The magnetic effect on strongly correlated system with two currents near the QCP from holography

We study the magnetic effect in strongly interacting system with two conserved currents near the quantum critical point (QCP). For this purpose, we introduce the hyper-scaling violation geometry with the blackhole. Considering the perturbation near the background geometry, we compute the transport coefficients using holographic methods. We calculated the magneto-transport for general QCP and discuss the special point $(z,\theta)=(3/2,1)$ where the data of Dirac material were well-described previously.


Introduction
For the strongly correlated system, the particle nature often is absent so that theories based on quasi-particles such as Landau-Fermi liquid theory fail. The strong correlation can happen even for weakly interacting system when the Fermi surface can be tuned to be very small, because then the electron-hole pairs which screen the Coulomb interaction are not sufficiently created due to the smallness of the Fermi surface. Therefore, any Dirac fluid can be strongly correlated as far as it has the small Fermi surface, which is already shown in the clean graphene [1,2] and in the surface of the topological insulator with magnetic doping [3][4][5]. To describe such system, we need to find a new way. We consider the quamtum critical point (QCP), where the microscopic details in UV are irrelevant and most of the informations in UV are lost. This loss of information is very similar to black hole system and this similarity between a QCP and a blackhole is the important motivation to use the holography to analyze the strongly correlated electron system. A QCP can be characterized by (z, θ) which is defined by the dispersion relation ω ∼ k z and the entropy density s ∼ T (d−θ)/z . We can use a geometry with the same scaling symmetry respected (t, r, x) → (λ z t, λ −1 r, λx): which is called hyper-scaling violation (HSV) geometry. In our previous works [6][7][8][9], we described the clean graphene and the topological insulator with magnetic doping in some parameter regions by using holographic method. For the surface of the topological insulator, we introduced just one current with a interaction to encode the magnetic doping [8,9]. We calculated magneto-conductivity and investigated the phase transitions from weak localization to weak anti-localization. As in the case of the graphene, it turns out that we can have the better fit for (z, θ) = (3/2, 1) than (1, 0).
For the graphene, we need two currents model [6,7]: when the electron and hole densities fluctuate from their equilibrium states, the system is supposed to reduce the difference by creating or absorbing electron-pair: In this process, the energy and momentum should be conserved. For the graphene, however, the kinematically available states are severely reduced [10] due to the geometry of the Dirac cone and this constraints makes the two currents J e and J h independently conserved. Hence, we need the two independent currents to describe the graphene. In [7], we analyzed the two currents model for hyperscaling violation geometry (HSV), and we found that the theory with a QCP at (z, θ) = (3/2, 1) than (1, 0) studied in [8]. The value θ = 1 is important because the holographic background with dual Fermi surface has the effective dimension d ef f = d − θ and the system with fermi surface should have d ef f = 1 so that theta = 1 can describe the character of fermion in this aspect. Indeed, we found that (z, θ) = (3/2, 1) can fit the data better therefore qualified as more proper critical exponent both in graphene and topological insulator.
In this paper, we study the holographic model with the two currents and a particular interactiion which is shown to describe magnetically doped material [8,9]. We calculated all transport coefficients and demonstrated some typical behavior of the magneto-transports. Although we don't have any experimental results for magnetically doped graphene, our work can be considered as predictions for the magnetic effect for graphene or other material which need two or more currents due to the presence of two layers or two valleys.
2 The two currents model with magnetic impurity in hyperscaling violating geometry We start from a 4-dimensional action with asymptotically hyperscaling geometry g µν , which includes a dilaton field φ, a gauge fields A µ to complete the asymptotic hyperscaling violating geometry, two extra gauge fields B (a) µ which are dual to two conserved currents, and the axion fields χ 1 , χ 2 to break the translational symmetry.
We use the ansatz where β denotes the strength of momentum relaxation. The equations of motion for gauge fields and gravity are given by The solution for the dilaton field is given by By solving the equations of motion, we can get the gauge couplings and dilaton coupling Z A , Z a , and Y as followings: Other exponents and potentials are given by where H is a constant magnetic field. Finally, we have the following background solutions: where a = 1, 2 and c 2 , c 3 , c 4 are given by We can define the conserved charge from the equations of motion for the gauge fields B a as the constants of integration The entropy density and the Hawking temperature are given by where Θ a = q χa β 2 r θ−2 H andZ = a=1,2Z a .

