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arXiv:1512.00791 · Journal of High Energy Physics

Entanglement and RG in the O(N ) vector model

Chris Akers, Omer Ben-Ami, Vladimir Rosenhaus, Michael Smolkin, Shimon Yankielowicz B

Published Mar 1, 2016• 14 citations
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Abstract

We consider the large N interacting vector O(N ) model on a sphere in 4 − ϵ Euclidean dimensions. The Gaussian theory in the UV is taken to be either conformally or non-conformally coupled. The endpoint of the RG flow corresponds to a conformally coupled scalar field at the Wilson-Fisher fixed point. We take a spherical entangling surface in de Sitter space and compute the entanglement entropy everywhere along the RG trajectory. In 4 dimensions, a free non-conformal scalar has a universal area term scaling with the logarithm of the UV cutoff. In 4 − ϵ dimensions, such a term scales as 1/ϵ. For a non-conformal scalar, a 1/ϵ term is present both at the UV fixed point, and its vicinity. For flow between two conformal fixed points, 1/ϵ terms are absent everywhere. Finally, we make contact with replica trick calculations. The conical singularity gives rise to boundary terms residing on the entangling surface, which are usually discarded. Consistency with our results requires they be kept. We argue that, in fact, this conclusion also follows from the work of Metlitski, Fuertes, and Sachdev, which demonstrated that such boundary terms will be generated through quantum corrections.

DOI
10.1007/jhep03(2016)002

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We consider the large N interacting vector O(N ) model on a sphere in 4 − ϵ Euclidean dimensions. The Gaussian theory in the UV is taken to be either conformally or non-conformally coupled. The endpoint of the RG flow corresponds to a conformally coupled scalar field at the Wilson-Fisher fixed point. We take a spherical entangling surface in de Sitter space and compute the entanglement entropy everywhere along the RG trajectory. In 4 dimensions, a free non-conformal scalar has a universal area term scaling with the logarithm of the UV cutoff. In 4 − ϵ dimensions, such a term scales as 1/ϵ. For a non-conformal scalar, a 1/ϵ term is present both at the UV fixed point, and its vicinity. For flow between two conformal fixed points, 1/ϵ terms are absent everywhere. Finally, we make contact with replica trick calculations. The conical singularity gives rise to boundary terms residing on the entangling surface, which are usually discarded. Consistency with our results requires they be kept. We argue that, in fact, this conclusion also follows from the work of Metlitski, Fuertes, and Sachdev, which demonstrated that such boundary terms will be generated through quantum corrections.
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We explore entanglement entropy of a cap-like region for a generic quantum field theory residing in the Bunch-Davies vacuum on de Sitter space. Entanglement entropy in our setup is identical with the thermal entropy in the static patch of de Sitter, and we derive a simple relation between the vacuum expectation value of the energy-momentum tensor trace and the RG flow of entanglement entropy. In particular, renormalization of the bare couplings and logarithmic divergence of the entanglement entropy are interrelated in our setup. We confirm our findings by recovering known universal contributions for a free field theory deformed by a mass operator as well as obtain correct universal behaviour at the fixed points. Simple examples of entanglement entropy flows are elaborated in d=2,3,4. Inthreedimensionswefindthatwhiletherenormalizedentanglemententropy is stationary at the fixed points, it is not monotonic. We provide a computational evidence that the universal ‘area law’ for a conformally coupled scalar is different from the known result in the literature, and argue that this difference survives in the limit of flat space. Finally, we carry out the spectral decomposition of entanglement entropy flow and discuss its application to the F-theorem.
We compute the entanglement entropy of the Wilson-Fisher conformal field theory (CFT) in 2+1 dimensions with O(N)\mathrm{O}(N)O(N) symmetry in the limit of large NNN for general entanglement geometries. We show that the leading large-NNN result can be obtained from the entanglement entropy of NNN Gaussian scalar fields with their mass determined by the geometry. For a few geometries, the universal part of the entanglement entropy of the Wilson-Fisher CFT equals that of a CFT of NNN massless scalar fields. However, in most cases, these CFTs have a distinct universal entanglement entropy even at $N=\ensuremath{\infty}$. Notably, for a semi-infinite cylindrical region it scales as N0{N}^{0}N0 in the Wilson-Fisher theory, in stark contrast to the NNN-linear result of the Gaussian fixed point.
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Following the ideas developed by Komargodski and Schwimmer in arXiv:1107.3987 and arXiv:1112.4538, we argue that the IR dilaton effective action on the cone computes the entanglement entropy of an even dimensional non-conformal field theory interpolating between a UV and an IR fixed point. We restrict our attention to theories which flow to trivial IR fixed point. We get exact non-perturbative results for the coefficients of the logarithmically divergent term of the entanglement entropy in these field theories in arbitrary even dimensions. The results match precisely with the weak coupling results available in the literature and also with the strong coupling results obtained via holography. We also write the universal terms for field theories which interpolate between a scale invariant but not conformally invariant UV fixed point and a trivial IR fixed point.