# Higher-curvature corrections to holographic entanglement with momentum dissipation

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## Abstract

We study the effects of Gauss–Bonnet corrections on some nonlocal probes (entanglement entropy, *n*-partite information and Wilson loop) in the holographic model with momentum relaxation. Higher-curvature terms as well as scalar fields make in fact nontrivial corrections to the coefficient of the universal term in entanglement entropy. We use holographic methods to study such corrections. Moreover, holographic calculation indicates that mutual and tripartite information undergo a transition beyond which they identically change their values. We find that the behavior of the transition curves depends on the sign of the Gauss–Bonnet coupling \(\lambda \). The transition for \(\lambda >0\) takes place in larger separation of subsystems than that of \(\lambda <0\). Finally, we examine the behavior of modified part of the force between external point-like objects as a function of Gauss–Bonnet coupling and its sign.

## 1 Introduction

*i*,

*j*label the boundary space-time directions. Noting that the Ward identity yields momentum conservation in the translationally invariant solution, its modification in this sense results in breaking the translational invariance of the theory.

Inserting scalar fields into the theory in fact leads to a deformation of states at the corresponding dual field theory and it would be a relevant question what happens to some specific concepts coming from holographic computations. For example some nonlocal measures of entanglement in such a model have recently been studied in [15] via the holographic methods.

^{1}The above formula only works for CFTs dual to Einstein gravity. In such theories, the central charges are the same, since on the gravity side there are no extra parameters to distinguish the central charges. By expanding the parameter space of the couplings one can address this problem, which can be done by introducing higher-derivative corrections in the action [18, 19, 20, 21, 22, 23]. Thus, to study general field theories in the context of holography, higher-derivative terms are in fact needed at the gravity side. In general, higher-derivative terms could potentially introduce ghost degrees of freedom; however, it is well known that a special combination of curvature squared terms, namely the Einstein Gauss–Bonnet theory, leads to second-order equations of motion and the theory is free of ghosts. Holographically, the Gauss–Bonnet (GB) term plays the role of leading-order corrections to the Einstein gravity and in the context of AdS/CFT, the GB background is dual to a theory with different central charges, i.e.,

*a*- and

*c*-functions; it is noted that AdS solutions resulting from Einstein–Hilbert action yield the same

*a*and

*c*[24]. Motivated by the fact that adding higher-curvature terms into the action may help to investigate several new aspects of the theory, in this paper, we consider certain nonlocal probes of entanglement in momentum relaxation theories when the action contains GB term. More precisely, we study the holographic entanglement entropy (HEE), mutual and tripartite information; we also make a comment on the potential between external objects by computing the expectation value of Wilson loop. We find the semianalytic expression for the coefficient of universal term in HEE which could introduce a modified ‘

*c*’-type central charge in the corresponding dual quantum field theory.

In order to compute HEE in the semiclassical regime when some higher-order derivative terms are added into the Einstein gravity, RT proposal should be replaced by some other recipes [19, 25, 26, 27, 28]. Some related work on this subject can be found, for example, in [29, 30, 31, 32] and the references therein.

In this paper, we will follow the proposal of [26] to study the HEE which will be reviewed in Sect. 2. We will focus on GB gravity theory with momentum relaxation and compute the HEE for strip, spherical and cylindrical entangling regions in Sect. 3. In Sect. 4, other measurements of quantum entanglement in this setup will be considered, i.e., mutual and tripartite information and their quantum phase transitions and also the Wilson loop. In fact, we are interested in the effect of GB corrections to these quantities in holographic theories with momentum relaxation. The subject is concluded in Sect. 5. Finally, in a short appendix we present some mathematical details.

## 2 Entanglement entropy for black brane solutions: a short review

Entanglement entropy is an important nonlocal measure of different degrees of freedom in a quantum mechanical system [33]. This quantity similar to other nonlocal quantities, e.g., Wilson loop and correlation functions, can also be used to classify the various quantum phase transitions and critical points of a given system [34].

