New black holes with hyperscaling violation and transports of quantum critical points with magnetic impurity

We find new black hole solution with hyper-scaling violation which is dual to the quantum matters doped with magnetic impurities near the quantum critical points. Using this solution, we calculate all transport coefficients including electrical, thermo-electric and heat conductivities using the holographic method and investigated their properties by plotting the analytic results as a function of physical parameters $H,T,\beta,q_{\chi}$ which are magnetic fields, temperature, impurity density and magnetic coupling respectively. We also investigated the the phase transitions from weak localization to weak anti-localization as the critical exponents change.


Introduction
When many particles are strongly interacting, entire system is entangled and single particle picture can not be used for low energy system property. In such case, reducing the degree of freedom is the key issue of condensed matter physics and one such idea is to look at the quantum critical point (QCP), where almost all informations are lost except a few critical exponents and symmetries. This is because there is no scale there hence no shape dependence either, so that detailed knowledge of system can not be retained there. Such information loss leads to the universality, which is parallel to the black hole (BH) physics where no hair theorem states that BH can not have much characters. Such parallelism between QCP and BH is the underlying principle of applying holographic method to strongly correlated condensed matter [1,2]: if a QCP is characterized by z, θ defined by the dispersion relation ω ∼ k z and the entropy density s ∼ T (d−θ)/z , there exist a metric with the same scaling symmetry.
with (t, r, x) → (λ z t, λ −1 r, λx) and the method developed for the exact AdS/CFT [3][4][5] is assumed to hold for this case too. The hyper-scaling violation metric has been constructed and used in various context  after flurry of activity on Lifshitz metrics [31][32][33][34][35][36][37][38][39][40][41]. Such method has been applied to the simplest case z = 1, θ = 0, which is called Dirac material, [42][43][44][45] with quantitative agreement with experimental results. The Dirac material is related to the strongly correlated system because having a small fermi surface(FS) is a general way for a system to be strongly correlated: the Coulomb interaction in a metal is small only because the charge screening is effective in the presence of large fermi surface. Extremely clean graphene [46,47] and the surface of topological insulator (TI) [48][49][50] provide strongly interacting system showing anomalous transports that could be quantitatively explained by holographic method [42][43][44][45].
The surface of TI does not have small FS in general because its fermi level is defined by a bulk and it is not necessarily pass through the tip of the zero mode cone. However, one can open the surface gap by doping magnetic impurities and tune the size of the gap such that the band just touches the Fermi level. At that point, fermi surface is small and there smooth transition from weak anti-localization to weak localization as the temperature goes down. Previously we showed that such transition can be quantitatively explained by strong correlation and its holographic analysis [43,44].
In this paper, we extend such studies to the cases z = 1 and θ = 0, the general non-Dirac materials. Our analysis will be useful for multilayer graphene system where the excitations are known to have dispersion relation ω = k n [51] as well as Dirac materials with nonzero θ.
The rest of the paper consists of as follows. In section 2, we introduce the hyperscaling violation gravity model with a magnetic coupling and present its solution. In section3, we calculate various transport coefficients. In section 4, we give a detailed study of the magneto-transports without and with magnetic impurity by plotting various transports as functions of physical parameters like temperature and impurity density. We will find phase transitions from weak localization to weak anti-localization as the critical exponents changes. In section 5, we study the density dependence of transports in detail. In section 6, we conclude with summary.

The model and its black hole solution
To get the solution with the dynamical exponent z and the hyperscaling violating factor θ, we consider a 4 dimensional action.
where we will use the convention 2κ 2 = 16πG = 1. We use ansatz where χ i 's are d−massless linear axions introduced to break the translational symmetry and (β 1 , β 2 ) = (α, λ) denote the strength of momentum relaxation. The action consists of Einstein gravity, axion fields, and U (1) gauge fields and a dilaton field. For simplicity, we only consider two U (1) gauge F (1) rt and F (2) rt in which the first gauge field plays the role of an auxiliary field, making the geometry asymptotic Lifshitz, and the second gauge field is the exact Maxwell field making the black hole charged.
