Liouville mode in Gauge/Gravity Duality

We establish solutions corresponding to AdS4 static charged black holes with inhomogeneous two-dimensional horizon surfaces of constant curvature. Depending on the choice of the 2D constant curvature space, the metric potential of the internal geometry of the horizon satisfies the elliptic wave/elliptic Liouville equations. We calculate the charge diffusion and transport coefficients in the hydrodynamic limit of gauge/gravity duality and observe the exponential suppression in the diffusion coefficient and in the shear viscosity-per-entropy density ratio in the presence of an inhomogeneity on black hole horizons with planar, spherical, and hyperbolic geometry. We discuss the subtleties of the approach developed for a planar black hole with inhomogeneity distribution on the horizon surface in more detail and find, among others, a trial distribution function, which generates values of the shear viscosity-per-entropy density ratio falling within the experimentally relevant range. The solutions obtained are also extended to higher-dimensional AdS space. We observe two different DC conductivities in 4D and higher-dimensional effective strongly coupled dual media and formulate conditions under which the appropriate ratio of different conductivities is qualitatively the same as that observed in an anisotropic strongly coupled fluid. We briefly discuss ways of how the Liouville field could appear in condensed matter physics and outline prospects of further employing the gauge/gravity duality in CMP problems.


Introduction
This paper is motivated by recent progress in applying the AdS/CFT correspondence to Condensed Matter Physics. Since several universal bounds (momentum η/s ≥ 1/4π and charge σ DC /χ ≥ d/4πT (d − 2) transport bound relations in holographic hydrodynamics [1], [2], [3], [4], ω g /T c 8 in holographic superconductivity [5]) have been established for strongly coupled effective dual media, it is reasonable to pose the question: How robust these relations are? If one is limited to the standard gravitational theory setup no signs of violation of these universal bounds have been found. 1 However, different extensions of general relativity with higher order curvature terms revealed the violation of these universal relations (see, e.g., [7], [8], [9]). But still the question remains: may violations of the universal bounds of Gauge/Gravity duality be found within the Einstein theory? In fact, the positive answer to this question has known and it is 1 Within the assumptions made on the structure of the bulk metric (see, e.g., [6]).
Indeed, the origin of universality in Gauge/Gravity duality is closely related to properties of black holes. In further discussion two observations will be important: 1. Thermodynamics of static charged black holes is fully managed by (g tt , g rr ) parts of the metric 2 and depends on global geometry of the horizon surface, more precisely on its integral volume; 2. Transport coefficients are determined by local geometry of the horizon surface, as well as by the g rr part of the metric, which determines the radial coordinate value of the horizon location.
On account of these facts it is easy to see that either modifications in the (g tt , g rr ) parts of BH solutions due to the change of the bulk gravitational dynamics [7], [8], [9] (without changing the horizon geometry) or changing the horizon surface geometry to the anisotropic one [10], [11], [12], [13], [14] (without changing the standard dynamics of the bulk gravity) should lead to corrections to universal relations. Note however that the latter case requires introducing additional, in compare to the standard Einstein-Maxwell system, fields.
