AdS/CFT and Landau Fermi liquids

We study the field theory dual to a charged gravitational background in which the low temperature entropy scales linearly with the temperature. We exhibit the existence of a sound mode which is described by hydrodynamics, even at energies much larger than the temperature, and explain how this, and other properties of the field theory, are consistent with those of a (3+1)-dimensional Landau Fermi liquid, finely tuned to the Pomeranchuk critical point. We also discuss how one could engineer a higher-derivative gravitational Lagrangian which reproduces the correct low temperature behavior of shear viscosity in a generic Landau Fermi liquid.


Introduction and summary
Fermi liquid theory is one of the very few low energy effective theories at finite density that we understand well. Since the AdS/CFT correspondence provides a simple description of strongly coupled field theories at finite density, it is natural to ask whether a Fermi liquid description can be recovered at low energies in a holographic model. The possibility of observing a Fermi surface at leading order in the "1/N expansion" was first raised in [1]. More recently, the logarithmic violation of the entanglement entropy was suggested as signifying the appearance of a Fermi surface [2][3]. In [4], a certain singular behavior of the current-current correlator was suggested as an indicator of a Fermi surface. These models imply that the effective theory at low energy differs from the conventional Landau Fermi liquid theory, which is characterized (among other things) by a linear heat capacity at small temperatures and by the validity of Luttinger's theorem. 1 On the other hand, the discovery of a gapless mode in holographic models at finite density and zero temperature [9] raised the possibility that the Landau Fermi liquid is not too far off: it contains a similar excitation called zero sound (subsequent work on various aspects of holographic zero sound includes [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]). It was pointed out in [10] that the equality between the speed of holographic zero sound and the speed of hydrodynamic sound is consistent with the Landau parameters, which control the interaction strength in a Fermi liquid, being parametrically large. In this paper we consider a particular background (reviewed in Section 2) which exhibits a linear heat capacity at small temperatures and features a holographic zero sound mode. In Section 3 we show that linearized hydrodynamics is completely sufficient to describe the holographic zero sound excitation to second order in the derivative expansion: it should therefore be identified with the usual hydrodynamic sound mode. The situation here is similar to [27], where a different background was considered.
In Section 4 we consider Landau Fermi liquid theory and show that the observations in the holographic model can be explained by considering a regime of parametrically small Fermi velocity and taking the second Landau parameter to the stability bound F 2 → − 5 corresponding to the Pomeranchuk critical point. In particular, the quasiparticle lifetime τ ∼ (F 2 + 5) becomes parametrically small, and as a result, linearized hydrodynamics is valid for energies that are much larger than the temperature, as long as they are small 1 Constructing Fermi liquids by the explicit inclusion of charged fermions in the bulk is another interesting direction [5][6][7][8].
compared to the chemical potential. This parametrically small lifetime means that we are pushing the Landau Fermi liquid into a regime where it behaves like a non-Fermi liquid.
This choice of parameters also resolves an apparent paradox: the viscosity/entropy density ratio for any model that involves Einstein-Hilbert gravity is η/s = 1/4π [28][29][30][31]. At first sight, the viscosity/entropy ratio of a generic Fermi liquid diverges like η/s ∼ µ 3 /T 3 for large ratios of chemical potential µ and temperature T . However, as we explain in Section 4, the coefficient in front of the leading term in η/s vanishes for F 2 = −5, and so a Fermi liquid description may be compatible once we tune the Landau parameters accordingly.
This raises the question of whether we can have a holographic dual of a generic Landau Fermi liquid, where hydrodynamics breaks down at high energies and collisionless behavior takes over? One clear signature of any such model is that η/s ∼ µ 3 /T 3 . In Section 5, we describe a procedure by which one could construct a model with precisely this behavior, by adding higher derivative terms to the gravitational Lagrangian. We discuss our results in Section 6. The appendices contain some technical details related to Sections 3 and 5.

