The connectivity index, defined as the number of decoupled components of a separable quantum system, can change under deformations of the Hamiltonian or during the dynamical change of the system under renormalization group flow. Such changes signal a rearrangement of correlations of different degrees of freedom across spacetime and field theory space. In this paper we quantify such processes by studying the behavior of entanglement entropy in a specific example: the RG flow in the Coulomb branch of large-N superconformal field theories. We find evidence that the transition from the non-separable phase of the Higgsed gauge theory in the UV to the separable phase of deformed decoupled CFTs in the IR exhibits sharp features in the middle of the RG flow in the large-N limit. The entanglement entropy on a sphere with radius ℓ exhibits the formation of a separatrix on the co-dimension-two Ryu-Takayanagi surface in multi-centered brane geometries above a critical value of ℓ. We discuss how other measures of entanglement and separability based on the relative quantum entropy and quantum mutual information might detect such transitions between non-separable and separable phases and how they would help describe some of the key properties of the IR physics of such flows.
The connectivity index, defined as the number of decoupled components of a separable quantum system, can change under deformations of the Hamiltonian or during the dynamical change of the system under renormalization group flow. Such changes signal a rearrangement of correlations of different degrees of freedom across spacetime and field theory space. In this paper we quantify such processes by studying the behavior of entanglement entropy in a specific example: the RG flow in the Coulomb branch of large-N superconformal field theories. We find evidence that the transition from the non-separable phase of the Higgsed gauge theory in the UV to the separable phase of deformed decoupled CFTs in the IR exhibits sharp features in the middle of the RG flow in the large-N limit. The entanglement entropy on a sphere with radius ℓ exhibits the formation of a separatrix on the co-dimension-two Ryu-Takayanagi surface in multi-centered brane geometries above a critical value of ℓ. We discuss how other measures of entanglement and separability based on the relative quantum entropy and quantum mutual information might detect such transitions between non-separable and separable phases and how they would help describe some of the key properties of the IR physics of such flows.
Abstract: The connectivity index, defined as the number of decoupled components of a quantum system, can change under deformations of the Hamiltonian or during the dynamical change of the system under renormalization group flow. Such changes signal a rearrangement of correlations of different degrees of freedom across spacetime and field theory space. In this paper we quantify such processes by studying the behavior of entanglement entropy, relative quantum entropy and quantum mutual information in a specific example: the RG flow in the Coulomb branch of large-N superconformal field theories. We argue that in this context there is an interesting sharp large-N transition in the middle of the RG flow from a non-separable phase of the Higgsed UV gauge theory to a separable phase of deformed decoupled CFTs in the IR. The entanglement entropy on a sphere with radius ` detects this transition via the formation of a separatrix on the co-dimension-two Ryu-Takayanagi surface in multi-centered brane geometries above a critical value of `. Other measures of entanglement and separability based on the relative quantum entropy detect a transition to a phase where they vanish identically. From the IR point of view the effect is closely related to the resummation of an infinite set of irrelevant multi-trace interactions.
A bstract Entanglement entropy has proven to be a powerful tool for probing renormalization group (RG) flows in quantum field theories, with c-functions derived from it serving as candidate measures of the effective number of degrees of freedom. While the monotonicity of such c-functions is well established in many settings, notable exceptions occur in theories with a mass scale. In this work, we investigate entanglement c-functions in the context of holographic RG flows, with a particular focus on flows across dimensions induced by circle compactifications. We argue that in spacetime dimensions d ≥ 4, standard constructions of c-functions, which rely on higher derivatives of the entanglement entropy of either a ball or a cylinder, generically lead to non-monotonic behavior. Working with known dual geometries, we argue that the non-monotonicity stems not from any pathology or curvature singularity, but from a transition in the holographic Ryu-Takayanagi surface. In compactifications from four to three dimensions, we propose a modified construction that restores monotonicity in the infrared, although a fully monotonic ultraviolet extension remains elusive. Furthermore, motivated by entanglement entropy inequalities, we conjecture a bound on the cylinder entanglement c-function, which holds in all our examples.