The equations of motion are given by The solution for the dilaton field is given as The gauge couplings Z 1 and Z 2 can be solved to give where λ 1 = (θ − 4)/ν and λ 2 = ν/(2 − θ). Other exponentials and potential are where H is a constant magnetic field. Finally, the solution is given by I = (λx, λy), (2.11) where β 2 = α 2 + λ 2 and c 2 , c 3 , c 4 can be expressed in terms of θ and z The charge density is determined by the condition a i (r H ) = 0 at the event horizon: The the entropy density and the Hawking temperature read Notice that all three terms in eq. (2.18) contribute negatively and if both z, θ < 2 and at zero temperature, the solution near the horizon becomes AdS 2 × R 2 : We classify the plane of (z, θ) into several classes A, B, C depending on the behavior of the temperature as a function of r H , the radius of the black hole which again is a complicated function of parameters of the theory. See Figure 1. A is the region which allows negative T . B has non-negative T and can be further classified to B 1 , B 2 , B 3 . For B 2 , temperature is monotonic function of r H and cover the entire positive temperature. For B 3 , there is a minimum temperature regardless of the chemical potential µ. For B 1 , depending on µ, temperature has minimum like B 3 (real red curve in (b)) or monotonic like B 2 (dotted red curve in (b)). we have two subclasses which are represented by solid (dashed) red curve for the case with the finite (vanishing) µ. We do not have positive T in region C. the monotonicity of

Calculation of electrical and thermal conductivities
We calculate the transport following the idea of linear response theory. We consider the following perturbations δg tx = h tx (r) + tf 3x (r), δg ty = h ty (r) + tf 3y (r), δg rx = h rx (r), δg ry = h ry (r), (3.1) Requiring the linearised Einstein equations to be time-independent, we can get the explicit form of f 1i , f 2i , and f 3i as The thermo-electric transport can be calculated at the event horizon. One can take the Eddington-Finkelstein coordinates (v, r) such that the background metric is manifestly regular at the horizon, where v = t + dr V /U . In this coordinate, the perturbed metric becomes In order to ensure the regularity of the perturbed metric at the horizon, we ask the vanishing of last two terms at the horizon so that Similarly, we can expand the gauge fields in Eddington-Finkelstein coordinates to get regularity condition at the horizon: where E ai is constant probe electric fields with a = 1, 2 and i = x, y. The full gaugefield perturbation will have the regular expansion δA ai ∼ E ai v + · · · in the Eddington-Finkelstein coordinates, provided we demand To fix the perturbations, we need the linearized Einstein equations. The (r, x) and (r, y)-components of the Einstein equation are given by For simplification, it is fruitful to define following functions: where every functions is computed at r = r H . For example, W = W (r H ) = r 2−θ H . Regularity at the event horizon yields The radially 'conserved' electric current density J µ = (J t , J x , J y ) can be defined by We can get explicit form of electrical currents from (3.14) as follows: For the transport coefficients, we will take J 2i and E 2i only, because A 1i is introduced just for constructing HSV geometry. We use a two-form field [52] associated with the Killing vector field, k = ∂ t , to define the conserved heat current Q i as follows: (3.17) Then Q i is indeed heat current in that each term corresponds to T ti and µ J i respectively at the boundary. Since Q i is radially conserved, it is enough to compute Q i at the horizon.
Finally we can read off the transport coefficients from the following matrix: where the temperature gradient −(∇ i T )/T = ζ i and each transport coefficients is given by The resitivity is defined as the inversion of the conductivity matrix: where The thermal conductivity when electric currents are turned off is given by From these quantities, the Seebeck coefficient, S, and the Nernst signal, N , are given by The Hall angle is given by .
