Entanglement entropy, which is given by the von Neumann entropy SA=-TrAρAlnρA,(2.1)provides the measure of entanglement between pure states. Generally, one uses the nontrivial replica trick to calculate it in quantum field theories [72]. However, implementing the replica technique in nontrivial field theories, such as those containing interactions, is much more tedious. The holographic idea, on the other hand, provides another pathway to calculate the entanglement entropy in field theories [7,8]. For the case of AdSd+1/CFTd, the entanglement entropy SA of the subsystem A in CFTd is given by the Ryu-Takayanagi prescription [7,8] SA=A(ΓAmin)4GN(d+1),(2.2)where A(ΓAmin) is the area of (d-1)-dimensional static minimal surface (ΓA) in the bulk whose boundary is homologous to the boundary ∂A of the subsystem A. GN(d+1) is the (d+1)-dimensional Newton constant. Equivalently, we can also recast the above holographic entanglement entropy formula in the following way: SA=14GN(d+1)∫Γdd-1σGindd-1,(2.3)where Gindd-1 is the induced metric on the surface Γ, which further needs to be appropriately minimized [9,10]. Since in this work we are interested in computing the entanglement entropy and other information related quantities in top-down holographic confining theories, where the dual bulk spacetime metric is usually written down in string frame, it is also useful to write down the entanglement entropy expression in string frame metric [9,10]: SA=14GN(d+1)∫Γdd-1σe-2ϕGindd-1,(2.4)where the dilaton field ϕ arises because of the frame change.

Since the entanglement structure of one interval subsystem is known to display interesting behavior in the confined phases, it is also natural to ask and investigate the entanglement structure with many disjoint intervals. One such natural entanglement measure that appears with two disjoint intervals is the mutual information [73–75]. For two subsystems (A1 and A2), the mutual information is defined as the amount of information that A1 and A2 can share and is defined in terms of entanglement entropy as I(A1,A2)=SA1+SA2-SA1∪A2,(2.5)where SA1, SA2, and SA1∪SA2 are as usual the entanglement entropies of subsystems A1, A2, and their union, respectively. It is quite obvious from Eq. (2.5) that the mutual information is zero for two uncorrelated systems. Importantly, the mutual information does not suffer from the ambiguities associated with the entanglement entropy and can provide more information than the entanglement entropy alone. In particular, mutual information is UV finite and does not contain the usual UV divergences. Moreover, the subadditivity property of the entanglement entropy also ensures that the mutual information is non-negative; i.e., I(A1,A2) provides an upper bound on the correlation functions between operators in A1 and A2. For more discussion on the mutual information, see Refs. [76–83]. For the discussion on mutual information and two-strip entanglement phase diagram in top-down and bottom-up confining models, see Refs. [52,55]. Similarly, we can also define other entanglement measures, such as n-partite information, with more disjoint intervals In(A{i})=∑i=1nSAi-∑i<jnSAi∪Aj+∑i<j<knSAi∪Aj∪Ak-⋯-(-1)nSA1∪A2∪⋯∪An.(2.6)In this work, we will concentrate only on mutual information, as the results for the n-partite information can be analogously obtained.

Entanglement entropy serves as a good measure for the entanglement in the case of pure states, but it fails in the case of mixed states. Therefore, it would be an interesting question to investigate how the mixed state entanglement measures behave in confining theories. One such mixed state entanglement measure is the entanglement of purification, which in the holographic context is suggested to be given by the minimal area of the entanglement wedge cross section [29,30].¹

For some related discussion on the entanglement wedge cross section and entanglement of purification, see Refs. [84–97].

To compute the entanglement wedge cross section holographically, we follow the prescription suggested in Refs. [29,30]. We first consider two subsystems A and B with no overlap on the d-dimensional boundary. The Ryu-Takayanagi minimal surfaces for A and B and their union AB=A∪B are denoted by ΓAmin, ΓBmin, and ΓABmin, respectively. The d-dimensional entanglement wedge MAB in the (d+1)-dimensional bulk is then described as a region which is bounded by A, B, and ΓABmin; i.e., the entanglement wedge MAB is the bulk region whose boundary is ∂MAB=A∪B∪ΓABmin.(2.7)One should note that the entanglement wedge is a bulk codimension-zero region. However, here it turns out to be d dimensional as we are considering static bulk configuration. Also note that if A and B are very small or have large separation between them, then MAB will have a disconnected form.

