Holographic Mutual and Tripartite Information in a Non-Conformal Background

Holographic mutual and tripartite information have been studied in a non-conformal background. We have investigated how these observables behave as the energy scale and number of degrees of freedom vary. We have found out that the effect of degrees of freedom and energy scale is opposite. Moreover, it has been observed that the disentangling transition occurs at large distance between sub-systems in non-conformal field theory independent of l. The mutual information in a non-conformal background remains also monogamous.


I. INTRODUCTION
The gauge/gravity duality provides a significant framework to study key properties of the boundary field theory dual to some gravitational theory on the bulk side [1]. The most concrete example of gauge/gravity duality is the Anti − desitter/Conf ormal correspondence which proposes a duality between asymptotically AdS spacetimes in d + 1 dimensions and d−dimensional conformal field theories. This outstanding correspondance is indeed a strong-weak duality which makes it possible to study various aspects of the strongly coupled systems such as quantum chromodynamic, quark-qluon plasma and condenced matter [2][3][4]. Although, study of different observables may not seem simple in the feild theory side the duality indeed proposes applicable prescription in the gravity side which makes the calculations as much as simple.
The applicability of the gauge/gravity duality is not restricted to CF T 's. It is then important to develop our understanding of this duality for more general cases.
There are many different families of non-conformal theories which one can study the effect of the non-coformality on their physical quantities [5,6].
When one studies properties of a given quantum field theory, it is common to investigate behaviors of correlation functions of local operators in the theory. However, properties of non-local quantities are equally important. One example of important non-local physical quantities in field theory with a well known dual gravity description is the entanglement entropy which nicely characterizes quantum entanglement between two subsystems A and its complementĀ for a given pure state. Since the quantum field theories have infinitely degrees of freedom, the entanglement entropy is divergent. In [7], it has been shown that the leading divergence term is proportional to the area of the entangling surface. In the language of AdS/CF T , the entanglement entropy has a holographic dual given by the area of minimal surface extended in the bulk whose boundary coincides with the boundary of the sub-system [8,9]. The study of entanglement entropy has not also restricted to the CF T 's. For instance in [10,11] the authors have been used this quantity to probe non-conformal theories nicely.
Due to the U V divergence structure of entanglement entropy, it is better to introduce an appropriate quantity, which is just a linear combination of entanglement entropy and then remain finite, called mutual information. It is an important concept in information theory. For two sub-systems A and B, it is more natural to compute the amount of correlations (both classical and quantum) between these two sub-systems which is given by the mutual information [12,13]. Note that the subadditivity property of the entanglement entropy gurantees that mutual information is always non-negative. The tripartite information is another important quantity which is defined for three sub-systems and measures correlation between them. In fact, this quantity can be used to measures the extensivity of the mutual information. It is also free of divergence and can be positive, negative or zero [14][15][16].
The background we have considered in this paper is a holographic 5-dimensional model consisting of Einstein gravity coupled to a scalar field with a non-trivial potential, which is negative and has a minimum and a maximum for finite values of scalar field. Each of these extrema corresponds to an AdS 5 solution with different radii [5]. In the gauge theory the 4-dimensional boundary is not conformal and, at zero temperature, flows from an U V fixed point to an IR fixed point. This renormalisation group is dual on the gravity side to a geometry that interpolates between two AdS spaces. We are now interested in study holographic mutual and tripartite information in a non-conformal background and study the effect of field theory parameters such as energy scale and number of degrees of freedom on these quantum information quantities.

II. REVIEW
The holographic model we study here is a fivedimensional Einstein gravity coupled to a scalar field with a non-trivial potential whose action is given by where G 5 N is the five-dimensional Newton constant and R is the Ricci scalar of curvature corresponding to the metric g. Scalar field and its potential are also denoted by φ and V (φ), respectively.
