Holographic Floquet states in low dimensions (I)

We study the response of a (2+1)-dimensional gauge theory to an external rotating electric field. In the strong coupling regime such system is formulated holographically in a top-down model constructed by intersecting D3- and D5-branes along 2+1 dimensions, in the quenched approximation, in which the D5-brane is a probe in the AdS_5 x S^5 geometry. The system has a non-equilibrium phase diagram with conductive and insulator phases. The external driving induces a rotating current due to vacuum polarization (in the insulator phase) and to Schwinger effect (in the conductive phase). For some particular values of the driving frequency the external field resonates with the vector mesons of the model and a rotating current can be produced even in the limit of vanishing driving field. These features are in common with the (3+1) dimensional setup based on the D3-D7 brane model and hint on some interesting universality. We also compute the conductivities paying special attention to the photovoltaic induced Hall effect, which is only present for massive charged carriers. In the vicinity of the Floquet condensate the optical Hall coefficient persists at zero driving field, signalling time reversal symmetry breaking.


Introduction
The physics of periodically driven quantum systems has been the subject of intense study in recent years (see [1][2][3][4] for reviews with tons of citations). Several reasons back up this interest, a very relevant one being technological in origin: the possibility of manipulating quantum systems in a controlled way by using time-periodic external fields. This approach goes under the name of Floquet engineering [5,6], following from the Floquet theorem, a temporal analogue of the Bloch theorem. The artificial setup involves mainly irradiating the system with a circularly polarised laser, or shaking it around. With appropriate periodic drivings, new phases of quantum materials have been created and non-equilibrium phenomena have emerged. Examples include light-induced superconductivity [7,8] and Floquet topological insulators [9,11,[13][14][15][16][17]. Moving on to (3+1) dimensions, artificial Weyl semimetals have also been reported [18][19][20]. In this paper we study the behavior of a strongly-coupled (2 + 1)-dimensional gauge theory under the influence of a external electric field E rotating in the xy plane as (1.1) We will address this problem in the context of the holographic AdS/CFT correspondence [21] (see [22] for reviews), and will follow closely the pioneering work in Refs. [23,24] in their study of (3 + 1)-dimensional systems. In the first of these two references, the authors studied the massless D3-D7 flavour system and computed the artificially induced Hall conductivity.
In the second they extend part of the analysis to massive flavours where an interesting phase space shows up. Our paper streamlines the same analysis in (2+1) dimensions, extending the analysis to encompass the optical and static conductivities also for massive flavours. The (2 + 1)-dimensional theory in place is realised as a codimension-one defect in N = 4 supersymmetric SU (N c ) super Yang-Mills theory in four dimensions. The field content and lagrangian was explicitly constructed in [25]. It contains a matter hypermultiplet living in the defect which couples to the fields of the ambient four-dimensional theory. In String Theory, this model can be realized as the intersection of D3-and D5-branes along 2 + 1 dimensions. In this setup the 4d gauge theory on the D3-branes is holographically dual to the AdS 5 × S 5 geometry, while the D5-branes provide (2 + 1)-dimensional flavors, i.e., fields living in the fundamental representation of the gauge group [26][27][28]. We will work in the quenched probe approximation and will hence neglect the backreaction of the D5-branes on the AdS 5 × S 5 geometry. The fluctuation of the fields living on the brane are dual to the mesonic excitations of the gauge theory. In ref. [29] the complete analysis of these fluctations was performed for the D3-D5 system and the exact spectrum of mesons was found (see [31] for a similar analysis in the D3-D7 case). When an external electric field acts on a medium, a vacuum polarization due to virtual charged particles (quarks) is produced. As a consequence, an oscillating polarization current is induced. If the external field is weak enough, the medium will remain in an insulator gapped phase in which the induced current is always perpendicular to the applied field. There is no Joule heating and the system is dissipation-less. When the external field is large enough the vacuum is unstable against the creation of quark-antiquark pairs (Schwinger effect). In the conductive phase the current becomes dissipative, and the driving induces a Joule heating.
For an external oscillating electric field as in (1.1) the critical value E c at which the insulatorconductor transition takes place depends on the frequency Ω, i.e., E c = E c (Ω) [30]. Remarkably, for some values Ω = Ω c of the frequency, the transition occurs for vanishing external electric field, i.e., E c (Ω c ) = 0. At this point of the phase space the rotating current j is not zero, in spite of the fact that the external driving field vanishes. Physically, for these frequencies the driving field enters in resonance with the vector meson excitations of the gauge theory. Actually, this resonating states with zero electric field and non-zero current also occur in the conductive pase in a finite range of frequencies Ω c < Ω ≤ Ω m . Following [24], where this behaviour was first found in the D3-D7 model, we will call a configuration with E = 0 and j = 0 a zero field Floquet condensate of vector mesons, or simply a vector meson Floquet condensate. Such stable rotating state is a fixed point of time evolution, namely a non-thermal fixed point [32]. In our holographic setup the flavor degrees of freedom are modeled as the excitations of a probe D5-brane embedded in the AdS 5 × S 5 ten-dimensional background. The probe brane extends along an AdS 4 ×S 2 submanifold of the ten-dimensional background. The holographic duality in the D5-brane worldvolume relates the fields at the boundary of AdS 4 to those of the dual (2 + 1)-dimensional gauge theory. In particular, to model the rotating field (1.1) we have to switch on an electric gauge field on the worldvolume attaining the value (1.1) as we approach the AdS 4 boundary (the subleading term near the boundary determines the current j). The D5-brane with this world-volume gauge field has a non-trivial profile which can be found by solving the equations of motion of the probe brane. The leading value of the profile function at the boundary determines mass of the quarks, while the subleading term is related to the quark condensate. A probe brane with an electric field in his worldvolume can develop an event horizon in the effective (open string) metric on its worldvolume. This is analogous, but not the same, to the case in which the brane is embedded in a black hole background geometry with a non-zero Hawking temperature [34][35][36]. Accordingly, we can borrow the terminology of the non-zero temperature case and classify the configurations of the brane depending on whether or not it crosses or not the effective horizon. In the so-called Minkowski embeddings the brane reaches the origin of AdS 4 without developing the effective horizon. These Minkowski configurations are dual to the insulating phase of the defect gauge theory. On the contrary, in the conductive phase the induced horizon forms at a finite radial distance from the D3-brane system. The flavour mesons are deconfined and a current can set up in response to an external electric field. At the interface between these two cases we find the critical embeddings, in which the probe develops the effective horizon right at its IR endpoint. In this work we study both types of embeddings in order to determine the stable non-equilibrium phases of the theory. In this task we will employ both numerical and analytic techniques. The rest of this paper is organized as follows. In section 2 we present our setup, find the equations of motion for the probe D5-brane and study the boundary behavior of the different functions and their relation to the observables of the gauge theory. In section 3 we analyze the different types of embeddings and determine the IR boundary conditions that regular solutions must satisfy. We also determine in this section the effective metric and the effective Hawking temperature. The numerical integration of the equations of motion and the structure of the phase diagram of the theory is presented in section 4. When either the mass is zero or the driving frequency is large, the equations of motion can be solved analytically in the linearized approximation. The exact analytical solutions are presented in section 5. Section 6 deals with the analysis of the AC and DC conductivities of the model, which are obtained by studying the response of the brane to an additional probe electric field which is treated in a linearized approximation. We end up with a summary and some concluding remarks in section 7. The paper is supplemented with some appendices. In appendix A we give details about the derivation of the regularity conditions of the brane embeddings at the pseudo-horizon. In appendix B, we find approximate analytic solutions of the equations of motion when the mass is small. The different embeddings when the equations of motion are linearized are studied in appendix C. The exact conductivities for the massless case are obtained in appendix D.

