Aspects of Hyperscaling Violating Geometries at Finite Cutoff

Recently, it was proposed that a $T\overline{T}$ deformed CFT is dual to a gravity theory in an asymptotically AdS spacetime at finite radial cutoff. Motivated by this proposal, we explore some aspects of Hyperscaling Violating geometries at finite cutoff and zero temperature. We study holographic entanglement entropy, mutual information (HMI) and entanglement wedge cross section (EWCS) for entangling regions in the shape of strips. It is observed that the HMI shows interesting features in comparison to the very small cutoff case: It is a decreasing function of the cutoff. It is finite when the distance between the two entangling regions goes to zero. The location of its phase transition also depends on the cutoff, and decreases by increasing the cutoff. On the other hand, the EWCS is a decreasing function of the cutoff. It does not show a discontinuous phase transition when the HMI undergoes a first-order phase transition. However, its concavity changes. Moreover, it is finite when the distance between the two strips goes to zero. Furthermore, it satisfies the bound $ E_W \geq \frac{I}{2}$ for all values of the cutoff.


Introduction
In recent years some kinds of very rich irrelevant deformations of two-dimensional conformal field theories (CFT), dubbed T T deformations, have been introduced [1][2][3], under which the action is changed as follows where λ is the deformation parameter and T T (x) is an irrelevant operator of dimension (2,2) which is defined by [1][2][3][4] T T (x) = lim Here T αβ is the stress tensor of the CFT, A i (x, y) is a function which might be divergent in the limit y → x and O i are local operators. The remarkable feature of these deformations is that they are solvable, in the sense that some quantities of the deformed CFT such as energy spectrum can be calculated exactly [1][2][3]. Moreover, analogues of these deformations were recently explored in higher dimensions, i.e. d > 2, [4,9] and in one dimension [5,6].
In the context of AdS/CFT [7], it has been proposed in ref [8] (see also [4,9,10]) that the holographic dual of a T T deformed CFT is a gravity theory in an asymptotically AdS spacetime with a radial cutoff on which one imposes Dirichlet boundary conditions on the fields. Moreover, the deformation parameter λ in the deformed CFT is related to the cutoff r c on the radial coordinate in the bulk spacetime. Furthermore, different aspects of T T deformed CFTs both on the QFT and holography sides were studied extensively including correlation functions [10][11][12], entanglement entropy [13][14][15][16][17][18][19][20], mutual information [21] and entanglement wedge cross section [21]. Motivated by the proposal of [8], we consider Hyperscaling Violating (HV) geometries given in eq. (2.1) at finite radial cutoff and zero temperature. 2 It should be emphasized that, the energy spectrum and Action-Complexity of these geometries at finite cutoff and temperature were recently studied in ref. [23]. The aim of this paper is to explore the effects of a finite radial cutoff in the bulk spacetime on some of the quantum entanglement measures such as Holographic Entanglement Entropy (HEE), Mutual Information (HMI), and Entanglement Wedge Cross Section (EWCS) in these geometries when the entangling regions are in the shape of strips. In particular, we explore the dependence of these quantities on the cutoff, and compare their behaviors to those when the cutoff is very small. In the following, we briefly review these quantities. One of the remarkable and rigorous quantities which is able to measure quantum entanglement in a quantum system is entanglement entropy (EE). Suppose a subsystem A on a constant time slice of the manifold on which a quantum field theory (QFT) lives. By knowing the density matrix ρ of the QFT, one can assign a reduced density matrix ρ A = Tr A ρ to A, where the trace is taken over the Hilbert space H A corresponding to A. Then the EE between the degrees of freedom inside and outside A is defined as the Von-Neumann entropy S = −Tr(ρ A log ρ A ), which can be calculated by the replica trick [24,25]. In the context of the AdS/CFT correspondence [7], there is a very brilliant prescription known as the Ryu-Takayanagi (RT) proposal for the calculation of EE [24,26], which is