Charged Lifshitz black hole and probed Lorentz-violation fermions from holography

We analytically obtain a new charged Lifshtitz solution by adding a non-relativistic Maxwell field in Horava-Lifshitz gravity. The black hole exhibits an anisotropic scaling between space and time (Lifshitz scaling) in the UV limit, while in the IR limit, the Lorentz invariance is recovered. We introduce the probed Lorentz-violation fermions into the background and holographically investigate the spectral properties of the dual fermionic operator. The Lorentz-violation of fermions will enhance the peak and corresponds larger fermi momentum, which compensates the non-relativistic bulk effect of the dynamical exponent. For a fixed $z$, when the Lorentz-violation of fermions increases to a critical value, the behavior of the low energy excitation goes from a non-Fermi liquid type to a fermi liquid type, which implies a kind of phase transition.


I. INTRODUCTION
The Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence [1][2][3][4] opens a new avenue to disclose many different strongly interacting systems. The correspondence makes a connection between gravitational theories on AdS spacetime and a quantum field theory(QFT) that lives on the conformal boundary of the AdS spacetime. Due to its remarkable feature connecting a strong coupling QFT with a weak coupling gravitational theory, this useful tool has been attracting considerable interest in studying strongly coupled physics, especially, the possible applications to condensed matter physics(see [5][6][7][8] for reviews).
The AdS/CFT duality is first generated in [9] to study a boundary theories with dynamical critical exponents which show the dynamical scaling t → λ z t, x → λx with z = 1 (1) instead of the scale invariance with z = 1. It is pointed out that the background, which is holographically applied to study the nonrelativistic QFT with dynamical (Lifshitz) scaling (1), is given by the Lifshitz spacetime where the spacial index i runs from 1 to D − 2 and here we set the radius of curvature to be unit.
In [9], though the dual QFT is non-relativistic with Lifshitz scaling, the bulk gravity is relativistic which are invariant under the full spacetime diffeomorphism and it has to couple with matters to generalize the Lifshitz geometry. This situation has been improved by the Hořava-Lifshitz(HL) gravity [10,11] which is based on the perspective that Lorentz symmetry should appear as an emergent symmetry at long distances, but can be fundamentally absent at short distances [12,13]. HL theory is a non-relativistic power-counting renormalizable theory of gravitation. Since the HL gravity is itself anisotropic between space and time, it is natural to expect that the HL gravity provides a minimal holographic dual for non-relativistic Lifshitz-type QFT. This proposal has been realized and carefully studied in [14][15][16]. Especially, it was addressed in [14] that the Lifshitz spacetime is a vacuum solution to the HL gravity. Analytical vacuum solution with asymptotically Lifshitz geometry was also found in [17,18]. The holographic applications in the framework of the HL gravity were constructed (for instance, holographic superconductor models in the HL gravity were studied in [19][20][21] ).
In this paper, as a first step, we couple the gravitational action of HL gravity with a U(1) gauge field whose Lagrangian is consistent with the symmetry of HL gravity. We analytically solve the equations of motions for the gravity and the gauge field, and obtain a new charged Lifshitz black hole solution to the coupled theory. As we will show later, in the UV limit, our charged solution behaves as the geometry (2) while at the IR limit, the Lorentz invariance (with z = 1) is restored and the geometry is Then we introduce the Lorentz-violation fermions in the charged HL background and study the properties of fermionic spectral. The motivations we introduce the Lorentzviolation Dirac field stem from the following two aspects. On one hand, the authors of [22] proposed that the general low energy action of the HL type can be derived as the spectral action for Dirac operator with P-violating terms. They also addressed that the obtained general Lorentz violating fermionic action related with the Dirac operator is consistent with the fermionic action of Standard Model Extension (SME) studied in [23], and the Lorentz violating effects is very small in weak gravitational field. Besides, in HL theory, the Lorentz symmetry is kind of approximate symmetry at low energy, which motivates us to consider Dirac field with weak Lorentz violation in this background. On the other hand, holography has been widely applied to explore the strongly correlated fermionic system. In the pioneer works [24][25][26][27][28] on this topic, the authors proposed that by studying the bulk Dirac equation in the RN-AdS black hole with the ingoing behavior at the horizon, the fermionic correlation can be extracted at the AdS boundary. The proposal helps us further understand the mysterious behaviors of the existing (non-)Fermi liquid. Besides in relativistic UV theory, the fermionic response was also studied in Lifshitz theory in [29,30]. These works has inspired more and more efforts to be involved in the related topics 1 . However, the previous works usually started by adding relativistic fermions into the (non-)relativistic bulk theory. Since 1 The generalization of holographic fermions obtain many remarkable progress, such as the effects of different bulk theory on the fermionic correlation [31][32][33][34][35][36][37][38][39][40], holographic non-relativistic fermionic fixed points by imposing the Lorentz violating boundary condition [41][42][43][44][45], phase transition from the Fermi liquid to non-Fermi liquid and also to the Mott insulating phase due to the interaction between Dirac field and gauge field [46][47][48][49][50][51] and the fermionic spectral function effected by lattice effects [52][53][54][55][56], etc.. the anisotropic scaling (1) is built by construction in the HL gravity, it is natural to expect the coupling matters exhibit the same scaling in the same footing. This is the other motivation to introduce the probed non-relativistic fermions into the non-relativistic HL gravity and study the effect of the Lorentz-violation on the Fermionic spectral function.
Our results show that, different from the effect of dynamical exponent which suppresses the peak of the fermionic spectral function [36,45], the Lorentz-violation of fermions will enhance the peak and make the fermi momentum larger to compensate the non-relativistic bulk effect. Also, for fixed z, small Lorentz-violation of fermions corresponds the low energy excitation as non-Fermi liquid with nonlinear dispersion relation, when the Lorentz-violation becomes larger than a critical value, the low energy excitation always behaves as Fermi liquid with linear dispersion relation. This may imply that in the non-relativistic HL gravity, it is more natural to introduce Lorentz-violation of fermions to dually describe a fermionic system whose low energy excitation is Fermi liquid.
The remaining of this paper is organized as follow. In section II, we obtain a charged black hole solution by solving the coupled system of non-relativistic gauge field and HL gravitational theory. We show the holographic setup by introducing the Lorentz-violation fermions into the coupled background sector in section III. Then in section IV, we exhibit the numerical results of the Fermi momentum and dispersion relation by analyzing the spectral function. Section V contributes to our conclusion and discussion.