Setup and ansatz
In order to analyze the Floquet states in the strong coupling regime, we will engineer a holographic brane setup in which two sets of branes intersect along 2 + 1 dimensions. More concretely we will consider the intersection of two stacks of D3-and D5-branes according to the following array 1 2 3 4 5 6 7 8 9 In (2.1) the D3-and D5-branes share the spatial directions 1 and 2, whereas the direction 3 is parallel to the D3-brane and orthogonal to the D5-brane. The field theory dual to the brane setup (2.1) is well-known [25][26][27][28]. It consists of a supersymmetric theory with N f matter hypermultiplets (flavors) living on the (2 + 1)-dimensional defect and coupled to the ambient N = 4 theory realized by the stack of N c color D3-branes. We will adopt the approximation in which the D5-branes are probes in the geometry generated by the D3-branes, so N f N c . The latter is the standard holographic dual of four-dimensional N = 4 super Yang-Mills, namely AdS 5 × S 5 with flux. Let us start by recalling this geometry where dΩ 2 2 = dθ 2 +sin 2 θdφ 2 is the metric of a unit two-sphere and R is the common radius of the AdS 5 and S 5 factors (in what follows we will take R = 1). In order to embed a D5-brane wrapping a AdS 4 × S 2 submanifold we choose the following set of worldvolume coordinates ξ a = (t, x, y, ρ, θ, φ) , (2.3) and the following ansatz for the transverse scalars The induced metric on the D5-brane woldvolume now takes the form where the dot denotes derivative with respect to t and the prime with respect to the holographic coordinate ρ. The main objective to study the response of this system to the driving of an external circularly polarized electric fied cos Ωt − sin Ωt sin Ωt cos Ωt with E = E(t = 0). This will source of a worldvolume gauge field with one-form potential whose field strength is The action of the probe brane is given by the DBI action where T 5 is the tension of the D5-brane. In the following we will adhere to the notation and reasoning advocated in [23,24], with slight modifications. To start with, the following switch from vector to complex notation is useful Now, computing the determinant − det(g 6 + 2πα F ), the DBI action becomes Another convenient switch is to represent the complex gauge field a in the rotating frame In the new variables, (b, χ) the DBI action (2.11) becomes (2.13) Since the action does not depend on t explicitly, taking the functions b, χ and w as independent of the time is a consistent ansatz (2.14) By removing the terms with the time derivative from the action, we can rewrite it in a much more simplified fashion as S ∼ dρ L , where now We first notice that χ is a cyclic variable which means that q, defined as is independent of ρ. The Euler-Lagrange equations derived from L are The last equation in (2.17) is redundant, since we can use the first integral (2.16) to obtain the value of χ in terms of the other functions (2.18) We can next use this last equation to eliminate χ in the first two equations in (2.17). The resulting equations for w and b depend on q (alternatively one could Legendre transform and obtain the corresponding Routhian).

Boundary conditions
Let us now study the equations of motion of the gauge field at the UV ρ → ∞. It is convenient to define the new complex combination hence a(ρ, t) = c(ρ)e iΩt . In terms of c(ρ) the lagrangian density L gives , (2.20) and the equation of motion for c(ρ) derived from L is Near the UV at ρ → ∞ we can show that L ≈ ρ 2 and the equation of motion reduces to ∂ ρ (ρ 2 c ) = 0, whose general solution is of the form c(ρ) = a 1 +a 2 /ρ. Redefining the constants a 1 and a 2 , we can write asymptotically where E is the complexified electric field defined in (2.12), and j the related current. Proceeding similarly, the embedding function w(ρ) behaves near the UV boundary ρ → ∞ as where m and C are constants related to the quark mass and quark condensate respectively. We will actually refer to them as such. The relation between E, j, m and C and the electric field E Y M (t), the electric current J Y M (t), quark mass m q and quark condensate O m in the boundary theory are where λ = g 2 Y M N c is the 't Hooft coupling of the N = 4 theory. The integration constant q defined in (2.16) can be written in terms of the field c and c * and can be easily related to UV data. As w(ρ) ∼ m + C/ρ when ρ → ∞, it follows that L ≈ ρ 2 for large ρ. Moreover, from (2.22) c ∼ −j/ρ 2 and, therefore which endows q with the physical interpretation of the Joule heating. If we assume that the driving effectively heats the system we should expect that q ≥ 0, and from (2.16) this implies Ω χ ≥ 0. We can see that this result is consistent with the ingoing boundary condition at the pseudohorizon, which can be argued as follows. The wavefront of the complexified gauge potential a(t, ρ) in (2.12) is the surface χ(ρ) + Ω t = constant (we are in the case in which χ does not depend on t). By performing a generic variation of this condition, it follows that Requiring that our gauge potential represents an infalling wave at the pseudo-horizon implies δρ < 0 for δt > 0 which, following (2.28), only occurs when Ω χ ≥ 0 and, therefore, q ≥ 0. The equations of motion (2.17) and the UV boundary conditions (2.22) and (2.23) have the following scaling symmetry The Lagrangian and the action transform homogeneously L → α 2 L and S → α 2 S. By choosing α = 1/m in (2.29) we can make m = 1 and deal with the remaining quantities in units of (the appropriate powers of) m.