proved in ref. [27]. According to the RT proposal, for holographic CFTs the holographic dual of EE is given by 3 where G N is the Newton constant in the bulk spacetime. Moreover, Γ A is a codimension two, spcaelike minimal surface in the bulk spacetime which is homologous to the subsystem A, i.e. ∂Γ A = ∂A. The above formula is valid to the leading order in the number N of the degrees of freedom of the CFT, and there is a sub-leading quantum correction to eq. (1.3) (see [25,28]) where we omit it here. On the other hand, EE is a divergent quantity in QFT and depends on the UV cutoff of the theory. However, one can define other measures which are independent of the UV cutoff. One of these quantities is quantum mutual information (MI) which is a measure of both classical and quantum correlations in a bipartite system. For two subsystems A and B, MI is defined as follows where S AB = S A∪B . It has several interesting properties including [30][31][32][33][34][35][36][37]: it is always non-negative as a result of subadditivity. It is finite and independent of the UV cutoff. 4 It shows a first-order phase transition when the two subsystems becomes far enough from each other. Moreover, when the distance between the entangling regions goes to zero, it diverges. On the other hand, when the QFT is in a mixed state, EE is not an appropriate measure of quantum entanglement, and one should apply other measures. One of these quantities is Entanglement of Purification (EoP) [38], which measures both classical and quantum correlations. Suppose that the density matrix ρ AB of two subsystems A and B is mixed. By enlarging the Hilbert space to H AA ⊗ H BB in which A and B are two arbitrary subsystems, one can find a pure state |ψ ∈ H AA ⊗ H BB , such that ρ AB = Tr A B |ψ ψ|. In this case, |ψ is called a purification of ρ AB . Then one can define the EoP as follows [38] E P (ρ AB ) = min where ρ AA = Tr BB |ψ ψ| and the minimization is done over all possible purifications of ρ AB . On the other hand, for a bipartite system consisting of A and B, one can define another quantity called entanglement wedge cross section (EWCS) as follows [39,40] where Σ min AB is a minimal, codimension two, spacelike surface anchored on the RT surface Γ AB corresponding to the region A ∪ B (see figure 7). Moreover, Σ min AB divides the entanglement wedge M AB corresponding to A ∪ B into two parts (see section 4 for more details). In other words, E W measures the minimal cross section of the entanglement wedge M AB . EWCS has a variety of interesting properties including [39,40]: it is non-negative, finite and independent of the UV cutoff. When ρ AB is pure, one has E W = S A = S B . When the two entangling regions are far enough from each other, M AB becomes disconnected (see the right panel of figure 7). Consequently, EWCS undergoes a discontinuous phase transition when the HMI shows a first-order phase transition (see also [41,42]). Furthermore, it satisfies a variety of inequalities such as where the inequality is saturated, whenever ρ AB is pure. It was observed in refs. [39,40] that the properties of EoP [38,44] are exactly the same as those of EWCS. Consequently, it was proposed that EWCS is the holographic dual of EoP [39,40] This conjecture has been explored extensively in refs. [41,42,[45][46][47][48][49][50][51][52][53]. The organization of the paper is as follows: in Section 2, we numerically calculate the HEE for entangling regions in the shape of strips. Moreover, we find analytic expressions for very large and small entangling regions as well as very small cutoff. In Section 3, we first briefly review the holographic prescription for the calculation of MI. Next, we compute the HMI for two parallel disjoint strips and investigate its dependence on the cutoff r c . It is observed that both the HMI and the location of its phase transition depends on the cutoff. In Section 4, we first review EWCS which is believed to be the holographic dual of EoP. Next, we calculate it for two parallel disjoint strips, and study its behavior. It is observed that the EWCS is continuous at the point where the HMI undergoes a phase transition. However, the concavity of the EWCS changes at this point. Moreover, it is verified that eq. (1.7) is valid for all values of the cutoff. In Section 5, we summarize our results.

