Entanglement Wedge Cross Section with Gauss-Bonnet Corrections and Thermal Quench
EEntanglement Wedge Cross Section with Gauss-BonnetCorrections and Thermal Quench
Yong-Zhuang Li , ∗ Cheng-Yong Zhang , † and Xiao-Mei Kuang ‡ School of Science, Jiangsu University of Science and Technology, Zhenjiang 212003, China Department of Physics and Siyuan Laboratory,Jinan University, Guangzhou 510632, China and Center for Gravitation and Cosmology,College of Physical Science and Technology,Yangzhou University, Yangzhou 225009, China
Abstract
The entanglement wedge cross section (EWCS) is numerically investigated both statically anddynamically in a five-dimension AdS-Vaidya spacetime with Gauss-Bonnet (GB) corrections, fo-cusing on two identical rectangular strips on the boundary. In the static case, EWCS arises asthe GB coupling constant α increasing, and disentangles at smaller separations between two stripsfor smaller α . For the dynamical case we observe that the monotonic relation between EWCSand α holds but the two strips no longer disentangle monotonically. In the early stage of thermalquenching, when disentanglement occurs, the smaller α , the greater separations. As time evolving,two strips then disentangle at larger separations with larger α . Our results suggest that the higherorder derivative corrections also have nontrivial effects on the EWCS, so do on the entanglementof purification in the dual boundary theory. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ h e p - t h ] M a r ontents I. Introduction II. Review of HEE in Einstein-Gauss-Bonnet theory III. Setup of EWCS with Gauss-Bonnet correction IV. EWCS in static Gauss-Bonnet black brane V. EWCS in Vaidya Gauss-Bonnet black brane VI. Conclusions and Discussions Acknowledgments Appendix References I. INTRODUCTION
The interdisciplinary research among the quantum information, condensed matter andquantum gravity attracts plenty of attention in the last decade. Gauge/gravity duality [1–3]is an important bond among those connections. Moreover, holography is also a powerful toolto investigate various involved physical quantities, especially in strongly correlated systemsand quantum information theory.In this area, entanglement is one of the significant concepts in quantum field theory andinformation theory. For a pure state, entanglement entropy (EE) measures the quantumcorrelations between a subsystem A and its complement B . But in most cases, it is verydifficult to compute EE with the use of quantum field theory(QFT) techniques. Fortunately,holography indeed simplify this problem and it provides an elegant geometric duality ofEE from gravity side. Specially, for a spatial region A in the boundary field theory, thegeometrical description of EE between A and its complement B was proposed as Ryu-Takayanagi (RT) formula [4–6], S A = Area ( γ A )4 G N , (1)where G N is the bulk Newton constant of N − dimensional Einstein-Hilbert gravity, and γ A is the codimension − A . The RT formula is then extended as Hubeny-Rangamani-Takayanagi (HRT) formula in the covariant case[7, 8].However, when A and B are two disjoint subsystems, EE is not a convenient quantity tomeasure the correlation as the total system is not pure but mixed state. In this case, the wellknown physical quantity to describe both the classical and quantum correlations betweenthe subsystems A and B is the mutual information (MI), which is defined as I ( A, B ) = S A + S B − S A ∪ B . MI is closely related to EE and it is the linear combination of EE, butMI is free from UV divergences and subadditivity guarantees its non-negativity. Moreover,2n CFTs MI has power to extract more refined information than EE [9, 10]. Since MI isdefined in terms of EE, it is straightforward to employ the (H)RT surfaces to study MI inholographic framework.Besides EE and MI, more physical quantities have been constructed to measure the cor-relations between the subsystems A and B , especially for the mixed state, for instance,entanglement of purification( E P )[11], reflected entropy ( S R )[12], odd entropy( S O )[13] andlogarithmic negativity( ξ )[14]. Their definitions in quantum field theory are briefly reviewedin the Appendix. In the framework of holography, all these correlation measures are con-nected with the entanglement wedge cross section as E W = Area (Σ min AB )4 G N , (2)where Σ min AB denotes the minimal cross sectional area of the entanglement wedge. E P and S R are both good measures of total correlations between two disjoined subsystems. As shownin the Appendix, their definitions depends on the purification schemes of the mixed states.Their holographic connections with EWCS were