We next divide ΓABmin as follows: ΓABmin=ΓAB(A)∪ΓAB(B),(2.8)and we further define Γ˜A=A∪ΓAB(A),Γ˜B=A∪ΓAB(B).(2.9)Note that, as a result of Eqs. (2.8) and (2.9), the boundary of the entanglement wedge MAB is now divided into two parts: ∂MAB=Γ˜A∪Γ˜B.(2.10)A pictorial representation of the holographic entanglement wedge is shown in Fig. 1. Next, look for the minimum surface ΣABmin subjected to conditions (i)  ∂ΣABmin=∂Γ˜A=∂Γ˜B,(ii)  ΣABminis homologous toΓ˜AinsideMAB.(2.11)The holographic entanglement wedge cross section EW(ρAB) is then simply defined as the area of ΣABmin divided by 4GNd+1: EW(ρAB)=minΓAB(A)⊂ΓABmin[A(ΣABmin)4GN(d+1)].(2.12)Note that for given A and B subsystems, as we will also see later in this work, there can be many possible ΣABmin surfaces. In those cases, the entanglement wedge cross section is given by the surface that has the minimum area. To summarize, EW(ρAB) is the minimal surface area of the entanglement wedge MAB connecting A and B.

In the previous section, we talked about the entanglement wedge cross section (and its holographic definition) as a suitable measure for the mixed state entanglement. Another quantum information quantity which is also known to capture mixed state entanglement is the entanglement negativity. This is defined in the quantum many-body system as [23,24] N=∥ρT2∥-12,(2.13)where ∥ρT2∥ is the trace norm of the partially transposed reduced density matrix ρT2 and this trace norm is generally given by the sum of the absolute eigenvalues of ρT2. There also exists a close variant of entanglement negativity, called logarithmic negativity, which is also frequently used in the quantum information community. This is defined as E=ln∥ρT2∥=lnTr|ρT2|.(2.14)The (logarithmic) entanglement negativity has been computed in a variety of quantum many-body and field theory systems and has been widely used in the condensed matter and quantum information community; see, for example, Refs. [98–114]. The primary reason for this is that it provides an upper bound on the distillable entanglement. Holographically, to the best of our knowledge, two different (yet equivalent) proposals for entanglement negativity have been suggested. In the first proposal, the logarithmic negativity is suggested to be given by the area of an extremal cosmic brane that terminates on the boundary of the entanglement wedge [31,32]. In the second proposal, the logarithmic negativity is suggested to be given by certain combinations of the areas of codimension-two minimal bulk surfaces [33–42]. Both these proposals have been tested in diverse physical situations and have shown to reproduce exact known results for the negativity in CFTs. Since the first proposal is very closely related to the entanglement wedge cross section, which we will anyhow be going to investigate, it might be more informative, and at the time complementary as well, if the second holographic proposal is adopted for the computation of entanglement negativity. It is also computationally slightly more straightforward to implement the second proposal, as opposed to the first proposal, which requires nontrivial cosmic brane backreaction calculation. Moreover, as we will see later on, the second proposal also advocates a universal result for the entanglement negativity in all holographic confining theories, which might be possible to check via lattice calculations. For these reasons, we will take the second proposal for the entanglement negativity in this paper. It would certainly be interesting to explicitly compute the entanglement negativity in confining theories using the first proposal and find its similarities and differences with the second proposal. We leave this interesting exercise for future work.

To compute the holographic logarithmic negativity for a single interval using Refs. [33–35], first, a bipartition of the system into A and its complement Ac is considered in CFTd. Next, two finite length intervals B1 and B2 are taken adjacent on both sides of A such that B=B1∪B2. Refer to Fig. 2. If the corresponding codimension-two bulk static minimal surfaces in AdSd+1 are denoted by ΓA, ΓB1, and ΓB2, then the holographic entanglement negativity for the bipartite system (A∪Ac) is suggested to be given by the following combination of the areas of the minimal bulk surfaces: E=limB→Ac316GN(d+1)[2A(ΓA)+A(ΓB1)+A(ΓB2)-A(ΓA∪B1)-A(ΓA∪B2)],(2.15)where in the above equation it should be understood that both B1 and B2 are extending to infinity such that B=B1∪B2=Ac. In the AdS3/CFT2 context, the above formula was deduced from how the four-point twist correlation functions factorize in the large central charge limit. In particular, the entanglement negativity in CFT2 is given by a specific four-point twist correlator in the n-sheeted Riemann surface [98,99]. This four-point correlation function factorizes into various combinations of two-point functions in the large central charge limit, which can then be mapped, using the AdS/CFT dictionary, to the length of the geodesic anchored on the boundary points and extending into the bulk. In this way, the authors of Refs. [33,34] arrived at the above formula for the entanglement negativity in CFT2 (and its d-dimensional extension). This entanglement negativity formula, though an unproven conjectured proposal, in contrast with the Ryu-Takayanagi entanglement entropy proposal, however, does reproduce mixed and pure state results of the entanglement negativity in CFT. For more details on this entanglement negativity conjecture, see Refs. [33,34].