Holographic Entanglement Entropy for a Strip
In this section, we first review the calculation of HEE in Hyperscaling Violating (HV) geometries at zero temperature when the cutoff is very small. In the following, the case in which the cutoff is very small is dubbed the zero cutoff case. Next, we calculate the HEE at finite radial cutoff. It was shown in ref. [54][55][56] that by turning on an appropriate combination of scalar and gauge fields in the bulk spacetime, one can make geometries called HV geometries whose metrics are given by It should be emphasized that minimization over all possible purifications in eq. (1.5) is a difficult task. Therefore, other candidates for the CFT counterpart of EWCS were introduced in the literature, such as logarithmic negativity [57], odd entanglement entropy [58] and reflected entropy [59,60].
where R is the AdS radius and r F is a dynamical scale. 6 Moreover, we defined an effective hyperscaling violation exponent θ e = θ d . Here z and θ are the dynamical critical and hyperscaling violation exponents, respectively, whose values are restricted by the null energy conditions [61] It was observed in [61] that, when θ > d, the HEE scales faster than the volume of the entangling region which is not consistent with the behavior of EE in a QFT. Moreover, in this case the gravity theory is not stable and it does not have a well defined decoupling limit in its string theory realizations [61]. In the following, we restrict ourselves to the case d − θ ≥ 1, then from the null energy conditions one has z ≥ 1. On the other hand, it should be emphasized that this geometry is dual to a QFT in which the Lorentz and scaling symmetries are broken. Furthermore, for z = 1 and θ = 0, the Lorentz and scaling symmetries are restored and the dual QFT reduces to a CF T d+1 . In this case, the metric (2.1) becomes an AdS d+2 spacetime in Poincaré coordinates. Now we review the calculation of HEE for an entangling region in the shape of a strip in HV geometries, which has been studied extensively in refs. [56,[61][62][63]. One can parametrize a strip with width and length L as follows Due to the translation symmetry along the length of the strip, the profile of the RT surface is given by x 1 = x 1 (r), and hence the area functional is as follows where r t is the radial coordinate of the turning point at which x 1 (r t ) → ∞. In the above expression, we defined an effective dimension d e = d − θ, for convenience. Next, minimization of the area functional leads to To find the HEE, one should first find r t in terms of from eq. (2.6) and substitute it in eq. (2.7). When the cutoff r c is very small, i.e. r c = → 0, the calculation was done in ref. [61]. For d e = 1, the RT surface is a semicircle and r t can be obtained from eq. (2.6) as follows Moreover, the HEE is given by [61] which has a logarithmic UV divergent term. Consequently, the well known area law behavior of the HEE is violated in this case [61]. On the other hand, for d e = 1 from eq. (2.6), r t is simply given by [61] r t = 2Υ , (2.10) where Υ is defined as follows In this case, the HEE is given by [61] (2.12) In the following, we study the HEE for the finite cutoff case. To do so, we consider the two cases d e = 1 and d e = 1, separately. For d e = 1, one can find the HEE exactly. However, for d e = 1, finding r t in terms of is a difficult task and one has to do it either numerically or by making some approximations.

Finite Cutoff and d e = 1
For d e = 1, form eq. (2.6) one can simply find that the RT surface is again a semicircle such that Now in contrast to eq. (2.8), r t depends on the cutoff r c . On the other hand, from eq. (2.7), one obtains (2.14) Next, by plugging eq. (2.13) into eq. (2.14), one has Notice that the HEE is independent of the exponent z. When the cutoff is very small, i.e. r c , one might expand eq. (2.15) in powers of r c to obtain For the zero cutoff case, i.e. r c = → 0, one might neglect the second term, and hence the HEE shows the expected logarithmic behavior which is given in eq. (2.9). Therefore, imposing a finite radial cutoff, introduces sub-leading corrections to the HEE. As mentioned before, for z = 1 and θ = 0, the Lorentz and scaling symmetries in the dual QFT is restored and it becomes a CF T d+1 . In particular, for θ = 0 and d = 1, from eq. (2.16), one has where c = 3R 2G N is the central charge of the dual CFT [64]. The above expression agrees with the HEE of a CF T 2 at zero temperature and finite cutoff calculated in ref. [15].

Finite Cutoff and d e = 1
For d e = 1, from eq. (2.6) one obtains Here we set R = 1 and renormalized S asS where Υ is given by eq. (2.11). Notice that in contrast to the zero cutoff case, r t depends on the cutoff. In figure 1, r t is drawn as a function of for different values of the cutoff r c . From figure  1, it is straightforward to see that for very small and large entangling regions, one has r t ≈ r c and ≈ r t r c , respectively. On the other hand, from eq. (2.4) one has Next, one has to find r t from eq. (2.18) and plug it into eq. (2.19) which is a very tough task. However, one can easily find the HEE numerically. In figure 2, the HEE is drawn as a function of for different values of r c . Since finding an analytic expression for r t in terms of and r c is impossible for an arbitrary value of r c , in the following we consider very large and small entangling regions and find perturbative expressions for the HEE.