In this paper we will limit ourselves with the standard dynamics managed by the Einstein-Hilbert-Maxwell action and will take a look at universality in Gauge/Gravity duality from a different angle. It is clear from the discussion above the universality violation will require changing in the horizon geometry. In [10], [11], [12], [13], [14] the standard geometry of the horizon surface was changed to the anisotropic one, modifying the planar geometry of the horizon with factoring one of the horizon coordinates by a function of the radial coordinate H(r). By means of differential geometry it is easy to see that since such deformation of the horizon is not isometric (it preserves the orthogonality of the coordinate system, but it does not preserve the volume of the horizon), it does not change the geometry type: the external curvature of the horizon surface is still equal to zero. Therefore, one may apply the same computational scheme to calculate the transport coefficients (the AC/DC conductivity and the shear viscosity [10], [13], [14]) as it was done before [1], [2], [6]. Our proposal consists in performing another deformation of the horizon surface, which is also not isometric, but still keeps the planar geometry: horizon surfaces considered here are conformally flat. Unlike the deformation previously considered in [10], [11], we consider "inhomogeneity" on the horizon surface encoded in function(s) solely dependent on the horizon surface coordinates. But we do it in a way, which realises the conformal flatness of the horizon that in its turn justifies employing the technique of [1], [2] in computing transport coefficients.
The rest of the paper is organised as follows. In Section 2 we formulate out setup, which is a standard one in searching for the solutions corresponding to charged black holes in 4D AdS space-time. Then, on the ground of two-dimensional surface theory we write down the part of the metric ansatz, corresponding to geometry of the horizon surface, in terms of isothermal coordinates. Recall, written in the isothermal coordinates any two-dimensional surface of genus zero possesses the geometry of conformally flat space. The log of the conformal factor in exponential parameterisation (the so-called metric potential) solely depends on the horizon surface coordi- In Section 3 we calculate the charge diffusion coefficient and the DC conductivity on the inhomogeneous planar horizon of electrically charged black hole within the stretched horizon approach [18] and the hydrodynamic limit of AdS/CFT correspondence [1], [2]. Here we derive the Fick's law of diffusion in inhomogeneous media and observe the exponential suppression in the diffusion coefficient. The latter results in violation of the universal bound of [3] for suitable configurations of the metric potential. In Section 4 we extend our computations to the shear viscosity in the effective inhomogeneous fluid and establish the same bound value of η/s ratio as in [1], [2]. However, fulfilment of this relation in the considered case reveals the exponential suppression of the KSS [1], [2] bound value η/s 0 = 1/4π, computed for the trivial Liouville mode.
Section 5 contains our comments on the transport coefficients in the background of black holes with non-planar inhomogeneous horizons, on occurrence of the Liouville equation in models of Condensed Matter Physics, and on generalisation of the obtained solutions to higher-dimensional AdS spaces. In the latter case we observe two different conductivities on inhomogeneous horizon of 5D AdS electrically charged black hole. We establish conditions under which the corresponding to [13] ratio of different conductivities behaves qualitatively the same as for the strongly coupled anisotropic plasma model of [11], [12]. In this section we also give an example of the Liouville mode configuration in the planar black hole solution, which preserves the KSS universal bound and fits the experimentally observed upper bound value of η/s ratio.
In the last section we present summary of the results. For the reader convenience, we add Appendices containing the notation and useful information on solutions to the elliptic Liouville equation.

Setup
Let's consider the Reissner-Nördstom (RN) type solution to the Einstein-Maxwell system in AdS space-time with cosmological constant Λ, dynamics of which is described by the following Integration over M/∂M stands for the integration over the AdS/boundary manifold; h mn is the induced metric on the boundary in the r space-time foliation, n m = √ g rr δ mr is the outward normal to the boundary surface.
We will find the RN-type solution to the Einstein-Maxwell equations of motion over the AdS 4 background with inhomogeneous horizon metric which generalises the planar (AdS)BH solution [15], [16]. As usual Λ = −3/l 2 , where l is a characteristic length of the AdS space.

