A renormalisation group equation for transport co-efficients in (2+1)-dimensions derived from the AdS/CMT correspondence

Within the framework of the AdS/CMT correspondence asymptotically anti-de Sitter black holes in four space-time dimensions can be used to analyse transport properties in two space dimensions. A non-linear renormalisation group equation for the conductivity in two dimensions is derived in this model and, as an example of its application, both the Ohmic and Hall DC and AC conductivities are studied in the presence of a magnetic field, using a bulk dyonic solution of the Einstein-Maxwell equations in asymptotically AdS$_4$ space-time. The ${\cal Q}$-factor of the cyclotron resonance is shown to decrease as the temperature is increased and increase as the charge density is increased in a fixed magnetic field. Likewise the dissipative Ohmic conductivity at resonance increases as the temperature is decreased and as the charge density is increased. The analysis also involves a discussion of the piezoelectric effect in the context of the AdS/CMT framework.


Introduction
The holographic AdS/CFT or, more generally AdS/CMT, approach to condensed matter systems at strong coupling has been an active area of investigation for a number of years now, for a review see [1]. These ideas are still speculative, the strongest evidence for the validity of the AdS/CFT program comes from supersymmetric SU (N ) gauge theories at large N in higher dimensions, but a naïve application of similar ideas to low dimensional non-relativistic problems, while not as well supported mathematically, often gives qualitative results that are at least intriguing and merit further investigation.
In [2] and [3] the properties of transport functions in two spatial dimensions at finite temperature in the presence of a magnetic field were studied using this method. The general idea is to take a charged black-hole in a 4-dimensional asymptotically AdS space-time, with a conformal field theory on the (2 + 1)-dimensional asymptotic boundary. In the framework of the AdS/CFT formulation a classical solution of Maxwell-Einstein gravity in the bulk corresponds to a strongly coupled quantum system on the boundary. The black-hole event horizon acts as a one-way membrane that gives rise to dissipative phenomena on the boundary. The Hawking temperature of the black-hole provides a thermal background and its electric charge generates a chemical potential. A static magnetic field normal to the two space directions on the boundary can be incorporated by giving the black-hole a magnetic charge, promoting it to the status of a dyon in (3 + 1)-dimensions. This setup gives a promising framework in which to study the quantum Hall effect and there has been a number of papers examining this possibility [4]- [24].
In [2,3] a dyonic black hole with a planar event horizon was used to study transport properties, thermal and electrical conductivity and the thermoelectric co-efficient were investigated. The basic idea is that information on transport co-efficients can be obtained by studying the second variation of the action on-shell, when the first variation vanishes. In the present work a first order differential equation that the conductivity must satisfy, as a function of the bulk radial co-ordinate, is derived in this setting, giving a non-linear renormalisation group equation. The equation is singular at the event horizon in that the co-efficient of the derivative term vanishes there, reducing it to an algebraic equation, and the RG equation then dictates its own boundary conditions in the infra-red (this is essentially the attractor mechanism). In general the equation must be solved numerically to obtain both the Ohmic and the Hall AC and DC conductivities as functions of the radial coordinate, but a particular truncation of the equation has a simple solution, which is an approximation studied in [3]. In this approximation the conductivity has a pole in the complex frequency plane and the numerical solutions indicate that this pole persists in the full solution albeit with renormalised residue and position in the complex frequency plane. The infra-red and ultra-violet cases are studied in the full numerical solution, for various values of the electric and magnetic charges, and real and imaginary parts of both the Ohmic and the Hall conductivities are extracted at finite frequency. The Q-factor of the cyclotron resonance is studied as a function of the electric and magnetic charges and it is shown to decrease as the temperature increases and increase as the ratio of the charge density to the magnetic field increases.
The layout of the paper is as follows: in §2 we briefly summarise the Gubser-Klebanov-Polyakov-Witten (GKPW) hypothesis and how it can be used to determine response functions on the boundary of the bulk theory; in §3 the formalism is applied to a dyonic black brane in 4-dimensional asymptotically AdS Einstein-Maxwell theory and the non-linear RG equation for the conductivity on the 2 + 1-dimensional boundary is derived; solutions of the RG equation are studied, both numerically and in certain analytic approximations, in §4 and presented in a number of figures; finally the results are summarised in §5. Two appendices contain a discussion of the piezoelectric effect and technical details about Riccati equations.

