Entanglement of heavy quark impurities and generalized gravitational entropy

Published for SISSA by Springer Received: November 11, 2017 Accepted: January 2, 2018 Published: January 11, 2018 S. Prem Kumar and Dorian Silvani Department of Physics, Swansea University, Singleton Park, Swansea, SA2 8PP, U.K. E-mail: , Abstract: We calculate the contribution from non-conformal heavy quark sources to the entanglement entropy (EE) of a spherical region in N = 4 SUSY Yang-Mills theory. We apply the generalized gravitational entropy method to non-conformal probe D-brane embeddings in AdS5 ×S5 , dual to pointlike impurities exhibiting flows between quarks in large-rank tensor representations and the fundamental representation. For the D5-brane embedding which describes the screening of fundamental quarks in the UV to the antisymmetric tensor representation in the IR, the EE excess decreases non-monotonically towards its IR asymptotic value, tracking the qualitative behaviour of the one-point function of static fields sourced by the impurity. We also examine two classes of D3-brane embeddings, one which connects a symmetric representation source in the UV to fundamental quarks in the IR, and a second category which yields the symmetric representation source on the Coulomb branch. The EE excess for the former increases from the UV to the IR, whilst decreasing and becoming negative for the latter. In all cases, the probe free energy on hyperbolic space with β = 2π increases monotonically towards the IR, supporting its interpretation as a relative entropy. We identify universal corrections, depending logarithmically on the VEV, for the symmetric representation on the Coulomb branch. Keywords: AdS-CFT Correspondence, D-branes, Wilson, ’t Hooft and Polyakov loops ArXiv ePrint: 1711.01554 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP01(2018)052 JHEP01(2018)052 Entanglement of heavy quark impurities and generalized gravitational entropy Contents 1 2 Generalized gravitational entropy for probe branes 2.1 Conformal defects from D3/D5-branes and EE 2.2 From AdS to hyperbolic AdS 2.3 Warmup: a single fundamental quark 4 6 7 9 3 D5-brane impurity 3.1 AdS embeddings of the D5-brane 3.1.1 The constant embedding 3.1.2 The D5 flow solution 3.2 Comparison with hOF 2 i 11 11 12 13 17 4 D3-brane impurities 4.1 Poincaré patch D3-brane embedding 17 18 5 D3-brane entanglement entropy 5.1 Choice of C4 5.2 D3-brane action in hyperbolic AdS 5.3 Conformal D3-embedding: symmetric representation 5.4 Action on Sβ1 × H 3 with deformation a > 0 5.5 Action for D3-brane embedding with a < 0 5.6 EE for D3-brane impurity 5.7 Comparison with hOF 2 i 19 19 21 21 22 24 26 28 6 Discussion and summary 28 A Transformations for D3-brane embedding 31 B Evaluation of hOF2 iD3 32 1 Introduction The holographic correspondence [1–3] between gauge theories and gravity has revealed an intriguing link between quantum entanglement and geometry [4–7]. The prescription of [4– 6] relating the entanglement entropy of some subsystem within a quantum system to the area of an extremal surface in a classical dual gravity framework, was put on firm footing in [8], where the replica trick was implemented in the gravity setting dual to the subsystem of interest, by using the method of [9]. This involves identifying a circle in the asymptotic –1– JHEP01(2018)052 1 Introduction • We focus attention on heavy quark probes in the symmetric and antisymmetric tensor representations of rank k, with k ∼ O(N ) (within the N = 4 theory at large-N ), which are dual to D3 and D5-brane probes in AdS5 ×S5 . In the conformal case, the worldvolume of the probe contains an AdS2 factor, reflecting the conformal nature of the quantum mechanics on the impurity. We calculate the contribution to the generalized gravitational entropy from these probe branes using the proposal of [15] and find a match with the results of [16] deduced via independent arguments. A nontrivial aspect of the calculation and observed agreement is the role played by the background Ramond-Ramond (RR) flux and its associated four-form potential, specifically in the case of the D3-brane probe dual to the symmetric representation source. The generalised gravitational entropy receives a contribution from the coupling of this potential to the D3-brane probe, and matching with the CFT arguments of [16] picks out a special choice of gauge for the four-form potential. –2– JHEP01(2018)052 geometry, which could be a compact Euclidean time direction, varying its periodicity in a well-defined manner and calculating the resulting variation in the action so as to obtain a gravitational or geometric entropy. A natural extension of these ideas is to study the effect of excitations above the vacuum state or inclusion of new degrees of freedom in the form of flavours or defects. Here it was understood that even for flavours or defects in the quenched approximation, the application of the Ryu-Takayanagi prescription [4, 5] appears to require knowledge of the backreaction from the corresponding probe degrees of freedom in the dual gravitational description [10– 14]. It has been subsequently pointed out in [15] that this procedure can be circumvented by applying the gravitational entropy method of [8] to the quenched degrees of freedom propagating in the un-backreacted gravitational backgrounds. In this paper, we will study pointlike defects or “impurities” that have a simple interpretation, namely they are test charges or heavy quarks introduced into the vacuum state of a large-N QFT. The coupling of the heavy quark to the quantum fields affects the entanglement of any region that contains the impurity, with the rest of the system. Specifically, we are interested in the change in entanglement entropy (EE) of a spherical region of some radius R upon introduction of a test quark in the N = 4 supersymmetric gauge theory in 3+1 dimensions, with SU(N ) gauge group. This question becomes particularly interesting if one can deform the quantum mechanics of the pointlike impurity so that the system is not conformally invariant and the degree of entanglement is a nontrivial function of the deformation strength. Our goal will be to examine and identify general scale dependent properties of EE across different tractable examples of such impurities at strong ’t Hooft coupling in the large-N theory. In [16] the excess EE due to such heavy quarks in large rank symmetric and antisymmetric tensor representations were computed (both at weak and strong coupling) by exploiting conformal invariance and relating them to known results [17–21] for supersymmetric Wilson/Polyakov loops in the N = 4 theory. In this paper we will apply the method of [15] based on gravitational entropy contributions to obtain the EE excess due to the corresponding probes (D-branes) in the gravity dual, including the effect of deformations that trigger flows on the impurity. The main results of this paper are summarized below: We calculate the EE excess (...truncated)