Before exploring the various aspect of transports, it will be convenient to divide regions of (θ, z) as Figure 2. One can see from eq 2.10 that there's singularities at z = 2 and θ = 2 . Therefore, it must be phase transition near those regions.  (b) Region where Null energy condition is satisfied. Black dots indicate specific (z, θ) where we will discuss the typical behavior of transports from next section

Magneto-transport
In this section, we discuss the typical behaviors of magneto-transport by taking a point from each sector A,B,C,D of the Figure 2(a). We take two more points along the θ = 0 line. Notice that the transverse quantities σ xy , κ xy , and S vanish when q χ = µ = 0. We used α = 0 and β = λ = 1.3.

4.1
Classifying the magneto-conductivity in (z, θ) plane Figure 3 shows the temperature evolution of magneto-conductivity. There are several  . Temperature evolution for σ xx (H) for different (z, θ). Each curves corresponds to T = 0.04, 0.1, 0.24 for blue, green, and red respectively. We used q χ = 0.7 for all non-zero q χ case.
interesting features for longitudinal electric conductivity, which we will understand by analytic calculations later : • For the regions A and B, σ 0 xx , the longitudinal conductivity at H = 0, increases as a function of temperature while it decreases in phase C and D. Such opposite behaviors are characters for other transport coefficients also. See figure 3.
• For the region B, C, the longitudinal electric conductivity has finite values at H → ∞.
It happens only when magnetic impurity q χ = 0).
• At the NEC boundary in the region A, weak localization appear at low temperature, while it appears at high temperature at the NEC boundary in D.
Although the longitudinal electric conductivity is very complicated, we can understand a few important aspects in the limit of small and large H. For H = q 2 = 0, The first term in (4.2) is dominant for high temperature if θ < 2z, while the second term is dominant if θ > 2z. In the region A and B satisfying NEC (θ < 2z − 2), θ is always smaller than 2z so that T ∼ r z H , then which is increasing function of temperature. This explains the temperature dependence of σ 0 xx in Figure 3 (c) -(h). The eq. (4.2) suggest that the region C and D might be divided into two parts. When θ < 2z, the first term in (4.2) is dominant and hence σ 0 xx follows the form of (4.3) but the exponent is negative by NEC. For θ > 2z, we have whose exponent is again negative when θ > 2z. Therefore, in the region C, D, σ 0 xx decreases with temperature regardless of sign of θ − 2z. This explains the temperature dependence of σ 0 xx in Figure 3  So far, we discussed the behavior of the longitudinal conductivity in the absence of the external magnetic field. When the magnetic field is finite, the relation between temperature and r H is rather complicated as one can see in (2.18) and the analytic expression of r H in terms of other parameters is not available. But in the small H limit, H dependence of r H for fixed parameters (T, β, λ, θ, z) can be obtained using where r 0 is r H (H = q = 0). Now, we have small H expansion of σ xx as (4.6) Depending on the sign of the coefficient of H 2 , the system has weak anti-localization (WAL) for − sign or weak localization (WL) for + sigin. When(z, θ) parameters are off the NEC boundary, the coefficient of H 2 in (4.6) is complicated function of z, θ and other parameters, but there still exist competition between two terms which lead to the transition from to WL. Figure 4 is the phase diagram where yellow region denotes WL (positive sign) and gray region corresponds to WAL (negative sign). Dotted line is the boundary of NEC. As temperature increases, yellow regions in A, B shrink and finally disappear, which means that the longitudinal conductivity shows weak anti-localization(WAL) behavior at high temperature. On the other hand, the yellow regions in C, D expand and fill the whole region of C, D at high temperature. For special case where (z, θ) at the boundary of NEC (θ = 2z − 2), the (4.6) has simpler form: In the absence of q χ (4.7) shows weak anti-localization behavior, while it changes to weak localization behavior when q χ λ 2 is large. For the given value of q χ λ 2 , temperature behavior depends on z and θ. At the NEC boundary, the first term in (4.2) is dominant and hence with a positive constant c * , which means that weak anti-localization appears at low temperature at the NEC boundary in A region z < 2 but appears at high temperature at the NEC boundary in D (z > 2). See Figure 3 (b) and (l). Another interesting phenomena in the region B and C is that σ xx goes to constant for large magnetic field H as one can see in Figure 3 (d), (h) and (j). To understand this, we first notice that large H behavior of the conductivity is entirely determined by that of r H , because eq. (3.20) says that both numerator and denominator have explicit H 4 behavior so that in this limit the conductivity is determined only by the H dependence of r H . From the expression of eq.(2.18), in the relevant regions. Namely, if r ∞ is well defined, the noted behavior is explained.