*A*and

*B*complementing each other. Thus, the corresponding total Hilbert space can be written in a specific partitioning as \(\mathcal {H}=\mathcal {H}_A\otimes \mathcal {H}_{B}\). By integrating out the degrees of freedom that live in the complement of

*A*, the reduced density matrix for region

*A*can be computed as \(\rho _A=\mathrm{Tr}_{B}\;\rho \) where \(\rho \) is the total density matrix. The entanglement entropy is given by the Von Neumann formula for this reduced density matrix as follows:

*d*-dimensional quantum field theories, the entanglement entropy follows the area law and it is infinite; the structure of the infinite terms are generally as follows [19, 35, 36]:

*V*stand for the area and volume of the entangling region in the boundary,

*s*(

*V*) is the finite part of entropy and \(g_i(\mathcal {A}_A) \) are local and extensive functions on the boundary of entangling region, which are homogeneous of degree

*i*. The coefficient of the most divergent term is proportional to the area of the entangling surface and this is indeed the area law which is due to the infinite correlations between degrees of freedom near the boundary of the entangling surface. The coefficients of infinite terms are not physical whereas the coefficient of the logarithmic term is physical and universal in the sense that it is not affected by cutoff redefinitions.

Since we are specifically interested in studying the GB corrections in holographic theories with momentum relaxation, in what follows, we will limit ourselves to the five-dimensional GB gravity in the bulk with three specific scalar fields which are responsible for breaking the translational invariance in the dual field theory.

## 3 Gauss–Bonnet gravity with linear scalar fields

The GB gravity can indeed be obtained by setting \(a = c = - \frac{b}{4} \equiv \frac{\lambda }{2}L^2\) in (2.3), where \(\lambda \) is a dimensionless coupling constant that controls the strength of the GB term. The five-dimensional GB gravity is the simplest example of a Lovelock action and it is itself important because, in a given background, the equations of motion for a propagating perturbation contain only two derivatives.

*U*(1) charge to produce an extremal black brane solution in this case, the momentum relaxation parameter gives us such a feature similar to the case of the RN-AdS black brane.

In the model that we are considering there are two deformations in the field theory due to the momentum relaxation parameter and GB term. In the following, we will develop the behavior of HEE of a quantum field theory whose states are in fact under the excitation of both momentum relaxation and GB term.

### 3.1 HEE of a strip

*H*plays an infrared regulator distance along the entangling surface. The corresponding codimension two hypersurface in a constant time slice can be parametrized by \({x_1} = x ( \rho )\); therefore, the induced metric becomes

*prime*stands for the derivative with respect to \(\rho .\) After doing some computation which are partially given in the appendix, the entropy functional is found as follows

^{2}:

*C*is a constant which can be fixed by imposing the condition that at the turning point \(\rho _t\) of the hypersurface in the bulk one has \(x'(\rho _t)\rightarrow \infty \). After minimizing the functional of (3.9) and using the condition of the hypersurface turning point, one gets the following conserved quantity along the radial profile:

^{3}

*c*-function in arbitrary dimensions. In a CFT\(_4\) it goes as follows:

#### 3.1.1 Low-thermal excitation (\(m\ell ^4\ll 1\))

### 3.2 HEE of a sphere

^{4}:

### 3.3 HEE of a cylinder

*H*for the

*z*direction which is along the cylinder length. By taking the profile as \(r=F(\rho )\), the entropy functional becomes

*R*is the Ricci scalar. In principle there are two trace anomaly coefficients in four-dimensional CFTs, namely

*c*- and

*a*-functions

*a*while for a cylindrical entangling region it relates to

*c*-function [38]. However, for all AdS backgrounds, one obtains

## 4 Holographic *n*-partite Information and Wilson loop

In addition to entanglement entropy, the *n*-partite information and also the Wilson loop are in fact useful quantities developed in the framework of gauge/gravity duality. In the case of two and three entangling regions, the *n*-partite information is equivalent to holographic mutual and tripartite information, respectively. These quantities indicate the amount of shared information or, more precisely, the correlation, between the entangling regions [40]. On the other hand, the Wilson loop is in fact another nonlocal operator which can be used as an important probe for studying phase structures of gauge theories. Investigating the effect of higher-order terms and momentum dissipation on these quantities is the main task of this section.

