Non-equilibrium black hole horizons are considered in scaling theories with generic Lifshitz invariance and an unbroken U(1) symmetry. There is also charge-hyperscaling violation associated with a non-trivial conduction exponent. The boundary stress tensor is computed and renormalized and the associated hydrodynamic equations derived. Upon a non-trivial redefinition of boundary sources associated with the U(1) gauge field, the equations are mapped to the standard non-relativistic hydrodynamics equations coupled to a mass current and an external Newton potential in accordance with the general theory of [arXiv:1502.00228]. The shear viscosity to entropy ratio is the same as in the relativistic case.
Electrically-charged Lifshitz spacetimes are hard to come by. In this paper, we construct a class of such solutions in five dimensional Einstein gravity coupled to Maxwell and $SU(2)$ Yang-Mills fields. The solutions are electrically-charged under the Maxwell field, whose equation is sourced by the Yang-Mills instanton(-like) configuration living in the hyperbolic four-space of the Lifshitz spacetime. We then introduce a dilaton and construct charged and colored Lifshitz spacetimes with hyperscaling violations. We obtain a class of exact Lifshitz black holes. We also perform similar constructi…
We study the entropic destruction of heavy quarkonium in strongly coupled theories with an anisotropic scaling symmetry in time and spatial direction. We consider Lifshitz and hyperscaling violation theories which are covariant under a generalized Lifshitz scaling symmetry with the dynamical exponent $z$ and hyperscaling violation exponent $θ$. It is shown that the entropic force depends on the parameters of these theories. In particular, increasing $z$ decreases the entropic force thus reducing the quarkonium dissociation, while increasing $θ$ has opposite effect.
We present an overview of the construction of the general holographic dictionary for asymptotically locally Lifshitz and hyperscaling violating Lifshitz backgrounds with arbitrary dynamical exponents $z$ and $θ$, compatible with the null energy condition, which was recently developed in [1,2]. A concrete definition of asymptotically locally Lifshitz and hyperscaling violating Lifshitz backgrounds is provided in the context of a generic bottom-up Einstein-Proca-Dilaton theory, and a systematic procedure for solving the radial Hamilton-Jacobi equation via a covariant expansion in eigenfunctions …
We study scalar perturbations of nonlinear charged Lifshitz black branes with hyperscaling violating factor, and we find numerically the quasinormal modes for scalar fields. Then, we study the stability of these black branes under massive and massless scalar field perturbations. Also, we consider different values of the dynamical exponent, the nonlinear exponent and the hyperscaling violating exponent.
In this note we consider the logarithmic curvature correction to Lifshitz and hyper scaling violation geometries. We investigate the effect of this correction to the gauge kinetic function f and the effective potential V . For the case of hyper scaling violating we show that the coupling between dilaton and the correction terms exhibit the logarithmic behavior for dilaton like for the case of squared curvature correction. We find the unexpected form of gauge kinetic function and potential in the presence of logarithmic terms. We try to resolve the IR singularity of Lifshitz and hyperscaling vi…
We consider Einstein gravities coupled to a cosmological constant and multiple $SU(2)$ Yang-Mills fields in general dimensions and find that the theories admit colored Lifshitz solutions with dynamic exponents $z>1$. We also introduce a Maxwell field and construct exact electric charged black holes that asymptote to the $z=D-1$ colored Lifshitz spacetimes and analyse their thermodynamical first law. Furthermore, we introduce a dilaton to the system and construct Lifshitz spacetimes with hyperscaling violations. After turning on the Maxwell field, we obtain a class of hyperscaling violating Lif…
A Vaidya type geometry describing gravitation collapse in asymptotically Lifshitz spacetime with hyperscaling violation provides a simple holographic model for thermalization near a quantum critical point with non-trivial dynamic and hyperscaling violation exponents. The allowed parameter regions are constrained by requiring that the matter energy momentum tensor satisfies the null energy condition. We present a combination of analytic and numerical results on the time evolution of holographic entanglement entropy in such backgrounds for different shaped boundary regions and study various scal…