Types of embeddings
The equations of motion written in (2.17) are potentially singular when where b 0 = b(ρ = ρ c ) and w 0 = w(ρ = ρ c ). The point ρ = ρ c where the condition (3.1) holds will be referred to as the pseudohorizon. Indeed, we will show below that ρ = ρ c is the event horizon of the induced open string metric on the D5-brane. The IR behaviour of the bulk fields when approaching this pseudohorizon determines three types of embeddings (see figure  1). We can have black hole embeddings (which cross the horizon), Minkowski embeddings (which do not intersect the pseudohorizon and reach the point ρ = 0) and critical embeddings (for which the brane reaches the ρ c at ρ c = 0). In order to avoid the singularity at ρ = ρ c the first derivatives of the functions w(ρ), b(ρ) and χ(ρ) have to be adjusted properly for black hole and critical embeddings.

Black hole embeddings
Expanding the functions w(ρ), b(ρ) and χ(ρ) around ρ = ρ c as we obtain coefficients w 1 , b 1 and χ 1 whose explicit expressions are given in appendix A. Notice that χ 1 appears in the expression of the on-shell lagrangian density L 0 evaluated at the pseudohorizon ρ = ρ c L 0 (ρ c ) = ρ 2 We can use this expression to obtain the Joule heating q in (2.16). Indeed as q is constant, we can evaluate it at the pseudohorizon, with the result which only depends on b 0 . Using (3.1), we get

Minkowski embeddings
In this case there is no ρ c that solves (3.1) and the brane reaches the origin ρ = 0. We can therefore evaluate the Joule heating constant q by taking ρ = 0 in (2.18) and, since the right-hand side of this equation contains a ρ 4 factor we get q = 0. Thus, for these embeddings there is no net energy flux on the boundary theory. Moreover, from (2.18) one readily concludes that the phase function χ(ρ) is in fact a constant. Expanding b(ρ) and

Critical embeddings
When the effective horizon reaches the origin, ρ c = 0, the embedding is critical. The value of the b field at the horizon is b 0 = w 2 0 /Ω, χ(ρ) is also here constant and we can expand b(ρ) and w(ρ) around ρ = 0, with the result

Effective metric
The effective open string metric γ ab is defined as [37] where we now denote by h ab the induced metric (2.5). Explicitly, for our ansatz, we have with F (ρ) given by the following function and e ± are complex 1-forms e ± = e ∓ iΩt dx ± idy . The (t, ρ) part of the metric gets diagonalized by means of the following change of coordinates where A(ρ) and B(ρ) are the following functions: (3.13) The new coordinate ρ * is just the Eddington-Finkelstein tortoise coordinate. In these coordinates, the transformations in (2.29) act as a rescaling (τ, ρ * ) → λ −1 (τ, ρ * ), and the effective metric takes the form (3.14) The function F (ρ) vanishes at ρ = ρ c and, therefore, this function acts as a blackening factor for the effective metric (3.14), with ρ = ρ c playing the role on an event horizon. The effective Hawking temperature T H is given by (3.15) Using the values of γ tt and γ tρ found above, we get the T H in terms of near horizon data

Phase diagram
In the previous section we have addressed the different types of embedding according to the IR boundary conditions. On the other hand, the phase diagram is constructed out of the UV boundary data that contain information about the sources and the responses. The analysis of the generic case proceeds by numerically integrating the system of equations (2.17). In this section we will obtain the precise boundary of the phase space of solutions presented in Figs. 2 and 3. Although the qualitative results are the same as the ones found in [24], we need the precise values of these boundaries in order to locate interesting places where to look for conductivity properties in the next section. As usual, integration proceeds from the IR to the UV. Indeed, enforcing regularity conditions found in section 3 as initial conditions at the pseudohorizon ρ = ρ c and integrating out up to the boundary yields a unique solution. For a given frequency Ω we can specify b 0 = b(ρ = ρ c ) and w 0 = w(ρ = ρ c ) and use the values of w 1 , b 1 and χ 1 computed in appendix A(for Minkowski embeddings ρ c = 0 and χ(ρ) is constant). From the UV behavior of the solutions b(ρ), χ(ρ) and w(ρ) when ρ → ∞ one can extract E, j, m and C. Due to the scaling symmetry (2.29) these quantities are not independent. It is natural to measure all quantities in terms of the quark mass m. Accordingly, we will present our results for Ω/m, |E|/m 2 , j/m 2 and C/m 2 .   In figure 2 we show the phase diagram in the parameter space (Ω/m, |E|/m 2 ). The solid curve is the locus of critical embeddings. We can think of it roughly as a boundary between the conductive phase (above) made of black hole embeddings, and insulator phase (below) made of Minkowski embeddings 2 . In figure 4 we plot the electric current versus the electric field for driving frequencies around the firs resonance. In figure 5 we represent the quark condensate for Minkowski and black hole embeddings and different driving frequencies. We notice in Figures 2 and 3 that |E|/m 2 has a series of maxima whose values decrease as Ω/m increases. Moreover, for some discrete values Ω c /m of the frequency the electric field vanishes and, as shown in figure 4, the corresponding rotating current is non-zero.
Thus, for these frequencies we have a vector meson Floquet condensate of the type mentioned in the introduction. Actually, these E = 0 Floquet states exist also for Minkowski embeddings in a finite range of frequency Ω c < Ω ≤ Ω meson , where Ω meson is the mass of a vector meson in the defect theory. This mass spectra was computed in [29] and is given by a tower of values depending on a principal quantum number n (4.1) In section 5.2 and appendix C we obtain these frequencies by means of a linear analysis of the equations of motion. The first four values of the critical resonant frequencies and of the meson masses are In figure 6 we plot the Joule heating q as a function of the electric field, whereas in figure  7 the Joule heating for black hole embeddings is plotted versus the driving frequency for several values of the electric field.
To get further insight on the structure of the phase diagram of the model, it is interesting to analyze in detail the relation between the current j and the electric field E. Since j and E are vector quantities with two components in the xy plane, we expect to have a relation of the type: where the form of the matrix is dictated by the rotational symmetry in the xy plane. The matrix elements γ xx and γ xy are susceptibilities, defined in analogy with the longitudinal and transverse (Hall) conductivities. From the definitions (4.4) it is immediate to find γ xx and γ xy in terms of the components of E and j: where δ is the angle formed by the vectors (j x , j y ) and (E x , E y ) in the xy plane. The quantity γ xx is related to the Joule heating q. Indeed, one has γ xx = q/|E| 2 . Given this, and the fact that we have profusely illustrated the behaviour of q in Figs. 6 and 7, we shall focus on the study of γ xy . For massless black hole embeddings γ xy vanishes, since in this case j and E are colinear, as we demonstrate analytically in section 5.1 below. On the other hand γ xy = 0 for Minkowski embeddings since there is no Joule heating in this case and the vectors j and E are orthogonal. This behavior can be verified by studying γ xy for black hole embeddings as a function of ρ c In figure 9 we plot together the |E|/m 2 and j/m 2 curves as a function of Ω/m. We notice that the current curve also displays a lobe-shaped structure, displaced with respect to the |E|/m 2 curve, whose maxima (zeros) are very close to the zeros (maxima) of the |E|/m 2 curve. We have also checked that sin δ(ρ c → 0) = ±1 and changes discontinuously its sign at the discrete set of frequencies described above (see figure 9).