Response functions and the GKPW proposal
The GKPW proposal [25,26] relates a conformal field theory on the boundary of an asymptotically AdS space-time to a gravitational theory in the bulk. More specifically it states that the generating functional for the quantum field theory on the boundary, with sources h i for operators O i , is equal to the partition function in the bulk when bulk fields ϕ i take the boundary value h i . In the Euclidean version, with where ϕ i represents all the bulk fields, including the metric, and ∂M is the boundary, GKPW proposed that so, when external sources vanish, When quantum corrections in the bulk are negligible, as will be assumed here, the effective action equals the classical action (3) is equivalent to the classical equations of motion with fixed boundary conditions. Assuming the boundary conditions pick out a unique classical solution, φ h , the bulk partition function (1) can be approximated by (if the boundary conditions do not pick out a unique solution, we would need to sum over all solutions φ h compatible with the boundary conditions).
Although the variation of the action is zero for a classical solution, the variation of the Lagrangian L(φ i , ∂ µ φ i ) need not be -in general it will be a surface term. For a classical solution θ µ (φ, δφ) can be non-zero in the bulk but when φ i | ∂M = h i is fixed and δφ i | ∂M = 0 it must vanishes on the boundary. However if the boundary conditions are varied then θ can, and in general will, be non-zero on the boundary. Define where d 3 Σ µ is the volume element on the boundary. Then This can now be related to the boundary CFT response functions. These are defined by When quantum corrections in the bulk are negligible we can now use (4) together with the GKPW hypothesis to equate 4 This is how response functions can be calculated in the AdS/CFT framework, In the next section this formalism will be used to analyze condensed matter systems in two spatial dimensions in the presence of a background magnetic field.

The RG equation for the conductivity
The starting point is that of [2,3], a bulk theory which is 4-dimensional Einstein-Maxwell with a negative cosmological constant Λ = − 3 L 2 , where κ 2 = 8πG and c = 1, and we shall also use units with e 2 h = 1. Using co-ordinates t, r, x and y let z = L r be a dimensionless radial coordinate, z → 0 being asymptotic infinity. There is a Reissner-Nördstrom AdS black-hole solution of (7) with a flat event horizon and constant radial electric and magnetic fields (the factor of L ensures thatq andm have the same dimensions, mass 1/2 × length −3/2 ). The line element is where M is the black-hole mass. The event horizon could be either an infinite plane or a flat torus with a finite area (we shall use the latter to keep global quantities finite with non-zero densities) and it lies at the largest root, z h , of The event horizon has area A = 1 z 2 h T orus dxdy and the total charge is The charge density at a general value of z ≤ z h is (the minus sign is because the outward normal is defined as the direction of decreasing z). It will be convenient to re-scale t, z, x and y by a factor of z h to in terms of which with 0 ≤ u ≤ 1 outside the horizon and q 2 = κ 2q2 L 2 z 4 h /2 and m 2 = κ 2 L 2m2 z 4 h /2 dimensionless. The Hawking temperature is so q 2 + m 2 is constrained to be less than or equal to 3.
To study transport properties we perturb the solution by introducing oscillating transverse electric and magnetic fields which depend on u but are independent of x and y. These can be derived from a vector potential δA α (u, t) = e −iωt δ A α (u) (12) with α = x, y, where txuy = xy = 1, δȦ β = ∂ t (δA β ) and δA β = ∂ u (δA β ). For static fields we can use The analysis is initially identical to that of [2,3]. We require not only that δE α and δB α satisfy Maxwell's equations in the background metric but also take into account the back reaction on the metric to first order in the variation, i.e. δA α (u, t) must satisfy the equation of motion. The transverse electromagnetic field then back reacts on the metric producing metric variations encoded into two functions of t and u defined by This variation of the metric components leaves the orthonormal 1-forms unchanged while the transverse orthonormal 1-forms do change where we ignore terms of O(δG 2 α ). The vierbein matrix and its inverse for the new metric can be chosen as This change then feeds in to the electric field seen by an inertial observer, where 0 and i = 1, 2 are orthonormal indices. In an orthonormal basis the variation in the transverse electric field is while the transverse magnetic field merely acquires a multiplicative factor, The −m α β δG α contributions to δE i are a kinematic effect. The metric is not invariant under Galilean transformations, rather t → t, x → x − δv x t, y → y−δv y t, with δv α small, changes the metric components, δg tα = − δvα u 2 = 1 u 2 δG α , with δg tt ∼ O(δv 2 ), so δE α −m αβ δG β = δE α +m αβ δv β and an inertial observer sees a contribution to the electric field arising from the combination v α with the radial magnetic field. Following [3] define then the perturbation δE α will generate a current δJ α = σ αβ δE β with σ αβ the conductivity tensor. Assuming the transverse space is isotropic this will satisfy σ xy = −σ yx , σ xx = σ yy .
In complex co-ordinates x ± iy Note that δE − is not the complex conjugate of δE + in general, since δE α itself is complex for AC fields. The authors in [2,3] further define and show that Einstein's equations can be used to eliminate δG α from Maxwell's equations resulting in where ω = ωL. These can be re-arranged to obtain From these we can derive an RG equation for the electrical conductivity, describing how it changes as u is varied.

