A refinement of entanglement entropy and the number of degrees of freedom

By Hong Liu, M. Mezei. 2012

331 citations recorded by the source corpus.

A bstractWe introduce a “renormalized entanglement entropy” which is intrinsically UV finite and is most sensitive to the degrees of freedom at the scale of the size R of the entangled region. We illustrated the power of this construction by showing that the qualitative behavior of the entanglement entropy for a non-Fermi liquid can be obtained by simple dimensional analysis. We argue that the functional dependence of the “renormalized entanglement entropy” on R can be interpreted as describing the renormalization group flow of the entanglement entropy with distance scale. The corresponding quantity for a spherical region in the vacuum, has some particularly interesting properties. For a conformal field theory, it reduces to the previously proposed central charge in all dimensions, and for a general quantum field theory, it interpolates between the central charges of the UV and IR fixed points as R is varied from zero to infinity. We conjecture that in three (spacetime) dimensions, it is always non-negative and monotonic, and provides a measure of the number of degrees of freedom of a system at scale R. In four dimensions, however, we find examples in which it is neither monotonic nor non-negative.

Subfields: Astronomy and Astrophysics, Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/0aaadeb9b1c60cfd9ca6918c26e088e76102cc36

How researchers see this paper

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In Astronomy and Astrophysics

1. T. Takayanagi
2. V. Hubeny
3. R. Myers
4. T. Nishioka
5. M. Rangamani
6. T. Faulkner
7. Z. Komargodski
8. T. Dumitrescu

In Atomic and Molecular Physics, and Optics

1. T. Takayanagi
2. T. Nishioka
3. R. Myers
4. M. Rangamani
5. V. Hubeny
6. T. Faulkner
7. H. Casini
8. Huajia Wang

In Nuclear and High Energy Physics

1. T. Takayanagi
2. T. Nishioka
3. R. Myers
4. V. Hubeny
5. M. Rangamani
6. T. Faulkner
7. J. Maldacena
8. Thomas Hartman

Citations in the corpus

• Comments on holographic entanglement entropy and RG flows (2012, methodology): “We also thank Hong Liu for sharing a draft of [43] with us before it appeared on the arXiv.”
• Extremal surfaces as bulk probes in AdS/CFT (2012, background): “in [37]; see also [48, 49] for more recent discussion).”
• Entanglement entropy from a holographic viewpoint (2012, background): “It has been suggested that pd can be used as a measure of degrees of freedom in the odd-dimensional CFT, where there are no conformal anomalies and central charges [59, 54, 63].”
• Interpolating between a and F (2014, background): “1 In the most physically interesting case d = 3, a proof of the F -theorem has been presented [14], relying on its exact relation with the entanglement entropy across a circle [15, 16].”
• Contact terms, unitarity, and F -maximization in three-dimensional superconformal theories (2012, background): “) It has been proposed [7-11] that a similar inequality should hold in three dimensions,”
• A one-loop test of quantum supergravity (2012, methodology): “A similar trick was used in [27] for extracting the universal part of entanglement entropy in quantum fleld theories.”
• On the RG running of the entanglement entropy of a circle (2012, background): “as the relevant c-function [11] (see also [8, 13]).”
• Bulk curves from boundary data in holography (2013, background): “The appearance of “corners” reflects a richer structure of divergences in the entanglement entropy in higher dimensions [46], which are more difficult to cancel out than the simple logarithmic divergence of eq.”
• Shape dependence of entanglement entropy in conformal field theories (2015, unknown): “3 Note that (19) is true (even for the UV divergent terms) if we use a “covariant” regulator to define EE [38–40].”
• On the architecture of spacetime geometry (2012, background): “In fact, a closer examination shows that each of these terms has a precise geometric interpretation involving an integration of various factors over the boundary Σ [7].”
• Mutual information and the F-theorem (2015, methodology): “With these new insights, we can also understand why the numerical results using a radial lattice yield the correct constant provided that the circle radius is defined as R = (n+ 1/2) δ with n being the number of lattice points inside the circle, and δ, the lattice spacing [34, 66, 67].”
• Circuit complexity in quantum field theory (2017, methodology): “(6.2) We therefore see that that generically, if ω0δ 1 or ωδ 1 (which were advocated to be the natural choices in section 4.1), the “renormalized” energy density of |ψR〉 diverges as δ → 0.43 Hence this would not be a state that would be considered to be part of the standard 42Of course, we are inspi”
• Causal holographic information (2012, background): “25 We thank Hong Liu for useful discussions on this issue; see also [35].”
• Modular Hamiltonians for deformed half-spaces and the averaged null energy condition (2016, background): “Any divergences should be local to the entangling surface, and assuming our regulator is geometric [72–74] no such term which respects the S(A) = S(A) purity condition can generate a divergence for first order spatial shape deformations.”