Holographic entanglement entropy of a 1 + 1 dimensional p-wave superconductor

Journal of High Energy Physics, Sep 2017

We examine the behavior of entanglement entropy S A EE of a subsystem A in a fully backreacted holographic model of a 1 + 1 dimensional p wave superconductor across the phase transition. For a given temperature, the system goes to a superconducting phase beyond a critical value of the charge density. The entanglement entropy, considered as a function of the charge density at a given temperature, has a cusp at the critical point. In addition, we find that there are three different behaviors in the condensed phase, depending on the subsystem size. For a subsystem size l smaller than a critical size l c1, S A EE continues to increase as a function of the charge density as we cross the phase transition. When l lies between l c1 and another critical size l c2 the entanglement entropy displays a non-monotonic behavior, while for l > l c2 it decreases monotonically. At large charge densities S A EE appears to saturate. The non-monotonic behavior leads to a novel phase diagram for this system.

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Holographic entanglement entropy of a 1 + 1 dimensional p-wave superconductor

HJE Holographic entanglement entropy of a 1 + 1 dimensional p-wave superconductor Sumit R. Das 0 1 Mitsutoshi Fujita 0 1 Bom Soo Kim 0 1 0 Lexington , KY 40506 , U.S.A 1 Department of Physics and Astronomy, University of Kentucky , USA We examine the behavior of entanglement entropy SEE of a subsystem A in a A fully backreacted holographic model of a 1 + 1 dimensional p wave superconductor across the phase transition. For a given temperature, the system goes to a superconducting phase beyond a critical value of the charge density. The entanglement entropy, considered as a function of the charge density at a given temperature, has a cusp at the critical point. In addition, we nd that there are three di erent behaviors in the condensed phase, depending to increase as a function of the charge density as we cross the phase transition. When l lies between lc1 and another critical size lc2 the entanglement entropy displays a nonmonotonic behavior, while for l > lc2 it decreases monotonically. At large charge densities SEE appears to saturate. The non-monotonic behavior leads to a novel phase diagram for A this system. on the subsystem size; For a subsystem size l smaller than a critical size lc1; SEE continues A - 1 3 7 1 Introduction 3.1 A phase diagram Discussions Introduction The backreacted solution Holographic entanglement entropy and the phase transition A The entanglement entropy SEE of a subsystem A of a system is a useful non-local quantity in quantum eld theories [1{6]. This quantity often provides interesting probes of the physics of phase transitions, most notably for quantum critical transitions where it typically diverges [7]. Even for thermal transitions, the entanglement entropy provides additional diagnostics of the nature of the phases, since it is sensitive to the number of available degrees of freedom. In recent years, the Ryu-Takayanagi prescription has become a key tool to compute entanglement entropy of strongly coupled systems which have gravity duals [8{ 10]. Indeed, starting with [11], there have been several studies of the change of behavior of the entanglement entropy in holographic models of critical phase transitions. [11] examined a model of s-wave holographic superconducting transition, [12] considered a model of an insulator-super uid transition while [13] studies a model of holographic p-wave superconductor phase transition based on [14, 15]. Recently, [16, 17] extended the study of the holographic entanglement entropy near critical phase transitions. In all these models a suitable order parameter condenses in the super uid phase, reducing the number of degrees of freedom. Indeed these studies found that the magnitude of the entropy in the condensed phase is always lower than what it would have been in the absence of condensation. In all these studies, the entanglement entropy as a function of the temperature or the chemical potential displayed a cusp at the location of the transition where the relevant derivative becomes discontinuous. As one goes into the condensed phase the entanglement entropy sometimes shows a non-monotonic behavior close to the transition, while far from the transition it keeps decreasing. In this paper, we perform a calculation of the holographic entanglement entropy (HEE) in a toy model of a 1 + 1 dimensional p-wave