investigated in [15] and [12] where theauthors conjectured E P = E W and S R = 2 E W , respectively. S O as a new measure of totalcorrelations for mixed states has a holographic duality as S O ( A, B ) = S ( A, B ) + E W ( A, B ).However, ξ captures only the quantum correlation for mixed states, which is different from E P , S R and S O . The conjectured holographic relation between the logarithmic negativityand EWCS was studied in [14, 16], where it showed that for the vacuum state and ball-shaped subregions, ξ = χ d E W with χ d a constant determined by dimension of the theory.Besides, the authors of [17] associated the EWCS to entanglement distillation with the useof bit threads.So it is obvious that EWCS is a very important bulk geometry description of the corre-lations of mixed state in the dual boundary theory. More efforts of analytical or numericalstudies in this direction have bee made in [18–29] and therein. The main goal of this paperis to investigate E W with Gauss-Bonnet (GB) corrections. Einstein-Gauss-Bonnet gravitytheory is a well known higher-derivative gravity, and it has been generally accepted thathigher-derivative gravity theory provides corrections to general relativity in the frameworkof gauge/gravity duality. The presence of higher-order terms in the bulk theory implies anappearance of new couplings among quantum operators in a holographic boundary theory.We expect the present study shall shed light to the correlations of mixed state in the bound-ary theory dual to Einstein GB gravity theory. Practically, in this work we shall mainlyemploy the conjecture E P = E W and study the properties of the entanglement measure, E P ,of the boundary theory dual to the bulk Einstein-Gauss-Bonnet theory.On the other hand, people have great interest in non-equilibrium dynamics and thermal-ization of a system in quantum field theory, because they can be used to describe processesin various areas of physics. It is significant to have a deeper understanding of thermal-ization, through which the physical quantities of the system reach equilibrium values aftercertain perturbation acts upon its initial equilibrium state. The thermalization is alwaysimplemented by a quench process in the system, and it is holographically described by theblack hole formation from the gravitational collapse, which is widely employed as an effectivemodel to study thermalization process, see [30, 31] as a review. Moreover, a mixed stateshall remain mixed under time evolution, so it is of great interest to explore how the finalmixed state will resemble a thermal one. In conformal field theory, the dynamical evolutionof reflected entropy,logarithmic negativity and odd entropy after global/local quench and3heir holographic dual have been analyzed in [32–35]. Moreover, the time evolution of E W in Vaidya geometry has been investigated in [19, 24].The exact Vaidya type black brane of GB gravity has been constructed in [36], which pavesthe way to study the holographic thermalization of the dual boundary theory. The propertiesof holographic EE and mutual information during the thermalization with GB correctionwere explored in [37–39]. In this paper, besides studying E W in static GB gravity, we shallalso study E W in the equilibration process after a quantum quench which is implementedby GB Vaidya theory.The plan of this paper is as follows. In section II, we review the general Einstein GBblack brane solution and the computation of its HEE. Then the construction of HMI andEWCS of the theory is present in section III. In section IV and V, we show the numericalresults of EWCS and discuss the properties affected by GB corrections for static GB andVaidya GB background, respectively. The last section is our conclusion and discussion. II. REVIEW OF HEE IN EINSTEIN-GAUSS-BONNET THEORY
For d + 1 ( d ≥
4) dimension Einstein-Gauss-Bonnet gravity dual to a d − dimensionalboundary CFT, the action is given by I = 116 πG d +1 N (cid:90) d d +1 x √− g [ R −
