Entanglement of heavy quark impurities and generalized gravitational entropy

Entanglement of heavy quark impurities and generalized gravitational entropy

S. Prem Kumar    and Dorian Silvani s.p.kumar@swansea.ac.uk d.silvani.492808@swansea.ac.uk
Abstract

We calculate the contribution from non-conformal heavy quark sources to the entanglement entropy (EE) of a spherical region in SUSY Yang-Mills theory. We apply the generalized gravitational entropy method to non-conformal probe D-brane embeddings in AdSS, 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 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.

institutetext: Department of Physics,
Swansea University,
Singleton Park,
Swansea, SA2 8PP, U.K.

1 Introduction

The holographic correspondence maldacena (); witten (); gkp () between gauge theories and gravity has revealed an intriguing link between quantum entanglement and geometry Ryu:2006bv (); Ryu:2006ef (); Casini:2011kv (); Maldacena:2013xja (). The prescription of Ryu:2006bv (); Ryu:2006ef (); Casini:2011kv () 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 Lewkowycz:2013nqa (), where the replica trick was implemented in the gravity setting dual to the subsystem of interest, by using the method of Callan:1994py (). This involves identifying a circle in the asymptotic 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 Ryu:2006bv (); Ryu:2006ef () appears to require knowledge of the backreaction from the corresponding probe degrees of freedom in the dual gravitational description Chang:2013mca (); Jensen:2013lxa (); Kontoudi:2013rla (); Jones:2015twa (); Erdmenger:2015spo (). It has been subsequently pointed out in karchuhl () that this procedure can be circumvented by applying the gravitational entropy method of Lewkowycz:2013nqa () 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- 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 upon introduction of a test quark in the supersymmetric gauge theory in 3+1 dimensions, with 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- theory.

In Lewkowycz:2013laa () 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 drukkerfiol (); paper1 (); yamaguchi (); passerini (); paper2 () for supersymmetric Wilson/Polyakov loops in the theory. In this paper we will apply the method of karchuhl () 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 focus attention on heavy quark probes in the symmetric and antisymmetric tensor representations of rank , with (within the theory at large-), which are dual to D3 and D5-brane probes in AdSS. In the conformal case, the worldvolume of the probe contains an AdS 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 karchuhl () and find a match with the results of Lewkowycz:2013laa () 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 Lewkowycz:2013laa () picks out a special choice of gauge for the four-form potential.

  • We then study certain deformations on the probes which appear as simple one-parameter BPS solutions for the D-brane embeddings. The D5-brane solution, first found in Callan:1998iq (), interpolates between sources in the fundamental representation at short distances, and an impurity transforming in the antisymmetric representation at long distances. The deformation appears as a dimensionful parameter in the UV111This is a puzzling aspect of both the D3- and D5-brane non-conformal solutions we study, as both appear to be triggered by the VEV of a dimension one operator in the UV picture Kumar:2016jxy (), and implies spontaneous breaking of conformal invariance, which should not be possible in quantum mechanics (on the impurity)., and has the effect of screening the fundamental sources into the representation . This is most directly seen by examining the profiles of the gauge theory operators (e.g. ) sourced by the impurity where the strength of the source first increases on short scales, subsequently turns around and decreases monotonically (figure 4) at large distances to an asymptotic value determined by the representation .

    We calculate the EE excess due to this impurity within a spherical region of radius surrounding the source, by mapping the causal development of the spherical region to the Rindler wedge which is conformal to the hyperbolic space with temperature . The contribution of the probe to the gravitational entropy is obtained by varying the temperature of the dual hyperbolic AdS black hole. As a function of the dimensionless radius , we find that the EE excess displays the same qualitative behaviour (figure 3) as the profiles of gauge theory fields, namely an increase on short scales accompanied by eventual decrease at large radii towards the asymptotic value governed by the representation .

    We also find that although the EE is a non-monotonic function of the radius , the impurity free energy on which can be interpreted as a relative entropy, increases monotonically from the UV to the IR.