Analytic Solutions
There are two limits where the equations are amenable to analytic solution. These are the low mass limit m → 0, and the high frequency limit Ω → ∞. In the first case all embeddings are of black hole type. In the second, the three types of embeddings can occur.

Massless embeddings
Let us consider the particular case of massless black hole embeddings, which correspond to putting the embedding function w(ρ) = 0. By inspection one easily concludes that w = 0 is a consistent truncation of the equations of motion in (2.17). In this massless case, the lagrangian density (2.15) takes the form As in the general case, χ(ρ) is a cyclic variable which, according to (2.18), is now given by Let us now eliminate χ(ρ) by introducing the Routhian R 0 , defined as After some calculation, R 0 as a function of b(ρ) and q becomes As usual in this flavour brane setup, demanding reality of the Routhian fixes transport coefficients in terms of horizon data [47]. Indeed, at ρ c the two factors under the square root must change sign, and therefore vanish, simultaneously. This enforces b 0 = b(ρ c ) and q to satisfy both equations (3.1) with w 0 = 0, and (3.4). Combined together, these equation entail, in the massless case Let us look for solutions to the Euler Lagrange equations in which b(ρ) is constant. Writing these equations in the following form it's clear that there is a chance to have b(ρ) constant precisely if it is equal to b 0 related to q as in (5.5). Plugging this value of b = b 0 into the right-hand side of (5.2) we find that can be integrated right away with χ 0 constant. The minus sign is consistent with a positive Joule heating q, see (2.16), and infalling boundary conditions, see (2.28). Then c(ρ) = be iχ ends up being The extreme simplicity of this solution must be contrasted with the need for numerical integration in the D3-D7 setup [23]. By expanding near the UV as in (2.22), the electric field and current at the boundary are equal and given by Hence, in the massless case, the rotating electric field and the induced current are aligned. The effective Hawking temperature for this massless case can be obtained from (3.16) .
Using (5.7) and (3.1) this simplifies into Black hole embeddings with a small non-vanishing mass can be studied as perturbations of the exact massless solutions of this section. This analysis is carried out in detail in appendix B.

High frequency limit
If the frequency Ω is large, the boundary condition at the UV (2.22) fixes that |c(ρ → ∞)| = E/Ω 1 and this enforces c(ρ) to be small for all values of ρ. The gauge field can then be treated as a perturbation around the state with c = 0 and, as we shall see, admits an analytical solution. To start with, using the scaling symmetry (2.29) we can choose to linearize the equations of motion derived from (2.20) near the solution with c = 0 and w = 1. (5.14) where c and δw are small. Notice that, although at linear order the equations decouple, as in [24] we have included the lowest order contribution to the coupling of modes, O(|δc| 2 ), in the equation of δw. The general solution of the equation of δc is where c 1 and c 2 are constants and g 1 (ρ) and g 2 (ρ) are the functions with asymptotic behaviour as ρ → ∞ given by Similarly near ρ = 0, g 1 and g 2 behave as At the linearized level the Joule heating q of Eq. (2.26) is proportional to the Wronskian of c and c * Since q is independent of ρ, we could compute it by using the expansions (5.17) around ρ = ∞ or (5.18) around ρ = 0. The result one obtains is the same and given by The solution for w(ρ) can be written as