One can also write the above formula in terms of the entanglement entropy using the Ryu-Takayanagi formula [Eq. (2.2)]: E=limB→Ac34[2SA+SB1+SB2-SA∪B1-SA∪B2].(2.16)In the subsequent sections, we will take the above formula to compute the entanglement negativity in a variety of top-down as well as bottom-up confining backgrounds. As we will see, the above entanglement negativity formula reduces to a simpler expression and suggests a universal relation in all confining theories.

For two disjoint intervals A1 and A2, the negativity is similarly suggested to be given by the following combination of the minimal area surfaces: E=34[SA1∪AX+SAX∪A2-SA1∪A2∪AX-SAX],(2.17)where X is the separation between the two intervals. Refer to Fig. 2 for more details on the surfaces appearing in Eq. (2.17). Again, this is an unproven conjectured formula which is obtained by analyzing the factorization of a four-point twist correlation function in terms of two-point correlations in the large central charge limit [39,41]. This conjectured formula, however, again reproduces the desirable entanglement negativity results of the boundary CFT.

The entanglement entropy has already been computed for this system in Ref. [9]. Here, we will reproduce their results to set the stage for the later sections. To calculate the entanglement entropy, we consider a strip subsystem of length L and define the subsystem domain as -L/2≤x1=x≤L/2, 0≤x2≤L2, and 0≤x3≤L3. The parametrization U=U(x) leads to the following expression for the entanglement entropy (2.4): SA=L2L3ω4(2πR4)4GN(10)∫dxR3/2U5/2f(U)+(RU)3U′2,(3.2)where ω4 is the area of unit four sphere. It turns out that there are actually two surfaces which minimize the above entanglement action: a (U-shaped) connected and a disconnected surface. The expression of the entanglement entropy of the connected surface is simply given by SAcon=L2L32GN(10)32π3R9/29∫U*U∞dUU7/2U0f(U)U5f(U)-U*5f(U*),(3.3)where U* is the turning point of the connected surface at which U′(x)|U=U*=0 and is related to the strip length L in the following way: L(U*)=2R3/2∫U*U∞dUU*5/2U3/2f(U*)f(U)⁢1U5f(U)-U*5f(U*).(3.4)Similarly, the entanglement entropy for the disconnected surface is given by SAdiscon=L2L32GN(10)32π3R9/29∫U0U∞dUUU0.(3.5)Note that the disconnected entanglement entropy does not depend on the strip length. This result will have profound implications in the entanglement structure of the confined phase.

We now present the numerical results for the entanglement entropy. For the numerical purpose, we take U0=1. The results are shown in Figs. 3 and 4, where the variation of L with respect to U* and connected and disconnected entanglement entropy difference (△SA=SAcon-SAdiscon) with respect to L, respectively, are plotted.²

Here, and in the subsequent subsections, we have set L2L3/4GN(10)=1.

Importantly, a change in sign in △SA appears as the strip length L varies. In particular, △SA is negative for small L, indicating that the connected surface ③ has the lowest entanglement entropy for the small subsystem, whereas △SA is positive for large L, indicating that the disconnected surface has the lowest entanglement entropy for the large subsystem. Therefore, a phase transition takes place between connected and disconnected entanglement entropies as L increases. This phase transition occurs at L=Lcrit≃1.288R4. Since the entanglement entropy of the disconnected surface is independent of L, it gives us the following important result: ∂SA∂L∝1GN(10)=O(N2)for  L<Lcrit∝1[GN(10)]0=O(N0)for  L>Lcrit.(3.6)The above type of connected-disconnected phase transition, where the order of the entanglement entropy changes from O(N2) at small L to O(N0) at large L, was suggested to be reminiscent of the confinement-deconfinement transition in QCD [9]. Moreover, the fact that the order of colored gluon degrees of freedom [O(N2)] above the deconfinement temperature and color neutral confined degrees of freedom [O(N0)] below the deconfinement temperature match well with the connected and disconnected entanglement entropies further naturally led to the interpretation of the critical subsystem size as the inverse deconfinement temperature (TD), i.e., TD∝1/Lcrit. In Ref. [9], the value of Lcrit was computed for different confining models, and the relation Lcrit=O(ΛIR-1) was found. This further suggested that the entanglement entropy can act as a probe to diagnose confinement.³

It has been suggested recently that the entanglement entropy signals only the presence of a mass gap rather than the confinement [115].