The two-charge black hole
Here we briefly review the dilatonic black hole in AdS 5 recently explored in [32][33][34][35]. This black hole is usually referred to as the two-charge black hole, because it arises as a solution to the truncation of IIB supergravity on AdS 5 × S 5 where two of the three U (1) charges are equal and non-vanishing while the third one is zero [36].
After this truncation one obtains the action [32], [35] with φ a scalar field and F ab the field strength of the Maxwell field A b . The 2-charge black hole solution is [32], [35] where a(r) = log The temperature T , chemical potential µ, entropy density s, charge density σ, energy density ε and pressure P of the dual field theory are given in terms of the parameters of the 2-charge black hole (2.2) via (2.5) At large densities, this black hole is dual to a semi-local quantum liquid [37] which violates hyperscaling [38]. See [39][40][41][42][43][44][45] for related work.

Hydrodynamics
In common with other field theories, for small amplitude excitations around this equilibrium state with sufficiently small frequencies and momenta, local thermal equilibrium is maintained and one can describe the system using hydrodynamics [46]. In this hydrodynamic limit, the excitations of the system include hydrodynamic sound modes with dispersion relations [47] where the ellipsis denotes higher order terms in k. The attenuation of this hydrodynamic sound mode is controlled by the viscosity η of the field theory. For the two-charge black hole, η = s/4π [31], and thus the hydrodynamic sound mode dispersion relation is Note that the leading k 2 contribution to the attenuation vanishes linearly with T in the limit T → 0. This is because s ∼ T at low T .
We are interested in the two-charge black hole solution in the large density limit µ ≫ T, ω, k. Furthermore, we will assume |ω| ∼ k from now, since this is true for the sound mode in which we are interested. Within this large density limit, the ratio k/T is still arbitrary. We will now clarify our previous assertion of how "sufficiently small" k must be such that hydrodynamics is applicable. In a quasiparticle description, there is a mean free path l η between (thermal) collisions and the regime of applicability of hydrodynamics is then kl η ≪ 1. For the two-charge black hole, we can identify a natural expansion parameter from the dispersion relation (3.2) This suggests that hydrodynamics is valid provided that k/µ ≪ µ/T + 2π 2 T /µ . Within the large density limit, this inequality is always satisfied and hence the hydrodynamic result (3.2) should be true for arbitrary k within the large density limit. This is intimately related to the fact that η ∼ s for holographic theories, since from (3.1) we can write the range of applicability of hydrodynamics more abstractly as |ω| ∼ k ≪ l −1 and s ∼ T is always small (in units of µ) in the large density limit. In a Landau Fermi liquid, l η ∼ µ/T 2 is the length scale over which individual fermions interact such that a hydrodynamic state is formed. For a generic holographic theory with η ∼ s, the equivalent length scale is always small, indicating that a single-particle description is not applicable.
This applicability of hydrodynamics at arbitrarily low temperatures was shown analytically for the collective excitations of the Reissner-Nordstrom-AdS 4 black brane in [27].
To verify this for the two-charge black hole, we have calculated numerically the dispersion relation of the sound mode of the field theory dual to the two-charge black hole over a large range of temperatures. The details of this calculation are given in appendix A and the results are shown in figure 1.
The dispersion relation agrees well with the hydrodynamic prediction (3.2) down to very low T . In the limit T → 0, where the leading order hydrodynamic result (3.2) for the sound attenuation vanishes, the numerical result begins to differ from the (vanishing) hydrodynamic prediction. This difference is of order k 3 and thus the numerical results are consistent with the hydrodynamic prediction (3.2) to O k 2 for all T in the large density limit. At higher orders in k, we expect that they will differ due to logarithmic terms present in the correlators of the semi-local quantum liquid state [27].
We emphasise here that this behaviour is totally different to that in a generic Landau Fermi liquid, which has η ∼ T −2 (since η = s/4π). For Landau Fermi liquids, hydrodynamics breaks down at low temperatures |ω| ∼ τ −1 η ∼ η −1 ∼ T 2 /µ where the hydrodynamic sound attenuation becomes very large [48][49][50]. It is replaced by a collisionless regime where quantum interactions sustain a 'zero sound' mode with a different dispersion relation than (3.1). This has been confirmed experimentally [51].

Can Landau Fermi liquid theory describe the two-charge black hole?
In this section we explore the possibility that Landau Fermi liquid theory might explain the low energy physics of the collective excitations in the theory holographically dual to (2.2). As explained in the previous section, this can only happen for a very specific limit of Fermi liquid theory, since the gravity dual necessarily implies that η/s = 1/4π.