Very Large Entangling Regions
From figure 1, it is straightforward to see that for very large entangling regions, one has ≈ r t r c . In this limit, one can expand eq. (2.6) in powers of rc rt 1 as follows (2.20) By keeping the first two terms on the right hand side of the above equation, one can rewrite it as follows Then one can find r t as follows (2.22) In figure 3, r t is drawn as a function of l. It shows that r t → r c when → 0. Moreover, r t → ∞ when → ∞, as it was expected. On the other hand, by expanding eq. (2.19) in powers of rc rt , one . (2.23) Next, by plugging eq. (2.22) into the above expression and expanding it in powers of rc 1, one obtains S = S 0 + ∆S, (2.24) where S 0 is the HEE for the zero cutoff case given in eq. (2.12), and (2.25) In figure 4, the above expression for the HEE is drawn as a function of for different values of r c , and compared with the numerical result.

Very Small Entangling Regions
From figure 1, it is straightforward to see that for very small entangling regions, one has r t ≈ r c . In this limit, one can expand eq. (2.18) around r t = r c as follows Next, one can solve the above equation and find r t . Then one can expand it in powers of l rc 1, and obtain In figure 5, the above solution is drawn as a function of . On the other hand, by expanding eq.
(2.28) Next, by plugging eq. (2.27) into the above expression, and expanding it again in powers of rc 1, one obtains (2.29) In figure 6, the above expression for the HEE is drawn as a function of , and compared with the numerical result.

Very Small Cutoff
For finite and very small r c , one again has ≈ r t r c , and hence the HEE is the same as in section 2.2.1. From eq. (2.24) and (2.25), it is obvious that, when r c = → 0, one has ∆S = 0 and hence the HEE reduces to eq. (2.12). Before concluding this section, we should emphasize that EE is a quantity defined in the dual QFT and should be expressed in terms of the parameters of the QFT. In particular, one should be able to express the coefficient c = R d 4G N r θ F which appears in the HEE. Unfortunately, for arbitrary values of the exponents z and θ, we cannot interpret the coefficient c in the dual QFT. However, in ref. [65] the two-point function of stress tensor in the dual QFT was calculated for the case z = 1 as follows Moreover, the constant C T is defined as follows [65] (2.32) Having said this, one might rewrite the coefficient c in terms of C T as follows is a dimensionless constant depending on d and d e .
As mentioned before, for z = 1 and θ = 0, the Lorenz and scaling symmetries are restored in the dual QFT and it becomes a CF T d+1 . Notice that this CF T d+1 lives on R d+1 and is in its vacuum state. Therefore, all of our results can be applied for this CF T d+1 , if one sets θ = 0. In particular, for d ≥ 2 and very small cutoff, by setting θ = 0 in eq. (2.25), one can write ∆S as follows where for c 0 and C T one should set θ = 0 in eqs. (2.32) and (2.34).