### 4.1 Holographic mutual information

*h*, there are two different configurations which are schematically shown in Fig. 1. It is worth mentioning that we have restricted ourselves to the case in which the entanglement entropy is an increasing function of the entangling region. Therefore, the mixed configurations have not been considered [43].

*S*stands for HEE of the corresponding entangling region which is given by (3.15). Now the aim is to study the effect of \(\alpha \) and \(\lambda \) on the mutual information and its phase transition. In the presence of momentum dissipation and a GB term, let us write the mutual information as

*h*and \(2\ell +h\) regions from (3.15), it is obtained as follows:

*h*compared to the cases of \(\lambda \le 0\).

### 4.2 Holographic tripartite information

*h*, Fig. 4 shows schematically all possible diagrams for computing the union parts of the tripartite information. The rest of the configurations can be obtained by rearranging these.

Similar to the mutual information, one can investigate that, in the presence of momentum relaxation parameter, the transition curves show a decreasing behavior with respect to \(\alpha \) and \(\lambda \), and for positive (negative) value of \(\lambda \), the phase transition in tripartite information happens in a larger (smaller) ratio than the case of \(\lambda =0\).

^{5}In principle, it can be concluded that the holography leads to a constraint on this quantity and its sign might be employed in a variety of work (see for example [49, 50]). In Fig. 5, we have plotted the tripartite information as a function of momentum relaxation parameter and GB coupling. One observes that it always remains negative. This behavior also holds when one changes the length of entangling regions for the given (fixed) values of momentum relaxation and GB coupling parameters.

### 4.3 Wilson loop

## 5 Conclusion

In principle, in the holographic models, considering higher-curvature terms in the gravity action is well motivated for several reasons; in particular, addressing different types of central charges could be an example. The Lovelock gravity is indeed the simplest set of higher-derivative terms in which various Euler densities appear as higher-derivative interactions in the gravity theory.

In this paper, we studied the effect of higher-order derivative terms on some nonlocal probes in the theories with momentum relaxation parameter. There are in fact two kinds of deformation in the states of dual field theory in this model: the higher-curvature terms, which could address the low-energy quantum excitation corrections, and the deformation due to scalar fields, which are responsible for the momentum conservation breaking. We used holographic methods to obtain the corresponding changes due to these deformations in the coefficient of the universal part in the entanglement entropy. Higher-order gravity theories are interesting in the sense that they provide us with an effective description of quantum corrections and one may probe the finite coupling effects and the *a*- and *c*-theorems via making such corrections to the Einstein gravity theory in the bulk space. We used five-dimensional Einstein GB gravity together with three spatially dependent massless scalar fields to obtain the corrections to universal and finite parts of HEE for strip, spherical and cylindrical entangling regions. For an interval of length \(\ell \) on an infinite line, Myers and Singh introduced a candidate for the *c*-function in a *d*-dimensional CFT which is the coefficient of the finite term in entanglement entropy. This expression in \(d=4\) is given by (3.18) and it can be considered as a function of the anomaly coefficients in the underlying CFT. We showed that, in the presence of the momentum relaxation parameter and GB coupling, this expression has been modified as (3.20). Moreover, in computing the HEE for a strip, a universal logarithmic term appears due to the momentum relaxation parameter which has been modified by the GB coupling. This universal term vanishes at \(\lambda \backsimeq 0.66\); however, noting that the GB coupling is constrained to a small range, i.e. \(-0.194\lesssim \lambda \le 0.09\) [39, 52], one gets a positive valued universal term due to both the momentum relaxation and the GB term in the present range.^{6}

In the case of spherical entangling region, the coefficient of the universal term in HEE could potentially address the *a*-central charge of the corresponding dual conformal field theory whereas the *c*-central charge is related to the coefficient of the universal term in HEE for the cylindrical entangling region. For theories dual to Einstein gravity one obtains \(a=c\); however, in the case of GB gravity one obtains unequal *a* and *c*, this is indeed the main motivation of considering such a term in the gravity action. We obtained the modified coefficients of universal terms which can be interpreted as ‘*c*’-type central charge of dual field theory.