Solutions for charged AdS BHs
To complete the task let us recall geometric properties of two-dimensional surfaces, which lie in the basis of String Theory. The line element of any two-dimensional surface of genus zero can be presented in the following way where (x, y) are the isothermal coordinates (see, e.g., Chapter 9, Addendum I of [17]). A function Φ(x, y) is often called the potential of the metric.
On account of (2.5) we get the following BH solutions to the AdS 4 Einstein equation (2.3) with zero Maxwell field and constant curvature horizon manifolds: 3 • For a planar type BH with Φ(x, y) satisfying the elliptic wave equation • For a spherical type BH with Φ(x, y) satisfying the elliptic Liouville equation • For a hyperbolic type BH 4 with Φ(x, y) satisfying the elliptic Liouville equation Recall, the Liouville equation includes the Gauss curvature of 2D manifold K. 3 See Appendix A for details on the used notation. 4 In Schwarzschild coordinates the corresponding interval looks like Restoring the Maxwell field we get the following (electrically and magnetically) charged BH solutions to the Einstein-Maxwell system (2.2)-(2.3): Depending on the Gauss curvature of the horizon surface we choose K = 0, ±1; Q e,m are the values of the electric/magnetic charge densities. As for the neutral BHs, the metric potential Φ(x, y) has to satisfy the elliptic Liouville equation (2.12). The Bianchi identities hold on the vector field ansatz (2.14).
Having established the solutions for the charged AdS BHs with inhomogeneous horizons, let us turn to computations of the related charge/momentum transport coefficients. From now on we will focus on the solution for electrically charged BH with the planar type inhomogeneous horizon (i.e., on the solution (2.12)-(2.14) with K = 0 and Q m = 0).

Charge diffusion on a stretched horizon
A quick way to obtain the charge diffusion coefficient is to use the stretched horizon approach of [18] (see also [19], [20] for early papers on electrodynamics of BHs and [21] and Refs. therein for introduction to the Membrane Paradigm approach).
It is a well-known fact (see, e.g., [18], [6]) that the variation of the Maxwell action in a BH gravitational background fieldḡ mn leads to the boundary term at the BH horizon 5 r + , which is compensated by the following surface term added to the action (cf. action (2.1)): Hereh is the induced background metric on the stretched horizon r = r + + , 1 in the r space-time foliation, n m = √ḡ rr δ mr is the outward normal to the horizon surface, and j m is the conserved current induced on the horizon 6 Now we will treat the Maxwell field as a small perturbation over the gravitational background.
In the linear order of perturbations the Einstein-Maxwell system (2. The conformally flat structure of the horizon geometry makes possible to use the plane wave representation (see Section 3.7. in [22]) for the perturbed Maxwell field. Without loss of generality we can choose Calculations of the DC conductivity are performed at q → 0 (see [1], [6]).
Requirement of the Maxwell field regularity near the horizon imposes the following boundary Alternatively, this boundary condition follows from the solution to Maxwell equations in the nearhorizon limit [1]. Other assumptions lying in the basis of the charge diffusion law derivation and compatible with q 2 /T 2 1 vector field series expansion (here T is the BH temperature) are [1] and Adapting the computational scheme of [1] to the considered case one may check the validity of (3.7) and (3.8).
6 Indeed, and on account of (3.6) and (3.7) we derive Using the definition of (3.4a) we further get Finally, we obtain which is the covariantisation of the Fick's first law for inhomogeneous media (compare to the corresponding expression in [1]). The diffusion coefficient entering (3.10) becomes a function of the horizon coordinates x, y where we have explicitly marked out the part depending on x, y coordinates. The Fick's second law comes from the current conservation: The Ohm's law contains the expression for the DC conductivity on the horizon: (3.14) From the Einstein relation D = σ DC /χ we can read off the charge susceptibility χ. Clearly, χ = r+e Φ(x,y) , σDC ≡ σ xx = 1, hence D = σDC /χ holds. and the bound D ≥ d/4πT (d − 2) suggested in [3] (KR bound) is exponentially suppressed. The violation of this bound strongly depends on the choice of inhomogeneity distribution function Φ(x, y). 8 If the distribution function is positively valued in the selected domain, the damping strength is determined by the local inhomogeneity degree on the horizon surface.