Λ + α L ( d − d − L GB ] , (3)where Λ = − d ( d − L , (4) L GB = R − R µν R µν + R µνρσ R µνρσ . (5) α is the GB coupling constant and L is the AdS spacetime radius (hereafter we will set L =1). It is well-known that there exists a constraint − ( d − d +2)4( d +2) ≤ α ≤ ( d − d − d − +3 d +2]4[( d − + d +2] restricted by the causality of dual field theory on the boundary or the positivity of the energyflux in the CFT analysis [40–43]. The above action admits a d + 1 dimensional static blackbrane solution [44] ds = − r f ( r ) dt + 1 r f ( r ) dr + r L AdS d x , (6)where f ( r ) = 12 α (cid:34) − (cid:114) − α (1 − r dh r d ) (cid:35) , L AdS = (cid:115) √ − α . (7) r h is the event horizon radius and L AdS is the effective AdS radius. x = ( x , x , · · · , x d − )corresponds to the spatial coordinates on the boundary. With the Eddington-Finkelsteincoordinates, the above solution can be expressed as ds = 1 z (cid:20) − f ( z ) dv − dzdv + 1 L AdS d x (cid:21) , (8)4here f ( z ) = 12 α (cid:34) − (cid:115) − α (cid:18) − z d z dh (cid:19)(cid:35) , (9) dt = dv + dzf ( z ) , r = 1 z , (10)As in the traditional way, for a strip A in the d -dim boundary strongly field theorywith one dimension x of length l and the other d − H d − , with aGauss-Bonnet gravity dual its EE should be modified as S ( V ) = 14 G d +2 N (cid:20)(cid:90) Σ d x d − √ h (cid:18) α ( d − d − R Σ (cid:19) + 4 α ( d − d − (cid:90) ∂ Σ d x d − √ σ K (cid:21) , (11)where Σ indicates a minimal surface which extends into the bulk and shares the boundarywith A , i.e. , ∂ Σ = ∂A . V stands for the volume of the bulk inclosed by Σ and A . R Σ isthe induced scalar curvature of surface Σ, σ is the determinant of the induced metric of theboundary ∂ Σ. K is the trace of the extrinsic curvature of ∂ Σ [45, 46].The induced metric of Σ is thus given by ds = h ij dx i dx j = 1 z (cid:20) − f ( z )( v (cid:48) ) − z (cid:48) v (cid:48) + ( x (cid:48) ) L AdS (cid:21) du + 1 z L AdS d y . (12)The prime here indicates the derivative with respect to u which parameterizes the minimalsurface Σ. x and y stand for the dimension x and other d − R Σ is thus given by R Σ = − ( d − { ( d + 1) h uu ( u ) z (cid:48) ( u ) + z ( u )( h (cid:48) uu ( u ) z (cid:48) ( u ) − h uu ( u ) z (cid:48)(cid:48) ( u )) } h uu ( u ) z ( u ) , (13)while by choosing the normalized unit vector n a = (cid:112) h uu ( u ) δ ua , we have K = − d − (cid:112) h uu ( u ) z ( u ) z (cid:48) ( u ) . (14)Finally, Eq. (11) can be rewritten as S ( V ) = H d − G d +1 N (cid:90) Σ du (cid:34) (cid:112) h uu ( u ) z ( u ) d − L d − AdS + 2 αz (cid:48) ( u ) z ( u ) d L d − AdS (cid:112) h uu ( u ) (cid:35) , (15)where H d − indicates the volume of strip on other dimensions. We will consider the d = 4case and use x as the parameter, i.e., x (cid:48) ( u ) = 1. We consider the very large H such that the lengths of other dimensions will not affect the behavior of theentanglement entropy. C B Σ ABmin Γ ACB Γ A Γ B Γ C FIG. 1: The schematic configuration for computing E W with the minimal surface Σ minAB whenthe entanglement wedge is connected. The two strips A and B are identical with width (cid:96) . Theseparation between two strips is labeled as C with width s . Γ indicates the extremal surface foreach strip. III. SETUP OF EWCS WITH GAUSS-BONNET CORRECTION
We consider two identical strips A and B in the boundary with width (cid:96) . The distancebetween two strips is labeled as C with width s . The HMI is defined as I [ A, B ] = S [ A ] + S [ B ] − min ( S [ A ∪ B ]) , (16)In the following sections, we will build the coordinate system in such a way that A and B is symmetric about the z -axis.By the definition, the entanglement of purification is holographically modeled by thewedge cross section in an AdS gravity [15]. However, to investigate the wedge cross sectionin an AdS-GB gravity, one may have to consider the effects of the boundary term. Con-sequently, there are two possible definitions for the EWCS. Note the entanglement wedgecross-section is only non-vanish when two sub-regions are connected, which will be the casewe discuss.The first one is directly given by the standard definition, i.e., the extremal surface definedby the induced metric without the Gibbs-Hawking boundary term, which is mathematicallygiven by S ( V ) = (cid:90) m √ hdu, (17)with integral range from point p on Γ( C ) to point p on Γ( A + B + C ).The second choice is the standard definition modified by the Gibbs-Hawking term. Weclaim that this definition should be reasonable. Physically, the extremal surface corre-sponding to the entanglement wedge cross-section should be part of a HEE extremal surfacetruncated by point p on Γ( C ) and p on Γ( A + B + C ). The intersection conditions are givenby z ⊥ ( p ) ⊥ Γ( C ) and z ⊥ ( p ) ⊥ Γ( A + B + C ) where we use z ⊥ ( p ) indicates the orthogonalvector of Γ at point p . It is well known that to investigate HEE in an AdS-Gauss-Bonnetgravity one has to concern the effects from the boundary Gibbs-Hawking term. This isthe intrinsic sample properties of holography, i.e., the boundary field quantities are repre-sented by the bulk geometric quantities. So the entanglement of purification with higherderivative corrections in physical intuitions should be identified by the entanglement wedgecross-section with the boundary effects. 