Holographic Mutual Information
In this section, we calculate holographic mutual information (HMI) for two parallel disjoint strips and study its dependence on the cutoff r c . In ref. [34] it was proposed that, there are two types of RT surfaces which contribute in S AB : Connected) in which each RT surface is started from the boundary of one entangling region, say A, and ended on the boundary of the other entangling region B (see the left panel of figure 7). Disconnected) in which each RT surface is started from Figure 7: Illustrations of the connected and disconnected RT surfaces which contribute to the HMI as well as EW for a bipartite system A ∪ B. Left) Connected configuration: the two entangling regions A and B are close enough to each other such that the RT surface corresponding to S AB is given by Γ h ∪ Γ 2 +h which are shown by the solid purple curves. In this case, the EW is also connected and indicated by the shaded violet region. The minimal surface Σ min AB which measures the minimal cross section of EW is indicated by an orange dashed curve. Right) Disconnected configuration: the two entangling regions are far enough from each other, such that the RT surface corresponding to S AB is disconnected and given by Γ A ∪Γ B shown by purple solid curves. Moreover, the EW is disconnected and given by M A ∪M B which are indicated by the shaded violet regions. In this case, the minimal surface Σ min AB does not exist, and hence E W = 0. Notice that, these diagrams are schematic, and only for the d e = 1 case, the RT surfaces are semicircles.
the boundary of one of the entangling regions and is ended on the other boundary of the same entangling region (see the right panel of figure 7). It was proposed in ref. [34] that S AB = Min (S con. , S dis. ) , where S con. and S dis. are the HEE for the connected and disconnected configurations, respectively. Therefore, depending on the size of the entangling regions and the distance between them, always one of the connected or disconnected configurations dominates and contributes in the HMI. Consequently, there is a first-order phase transition in the HMI [34,35]. For convenience, we choose the width of both strips to be equal to and show the distance between them by h. One simply obtains In the following, we consider the d e = 1 and d e = 1 cases, separately. It should be pointed out that the HMI for HV geometries at zero cutoff was studied in refs. [37,62,66] which is independent of the UV cutoff, and for θ = 0 and d = 1, it reduces to that of a CF T 2 in its vacuum state [34,37]. Moreover, there is a first-order phase transition which happens at the critical value h crit. given by Furthermore, as it was shown in refs. [36,43], when the entangling regions have common boundaries, i.e. h → 0, one has I 0 → ∞. On the other hand, for the finite cutoff case from eq. (2.15), one obtains Therefore, in contrast to the zero cutoff case, h crit. depends on the cutoff r c . In particular, the left side of figure 9, shows that h crit is a decreasing function of r c . Another interesting point is that according to eq. (3.5), the HMI depends on the cutoff r c , which is in contrast to the zero cutoff case, i.e. eq. (3.3). Furthermore, it is observed from the right panel of figure 9 that the HMI is a decreasing function of r c , and goes to zero in the limit r c → ∞. Moreover, when h → 0 from eq. (3.5), one has Consequently, in contrast to the zero cutoff case, the HMI remains finite when the distance between the two strips goes to zero. In ref. [21], it was observed that for BTZ black holes at finite radial cutoff, the HMI shows the same behavior.

d e = 1
In figures 10, 11 and 12 the HMI is calculated numerically from eq. (2.19). In figure 10, the HMI is drawn as a function of , and one can see that by increasing d e the phase transition happens at smaller values of . In figure 11, the HMI is drawn as a function of h. It is evident that when the distance between the two strips goes to zero, i.e. h → 0, the HMI remains finite which is in contrast to the zero cutoff case (see also eq. (3.9)). Furthermore, comparison between the left and right panels of figure 11, shows that for a specific value of d e , h crit. decreases by increasing r c . 7 On the other hand, in figure 12, the HMI is drawn as a function of r c . It is observed that the HMI Moreover, by increasing d e , h crit becomes larger. Furthermore, at h = 0, the HMI is finite which is in contrast to the zero cutoff case. . It is obvious that the HMI depends on the cutoff, and is a decreasing function of r c , such that it goes to zero when r c → ∞.
depends on the cutoff and is a decreasing function of r c , and it goes to zero in the limit r c → ∞.
In the following, we find analytic results for very large and small entangling regions.

Very Large Entangling Regions
First we consider very large entangling regions, i.e. r c . From eq. (2.24), one has where I 0 is the HMI for the zero cutoff case and given by [37] , (3.9) in which I(n) is defined as follows (3.10) Notice that I 0 is independent of the cutoff r c . Moreover, it goes to infinity when h → 0. On the other hand, one has − d e (3d e + 1) 3(d e + 1) 2 Υ I(4d e + 2) (Υr c ) 2(de+1) + 3d e (3d e + 1)Υ I(5d e + 1) (Υr c ) 3de+1 + · · · . (3.11) Notice that the above expression is only valid for r c and h. From the above expression, one can see that the HMI depends on the cutoff in contrast to the zero cutoff case. In the left panel of figure 13, the perturbative expression (3.8) for the HMI is drawn as a function of r c , and compared with the numerical result based on eq. (2.19). It is observed that the HMI is a decreasing function of r c . Moreover, in the middle and right panels of figure 13, the perturbative expression (3.8) for the HMI is drawn as a function of and h, and compared with the numerical result based on eq. (2.19). On the other hand, by applying eq. (3.8) one can find the critical length h crit as follows: where Therefore, h crit. depends on the cutoff r c . In figure 14, eqs. (3.12) and (3.14) are drawn as a function of r c . It is observed that h crit decreases by increasing the cutoff. Furthermore, it should be pointed out that for finite and very small cutoff, again one has r c . Consequently, eqs. (3.8) to (3.14) are still valid when the cutoff is very small. From eq. (3.11), it is obvious that for r c = → 0, one has ∆I = 0 and the HMI becomes equal to that of the zero cutoff case I 0 , as can be seen in the left panel of figure 13.