AdS/CFT calculations of DC conductivity and charge diffusion on the horizon
Now let us compute the transport coefficient in the hydrodynamic limit of AdS/CFT correspondence. Here we follow [23], [24], [25], [26], [27].
Following the AdS/CFT prescriptions in computing the retarded Green function, let's consider a small perturbation of the Maxwell field over the RN background 9 We choose the perturbed field as Then, in the leading order of perturbations the vector field equation of motion (2.2) turns into   Φ(x, y)). 9 Though originally small perturbations of Maxwell field were considered over the AdS background (see [23], [24]), we will make our calculations maximally close to calculations of the shear viscosity (see below).

which we rewrite to
Solution to the latter equation in the ω → 0 limit is apparent; it is 10 with some constant α. This constant is fixed by the requirement of having the in-falling boundary condition near the horizon (see, e.g., Appendix A of [6]): According to the AdS/CFT recipe [25], [26], [27] we have to substitute the solution to the dynamical bulk equation into the boundary term 25) and to extract the retarded Green function: Applying the Kubo formula of the linear response theory leads to the following expression of the DC conductivity near the horizon: that coincides with eq. (3.14) obtained within the stretched horizon approach.
The same arguments as before (see footnote 7 to this end) lead to the charge diffusion on the horizon (3.15), so we arrive at the same conclusions on violation of the KR universal bound [3] as in the previous subsection. is the solution to (3.20) satisfying the in-falling boundary condition in the near horizon limit and the second boundary condition Ax → 1 at r → ∞.

Shear viscosity and η/s ratio
In this part of the paper we compute the η/s ratio for the planar BH solution with inhomogeneous horizon surface.
Expanding the Einstein equation R mn = Λg mn to the linear order in fluctuations we get where we have used eq. (2.7). Taking h xy (t, r, x) = h xy (r)e −iωt+iqx (4.2) one can notice that the resulted system of equations is compatible to the zero momentum limit q → 0, which is suitable for our aims. 11 Therefore, to compute the shear viscosity we have to solve the equation for h xy (r) first.
Plugging (4.2) into the xy part of the perturbed Einstein equation one gets which on account of the explicit expression for f (r) can be written down in the following form: Then, the resulting equation for h xy coincides with that of obtained in [28] (cf. eq. (2.19) therein). 11 To consider the case of non-zero but small momentum q one needs to extend the expansion to the second order in hxy terms.
Solution to eq. (4.5) satisfying the boundary condition h xy → 1, r → ∞ is Again, the constant α is fixed by the in-falling boundary condition at the horizon Comparing (4.6) to (4.7) in the near-horizon limit results in with the entropy density Here we have denoted the entropy of a BH with isotropic homogeneous horizon as s 0 . Clearly, the entropy density s becomes a function of the local distribution Φ(x, y).
The on-shell action for the perturbed gravity mode is as follows: Plugging the retarded Green function into the Kubo formula leads to the well-known expression [24] η/s = 1 4π . (4.12) Now, once we recover the KSS η/s 0 ratio [1], from (4.9) we get

Comments on transport coefficients for non-planar inhomogeneous horizons
Calculations performed in previous sections may be extended to the case of black hole solutions with non-zero constant curvature horizons (solutions (2.13)-(2.14) with K = ±1). In this cases one also gets the exponential suppression in formulae for the diffusion coefficient and for the η/s 0 ratio, eqs. (3.15) and (4.13). Formally the results are the same; however, there are differences in compare to the case of a planar inhomogeneous horizon.
Looking at solutions to the elliptic Liouville equation (some of which are borrowed from [29] and presented in Appendix B), one can notice that exp(±Φ(x, y)) contains in general different

Comments on the Liouville field in Condensed Matter Physics
The Liouville equation has been widely recognised in 2D QFT (see, e.g., [31], [32] for comprehensive reviews). On the Condensed Matter Physics side the appearance of the Liouville field theory may be found in description of disordered charged media at the strong coupling limit [33] and under consideration of particle motion in a random potential [34] or in diffusion process of a random walk particle in δ-potential [35], [36] (see also [37] for a review of diffusion processes in disordered media). These observations give us an evidence to interpret the parameter of inhomogeneity of the horizon Φ(x, y) as the inhomogeneity degree in the dual strongly coupled effective media related to its disorder and the degree of chaotisation. We hope it opens a new prospect in searching for the holographic description of such CMP models in terms of Gauge/Gravity duality. 12