6ince the two subsystems are identical, the minimal surface along the radial geodesic z (0)from Γ( C ) to Γ( A + B + C ) actually lies in ( z, v ) plane. So the induced metric is given by ds eop = 1 z (cid:20) − f ( z, v ) − dzdv (cid:21) dv + 1 z L AdS d y . (18)According to Eq. (15) the volume of Σ minAB can be presented as˜ S eop ( V ) = (cid:90) Σ du (cid:34) (cid:112) h uu ( u ) z ( u ) d − L d − AdS + 2 αz (cid:48) ( u ) z ( u ) d L d − AdS (cid:112) h uu ( u ) (cid:35) , (19)where we have absorbed the constant coefficient 4 G d +1 N H − d into ˜ S eop ( V ). Considering therelation (10), for static case one has˜ S steop ( V ) = (cid:90) dzz d − L d − AdS (cid:104)(cid:112) f ( z ) − + 2 α (cid:112) f ( z ) (cid:105) . (20)For dynamical case we have˜ S dyeop ( V ) = (cid:90) v ACB v C dvz d − L d − AdS (cid:114) − f ( z, v ) − dzdv + 2 α (cid:18) dzdv (cid:19) (cid:32)(cid:114) − f ( z, v ) − dzdv (cid:33) − , (21)where v C and v ACB should be chosen within the same time slice. For α = 0, this equationreduces to the standard representation of holographic EoP [19]. IV. EWCS IN STATIC GAUSS-BONNET BLACK BRANE
For static Gauss-Bonnet black brane, the re-normalized HEE is given by S ( V ) = (cid:90) Σ dx (cid:20) Φ( x ) z ( x ) L AdS + 2 αz (cid:48) ( x ) z ( x ) L AdS Φ( x ) (cid:21) − S div ( V ) , (22)Φ( x ) = (cid:115) L AdS + z (cid:48) ( x ) f ( z ( x )) , (23)where S ( V ) ≡ G d +1 N S ( V ) /H d − and S div ( V ) is the divergent term due to the divergence ofthe extremal surface near the boundary.By extremizing this action, we could obtain a cumbersome equation of motion which wewill not present here. The numerical method has to be admitted as it will be impossible tofind an analytical solution for such an equation. The initial conditions should be set in sucha way that z | boundary → (cid:15) with (cid:15) an UV cut-off, and z ( P ) → z ∗ , z (cid:48) ( P ) → P indicatingthe twist point of the extremal surface. Benefiting from the translation invariant along the x direction, we only need to numerically find the relation between the width of strip and z ∗ once for all.Following Fig. 1, the renormalized HMI (RHMI, hereafter) is then defined by I ( A, B ) = 2 S ( (cid:96) ) − min [2 S ( (cid:96) ) , S (2 (cid:96) + s ) + S ( s )] . (24)7 .10 1.12 1.14 1.16 1.18 1.20 1.22020406080100120 ℓ / R H M I s / R E W C S FIG. 2: (Left) The renormalized HMI with the width of strip for different α , which shows amonotonic relation between RHMI and α . From top to down α = 0 . , , − . , − . , − .
19. (Right)The renormalized EWCS with the distance of two strips for different α , which shows a monotonicrelation between EoP and α . From right to left α = 0 . , , − . , − . , − . To perform the numerical calculation, we will first set the fixed 2 (cid:96) + s but with free (cid:96) . Theinitial conditions are chosen as 2 (cid:96) + s = 2 . , z = v = 0 .
05. In Fig. 2 we plotted RHMIand renormalized EWCS (REWCS) with different α . For both cases, a monotonic relationshows that with larger α the corresponding RHMI or REWCS is larger. REWCS showsa discontinuous phase transition at certain separation ( s c ) between trips, and for s < s c ,REWCS is positive while it is zero for s > s c . The vanishing of REWCS stems from thedisconnection between the RT surfaces of A and B such that the related entanglement wedgeis empty. Especially, the phase transition between entangling phase and disentangling phaseoccurs at larger s c with larger α .To demonstrate the effects of coupling constant on the discontinuous phase transition ofRHMI (and thus REWCS) more clearly we need to go further. In Fig. 3 we plot the criticalcurves where RHMI is zero for different α . According to the plot, for fixed separation s the nonzero RHMI and REWCS only exists for the shadow region above the critical curve.Comparing the critical curves with different α , the curves are almost coincident for smallseparation and width, and they become different as s and (cid:96) become larger. It is obviousthat for larger α , the parameters range with nonvanishing REWCS is wider.Another feature shows that for each α there exists a special s ∗ where for s > s ∗ thetwo subregions are disentangling no matter how large the subregions are. Limited by theaccuracy of our numerical method we can roughly claim that s ∗ (cid:38) (0 . , . , . , . , .
21) for α = (0 . , , − . , − . , − . . (25)An relation between s ∗ and α can be found. With the monotonicity of HEE, if HMI iszero for very large (cid:96) then it must be zero for strips with smaller widths, too. Assuming During the numerical calculation we have chosen the step size of separation as δs = 0 .