Conductivities
One of the main uses of the Floquet driving is the artificial engineering of materials exhibiting transport phenomena with topological character. In this sense, the obtention of a Hall conductivity without a magnetic field is an important hallmark. In the present context, this has been termed photovoltaic Hall effect [9] where it's origin has been shown to be topological, like the Thouless pumping. The time-reversal symmetry breaking and hence, the nontrivial band topology, is induced by the circularly polarized light. This proposal has been the subject of both theoretical and experimental studies during the last decade [52][53][54][55][56].
In [23] the photovoltaic Hall conductivity for massless carriers was obtained in the D3-D7 model, and speculated to behave consistently with the picture of a strongly coupled version of a (3+1) Weyl semimetal. Our findings show similarities and departures from their results. For example, the massless case will be shown to be trivial. However, in the massive case, we will indeed find an intricate Hall effect. Following the proposal in [10] we proceed to study the response of our system to a small additional electric field on the boundary pointing in a fixed direction. From this response we will be able to extract the AC and DC conductivities. We will closely mimic the strategy in [23] whose main steps will be reviewed and extended here for completeness, as our case involves three coupled field perturbations. For the analysis it is most convenient to stick to the cartesian basis used in (2.6). On top of this background field, the idea is to place another fixed direction electric field with | | | E|, and a harmonic (AC) time dependence Our task is to extract the response of the system out of the non equilibrium steady state (NESS) solutions spelled out in the previous sections. In particular, the currents will also suffer a perturbation J (t) → O(t) j + δ j(t) and one of our targets will be to extract the effective conductivity response matrix σ, such that δ j(t) = σ · (t). From (6.1) we expect a perturbation of the bulk gauge field, which we also write now in cartesian components, a(t, ρ) → a(t, ρ)+δ a(t, ρ). The idea is now to refine the ansatz, taking into account the new driving data (6.1) and (6.2). First of all, let us reconsider the boundary conditions of the unperturbed gauge field, a(t, ρ) (2.7), now in cartesian coordinates. The near boundary expansion will be where ε stands for the 2 × 2 antisymmetric matrix The leading term matches the vector potential of the rotating electric field through E(t) = −∂ t a(t, ρ = ∞). This easily follows from the relation ∂ t O ε = −Ω O. The boundary condition (6.3) suggests the following ansatz for the bulk solution Considering the perturbations, now the bulk gauge potential a + δ a has to match the full electric field (6.1) at the boundary, Induced by this new boundary conditions, the ansatz will assume that the static field, c(ρ), develops a time dependent perturbation When the flavour branes are massive, these fluctuations couple to a perturbation of the brane embedding functions w(ρ) → w(ρ) + δw(t, ρ). In this case we will be dealing with a 3 component vector of fluctuations δ ξ(t, ρ) = (δc x , δc y , δω). The perturbed equations of motion, linear in δ ξ, will assume the following form, in terms of the (τ, ρ * ) coordinates (see Eq. (3.12)) Here, A, B and C are 3 × 3 matrices depending on the radial coordinate ρ(ρ * ) and, parametrically, on the rotating frequency Ω. At the pseudo-horizon ρ = ρ c (ρ * → −∞) one finds Thus, in this limit, ρ * → −∞, the fluctuation equations become whose general solution takes the form with f and g arbitrary vector functions. Imposing that g = 0 selects the ingoing wave boundary condition at the pseudohorizon.
Let us now work out the boundary conditions for the fluctuations. First, we want to keep the mass of the flavor brane fixed. After inserting (6.9) into (6.8) and multiplying by O −1 , the boundary conditions for δ ξ(t, ∞) turn out to be , δω(t, ρ = ∞) = 0 . (6.14) Following [23], we expand the matrix O(t), defined in (2.6) as where M ± are orthogonal projectors given by Finally, defininig ω ± = ω ± Ω , the boundary UV condition (6.14) of δ c can be written as It is natural to assume that from these boundary condutions the bulk fluctuations δ c (t, ρ) will oscillate with frequencies As a consequence of the coupling in the linearized equations, we will assume likewise that the embedding fluctuations will resonate with the same frecuencies In summary, setting δ ξ(t, ρ) = ξ ± (ρ)e −iω ± t , the 3 component vectors ξ ± (ρ) = (β ±,x , β ±,y , γ ± ) will satisfy the linearized ordinary differential system From (6.14), the brane fluctuations have vanishing boundary source γ ± (∞) = 0 . Inserting the boundary expansions for the fields in (6.18) and comparing we get Comparing the first of these equations with (6.17) we conclude that The subleading vectors β ± will no more be independent. From the linearity of the equations of motion it follows that this relation is also linear β (1) where X ± are 2 × 2 matrices to be computed and (6.23) has been used. Plugging this into (6.24) allows to find a relation between the current δJ and the applied electric field which defines σ, σ + and σ − as the 2 × 2 conductivity matrices of the three current modes, one with frequency ω and two with ω ± 2Ω (the heterodyning mixing modes [50]) The procedure to compute X ± starts by solving the differential equation system (6.20) from the pseudohorizon ρ * → −∞ to the AdS boundary at ρ * = 0. At the horizon we impose the ingoing boundary condition ξ ± (ρ * ) c e −iω ± ρ * , where c are constant vectors (1, 0, 0), (0, 1, 0) and (0, 0, 1). Due to linearity of the equations we obtain in this way two complex 3 × 3 matrices P ± and Q ± such that ξ (0) ± = P ± c ± , and ξ (1) ± = Q ± c ± . In this way we can solve for ξ (1) ± . Since we will be setting the source for the embedding fluctuations to zero, i.e. ξ ±, 1,2 , the searched for matrices X ± in (6.25) will be given by the 2 × 2 submatrix of the product Q ± P −1 From them, the conductivities are extracted using (6.27).

AC conductivities
The results for the photovoltaic optical Hall conductivity, σ xy , as well as for the photovoltaic optical absortion spectrum, σ xx , can be seen in Fig. 10. We have plotted everything in units of the carrier mass m. 4 The driving laser has been set to the first resonant frequency Ω/m = 1.496 for a discrete set of values of |E|/m 2 (see the coloured dots along the vertical dashed line in Fig. 12). The behaviour in all components of σ shows oscillations whose amplitude grows for decreasing |E|/m 2 . All conductivities oscillate around 0 except for Re(σ xx ) which does it around a saturation value 1.
The departure in (2+1) dimensions, from the analog (3+1) dimensional holographic system is significant (see Fig. 7 and 8 in [23]). In both cases, in the large frequency limit, the conductivity becomes proportional to the identity matrix σ(ω → ∞) = c I. In the D3/D7 case, c is a complex linearly growing function of ω, whereas in our case c is a constant (equal to 1 in our normalization in this section, see later however). This asymptotic behaviour is also attained in the limit of high |E|/m 2 1, as controlled by the massless limit m → 0 which is analytically solvable. Not only the amplitude of the maxima, but also their movement is different. In the D3/D7 case their positions shift to lower frequency for decreasing external field. Here we see that they tend to approach the critical ω = 1.496. As |E|/m 2 becomes smaller, strong oscillations develop, with the most prominent one sharpening around the resonance value for the optical and driving frequencies ω = Ω (signalled with a dotted vertical line in all the plots). This feature bears resemblance with the one obtained in [10] (Fig 1b), although the behaviour of the peak amplitude with the intensity is the opposite.
In the limit |E|/m 2 → 0 these oscillations become very sharp peaks as shown in the insets in  [12], where also the resonance peak is visible as well as secondary peaks. Importantly, in the limit E → 0 the photoelectric Hall conductivity does not vanish. This signals spontaneous time reversal symmetry breaking and might be a consequence of the underlying vector meson Floquet condensate state. 5 The enhancement of the oscillations and peak formation observed in Fig.10 is a universal feature as we approach the line of critical embeddings. We have followed the movement of the peaks for a sequence of configurations very closely above the critical line, for background frequencies in the range Ω/m ∈ (1.3, 2.32) (see Fig. 2). The resulting curves are plotted in Fig. 11. A remarkable fact is the presence of one peak whose position barely changes and is confined to the interval ω/m ∈ (1.4965, 1.73), i.e. among the vector meson Floquet condensate frequency and the meson mass resonance (see Fig. 2 and eqs. (4.2)(4.3)). Very slightly above the critical line there are two sidewise peaks which appear within a short interval in E/m 2 . They all can be seen also in Fig. 11. The remarkable fact about these lateral peaks is that they shift position with the driving in such a way that their mean value coincides with the background frequency Ω, while their splitting is almost constant ∆ω/m ∼ 1.7 . This curious result is very neatly seen on embeddings to the right of the vector meson Floquet condensate, i.e. in the segment Ω/m ∈ (1.496, 2.32) and, very much damped, nearby to its left. The blue curve at Ω/m = 1.496 is the same one that can be observed in the insets of Fig. 10, which has E/m 2 = 0.0175. At this frequency Ω = Ω c , there are embedding solutions down to E = 0 and this gives the magenta curves in those insets at values of E/m 2 = 0.001 and even lower.
We also computed the heterodyning conductivities σ ± (ω). They bear the same features as for the normal conductivity, and the reader can found further information in Appendix E. Roughly stated, the curves for these exotic conductivities share, each of them, half of the peaks visible in σ(ω).

