Abstract
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 $\ell$ exhibits the formation of a separatrix on the co-dimension-two Ryu-Takayanagi surface in multi-centered brane geometries above a critical value of $\ell$. 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.
Highlights
In a well studied example we take a system defined in p spatial dimensions and separate the degrees of freedom inside a spatial region A from the degrees of freedom in the complement Ac
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 renormalization group (RG) flow in the large-N limit
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
Summary
There are several common mechanisms in quantum field theory that change the connectivity index. In strongly coupled gauge theories an operator will frequently hit the unitarity bound and decouple from the remaining degrees of freedom as a free field.. The entanglement entropy of large-N conformal field theories with a weakly curved gravitational dual can be computed efficiently using the RyuTakayanagi prescription in the AdS/CFT correspondence. This computation, which will be performed in the four sections, involves the analysis of a minimal co-dimension-2 surface in multi-centered brane geometries in ten- or eleven-dimensional supergravities. The general discussion in the beginning of this section suggests that this increase is suppressed in the large-N limit at low energies because the effects of inter-system interactions mediated by multi-trace operators are suppressed. An efficient computational method for the relative quantum entropy would be helpful in addressing these issues, but goes beyond the immediate goals of this paper
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