We study the quasinormal modes of scalar field perturbations in the background of non-Abelian hyperscaling violating Lifshitz black holes. We find that the quasinormal frequencies have no real part so there is no oscillatory behavior in the perturbations, only exponential decay, that is, the system is always overdamped, which guarantees the mode stability of non-Abelian hyperscaling violating Lifshitz black holes. We determine analytically the quasinormal modes for massless scalar fields for a dynamical exponent $z=2$ and hyperscaling violating exponent $\tildeθ>-2$. Also, we obtain numericall…
We explore string/M-theory constructions of holographic theories with Lifshitz scaling exponent $z$ and hyperscaling violation exponent $θ$, finding a range of $z,θ$-values. Some of these arise as effective metrics from dimensional reduction of certain kinds of null deformations of $AdS$ spacetimes appearing in the near horizon geometries of extremal D3-, M2- and M5-brane theories. The $AdS_5$ solution in particular gives rise to $θ=1$ in $d=2$ (boundary) space dimensions. Other solutions arise as the IIA D2- and D4-brane solutions with appropriate null deformations, and we discuss the phase s…
In this note we describe several procedures to construct, from known black-hole and black-brane solutions of any ungauged supergravity theory, non-trivial gravitational solutions whose "near-horizon" and "near-singularity" limits are Lifshitz-like spacetimes with dynamical critical exponent z, "hyperscaling violation" exponent θand Lifshitz radius \ell that depends on the physical parameters of the original black-hole solution. Since the new Lifshitz-like solutions can be constructed from any black-hole solution of any ungauged supergravity, many of them can be easily embedded in String Theory…
Considering the cubic theory of gravity which has been constructed in \cite{myers}, we study the existence of Lifshitz and hyper scaling violating Lifshitz solutions. We firstly extend the black hole solution of \cite{myers} and find that such extended solutions are valid for any value of dynamical exponent $z$. Next, we examine the existence of the AdS black hole solution with non-zero hyperscaling-violating exponent $θ$ and general dynamical exponent $z$. We find that the solutions do exist only for $θ=0,3$ in 4 dimension and $θ=0,4$ in 5 dimension. In particular, when $θ=3(4)$ in $4(5)$ dim…
We study Rényi entropies for geometries with Lifshitz scaling and hyperscaling violation. We calculate them for specific values of the Lifshitz parameter, and analyze the dual spectrum of the ground state. In the large $d-θ$ limit they show that the ground state is unique in specific parameter ranges. We also calculate the Rényi entropies perturbatively around $n=1$, and derive constraints using the Rényi entropy inequalities, which correspond to the thermodynamic stability of the black holes.
The derivation of Lifshitz-invariant hydrodynamics from holography, presented in [arXiv:1508.02494] is generalized to arbitrary hyperscaling violating Lifshitz scaling theories with an unbroken U(1) symmetry. The hydrodynamics emerging is non-relativistic with scalar "forcing". By a redefinition of the pressure it becomes standard non-relativistic hydrodynamics in the presence of specific chemical potential for the mass current. The hydrodynamics is compatible with the scaling theory of Lifshitz invariance with hyperscaling violation. The bulk viscosity vanishes while the shear viscosity to en…
In our previous papers we have shown how Lifshitz-like space-time (space-time having a Lifshitz scaling along with a hyperscaling violation) can arise by taking near horizon limits of certain intersecting solutions (F-string with D$p$-branes and also with two D-branes) of string theory. In this paper we construct intersecting bound state solutions in the form of NS5-D$p$-branes (with $1\leq p \leq 6$) of type II string theories. These are 1/4 BPS and threshold bound states unlike the known NS5-D$p$ bound states which are 1/2 BPS and non-threshold. We show that the near horizon limits of these …
We explore in greater detail our investigations of shear diffusion in hyperscaling violating Lifshitz theories in arXiv:1604.05092 [hep-th]. This adapts and generalizes the membrane-paradigm-like analysis of Kovtun, Son and Starinets for shear gravitational perturbations in the near horizon region given certain self-consistent approximations, leading to the shear diffusion constant on an appropriately defined stretched horizon. In theories containing a gauge field, some of the metric perturbations mix with some of the gauge field perturbations and the above analysis is somewhat more complicate…
