Plasmons in Holographic Graphene

For strongly correlated systems, holography is an ideal framework for computing plasmon properties. We identify all these self-sourced collective modes in holographic Maxwell theories via a specific choice of boundary condition. This method's potential to improve holographic models of e.g. graphene is demonstrated by computing the 2D-plasmon dispersion, which naturally connects to regions in parameter space accessible to conventional methods. Beyond that, this method also allows to compute the dynamical charge response of strange metals, which could be compared to experiments using (M-EELS).


Introduction
Holography is a powerful framework for computing the response functions of strongly correlated matter to all orders in perturbation theory. Since the dielectric function , whose zeroes give the plasmon modes, is fundamentally related to the complex conductivity of a material it is an ideal candidate to compute holographically. This way we can both compare holographic models to experiments, as well as predict novel behavior of strongly correlated matter. Of particular interest is the little understood 'strange-metal' phase appearing e.g. above the critical temperature in high-temperature superconductors, and also in graphene in the form of the Dirac fluid. The dynamic charge response of strange metals, including plasmonic properties, has recently become experimentally accessible using the new method of momentum-resolved electron energy-loss spectroscopy (M-EELS) [1]. Note that we use the term "plasmon" rather loosely to describe propagating self-sourced plasma oscillations, even in the absence of long lived modes. In recent work [2], we demonstrated that one can describe plasmons holographically with a very specific choice of boundary conditions. In this paper, we generalize these findings further.
When considering linear response in holography, one generally obtains a set of coupled PDEs. Modes of the boundary theory correspond to specific solutions of these equations with a set of boundary conditions that would make the resulting boundary value problem over-determined, that is, while explicitly demanding non-trivial solutions. This is to be expected, however, since modes, by definition, correspond to particular intrinsic properties of the system. The conditions at the horizon are determined by regularity of the solution, and are thus inviolable. At the boundary, conditions are related to the holographic dictionary, with different choices corresponding to different responses one wishes to study. What has been well-established is that Dirichlet conditions give quasi-normal modes (QNMs), which are identified with poles of the holographic Green function. The knowledge of only these modes, however, is not sufficient to characterize all electromagnetic properties, as they just give the response to a screened electric field E, which is 'internal' to the system. To obtain the 'genuine', physical, response function [3], one needs to know the response to the, external, electric displacement field D. In this work, we aim to fill this gap and, based on that, draw conclusions on what conditions holographic models need to satisfy in order to yield physically relevant results. This paper is structured as follows. In sec. 2 we will summarize how internal and physical properties are related through the dielectric function. In sec. 3 we will then refine previous work [2] to illustrate how physical modes correspond to a specific choice of boundary conditions. This identification is consistent with the notion that these modes are, in condensed matter physics, usually identified with the vanishing of the dielectric function. Building up on this correspondence, we then continue in sec. 4 to lay out how these insights have to be incorporated into an effective holographic description of a strongly correlated codimension one system, like graphene, since the relative number of dimensions in which the charge carriers can move compared to the number of dimensions the potential permeates is an essential point. Based on these considerations, we suggest a toy model that yields the characteristic ω ∝ √ k dispersion for surface plasmons, as we demonstrate in sec. 5. There, within the large parameter space of the holographic model, we will elaborate on the detailed agreement with results in regions where conventional condensed matter approaches are applicable.

Physical Modes
Below, we summarize the notion of a 'physical mode' in the density-density response function. 1 In a medium, response to external electromagnetic fields is entirely described by Maxwell's equations, (2.1) The field strength F and induction tensor W are decomposed as, Thus, W describes the 'external' electric displacement field D and magnetic field strength H, which are sourced by an external current J ext . In contrast, F describes the electric field strength E and magnetic flux density B inside the system, being the sums of the external fields and contributions from screening due to effects like polarization and magnetization in the material. This distinction is relevant in order to decide which modes are to be considered 'physical'. The Green function gives the response to the induced current, when the gauge field A is varied. This function encodes current-current correlations, and can be used to obtain conductivity or the 'screened' density-density response, And due to functional identities of the Green function, it is also straightforward to derive elementary identities like, where σ L is the longitudinal conductivity. One has to keep in mind though, that these functions are 'constructs', as they describe the response to the screened fields F, not the external fields W. For an electric response, the relation between the screened and unscreened response is encoded in the dielectric function In particular, for the physical density-density response, Thus, a priori, physical modes, i.e. poles of the response function χ, would be given by the poles of χ sc , as well as by the zeros of L . However, due to Maxwell's equations (2.1) there is a relation between the dielectric tensor and the conductivity, In isotropic electromagnetic media in particular, this implies that poles of σ L are also poles of L , meaning that it is only the zeros of the latter that characterize the poles of the physical response χ.