Having discussed the entanglement structure with one strip, we now move on to discuss it with two strips. Here, for simplicity, we concentrate only on equal size strips (L1=L2=L). The entanglement structure with two strips is much more interesting than with one strip. In particular, depending upon the strip length L and the distance between them X, there are now four different entangling surfaces which compete with each other. These surfaces are shown in Fig. 5. With two strips, both connected (S1 and S2) and disconnected (S4) as well as a combination of connected and disconnected (S3) surfaces can appear.

The entanglement entropy expressions of these surfaces are given as S1(L,X)=2SAcon(L),S2(L,X)=SAcon(X)+SAcon(2L+X),S3(L,X)=SAcon(X)+SAdiscon,S4(L,X)=2SAdiscon,(3.7)where SAcon and SAdiscon are as usual the entanglement entropies of the connected and disconnected surfaces, respectively, with one strip.

Depending on the magnitude of L and X, there appear various phase transitions between S1, S2, S3, and S4. In Fig. 6, the complete phase diagram in the phase space of (L,X) is shown. For small X,L≪Lcrit, it is usually the connected S1 surface which has the lowest area (and entanglement entropy). The area of the connected S2 surface, however, becomes smaller as L increases. With a further increase in L, keeping X(≪Lcrit) small, the area of the S3 becomes smaller and a phase transition from S2 to S3 takes place. For X=0, this phase transition occurs at L=Lcrit/2. For a general value of X, the phase transition line between S2 and S3 is given by 2L+X=Lcrit, as is obvious from Eq. (3.7). Finally, the S4 configuration becomes more favorable with a further increase in both X,L>Lcrit.

Interestingly, with two strips, two tricritical points appear where three minimal area entangling surfaces coexist. These are denoted by black dots in Fig. 6. At the first tricritical point Lc1, the phases (S1, S2, and S3) coexist, whereas at the second tricritical point Lc2, the phases (S1, S3, and S4) coexist. The coordinates of these tricritical points are (L/R4=0.454, X/R4=0.381) and (L/R4=1.288, X/R4=1.288), respectively. Importantly, the tricritical points and two-strip phase diagram again suggest nonanalyticity in the entanglement structure.

Let us now probe the structure of mutual information in the above mentioned four entangling phases. In these phases, the mutual information simply reduces to I1(L,X)=SAcon(L)+SAcon(L)-2SAcon(L)=0,I2(L,X)=SAcon(L)+SAcon(L)-SAcon(X)-SAcon(2L+X)≥0,I3(L,X)=SAcon(L)+SAcon(L)-SAcon(X)-SAdiscon≥0,I4(L,X)=SAdiscon+SAdiscon-2SAdiscon=0.(3.8)Here, the range of X and L in I2 and I3 should be understood to be restricted in their respective phases. The above equations also imply ∂I1∂L∝1GN0=O(N0),∂I2∂L∝1GN=O(N2),∂I3∂L∝1GN=O(N2),∂I4∂L∝1GN0=O(N0).(3.9)

As expected, the mutual information is zero for the phases containing two largely separated subsystems, i.e., for S1 and S4 phases, whereas for the phases S2 and S3 it is nonzero and finite. The variation of mutual information with respect to strip length L and separation length X are shown in Figs. 7 and 8. It turns out that the mutual information is not only a smooth function of L and X but also behaves smoothly as it passes from one phase to another. For instance, as shown in Fig. 7, it connects smoothly between S2 and S3 phases and no discontinuity arises as the S2/S3 critical line is approached. Similarly, the mutual information also smoothly goes to zero as S1 (or S4) phase is approached from the S2 (or S3) side. This is shown in Fig. 8.

An important point to note is that, unlike the entanglement entropy, the order of the mutual information may or may not change as we go from one phase to another. For instance, a change in the order of mutual information [O(N2) to O(N0)] occurs as we go from S2 to S1 phase (by increasing X); however, no such change in the order occurs from S2 to S3 phase (by increasing L). It all depends on the particular phases involved in the transition.