We study the divergent terms and the finite term in the expansion of the holographic entanglement entropy as the ultraviolet cutoff vanishes for smooth spatial regions having arbitrary shape, when the gravitational background is a four dimensional asymptotically Lifshitz spacetime with hyperscaling violation, in a certain range of the hyperscaling parameter. Both static and time dependent backgrounds are considered. For the coefficients of the divergent terms and for the finite term, analytic expressions valid for any smooth entangling curve are obtained. The analytic results for the finite te…
We construct two interpolating solutions in type II string theory which interpolate between an AdS$_5$ in the UV and a hyperscaling violating three (spatial) dimensional Lifshitz space-time in the IR. The first solution is non-supersymmetric and is obtained from a known intersecting non-supersymmetric D3-brane with chargeless D0-brane solution of type IIB string theory, by restricting some parameters characterizing the solution and going to a new coordinate. In the IR the dilaton is non-constant in general and the metric is three (spatial) dimensional hyperscaling violating Lifshitz with dynam…
In this paper, we investigate the thermodynamics of hyperscaling violating Lifshitz black branes in the presence of a nonlinear massless electromagnetic field. We, first, obtain analytic nonlinear charged black brane solutions with hyperscaling violating factor in dilaton gravity and give the condition on the parameters of the metric for having black brane solutions. Second, we introduce the appropriate finite action in grand-canonical and canonical ensembles for nonlinear electromagnetic field. Next, by generalizing the counterterm method for the asymptotic Lifshitz spacetimes with hyperscali…
It is observed that the (intersecting) branes of M/string theory, which are known to give AdS geometry (directly or upto a conformal transformation) in the near horizon limit, do also lead to Schr\" odinger/Lifshitz dual space-times (without or with hyperscaling violation) upon using appropriate solution generating transformation and dimensional reduction. We show that the dynamical exponents of the Schr\" odinger and the Lifshitz space-times obtained in this way always add upto 2. We illustrate this by several examples, including M2-, M5-branes of M-theory and D$(p+1)$-branes ($p\neq 4$, sinc…
We present a novel family of asymptotically Schrodinger hyperscaling violating black holes with a generic dynamical critical exponent and an arbitrary number of spacetime dimensions. This black hole family represents a solution within the Einstein-Maxwell-scalar setup with a self-interaction scalar potential where the Maxwell field is coupled to the scalar field. Through an analysis of the curvature invariants it is observed that this configuration is asymptotically regular for different ranges of the hyperscaling violating exponent. Furthermore, the above mentioned solution constitutes a grav…
We obtain two types of Yang-Mills-dilaton black hole solutions in both Lifshitz and hyperscaling violation spacetimes. We must consider at least three Yang-Mills gauge fields that interact with a scalar field and either $SO(n)$ or $SO(n-1,1)$ gauge symmetry groups, where $n+1$ denotes the dimension of the spacetime. They lead to the spherical and hyperbolic solutions. The obtained solutions in the hyperscaling violation spacetime fall into two categories for $z\neq n-3-\frac{n-5}{n-1}θ$ and $z= n-3-\frac{n-5}{n-1}θ$, where $θ=0$ represents the Lifshitz spacetime. In order to have a real asympt…
We analytically compute the thermoelectric conductivities at zero frequency (DC) in the holographic dual of a four dimensional Einstein-Maxwell-Axion-Dilaton theory that admits a class of asymptotically hyperscaling violating Lifshitz backgrounds with a dynamical exponent $z$ and hyperscaling violating parameter $θ$. We show that the heat current in the dual Lifshitz theory involves the energy flux, which is an irrelevant operator for $z>1$. The linearized fluctuations relevant for computing the thermoelectric conductivities turn on a source for this irrelevant operator, leading to several nov…
We consider dyonic black hole in hyperscaling violating Lifshitz theories arised in a four dimensional Einstein-Maxwell-dilaton system along with axion fields. Considering the linearised equation of relevant fluctuations in metric and gauge fields, we analytically compute thermoelectric conductivity of the dual theory using Dirichlet boundary condition and find agreement with conductivities obtained in near horizon analysis. We also study temperature dependence of the conductivities.