02 which is notsmall enough to obtain an accurate numerical solution for s ∗ . But such a step size is good enough todescribe the behaviors of RHMI and REWCS. These numerical results are consistent with Ref.[19]. .05 0.10 0.15 0.20 0.250.00.20.40.60.8 s / ℓ / FIG. 3: Nonzero region of RHMI for α = − . , − . , − . , , . s , the RHMI shows a discontinuous phase transition when (cid:96)/ that the width of strip (cid:96) → ∞ , then the area of the corresponding extremal surface can berepresented as Γ A = ˜Γ A + 2 Γ → where ˜Γ A presents the area around the horizon ( z h = 1)and Γ → indicates the area of the radial subsection given by Eq. (20). The area Γ ACB canbe expressed as Γ
ACB = 2 (cid:16) ˜Γ A + Γ → (cid:17) + ˜Γ C where we use ˜Γ C to present the area aroundthe horizon for C with width s ∗ . So the condition HMI= 0 leads us to the following relation:˜Γ C + Γ C = 2 Γ → . (26)One can easily obtain an expression for ˜Γ C . Since it indicates the area around horizoncorresponding to boundary width s ∗ so for boundary parameter x ∈ [ − s ∗ / , s ∗ / z ( x ) = 1, z (cid:48) ( x ) = 0, ˜Γ C = (cid:90) s ∗ / − s ∗ / dx L AdS = s ∗ L AdS . (27)Unfortunately, Γ C is still a complicated function about α and s ∗ which is too challengingto find its analytical expression. V. EWCS IN VAIDYA GAUSS-BONNET BLACK BRANE
To investigate the evolutionary properties of the EoP via EWCS with GB corrections,we will adopt the so-called the thermalization process, which is conventionally modeled bya homogeneous falling thin shell of null dust in the bulk. We invoke a Vaidya-type solution[36, 44]: ds = 1 z (cid:20) − f ( v, z ) dv − dzdv + 1 L AdS d x (cid:21) , (28)9here f ( v, z ) = α [1 − (cid:112) − α (1 − m ( v ) z d )]. The mass function is m ( v ) = M (cid:20) (cid:18) vv (cid:19)(cid:21) , (29)where M denotes the mass of the black brane outside the shell, i.e. , v > v . v represents afinite shell thickness. For the sake of numerical calculation we hereafter choose M = 1 and v = 0 . v [30, 31].In this sense, the mass function should be chosen as an unit step function, leading to ∂ v f = 0except at the point v = 0. Suppose that the Euler-Lagrange equation from the action (21)has a solution given by Q ( v, z ( v )) = dzdv . (30)Subtracting this solution into the Euler-Lagrange equation leads to( f + 2 Q ) ( P − P ) = 0 , (31) P = ( f + Q )[2( d − f ( f + 2 Q + 2 Q α ) − ( f + 2 Q + 6 Q α ) z ∂ z f ] ,P = [ f + 2 Q + 2 α Q (2 f + 3 Q )] z ∂ v f. Since theoretically ∂ v f = 0 except the points near v = 0 so the above equation holds withthe following conditions: f + 2 Q = 0 , or P = 0 . (32)Note there is no “and” between two conditions as if they both hold then f will no longer afunction of v .Subtracting the first condition into the action then one gets a zero denominator, so thiscondition does not hold. The second condition is composed of two more conditions, f + Q = 0 , or 2( d − f ( f + 2 Q + 2 Q α ) − ( f + 2 Q + 6 Q α ) z ∂ z f = 0 . (33)However, it is easy to check that the latter condition could never be fulfilled in our setup.Finally, we have a solution of the Euler-Lagrange equation given by − f ( v, z ( v )) = dzdv . (34)So with the mass function given by (29) the action (21) only gives a solution z ( v ) closeto the real extramal surface , so does Eq. (34). It is noted that as v →
0, the contributionof the points near v = 0 to the HEE (21) could be negligible because the integral functionaround v = 0 is finite. With smaller v , one gets a z ( v ) closer to the real extramal surface.Hereafter, we argue that it is meaningful to use solution (34) as an approximate extramalsurface. Finally, the holographic EoP can be represented as˜ S dyeop ( V ) = (cid:90) z u ∗ z b ∗ dzz d − L d − AdS (cid:104)(cid:112) f ( v, z ) − + 2 α (cid:112) f ( v, z ) (cid:105) , (35) Here, we use the word “real” to present the extramal surface given by the limit v → .0 0.5 1.0 1.5 2.069.570.070.571.0 t H M I s = t H M I s = t H M I s = FIG. 4: The evolutions of HMI for different α with fixed s = 0 . , s = 0 . , s = 0 .
46 (from leftto right), respectively. For each panel, from top to bottom the coupled constant α is chosen as α = 0 . , . , . , , − . , − . , − .