Very Small Entangling Regions
For very small entangling regions, i.e. r c , from eq. (2.29), one can simply write and K(n) is defined as follows In figure 15, the perturbative expression (3.16) for the HMI is drawn as a function of r c , and h, and compared with the numerical results based on eq. (2.19).

Entanglement Wedge Cross Section
In this section, we study entanglement wedge cross section for two strips. Consider a spacelike region A in a holographic CFT and its density matrix ρ A . It was proposed in refs. [67][68][69] that the holographic dual of ρ A is a codimension zero region M A in the bulk spacetime which is called Entanglement Wedge (EW). More precisely, EW is the domain of dependence of a spacelike region in the bulk spacetime which is enclosed by the corresponding RT surface Γ A and the region A. In the following, when we talk about EW, we mean a canonical time slice of it which on the boundary coincides with the region A. Now consider a bipartite system consisting of two disjoint subsystems A and B in the CFT whose RT surface and EW are shown by Γ AB and M AB , respectively. Next, one can decompose Γ AB as follows [39] and define two regionsΓ AB . Now one can find a RT surface Σ min AB in the bulk spacetime whose area gives the HEE ofΓ A andΓ B [39] In this case, it is evident that Σ min AB is homologous toΓ A andΓ B . Then one can define the entanglement wedge cross section (EWCS) as in eq. (1.6) [39,40]. In other words, EWCS measures the minimal cross section of the corresponding EW. In the following, we calculate the EWCS for two strips with equal widths which are separated by the distance h (see figure 7). When the two strips are close enough to each other, the EW is connected (see the left panel of figure 7). In this case, it is straightforward to see that where r t (h) and r t (2 + h) are the radial coordinates of the turning points for the RT surfaces Γ h and Γ 2 +h , respectively. Notice that for the connected configuration, one has Γ AB = Γ h ∪ Γ 2 +h . On the other hand, when the strips are far enough from each other, the EW is disconnected (see the right panel of figure 7). In this case, there is no minimal surface Σ min AB , and hence E W = 0.

d e = 1
For d e = 1 and the zero cutoff case, by applying eq. (2.10) and (4.3) one easily obtains where for θ = 0 and d = 1 it reduces to that of AdS 3 in Poincaré coordinates (see ref. [39]). On the other hand, for the finite cutoff case, from eqs. (2.13) and (4.3) one can simply find E W as follows (4.5) From the above expression, one can see that E W is independent of the dynamical exponent z and depends on the hyperscaling violation exponent θ only through the coefficient 1/r θ F . Moreover, the comparison of eqs. (4.4) and (4.5) shows that only in the finite cutoff case, E W depends on the cutoff r c . In the left panel of figure 16, E W is plotted as a function of r c . It is observed that E W satisfies the inequality given in eq. (1.7) for all values of r c . Furthermore, both of the E W and HMI are decreasing functions of the cutoff, and E W goes to zero when r c → ∞. In the middle and right panels of figure 16, E W is drawn as a function of and h, respectively. It is observed that at the point h crit. where the HMI undergoes a first-order phase transition, E W is smooth and its concavity changes. In other words, there is a point of inflection in E W which coincides with h crit . It is in contrast to the zero cutoff case, where E W shows a discontinuous phase transition at h crit. . Moreover, similar to the zero cutoff case [39,41,42], by increasing the distance h, E W goes to zero. On the other hand, from eqs. (4.4) and (4.5), one can see that in the limit h → 0, E 0 W diverges. However, E W remains finite in this limit. Furthermore, in the limit h → ∞, E W becomes zero which has the same behavior as in the zero cutoff case. At the end, when the In this case, from eq. (4.3), one can easily write (4.7) For the zero cutoff case, by plugging eq. (2.10) into the above equation, one can simply obtain (see also [42]) where E(n) is defined as follows (4.9) One can see that similar to the d e = 1 case (see eq. (4.4)), E 0 W is infinite when h → 0. Furthermore, in the limit h → ∞, it becomes zero. Moreover, it is independent of the cutoff. On the other hand, one can numerically calculate eq. (4.7). In figure 17, E W is drawn as a function of and h. The behaviors of E W are very similar to those of the d e = 1 case in the previous section. In particular, it does not show a discontinuous phase transition when the HMI undergoes a phase transition. Instead, E W has a point of inflection which coincides with h crit. . Moreover, it is finite Moreover, in figure 18, E W is drawn as a function of r c . It is observed that it is a decreasing function of the cutoff. In particular, it is finite when r c → 0, and goes to zero when r c → ∞. Furthermore, the inequality (1.7) is valid for all values of the cutoff. On the other hand, one can find analytic expressions for E W in some limits. In the following, we consider the very small cutoff, i.e. , h r c , and very large cutoff, i.e. , h r c , cases, respectively.
• Very large cutoff ( , h r c ): In this limit, one can use eq. (2.27), and find E W as follows  Therefore, when r c → ∞, one has E W → 0.

