Entanglement entropy: Holography and renormalization group

Abstract

Entanglement entropy plays a variety of roles in quantum field theory, including the connections between quantum states and gravitation through the holographic principle. This article provides a review of entanglement entropy from a mixed viewpoint of field theory and holography. A set of basic methods for the computation is developed and illustrated with simple examples such as free theories and conformal field theories. The structures of the ultraviolet divergences and the universal parts are determined and compared with the holographic descriptions of entanglement entropy. The utility of quantum inequalities of entanglement are discussed and shown to derive the C-theorem that constrains renormalization group flows of quantum field theories in diverse dimensions. 42 4. Higher dimensions 42 C. The entropic c-theorem in (1 + 1) dimensions 43 D. The F -theorem in (2 + 1) dimensions 44 1. A sketch of the proof 44 2. Constraints from the F -theorem 45 3. Large mass expansion 45 4. Stationarity at UV fixed points 47 E. The entropic C-theorems in d ≥ 4 dimensions 48 F. Holographic RG flow 49 1. Domain wall and gapped RG flows 49 2. Topology change in a gapped phase 50 3. Testing the F -theorem in holography 51 G. Supersymmetric Rényi entropies 52 IX. Outlook 53 Acknowledgments 53 A. Real time formalism for fermions 53 1. Fermionic system on lattice 53 2. Free massive fermionic fields 54 References 55 arXiv:1801.10352v3 [hep-th] 25 Dec 2019 2We normalize the ground state as Ψ|Ψ = 1 so that trtot(ρtot) = 1.

Renormalized AdS gravity and holographic entanglement entropy of even-dimensional CFTs

“…Note that in the general case, for W (0) = 0, W (2) ij kl can be further simplified and rewritten in a more compact form. In particular, we have that 26) or equivalently,…”

“…For reviews on developments in these two directions, see e.g. [25,26] and [27][28][29], respectively. In more detail, given a state of a quantum system, and a subset A of its degrees of freedom, entanglement between A and its complement A c is quantified by the entanglement entropy S ≡ − Tr A ( A ln A ), with A ≡ Tr A c .…”

Entanglement entropy in generalised quantum Lifshitz models

“…One also finds a universal sub-leading term in odd-dimensional CFTs but in this case it is finite rather than logarithmic, see e.g. [10] and references therein. The coefficient of the universal term is a function of topological and geometric invariants, such as the Euler density and Weyl invariants constructed from the entangling (hyper)-surface [11,12].…”

“…In recent years considerable effort has been devoted to investigating such universal terms in the entanglement entropy of CFTs in arbitrary dimensions (see [9,10] for reviews). What will be important for us is the following general UV behavior of entanglement entropy in even D-dimensional CFTs,…”

Localized thermal states and negative energy

“…where T = 2πα/β, Vol(S d−1 ) = 2π d/2 /Γ(d/2), and a * d is the generalized central charge given by [76] 20) with A d the A-type trace anomaly of the stress tensor (see Ref. [77] for conventions) and Z(S d ) the regularized vacuum partition function of the CFT placed on a unit d-dimensional sphere (see Ref.…”