Establishing the N -dependence of the Fermi liquid parameters
The low temperature heat capacity of the field theory holographically dual to (2.2) is given by where we have used L 3 /G = 2N 2 /π (see e.g. [52]). This should be compared to the Landau Fermi liquid result Let us assume that Luttinger's theorem holds and therefore that the charge density is We assume that the numerical coefficient α is O(N 0 ) and counts the number of charged operators as well as their charge. This would be true, for example, if the charge in the state is carried by an O(1) number of fermionic operators. The validity of Luttinger's theorem has been widely debated in the literature (see e.g. [8,53,54]). In [54] it was shown that, by including magnetic monopole contributions, Friedel oscillations in the charge density are seen at a wavevector satisfying Luttinger's theorem for a (1+1)-dimensional theory. The validity of this result in higher dimensions (see [55] for progress in this direction) would be a useful check of our assumption.
With this assumption, we can deduce the N -dependence of the various Landau Fermi liquid parameters. We can rewrite one of the expressions in (2.5) as at T = 0. Comparing this to (4.3), we deduce that Now comparing (4.1) and (4.2) and using (4.5) we obtain the following expressions for the effective quasiparticle mass and the Fermi velocity We are therefore dealing with an extremely massive Fermi liquid whose Fermi velocity is parametrically small. We will now compare (4.6) with the formula for the effective quasiparticle mass in a relativistic Landau Fermi liquid to deduce the Landau interaction parameter where we have neglected terms that are O(N −2/3 ) and introduced an expansion parameter We next consider the Landau Fermi liquid expression for the speed of hydrodynamic sound where the last equality follows from the fact that we are dealing with a conformal field theory where the expectation value of the trace of the stress-energy tensor vanishes. From (4.10) we deduce that The main lesson of this equation is that With these N -dependences, we can explain the results of various holographic computations, as we will show shortly. To explain the hydrodynamic results of the previous section, one more assumption is needed. We must tune F 2 → −5 so that the quasiparticles are short-lived. This means that a hydrodynamic, rather than a single-particle, description is applicable at low T , as we found for the holographic theory in the previous section.
In this limit, the speed of the zero sound mode c 0 is equal to the hydrodynamic value To compute the speed of zero sound, one needs to solve the integral equation where ν(θ, ϕ) parametrizes the deviation of the Fermi surface from the spherical form. For zero sound, this deviation is ϕ-independent and we can expand in Legendre polynomials, Equation (4.12) then becomes where We now substitute into (4.14) and (4.15) and expand in powers of s. This is an expansion in inverse powers of N . We start by computing the determinant of For (4.14) to have a non-trivial solution, it must be that detA = 0.
At this point we will make an important assumption that which will simplify things considerably. We will only keep terms up to O( s 2 ) in the matrix A ll ′ (4.17). After some algebra, we arrive at In (4.19), we have used (4.18) and F 1 , to write only terms that can possibly be O(1) (or smaller). It is now clear that keeping terms subleading to O( s 2 ) in (4.15) and (4.17) will only modify subleading terms in (4.19).
We now substitute into (4.19) (i.e. impose that c 0 = c 1 ) and demand detA = 0, which gives rise to 4 25 which implies We also need to determine the values of ν l . This computation is technically more involved than calculating the determinant: we now need to compute the eigenvector of A associated with the vanishing eigenvalue. We have done this calculation with F n = 0, n ≤ 3.
The result is

Observables
We start by considering observables that present some obvious contradictions with a Landau Fermi liquid description and we will see how these are resolved by counting powers of N . We will then consider the fate of collective excitations at small temperatures, where the Fermi liquid description predicts a particular low temperature behavior.

Friedel oscillations
It has been argued that the two-point functions of the currents should develop a nonanalytic behavior at k = 2k F and ω→0. As is clear from (4.5), such non-analytic structure cannot be observed from the usual linear fluctuations in supergravity, since the equations of motion for these fluctuations have k ∼ O(N 0 ). To detect non-analytic behavior at , one should include the effects of magnetic monopoles along the lines of [54], [55].

Particle-hole continuum; Landau damping
A related problem is a non-trivial structure of the spectral function associated with the particle-hole continuum. On the ω axis, the upper edge of the continuum is defined, for small ω and k, by ω = υ F k. The physics of this has been recently reviewed in [20], where it was proposed that the continuum is not seen because of a parametrically small υ F . Precisely such a situation is implied by eq. (4.6). See [4] for related work.