We now move on to discuss the entanglement wedge cross section EW in the current confining background. For this purpose, note that the surface which divides the entanglement wedge into two parts, associated with A and B (see Fig. 1), can be identified by symmetry consideration to be a vertical flat surface Σ. Therefore, the symmetry of the strip configuration ensures that the entanglement wedge cross section in the confining background is given by the area of a constant-x hypersurface located in the middle of the strips (see Fig. 5). The induced metric on Σ (a surface defined with t=const and x=const) is then given by (ds2)Σind=(UR)3/2[(RU)3dU2f(U)+dxidxi]+R3/2U1/2dΩ42+(UR)3/2f(U)(dx4)2,(3.10)where i=1, 2. Correspondingly, the entanglement wedge cross section for the dual confining theory is given by EW=14GN(10)∫d8σe-2ϕgΣind,(3.11)where we have again used the fact that the induced metric in Eq. (3.10) is in the string frame. As mentioned above, there are, in fact, four different minimal area surfaces for two parallel strips. Since there is no correlation between the two disjointed strips in S1 and S4 phases, the corresponding entanglement wedge cross section is trivially zero. For the connected S2 and S3 phases, the entanglement wedge cross section is nonzero. For the S2 phase, it is given by EW2=L2L3ω4(2πR4)4GN(10)∫U*(2L+X)U*(X)dUR3U=L2L34GN(10)(12π3R43U0)2[U*2(X)-U*2(2L+X)].(3.12)Similarly, for the S3 phase it is given by EW3=L2L3ω4(2πR4)4GN(10)∫U0U*(X)dUR3U=L2L34GN(10)(12π3R43U0)2[U*2(X)-U02].(3.13)From the above definition, it is clear that EW2 and EW3 are both positive as U*(x)≥U*(2L+X)≥U0.

In Fig. 9, we have shown the variation of entanglement wedge as a function of X for a few values of L. Here, near-critical values of L are chosen so that EW behavior near the S1/S3 and S3/S4 phase transition points can be seen. We find that EW is a monotonic function of X and is discontinuous at the transition point. In particular, EW does not go to zero as the S3/S4 critical line is approached from the S3 side (blue curve). The discontinuous behavior can also be seen mathematically from Eq. (3.13). Notice that, for EW3 to be zero at the S3/S4 critical line, one requires U*(X=Lcrit)=U0. However, U*(Lcrit) is always larger than U0 in the current confining background, which implies that EW3 is greater than zero at Xcrit. Similarly, EW3 also does not go to zero at the S1/S3 transition line (red curve). This analysis suggests that, in the confining phase, the entanglement wedge cross section vanishes discontinuously for large values of X and L.

It is also interesting to investigate the behavior of EW near the S2/S3 critical line, considering that EW is nonzero in both these phases. The results are shown in Fig. 10. Here, we have considered a particular line, X=0.7L, so that the behavior of EW in S2, S3, and S4 phases can be probed simultaneously. EW again turns out to be discontinuous at the S2/S3 transition line. This can also again be seen mathematically from Eqs. (3.12) and (3.13). In particular, the condition U*2(2L+X)≠U0 ensures that EW2 and EW3 do not correspond to the same value at the S2/S3 critical point. Moreover, as the S2/S3 critical point is approach from the S2 side, there is an upward jump in the EW value, i.e., EW3>EW2, suggesting that the area of the entanglement wedge grows at the critical point. This result can again be traced back to the fact that U*(2L+X)>U0 and is also clear from Fig. 5. Therefore, it is clear that EW shows nontrivial features each time a phase transition between different phases occurs.

The entanglement wedge is known to satisfy a few inequalities in holographic settings. In particular, it has been proved that EW always at least exceeds half the mutual information, i.e., EW≥I/2 [29,86]. In Figs. 11 and 12, we made a comparison between I/2 and EW for diverse values of X and L and find that this inequality is always satisfied in the current holographic confining model. As we will show in later sections, this inequality will remain true for other holographic confining models as well, including a phenomenological bottom-up model. Therefore, the inequality EW≥I/2 appears to be a generic feature of all holographic models. It would certainly be interesting to see whether such an inequality exists in real QCD or not.