8
The two complex functions σ ± have complete information about both the real (dissipative) and imaginary (refractive) parts of σ xx as well as the real and imaginary parts of σ xy . In the notation of the previous section, for this solution and this specific form of variation, (with the variations specified in (12) and (13) the Einstein scalar itself gives no contribution to the variation at first order). Choose space-time to be bounded by the event-horizon, u = 1, an outer cylinder at some value of u < 1 together with two constant t space-like hypersurfaces Σ t ± at some future time t + and past time t − . If we choose t + − t − to be an integer multiple of 2π ω the contributions from Σ t ± cancel and where the contribution from u = 1 has been discarded since F uα = u 4 f (u) L 2 F uα vanishes there. With (12) and (13) The current is The contribution of δG α to the current is a manifestation of the piezoelectric effect (see appendix A) but the conductivity is defined as the response to a variation of the electric field, not the metric -while theq δG α term contributes to the current it does not contribute to the conductivity. In linear response the conductivity (14) is therefore 1 as in [3]. Equations (17) and (20) can now be used to derive an RG equation for σ ± (u). Differentiating (20) with respect to u and using (17) gives This can be re-arranged to write the RG equation for the conductivity as a Riccati equation (the relevance of Riccati equations to holographic 2-point functions was observed in [28,29]) and matrix version of coupled Riccati RG equations was presented in [30]- [33].

Solutions of the conductivity RG equation
To study solutions of (21) first note that it is singular at the event horizon, since f (u) → 0 as u → 1 and, if ω = 0, we must necessarily use boundary conditions σ 2 ± = −1 at u = 1. Next, as observed in [3], which allows us to extract At the event horizon σ 2 ± = −1 is independent of m so σ xy = 0 and σ ± = ±iσ xx . Since Re(σ xx ) is necessarily positive σ xx = 1 at the event horizon and we must use boundary conditions These boundary conditions correspond to matter falling into the event horizon and, as emphasised in [1], this is the source of dissipation in holography. In particular these boundary conditions imply that it is not the case that the solutions of (21) are related by Since Re(σ xx ) = 1 2 Im(σ + − σ − ) changing the sign of ω corresponds to sending Re(σ xx ) → −Re(σ xx ), as one expects from time reversal. The sign of ω should always be chosen so that the dissipative Ohmic conductivity, Re(σ xx ), is non-negative.
Equation (21) is also invariant under electromagnetic duality in the bulk, a property of the conductivity noted in [2]. Of course the real and imaginary parts of (22) and (23) are not independent, they are related by the Kramers-Kronig relations and similarly for σ xy ( ω). Some consequences of equation (21) in certain limits are immediate:

AC conductivity
At the event horizon, as u → 1, f → 0 and, if ω = 0, This is the attractor mechanism.

Large ω or small u
When ω is large or u is small the first term on the right-hand side of (21) can be ignored and As ω → ∞ (or as u → 1) the solution becomes a constant σ + = i and On the other hand, as u → 0 for large but finite ω, f → 1 and This is independent of m and hence, from equations (22) and (23), where σ xx (0, ω) must be determined by numerically integrating (21) from u = 1 to u = 0 with the boundary condition σ ± = ±i at u = 1. The classical Drude form of the conductivity, allowing for the cyclotron resonance, would be with cyclotron frequency ω 0 and relaxation time τ .