Starting with the H expansion of eq.(2.18), In order for the left hand side to be finite, the coefficients of H 2 should vanish. Therefore which is precisely the equation defining the region B and C.
Now we consider transverse magneto-conductivity. Figure 5 shows external magnetic field dependence of σ xy (H). Because we are considering only zero chemical potential case, the transverse conductivity vanishes in the absence of q χ , which means that the transverse conductivity is generated by the q χ interaction term. This can be understood by the fact that q χ term generates finite magnetization as well as the effiectve charge carrier by (2.16) and therefore transverse movement of charge carriers is also generated.
We can also expand σ xy in small magnetic field limit as previous discussion, where A 1 is defined in (4.5). σ 0 xy is called 'anomalous Hall conductivity' which is the transverse conductivity in the absence of the external magnetic field. It is given by (4.14) Using the similar analysis as before, we found that that σ 0 xy is a decreasing function of temperature in the region A, B but increasing function in the region C and D.
The second term in (4.13) determines curvature of the transverse conductivity near H = 0 which is not easy to analyze. We checked numerically that the sign of the second term is always negative in all region. The result is given by the Figure 6.

Classifying the magneto-thermal conductivity in (z, θ) plane
In this section, we will discuss magneto-thermal conductivities. Figure 7 shows the external magnetic field dependence of the thermal conductivity given in (3.29). In the small H limit, the longitudinal thermal conductivity can be expanded as  where A 1 is defined in (4.5) and The temperature dependence of κ 0 xx can be understood using the same analysis we used for the electric conductivity. In the region A and B, horizon radius ∼ T 1/z and z > θ. Therefore, κ 0 xx increases at high temperature. One can easily check that the exponent becomes negative in the region C and D and hence κ 0 xx is suppressed at high temperature. The coefficient of H 2 in (4.15) determines a shape of the thermal conductivity near H = 0. However, the coefficient is too complicated function for analytic approaches. The numerical result for the sign of the coefficient of H 2 is drawn in Figure 8, where yellow region denotes the positive sign corresponding to the weak localization. Gray region corresponds to the negative sign and weak anti-localization. In region B and C, the coefficient is always negative but in region A and D, there are domains of positive sign. The sign of the coefficient is positive near z = 2 and turns to negative at other region. See Figure 8.  The transverse thermal conductivity can be expanded for small H as (4.17) When q χ = 0, the sign of coefficient is only depends on the sign of −(2 + z − 2θ) because all other terms are positive definite. This sign is negative in region A, B and positive in C, D. The full magnetic field dependence of the transverse thermal conductivity is drawn in Figure 10. When we turn on q χ , the second term can change the value of (z, θ) where the sign of the coefficient flipped. Numerical calculation indicates that the sign of H 2 changes near (z, θ) = (2, 2) in the region A, B. This region increase as q χ increases, see Figure 11 (a). Figure 12 and Figure 13 show the external magnetic field dependence of the Seebeck coefficient and Nernst signal which are related to the thermoelectric conductivity α ij . These quantities have different behavior from the other transport coefficients. They are odd function of the magnetic field. The small field expansion of the Seebeck coefficient and    Nernst signal become 18) which are enough to explain the linearity of N and S near H = 0 in the figures 12 and 13. In the absence of q χ , the slop of Nernst signal is alway positive, but it can be negative depending on the value of q χ . The coefficient of H is same as the coefficient of H 2 of the transverse thermal conductivity (4.17) with opposite sign. Therefore, the region where the sign changes should be the same as κ xy . See Figure 11 (b). The sign of the slope of Seebeck coefficient only depends on (2 + z − 2θ) in the denominator. Together with NEC, the sign is positive in the region A, B and negative in C, D.  Temperature evolution for N (H) for for each region of (z, θ). Curves corresponds to T = 0.04, 0.1, 0.16, 0.24 for blue, green, yellow, and red respectively. We used q χ = 0.7 for all non-zero q χ case.