Abstract We quantitatively address the following question: for a QFT which is partially compactified, so as to realize an RG flow from a D-dimensional CFT in the UV to a d-dimensional CFT in the IR, how does the entanglement entropy of a small spherical region probing the UV physics evolve as the size of the region grows to increasingly probe IR physics? This entails a generalization of spherical regions to setups without full Lorentz symmetry, and we study the associated entanglement entropies holographically. We find a tight interplay between the topology and geometry of the compact space and the evolution of the entanglement entropy, with universal transitions from ‘cap’ through ‘bridge’ and ‘corner’ phases, whose features reflect the details of the compact space. As concrete examples we discuss twisted compactifications of 4d N N = 4 SYM on T2, S2 and hyperbolic Riemann surfaces.
Entanglement is a fundamental feature of quantum mechanics and plays an important role in characterizing many-body systems and quantum phase transitions. A quantitative measure of quantum correlations between subsystems is entanglement entropy. In this thesis, we study the entanglement entropy of an interacting quantum field theory, the O(N) model. In local quantum field theories, the leading-order contribution in the scaling of entanglement entropy obeys an area law. This dominant term is governed by short-distance physics near the entangling surface and depends on the ultraviolet regularization, making it non-universal. Beyond this lead- ing scaling behavior, entanglement entropy can contain subleading universal contributions, which encode intrinsic information about the underlying quantum field theory. In this thesis, we calculate one such universal term: the correlation length coefficient r, which appears as a correction to the entanglement entropy evaluated slightly away from the quantum critical point. The analysis is carried out in D = 2 + 1 spacetime dimensions for an infinite cylindrical geometry. We study the model in the vicinity of criticality using the large-N expansion, which enables controlled analytical calculations. Our main result is that the coefficient r scales linearly with the number of field components N . This behavior is consistent with the interpretation of r as a universal quantity associated with the degrees of freedom of the theory. The calculation is based on the replica method and renormalization group arguments, which are presented in detail. Finally, we discuss the physical meaning of the result and possible methods for verifying it, and we review related results for the O(N) model in different geometries.
Entanglement entropy has proven to be a powerful tool for probing renormalization group (RG) flows in quantum field theories, with c-functions derived from it serving as candidate measures of the effective number of degrees of freedom. While the monotonicity of such c-functions is well established in many settings, notable exceptions occur in theories with a mass scale. In this work, we investigate entanglement c-functions in the context of holographic RG flows, with a particular focus on flows across dimensions induced by circle compactifications. We argue that in spacetime dimensions d≥4, standard constructions of c-functions, which rely on higher derivatives of the entanglement entropy of either a ball or a cylinder, generically lead to non-monotonic behavior. Working with known dual geometries, we argue that the non-monotonicity stems not from any pathology or curvature singularity, but from a transition in the holographic Ryu--Takayanagi surface. In compactifications from four to three dimensions, we propose a modified construction that restores monotonicity in the infrared, although a fully monotonic ultraviolet extension remains elusive. Furthermore, motivated by entanglement entropy inequalities, we conjecture a bound on the cylinder entanglement c-function, which holds in all our examples.
We study entanglement transitions for pairs of disjoint intervals in two-dimensional holographic interface conformal field theories. We consider several holographic models with equal central charges on the two sides of the interface: thin-wall geometries with one and two branes, the Janus solution, and the super-Janus solution. We first discuss the finite interface contribution to the entanglement entropy of a single interval, distinguishing intervals that cross the interface from those that lie entirely on one side. We then use the mutual information to locate the transition between connected and disconnected Ryu--Takayanagi geodesic configurations for two intervals. When the interface lies between the two intervals, increasing the boundary entropy logg, which provides a measure of interface strength in the models considered here, generally decreases the critical separation at which the connected configuration ceases to dominate. When both intervals lie on the same side of the interface, the behavior is model dependent: the connected phase persists to larger separations in the Janus and super-Janus models, while the critical separation is unchanged in the thin-wall models. Thus, as diagnosed by the two-interval transition, the interface weakens correlations across the interface while preserving or enhancing correlations between regions on the same side. We also identify a universal ``Silver Blaze'' configuration, in which the interface lies inside one of the intervals and the transition occurs at the same location as in a CFT without an interface, independently of the interface parameters and of the holographic model. This universality follows from a cancellation between crossing and non-crossing interface contributions to the mutual information.