Entanglement entropy
It has been argued that the presence of a Fermi surface leads to a logarithmic violation of the area law Such a violation has not been observed (see e.g. [56]). The resolution of this puzzle is simple: according to (4.5), the L 2 log L term in the entanglement entropy must multiply a factor of N 4/3 , while the tree level result [57], [58] only gives the leading term which is

Collective modes with azimuthal dependence
As well as zero sound, a Landau Fermi liquid can support other collective modes due to more complicated fluctuations of the Fermi surface. These have not yet been seen in any holographic theory. Including azimuthal dependence in the deformation of the distribution function (4.13) leads to simple modifications of eqs. (4.14) and (4.15). The analog of the matrix (4.17), whose determinant must vanish, is now where and m ≤ l, l ′ . Repeating the steps that led to (4.21), we obtain where S 1 is the speed of the m = 1 collective mode c (m=1) measured in units of c 0 , with c (m=1) of order one. We have also checked that for higher modes (m > 1) the analog of eq. (4.27) has no solutions with c (m>1) = O(1).

Shear viscosity
In the holographic model that we consider, the shear viscosity satisfies the universal This, at first sight, presents a puzzle, since the Fermi liquid result is and λ η = O(1). The quasiparticle lifetime, τ in (4.29) is related to the scattering probabilities of the quasiparticles W (θ, ϕ) via The probabilities are related to the scattering amplitudes which, in turn, can be related to the Landau parameters, if assumed to be ϕ-independent: where .
, which gives η ∼ N 2 µ 5 /T 2 and η/s ≃ µ 3 /T 3 in sharp contradiction with (4.28). However the value of F 2 in (4.22) is very non-generic! Precisely for this value, the denominator in (4.34) vanishes (as O(ǫ 2 )) and τ receives an extra factor of ǫ 4 . Hence, once (4.22) is taken into account, the leading order result (in powers of N 2 ) for η/s is no longer µ 3 /T 3 , and one needs to consider higher order terms in T /µ which might well give rise to the observed value (4.28).

The applicability of hydrodynamics
In Section 3 we observed that the characteristic length scale where a hydrodynamic description of the sound mode breaks down is given by eq. (3.3). As explained above, this is consistent with Landau Fermi liquid theory with F 2 = −5, in which the divergence of the viscosity at low T -which ensures the breakdown of hydrodynamics -is suppressed.

Higher derivative corrections
In the previous section we have shown that in the limit F 2 → −5, many of the generic properties of a Landau Fermi liquid are no longer realised. It is important that we can recover these generic properties by altering the gravitational theory such that F 2 is no longer fine-tuned in this way. In this section we will discuss a procedure by which one may be able to recover the expected η/s ∼ µ 3 /T 3 behavior of a Landau Fermi liquid, by including specific higher-derivative terms in the gravitational Lagrangian.

Entropy
To compute the entropy of the field theory dual to a gravitational solution, we use the Wald formula where L is the Lagrangian, dΩ = d 3 x √ g 3 is the horizon volume element, E ab = |g 2 |ǫ ab is the antisymmetric tensor with indices taking values t, r, and g 2 = g tt g rr . We will consider a set of three different four-derivative gravitational terms with coefficients c i , i = 1, 2, 3.
After setting the gravitational constant G = 1 and the AdS radius L = 1, the terms in the Lagrangian which contribute to the Wald formula are where β 1 , β 2 , γ 1 and γ 2 are constants. We wish to fix the various constants to satisfy two requirements: that the viscosity is that of a Landau Fermi liquid in the low-temperature limit r H /Q ≪ 1, as discussed in the next subsection, and that the metric and dilaton have the same near-boundary asymptotics as in the two-derivative theory, as discussed in Appendix B.
The Wald formula (5.1) gives the following expression for the entropy: where A is the area of the horizon. For the two-charge black hole (2.2) we obtain To retain the Fermi liquid result s ∼ T , we do not want the entropy to get corrections from these higher-derivative terms. For this purpose we choose which corresponds to the Gauss-Bonnet term. The entropy density is therefore given by (2.5).
Note that in addition to the explicit corrections to the entropy density due to the higher-derivative contributions to the Wald formula, there is also a correction ∆s coming from the O(β) correction to the classical background (2.2). The two-derivative term gives a contribution s = (g xx (r H )) 3/2 /4 to the entropy, and therefore ∆s = 3 g xx (r H )∆g xx (r H )/8. However, we can define a new radial coordinate ρ (see Appendix B) such that, as in [59], g xx (ρ) = −ρ 2 . Then we can choose the constants of integration of the higher-derivative equations of motion for the metric such that the horizon radius ρ H is unchanged, and therefore ∆g xx (ρ H ) = 0 and ∆s = 0.