In the figure 12, we draw the temperature evolution for N (H) for different (z, θ) which are denoted as big dots in the figure 2(b). In the figure 13, we draw the temperature evolution for S(H) for different (z, θ) which are denoted as big dots in the figure 2(b).

Density dependence of Transports
In this section, we discuss the q 2 , density, dependence of transport coefficients. For the ease of the analysis we consider only only zero external magnetic field cases. Notice that ρ yx , κ xy , and N vanish when q χ = 0 and H = 0. In the region A and B, the longitudinal electric conductivity has a scaling behavior for large density, namely, γ = ∂(log σ xx )/∂(log q 2 ) is a constant at q 2 → ∞. To see this, we consider the temperature at H = 0: In region A and B satisfying NEC, θ < 2z so that we can neglect the third term for large q 2 . For the region A and B where θ < 2, we have r H ∼ q 1 z−θ+1 2 . Rewriting the longitudinal conductivity from eq (3.20) with this horizon behavior for large density, Notice that the first term in (5.2) is subleading so the behavior is governed by the explicit q 2 dependent term. Fig. 14 demonstrate the scaling behavior of σ xx (q 2 ). Notice that when z = θ, r 0 has no dependence on q 2 which means γ is fixed as 2. Now we consider the density dependence of the longitudinal resistivity. Figure 15 shows the density dependence of ρ xx (q 2 ). When q 2 = 0, the longitudinal resistivity is given  Figure 14. Scaling behavior of σ xx (q 2 ) at large q 2 and T = 10. (a) z-evolution of σ xx (q 2 ) at θ = 0. (b) log-log plot of (a), (c) θ-evolution of σ xx (q 2 ) at z = 1. (d) log-log plot of (c), (e) The exponent γ of σ xx (q 2 )| q2→∞ ∼ q γ 2 for various (z, θ). Whenever z = θ, the value of γ is discontinuous, which is indicated by empty circles in the figure. At all the empty circle points, γ = 2 which is discontinuous value from its neighboring point. Here, we analyzed this only in Phase A and B. (f) When z = 2, γ = 2 independent of θ, which is indicated by yellow line in (e) by where r 0 is r H (q 2 = 0). Different from electric conductivity, the resistivity does not have the scaling behavior in T , due to the q 2 χ term in denomenators. For the finite density case, the relation between r H and T is still complicated so we will take similar way to previous section. In the small q 2 limits, q 2 dependece of r H for fixed parameters (T, β, λ, θ, z) can be obtained from Now, we have small q 2 expansion of ρ xx as where D = r 4z 0 + q 2 χ λ 4 r 4θ 0 . The coefficient of q 2 2 in (5.5) determines a shape of the resistivity    Figure 16. Sign of the coefficient of q 2 2 of the longitudinal resistivity near q 2 = 0. Yellow region denotes positive sign and gray region is for negative sign. Dotted line is NEC. Here we use the same parameters as Figure 15 and T = 0.04. near q 2 = 0. Roughly speaking, q χ can flip the sign of curvature flipped in (z, θ) dependent way, which is difficult to analyze. We calculated the the sign of the second term numerically and the result is in Figure 16. When q χ = 0, the sign of coefficients of q 2 2 is always negative. In region A,B, q χ interaction suppress the resitivity near q 2 = 0, but the effect of q χ diminishes as θ increases. In region C,D, the effect of q χ terms become negligible due to the large θ.  