Comments on higher-dimensional generalisation of the solutions
Higher dimensional generalisation of the solutions (2.13)-(2.14) is easy to derive on account of the previously found solution for an electrically charged AdS n+1 black hole [43]. Adapting to 12 See also [38], [39] as an interplay between the Bose-Einstein condensate, BH physics and the Liouville theory.
our case this solution transforms into where the metric potential Φ(x, y) satisfies the elliptic Liouville equation Applying the technique has been used in computing transport coefficients, one may found the effect of anisotropy in n-dimensional effective dual media. For instance, in 5D case we get two different conductivities: and two different diffusion coefficients: one of which is at the KR [3] bound value; the other one is exponentially suppressed. The ratio σ zz /σ xx = e Φ depends on the degree of inhomogeneity, determined by the metric potential, and it is always smaller than one inside domains of exp(Φ(x, y)) < 1, where the KR bound holds.

Fitting to RHIC and LHC data
The combined analysis of all data of high-ion collisions measured at RHIC and at LHC gives the following experimental restrictions on η/s value [41], [42]: In what follows we will focus on the LHC result. Also we will accept the KSS value for the lower bound η/s 0 ∼ 0.08.
According to our calculations, the η/s ratio, measured in s 0 units, is not a constant anymore; it is a function of the Liouville mode Φ(x, y) (cf. (4.13)) with natural "boundary condition" Hence, in our interpretation a wide range of experimentally fixed values of η/s (5.5) is an impact of the local inhomogeneity distribution in quark-gluon plasma. Then the KSS bound value corresponds to the QGP near-equilibrium isotropic homogeneous state. Now our aim is to find a shape of the metric potential Φ(x, y) which will satisfy: 1) the By trials and errors method we found the following trial form of Φ(x, y), which falls into the above mentioned criteria: To satisfy the upper bound η/s 0.2, the BH horizon should be restricted in x, y directions to Fig.1). Then, depending on the local value of the having unfixed functions converts fitting to experimental data into a merely technical task.

Summary and conclusions
To summarise, we have obtained solutions to the Einstein-Maxwell dynamical system, which correspond to the static charged AdS black holes with inhomogeneity distribution function on the black hole horizon surface. The inhomogeneity of 2D horizon surface is encoded in the conformal factor entering the metric ansatz, which depends on the horizon coordinates and whose dynamics obeys the Liouville equation. That is why we have called such ingredient of the metric as the Liouville mode.
Focusing on the AdS 4 space-time we have computed the charge diffusion coefficient and the DC conductivity on the horizon within the stretched horizon approach and have observed that: 1. The resulted Fick's laws describe diffusion in inhomogeneous strongly coupled dual media that is natural to expect; 2. The diffusion coefficient is exponentially suppressed, that may in principle violate the previously suggested KR universal bound for the diffusion constant [3]. In all possible cases the violation degree is proportional to the strength of local inhomogeneity.
We have also calculated the charge diffusion and transport coefficients in the hydrodynamic limit of AdS/CFT correspondence and have realised that the KSS shear viscosity-per-entropy density universal bound [1], [2] is also exponentially suppressed. Hence, we have observed the anisotropic strongly coupled plasma [13].
Turning back to occurrence of the Liouville equation in Condensed Matter Physics problems, we recall that the Liouville field theory naturally appears in CMP models related to diffusion processes in random media, or to the description of strongly coupled disordered media [33], [34], [35], [36]. We believe that our results open a new prospect in searching for holographic description of physical processes in disordered media at the strong coupling constant regime. We hope to report on progress in this and other directions in the future.