We investigate an Einstein-Maxwell-Dilaton-Axion holographic model and obtain two branches of a charged black hole solution with a dynamic exponent and a hyperscaling violation factor when a magnetic field presents. The magnetothermoelectric DC conductivities are then calculated in terms of horizon data by means of holographic principle. We find that linear temperature dependence resistivity and quadratic temperature dependence inverse Hall angle can be achieved in our model. The well-known anomalous temperature scaling of the Nernst signal and the Seebeck coefficient of cuprate strange metals…
Lifshitz and hyperscaling violating geometries, which provide a holographic description of non-relativistic field theories, generically have a singularity in the infrared region of the geometry, where tidal forces for freely falling observers diverge, but there is a special class of hyperscaling violating geometries where this tidal force divergence does not occur. We explicitly construct a smooth extension of the spacetime in this case, and explore the structure of the spacetime. We find that the spacetime has two disconnected boundaries, as in AdS$_2$. We also consider the behaviour of finit…
We present an analytic solution of a charged black hole with hyperscaling violating factor in an Einstein-Maxwell-Dilaton model where the scalar potential is key to the existence of a solution. This solution provides a candidate gravitational description of theories with hyperscaling violation at both finite temperature and finite charge density. Using this background we explore certain features of these theories via AdS/CFT correspondence. Finally, we discuss embeddings based on the well-known sphere reductions of ten and eleven-dimensional supergravity.
We analytically consider the spontaneous formation of a fermionic crystalline geometry in a gravity background with Lifshitz scaling and/or hyperscaling violation. Fermionic vortex lattice solution sourced by the lowest Laundau level has been obtained. Thermodynamic analysis shows that the fermionic vortex lattice favors a triangular configuration, regardless of the values of the Lifshitz scaling $z$ and the hyperscaling violation exponent $θ$. Our results also show that the larger $z$ or lower $θ$ leads to more stable lattices thermodynamically.
Using complexity=action proposal we study the growth rate of holographic complexity for Lifshitz and hyperscaling violating geometries. We will consider both one and two sided black branes in an Einstein-Maxwell-Dilaton gravitational theory. We find that in either case Lloyd's bound is violated and the rate of growth of complexity saturates to a value which is greater than twice the mass of the corresponding black brane. This value reduces to the mass of the black brane in the isotropic case. We show that in two sided black brane the saturation happens from above while for one sided black bran…
We investigate the holographic dc conductivity of (2+1) dimensional systems while considering hyperscaling violating geometry in the bulk. We consider Einstein-Maxwell-Dilaton system with two gauge fields and Liouville type potential for dilaton. We also consider axionic fields in the bulk in order to introduce momentum relaxation in the system. We apply an external magnetic field to study the response of the system and obtain analytic expressions for dc conductivity, Hall angle and (thermo)electric conductivity.
Considering a model with Coupled Dilaton-Squared Curvature terms and a dilatonic potential which replace the cosmological constant, we present an analytic hyperscaling violating solution which is the generalization of arXiv:1305.3279 for non zero $θ$. We show that a specific coupling between dilaton and the squared curvature terms produces a simple logarithmic behavior for the dilaton. By imposing the null energy condition and stability constraints, we investigate the allowed regions for the parameters of this model. We then study the perturbations around $\mathrm{AdS_4} $ in the UV and $\math…
Using the complexity equals action proposal we study holographic complexity for hyperscaling violating theories in the presence of a finite cutoff that, in turns, requires to obtain all counter terms needed to have finite boundary energy momentum tensor. These terms could give non-trivial contributions to the complexity. We observe that having a finite UV cutoff would enforce us to have a cutoff behind the horizon whose value is fixed by the UV cutoff; moreover, certain counter term should be defined on the cutoff behind the horizon too.
We show that many results about holographic conductivities in geometries with hyperscaling violating scaling can be reproduced from simple scaling laws in the dual field theory. We show that the electro-magnetic response of probe branes in these systems require at least one additional scaling parameter Phi beyond the usual dynamical exponent z and hyperscaling violating exponent theta, as also pointed out in earlier work. We show that the scaling exponents can be chosen in such a way that the temperature dependence of DC conductivity and Hall angle in strange metals can be reproduced.
This short note is devoted to deriving scaling but hyperscaling violating solutions in a generalised Einstein-Maxwell-Dilaton theory with an arbitrary number of scalars and vectors. We obtain analytic solutions in some special case and discuss the physical constraints on the allowed parameter range in order to have a well-defined holographic ground-state solution.
We present an analytic solution of a Vaidya-charged black hole with a hyperscaling violating factor in an Einstein-Maxwell-dilaton model, where the scalar potential plays a key role in the existence of the solution. By making use of this result, we study the process of thermalization after a global quench in a theory which its gravitational description is provided by the resultant solution in the case of zero charge. In particular, we probe the system by entanglement entropy and show that it exhibits certain scaling behaviors during the process.