In order to have a bottom-up model the following potential has been choosen [5] This potential possess a maximum at φ = 0 and a minimum at φ = φ M > 0, each of them corresponds to an AdS 5 background with different radii. In the language of the gauge theory, each of these solutions is dual to a fixed point of the RG flow from the U V fixed point at φ = 0 to the IR fixed point at φ = φ M > 0. It is easy to see that the radii of these asymptotically AdS 5 take the form [5] Following the fact that L IR < L U V = L and according to gauge-gravity dictionary, indicated that the number of degrees of freedom in the gauge theory is related to the radius of the background, a smaller number of degrees of freedom lives in the IR limit, i.e. N IR < N U V . Furtheremore, as one increases φ M the difference in degrees of freedom between the U V and IR fixed points increases.
If one interested in domain-wall solutions which are interpolating between the two underlying AdS 5 backgrounds, the vacuum solutions to the Einstein equations can be obtained from the action (1) . The parametrized metric for arbitrary φ M can be read where where Λ is the energy scale that break the conformal symmetry in the dual gauge theory. It is also related to the asymptotic value of the scalar field, i.e. φ(r → ∞).
For more details about the background see [5]. Entanglement entropy is one of the most significant features of quantum physics and plays a significant role in understanding quantum many-body physics, quantum field theory, quantum information and quantum gravity. Consider a constant time slice in a d−dimensional quantum field theory and divide it into two spatial regions A andĀ where they are complement to each other. In quantum field theory, the reduced density matrix for region A can be computed by integrating out the degrees of freedom living inĀ, i.e. ρ A = T rĀ ρ where ρ is the total density matrix. The entanglement entropy measures the entanglement between an arbitrary subregion A and its complementĀ. It is defined as the von Neumann entropy of the reduced density matrix where ρ A is the reduced density matrix of A. In general, it is difficult to compute entanglement entropy directly due to the infinite degrees of freedom in a field theory. Motivated by this and by applying AdS/CF T correspondence, a holographic prescription, known as Ryu and Takayanagi (RT ) formula, has been proposed to compute entanglement entropy through the following area law relation [8,9] where γ A is the d−dimensional extremal surface in AdS d+2 whose boundary is given by ∂A and G d+2 N is the d + 2− dimensional Newton constant. Intrestingly, one can extend this formula to any asymptotically AdS spaces.
For two disjoint subregions A and B one can define the mutual information, I(A, B), which measure the amount of correlation between them [13] where S(Y ) denotes the entanglement entropy of the region Y . It is finite, regulator-independent and nonnegative. When the two subregions are close to each other, there is a finite correlation between them, i.e. I = 0, however as the separation between them increases mutual information vanishes, i.e. I = 0. we call the distance at which disentangling transition occurs as x DT . Another interesting quantity that can be defined from the entanglement entropy is the tripartite information where A, B and C are three disjoint subsystems. It is obvious from the third line in (10) that tripartite information is a measure of the extensivity of the mutual information. According to the RT formula, the mutual information is always extensive or superextensive, I [3] = 0 and I [3] < 0 respectively. Hence holographic mutual information is monogamous [14]. In this paper we are interested in studying the effect of the energy scale Λ and the number of degree of freedom on the mutual and tripartite information. In the following we represent the numerical results corresponding to study the effects of parameters such as energy scale Λ and the length of two subsystems l on the mutual and tripartite information.

CFT
In figures 1 and 2 we have plotted holographic mutual information as a function of x for two subregions of length l = 2 and l = 0.2, respectively and in   For larger values of Λ, one might guess that x DT in the non-conformal field theory (N CF T ) should be smaller than x c DT since N ≡ (N U V − N IR ) > 0. However, our results, in table I for l = 0.2, are opposed and indicate that x DT > x c DT especially for larger values of Λ. Therefore we do beleive that the effect of energy scale Λ overcomes the decrease of degrees of freedom and causes the two sub-systems become disentangled at further distances. In other words, our intution, i.e smaller number of degrees of freedom smaller mutual information, is correct when Λ is small enough.
In the case of large l, i.e. l = 2, for large Λ the two sub-systems do not meet energy scale Λ effectively and become uncorrelated at distances just like the CF T case, x DT x c DT . In other words, on the gravity side the turning point of the extremal surface is far away from the position of the energy scale Λ and therefore it does not change the shape of this surface substantially. However, for small Λ the energy scale plays the key role and x DT > x c DT , in agreement with figure 2. Up to now our results can be classified as follows: • The energy scale Λ and decrease of degrees of freedom have opposite effect on the mutual information.