05. Furthermore, in this figure we have rescaled HMI inthe first and second panel as HMI = HMI α − (HMI consα − HMI cons − . ) + 5 α . HMI α indicates thenon-rescaled values of HMI with the coupled constant α , while HMI consα is its equilibrium value. where “ u ” and “ b ” stand for z ∗ at Γ ACB and Γ C , respectively. Again, both z b ∗ and z u ∗ shouldbe calculated at the same boundary time.Labeling the radial position of the shell as z v = z v ( t ), then the radial EoP in this caseshould be expressed according to z v :1). For 0 < z v < z b ∗ , the shell has not reach point z b ∗ , the radial EoP should be given byEoP t = (cid:90) z t ∗ z b ∗ dzz d − L d − AdS (cid:18) αL AdS (cid:19) ; (36)2). For z b ∗ < z v < z t ∗ , the location of shell is between ( z b ∗ , z u ∗ ), and the EoP is given byEoP t = (cid:90) z v z b ∗ dzz d − L d − AdS (cid:104)(cid:112) f ( z ) − + 2 α (cid:112) f ( z ) (cid:105) + (cid:90) z t ∗ z v dzz d − L d − AdS (cid:18) αL AdS (cid:19) ; (37)3). For z t ∗ ≤ z v , the system has reached the equilibrium, and the radial EoP is directlygiven by Eq. (20).We first present the qualitative evolutionary of the unrenormalized HMI and EWCS. Bynumerical method, we set the initial conditions as z (0) = z ∗ , v (0) = v ∗ , z ( ± (cid:96) z , v ( ± (cid:96) t , z (cid:48) (0) = v (cid:48) (0) = 0 , (38)where z is the UV cut-off and t is the boundary time. Hereafter, we set z = 0 .
02. Theequations of motion are given by Eq. (15) with f ( z ) replaced by f ( z, v ). The couplingconstant is chosen as α = ± . , ± . , ± . , , respectively.Fig. 4 presents the evolutionary of HMI with different separations s between strips, show-ing a monotonic behavior between HMI and the coupling constant α . Such behaviors havebeen discussed in pioneering literature [38]. Besides, with time evolving, for small separa-tions the two strips are always entangled till the equilibrium. For large enough separations11he two strips are initially entangled then being disentangled, see the right panel in Fig. 4.Especially, the first two panels show that the equilibrium time increasing as α decreasingfrom 0 .
05 to − .
05, while the third panel shows the contrary behavior.The evolution of EWCS for different α with fixed separations is shown in Fig. 5. For theleft and middle panels the strips are entangled with small separations during the thermalquench. EWCS behaves as a constant at the early stage of evolution, and monotonouslyincreases as time evolving in the middle region, then enters an “unphysical region” (braydashed lines) -which we will explain later- before becoming constant again as the thermalequilibrium being reached. The left panel of Fig. 6 presents the constant behavior at earlystage by taking α = − .
05 as an example. The right panel of Fig. 5 displays behaviorsbeing consistent with the right panel in Fig. 4, i.e., the two strips are only entangled beforea transition time for adequate range of s . For larger enough s the two strips will completelydisentangled. Furthermore, such a transition time is shortened as α decreasing.To explain the unphysical region during the evolution, we plotted the position of shellas time evolving by taking α = − .
05 as an example in the right panel of Fig. 6. Theboundary region is chosen as 2 (cid:96) + s = 2 .
02. The brown line indicates the equilibrium z equ ∗ ofthe extreme surface, i.e., the turning point of Γ ACB in Fig. 1. The curve of z v intersects z equ ∗ at t ≈ .
2, which then enters the unphysical region. Mathematically, after z v ≥ z v intersects z equ ∗ is different with the time when thecurve of EWCS enters unphysical region. This is the consequence of the mass function m ( v )we have chosen. Theoretically, the shell should be zero thickness with certain mass, as theenergy injected instantaneously. But in the conventional numerical calculations we have tochoose v (cid:54) = 0. In our case v = 0 .
02, so by choosing the mass function as Eq. (29) one has m ( v ≤ − . ≈ m ( v ≥ . ≈
1. However, v = 0 presents the position of the shell,then m (0) = 1 /
2, which leads f ( z, v ) to be a real valued function between z ∈ [ z , . z v = (cid:39) . t (cid:39) .
58, making two time points coincide.We also investigated the relation between EWCS and the separation s at different timeslices t = 0 . , . , . , .
99 for different α , see Fig. 7. The first case, t = 0 .