We study the holographic entanglement entropy for singular surfaces in theories described holographically by hyperscaling violating backgrounds. We consider singular surfaces consisting of cones or creases in diverse dimensions. The structure of UV divergences of entanglement entropy exhibits new logarithmic terms whose coefficients, being cut-off independent, could be used to define new central charges in the nearly smooth limit. We also show that there is a relation between these central charges and the one appearing in the two-point function of the energy-momentum tensor. Finally we examine…
We examine spacetimes which generalize Lifshitz scaling to allow hyperscaling violation invariance (i.e. a constant conformal transformation) for the types of singularities frequently found in the Lifshitz case. We find that most of these spacetimes suffer either from a traditional naked curvature singularity or a null curvature singularity robust against stringy higher derivative corrections. We do find a one-parameter family that evades these issues but this family requires exotic matter, in particular involving a negative energy density that may not be interpreted as due to a cosmological c…
We propose a formalism for counting the microstates of a class of three-dimensional black holes which are not asymptotically AdS. The formalism rests on the invariance of a dual field theory under a generalized modular transformation and is extended to Rindler horizons by a singular limit. We also obtain logarithmic corrections.
We study quasinormal modes of shear gravitational perturbations for hyperscaling violating Lifshitz theories, with Lifshitz and hyperscaling violating exponents $z$ and $θ$. The lowest quasinormal mode frequency yields a shear diffusion constant which is in agreement with that obtained in previous work by other methods. In particular for theories with $z< d_i+2-θ$ where $d_i$ is the boundary spatial dimension, the shear diffusion constant exhibits power-law scaling with temperature, while for $z=d_i+2-θ$, it exhibits logarithmic scaling. We then calculate certain 2-point functions of the dual …
We search for a possible relation between weak gravity conjecture and conformal field theory in the hyperscaling violating and Kerr-Newman-AdS black holes. We deal with the critical points of the black hole systems through the correlation function introduced in conformal field theory. We discuss the weak gravity conjecture conditions by the imaginary part of the energy obtained from the critical points and their poles. Under the assumptions $z=1$, $d=1$ and $θ\rightarrow0^{-}$, we link weak gravity conjecture to the hyperscaling violating black holes due to the existence of $r_{H}$ larger and …
In this paper we investigate the ratio of shear viscosity to entropy density, $η/s$, in hyperscaling violating geometry with lattice structure. We show that the scaling relation with hyperscaling violation gives a strong constraint to the mass of graviton and usually leads to a power law of temperature, $η/s\sim T^κ$. We find the exponent $κ$ can be greater than two such that the new bound for viscosity raised in arXiv:1601.02757 is violated. Our above observation is testified by constructing specific solutions with UV completion in various holographic models. Finally, we compare the boundedne…
We compute the bulk viscosity in holographic models dual to theories with Lifshitz scaling and/or hyperscaling violation, using a generalization of the bulk viscosity formula derived in arXiv:1103.1657 from the null focusing equation. We find that only a class of models with massive vector fields are truly Lifshitz scale invariant, and have a vanishing bulk viscosity. For other holographic models with scalars and/or massless vector fields we find a universal formula in terms of the dynamical exponent and the hyperscaling violation exponent.