Viscosity to entropy ratio
The total action we study I = I 0 + I hd + I a + I pot is a sum of the two-derivative terms I 0 , given by (2.1), the higher-derivative Gauss-Bonnet term with dilaton coupling (see (5.2) and (5.5)) the extra dilaton terms and the extra dilaton potential terms Consider the fluctuation h y x (r, t, x) of the metric tensor in the momentum representation The action I = I 0 + I hd , expanded up to second order in fluctuations, has the general form [59][60][61] I= 1 16π where where a prime denotes a derivative with respect to r, and where the momentum k and frequency ω are set to zero. The terms I a and I pot , given in (5.7) and (5.8), do not involve derivatives of h y x and so do not affect the viscosity. We can separately compute the contribution to the viscosity η 0 from the two-derivative terms in the action I 0 , and the contribution to the viscosity η 1 from the higher-derivative terms in the action I hd . The total viscosity is given by η = η 0 + η 1 .
Consider first the higher-derivative term I hd . We can expand I hd up to second order in fluctuations and extract the corresponding coefficients A 1 , B 1 , E 1 , F 1 , before using equations (5.13) and (5.14) to give Now we will determine the contribution to the viscosity from the two-derivative term I 0 . The most general two-derivative term which contributes to the entropy density and to the shear viscosity is of the form where L m is the Lagrangian for matter. We denote the corresponding viscosity as η 0 and the corresponding entropy density as s 0 . If the classical background is corrected by higher-derivative terms, then the viscosity receives a correction ∆η and the entropy density receives a correction ∆s. In addition to these, the higher-derivative terms, evaluated on the classical background, contribute η 1 and s 1 to the viscosity and the entropy density respectively, as previously explained.
The question is whether we need to compute ∆η and ∆s to determine η/s. To leading order in β, the effect of these terms is But it is known [28][29][30][31] that η/s, when computed from the two-derivative action (5.16), does not depend on the background. In fact since η = g 3/2 xx /(16π) (one can derive this in the formalism of [59], that is using equation (5.13) for the action (5.16) with arbitrary background metric), and s = g 3/2 xx /4, one always has s 0 ∆η −η 0 ∆s = 0 for any correction to the background. Therefore as long as we are interested in η/s, we do not have to compute ∆η and ∆s [59,61]. In fact, as we noted in the previous subsection, we have ∆s = 0 and ∆η = 0 satisfied separately, simply because of the ∆g xx (r H ) = 0 equation.
We choose The parameters β 1,2 and γ 1,2 then satisfy which, as shown in Appendix B, is crucial to obtain the correct asymptotic behavior of g tt and the dilaton. Therefore in the limit r H /Q ≪ 1, we obtain This behavior agrees with the low-temperature result of Landau Fermi liquid theory, provided that the black hole solution has T ∼ r H and µ ∼ Q when r H /Q ≪ 1, as in the two-derivative theory. This is a non-trivial requirement which needs fine tuning of the coefficients c 1 , d 1,2,3 and w 1,2,3 . In Appendix B, we describe a numerical procedure by which one can tune these coefficients and change the leading functional dependence of T and µ on r H and Q. It is clear from this analysis that it is not easy to produce a gravitational dual of these generic Landau Fermi liquid properties. This further underlines how different holographic metals are, compared to their conventional counterparts.