Next, we consider the density dependence of the Hall conductivity. Figure 17 shows the density dependence of ρ yx (q 2 ). When q 2 = 0, the longitudinal resistivity is given by where r 0 is r H (q 2 = 0). Also, we can expand ρ yx in small density limit as previous discussion, where D = (r 4z 0 + q 2 χ λ 4 r 4θ 0 ). Roughly speaking, q χ can flip the sign of curvature (z, θ) dependent way, which is difficult to analyze analytically. We investigated it numerically and the result is in Figure 18. In region A,B, q χ -term suppress the resistivity near q 2 = 0, but the effect of q χ is reduced as θ increases. In region C,D, the effect of q χ terms becomes negligible due to the large θ. Now let's turn to the density dependence of the thermal conductivity. We can perform the same analysis as the resistivity:  Figure 18. Sign of the coefficient of q 2 2 of the transverse resistivity near q 2 = 0. Yellow region denotes positive sign and gray region is for negative sign. Dotted line is NEC. Here we use the same parameters as Figure 15 and T = 0.04.  Figure 19. Temperature evolution for κ xx (q 2 ) for different (z, θ). Each curves corresponds to T = 0.04, 0.1, 0.160.24 for blue, green, yellow, and red respectively. In all figure, q χ = 5. density limit is expressed as  Figure 20. Sign of the coefficient of q 2 2 of the κ xx near q 2 = 0. Yellow region denotes positive sign and gray region is for negative sign. Dotted line is NEC. Here we use the same parameters as Figure 19 and T = 0.04.
Notice that κ xy (q 2 = 0) = 0. The coefficient of q 2 2 in (5.9) is positive definite for every (z, θ). See Figure 21.  . Temperature evolution for κ xy (q 2 ) for different (z, θ). Each curves corresponds to T = 0.04, 0.1, 0.16, 0.24 for blue, green, yellow, and red respectively. We used q χ = 5. Figure 22 and Figure 23 show the density dependence of Nernst signal and Seebeck coefficient respectively, which are related to the thermoelectric conductivity α. We can expand S(q 2 ) and N (q 2 ) in the small density limit: S ∼ 4πr 4z 0 β 2 D q 2 + · · · , (5.10) where D = r 4z 0 +q 2 χ λ 4 r 4θ 0 . The stepping feature near the zero q 2 in the figures 23 for nonzero q χ case is due to the suppression of the linear term by the presence of D that contains q 2 χ which is chosen to be 5, which is rather large.  (l) PD Figure 23. Temperature evolution for S(q 2 ) for different (z, θ). Each curves corresponds to T = 0.04, 0.1, 0.160.24 for blue, green, yellow, and red respectively. q χ = 5 if q χ = 0.

Conclusion and discussions
In this paper we reported a new black hole solution with hyperscaling violation which is relevant to an impurity doped quantum materials. We calculated all transport coefficients including electrical, thermo-electric and heat conductivities. We investigated their properties in detail by plotting the analytic results as a function of physical parameters. We also investigated the the phase transitions from weak localization to weak anti-localization as the critical exponent changes. One interesting question is whether we can find a strange metal properties where resistivity ∼ T and Hall angle Θ ∼ 1/T 2 for some exponent. It turns out that the relevant point is (z, θ) = (1, 1), which turns out to stay out of the stability regime requested by the null energy condition. See appendix. However, it is too early to abandon such regime, since we did not in-cooperate the presence of the neutralizing ionic charges into gravitational context.
Finally the most interesting project would be the matching the gravity solutions with real systems, which would request collecting all available data. We wish to come back to this issue soon.
-25 - On the other hand, after evaluating the Ricci tensor for our ansatz, we obtain