  • For the D3-brane probes, a simple BPS deformation exists which was discussed relatively recently in Schwarz:2014rxa () and Kumar:2016jxy (). There are two categories of these solutions (figure 1): One yields a symmetric representation source in the UV “dissociating” into coincident quarks in the IR, while the second category describes a heavy quark in representation on the Coulomb branch of the theory with broken to .

    Figure 1: The two types of D3-brane embeddings in . Shown above are the proper sizes of the two-sphere wrapped by the D3-branes as a function of AdS radial coordinate . The figure on the left represents an interpolation between the symmetric representation in the UV () and an IR spike of strings, while the one on the right is the symmetric representation source ending on a Coulomb branch D3-brane.

    We apply the gravitational entropy method to these sources taking care to employ the correct gauge for the RR potential which yields the expected result for the undeformed conformal probe. In both cases the EE excess displays non-monotonic behaviour over short scales - first increasing as a function of , and reaching a maximum. At large distances, however, the two categories display qualitatively distinct features. The EE excess for the first class of solutions saturates in the IR (figure 8) at a higher value (that of fundamental sources) than in the UV (corresponding to the representation ). For the Coulomb branch solution, we find the the EE excess decreases monotonically in the IR without bound with some universal features (figure 9).

    In all cases however, the free energy on for each of the probes increases monotonically from the UV to the IR, consistent with the interpretation as a relative entropy. The IR asymptotics of this free energy for the Coulomb branch solution exhibits certain universal features, namely, quadratic and logarithmic dependence on the Coulomb branch VEV with the coefficient of the logarithmic term being universal.

    We further confirm that D3-brane impurities with the deformations turned on, display a screening of the source in the representation . We see this for both categories of solutions by calculating the spatial dependence of gauge theory condensates sourced by the heavy quark impurities.

The paper is organized as follows: In section 2 we review the argument of karchuhl () for calculating the EE of probes without backreaction. We also review known results for the EE of conformal probes, and for completeness, we also explictly write out the trasnformations from AdS to AdS-Rindler and hyperbolic-AdS spacetimes. Section 3 is devoted to the analysis of the D5-brane probe embeddings and their entanglement entropies. In Section 4 we review the D3-brane BPS solutions. All details of the EE calculation for the D3-brane impurities are presented in Section 5. We summarize our results and further questions in Section 6. Certain technical aspects of the calculations including transformations of D3-brane worldvolume integrals from one coordinate system to another and evaluation of certain integrals are relegated to the Appendix.

2 Generalized gravitational entropy for probe branes

It was argued in karchuhl () that the entanglement entropy contribution from a finite number of flavour degrees of freedom, introduced into a large- CFT (with a holographic gravity dual), can be computed without having to consider explicit backreaction of flavour fields. A key element in this approach is the method of Lewkowycz:2013nqa () which can, in principle, be adapted to include the backreaction from flavour fields. However, this turns out to be unnecessary as the leading contribution at order is determined completely by an integral over the flavour branes in a geometry without backreaction.

The entanglement entropy of a spatial region in a CFT in spacetime dimensions can be calculated by a holographic version of the replica trick in Euclidean signature. This is performed by considering smooth, asymptotically AdS geometries with a finite size Euclidean circle at the conformal boundary of period (and ) going around the boundary of the spatial region of interest. The classical action for these geometries then yields the holographic entanglement entropy via,

(1)

This quantity only receives non-zero contribution from a boundary term within the bulk, arising from the locus of points where the circle shrinks. This corresponds to the Ryu-Takayanagi minimal surface Ryu:2006bv (). Upon introducing probe branes (defects or flavours) the complete action for the gravitational system can be separated into ‘bulk’ and ‘brane’ components:

(2)

where the brane contribution is parametrically smaller by a factor of . To paraphrase the argument of karchuhl (), if one views the backreacted metric as a small perturbation about (the -fold cover of) AdS, the deviation of the bulk action from AdS only appears at order . Then the probe contribution to the gravitational entropy at order is completely determined by an integral over the brane worldvolume alone. Furthermore, the brane embedding need only be known in ordinary AdS spacetime (with ), since the inclusion of backreaction will only affect the probe action at order and deviations of the embedding functions at order will also contribute to the action at order , since the embedding solves the equations of motion.