DC conductivities
The zero frequency limit ω → 0 is formally singular (c.f.(6.14)). However the numerical value of the AC conductivity is well behaved and stable down to values of ω/Ω well below 10 −3 . Hence we will use this value to define our DC conductivity. The authors of [23] performed a numerical analysis in the time domain and showed consistent agreement between the two prescriptions. From the plots in Figs. 10, it is apparent that the imaginary parts vanish in the limit ω → 0. Hence the DC conductivities are real, as follows from the relation σ(−ω) = σ(ω) * .  Figures . 13 and 14. In all cases, the same color code has been used.  Fig. 13 we exhibit σ(ω = 0) for varying Ω/m along lines of constant |E|/m 2 . For high enough E/m 2 the limit Ω → 0 stays in the conducting phase and the direct conductivity σ xx has the correct static limit obtained in [47] where it becomes a constant that only depends on m, and saturates to a maximum value equal to one (see section 6.3 below for the correct normalization). In Fig. 14 we plot σ(ω = 0), varying |E|/m 2 along lines of constant Ω/m. Our results should be compared with the condensed matter calculation in [9] (see the discussion after Fig.2). For the Hall conductivity we see qualitative agreement in that, in absolute value, |σ xy | first increases with |E|/m 2 and then decreases. For high Ω/m(∼ 3.53) we find values where this conductivity has zeros, this being a remarkable feature. For σ xx we find a monotonic growth that reaches a limiting value for high |E|/m 2 . In contrast, authors of [9] (see Fig.2b) obtained an intermediate valley which they proposed to be a reflection of the gap opening at the Dirac points. If so, our model is too crude to capture this feature. The comparison with the (3+1) case is complicated because the authors of [23] only considered the massless case, which in our setup turns out to be analytic and trivial. Indeed, taking (|E|/m 2 , Ω/m) → ∞ the Hall conductivity vanishes whereas the DC conductivity asymptotes to a constant value. One important difference is the sign of σ xy which in our case comes out negative, hence opposite to the one in the (3+1) dimensional setup. This is so, despite the fact that we have been careful in keeping all the conventions the same, and the sign in σ xx is positive as well. 6 The region in phase space where a significant photoelectric Hall conductivity is obtained is rather small. It happens for values of the parameters in the region of Ω/m ≤ 2 and also small |E|/m 2 ≤ 0.6. This corresponds roughly to the close vicinity of the first dome to the left of the critical value Ω c = 1.496 in Fig. 12.

Massless limit
When the unperturbed embedding of the probe brane in massless, i.e., when w = 0, the equations for δw and δ c decouple and we can take safely δw = 0. The unperturbed configuration in this case is the analytic solution discussed in section 5.1. The equations of motion for δc x and δc y greatly simplify for this solution. Amazingly, the general solution of these coupled equations can be obtained analytically, as shown in detail in appendix D. Using these results one can extract the exact value of the conductivity matrices, which turn out to be σ(ω) = c I , σ + (ω) = σ − (ω) = 0 . Of course, the coefficient c in Eq. (6.28) is not equal to 1. Using (2.24) it is easy to derive the correctly normalised physical conductivity, relating the the physical electric field and the physical current, as Up to a convention factor of √ 2 this result matches the one obtained in [47], see eq. (5.8), were the observation is made that this conductivity is nothing else than 1/g 2 in terms of the effective defect field theory gauge coupling on the D5-brane. This normalisation should affect all the conductivities obtained previously in this paper.

Summary and outlook
In this paper we employed holographic techniques to analyze the response of a (2+1)dimensional gauge theory to an external rotating electric field. The system can be in two phases (insulator and conductor) depending on the type of embedding of the probe brane. In agreement with the (3+1)-dimensional results of [23,24] we have found that for certain frequencies the driving field resonates with the vector meson excitations of the gauge theory and gives rise to zero field Floquet resonances. The lobbed structure shown in Fig. 2 may be a universal feature. Transport properties are an important benchmark, and they exhibit substantial differences in the two dimensionalities. In (2+1) dimensions, the massless case turns out to be trivial. In the massive situation, however, the conductivity tensor exhibits a rich structure. We have investigated the AC and DC conductivities, paying special attention to the first resonant frequency Ω = Ω c = 1.4965 which is close to the meson mass frequency. In this situation the system creates efficiently charged carriers and holes. In experimental setups, the asymmetry in the electron-hole pairs for different chiralities gives rise to the so called "optically induced valley polarization". This might be the closest physical realization to our setup at the resonant point. Weather the double peak structure observed in Fig: 11 may have some meaning in this direction is something that deserves further scrutiny. Otherwise, it would probably mean this model is too crude to capture the chiral asymmetry in the splitting of the Dirac point. Many distinguishing features are present for configurations approaching the critical line separating the conducting from the isolating phase. The strong enhancement of the oscillations deforming into sharp peaks is one such. These peaks seem to point to bound states that are shifted from the meson masses. At this point, a very relevant question is that of stability of the embeddings. The exact point where, say, Black Hole embeddings decay into Minkowski embeddings and viceversa is a problem that can be addressed in equilbrium thermodynamics, and has a practical answer in terms of the equal area law. Here however, the similar tools are not available from a would be periodic thermodynamics [51], and this certainly an important aspect that will need further study in the future. Let us now discuss possible related lines of research for the future. An interesting avenue would be to complete the phase diagram of the D3-D5 model by studying the system at (real) non-zero temperature and/or finite chemical potential. Moreover, we have treated the system in the quenched approximation. Dealing with the backreaction of the D5-branes is, in general, a tough problem. However, we could adopt some approximation, as it is done in the smearing approach (see [38] for a review). Interestingly, the smeared backreacted D3-D5 background has been obtained in [39] (see also [40][41][42]) and one could try to generalize these results to the case in which an external rotating electric field is acting on the flavor brane. The comparison of our results with the D3-D7 model points towards some universality in the phase space. To strengthen this point, a natural strategy is to extend the study to other brane setups. Most prominently, the (2+1) dimensional ABJM model is an example of superconformal QFT with a well stablished gravity dual [43]. The flavor branes in this case are D6-branes [44,45] (see [46] for an analysis of the thermodynamics of the flavor branes in ABJM). The ABJM supergravity background has a non-trivial topological structure which allows to switch on fluxes of the gauge fields and realize the quantum Hall effect [48]. This suggests that it could be the right place to find non-trivial topological effects induced by an external driving in an holographic setup like the ones we are missing here. Another direction worth pushing is to consider other types of drivings. The brane picture of gauge theories allow for several possibilities. For example, instead of an electric field we could consider a magnetic field, or let the brane oscillate periodically. Actually, this last scenario has been considered in [49] in connection with the realization of the chiral magnetic effect in holography. It would be good to revisit this setup and search for possible stable non-equilibrium phases and condensates. The natural expectation is to observe resonances with the other type of mesons of the theory, like scalar mesons. 2022) and the European Union (European Regional Development Fund -ERDF "María de Maeztu" Units of Excellence MDM-2016-0692, and the Spanish Research State Agency. The work of A.G. has been supported by the Xunta de Galicia action under the grant ED481A-2019/054.