Cyclotron resonance
The presence of a resonance is straightforward to show when the second term on the right hand side of (21) is small relative to the first and can be ignored. We then have With the boundary condition σ + | u=1 = i the solution is In the complex σ + -plane, plotting Im(σ + ) against Re(σ + ) using ω as a parameter, this is a perfect circle lying entirely in the upper-half complex plane and tangent to the real axis. In terms of the filling factor ν = q m the radius is 1+ν 2 2 and the centre is at Re(σ + ) = −ν, Im(σ + ) = 1+ν 2 2 . The approximate form (31) for σ + was found in [2] for small q 2 and m 2 , both of the order of ω, and indeed it works best in this range.
When u < 1 there is a pole in the complex ω-plane at and near ω * equation (31) has the form of a resonance with resonance frequency, width and amplitude at resonance In this approximation the corresponding Q-factor is one-half the filling factor, independent of u, and the residue is The resonance disappears at u = 1 and is strongest at u = 0 where When σ + (m) and σ + (−m) are combined to construct the Ohmic and Hall conductivities in (22) and (23) one might naïvely be led, at u = 0, to but these are not the same as the maximum values of Re(σ xx ) and Im(σ xy ) . Using (31), and setting u = 0, gives In these expressions the maximum of Re(σ xx ) occurs when where Since ω xx 0 is only real for m 2 < 3 q 2 , we would expect that m must be restricted to this range and σ xx M ax = 1 when the bound is saturated and ω xx 0 → ∞. However although ω xx 0 diverges when m 2 = 3q 2 , and becomes imaginary for m 2 > 3q 2 , σ xx M ax is well defined for all 0 < m 2 < 3 − q 2 and indeed the approximation (41) is remarkably close to the numerical result shown in Fig. 7(a) when q = 0.1 for the whole of the allowed range of m.
The full numerical solutions presented below indicate that near the cyclotron frequency, and for a significant band on either side of it, Im σ xy ( ω) behaves very like the dissipative part of the Ohmic conductivity while the real part displays the properties of a refractive conductivity (σ xy changes sign when either m or q change sign). The maximum of Im σ xy ( ω) is at where σ xy M ax = Im σ xy ( ω xy 0 ) is These expressions can be more succinctly written in terms of the filling It is not immediately obvious what the widths of the peaks are in (38) and (39). We shall define the widths by assuming the resonance form (32), e.g. for σ xx ( ω) M ax Γ xx , for which the width would be given by For any σ xx ( ω) with a maximum of Re σ xx ( ω) at ω 0 , where Re σ xx ( ω xx 0 ) = σ xx M ax , equation (44) will be used as the definition of Γ xx . Similarly 14 With these definitions the widths arising from the analytic approximations (38) and (39) are not very illuminating but are given here for completeness, These expressions are compared to the full numerical solutions graphically below.

m = 0
When m = 0 equation (21) can also be solved analytically in the approximation that ω is small. It is a Riccati equation which can easily be recast as a second order, linear, homogeneous equation. Details are given in appendix B and in this section we examine the case m = 0. Define the function Then, from (57) with m = 0, equation (21) can be re-expressed as which, for ω = 0, reduces to with corrections of O( ω 2 ). Since f (1) = 0 and f (1) = −3 + q 2 a boundary condition on X(u) at u = 1 is Up to an overall normalisation equation (46) determines its own boundary conditions because u = 1 is a singular point. The solution is immediate with c constant and we arrive at Hence, from (22) and (23), as u → 0 the refractive Ohmic conductivity has a pole at ω = 0, with zero width and the Q-factor diverges. This is a better approximation than (31), with m = 0 and u = 0, because (31) is only valid for q 2 ∼ O( ω) when ω 1.

DC conductivity
Away from the event horizon f = 0 and, when ω = 0, the Ohmic conductivity vanishes and the Hall conductivity is − q m , as found in [3].

u → 1
We can go all the way to u = 1 for the DC conductivity with the solution where H(u) = 0, u < 1; 1, u = 1 is the step function. Thus again σ xy = 0 and σ xx = 1, again the same value that one gets from infalling boundary conditions at the horizon for a 2-dimensional system [1]. As observed earlier for any ω the conductivity at u = 1 is always σ xy = 0, σ xx = 1, an attractive fixed point in the infra-red. A universal critical point at σ xx = 1 was predicted in [34,35].