superconductor at non-zero charge density and temperature. The dual theory is a 2 + 1 theory of gravity and SU(2) Yang-Mills eld with a negative cosmological constant. The action is given by IG = 1 Z In this convention, A~ is dimensionless, while gYM and 1= 2 have the dimension 1. This model has been introduced in [18] following earlier work in higher dimensions [14]. An igYM[A~ ; A~ ]. important well known aspect of this 1 + 1 dimensional model is that the gauge eld has to be treated in alternative quantization [19]: the non-normalizable part of the eld can be interpreted as the expectation value of a dynamical gauge eld living on the boundary, which makes it morally closer to true superconductors, as opposed to super uids. In the normal phase the dual geometry is a AdS3 charged BTZ black hole, with the role of the usual Maxwell eld played by the time component of the diagonal gauge eld At( 3 ). The boundary theory then has a nonzero temperature and charge density. As we increase the charge density for a given temperature, there is a critical value c beyond which the black hole acquires a vector hair, e.g. a spatial component A(x1) becomes nonzero:1 its boundary value then becomes the expectation value of a vector order parameter in the dual eld theory. In the probe approximation the corresponding bulk solutions have been obtained in [18]. The dual eld theory lives on a circle with circumference 2 L~ (where L~ is the AdS scale as in equation (1.1), and we calculate the holographic entanglement entropy of an interval of size l in this circle. In the normal phase the exact gravity solution is of course known. In the condensed phase the solution is known only in the probe approximation. To obtain the holographic entanglement entropy we rst obtain the fully backreacted solution for the hairy charged BTZ black hole by numerically integrating the bulk equations of motion. A Using this backreacted solution we compute the Bekenstein-Hawking entropy SBH = 2 2 (where A is the horizon area and is the gravitational constant) and the entanglement entropy SEE using the Ryu-Takayanagi prescription SEE = ( A); 2 2 (1.2) where A is the length of the one-dimensional bulk geodesic whose end points coincide with the two end points of the interval on the boundary. We focus on the behavior of these quantities at some xed temperature as a function of the charge density for various values of l. We nd that the Bekenstein-hawking entropy SBH has a cusp at the critical charge density q = qc where @SBH is discontinuous. For q > qc this monotonically decreases, @q approaching a constant value. This decrease is essentially due to the reduction of degrees of freedom due to condensation. For a given temperature and q < qc the entanglement entropy monotonically increases as a function of q and its derivative has a discontinuity at q = qc. This cusp-like behavior has been observed earlier in higher dimensional models. However the behavior for q > qc appears quite di erent from that in the higher dimensional examples. This depends on the subsystem size l and there are two critical sizes lc1; lc2 where the behavior changes. For small enough size, l < lc1 the entanglement entropy keeps increasing monotonically. When lc2 < l < lc1 this quantity displays a non-monotonic behavior, decreasing at rst to 1This is di erent from vector hair in addition to a scalar condensate which represents a current [20]. { 2 { a local minimum, and then increasing to a maximum value and nally decreasing again. The local minimum moves closer to = c as l approaches lc1. At l = lc2 the local minima and maxima merge to a point of in ection, so that for l > lc2 the entanglement entropy decreases monotonically with increasing charge density. While we do not have a complete understanding, it appears that this complex behavior arises from a competition between charge density and condensation. If there was no condensation, the entanglement entropy would have kept increasing with charge density for any subsystem size. Condensation reduces the number of e ective degrees of freedom. When the subsystem size is small enough, the e ects of condensation are small, and the tendency of the entanglement entropy to increase with charge density wins over. When the ferent \phases". In section (2) we describe the system and the associated critical transition, and calculate the backreaction by numerically solving the equations of motion. The backreacted solution is then used to calculate the Bekenstein-Hawking entropy. In section ( 3 ) we use the solution above to calculate the minimal geodesics and hence the holographic entanglement entropy. ( 4 ) contains a brief discussion. 