01, indicatesthe beginning of the thermal quench, where the spacetime is pure AdS with GB corrections.The metric (6) then can be rescaled as ds = − ˜ r dt + L AdS d ˜ r ˜ r + ˜ r d x , (39)where ˜ r = r /L AdS . Such rescaling suggests that the GB corrections will not affect the phasetransition of HMI in the pure AdS-GB spacetime. Consequently, for different α EWCS willtransition from entangled phase to disentangled phase at same separation s .For cases t = 1 .
01 and t = 1 .
51, the EWCS is divided into two parts as the shell falling.The significant difference is that the transition separations are no longer monotonic with α , which is coincident with the evolutionary properties of HMI presented in Fig. 4. Forexample, when s = 0 .
46 the HMI becomes zero for different α around t ≈ .
5. While when t = 1 .
51 the EWCS becomes zero around s/ ≈ . .0 0.5 1.0 1.5 2.081.281.481.681.882.082.2 t E W C S s = t E W C S s = t E W C S s = FIG. 5: The evolutions of EWCS for different α with fixed s = 0 . , s = 0 . , s = 0 .
46 (from leftto right), respectively. For each panel, from top to bottom the coupled constant α is chosen as α = 0 . , . , . , , − . , − . , − .
05. We have also rescaled EWCS in the first and secondpanel as EWCS = EWCS α − (EWCS consα − EWCS cons − . ) + 5 α . EWCS α indicates the non-rescaledvalues of EWCS with the coupled constant α , while EWCS consα is its equilibrium value. s = = = t E W C S α =- t z v ℓ + s = α =- FIG. 6: (Left) The evolutions of EWCS at early stage for α = − .
05. To compare the differentbehaviors of three cases ( s = 0 . , . , .
46) we have rescaled the EWCS by reducing a constant foreach curve without changing its behavior. (Right) The evolution of the shell position for α = − . (cid:96) + s = 2 .
02 .
As time evolving, the system reaches the equilibrium, becoming static case discussed inSec. IV. Comparing the forth panel ( t = 1 .
99) and the right panel of Fig. 2, the relationsamong EWCS, s and α are same, only with numerical discrepancies. Such discrepancies arisefrom the different numerical initial conditions we have chosen in the static and dynamicalcases, which will not affect the final results. VI. CONCLUSIONS AND DISCUSSIONS
To explore the properties of entanglement of purification with higher order corrections,we studied its holographic dual, i.e., the entanglement of wedge cross section using theAdS/CFT correspondence and the GB-Vaidya model. For the sake of simplification we13 .20 0.22 0.24 0.26 0.280.00.51.01.52.02.53.03.5 s / E W C S t = s / E W C S t = s / E W C S t = s / E W C S t = FIG. 7: The EWCS with the width of strip for different α at different time. For each panel, fromtop to bottom α = 0 . , . , . , , − . , − . , − . consider two identical and separate rectangular strips in the boundary CFT, and thus onlythe radial EWCS is concerned. By the numerical method, we investigated the characteristicsof EWCS and its relation with the coupling GB constant α in static and dynamics case,respectively.In the static case, the calculations suggest that a monotonic relation between the renor-malized EWCS and α , i.e., the larger α , the larger renormailized EWCS. Besides, for large α the phase transition between entangling phase and disentangling phase occurs at largeseparations between two strips. Note that here we focused on the symmetric case withtwo identical subsystems. It would be interesting to study whether the rule still holds inasymmetric case. We hope to address the answer after improving the numerical skill in thefuture.For dynamical case, we have adopted the so-called holographic thermalization procedureto explore the evolution of EWCS. The monotonic relation between EWCS and α holdsduring the whole evolution process. Nonetheless, the separations where the phase transitionoccurs are nonmonotonic with α , and can be summarized into three stages. First, at thebeginning of thermalization, the phase transition occurs at the same time for all α . Insecond stage, the entangled strips disentangle at small separations for large α . Finally, inthird stage, the entangled strips transform into the disentangling phase at large separationsfor large α . Such behaviors state that the higher order corrections have nontrivial effects onthe entanglement of purification.Besides the main results of this paper, we also noted that an unphysical region appearsin the evolution of EWCS when the separation is small but 2 (cid:96) + s is large enough. We haveargued that such behaviors may arise from the model we have chosen. Checking the influenceof different boundary region on the evolution of EWCS or HMI with GB corrections may14hed light on the deep physical prescription of EoP evolution. Acknowledgments
This work is partly supported by the Natural Science Foundation of China under GrantsNos. 11947067, 12005077 and 11705161. Y-Z. Li is also supported by fund No.1052931902for doctoral research of Jiangsu university of science and technology. X-M. Kuang is alsosupported by Fok Ying Tung Education Foundation under Grant No.171006.