u → 0
The u → 0 and ω → 0 limits do not commute and we cannot trust (48) at u = 0. Instead consider the conductivity (29) in the limit σ 0 → ∞, τ → ∞, with the ratio r 0 L = σ 0 τ finite, The Kramers-Kronig relations require that a pole in Im σ xx (ω) is associated with a δ-function singularity in Re σ xx (ω) , (49) ω 0 = 0 when m = 0 and a δ-function in the DC conductivity is a signal of a superconductor. Equation (47) gives

Numerical solutions
In this section we shall present some conductivity profiles obtained by integrating (21) numerically. Some conductivities obtained from numerical integration of asymptotically AdS solutions of a bulk theory consisting of Einstein-Maxwell with a negative cosmological constant have been calculated before, [29,36]. A more thorough investigation is made here as an example of the application of equation (21). In Fig. 1 the conductivities are plotted for q = m = 1 as functions of u for various values of ω . The approach to the analytic form (48) as ω approaches zero is evident. The negative of the imaginary part of the Hall conductivity (purple curve) is plotted to emphasize the similarity (and difference) with the real part of the Ohmic conductivity (red curve) -the same flip in sign would be achieved by changing the sign of either q or m.
A general numerical analysis is rather involved as we are studying four functions, Re σ xx (u) , Im σ xx (u) , Re σ xy (u) and Im σ xy (u) , in a 3dimensional parameter space ( ω, q, m). Since u → 1 is fixed by regularity conditions we shall simplify the analysis by focusing on the UV limit, u → 0, and consider Re σ xx ( ω) , Im σ xx ( ω) , Re σ xy ( ω) and Im σ xy ( ω) in a 2parameter space (q, m).
The u → 0 conductivity in zero magnetic field was studied in the context of a holographic superconductor in [36]. In the presence of a magnetic field a pole in σ + ( ω) in the complex ω plane was found in [3], associated with a cyclotron resonance. Near such a pole with resonance frequency ω 0 , width Γ and resonance value σ 0 the conductivity is of the form (32), Normally Re σ( ω) is the dissipative conductivity, with a peak at ω 0 , while Im σ( ω) is the refractive conductivity, with a zero at ω 0 when σ 0 is real. That interpretation is not valid for σ + as its real and imaginary parts contain information about both the Ohmic and the Hall conductivities and these Figure 1: The conductivities as a function of u for q = m = 1 and various values of ω. The real part of the Ohmic conductivity is red, the imaginary part is green. The real part of the Hall conductivity is blue and the negative of the imaginary part is purple. The approach to the analytic form (27) when ω is large is evident in (a) and to the form (48) as ω → 0 is evident in (h). must be extracted separately. For the Ohmic conductivity, assuming the resonance form (50) with σ 0 real and positive, Im σ xx ( ω) vanishes at the same frequency as that for which Re σ xx ( ω) is maximised. For the Hall conductivity it is the opposite way round, the peak is in Im σ xy ( ω) and the zero is in Re σ xy ( ω) . As noted above when q and m are both positive the Hall maximum peak is in −Im σ xy ( ω) rather than Im σ xy ( ω) , but the sign could be changed by changing the relative sign of q and m.
The three parameters σ 0 , ω 0 and Γ will in general depend on q and m and the numerical analysis below shows that, at fixed q, they differ significantly as functions of m for the Ohmic and Hall conductivities when q < 1 but are remarkably similar when q > 1.
The numerical results are displayed in a number of figures and we first summarise the figures before describing them in more detail.

Summary of figures:
• Fig. 2: conductivities for q = 1, m = 0.5. • Figs. 6 and 7: cyclotron frequencies and conductivity maxima at fixed q, as a function of m.
• Fig. 8: the inverse resonance widths at fixed q, as a function of m.
• Fig. 9: the Q factor for the cyclotron resonances at fixed q as a function of m.
• Fig. 10 and Fig. 11: further details of the resonances for a small value of q, (q = 0.1).
• Fig. 12: further details of the resonances for a large value of q, (q = 1.5).
• Fig. 14: some resonance properties as a function of temperature.
• Fig. 15: zero temperature conductivities and Q-factors as functions of the filling factor ν.
These figures are now described in more detail.

Conductivities for
The asymptotic, u = 0, conductivities are plotted in Fig. 2 for q = 1 and m = 0.5 as a function of ω (similar plots appeared in [29]). The resonance found numerically in [3] is clearly visible. For these values of q and m the resonances for σ xx and σ xy are very similar, and the plots only differ significantly near ω = 0 (though we know from (27) that they also differ for large ω, but this is not clearly visible on the scale in the plots).