To compute the entanglement entropy of the region one applies the well known method of Casini:2011kv () for the specific case when is a sphere . This maps the causal development of the region within the sphere to a Rindler wedge. The spherical boundary of the entangling region is mapped to the origin of the Rindler wedge. In this process the reduced density matrix for the degrees of freedom inside the sphere then corresponds to the Rindler thermal state with inverse temperature . The latter is also conformal to a spacetime with hyperbolic spatial slices , so that Casini:2011kv (). The entanglement entropy of the region is then given by the thermal entropy of the CFT on :

(3)

For theories possessing a holographic dual, the computation of requires a bulk (AdS) extension of the boundary Rindler wedge away from the Rindler temperature . This becomes possible for the case of a CFT where we may transform the bulk extension of the wedge to hyperbolically sliced geometry. The thermal partition function on is computed holographically by the classical action of the bulk Euclidean geometry with hyperbolic slices, the replica trick is implemented by allowing the inverse temperature of the hyperbolic black hole to deviate from the value :

(4)

The unique extension of the bulk hyperbolically sliced geometry, away from , is related to the replica method via the observation of Lewkowycz:2013nqa (). In particular, the value of the replicated partition can be replaced by times the replicated partition function with the time interval restricted to the domain , and eq.(3) reduces to,

(5)

The method reproduces the vacuum EE area formula of Ryu:2006bv () and will allow us to extract the EE excess due to the insertion of defects without the need for backreaction on either the background or the defect itself.

2.1 Conformal defects from D3/D5-branes and EE

A point-like impurity in gauge theory arises most naturally upon the introduction of a Wilson line or heavy quark transforming in some representation of the gauge group. Wilson lines in fundamental , rank- antisymmetric and symmetric representations of are particularly nice from the perspective of gauge/gravity duality as they have simple realisations in terms probe string and brane sources malpol (); yamaguchi (); paper1 (); paper2 (); passerini (). Such sources compute BPS Wilson lines in different representations in the supersymmetric gauge theory at strong coupling and large-, and are introduced as probes in the dual background. In the absence of any probe deformations, the world volume metric on such probes includes an factor, so that the dual impurity theory is a (super)conformal quantum mechanics.

The excess contribution from such an impurity to the EE of a spherical region in SYM was calculated in Lewkowycz:2013laa () using the method described above, leading to eq.(3) but where is replaced by the impurity partition function in hyperbolic space, computed by a Polyakov loop or circular Wilson loop . One way to understand the appearance of the circular Wilson loop is to note that upon mapping the causal development of a spherical region to the Rindler wedge, the worldline of the heavy quark maps to the hyperbolic trajectory of a uniformly accelerated particle. Upon Euclidean continuation, the hyperbolic trajectory turns into a circle. Therefore,

(6)

where in the final expression we are required to compute the expectation value of the field theory stress tensor on , in the presence of the Wilson/Polyakov loop insertion. As argued in Lewkowycz:2013laa (), conformal invariance fixes the form of the stress tensor, and the expectation value of the energy density integrated over depends on a single normalisation constant :

(7)

The normalisation constant for SYM was calculated in Gomis:2008qa () by relating it to the expectation of a dimension two chiral primary field, with net result,

(8)

While localization results can, in principle, be used to determine the circular Wilson loop in various representations for any and gauge coupling, we will focus attention on the strict large- limit at strong ’t Hooft coupling paper1 (); drukkerfiol (). In this limit, the following results can be deduced for the EE contributions from the conformal impurities in the three different representations described above222The results quoted here differ from those of Lewkowycz:2013laa () by an overall factor of . We clarify the reason for this normalization below eqs.(14) and (25). :

(9)

Our aim will be to reproduce these results for the conformal impurities using the method of karchuhl () and then apply the same to the case of the non-conformal impurity flows that were discussed in Kumar:2016jxy ().

2.2 From AdS to hyperbolic AdS

Now we review the maps that take the AdS-extension of the causal development of the spatial sphere in to hyperbolically sliced . This will help set our conventions, and will be important subsequently since the evaluation of EE for non-conformal impurities will involve computation of integrals over specific brane embeddings in hyperbolic-AdS geometry, and the explicit calculation of these will require us to go back and forth between different coordinate systems.