2 The backreacted solution The action of Einstein-Yang Mills system we consider in this paper is IG = 1 Z igYM[A~ ; A~ ]. One of the spatial directions (called y below) is compact, y In (2.1), A~ is dimensionless, while gYM and 1= 2 have the mass dimension 1. A standard eld rede nition A~ while the Yang-Mills kinetic term becomes proportional to 1=( 2gY2M). In this section, we analyze the backreaction (whose strength is controlled by 1=gY2M ) of the SU(2) YangMills term into the metric of the 2+1 dimensional gravity to compute the holographic ! A =gYM the eld strength becomes F The equations of motion of the gauge eld and the metric derived from the above ( p gF p ) g[A ; F { 3 { where the energy momentum tensor is T = 2 2 tr F~ ~ F 1 4 g F ~ ~ F = 2 2gY2M 1 4 tr F F g F F : (2.4) q 2 gYM z2 log z z0 ; h2(z) = 1; (z) = q log : (2.8) z z0 The last equation is a constraint. Using the ansatz the equations of motion for the Yang Mills elds become ph2f2(zph2 0)0 + zw2 = 0; ph2f2(zph2f2w0)0 + z 2w = 0: Note that the equations of motion depend only on the product of AdS radius L~ and gYM. This can be used to set the scale L~ = 1. In the uncondensed phase the y component of the gauge eld w(z) vanishes. In this case, nonlinear terms are absent in the eld strength and the system reduces to an EinsteinMaxwell system. The solution is then well known: it is a charged AdS3 black hole considered in [19, 21] and described in (2.6) with If we set the Yang-Mills eld to zero, the solutions to Einstein equations are either pure AdS3 or an uncharged BTZ black hole. In the presence of a nontrivial gauge eld which is assumed to depend only on the radial AdS direction, the Einstein equations (2.2) and (2.3) can be solved with an ansatz: where a (a = 1; 2; 3) are the Pauli matrices. The function f2(z) vanishes at z = zh, the location of the black hole horizon. The Einstein equations can be re-written as the following three equations: The black hole horizon is located at z = z0. The Hawking temperature is given by Furthermore, we have chosen (z) to vanish at the horizon, as required by regularity of the solution in the euclidean domain. The regularized mass of the black hole, M0 = (L~=z0)2, is inversely proportional to the squared horizon position and satis es the BPS-like bound { 4 { (2.5) = 0; = 0; = 0; (2.6) (2.7) (2.9) (q=p2gYM)2 [22] which is saturated at TH = 0. At the equality, the horizon position zh is equal to the value of z where f (z) has its extremal value. As explained in [19] the presence of a logarithmic term implies that the gauge eld has to be treated in alternative quantization, which requires us to x boundary conditions which specify the charge density q. The chemical potential is then given by the constant part = q log z0. The mass dimension of q and are both 1. When the charge density is large enough there is another solution which has w(z) 6= 0, with a free energy lower than a charged AdS3 black hole. The behavior of this solution near the AdS boundary z = 0 is (2.10) HJEP09(217)6 (2.11) (2.12) (z) q log w(z) f2(z) h2(z) z zp ; 1 h0; wc + Jw log(z); gYM z2 log(z); (z) = a1(zh z) + : : : ; w(z) = b1 + b2(zh z) + : : : ; f2(z) = d2(zh z) + : : : ; h2(z) = c1 + c2(zh z) + : : : : where once again q is the charge density of the boundary eld theory while the chemical potential is given by q log(zp).2 The parameter wc is now interpreted as the expectation value of a vector order parameter in the boundary theory while Jw is the source for this order parameter. Both these parameters have mass dimension 1. h0 and r0 are constants. The black hole horizon is now located at z = zh as speci ed by the condition f2(zh) = 0. Regularity at the horizon requires as usual (zh) = 0. The elds can be now expanded near the horizon as follows: The coe cients (a1; b1; b2; c1; c2; d2) are constants. In terms of these parameters the Hawking temperature is Other parameters like b2; c2 and d2 are then represented by these 4 parameters and gYM: b2 = 0, d2 = 2=(c1zh) a21zh3=gY2M, and We now obtain a backreacted solution by numerically solving the equations of motion. Recall that the last equation of (2.6) is a constraint equation. This can be solved by choosing regularity conditions at the horizon. We then solve the rst two Einstein equations 2zp inside log is normalized by L~ to be dimensionless [23]. { 5 { of (2.6) and two equations (2.7) numerically, starting from the horizon and proceeding to the AdS boundary. To look for a superconducting