Discussion
In Section 4 we described a scenario where a Landau Fermi liquid with specific parameters was responsible for the properties of the two-charge black hole. The two crucial assumptions were the existence of a single or, at most, O(1) Fermi surfaces and the applicability of Luttinger's theorem. There is some tension with the observation of singularities in the two-point function of the gauge-invariant fermionic operators [33], which would signify that k F ∼ µ (as opposed to (4.5)). If one takes the singularities at k F ∼ µ seriously, two logical possibilities present themselves: i) the Fermi surface is formed by the N 2 species of gauginos, and Luttinger's theorem is almost satisfied [33], or ii) Luttinger's theorem is strongly violated in holographic models where charge is sourced by black hole horizons due to fractionalization in the dual field theory [3,53].
In this paper we raise a third possibility, which explains all other observable data (such as the absence of Friedel oscillations, the particle-hole continuum and the logarithmic violation of entanglement entropy, as well as the behavior of collective excitations). To make this picture more precise, it would be interesting to identify possible effects responsible for (4.5), and also to check more thoroughly whether one can depart from the F 2 = −5 limit and recover a generic Landau Fermi liquid. In this case, there are precise predictions for the behavior of the sound mode outside of the hydrodynamic regime. 2 We took the first steps in this direction in Section 5 by describing how one could, in principle, construct a higher derivative gravitational theory which reproduces the viscosity of a Landau Fermi liquid.
It would be interesting to check our assumption of Luttinger's theorem by a calculation along the lines of [54] (work in this direction was undertaken in [55]). We do not have anything to say at the moment regarding the microscopic origin of the parametrically large k F . It is known that the naive large N counting breaks down in vector models at finite chemical potential [63,64], and the recent works [65][66][67][68][69] have suggested other ways to construct controlled microscopic descriptions of related systems. It would be interesting to see whether the power law in (4.5) can be naturally reproduced. Finally, let us comment on the nature of the F 2 = −5 limit. It is known that for F l < −2l − 1 the Landau Fermi 2 We remind the reader that hydrodynamics fixes the speed of the sound mode c 1 to be equal to the thermodynamic value, c 2 1 = ∂P/∂ε, which is different from the speed of zero sound in a generic Fermi liquid. liquid becomes unstable. If our scenario is correct, it would be interesting to identify this instability from a bulk perspective (see [70] for related work). 2) which are infalling at the horizon r = r H and whose leading term vanishes at the boundary r → ∞. It is convenient to go to Fourier space and perturb the background fields as follows

Acknowledgements
The equations of motion for these perturbations can be obtained from the action (2.1) but they are very lengthy and we will not present them here.
The sound mode in which we are most interested is associated with longitudinal density oscillations in the field theory and thus is realised in the gravitational dual as a quasinormal mode of h tt (which is dual to the component of the field theory energy-momentum tensor T tt ). At linear order, h tt couples to h zz , h xx +h yy , h rr , h tr , h zr , h zt , a t , a z , a r , and ϕ. Within this set of fields there are, in fact, only three independent degrees freedom which are invariant under infinitesimal diffeomorphisms and U(1) gauge transformations. We choose the basis and write the equations of motion in the form where the coefficients C i will be given shortly. Near the spacetime boundary r → 0, the solutions of these equations are of the form where the ratios Z (0) /Z (1) are fixed by imposing infalling boundary conditions at the horizon. Quasinormal modes are solutions to these equations of motion which are infalling at the horizon and for which Z (0) /Z (1) vanishes for all three fields. These modes exist only for a discrete set of quasinormal frequencies ω (k, T, µ). To determine these quasinormal frequencies numerically we follow the 'determinant method' described in [72] with one difference. Since the source term in the near-boundary expansion of ϕ is the logarithmic term [73], we extract it by fitting the near-boundary expansion of the numerical solution to the form in (A.4) and construct the determinant with this. We note that this method does not give any information about the residue of the pole, only its location in the complex frequency plane. It would be interesting to determine the corresponding residue but we will not address this issue here.

B.1. Adding Gibbons-Hawking boundary terms
The total action is I = I 0 + I hd + I a + I pot , (B.1) and must be supplemented by boundary terms which compensate for the variations of derivatives of the fields on the boundary of AdS, r = ∞. For the two-derivative action I 0 , this is the Gibbons-Hawking term where h ab is the metric on the boundary, K is the trace of the extrinsic curvature tensor K b a = ∇ a n b , and the unit vector normal to the boundary is given by For the Gauss-Bonnet term I hd , the necessary boundary term is (see e.g. [74,75]) Consider first the two-derivative action I 0 . Evaluating it on the ansatz with dilaton φ(r) and gauge potential A t (r), we obtain Despite the presence of the second derivatives of the fields h ′′ and a ′′ in the action (B.6), we have a well-defined variational problem due to the boundary term (B.2). Now, evaluating the Gauss-Bonnet term I hd on the ansatz (B.5), we obtain Again, the action depends on the second derivatives of the fields. Thanks to the boundary terms (B.4), we have a well-defined variational problem.