In the context of quantum information theory and also quantum many-body systems, for two disjointed systems, the mutual information is usually used as a measure of the quantum entanglement that these two systems can share; the mutual information can also be utilized as a useful probe to address certain phase transitions and critical behavior in these theories. For example, it is well known that mutual information undergoes a transition beyond which it is identically zero; this kind of transition, which is called a disentangling transition, is in fact a universal qualitative feature for all classes of theories with holographic duals [54]. In this paper, we considered the effect of the GB term on such a phase transition in both the mutual and the tripartite information and it was shown that the behavior of such a phase transition is different, depending on the sign of GB coupling. For two strips with the same length separated by distance *h*, we showed that, for a fixed momentum relaxation parameter, the phase transition of holographic mutual information takes place at a larger distance by increasing the GB parameter. The general behavior of the phase transition is decreasing by \(\alpha \), though for \(\lambda >0\) the phase transition occurs in larger *h* compared to the cases of \(\lambda \le 0\). For \(\lambda >0\) this transition happens at a larger value than the case of \(\lambda <0.\) We also showed that the tripartite information has a negative value in our setup, which means that mutual information is monogamous.

Moreover, by considering the holographic Wilson loop, we found that the sign of \(\lambda \) plays a crucial role in the computation of the effective potential and its corresponding force between point-like external objects. The result shows that both momentum dissipation and GB coupling parameters can lead to a correction of the potential and the corresponding force between quark and antiquark. Noting that from the regularized part of AdS one receives an attractive force between these external particles, the correction due to the momentum relaxation is always repulsive and independent of \(\ell \), which is in agreement with the results in [15]. However, the contribution of GB coupling is somehow nontrivial. The \(\lambda \)-correction part depends on the separation \(\ell \) and according to the sign of GB coupling could be either positive or negative, which results in a decreasing or an increasing attractive force between quark and antiquark, respectively.

## Footnotes

- 1.
In the extended version of RT proposal named the HRT proposal, for time-dependent geometries, one should use the extremal surface [17].

- 2.
GB gravity is a special form of curvature squared action and it was shown that for five-dimensional GB gravity, the proposal of computing HEE presented in [26] reduces to [19, 25] and the results are the same. Note that, taking into account the boundary term, only modifies the coefficient of the leading UV-divergent term.

- 3.
- 4.
In the case of spherical and cylindrical entangling regions, we will consider the terms up to \(\mathcal{O}(\lambda ^2,\alpha ^4,\lambda \alpha ^2,m)\).

- 5.
In the context of quantum information theory, the inequality of the form \(F({A_1},{A_2})+F({A_1},{A_3})\le F({A_1},{A_2}U{A_3})\) is known as the monogamy relation. This feature of measurement is related to the security of quantum cryptography indicating that entangled correlations between \(A_1\) and \(A_2\) cannot be shared with a third system \(A_3\) without spoiling the original entanglement [45].

- 6.
It is worth mentioning that considering the GB terms non-perturbatively leads to the violation of causality in any pure Gauss–Bonnet gravity [53]. Moreover, we assume that momentum relaxation does not change the constraints on the GB coupling. We thank the referee for his/her useful comment on this point.

## Notes

### Acknowledgements

The author would like to thank Mohsen Alishahiha, M. Reza Mohammadi-Mozaffar and Ali Mollabashi for their helpful comments and discussions. MRT also wishes to acknowledge A. Akhvan, A. Faraji, A. Naseh, A. Shirzad, F. Omidi, F. Taghavi and M. Vahidinia for some their comments. We also thank the referees of this paper for their useful comments. This work has partially been supported by IAUCTB.

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