phase we need to nd solutions with a vanishing source in the boundary theory. This implies that at the AdS boundary, Jw has to vanish. A nontrivial solution for w(z) then signi es spontaneous symmetry breaking in the boundary theory. This also breaks residual bulk U( 1 ) gauge symmetry generated by A3 spontaneously. The parameters wc; Jw(= 0); h0 are then speci ed by using 4 parameters (a1; b1; c1; zh) of the horizon expansion. To go further, consider the scaling symmetry in the equations of motion as follows: (t; y; z) ! s 1(t; y; z); f2 ! s2f2; ! h2 ! s ; s 2h2; w ! ! sw; s : (2.13) (2.14) HJEP09(217)6 The rst symmetry can be used to x zh = 1. Using second symmetry, the leading coe cient of h2(z) in (2.10) can be xed to set (h0 = 1): this yields the standard AdS3 metric near the boundary. Under scaling symmetry (2.13), the parameters of the solution transform as follows: q ! Jw ! sq; sJw; = q log(zp) ! sq log(zp) + sq log s ; wc ! swc + sJw log s = swc; ! s 1 ; (2.15) for gY2M = 5 wc 1:18qp1 where we have used Jw = 0 becomes zero in the last equality. Note that the scaling transformation of log(z) in (2.10) produces the shift log s in the chemical potential . The scaling transformation is important to x parameters like the temperature or the charge. We can not x these parameters without changing zh. When we x the inverse temperature to be , we need the scaling transformation s = jzh=1= and then the horizon position zh is changed into zh = =( jzh=1) by using the above transformation. To consider the system of the nite charge q0, instead, we need to perform the scaling transformation s = q0=qjzh=1. The horizon position is then changed into zh = qjzh=1=q0. We nd that with a vanishing Jw a solution with non-zero wc exists only when the charge density q exceeds a critical value qc. This solution is a hairy black hole. As discussed above wc is the expectation value of a vector order parameter in the boundary eld theory: the hairy black hole is then the gravity representation of a superconducting phase. In gure 1 we plot the behavior of wc=q as a function of qc=q at a xed temperature TH = 0:15 with varying gY2M. The critical charge qc becomes 21:7TH , 33:5TH , and 45:1TH 105, 10, and 6, respectively. The condensate vanishes for q < qc. The point q = qc is a critical phase transition. Near the critical charge qc, the condensate behaves as qc=q so that we get a mean eld critical exponent. The probe approximation corresponds to large gY2M: this is the situation when the gravity backreaction can be ignored. Our results clearly shows that backreaction decreases wc=q at large charge q, while near the critical point the results approach those of the probe approximation. This is expected since near the critical point, wc is small so that the backreaction is small as well even for nite gY2M. { 6 { 25S 20 15 10 5 0 0 0.6 0.4 0.2 with varying gY2M. The dashed line shows the analytic curve 1:18p1 is 21:7TH and 33:5TH for gY2M = 5 10 5 and 10, respectively. 2 (= c2d=6) when gYM = 10. The dashed line gives the entropy in the AdS3 charged black hole. The solid line gives the entropy in the condensed phase T > Tc or q < qc. Left: the normalized BH entropy is plotted for xed temperature as a function of q=qc = q=(33:5TH ). Orange, green, and purple curves denote the normalized entropy for TH = 3=10, 3=20, 1=100, respectively. Right: the normalized entropy S = 2 =zh is plotted for xed charges as a function of TH =Tc. Orange, green, and purple curves denote the normalized entropy for q = 10, 4, and 1=2, respectively. In gure 2, we plot the Bekenstein-Hawking entropy of the hairy black hole S = (2 )2=( 2zh) in units of 2 = 2(= c2d=6), where c2d = 12 = 2 in units of L~ = 1 is the central charge of the CFT. In the uncondensed phase, the entropy grows like q2 when q=(gYMTH ) 1 (the left gure) and grows like a linear function of TH when q=(gYMTH ) 1 (the right gure). We nd a cusp at the critical phase transition where q = qc(= 33:5TH ). 