A Regularity conditions
In this appendix we derive the regularity conditions for black hole embeddings. The righthand side of the equations (2.17) must vanish at the pseudohorizon ρ = ρ c in order to avoid the potential singularity at this point. These conditions yield the following three equations From the first equation in (A.1), we get .

(A.2)
Using this result in the second equation in (A.1) we arrive at the following relation between b 1 and w 1 Let us next plug (A.2) into the third condition in (A.1). We get Using now (A.3) we arrive at the following quadratic equation for w 1 Let us write the solution of (A.5). First of all we define the quantities s and D as s ≡ 4(ρ 2 c + w 2 0 ) + Ω 2 4(ρ 2 c + w 2 0 ) 2 + Ω 2 ρ 2 c , Then, we can write Let us now analyze some limiting cases of the relations found above.

A.1 Large frequency
Let us expand in powers of 1/Ω for large frequency Ω. Taking the limit Ω → ∞ in (A.6) we get that s ≈ ρ c Ω 2 . and D ≈ (w 2 0 + 2ρ 2 c ) Ω 2 . Plugging these values in (A.7) we get that w 1 is given at leading order by Using this value of w 1 to evaluate the rhs of (A.3) we get that b 1 → 0 as Ω → ∞. Actually, one can prove that Taking into account the values of b 1 and w 1 found in this large Ω limit, we get the following value of χ at leading order (A.10)

A.2 Small frequency
When Ω → 0 the quantities s and D attain the values and one can verify readily that

B Small mass solutions
We now consider a small perturbation around the analytic massless solution of section 5.1. We write b(ρ) and w(ρ) as b(ρ) = b 0 + β(ρ) , w(ρ) = λ(ρ) , (B.1) and we consider the equations of motion (2.17) at first-order in β and λ. Using (2.18) to eliminate χ , we arrive at the following decoupling linear equations for λ(ρ) and β(ρ) (We are assuming that q > 0). Let us consider from now on the equation for λ in (B.3). This equation can be solved analytically in terms of hypergeometric functions where c 1 and c 2 are constants. The hypergeometric functions written in (B.4) have a logarithmic singularity as we approach the pseudohorizon at ρ = q 1 4 . Indeed, from the general equation Let us now look at the UV behavior ρ → ∞. In general, for large z, one has Therefore, the coefficient of the leading term in λ(ρ) as ρ → ∞ is The imaginary part of this leading term in the UV is zero if the ratio of c 1 and c 2 is which can be seen to be equivalent to (B.7). If this ratio holds, the leading term λ(ρ = ∞) is finite and given by Let us now look at the UV subleading term, which is of the form C/ρ. The coefficient C (i.e., the condensate), is given by The imaginary part of C vanishes if which again can be proven to be equivalent to (B.7). Then, the condensate C is given by and the two functions F (ρ) and G(ρ) as Then, the general solution for the function β is where c 1 and c 2 are real constants. In order to find E and j for this solution, we must study the UV behavior of the complex potential c = b e iχ . The equation for the perturbation δc = c − b 0 e −iΩ/ρ at first order is It is difficult to solve this equation directly. However, the equation for δχ = χ + Ω ρ can be obtained form (2.18) and is rather simple Notice that δc is related to β and δχ as with d being a complex constant. Then, we can show that δχ is with λ ∈ R. Plugging β and δχ on the right-hand side of (B.24), we get Notice that the function multiplying d is an outgoing wave, whereas that multiplying λ is an incoming wave. It is also interesting to notice that the first order variation of the Joule heating q is zero for this solution, as it can be checked by using (2.16).

C Embeddings in the linearized approximation
In this appendix we study in detail the three types of embeddings in the linearized approximation of section 5.2 (valid for large driving frequency Ω or small electric field E).

C.1 Minkowski embeddings
To obtain the Minkowski embeddings that reach ρ = 0 we impose regularity of the function c(ρ) at ρ = 0. This implies that the constant c 2 must vanish and, therefore the gauge field c(ρ) is given by From the boundary expansion (2.22) we find the electric field and current hence Eliminating the constant c 1 we obtain the current in terms of the electric field E We can also eliminate c 1 in (C.1) using (C.2) Let us now look at the embedding function δw given in (5.21). Imposing δw(ρ = 0) = 0 fixes the value of the integration constant c 3 Then, the resulting δw(ρ) can be written as From the UV expansion (2.23) we find the mass and the condensate In terms of the electric field at the boundary, using (C.2) we obtain To find the resonant frecuencies of the Floquet condensates we impose that E = 0 in (C.2), which means that the Ω's must satisfy 1 + Ω 2 n = 2(n + 1) , (C.10) for n = 0, 1, 2, · · · . This gives the following discrete set which are the same as the masses of the vector mesons in the D3-D5 model [29].

