These chapters are based heavily upon the papers [1] and [2, 3], respectively. In the introduction to chapter 1 we take the opportunity to recall the definition of bipartite entanglement entropy in quantum field theory and describe, briefly, how the entropy associated to connected spatial subregions provides a signature of topological order at the level of the wave-function. In particular, for a gapped phase with topological order, we recapitulate how the sub-leading correction to the “area-law” of the entanglement entropy contains coarse information about the topological phase while also being insensitive to the short-distance details of the underlying theory. We then present results that show that this sub-leading correction can also detect the class of interactions that glue a subregion to its complement. In principle, the subregion could be a different topological phase than its complement, and so we say that the entanglement entropy provides a signature of the interface between topological phases. In showing this, we discuss the classes of interactions between gapped topological phases and how they are represented as boundary conditions in the low-energy effective topological field theory, which in this case is an Abelian Chern-Simons theory. Using these boundary conditions, we perform path-integral calculations of the entanglement entropy using the replica trick and show that the universal contribution does indeed depend on the class of boundary conditions, and the gapped topological phases of the subsystem and its complement. To finish this chapter we reformulate this problem at the level of the Hilbert space and show how these universal contributions are related to resolving a generic ambiguity in defining subsystems of field theories with gauge invariance. We continue in the theme of topological field theories for chapter 2. Here we will shift focus from subregions of a connected spatial slice to disconnected spatial regions in an effort to explore, firstly, how long-range entanglement is encoded in topological field theory, and secondly, how that entanglement is encoded amongst multiple parties. We begin with a brief introduction of multi-party entanglement and recall the classification of three-party entanglement. Moving on to our specific setup, we describe the construction of states on multiple non-intersecting torus boundaries. Generically, these are states on link complements where the torus boundaries can be thought of as the edges of thickened closed strings entwined and knotted inside a compact manifold, which we will take to be the three-sphere. The wave-functions of these states are represented by various link invariants (such as the Gauss linking number for Abelian links, and Jones polynomial for SU(2)). We will exhibit several examples of multi-component links in both the Abelian and the SU(2) theories, and afterwards discuss generic features of the entanglement of link states in Chern-Simons theories with a compact group. In this discussion, we will conjecture a classification (albeit broad) of links via multi-partite entanglement of their corresponding states. We will end the chapter moving away from compact groups and discuss hyperbolic links in SL(2,C) Chern-Simons theory in the semi-classical limit. In particular, utilizing the generalized volume conjecture, we will show that the entanglement structure of hyperbolic links is greatly constrained in this semi-classical limit. Finally, in chapter 3 we switch gears and discuss renormalization of free bosonic field theory. The content of this chapter is drawn from the work of [4]. Motivated by work establishing duality between the singlet sector of O(N) vector models at large N and higher spin gauge theory on Anti de Sitter (AdS) space, we develop an exact renormalization group (ERG) framework for discussing the renormalization of a large class of O(N) singlet states. We will show that for the ground state and the excited states in this class the ERG is implemented by the action of unitary operators (that can be taken to be local). Consequently, the ERG framework can be regarded as a continuum tensor network. We contrast this tensor network with some well known tensor network renormalization schemes such as the muti-scale entanglement renormalization ansatz (MERA) and its proposed continuum version (cMERA). We also comment on the nature of this network with ordinary Wilsonian renormalization. One of the central features of the ERG network is how it acts on the momentum space entanglement of the field theory. In particular, we argue that for excited states, the ERG disentangles modes that lie just above and below the UV cutoff.
We explore interfaces and junctions joining multiple two-dimensional conformal field theories, with the goal of calculating entanglement entropies in their presence and exploring their holographic duals. In chapter 1 we start with an overview of the three subjects, collecting various well-known results and reviewing some foundational works.In chapter 2 we calculate the holographic entanglement entropy in the presence of a conformal interface for a geometric configuration in which the entangling region lies on one side of the interface. For the supersymmetric Janus solution we find exact agreement between the holographic and conformal field theory calculation of the entanglement entropy.In chapter 3 we calculate the entanglement entropy for topological interfaces in rational conformal field theories for the case where the interface lies at the boundary of the entangling interval and for the case where it is located in the center of the entangling interval. We compare the results to each other and also to the left/right entropy of a related boundary conformal field theory. We also comment on the entanglement entropies for topological interfaces in Liouville theory.In chapter 4 we consider entanglement through permeable junctions of N free boson and free fermion conformal field theories. We constrain the form of the general boundary state and calculate the replicated partition functions with interface operators inserted, from which the entanglement entropy is calculated. We find the functional form of the universal and constant terms to be similar to the N=2 case, depending only of the total transmission of the junction and the unit volume of the zero mode lattice. For N>2 we see a subleading divergent term which does not depend on the parameters of the junction. For N=3 we consider some specific geometries and discuss various limits.In chapter 5 we investigate topological interfaces between three-dimensional Abelian Chern-Simons theories in the context of the AdS_3/CFT_2 correspondence. We show that it is possible to connect the topological interfaces in the bulk Chern-Simons theory to topological interfaces in the dual conformal field theory on the boundary. In addition to the [U(1)]^{2N} Chern-Simons theory on AdS_3, we show that it is possible to find boundary counter terms which lead to the N conserved currents in the dual two-dimensional conformal field theory.
We develop a renormalization method for holographic entanglement entropy based on area renormalization of entangling surfaces. The renormalized entanglement en-tropy is derived for entangling surfaces in asymptotically locally anti-de Sitter spacetimes in general dimensions and for entangling surfaces in four dimensional holographic renor-malization group flows. The renormalized entanglement entropy for disk regions in AdS 4 spacetimes agrees precisely with the holographically renormalized action for AdS 4 with spherical slicing and hence with the F quantity, in accordance with the Casini-Huerta-Myers map. We present a generic class of holographic RG flows associated with deforma-tions by operators of dimension 3/2 < Δ < 5/2 for which the F quantity increases along the RG flow, hence violating the strong version of the F theorem. We conclude by explaining how the renormalized entanglement entropy can be derived directly from the renormalized partition function using the replica trick i.e. our renormalization method for the entangle-ment entropy is inherited directly from that of the partition function. We show explicitly how the entanglement entropy counterterms can be derived from the standard holographic renormalization counterterms for asymptotically locally anti-de Sitter spacetimes.