• x DT is always bigger than x c DT in the presence of energy scale Λ. Put it in different words, the two sub-systems in N CF T become disentangled in larger separation distance than CF T case, independent of l.
• These two regims reveal by our numerical caculation: It can be seen from figure 3 that for fixed values of l and φ M by increasing Λ the two sub-systems feel the appearance of the energy scale tangibely or equivantly difference between mutual information of CF T and N CF T become larger up to a maximum at a specific value of x, let's say x max . This value seems depend on the energy scale Λ mildly. For x > x max , the difference decreases and there is a point at which I N CF T = I CF T . It is interesting since the effect of decrease of number of degrees of freedom and of energy scale Λ cancel out each other at this point. There are some values of x where I N CF T > I CF T . Note that all curves reach one since x DT > x c DT and therefor ∆I I N CF T = 1.
In figure 4 we have pictorially shown phase space diagram corresponding to the two sub-systems with same length l. We set φ M = 10 and consider different values of Λ. The diagram shows the regions where the two sub-systems have either non-zero or zero mutual information. All curves stand for zero mutual information and the area below them represents the regime of parameters where there is non-zero correlation between the two sub-systems. It can be observed from this plot that the region of the phase space where the mutual information has non vanishing value in non-conformal field theory is wider than that of the CF T 's one which coincided with the results reported in table I. Furtheremore, on the field theory side the non-conformal field theory behaves conformally in the U V regime which is probed by very small l, the length of two sub-systems. It is evident from the figure 4 that for small Λ and with small values of x and l, the region where the correlation between two subsystems is non-zero, is closer to the conformal result than the large Λ which is in perfect agreement with results (11). In other words, on the gravity side for very small value of l one may argue that the turning point of the extremal surface gets closer to the boundary region which is asymptotically AdS 5 and then the deviation from this geometry result vanishes approximatelly for the smaller Λ. This is in agreement with the table I. By incresing l and x the role of energy scale Λ exchanges and hence the region of entanglement in phase space become more limited. For smaller Λ, In other words, sufficiently deep in the IR, as one can observe from the figure 4 that the nonzero mutual information region approaches to the AdS 5 result which is consistent with (12).
In figure 5 we have fixed the energy scale Λ = 2 and shown how changing the value of φ M can affect the phase space diagram of two sub-sytems. It is evident that as φ M increases the region of the parameter space where two sub-systems have non-zero correlation begins to be more broader. In other words, the more difference in number of degree of freedom between U V and IR regime, the wider region of entanglement. Furthermore, for small l changing values of φ M has no role and the results are approximately the same as CF T case since small values of l probes the U V of the field theory.  In figure 6 holographic mutual information as a function of x for two subregions of length l = 2 has been plotted. In the table II we have listed different values of x DT corresponding to l = 2 and l = 8 as small and large length. We fix Λ = 1 and consider differentt values of φ M . From the above tables, the interesting point is that whether l is small or large the two sub-systems become disentangled for larger distances in the N CF T . In fact, as φ M increases the difference in degrees of freedom between the U V and IR fixed points increases and hence one may expect that x c DT < x DT but the energy scale Λ has dominant effect and causes x c DT > x DT . Similar to the previous result this can be easilly seen from figure 6. We have plotted in figures 7 and 8 the results of tripartite information I [3] as a function of the distance between sub-systems x for different values of φ M and Λ. It is obvious that the tripartite information starts initially at the negative values and ends finally at less negative values passing through an intermediate phase where it is absolutely negative. Noticebly, depending on the x l , these values can be non-positive interchangeably. As one expects tripartite information is generically non-positive in both CF T and N CF T and hence mutual information in N CF T respects also the strong subadditivity of the holographic entanglement entropy and the monogamy of the holographic mutual information. In both plots, independent of how Λ and φ M vary, tripartite information for CF T is more negative than N CF T . From figure 7 it is observed that the energy scale has pushed tripartite information towards the extensive mutual information, i.e. I [3] = 0 and this process has been catalyzed by increasing Λ. On the other side, from figure 8 one can observe that increasing φ M has the same story as the energy scale's one.