Cyclotron frequencies and conductivity maxima at fixed q as a function of m.
The resonance frequencies and resonance peaks of the conductivities are shown in Figs. 6 and 7, as a function of m at q = 0.1, 0.5, 1 and 1.5. For a sharp resonance it can be difficult to obtain an accurate value for the maximum of the conductivity numerically, as to do so requires a very fine discretisation of the frequency. In that case we use the fit (50) for a well-defined resonance, and assume that, for the Ohmic conductivity, the peak in Re σ xx ( ω) occurs at the same frequency, ω 0 , as that at which Im σ xx ( ω) −1 vanishes (and vice versa for the Hall conductivity) while at the same time σ 0 = Re (σ xx ( ω 0 )) −1 −1 . For a sharp peak it is much easier numerically to determine the zero of Im σ xx ( ω) −1 than it is to determine the maximum of Re σ xx ( ω) and for this reason when the Qfactor is greater than 10 (i.e. q = 1 and 1.5 in the plots) the zero of Im σ xx ( ω) is used to define ω 0 and σ xx 0 = Re σ xx ( ω 0 ) −1 then gives Re(σ xx ) M ax . If the Q-factor is less than 10 (50) is not a good fit, but the peak is broad so it easier to ascertain the height and resonance frequency just by examining Re σ xx ( ω) directly and determining where its maximum lies, and this is the method used for analysing for q = 0.1 and 0.5. It is somewhat surprising that the analytic approximations (41) is such a good fit for the Ohmic conductivity in Fig. 7(a) (q = 0.1) for the full range of m, as the corresponding fit to the resonance frequency (40) (Fig. 6(a)) diverges when m = √ 3 q = 0.173 (red dotted line) and is imaginary for m >

4.3.4
The resonance widths at fixed q as a function of m.
In Fig. 8 the inverse widths Γ −1 of the resonances at fixed q as a function of m are shown for a selection of values for q. The widths are extracted from the numerical solutions by determining the resonance frequencies and peak values σ xx 0 = Re( σ xx ( ω 0 ) and σ xy 0 = Im σ xy ( ω 0 ) as described in 4.3.3 above and assuming the forms The widths Γ xx and Γ xy are then obtained by differentiating the numerical solutions. Again the widths for σ xx and σ xy are indistinguishable on this scale for q = 1 and q = 1.5.

4.3.5
The Q factor at fixed q as a function of m.
The Q-factor for the resonances is shown as a function of m at fixed q in Fig. 9. For small q the system is heavily damped for filling factor ν ≈ 1, as anticipated in equation (34), as q increases the damping is reduced but is greatest when ν is just below 1 (once q is greater than 3 2 = 1.225, ν can no longer attain the value 1).

4.3.6
Details of the resonances for q = 0.1.
In Fig. 10 more detailed characteristics of the resonance are shown for q = 0.1 and low m, 0 < m < 0.2. When σ 0 in (50) is real the peak in Re σ( ω)) coincides with a zero in Im σ( ω) and this relation is obeyed well for q ≥ 0.5 but it is does not apply in the approximation (38) and (39) and numerically it is not true for q = 0.1. The notion of a resonance frequency is ambiguous for low Q and, as described above, the graphs in Fig.10 use the maximum of Re σ( ω) to define ω 0 rather than the zero of Im σ( ω) .
An instructive way of viewing the conductivities is to plot the real parts of the conductivities against the imaginary parts. At fixed q this can be done using a parametric plot with ν = q m as a parameter. To keep the temperature positive m is restricted to the range 0 ≤ |m| < 3 − q 2 , so there is a lower bound on |ν| when q is fixed, namely |q| √ 3−q 2 ≤ |ν|. The range of |ν| is therefore greater for smaller q and this kind of plot is more useful . In (d) the the numerical data are shown as dots and the linear fit 0.41qm is shown in grey. A more detailed version of (a) at small m is shown in Fig. 10(a) and a more detailed version of (d) for small m is shown in Fig. 12(a). A more detailed version of (a) at small m is shown in Fig. 10(b) and a more detailed version of (d) for small m is shown in Fig. 12(b).