3 Holographic entanglement entropy and the phase transition In this section we will calculate the entanglement entropy of an interval on the boundary in the y direction of length l, using the Ryu-Takayanagi formula [8{10]. The arc on the boundary is chosen to be l=2 y l=2. We then need to calculate the length of a geodesic with minimum length in the bulk metric on a constant time slice which joins the { 7 { two endpoints of the interval.3 Z IEE = dyIEE = Z dy 1 z s is useful to consider the action IEE as the integral of a Lagrangian with y being considered as the time. Since translations of y is a symmetry, the corresponding Hamiltonian is conserved. Thus, the equation of motion which follows from this action becomes a rst order di erential equation for z(y): HJEP09(217)6 s where z = z is the turning point, i.e. the point where z0(y) vanishes. The curve z(y) is assumed to be smooth everywhere. The length l of the interval can be now easily calculated We have assumed that the curve is symmetric under re ections about the turning point this gives the factor of 2 in front of lcurve A solution to the equation (3.2) with speci ed end points is a minimal length geodesic joining those points. According to the Ryu-Takayanagi formula the holographic entanglement entropy is the on-shell action of IEE divided by the gravitational constant SAEE = 2 2 IEEjon-shell = 4 Z 2 z dz z r 1 f2(z)h2(z) 1 z2 z2 : One can check that l ! s 1l under the rescaling (z; zh; z ; ) ! s 1(z; zh; z ; ). It is then convenient to introduce the scale invariant quantities lq or lT . Note that the holographic entanglement entropy is invariant under the scaling transformation varying (z; zh; z ; ). In equation (3.4) the boundary z = 0 has been replaced by a cuto boundary z = (which is essentially a UV cuto of the boundary eld theory). This regulates the UV divergence of the entanglement entropy, as discussed below. A We are interested in calculating SEE as a function of the charge q for a given temperature. As discussed above, for q < qc the relevant bulk geometry is a charged black hole, while for q > qc the background is given by a hairy black hole with a nonzero condensate wc, with the metric (2.5) in section 2. In both cases, the UV divergent (area law) term of IEE behaves like IEE minimal length, 2 log( ). This motivates the de nition of the nite part of the Im;fin IEE + 2 log( ) (3.5) 3For time dependent situations, one needs to consider an extremal geodesic [24]. { 8 { (3.1) (3.2) (3.3) (3.4) -3.8 -3.9 -4.1 curve is for a AdS3 charged black hole corresponding to the uncondensed phase, q < qc. The solid curve is for the condensed phase described by a hairy black hole. Im;fin is independent of the cuto . Under the scaling transformation discussed above, Im;fin changes only by the additive constant 2 log( s), which will not a ect our analysis qualitatively. We now use the metric obtained in the previous section for both the condensed and uncondensed phase to calculate Im;fin. This involves evaluation of the integral (3.4), which we perform numerically. In the uncondensed phase T > Tc or q < qc, we nd that the holographic entanglement entropy is proportional to the subsystem size l (a volume law) when the length l is large (lTH 1) exactly as in higher dimensions [25, 26]. In the opposite limit TH l 1 and ql 1, Im;fin approaches the value in pure AdS3, namely, SEE A 4 = 2 log(l=a) = c=3 log(l=a). These behaviors in the extreme limits are of course what is expected. Interesting non-trivial behavior is expected for T or q in the intermediate range [27], particularly when they are close to their values at the critical superconducting transition. length l as a function of q=qc for gY2M = 10 for a dimensionless length lTH = 0:019. The dashed curve is Im;fin in the uncondensed phase described by the AdS3 charged black hole. The solid curves is the result in the condensed phase. In both phases the nite part of the entanglement entropy increases with increasing charge. However there is a cusp at the critical point q = qc where dI(l;fin) is discontinuous. Note that in the condensed phase dq Im;fin is always smaller than the value of this quantity in the background of a charged black hole. This is consistent with the fact that condensation results in a depletion of the number of degrees of freedom. The monotonic increase of Im;fin with increasing charge continues to larger interval lengths till we reach a critical interval size lc1. For gY2M = 10 we have lc1TH 0:15. For l > lc1 there is a non-monotonic behavior: typically, Im;fin rst decreases beyond the cusp at the critical point, reaches a minimum and increases to reach a maximum, and then approaches a plateau. This kind of non-monotonic behavior is shown in gure ( 4 ) for an { 9 { HJEP09(217)6 2 gYM = 10 depicted in the left plot. Right: its derivative with respect to charge. One can see clearly two zeros: the rst one is minimum at q = 1:8qc, while the second one is maximum at q = 3qc. interval size lTH = 0:219. There are several notable aspects of the behavior. First, as we increase lTH the minimum of Im;fin is pushed to values of q away from the critical point q = qc. In fact the critical