II. ANALYTICAL CHARGED LIFSHITZ SOLUTION FROM HORAVA-LIFSHITZ GRAVITY
It was studied in Ref. [14] that the HL gravity can provide solution dual to Lifshitz-type field theories, where the following gravitational action is proposed It will return to Einstein gravity as λ = β = 1 and α = 0. In Ref. [14], the authors found that for the requirement of stability and perturbative unitarity around flat spacetime, constraints should be imposed on the couplings, i.e., β > 0, α ≤ 2βd d−1 and λ ≥ 1 or λ 2 ≤ 1 d−1 for Λ = 0. When Λ < 0, if we take the Lifshitz metric (2) is also a vacuum solution of the equation (3). In our paper, we will use the condition with Λ < 0. Besides, the authors worked with the Arnowitt-Deser-Misner metric ansatz, Here, N, N i and g ij are the lapse function, the shift vector and the metric of the spacelike hypersurface, respectively. Thus, in the action, one has the formula √ −G = √ gN, where R is the scalar curvature of the metric g ab . Later, the authors of [17] analytically studied the neutral Lifshitz black hole solutions to the gravitational thoery.
Here we intend to explore the possible charged black hole solutions in HL gravity. We consider Lorentz-violating electromagnetism field coupling to the HL gravity with the action where Then the total action of the background we will take account into is We consider the electromagnetic field with the only non-vanishing component A t (r) and also β µ = 0(µ = 0, 1, 2). Subsequently, the Maxwell equation reads as which gives us the solution with the integral constant Q e , which can be interpreted as the charge of the Lifshitz black hole.
In order to get the complete solution to the equations of motion deduced from the action (7), we will borrow the analytical process shown in Ref. [17]. To proceed, we set the metric components in (5) as By substituting the formulas in (10) into the action (7), we obtain where υ = dtd d x and Considering the above S t as one dimensional action of the functions f (r), l(r) and h(r) and then variating the action, we can get two independent equations which can be further simplified as It seems not so direct to find the solutions of the three functions f (r), l(r) and h(r), but we can assume the following forms so that the metric becomes The issue now reduces to find the solution of ξ(r). Inserting the formulas (16) into equation (15), gives us the equation of motion for ξ(r) It is obvious that if we take the constraints (4), then we will get the solution of ξ(r) where M and Q e are integral constants which can be understood as the mass and charge of the Lifshitz black hole. At the horizon r 0 , ξ(r 0 ) should be zero, which implies that . Substituting the solved metric into (8), we can fix the Maxwell field The Hawking temperature of the charged black brane is We will move on to give some comments on our black hole solution.
Firstly, the solution (17), (19) and (20) is a new charged Lifshitz solution which is different from the solutions proposed in [58][59][60][61][62][63][64]. However, it can recover AdS RN black hole solution when z = 1. It is worthy to point out that from the expression of δ in equation (12), α = 0 will force us to keep z = 1 which is the AdS RN solution. Thus we can only expect the above Lifshitz solution when α = 0, which also happens in neutral case studied in [17].
Secondly, from the expression of the temperature (21), it is straightforward that the extremal case is fulfilled when Then it is explicit that in order to have real chemical potential, the condition β > 0 should be satisfied. Furthermore, recalling the expression of δ in equation (12), we should have the range of the Lifshitz exponent as z > 1 with α = 0.
Thirdly, we will check the asymptotical behavior of our solution. In asymptotical boundary r → ∞, since ξ(r → ∞) → 1, so that the metric (17) reduces to the asymptotical Lifshitz metric (2). While, near the horizon, the geometry is a bit subtle. For simplicity, we will set the horizon r 0 = 1 in the following discussion, then the redshift factor at zero temperature can be rewritten as whose behavior near the horizon is ξ . Therefore, at the zero temperature, the near horizon geometry is AdS 2 × R d with the AdS radius L 2 . This is more explicit by considering the transformation Consequently, the metric (17) can be deduced into the AdS 2 × R d form and the gauge field is Since the near horizon geometry is AdS 2 × R d and the solution is Lifshitz in UV limit, we can use the Lifshitz holography [9] to study its application in strongly coupled physics.
In the next section, as an attempt, we will study the fermionic spectral of the operator dual to the probed fermions in this background .