29
for smaller q values than for larger ones. 2 In Fig. 11 the real parts of the Ohmic and Hall conductivities are plotted against their respective imaginary parts for q = 0.1 (0.058 ≤ ν < ∞) for six different frequencies. The Ohmic conductivity is red and the Hall conductivity is blue: for σ xx the vertical axis is Re(σ xx ) and the horizontal axis is Im(σ xx ), for σ xy the vertical axis is −Im(σ xy ) and the horizontal axis is Re(σ xy ). The dotted curves are numerical data, the red and blue solid lines are the approximations (38) and (39) respectively. The Ohmic conductivities are very close to being circular arcs, particularly for the larger values of ω, 0.64 and 1.12. The numerical data are well approximated by circles in the complex Ohmic conductivity plane which is a generalisation of the the Drude form with ω 0 monotonic in ν. Similar plots for the Hall conductivity from experimental data on Ga/As heterojunctions at THz frequencies, were given in [37].
for both conductivities, at least for m in the lower part of its allowed range. This is the same functional from as (33) but with different co-efficients. Indeed the four numbers in (52) can be obtained from only three parameters in the analytic expression This suggests that perhaps it may be possible to justify some similar analytic approximation when |q| is close to its maximum value of √ 3, but we have not found an analytic argument for this. ν=0.058 (f) ω=1.12 Figure 11: The real parts of the conductivities plotted against the imaginary parts for q = 0.1 at various frequencies (Ohmic conductivity in red, Hall conductivity in blue). For the Ohmic conductivity the vertical axis is Re σ xx ( ω) and the horizontal axis is Im σ xx ( ω) , for the Hall conductivity the vertical axis is −Im σ xy ( ω) and the horizontal axis is Re σ xy ( ω) . The dots are numerical data, red and blue solid lines are the approximation (38) and (39). For the higher values of ω (0.64 and 1.12) the purple curves are fits to the Ohmic conductivity using a perfect circle. Increasing ν is clockwise in the figures. (c) The inverse widths Γ −1 for the Ohmic (red) and Hall (blue) conductivities at q=1.5 for 0<m<0.2. The fit Γ −1 = 435 m 2 is shown in grey.
(d) The Q-factor for the Ohmic (red) and Hall (blue) conductivities at q=1.5 for 0<m<0.2. The hyperbolic fit Q=90 q m is shown in grey.

4.3.8
The residues at q = 0.5, q = 1 and q = 1.5 as functions of m Inspection of Figs. 7 and 8 shows a possible correlation between σ 0 and Γ −1 , they are strikingly similar. In Fig. 13 the product Γ xx Re(σ xx 0 ) (red) and Γ xy Im(σ xy 0 ) (blue) are plotted for q = 0.5, 1 and 1.5. For a well-defined resonance of the form (50) this the residue of the pole at ω * = ω 0 − iΓ. For q = 0.5 the plots for σ xx and σ xy are rather different but for q = 1 and q = 1.5 they exhibit a remarkable correlation. Indeed for q = 1.5 the residue appears to be a constant, independent of m, suggestive of the Drude -from (51) with σ 0 and Γ independent of m and ω 0 linear in m.

Resonance properties as functions of temperature.
To get an intuitive understanding of the physics it is better to transform from the geometric parameters q, m and z h to physical parameters, such as charge density ρ, magnetic field B and temperature T , [3]. We have Fixing ν, B and L gives a monotonic relation between T and m. Fig. 14 shows the peak conductivities at resonance and Q-factors as a function of the reduced temperature for three different values of ν. We see that the conductivity decreases and the damping increases as T increases, which is what one would expect from electron-phonon scattering.
4.3.10 Zero temperature conductivities as functions of the filling factor ν.
To investigate the low temperature regime the conductivities are plotted in Fig. 15 for various values of ν with q 2 + m 2 = 3, corresponding to T = 0 regardless of the value of z h . The resonance becomes sharper as |ν| increases, i.e. as the charge density increases at fixed B or as B decreases at fixed charge density. The DC Ohmic conductivity has the δ-function behaviour of a superconductor as B → 0 at fixed ρ. There is significant damping at smaller values of |ν| but the resonance persists all the way down to ν = 0, where the Q-factor is finally reduced to 3.15. Thus increasing B at fixed ρ damps the system from a superconductor to an ordinary conductor but it is not clear what the physical origin of this damping is. It may be indicative of a strengthening magnetic field destroying superconductivity but this merits further investigation.     Re σ xy ( ω) , blue. There is a δ-function singularity in Re σ xx ( ω) at ω = 0 for the case ν → ∞,