Conserved currents and DC transports
We consider following perturbations to compute the transport coefficients based on the idea of linear response theory [11]: δg ti = h ti (r) + tf 3i (r), δg ri = h ri (r), δB ai =b ai − tf ai , δχ i = ϕ i (r). (3.1) We take the functions f i (r) as to make the linearised Einstein equations time-independent. Here, E ai are the external electric fields acting on J ai respectively and ζ i is thermal gradient which is defined as ζ i = −(∇ i T /T ). In the final expression for the each conserved currents, we will set E 1i = E 2i = E i . Since all the transports can be computed at the event horizon, we need to find the regularity condition at the horizon. We take the Eddington-Finkelstein coordinates (v, r) where the background metric is regular at the horizon, where v = t + dr V /U . In this coordinates, the metric perturabation is given by To guarantee the regularity of the metric with perturbation at the horizon, we demand the last term to vanish at the horizon so that The gauge fields can be reexpressed in the Eddington-Finkelstein coordinates to get the regularity condition at the event horizon: Then, the full gauge field have the regular form of δB ai ∼ E ai v + · · · in the Eddington-Finkelstein coordinates by demanding We can define the radially conserved currents which is defined by where the index a = 1, 2 denotes the two currents which are dual to the two gauge fields B a . Since J a and Q i are the conserved quantities along the radial direction so that they can be evaluated at arbitary value. Hence, it is enough to compute at the horizon [11]. Finally, we can express the boundary current in terms of the external sources and transports coefficients: Before we express the transport coefficients explicitly, it will be useful to define following functions for the simple expression: where Θ a = q χa β 2 /W . One can define the total electric current as J i = a J ai and identify the external electric field as E ai = E i . Then, each transport coefficient based on this total current and electric field is given by where Z = Z 1 + Z 2 and Θ = Θ 1 + Θ 2 . Notice thatᾱ ij = α ij . The resisitivity is defined as the inverse of the conductivity matrix: The thermal conductivity κ is defined by the response of the temperature gradient T ζ i to the heat current Q i in the absense of the electric currents J ai . Setting J ai = 0 in 3.9, we can write E bj in terms of ζ j to substituting to the expression of the heat current in 3.9. Then, we can get where δ = σ 12 = σ 21 . Notice that this expression is very similar to that in [6], but it is 2 × 2 matrices multiplication which is different from the simple scalar multiplication in [6]. The Seebeck coefficient S and the Nernst signal N are given by In figures 1, 2, we show the typical behaviors of each magnetotransport when z = 1.5, θ = 1 and µ 1 = µ 2 = 0 where we are interested. Notice that κ xy = 0 when there's no conserved electric charge. As one can see from Figure 1 Figure 1. Magnetotransport for z = 3/2, θ = 1 without the magnetic impurities. We choose the parameters asZ 1 =Z 2 = 1, q χ1 = 0, q χ2 = 0, and β = 1.5.
difference between a single current model and two currents model in the qualitative sense compared to the results in [9]. But, as in [6,7], the two currents model can give the physical implication if there is experimental data to compare with this model. Unfortunately, we don't have any relevant experiments to be conducted so that we leave our results as a qualitative prediction for experiment for the graphene with magnetic doping.  Figure 2. Magnetotransport for z = 3/2, θ = 1 with the magnetic impurities. We choose the parameters asZ 1 =Z 2 = 1, q χ1 = 1, q χ2 = 2, and β = 1.5.

Conclusion
In this paper, we investigated the two currents model with magnetic doping in the presence of the magnetic field, which is based on [6][7][8][9]. From this model, we calculated all transport coefficients. Although we do not expect qualitative difference between a single current and two currents model, the presence of two current are definitely necessary to describe the quantitative data fitting for the material which involve two independent electron system that are very weakly coupled, like graphene [6,7] or other multi-valley systems or multilayer systems which will be studied in a future.