We now move on to discuss the entanglement negativity in the current confining theory using the holographic prescription suggested in Refs. [33–35]. For a single interval, the entanglement negativity is given by Eq. (2.16). Note that, in the confined phase where the disconnected surface is more favorable for large strip length, the limiting condition B→Ac→∞ ensures that AB1=AB2=AA∪B1=AA∪B2=Adisconn,(3.14)the above four terms cancel out in Eq. (2.16), and we are left with E=limB→Ac34[2SA+SB1+SB2-SA∪B1-SA∪B2],E=32SA.(3.15)Interestingly, the entanglement negativity is just 3/2 times of the entanglement entropy. Since SA depends on L and exhibits a discontinuous behavior at Lcrit, this discontinuity of SA manifests itself in E as well. Therefore, interestingly, just like the entanglement entropy, the order of the entanglement negativity also changes from O(N2) to O(N0) at Lcrit: ∂E∂L=O(N2)for  L<Lcrit,∂E∂L=O(N0)for  L>Lcrit.(3.16)The discontinuous behavior of the entanglement negativity in the confined phase is a genuine new prediction from holography (strictly speaking, a prediction from the entanglement negativity proposal of Refs. [33–35]) and should be independently tested. Unfortunately, there are no lattice results for the negativity in real QCD to compare with yet. It would really be interesting if this discontinuous nature could be tested via lattice calculations in the near future.

Let us now discuss the entanglement negativity for two disjoint intervals [39,41]. Compared to Refs. [39,41], we have ls=X and l1=l2=L. So E is given as E=34[SA(L+X)+SA(L+X)-SA(2L+X)-SA(X)],(3.17)where SA is again the entanglement entropy of a single strip of length L. Notice that, for X>Lcrit, E is trivially zero, as every term in Eq. (3.17) corresponds to SAdiscon. This implies that, just like the mutual information and entanglement wedge, E is zero in the S4 phase as well. However, the same is not true for the S1 phase. In particular, E is nonzero and a monotonic function of X and L in the S1 phase. This is in sharp contrast with the mutual information and entanglement wedge behavior, which are zero in the S1 phase as well. The results are shown in Figs. 13 and 14. Note that E smoothly goes to zero at X=Lcrit, whereas it remains finite for L>Lcrit. Moreover, unlike the entanglement wedge, E is continuous across the various phase transition points but has a cusp (as is visible for Fig. 13). This is again a new prediction form holography and should be tested in lattice settings.

The entanglement entropy for this top-down model has also been computed previously in Ref. [9]. Here, we first review their calculation. To compute the entanglement entropy, we consider the previous D4-brane setup by taking a strip subsystem of length L, with subsystem domain -L/2≤x1=x≤L/2 and 0≤x2≤L2. Using U=U(x), we get the entanglement entropy (2.4) as SA=L2ω5(2πR3)4GN(10)∫dxR2U3f(U)+(RU)4U′2,(4.3)where ω5 is the area of the unit five sphere. In this case also, there are two minimal area surfaces: a (U-shaped) connected surface and a disconnected surface. The entanglement entropy of the connected surface turns out to be SAcon=L22GN(10)(π4R6)∫U*U∞dUU4U0f(U)U6f(U)-U*6f(U*),(4.4)where U* is the turning point for the above connected surface and U′(x)|U=U*=0. Furthermore, the strip length L as a function of U* is L(U*)=2R2∫U*U∞dUU*3U2f(U*)f(U)1U6f(U)-U*6f(U*),(4.5)and the expression of the entanglement entropy of the disconnected surface is SAdiscon=V22G10(π4R6)∫U0U∞dUUU0.(4.6)It is worthwhile to note here that the entanglement entropy of the disconnected surface is again independent of the strip length. This information will be relevant and useful when studying the entanglement structure of the current D3-brane confined setup.

We now show the numerical results of the entanglement entropy in the current confining model. For numerical purpose, we again considered U0=1. These numerical results are presented in Figs. 15 and 16, where in the first case we have plotted L as a function of U* and in the second case we have plotted the difference between connected and disconnected entanglement entropies (△SA=SAcon-SAdiscon) as a function of the strip length L, respectively.⁴

We have used L2/4GN(10)=1 in the numerical calculations.