Conclusions
Equation (21) is a renormalisation group equation for the conductivity of a two-dimensional system whose solutions can be used to extract AC and DC Ohmic and Hall conductivities, in the presence of a magnetic field. The infrared conductivities at the event horizon are input as boundary conditions and are fixed by the singular nature of the differential equation there. Both the real and the imaginary parts of the Ohmic and Hall conductivities can be obtained and the cyclotron resonance peaks and corresponding Q-factors show qualitatively reasonable behaviour as functions of temperature and charge density at fixed magnetic field. As q 2 approaches 3 the resonance peaks of the Ohmic and Hall conductivities become very nearly equal and the Q-factor increases at small m giving very sharp resonance peaks with high conductivities, tending to superconductor behaviour as m → 0. On the other hand the resonance forms of the Ohmic and Hall conductivities are somewhat different when q and m are both low, for q ≈ m ≈ 0.1 the Q-factor is so small that the cyclotron peaks can hardly be called resonances.
The RG equation has a simple truncation (30) when q 2 ≈ m 2 ≈ ω 1 for which analytic solutions exist, corresponding to the approximate expressions for the conductivity found in [3]. Near extremality for the black hole, when q 2 approaches 3, numerical results indicate that a similar analytic form to (30), but with different co-efficients, is again a good approximation.
A notable feature of the numerical analysis presented above, particularly evident in Fig. 12, is the regularity of the resonance peaks for q ≥ 1. The characteristics of the resonances for the Ohmic and Hall conductivities are almost identical for q > 1 but differ substantially for q < 1. When q > 1 the cyclotron frequency shows a linear dependence on the magnetic charge m, at least for small m, Fig. 6(d), as it does for q and m small, but with a reduced slope as if the effective mass of the charged particle has increased, or its effective charge decreased. Within numerical accuracy the residue becomes independent of m.
The present analysis only produces one peak in the dissipative Ohmic conductivity, there is no sign of Shubnikov-de Haas oscillations or the quantum Hall effect. This may be expected as there is no fermionic matter in the bulk, but there is no charged matter at all in the bulk -it is interpreted a classical effective theory for the electro-magnetic field after charged matter is integrated out. The quantum Hall effect has been related to an emergent Sl(2, R) symmetry in the infra-red [38,39,40,41] and, while the bulk theory used here does enjoy an electromagnetic duality symmetry, giving rise to S-duality of the conductivity σ ± → −1/σ ± as pointed out in [3], this does not extend to Sl(2, R) symmetry in the bulk unless the bulk action is changed. A bulk theory with near horizon Sl(2, R) electro-magnetic duality was studied in [7], and in a subsequent paper [42] we shall extend these ideas to study a different action which has full Sl(2, R) symmetry in the bulk and exhibits features that show strong parallels with the experimental situation with the integer and fractional quantum Hall effects.

A The piezoelectric effect
Non-relativistically the piezoelectric effect occurs when a distortion of a medium generates an electric induction [43], where S βγ = ∇ β v γ + ∇ γ v β is the strain tensor generated by a displacement v α and γ α,βγ is the piezoelectric tensor. As emphasised in [43] the definition the electric permittivity αβ in such situations is somewhat conventional, one could equally use the stress tensor on the right-hand side, rather than the strain tensor, and defineˆ αβ andγ α,βγ using D α =ˆ αβ E β +γ α,βγ T βγ and αβ andˆ αβ would in general be different. For our purposes the first definition is the more appropriate. Consider for example a distortion of the metric components due to a diffeomorphism generated by a vector field v α , δg αβ = ∇ α v β + ∇ β v α , clearly δg αβ plays the role of the stress tensor. Now consider the case where the electric field and electric induction are time dependent infinitesimal variations with an oscillatory phase, δD α = e −iωt δ D α , and δE α = e −iωt δ E α , and the diffeomorphism is due to a vibration e −iωt δṽ α with δṽ α a constant displacement. Then δg tα = −iωδv α and differentiating equation (54), then using Maxwell's equations with zero magnetic field, implies δj α = δḊ α = −iω( αβ δE β + γ α,tβ δv β ).
where the second term is a contribution to the current generated by the forced oscillation, completely independent of any microscopic properties of the material, apart from the charge density (see equation (19)). It would not be appropriate to include it in the conductivity tensor σ αβ .