III. HOLOGRAPHIC SETUP OF LORENTZ VIOLATION FERMIONS
We will introduce the fermions into the above theory as a probe and apply the holography to study the fermionic correlation. We consider the Lorentz-violation action for the probed fermionic sector [23] The first term in Γ a and the first two terms in M respectively lead to the usual Lorentzinvariant kinetic term and mass term for the Dirac field. In order to take Lorentz-violation Γ a and M, we will choose the setting of parameters as c µν = ρh µν where h µν = g µν + n µ n ν Redefining the field as ψ = (−gg rr ) and setting φ = (φ 1 , φ 2 ) T with φ I the two-component spinors, then choosing k i = kδ i x because of the rotation symmetry in the spatial directions, we reduce the Dirac equation (29) into the form with a = (1 − (d + 1)ρ) after we choose the gamma matrices as [65] Further setting φ I = (X I , Y I ) T , we divide the above equation (31) Not that we have shifted η 2a → η in the equations. Later, we will use η to denote the strength of the Lorentz-violation of the fermions, and η = 0 means that the fermions are Lorentz-invariance. With the above equations in hands, we further introduce the new ratios Hereafter, the Dirac equation can be reduced into (r 2 ξ(r)∂ r + 2mr)χ I = 1 The boundary condition of χ I near the horizon r → 1 is

UV behavior
In UV limit, the metric (17) becomes with z > 1 and the equation (31) becomes of which the leading order 2 is and its solution near the boundary is As it is addressed in [28], choosing the normal boundary term we can suppose that the source and the response in (40) are related by and the boundary Green's function G(ω, k) is Recalling the definition of χ I = X I Y I , we can rewrite the Green function as

IR behavior
In near-horizon geometry AdS 2 × R d , the Dirac equation (31) in the zero frequency limit or low energy limit, can be rewritten as 2 When z → 1, the leading order is too stiff to say (39) is the leading order of equation (38). In fact, later in our numerical calculation, we always choose z = 1 + ǫ where ǫ is not a small number, it cannot be ignored. So when r → ∞, it will be general to choose the leading order like equation (39).
where Γ ς = −Γ r and others are the same as (32) when we reflect the change between the radial coordinate r to the coordinate ς. Moreover, near the AdS 2 boundary with ς → 0, the leading term of the above equation is which gives us the leading behaviour of φ I as where υ ± and ±ν(k) are real eigenvectors and eigenvalues of the matrix in the right side of equation (46) with the form Subsequently, according to the discussion in [26], in the IR CFT the conformal dimension of the operator O k will be δ k = 1/2 + ν(k), which is dependent of the boundary dimension and Lifshitz dynamical critical exponent because they have information in the IR geometry.