We first consider the transformation,

(10)

where . Here is the radial AdS coordinate, with the conformal boundary at . This is the extension of the boundary CFT special conformal transformation to an isometry of AdS. The map has the following actions:

  • On the conformal boundary at , the ball : at is mapped to the half-line . The causal development of is mapped to the Rindler wedge .

  • The world line of the impurity on the boundary, located at the spatial origin , is mapped to the trajectory of a uniformly accelerated particle, , with . In Euclidean signature this maps to one half of the circular Wilson loop with .

  • The transformation acts on the Poincaré patch metric as an isometry:

    (11)

    while the boundary metric itself transforms by a conformal factor. The holographic extension of the causal development of the ball into the AdS bulk (entanglement wedge) is given by the causal development of the hemisphere (defined at ). This is mapped by the above isometry to the Rindler-AdS wedge .

The Rindler-AdS wedge is further mapped to hyperbolically sliced AdS by the transformations listed below. First we parametrize the Rindler-AdS wedge by defining the coordinates,

(12)

so that

(13)

The wordline of the heavy quark on the boundary is given by . In order to perform the replica trick it is crucial that we move to Euclidean signature, via the replacement , so we obtain AdS in “double polar” coordinates, and the heavy quark impurity then traces out a Polyakov loop at ,

(14)

The Euclidean time must be restricted to the domain where is positive, so that . The -coordinate is periodic under the shifts which also ensures that the “double polar” geometry is free of conical singularities. The map to hyperbolically sliced AdS is achieved by the transformations

(15)

which yield the Euclidean AdS black hole with hyperbolic horizon,

(16)

Once again we have the restriction on the range of the Euclidean time which has periodicity , guaranteeing that the space caps off smoothly at . Finally, it will be useful to to recall the coordinate transformations which directly map the entanglement wedge in the original AdS spacetime,

(17)

to the hyperbolic AdS black hole (16) with inverse temperature . The relevant coordinate transformations are (in Lorentzian signature):

Upon continuation to imaginary time , we must restrict to the domain of to . It can be shown that the pre-image of the Euclidean hyperbolic AdS black hole, given this domain, is the interior of the hemisphere in the original (Euclidean) AdS geometry,

(19)
Hyperbolic AdS and replica method:

The replica method requires that we consider a hyperbolic AdS black hole in which the Euclidean time has period where , so that

(20)

The Hawking temperature of the black hole is

(21)

It is clear that implementation of the replica trick is equivalent to varying the Hawking temperature of the black hole, ensuring as usual, the absence of a conical singularity in the Euclidean geometry. In this approach the entanglement entropy is given by the thermal entropy evaluated in the hyperbolic AdS geometry. In particular, using eq.(5), we have

(22)

Here is the action of the hyperbolic AdS geometry including any probes dual to the impurities or defects under consideration, and where the integration over Euclidean time is restricted to the domain .

2.3 Warmup: A single fundamental quark

As a warmup, we compute the EE excess due to the insertion of a single fundamental quark into the spherical entangling region. In the AdS dual, this is achieved by inserting a probe fundamental string (F1) into the hyperbolic AdS geometry and computing the thermal entropy from the Nambu-Goto worldsheet action in this geometry. The F1-string worldsheet is placed at , and stretches from the hyperbolic horizon at to the conformal boundary at . The tension for the fundamental string, in units of the AdS radius is

(23)

where is the ’t Hooft coupling for the theory. Then the action for the static F-string embedding stretched along the radial AdS coordinate is

(24)

The determinant of the induced metric on the worldsheet for this embedding is unity, and the boundary counterterm which regularises the worldsheet action is independent of the temperature as it is only sensitive to UV details. Varying with respect to , we thus obtain

(25)

Our result differs by a factor of two from that of Lewkowycz:2013laa (), as the range of integration over Euclidean time is restricted to , which corresponds to one half of the Polyakov loop on .

