Entanglement entropy for singular surfaces in hyperscaling violating theories

Published for SISSA by Springer Received: July 27, 2015 Accepted: September 2, 2015 Published: September 24, 2015 Mohsen Alishahiha,a Amin Faraji Astaneh,b Piermarco Fondac and Farzad Omidid a School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran b School of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran c SISSA and INFN, via Bonomea 265, 34136, Trieste, Italy d School of Astronomy, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran E-mail: , , , Abstract: We study the holographic entanglement entropy for singular surfaces in theories described holographically by hyperscaling violating backgrounds. We consider singular surfaces consisting of cones or creases in diverse dimensions. The structure of UV divergences of entanglement entropy exhibits new logarithmic terms whose coefficients, being cut-off independent, could be used to define new central charges in the nearly smooth limit. We also show that there is a relation between these central charges and the one appearing in the two-point function of the energy-momentum tensor. Finally we examine how this relation is affected by considering higher-curvature terms in the gravitational action. Keywords: Gauge-gravity correspondence, AdS-CFT Correspondence, Holography and condensed matter physics (AdS/CMT) ArXiv ePrint: 1507.05897 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP09(2015)172 JHEP09(2015)172 Entanglement entropy for singular surfaces in hyperscaling violating theories Contents 1 2 Entanglement entropy for a higher dimensional cone 4 3 New divergences and universal terms 3.1 dθ = 1 3.2 dθ = 2 3.3 dθ = 3 3.4 dθ = 4 3.5 dθ = 5 9 11 11 12 12 13 4 New charge 14 5 Conclusions 17 A Backgrounds with a hyperscaling violating factor 18 B Explicit expressions for ϕ2i and a2i for i = 1, 2, 3 22 1 Introduction It is well known that the central charge of a two dimensional conformal field theory is an important quantity characterizing its behaviour: it is ubiquitous in many expressions such as the central extension of the Virasoro algebra, the two point function of the energymomentum tensor, in the Weyl anomaly and is the coefficient of the logarithmically divergent term in the entanglement entropy [1]. It also appears in the expression of Cardy’s formula for the entropy. Actually the corresponding central charge may be thought of as a measure of the number of degrees of freedom of the theory. Moreover Zamolodchikov’s c-theorem in two dimensions indicates that in any renormalization group flow connecting two fixed points, the central charge decreases along the flow, thus indicating that IR fixed points are characterized by fewer degrees of freedom. In higher dimensional conformal field theories the situation is completely different. First of all the conformal group in higher dimensions does not have a central extension and thus it is finite dimensional. Moreover the parameter which appears in the two-point function of the energy-momentum tensor is not generally related to the one multiplying the Euler density in the Weyl anomaly in even dimensional conformal field theories, 1 nor is 1 It was conjectured [2] that, in four dimensional space-times, the coefficient that multiplies the Euler density always decreases along RG flow and may naturally define an a-theorem in four dimensions. This conjecture has been proved in [3]. –1– JHEP09(2015)172 1 Introduction it directly related to the cut-off independent terms of the entanglement entropy computed for a smooth entangling region. Indeed if one computes entanglement entropy for a given smooth entangling region in a d + 1 dimensional conformal field theory, one finds [4, 5] [ d2 ]−1 SE = X i=0 A2i 1 H + δ2[ d ]+1,d A2[ d ] log + finite terms, d−2i−1 2 2 d − 2i − 1 ε ε (1.1) L + a(ϕ) log  + S0 (1.2)  where the cusp is specified by an angle defined such that ϕ = π/2 corresponds to a smooth line. Here L is the length of the boundary of the entangling region and S1 is a constant which depends on the UV cut off, while a(ϕ) and S0 are universal parameters. More recently based on early results [11–13] it was shown that “the ratio a(ϕ) CT , where CT is the central charge in the stress-energy tensor correlator, is an almost universal quantity” [15, 16](see also [17]). Indeed it was conjectured in those works that in a generic three dimensional conformal field theory there is a universal ratio [15] S = S1 σ π2 = , CT 24 (1.3) where σ is defined through the asymptotic behaviour of a(ϕ), i.e. a(ϕ → π/2) ≈ σ(ϕ−π/2)2 . –2– JHEP09(2015)172 where ε is a UV cut off, Ai ’s are some constant parameters (in particular A0 is proportional to the area of the enclosed entangling region) and [x] denotes the integer part of x. H is a typical scale in the model which could be the size of entangling region. For an even dimensional field theory (odd d in our notation) the coefficient of the logarithmic term, A2[ d ] , is a universal constant in the sense that it is independent of the UV cut off: in other 2 words it is fixed by the intrinsic properties of the theory. Two dimensional CFTs fall in this case since the central charge is indeed a universal quantity. In general for an even dimensional conformal field theory it can be shown that the coefficient of the universal logarithmic term is given in terms of the Weyl anomaly (see for example [6–8]). In particular, when the entangling region is a sphere the coefficient is exactly the same as the one multiplying the Euler density. For odd dimensional spacetimes (even d) one still has a universal constant term which might provide a generalization of the c-theorem for odd dimensional conformal field theories [9, 10]. Having said this, it is natural to pose the question whether one could find further logarithmic divergences in the expression of the entanglement entropy whose coefficients, being universal in the sense specified above, could reflect certain intrinsic properties of the theory under consideration. Moreover, if there is such a universal term, it would be interesting to understand if any relation between it and other charges of the theory is present. Indeed these questions, for some particular cases, have been addressed in the literature (see for example [11–13]). In particular, it was shown that there is also a logarithmic term in three dimensions for sets of entangling regions with non-smooth boundary. In [14] it was shown numerically that the same logarithmic term arises for finite-sized entagling regions. More precisely, for an entangling region with a cusp in three dimensions one has [11–13] The aim of the present paper is to extend the above consideration to higher dimensional field theories.2 Nonetheless, we will consider cases where the dual field theory does not even have conformal symmetry. More precisely in this pape (...truncated)