C.2 Black hole embeddings
We now determine c(ρ) for a black hole embedding by imposing the appropriate in-falling boundary condition at the pseudohorizon. Now the phase χ(ρ) is not constant and, in fact, χ is related to the Joule heating q. To obtain this relation in the linearized approximation, let us take into account that c c * − c * c = −2i χ b 2 . Since the lagrangian density L = ρ 2 in the linear approximation, we get from (2.16) that . (C.12) Let us now take ρ = ρ c in this last expression and use (3.1) and (3.5) to relate b 0 = b(ρ c ) and q to w 0 = w(ρ c ) and ρ c . We immediately get which is an initial condition for χ(ρ). We begin by introducing a new variable y, related to ρ as 7 y = arctan(ρ) , (C.14) and let y c be the y coordinate of the pseudohorizon y c = arctan(ρ c ). In terms of the variable y, the initial condition (C.13), for w 0 = 1, takes the form We will impose the following infalling condition at the pseudohorizon dc dy y=yc = i Ω c(y c ) . (C. 16) Taking into account that, in general this is equivalent to requiring that db dy y=yc = 0 (C. 18) or, equivalently, that the infalling frequency is real. To simplify the calculation, let us now introduce a new complex field As c = ψ(y)/ sin y, the infalling boundary condition (C.16) takes the following form in terms of ψ dψ dy y=yc = i Ω + cot y c ψ(y c ) . (C.20) Taking into account that cot y c = 1/ρ c , we finally get The general solution (5.15) can be written as where d 1 and d 2 are complex constants and Ω is defined as We now impose the pseudohorizon boundary condition (C. 16). It is easy to demonstrate that (C.16) is satisfied if the ratio of the constants d 1 and d 2 is given by To rewrite this condition in a more convenient form, define the phases Λ c and Λ c in terms of which we can write (Ω + Ω)ρ c + i = (Ω + Ω) 2 ρ 2 c + 1 e iΛc , (Ω − Ω)ρ c + i = (Ω − Ω) 2 ρ 2 c + 1 e iΛc , (C. 26) and d 1 /d 2 takes the form It is now straightforward to relate the value of ψ at y = y c with the constant d 1 and to obtain the value of c at the pseudohorizon. We get c(y c ) = 2Ω 1 + ρ 2 c (Ω + Ω) 2 ρ 2 c + 1 We choose the phase of c(y c ) to vanish at ρ = ρ c . This requirement determines the phase of d 1 , which must be of the form Moreover, since Finally, we can expand this result near ρ = ∞ to obtain the electric field E 38) and the current j As a check one can verify that these results reduce to the ones obtained for Minkowski embeddings when we take ρ c = y c = 0.

C.3 Critical embeddings
According to our general analysis of section 3.3, the critical solutions satisfy Ω b(ρ = 0) = w 2 0 . Therefore, if we define the parameter as these embeddings must satisfy = w 2 0 . In our linearized analysis we have w 0 = 1. Thus, we are tempted to describe the critical embeddings by means of the solution g 1 (ρ) of (C.1) with = 1. Indeed, contrary to the other solution g 2 (ρ), the function g 1 (ρ) is regular at ρ = 0 and one can easily find as a function of Ω and the constant c 1 Plugging this relation in (C.1), we get c(ρ) as It turns out, however, that this linearized solution cannot describe accurately the critical embedding near ρ = 0. Indeed, we found in section 3.3 that the b field behaves linearly in ρ around its value at ρ = 0, whereas the function g 1 (ρ) behaves quadratically (see (5.18)). As argued in [24] for the analogous D3-D7 case, the critical solutions behave highly non-linearly around ρ = 0. Nevertheless the linear solution can describe rather accurately the true nonlinear solution for ρ 1 if we change the value of from its naive value = 1. To illustrate this we compare the linear analytic solution for = 1/2 with the numerical results. In figure  15 we plot the values of c(ρ) calculated by these two procedures. For a general value of , the electric field E, the current j and the condensate C in the linear approximation are In figure 16 we compare the approximate linear expressions (C.43) for E and C (again for = 1/2) to the numerical result of the full non-linear calculation. A reasonable agreement for large Ω is found. The discrepancies can be attributed to the intrinsic non-linearity of the critical solutions. In particular, eq. (C.43) implies that the critical electric field decreases as (Ω/m) −1 for large Ω/m. We can also use (C.43) to estimate the susceptibility parameter γ xy (defined in (4.4)) for critical embeddings: This expression explains qualitatively well the change of sign of γ xy pointed out in section 4 and reproduces quite accurately the numerical results of γ xy for large values of the driving frequency Ω.

D Conductivities in the massless case
In the massless case the fluctuations δw of the embedding function decouple from those of the gauge field δ c. Therefore, since we are interested in computing conductivities, we can concentrate in studying the equations for δc x and δc y . In order to write these equations in a more convenient form, let us define the following differential operators (D.1) Then, δc x and δc y satisfy the following system of coupled second-order differential equations To decouple these equations, let us consider the following complex combinations of δc x and δc y η(t, ρ) ≡ δc x (t, ρ) + iδc y (t, ρ) ,η(t, ρ) ≡ δc x (t, ρ) − iδc y (t, ρ) .

(D.3)
Notice thatη is not the complex conjugate of η since δc x and δc y are not necessarily real. It is straightforward to verify that the equations for η andη are indeed decoupled and given by Let us now separate variables as η(t, ρ) = β(ρ) e −iω t ,η(t, ρ) =β(ρ) e −iω t , (D.5) for some frequency ω. Then, remarkably, one can find the following general solutions where A, B,Ã andB are complex constants which are determined by imposing boundary conditions both at the IR and UV. First of all, let us look at the regularity conditions at the pseudo-horizon ρ = ρ * . These conditions are better studied by using the tortoise coordinates (τ, ρ * ), related to (t, ρ) in this massless case by the following differential relations which can be integrated as Notice that the second of these equations can be simply rewritten as The new radial coordinate ρ * varies from ρ * = −∞ at the pseudohorizon to ρ * = 0 at the UV boundary. Actually, one can prove that in these regions it can be related to ρ as (D.14) Let us now impose the boundary conditions at the UV. These conditions are those written in (6.17) and can be fulfilled if we add two solutions of the form (D.14) with frequencies ω + = ω + Ω and ω − = ω − Ω with amplitudes β Let A ± andÃ ± denote the constants in (D.14) for the frequency ω ± . From the leading UV term of our solution, we get that where M + is the matrix defined in (6.16). Thus, the matrix X + introduced in (6.25) in this massless case is given by X + = i ω I . (D.20) Proceeding similarly for the frequency ω − , we obtain that the constants A − andÃ − must be 21) and the corresponding amplitude for the subleading term takes the form −,y = − Using these values of X + and X − we can now obtain the conductivities from (6.27). As

E More on optical conductivities
In Fig. 17 we show some plots for the heterodyning optical conductivities σ ± (ω). As proven in [23], a set of non-trivial relations among the components of σ and σ ± allows to take as independent components σ xx , σ xy and σ ± xx . The relation σ(−ω) = σ(ω) * implies that the DC conductivity will be real. For the heterodyning conductivities, in contrast, σ ± (−ω) = σ ∓ (ω) * and, hence, the associated DC limits will be complex. They exhibit a similar pattern of oscillations whose amplitude increases as we approach the critical point. Eventually they sharpen into peaks as well. Comparing for example the peaks in σ ± xx with those in σ xx in Fig. 10 it seems apparent that they share half of them each. This is apparent in the heterodyning version of Figs.11 shown below in Fig.18. Namely, while σ − xx has only the central stable peaks, σ + xx has the external peaks whose average value is Ω.