EE from stress tensor evaluation:

For this simple example it is instructive to verify how the above result can be reproduced holographically, using eq.(8) which relies on the expectation value of the stress tensor in the presence of the temporal Wilson line in Rindler frame. This computes the expectation value of the entanglement Hamiltonian which generates time translations along the compact time direction. In particular, the EE for the impurity is given as

(26)

The ingredients in the computation can be calculated either directly in the AdS Poincaré patch, or after translating to the hyperbolic AdS picture. In the Poincaré patch, we need to ensure that all integrals over the Euclidean string worldsheet are restricted to the domain,

(27)

Therefore, the impurity action in hyperbolic space is given by integrating the (Euclidean) Nambu-Goto action in the Poincaré patch of AdS over :

(28)

The second term is the worldsheet counterterm induced on the conformal boundary at , as is taken to zero. The stress tensor expectation value333The worldsheet stress tensor for the string embedding is obtained by varying with respect to the spacetime metric, so that , in Lorentzian signature. for the heavy quark source in the Unruh state, or equivalently, in hyperbolic space would normally be computed by reading off the normalizable mode of the metric sourced by the probe string in the bulk. Alternatively, from the Hamiltonian formulation of the AdS/CFT correspondence, the (regularized) energy of the probe should directly yield the energy of the corresponding source (impurity) in the boundary CFT Karch:2008uy (). The result for the energy of the probe string is thus of the form

(29)

where is the UV cutoff. Keeping only the finite terms, we find

(30)

so that the contribution to the EE of the spherical region from the heavy quark is

(31)

3 D5-brane impurity

In this section we will focus our attention on the D5-brane embedding which computes the BPS Wilson loop in SYM, in the antisymmetric tensor representation. The embedding admits a deformation which can be interpreted as an RG flow on the worldvolume of the impurity Kumar:2016jxy (). Our goal will be to extract the behaviour of the impurity EE along this flow.

3.1 AdS embeddings of the D5-brane

The D5-brane embedding, dual to a straight Wilson line in the theory, preserves an subgroup of the global R-symmetry. This is realized geometrically, by having the D5-brane wrapping an latitude of the five-sphere in AdS. In the non-conformal “flow” solution described in Kumar:2016jxy (), the polar angle associated to this latitude varies as a function of the radial position in AdS. We can choose the worldvolume coordinates to be , where parametrises the non-compact spatial coordinate on the brane. We will eventually choose the gauge . The induced metric for such an embedding in (Euclidean) AdS is,

(32)

The action for the D5-brane consists of the standard Dirac-Born-Infeld (DBI) and Wess-Zumino (WZ) terms. The latter supports the configuration when a non-zero, radial world-volume electric field is switched on. In Euclidean signature this is purely imaginary and will be denoted in terms of the real quantity :

(33)

The Wess-Zumino term for the D5-brane embedding is induced by the pullback of the RR four-form potential determined by the volume form on AdS. In particular, the relevant component of is

(34)

where is the volume form of the unit four-sphere. The four-form potential is chosen so that the five-form flux comes out proportional to the volume form of :

(35)

The D5-brane embedding is then determined by the equations of motion following from the action

(36)

The action is regularized by counterterms . The dilaton vanishes in the AdS background dual to the theory, and the D5-brane tension can be expressed in terms of gauge theory parameters as

(37)

The counterterms can be split in two pieces: one which regulates the UV divergences in the action and another which fixes the number of units of string charge carried by the embedding to be drukkerfiol (); Kumar:2016jxy (),

(38)

The counterterm enforces a Lagrange multiplier constraint that fixes the number of units of string charge carried by the configuration. Putting together all these ingredients, choosing the gauge , the final form for the D5-brane action is

(39)

with

(40)

3.1.1 The constant embedding

It is easy to check that the equations of motion yield a constant solution:

(41)

This solution is BPS and has vanishing regularized action in Poincaré patch. It yields the straight BPS Wilson loop in the antisymmetric tensor representation yamaguchi (); passerini (); paper1 (). In all respects the constant solution is identical to the F-string solution for a fundamental quark, except for the normalization of the action which is controlled by .