We see from Fig. 16 that △SA changes sign with L. For small L, △SA is negative. This reflects the fact that for a small subsystem length the connected surface ① has the smallest area. On the other hand, for large L, △SA is positive, reflecting the fact that for a large subsystem length the disconnected surface has the smallest area. Correspondingly, we have a phase transition at L=Lcrit≃1.23R3 between connected and disconnected entanglement entropies. As the minimal surface area for the disconnected surface is independent of L, its entanglement entropy is also independent of L. This connected-disconnected phase transition, therefore, leads to the following result (similar to the previous D4-brane setup): ∂SA∂L∝1GN(10)=O(N2)for  L<Lcrit∝1[GN(10)]0=O(N0)for  L>Lcrit.(4.7)Importantly, once again the order of the entanglement entropy changes as the subsystem size is varied in the confined phase.

We now move on to discuss the entanglement structure of two disjoint strips in the current confining system. The two-strip phase diagram of this system has been studied previously in Ref. [55]. Here, we first reproduce their results to set the stage for the discussion of mutual information and entanglement wedge cross section in this system later on. For our purpose, we again consider equal strip lengths, i.e., (L1=L2=L), for simplicity. This leads to four entangling surfaces {S1,S2,S3,S4}, the expressions of which are given in Eq. (3.7). See Fig. 5 for the pictorial representation of these surfaces.

The two-equal-strip entanglement phase diagram is shown in Fig. 17. The four phases {S1,S2,S3,S4} again compete with each other, leading to a similar phase diagram as in the case of D4 branes on a circle confining system. For instance, for small values of X,L≪Lcrit, the S1 configuration again has the lowest entanglement entropy, whereas for large values of X,L>Lcrit, it is the S4 configuration which has the lowest entanglement entropy. In between, we have S2 and S3 configurations. The S2 to S3 phase transition happens at L=Lcrit/2 for X=0, whereas this phase transition is governed by the equation 2L+X=Lcrit for a generic value of X.

The two tricritical points, indicated by black dots in Fig. 17, reflect the coexistence of three configurations together. The first tricritical point Lc1 denotes the coexistence of three configurations {S1, S2, S3}, and its coordinates are (L/R3=0.45, X/R3=0.331), whereas the second tricritical point Lc2 denotes the coexistence of {S2, S3, S4} configurations, and its coordinates are (L/R3=1.23, X/R3=1.23). The two-strip phase diagram, along with the presence of tricritical points, again indicates the nonanalytic nature of the entanglement structure in confining theories.

Similarly, one can compute the mutual information in these four phases. The relevant expressions are given in Eq. (3.8). The mutual information is again zero when the two strips have a relatively large separation, i.e., for the S1 and the S4 configurations, whereas it is nonzero in the remaining S2 and S3 configurations. The behavior of mutual information as a function of L and X is presented in Figs. 18 and 19. Here, the specific values of L and X are taken in order to move through the S2 and S3 configurations. It turns out that the mutual information is a monotonic function of L and X and varies smoothly as these parameters are changed. Furthermore, the mutual information also exhibits a smooth behavior while passing from one configuration to another. An interesting aspect of this analysis is that there may or may not be a change in the order of mutual information. To be clearer, we see a change in the order of mutual information from O(N2) to O(N0) as we move from S2 to S1 by varying X. On the other hand, there is no such change in the order as we move from S2 to S3 by varying L. Therefore, we again have the following relations in the current confined phase: ∂I1∂L∝1GN0=O(N0),∂I2∂L∝1GN=O(N2),∂I3∂L∝1GN=O(N2),∂I4∂L∝1GN0=O(N0).(4.8)

Similar to the previous D4-brane setup, the symmetry of the strip configuration again dictates that the entanglement wedge cross section in the current confining background is given by the area of a constant-x hypersurface Σ, located in the middle of the strips (as depicted in Fig. 5). From the following induced metric on Σ: (ds2)Σind=(UR)2[(RU)4dU2f(U)+dx2dx2]+R2dΩ52+(UR)2f(U)(dx3)2,(4.9)we can get the expression of the entanglement wedge cross section as EW=14G10∫(d8σ)(e-2ϕ)gΣind.(4.10)When considering the two-disjoint-strip setup, similar to the previous D4-brane case, the entanglement wedge cross section is zero for the S1 and S4 configurations, while it is nonzero for the S2 and S3 configurations. In particular, for the S2 configuration it is given by EW2=L24G10(π4R6)∫U*(2l+x)U*(x)dUUU0=L24G10(π4R62U0)[U*2(x)-U*2(2l+x)],(4.11)whereas for the S3 configuration it is given by EW3=L24G10(π4R6)∫U0U*(x)dUUU0=L24G10(π4R62U0)[U*2(x)-U02].(4.12)From the above equations, one can again conclude that both EW2 and EW3 are positive, as U*(x)≥U*(2L+X)≥U0.