length lc1 is the value of l where this local minimum is exactly at the critical point. Secondly, Im;fin approaches a plateau at large q. The value of Im;fin at the plateau is larger than the value of Im;fin at q = qc for small interval lengths (still larger than lc1) whereas at larger interval lengths the value of Im;fin at the plateau is smaller than the value of Im;fin at q = qc. Finally, the di erence of the value of Im;fin and its minimum value decreases as we increase l. Consequently, there is another critical interval size lc2. For l > lc2 the quantity Im;fin decreases monotonically for q > qc. For gY2M = 10 we get lc2TH 0:22. At this value of l the minimum of Im;fin is pushed beyond the largest value of q for which we could do the numerics, while the plateau value becomes very close to the minimum value. This complex behavior is possibly a result of two competing trends. First, an increasing charge density tends to increase the entanglement entropy. This is clear in the uncondensed phase for all values of l. However condensation beyond q = qc results in a depletion of the number of degrees of freedom, and therefore tend to decrease the entanglement entropy. For small enough interval sizes, the e ect of condensation is not very pronounced, so that the rst trend wins leading to a monotonically increasing SAEE. In the dual gravity picture the condensate is pronounced near the horizon - the minimal geodesic stays far from the horizon for small intervals. This is consistent with the fact that at length scales which are signi cantly smaller than the scale of symmetry breaking, e ects of symmetry breaking are invisible. However as the interval size increases, the minimal geodesic goes deep into the bulk and the e ect of the condensate becomes pronounced. The intermediate region lc1 l lc2 is possibly characterized by the regime where these two e ects are of the same order and therefore compete. Finally, for l > lc2 the e ect of condensation overcomes the e ect of increasing charge density. The degrees of freedom keep decreasing as we increase the charge density resulting in a monotonic decrease of the SAEE. While we do not have a quantitative calculation to back the above scenario we can check its consistency by calculating Im;fin for di erent values of gY2M and therefore changing the strength of backreaction. As we will see, this turns out to give further insights into the nature of the critical values of the critical sizes of lc1 and lc2 and the mechanism of the competition. We present the gY2M = 10 case rst, followed by gY2M = 6 and gY2M = 50. The smaller the value, the stronger the back-reaction and the e ects of the competition mentioned above become more pronounced. The entanglement entropy SAEE , normalized as Im;fin(I; q)=Im;fin(I; qc), for gY2M = 10 with 6 di erent sub-system sizes is presented in gure 5, so that all the SEE coincide at the critical point q = qc. In the left gure of gure 5, we show the behavior of entanglement entropy as we increase the sub-system size from the top plot to the bottom plot. The right A gure is the slope of the entanglement entropy with respect to the charge, dIm;fin(I; q)=dq. In the range q > qc, the slope is positive for lTH = 0:14 and thus entanglement entropy increases monotonically, while the slope is negative for lTH = 0:23 with monotonically decreasing entanglement entropy. The entanglement entropy for 0:21 . lTH . 0:22 have non-monotonic behaviors revealing two zeros of its derivative. This is shown in the right panel of gure 6. As we increase the charge, entanglement entropy decreases, hits a minimum ( rst zero of the derivative), increases, arrives at a maximum (second zero of the derivative), and eventually reaches a plateau. Typically, the second zero corresponding to a local maximum happens at a very large value of q=qc. Finally, we clearly see that the minimum of the entanglement entropy moves to q = qc for lTH = 0:15, while for lTH = 0:22 the minimum and the maximum of the entanglement entropy coincide, forming a point of in ection. Similarly, we present entanglement entropy for gY2M = 6 in gure 7 and for gY2M = 50 in gure 8 with di erent sub-system sizes. The entnaglement entropy shows behaviors similar to those at gY2M = 10. For gY2M = 6, non-monotonic behavior can be observed for 0:09 . lTH . 0:18 with two critical points located at lc1TH = 0:09 and lc2TH = 0:18. For gY2M = 50, non-monotonic behavior exists for 0:22 . lTH . 0:29 that is sandwiched by two critical points lc1TH = 0:22 and lc2TH = 0:29. Combining together we see the following pattern. 