Embedding in hyperbolic AdS:

The contribution to the EE of a spherical region can be calculated by applying the formula eq.(22) to the constant embedding in hyperbolic AdS space (20). Repeating the above excercise for the solution which yields , we obtain the regularized action as a function of the temperature of the hyperbolic AdS black hole:

(42)

where . The entanglement entropy contribution from the impurity in the antisymmetric tensor representation is then,

(43)

3.1.2 The D5 flow solution

The Poincaré patch action for the D5-brane embedding permits a non-constant zero temperature BPS solution Callan:1998iq (). This solution interpolates between a spike or bundle of coincident strings in the UV and the blown-up D5-brane configuration corresponding to the antisymmetric representation reviewed above. In the boundary gauge theory, the flow can be interpreted as the screening of coincident quarks in the fundamental representation to a source in the antisymmetric tensor representation Kumar:2016jxy (). As seen in Kumar:2016jxy (), the flow appears as a result of a condensate for a dimension one operator in the UV worldline quantum mechanics of the impurity. The Poincaré patch BPS embedding solves the first order equation,

(44)

and is explicitly given by the solution,

(45)

where is an integration constant with dimensions of inverse length. For small , the polar angle approaches , so that the wrapped by the D5-brane shrinks to zero size and the collapsed configuration must be viewed as -coincident strings. In the IR limit on the other hand, when , approaches which yields the blown-up D5-brane embedding.

In order to calculate the excess EE contribution from this non-conformal impurity in the boundary CFT, we first need to map the configuration to hyperbolically sliced AdS (20). The internal angle of the ten dimensional geometry is unaffected by the map. The only other active coordinate in the D5-brane embedding is the radial position in AdS spacetime which, upon rewriting in terms of hyperbolic Euclidean AdS coordinates (LABEL:poinctohyp), yields the transformed solution:

(46)

with the restriction . The impurity is placed at the spatial origin, in , which corresponds to in . Since is a function of and , the induced metric on the D5-brane is,

(47)

where is given in eq.(20). The D5-brane embedding, mapped to hyperbolic AdS, must also have a non-trivial background worldvolume electric field. Since the embedding shares only the temporal and radial directions with the bulk AdS geometry, there is only one component of the field strength to switch on:

(48)

For the case with , can be obtained directly by transforming the field strength in the Poincaré patch solution. To implement the replica trick, however, we first need to consider general temperatures of the hyperbolic black hole. Using the above ansatz for the D5-brane embedding, the action in the hyperbolic AdS background is,

(49)

Solving for using its equation of motion and plugging it back in,

In order to extract entanglement entropy excess due to the impurity, we need to vary this action with repect to and set , whilst keeping fixed as the BPS solution at . The latter is justified because the first variation of the action with repect to vanishes by the equations of motion at .

Once the variations with respect to are performed, the remaining integrals are most easily evaluated in Poincaré patch coordinates, in which the D-brane embedding function is simpler. The transformations (LABEL:poinctohyp) when restricted to the location of the heavy quark at imply,

(51)

The Jacobian for the transformation on the worldvolume back to Poincaré patch coordinates is,

(52)

We also note that the kinetic terms for a static Poincaré patch configuration satisfy,

(53)

We first evaluate the action (or free energy) of the BPS embedding in the hyperbolic AdS background with , by recasting in Poincaré patch coordinates:

(54)

where is defined in eq.(44). Although the integrand is a total derivative, the fact that the integration region is limited to the half-disk (eq.(27)), renders the evaluation nontrivial. In particular, the integration over is performed first since the integrand is independent of time. Following this, the remaining integral can be performed numerically after exchanging the integration variable for , which is more convenient as the solution is known explicitly for as a function of .

Figure 2: The free energy of the non-conformal D5-brane impurity (blue) on with , as a function of the deformation parameter . It interpolates between the values of the circular Wilson lines for coincident quarks (dotted black) in the fundamental representation, and the antisymmetric tensor representation (dashed red line).

The values of the (regularized) actions for the two types of conformal sources, fundamental and antisymmetric tensor in hyperbolic space are:

(55)

The partition function of the heavy quark source in hyperbolic space with inverse temperature is plotted in figure 2 as a function of the deformation parameter . It is a monotonically decreasing function of the size of the entangling region and interpolates between the value for coincident fundamental quarks in the UV and that for a source transforming in the antisymmetric tensor representation in the IR.