Appendix
In this appendix, we shall briefly review the definitions of entanglement of purification( E P ), reflected entropy( S R ), odd entropy( S O ) and logarithmic negativity( ξ ). Let us considertwo subsystems A and B in the Hilbert space H A and H B , respectively. And ρ AB is themixed density matrix for A ∪ B living in the total Hilbert space H = H A ⊗ H B . • Entanglement of purification E P is a measure of total (classical and quantum) correlations between two subsystems.In order to this quantity, one shall add some auxiliary degrees of freedom to H , afterwhich the total enlarged Hilbert space is ¯ H = H A ⊗H B ⊗H A ⊗H B . Then one can purifythe mixed state by constructing a pure state | ψ (cid:105)(cid:104) ψ | such that ρ AB = Tr A (cid:48) B (cid:48) | ψ (cid:105)(cid:104) ψ | and | ψ (cid:105) ∈ H AA (cid:48) ⊗ H BB (cid:48) , though this purification is not unique. Then E P is defined bythe minimum EE between A and its auxiliary partner A (cid:48) for all possible purification,which is[11] E P ( A, B ) = MIN ρ AB =Tr A (cid:48) B (cid:48) ( | ψ (cid:105)(cid:104) ψ | ) S ( A, A (cid:48) ) . (40)It is noticed that the above definition recovers EE when ρ AB is pure. • Reflected entropy S R is a new measure of the total correlation between two disjoined subsystems. Todefine S R , one could double the initial Hilbert space H into H ⊗ H (cid:48) , and then canon-ically purify the mixed state ρ AB = Σ i P i | ρ (cid:105)(cid:104) ρ | such that √ ρ = Σ i √ P i ρ i ⊗ ρ i is apure state in H ⊗ H (cid:48) . Then S R is defined as the EE between A and A (cid:48) as[12] S R ( A, B ) = − Tr( ρ AA (cid:48) ln ρ AA (cid:48) ) , √ ρ AA (cid:48) = Tr BB (cid:48) | √ ρ (cid:105)(cid:104)√ ρ | . (41)It is obvious that the above reflected entropy also reduces to EE if ρ AB is pure state. • Odd entropy S O is also a new measure of correlations for the mixed states. The definition is [13] S O ( A, B ) = Lim n O → − n O (cid:0) Tr( ρ T BAB ) n O − (cid:1) (42)where ρ T BAB denotes the partial transpose of ρ AB with respect to B . It was proved thatsimilar to the E P and S R , S O also reduces to the EE when the state is pure.15 Logarithmic negativityDifferent from the entanglement of purification, reflected entropy and odd entropy, thelogarithmic negativity captures only the quantum correlations for mixed states. Itsdefinition is [14] ξ ( A, B ) = ln Tr( ρ T BAB ) . (43) [1] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int.J. Theor. Phys. , 1113 (1999) [Adv. Theor. Math. Phys. , 231 (1998)] [hep-th/9711200].[2] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators from noncriticalstring theory,” Phys. Lett. B , 105 (1998) [hep-th/9802109].[3] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. , 253 (1998)[hep-th/9802150].[4] S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,”Phys. Rev. Lett. , 181602 (2006) [hep-th/0603001].[5] T. Takayanagi, “Entanglement Entropy from a Holographic Viewpoint,” Class. Quant. Grav. , 153001 (2012) [arXiv:1204.2450 [gr-qc]].[6] A. Lewkowycz and J. Maldacena, “Generalized gravitational entropy,” JHEP , 090 (2013)[arXiv:1304.4926 [hep-th]].[7] V. E. Hubeny, M. Rangamani and T. Takayanagi, “A Covariant holographic entanglemententropy proposal,” JHEP , 062 (2007) [arXiv:0705.0016 [hep-th]].[8] X. Dong, A. Lewkowycz and M. Rangamani, “Deriving covariant holographic entanglement,”JHEP , 028 (2016) [arXiv:1607.07506 [hep-th]].[9] P. Calabrese, J. Cardy and E. Tonni, “Entanglement entropy of two disjoint intervals inconformal field theory,” J. Stat. Mech. (2009), P11001 [arXiv:0905.2069 [hep-th]].[10] P. Calabrese, J. Cardy and E. Tonni, “Entanglement entropy of two disjoint intervals inconformal field theory II,” J. Stat. Mech. (2011), P01021 [arXiv:1011.5482 [hep-th]].[11] B. M. Terhal, M. Horodecki, D. W. Leung and D. P. Di-Vincenzo, The entanglement ofpurification,” J. Math. Phys. 43 (2002) 4286, [arXiv:quant-ph/0202044 [quant-ph]].[12] S. Dutta and T. Faulkner, “A canonical purification for the entanglement wedge cross-section,”[arXiv:1905.00577 [hep-th]].
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