2 gYM = 6 2 gYM = 10 2 gYM = 50 2 gYM = 250 lc1TH 0:09 0:15 0:22 0:25 lc2TH 0:18 0:23 0:29 0:30 lTH = lc2TH lc1TH 0:09 0:08 0:07 0:05 As we increase the strength of the back reaction, decreasing gY2M, both critical lengths lc1TH and lc2TH decrease while the invariant distance between them, lTH = lc2TH lc1TH , increases. 3.1 A phase diagram As discussed above, the non-monotonic behavior of the entanglement entropy is a result of a competition between two e ects: the e ect of charge density and the e ect of condensation. It is useful to re-cast this in terms of two length scales which emerge in the problem. Im1.,f1in5(l, qc) 1.10 0.9 1 2 3 4 5q/qc 2 3 4 5q/qc system sizes lTH (from top to bottom, lTH = 0:14; 0:15; 0:18; 0:21; 0:22; 0:23) with a xed 2 temperature TH = 0:15 and gYM = 10. The dashed and solid curves are for uncondensed and condensed phases, respectively. HJEP09(217)6 lTH=0.14 lTH=0.15 lTH=0.18 lTH=0.21 lTH=0.22 for di erent invariant sub-system sizes lTH . Right: more details of the second zeros by changing 2 3 4 5q/qc 1.0 1.5 2.0 2.5 3.0q/qc -0.1 qc / TH = 45.1 entanglement entropy for 6 di erent invariant sub-system sizes lTH (from top to bottom, lTH = 0:09; 0:11; 0:12; 0:17; 0:18; 0:19) with a xed temperature TH = 0:15. The dashed and solid curves are for uncondensed and condensed phases, respectively. dq 0.10 lTH=0.21 lTH=0.23 lTH=0.26 lTH=0.27 lTH=0.29 lTH=0.3 HJEP09(217)6 1Im.0,f1in5(l, qc) 1.010 dq entanglement entropy for 6 di erent invariant sub-system sizes lTH (from top to bottom, lTH = 0:21; 0:23; 0:26; 0:27; 0:29; 0:3) with a xed temperature TH = 0:15. The dashed and solid curves are for uncondensed and condensed phases, respectively. Let us recapitulate the salient features of our results in Fig 3 - Fig 8. These are all plots of the entanglement entropy as a function of the charge density, the di erent curves being results for di erent values of (lTH ). For a given value of gYM we have the following: For l > lc2 the entanglement entropy decreases monotonically as a function of q=qc At l = lc2 there is a point of in extion at q = q? For lc2 > l > lc1 the entanglement entropy has a minimum at some value qmin(l) and a maximum at a larger value qmax(l). The minimum qmin(l) increases as a function of l. At l = lc1 one has qmin(lc1) = qc. On the other hand, qmax(l) decreases with increasing l, reaching its minimum value q? at l = lc2. Thus at l = lc2 the maximum and the minimum merge into a point of At l = lc1 the minimum is at q = qc For l < lc1 the entanglement entropy increases as a function of q at least upto q 5qc which is the maximum value of q used in our calculation. It is then clear that the minimum of the entanglement entropy as a function of the charge can occur only for q < q?, while the maximum of the entanglement entropy as a function of the charge can occur only for q > q?. To rephrase the behavior in terms of length scales it is useful to plot the quantity We now de ne two length scales Lc; Lq, which depend on the charge density, as follows. q < q? q > q? (3.6) 0.10 0.05 -0.05 l TH l c2 TH l c1TH q=qc. The left panel shows the results for q < q?, where q? is in between q=qc = 2 and q=qc = 3. The zero of @Im;fin=@q in this regime of charge is a minimum (as a function of q), and the length scale Lc is de ned as the value of l at which @Im;fin=@q = 0. Clearly, Lc increases as the charge density increases. The right panel shows the results for q > q?. Now the zero of @Im;fin=@q is a maximum (as a function of q). The length scale Lq is de ned as the value of l at which @Im;fin=@q = 0. Clearly Lq decreases with increasing q. HJEP09(217)6 Lc Lq q = q c q * q c q q c the tent like region below these curves the derivative (@Im;fin=@q)l > 0, outside the tent this is negative. From gure (9) it is clear that Lc increases with the charge density while Lq decreases with the charge density. At q = q? one has Lc(q?) = Lq(q?) = lc2, which is the maximum possible value of either of Lc; Lq. These two length scales and their dependence on the charge density can be used to chart out a \phase diagram" for the entanglement entropy. Figure ( 10 ) shows the general nature of the phase diagram, parameterized by the length of a subsystem (vertical axis) and the charge density q=qc (horizontal axis). The solid lines denote the functions Lc(q=qc) and Lq(q=qc) - as discussed above the former quantity exists for q < q? while the latter quantity exists for q > q?. Note these curves are not real data - they are drawn to provide an impression of the general behavior. The two curves meet at q = q?, forming a \tent". At all points above this tent we have (@Im;fin=@q)l < 0, while at points inside the tent we It is natural to associate the scale Lc(q) with the condensation and the scale Lq(q) with the charge density. As remarked earlier, when the size of the subsystem is small, the entanglement entropy does not feel the e ect of condensation which increases as the charge density increases. This explains why the region for small l has (@Im;fin=@q)l > 0. When l is large enough, there are two competing e ects - condensation tends to reduce the entanglement entropy while the charge tends to increase it. For very large l the e ect of HJEP09(217)6 condensation dominates the physics and the entanglement entropy decreases as a function of charge since increasing charge leads to more condensation. As remarked at the end of the last section, the association of Lc(q) with condensation is supported by the fact that for smaller gYM the value of lc1 decreases so that smaller subsystems can feel the e ect of condensation. The phase diagram gure ( 10 ) is a way to express the competition of two physical e ects: the sign of (@Im;fin=@q)l is a kind of \order parameter" which distinguishes two di erent physical behaviors. It would be interesting to obtain a quantitative physical understanding of this rather novel phase diagram. 4 Discussions In this paper we have used holographic methods to examine how the entanglement entropy in a 1+1 dimensional eld theory behaves as we change parameters such that the system crosses a critical point. The critical point in question separates a normal phase and a pwave superconducting phase. Speci cally, we studied the dependence of the entanglement entropy on the charge density for a given temperature. We found a rich behavior depending on subsystem size, with regimes of non-monotonicity. While we do not have an analytic understand of this behavior we have o ered a qualitative explanation based on the competition between two opposing factors: the tendency of the entanglement to increase with increasing charge density and the depletion of degrees of freedom with increasing charge density in a condensed phase. It will be interesting to consider the entanglement entropy of the phase transition for the s-wave models with Maxwell and scalar elds. We expect that some of the characteristic properties we have found for our p-wave model will present there as well. However to say anything de nite we need to perform detailed calculations. It would also be interesting to see how the entanglement entropy behaves when we dynamically go across the critical point as in a quantum quench. In particular we would like to understand possible universal scaling of the entropy as a function of the quench rate. While some results about such scaling behavior are known in solvable and integrable eld theories [28{32] , very little is known in strongly coupled systems. In the past, holographic methods have been useful to understand scaling of one point functions in various regimes of the quench rate [33]. The setup used in this paper should be useful in extending the discussion to entanglement entropy. Finally, it will be interesting to explore in detail the stringy embedding of the 3d toy model (2.1). In the probe limit of avor branes, the toy model (2.1) corresponds to a D3 D3 system of type IIB string theory [18] without backreaction. In this limit the dilaton is a constant, which is why it is omitted in (2.1). In the presence of backreaction, the dilaton runs, making the problem much more di cult. It would be interesting to see if the strategy of [34, 35] which expresses the leading order correction to the entanglement entropy in terms of the energy momentum tensors of the probe brane and the minimal surface and the bulk graviton propagator can be used to calculate this. Such a calculation would also need to include the backreaction of the four form gauge eld sourced by the D3 branes. We hope to pursue this in the near future. Acknowledgments We would like to thank for Matteo Baggioli, Akikazu Hashimito, Song He, Matthias Kaminski, and Elias Kiritsis for helpful discussions. This work is partially supported by the grants NSF-PHY-1521045. SRD would like to thank Tata Institute of Fundamental Research for hospitality during the completion of this paper. Open Access. 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Sumit R. Das, Mitsutoshi Fujita, Bom Soo Kim. Holographic entanglement entropy of a 1 + 1 dimensional p-wave superconductor, Journal of High Energy Physics, 2017, 16, DOI: 10.1007/JHEP09(2017)016