We note that is like a relative entropy relative (). It is the free energy difference between the embeddings with non-zero and vanishing deformations in the thermal state with associated to the modular Hamiltonian. This explains the monotonic increase of with , and the vanishing slope in figure 2 for arbitrarily small deformations. By expanding the solution for the embedding function , the deformation can be interpreted as the expectation value of a dimension one operator in the UV quantum mechanics of the boundary impurity Kumar:2016jxy ().

The EE contribution from the impurity is obtained by varying the “off-shell” action (3.1.2) with respect to and evaluating the first variation on the BPS solution,

(56)

We have made use of the BPS formula (44) and that when . Recasting the result in terms of the integral over the domain in Poincaré patch, we find:

(57)

As in the case of the free energy above, the integration over the domain must be performed numerically. The integral over the coordinate can once again be obtained analytically, and the final integration is achieved numerically after exchanging for . The result for the entanglement entropy excess is a function of the dimensionless combination , as plotted in fig.(3).

Figure 3: The ratio of the EE contribution due to the D5-brane impurity (thick blue) and that of fundamental quarks (dotted black line), as a function of the radius of the spherical entangling region. The deformation drives the flow and the dimensionless tunable parameter is . The red dashed line is the ratio .

For every value of , we see that the entanglement entropy contribution interpolates between that of coincident fundamental quarks and a source in the antisymmetric representation :

(58)

The main notable feature of the results is that the variation of the EE with size of the entangling region (or equivalently the deformation ) is non-monotonic, exhibiting a maximum at a special value of of order unity, and decreasing monotonically subsequently.

3.2 Comparison with

The D5-brane is a source of various supergravity fields in and the falloffs of these fields yield the VEVs of corresponding operators in the boundary gauge theory. In particular, the dilaton falloff was used in Kumar:2016jxy () to infer the VEV of the dimension four operator , equal to the Lagrangian density of the theory, in the presence of the non-conformal D5-brane impurity. Since is a dimension four operator, for conformal impurities the VEV of this operator scales as where is the spatial distance from the heavy quark on the boundary:

In fig. (4), we plot the dimensionless ratio as a function of the dimensionless distance from the impurity .

Figure 4: The VEV of the dimension four operator dual to the dilaton, in the presence of the non-conformal impurity, divided by the corresponding VEV due to coincident quarks (dotted black), plotted (solid blue) as a function of . The red dashed line denotes the VEV of the same operator in the presence of the source in representation .

The qualitative features of the plots are similar to those of the entanglement entropy contribution from the defect. The sources in the fundamental representation are screened into the antisymmetric representation, but the effect is non-monotonic as a function of the distance from the source.

4 D3-brane impurities

The D3-brane embedding with worldvolume found by Drukker and Fiol drukkerfiol () computes BPS Wilson lines in the rank symmetric tensor representation passerini (); paper1 (). In Kumar:2016jxy (), a D3-brane (BPS) embedding was analyzed which interpolates between the representation in the UV and coincident strings in the IR. We will first review the properties of this zero temperature solution in Poincaré patch and subsequently analyze its geometric entropy.

4.1 Poincaré patch D3-brane embedding

The D3-brane wraps an AdS subset of AdS and is supported by units of flux. Since the internal five-sphere plays no role we will suppress it in the discussion below. The D3-brane impurity preserves the same symmetries as a point at the spatial origin of the boundary CFT on . In particular, choosing the worldvolume coordinates to be the induced metric for the relevant embedding takes the form (in Euclidean signature),

(60)

Eventually we will set after discussing the counterterms and UV regularization. The background five-form RR flux, and its associated four-form potential play a crucial role in stabilizing the D3-brane configuration. In particular, the pullback of the four-form potential onto the D3-brane worldvolume is

(61)

where is the volume-form on the unit two-sphere. We also recall that is only defined up to a gauge choice. The choice of gauge will be important when we proceed to the calculation of the entanglement entropy contribution from the defect. The expanded D3-brane configiuration also has a worldvolume electric field and the a tension . Putting all ingredients together, we find,