IV. RESULTS OF FERMI SURFACE AND DISPERSION RELATION
In this section, we will numerically solve the Dirac equation (35) to investigate the momentum of Fermi surface and the dispersion relation of the excitation near the Fermi surface.
In the previous works [36,45], the authors claimed that as the Lifshitz exponent z increases, the peak of the Fermi-like peak suppresses and for large enough z, the peak is too small to be observed. In our model, the same phenomena is observed. Thus, we will fix a small z = 1.02 3 and mainly focus on the effects of the Lorentz violation parameter η on the holographic fermionic systems. Meanwhile, we will fix the mass of fermions m = 0 and charge q = 1 in the 3 + 1(d = 2) dimensional bulk geometry 4 .
Then from equation (35), the Green function satisfies the following symmetry so that without loss of generality, we can only study the component G 2 with positive k for the Fermi surface and dispersion relation. Note that we will study at zero temperature with µ having the form in equation (22).  In figure 1, we show how the Green function G 2 behaves as the momentum and frequency for different η. The left plot corresponds to z = 1.02 and η = 0, which shows a sharp Fermilike peak at zero frequency implying a Fermi surface. And, this peak becomes sharper and its location of momentum is larger when we increase the Lorentz violation parameter η, as shown in the middle and right plots. This phenomena is more obvious in figure 2 where we fix a tiny frequency ω = 10 −5 and plot the behavior of G 2 by changing the momentum. It means that η compensates the effect of the Lifshitz exponent which suppresses the height of peak. From this perspective, this observation may explain why it is more reasonable to introduce Lorentz violation fermions to our black hole solution to holographically study the spectral in the boundary theory since we always expect sharp peak at ω = 0 to define the After we determine the fermi momentum, we can directly calculate the dispersion relation which is defined as [26] ω(k) ∝k θ , with θ = wherek = k − k F andω denote the shift away form the fermi surface with (k = k F , ω → 0), and ν(k) is defined in equation (48). Substituting the determined Fermi momentum into (50), we compute the exponent θ of the dispersion relation and the results are shown in figure   4. When η is smaller than 0.033, the exponent θ decreases with the increasing of η and it is always higher than unit, which implies that the low energy excitation is non-Fermi Liquid.
On the other side, when η is larger than 0.033, the low energy excitation may behave as Fermi Liquid because θ keeps at unit. Our results imply that when the parameter of Lorentz violation of fermions reach strong enough, there may be a phase transition from non-Fermi liquid to Fermi liquid in boundary fermionic system. Thus, in order to holographically describe the Fermi liquid in non-relativistic HL gravity, it is more reasonable to introduce Lorentz-violation fermions.

V. CONCLUSION AND DISCUSSION
In this paper, we analytically solved the equations of motion for the coupling system between non-relativistic gauge field and HL gravitational action. We obtained a new charged Lifshitz black hole solution, which in the UV limit behaves the Lifshitz geometry (2) with a dynamical scaling between space and time, while at zero temperature, the near horizon geometry is AdS 2 × R D−2 . Here we did not explore more detailed features of our new black hole, such as thermodynamics and various stabilities, etc., but we will present this study somewhere else in the near future. Moreover, it would be very interesting to study whether our bulk theory admits the hyperscaling violating black hole solution [66,67], which has been holographically applied in condensed matter system, please see [68] and thereby in.
Then, by introducing the probed Lorentz-violation Fermions into the background, we studied the properties of the fermionic sector dual to the charged HL gravity theory. We fixed z = 1.02 and focused on the effect of the Lorentz-violation feature on the Fermi momentum and the dispersion relation. We found that for stronger Lorentz-violation of the fermions, the peak in the spectral function, which is suppressed by the dynamical exponent, is enhanced and the Fermi momentum is larger. Meanwhile, the stronger violation makes the exponent of dispersion relation lower until it reaches a critical value η ≃ 0.033, the dispersion relation is kept linear even though we further enhance the violation, which means that at the critical value η ≃ 0.033, the type of low energy excitation will transit from non-Fermi liquid to Fermi Liquid. We also argued that in order to describe a Fermi type Liquid of the low energy excitation, it is more reasonable to add the Lorentz-violation fermions in the non-relativistic charged HL gravity. It would be very interesting to further find an interpretation for the violation parameter η consistent with the Luttinger theorem, which states that the charge density is equal to the volume enclosed by the Fermi surface.
Note that in the fermionic sector we studied, we chose the boundary condition (40) for the Dirac field. Since all fields in our system are non-relativistic, we have strong motivation to choose the Lorentz violation boundary condition for the bulk fermion field With the above boundary condition, it was first addressed in [41] that the spectral function of the dual holographic fermionic system presented a flat band of gapless excitation.
Then we follow the discussion of [41], in this case, the retarded Green function is a kind of recombination of Green functions of the relativistic fixed point as which was firstly obtained in [42]. Due to the symmetry (49), the spectral function is However, we did not find Fermi surface in this case of our system. The physics behind this phenomena calls for further research, and we hope that further study on the features of our black hole solution may give some insight.