B.2. Scaling charge
Following [32], we note that the total action I tot possesses a scaling symmetry The corresponding conserved charge is given by where we have used the equation of motion for the gauge field and where C 0 is a constant. We have also denoted β 1 = β and β 2 = χβ. For our choice (5.18), χ = 1/2. We choose the integration constant, which is related to the field theory charge density, to be which is consistent with the two-derivative solution.
To Our strategy will be to evaluate numerically this equation at O(β) and solve for A t (∞) at this order. This allows us to determine how the chemical potential depends upon r H and Q at O(β).

B.3. Temperature and chemical potential
In the remainder of this appendix we are going to describe how to numerically construct the black hole solution to O(β), and then explain how, by fine tuning the parameters c 1 , d 1,2,3 and w 1,2,3 in the extra dilaton potential terms I pot , one can change the functional dependence of T and µ on r H and Q. It is convenient to choose the ansatz (as in [76]) where ρ is a new radial coordinate. In this ansatz the total Lagrangian, including the Gibbons-Hawking terms (B.2) and (B.4), is It is convenient to work with the dimensionless parameter θ = r H /Q.
First of all, we solve for the gauge field with C 0 given in (B.11). Then we solve the equations of motion for δ, B and φ perturbatively up to O(β). Denoting the background (two-derivative) solution as δ 0 , B 0 and φ 0 , the equations for the O(β) corrections to the background, which we denote as δ 1 , B 1 and Expanding near the boundary, ρ ≫ 1, we find φ 1 = 1 ρ 2 (C f 1 + C f 2 log ρ + C f p log 2 ρ) + . . . , (B.24) After we have solved for B 1 and φ 1 , we can use this solution to integrate (B.20) to find δ 1 . We fix the constant of integration in such a way that g tt /ρ 2 = Be −2δ /ρ 2 asymptotes to one near the boundary, so that the speed of light is equal to one in the dual field theory. Therefore the near-boundary expansion of g tt = Be −2δ is given by g tt ρ 2 = 1 + −1 − 2θ 2 − θ 4 ρ 4 + −1917 + 12b 1 + 6b 2 + 9C b2 − 2 √ 6C f 1 + 648θ 2 + 324θ 4 β 9ρ 4 . (B.28) We want to extract δg (4) tt , the coefficient of the sub-leading 1/ρ 4 term at O (β), as this will allow us to determine the chemical potential A t (∞) from equation (B.12). 3 At O β 0 , this term, which controls the energy density of the dual field theory, is proportional to Q 4 at low T . It follows from (B.28) that, to find the O(β) correction to it, we need to find the Our numerical procedure is as follows. We solve equations (B.18) and (B.19) for φ 1 and B 1 , with the horizon boundary conditions given by We then fix numerically the value of c such that C f 2 = C b3 = 0 (we have verified numerically that both of these coefficients go to zero at the same time for the particular value of c). This ensures that φ 1 and B 1 behave, near the boundary, in the same way as φ 0 and B 0 respectively. For this value of c, we then extract numerically the coefficients C f 1 and C b2 in the near-boundary expansions.
After fixing w 1 , w 2 , w 3 , we find numerically that, for generic c 1 , d 1,2,3 , one finds (for θ ≪ 1), δg By varying c 1 , d 1 , d 2 , d 3 , the four coefficients C 1,2,3,4 change. It is possible to make these coefficients numerically small, and we expect that by suitably fine tuning these coefficients, it is possible to produce a black hole solution with C 1,2,3,4 = 0. However, this is a very expensive procedure to implement numerically and we have not been able to determine the required values of c 1 and d 1,2,3 . Supposing that this solution exists, it would have T ∼ r H and δg (4) tt ∼ Q 4 to O (β) in the small T /µ limit. Equation (B.12) then implies that µ ∼ Q in this limit, as required.