We examine the holographic entanglement entropy in hyperscaling violating backgrounds. Precisely in such theories by semi-analytic computation, we use holographic methods to derive the universal terms of entanglement entropy in various dimensions. We also find such terms when the bulk action includes higher-order curvature terms. We make further comments on the mutual information in hyperscaling violating backgrounds with higher curvature interactions. We explore the sign of tripartite information which results in a monogamy property on mutual information. It is shown that for various hypersca…
We present a new family of charged black holes with hyperscaling violating asymptotics and non-trivial horizon topology, for arbitrary Lifshitz exponent $z$ and hyperscaling violation parameter $θ$. They are $(d+2)$-dimensional analytic solutions to a generalized Einstein-Maxwell-Dilaton theory with an additional vector field supporting spherical or hyperbolic horizons. The hyperbolic solutions are only consistent with the null energy condition for $1 \le z < 2$ and $θ= d(z-1)$, which explains why they are absent in the pure Lifshitz case. We further study the (extended) thermodynamics of thes…
We study a gravity theory where a scalar field with potential, beyond its minimal coupling, is also coupled through a non-minimal derivative coupling with the torsion scalar which is the teleparallel equivalent of Einstein gravity. This theory provides second order equations of motion and we find large-distance non-perturbative static spherically symmetric four-dimensional solutions. Among them a general class of black hole solutions is found for some range of the parameters/integration constants with asymptotics of the form of hyperscaling violating Lifshitz spacetime with spherical horizon t…
We derive new black hole solutions in Einstein-Maxwell-Axion-Dilaton theory with a hyperscaling violation exponent. We then examine the corresponding anomalous transport exhibited by cuprate strange metals in the normal phase of high-temperature superconductors via gauge/gravity duality. Linear temperature dependence resistivity and quadratic temperature dependence inverse Hall angle can be achieved. In the high temperature regime, the heat conductivity and Hall Lorenz ratio are proportional to the temperature. The Nernst signal first increases as temperature goes up but it then decreases with…
We study the possible boundary conditions of scalar field modes in a hyperscaling violation(HV) geometry with Lifshitz dynamical exponent $z (z\geqslant1)$ and hyperscaling violation exponent $θ(θ\neq0)$. For the case with $θ>0$, we show that in the parameter range with $1\leq z\leq 2,~-z+d-1<θ\leq (d-1)(z-1)$ or $z>2,~-z+d-1<θ\leq d-1$, the boundary conditions have different types, including the Neumann, Dirichlet and Robin conditions, while in the range with $θ\leq-z+d-1$, only Dirichlet type condition can be set. In particular, we further confirm that the mass of the scalar field does not p…
We study in detail a variety of gravitational toy models for hyperscaling-violating Lifshitz (hvLif) space-times. These space-times have been recently explored as holographic dual models for condensed matter systems. We start by considering a model of gravity coupled to a massive vector field and a dilaton with a potential. This model supports the full class of hvLif space-times and special attention is given to the particular values of the scaling exponents appearing in certain non-Fermi liquids. We study linearized perturbations in this model, and consider probe fields whose interactions mim…
We study certain features of scaling behaviors of the mutual information during a process of thermalization, more precisely we extend the time scaling behavior of mutual information which has been discussed in \cite{Alishahiha:2014jxa} to time-dependent hyperscaling violating geometries. We use the holographic description of entanglement entropy for two disjoint system consisting of two parallel strips whose widths are much larger than the separation between them. We show that during the thermalization process, the dynamical exponent plays a crucial rule in reading the general time scaling beh…
In this work, we consider a class of hyperscaling violating Lifshitz-like black branes with metric scaling components $z=2$ and $θ=-1$ whose corresponding holographic model can be treated as a non-relativistic fluid exhibiting Lifshitz-type symmetry. Having performed analytical calculations via the Klein-Gordon equation and the linear response theory, the experimental realizations of the concerned model, namely the transport coefficients, are found to behave as $η\propto T^{3/2}$, $σ_{DC} \propto T^{3/2}$, and $ρ_{DC} \propto T^{-3/2}$. The associated metric scaling exponents from the bulk the…
We present a semi-analytic method for constructing holographic black holes that interpolate from anti-de Sitter space to hyperscaling-violating geometries. These are holographic duals of conformal field theories in the presence of an applied chemical potential, $μ$, at a non-zero temperature, $T$, and allow us to describe the crossover from `strange metal' physics at $T \ll μ$, to conformal physics at $T \gg μ$. Our holographic technique adds an extra gauge field and exploits structure of the Einstein-Maxwell system to manifestly find 1-parameter families of solutions of the Einstein-matter sy…
We provide a general algorithm for constructing the holographic dictionary for any asymptotically locally Lifshitz background, with or without hyperscaling violation, and for any values of the dynamical exponents $z$ and $θ$, as well as the vector hyperscaling violating exponent, that are compatible with the null energy condition. The analysis is carried out for a very general bottom up model of gravity coupled to a massive vector field and a dilaton with arbitrary scalar couplings. The solution of the radial Hamilton-Jacobi equation is obtained recursively in the form of a graded expansion in…
We investigate two transport coefficients, shear viscosity and conductivity, in a non-relativistic boundary filed theory without hyperscaling symmetry, which is dual to a bulk charged hyperscaling violating black brane. Employing matching method, we obtain that the ratio of shear viscosity and the entropy density is alway $1/4π$ at any temperature, which satisfies the Kovtun-Starinets-Son (KSS) bound. Besides, we also present the universal formulas of AC conductivity, which is closely dependent on the relation between geometrical parameters $z$ and $θ$. The optical conductivity at high frequen…