The behavior of the entanglement wedge as a function of X for specific values of L is shown in Fig. 20. These specific values of L are again chosen so that we can see the nature of EW close to the S1/S3 and S3/S4 phase transition points. From this analysis we see that EW changes monotonically with respect to X and shows discontinuity at the transition point. The discontinuity in EW at the S3/S4 transition line can also be inferred from Eq. (4.12). In particular, the condition U*(X=Lcrit)>U0 ensures that EW3>0 at Xcrit (as opposed to EW4, which is zero at Xcrit). In the same fashion, EW3 picks up a nonzero value at the S1/S3 transition line (as indicated by a red curve). Our entire exercise suggests that the entanglement wedge cross section disappears in a discontinuous fashion when the values of X and L are large in the case of confining backgrounds.

We have also studied the nature of EW close to the S2/S3 critical line. This analysis is shown in Fig. 21. We have chosen a specific value of X=0.5L in order to study EW for S2, S3, and S4 configurations simultaneously. We find a discontinuity in EW at the S2/S3 transition line as well. This discontinuous behavior can also be inferred from Eqs. (4.11) and (4.12). In particular, the condition U*(2L+X)≠U0 ascertains that EW2 and EW3 cannot have the same value at the S2/S3 critical point. There is also a positive jump in the value of EW as the S2/S3 critical point is approached from the S2 side, indicating an increase in the area of entanglement wedge at the critical point. This increment in area can also be extracted from the condition U*(2L+X)>U0. To conclude, we see that EW shows a nontrivial behavior whenever a phase transition happens.

We further tested the inequality EW≥I/2 in the current confining setup. We have studied this inequality numerically for various values of X and L and found it to be always true. The results for this exercise are presented in Figs. 22 and 23.

The holographic entanglement negativity for a single interval is given by Eq. (2.16). In the limit B→Ac→∞, we again have AB1=AB2=AA∪B1=AA∪B2=Adisconn,(4.13)which leads to the following result for the negativity: E=limB→Ac34[2SA+SB1+SB2-SA∪B1-SA∪B2],E=32SA.(4.14)This again tells us that the entanglement negativity is discontinuous at Lcrit. Consequently, in this confining model as well, a change in the order of entanglement negativity [from O(N2) to O(N0) or vice versa] appears at Lcrit.

The entanglement negativity for the two disjoint strips is again given by [39,41] E=34[SA(L+X)+SA(L+X)-SA(2L+X)-SA(X)],(4.15)where SA is the entanglement entropy of a single interval. For the S4 configuration, where X>Lcrit, E is zero, as all the terms in Eq. (4.15) become SAdiscon, whereas E is nonzero in all other configurations, in particular, in the S1 configuration. This behavior is different from the mutual information and entanglement wedge cross section behavior, which were in the S1 configuration as well. From Figs. 24 and 25, we can infer that E, in general (except for the S4 case), has a nonzero value and behaves monotonically with respect to X and L. Furthermore, E is finite for L>Lcrit, and it goes to zero in a smooth fashion at X=Lcrit. In contrast to the entanglement wedge, E shows a continuous behavior during the phase transition but exhibits a cusp (see Fig. 24).

Before we end our discussion on top-down models, we further add that we have performed similar analysis in other top-down confining models as well. In particular, we have also computed entanglement entropy, mutual information, entanglement wedge cross section, and entanglement negativity in the Klebanov-Strassler and Klebanov-Tseytlin confining backgrounds [116], and results similar to those presented here are found for these entanglement measures.

The behavior of entanglement entropy in this confining model has already been studied in Refs. [51,52]. Here, we briefly mention their results. For a strip subsystem A of domain {-L/2≤y1≤L/2,0≤y2≤L2,0≤y3≤L3}, the entanglement entropy of the connected surface is given by SAcon(L)=L2L2R34GN(5)∫0z*dz2z*3z3e3A(z)-3A(z*)[z*6e-6A(z*)-z6e-6A(z)],(5.2)where z* is the turning point of the connected minimal area surface and is related to the strip length L in the following way: L=2∫0z*dzz3e-3A(z)[z*6e-6A(z*)-z6e-A(z)],(5.3)whereas the entanglement entropy for the disconnected surface is given by SAdiscon=L2L3R34GN(5)∫0∞dz2e3A(z)z3.(5.4)Note that the entanglement entropy of the disconnected surface is again independent of the strip length L.