Discussion
In ref. [4,8,9], it was proposed that the holographic dual of a T T deformed CF T d is a gravity theory in an AdS d+1 spacetime with a radial cutoff. Motivated by this proposal, we considered Hyperscaling Violating (HV) geometries at zero temperature and finite radial cutoff, which one might expect to be dual to a T T deformed QFT in which the Lorentz and scaling symmetries are broken. We calculated holographically some measures of quantum entanglement in these geometries including holographic entanglement entropy (HEE), mutual information (HMI) and entanglement wedge cross section (EWCS) which is proposed to be the holographic dual of entanglement of purification [39]. In particular, we considered entangling regions in the shape of strips, and calculated the HEE numerically. It was observed that the turning point depends on the cutoff, in contrast to the zero cutoff case. Moreover, we found analytic expressions for very small and large entangling regions as well as very small cutoff. Furthermore, we studied the HMI between two disjoint parallel strips, and its dependence on the cutoff. The HMI shows interesting behaviors in comparison to the zero cutoff case: • It is a decreasing function of the cutoff r c , and goes to zero in the limit r c → ∞ (see figure 9 and 12). It is in contrast to the zero cutoff case where the HMI is independent of the cutoff (see eqs. (3.3) and (3.9)).
• It still shows a first-order phase transition, and the critical length h crit. becomes larger by increasing d e (see figure 11). Moreover, h crit. depends on the cutoff and decreases by increasing r c (see figure 9 and 14). It is in contrast to the zero cutoff case where h crit. is independent of the cutoff (see eqs. (3.4), (3.13) and (3.15)).
• When the distance between the two entangling regions becomes zero, i.e. h → 0, the HMI does not diverge and remains finite (see figures 8 and 11). This behavior is in contrast to the zero cutoff case where the HMI diverges in the limit h → 0 (see eqs. (3.3) and (3.9)).
• Since the HEE is independent of the dynamical exponent z, the HMI also shows the same behavior.
On the other hand, we considered EWCS for two disjoint parallel strips. It was observed that it has the following properties: • It is a decreasing function of the cutoff and goes to zero in the limit r c → ∞ (see figure 16 and 18). It is in contrast to the zero cutoff case where E W is independent of r c (see eqs. (4.4) and (4.8)).
• It is a smooth function of both and h (see figures 16 and 17), and at the point h crit. where the HMI undergoes a phase transition, it does not show a discontinuous phase transition. It is in contrast to the zero cutoff case. However, at h crit. the concavity of E W changes.
• By increasing the distance h between the two strips enough, E W goes to zero, similar to the zero cutoff case (see figures 16 and 17). However, in the limit h → 0, E W remains finite, which is in contrast to the zero cutoff case where E W diverges in this limit (see eqs. (4.4) and (4.8)).
• The inequality given in eq. (1.7) is satisfied for all vales of the cutoff r c as well as and h.
• It is independent of the dynamical exponent z.
Furthermore, for z = 1 and θ = 0, the Lorentz and scaling symmetries are restored and the background in eq. (2.1) becomes an AdS d+2 spacetime in Poincaré coordinates. Since all of the aforementioned quantities are independent of the dynamical exponent z, all of our results can be applied for an AdS d+2 bulk spacetime in Poincaré coordinates if one sets θ = 0.