Holographic Boundary Conditions for Physical Modes
In the following, we will use the term 'physical modes' to refer to poles of physical response functions. That is, functions describing a response to an external, physical, source -in contrast to a response to the screened fields E and B. As argued in the last section, these physical modes correspond to the zeros of the longitudinal dielectric function L . These correspond to modes for plasma oscillations inside the medium, so-called plasmons. It has been recently shown how to identify them holographically [2]. In that framework, these modes are again given by linear response in the electromagnetic sector, but one has to consider boundary conditions which are fundamentally different from Dirichlet conditions -as one would have when calculating QNMs, which would correspond to poles of the screened response χ sc . This is because these modes characterize a situation where there are no external fields, i.e. D = 0 and ρ ext = 0, but there are effects like plasma oscillations, and thus E = 0, in the interior of the material.
In the case of an isotropic system, there is only one transverse counterpart T , and together with L , they provide all necessary information to identify the physical modes of the system. We proceed by deriving the boundary conditions for corresponding bulk fields. Without loss of generality we impose δA t ≡ φ = 0 and study a harmonic perturbation with momentum in the x-direction, and y denoting any transverse direction. Then, after Fourier transforming, from Maxwell's equation (2.1) and the defining relation (2.3) in the absence of external fields, follows These conditions can be turned into boundary conditions in the holographic dictionary, where we can relate the field strength on the boundary to the boundary value of the potential for the corresponding field in the bulk, as well as the current to the bulk induction tensor. Therefore, the explicit formulation is model-dependent, but in the class of Lagrangians usually studied in holography, the current is generally related to the normal derivative of the corresponding potential at the boundary, such that, These provide a unified description of all self-sourced solutions to holographic Maxwell theories. The functions p L/T are determined by the specific bulk theory at hand and the holographic dictionary [2]. They are generally bounded, but may depend on ω and k. In the case of a standard Maxwell action in the bulk, p is constant. The type of mixed boundary conditions we arrive at are related to a double trace deformation in the QFT [4,5], corresponding to the RPA form of the Green function [6]. Therefore, our approach is consistent with conventional CMT and in fact quite natural for computing the dielectric function. It is also straightforward to reformulate the boundary conditions in term of the dielectric function. This essentially follows from basic definitions and the relation (2.8), as well as noting that in the absence of external fields we have M = B. Then, a bit of algebra reveals that (3.1) and (3.2) are nothing else but The first one, in accordance with the nomenclature in [2], we will refer to as the plasmon condition, and the modes satisfying it as plasmons. The second one, i.e. the counterpart in the transverse direction, are transverse symmetric waves [3] involving transverse current fluctuations.
We emphasize that the modes corresponding to (3.3) and (3.4) are complementary to QNMs. The latter correspond to poles of the Green function for internal correlators, the former being necessary to relate aforementioned to external, physical, quantities that can actually be accessed via experiment. Thus, identifying all physical modes is a crucial step to relate holographic results to actual data and, for the holographic Maxwell theories we consider, all physical modes correspond to (3.3) and (3.4), as explained above.

Holographic Graphene as Codimension One Boundary Theory
All electromagnetic phenomena can be described through the field strength F and the induction tensor W. While it has been known for a while how to identify the former in the boundary theory, the correspondence for the latter seems to have gone unmentioned until recent [2]. In the following we will examine consequences of this identification with regard to codimension one systems, like graphene. For a different holographic bottom-up model of graphene see [7].
Holography has been established as an effective description of strongly correlated systems. However, while the strength of interaction is large in these systems, one does not wish to change the fundamental nature of the interaction, respectively test charges, itself. And the latter is intricately connected with the dimension in which particles interact, as creation and annihilation operators are determined as distribution-valued operators for point-like sources. This becomes relevant in systems like graphene, certain high-T c superconductors and other compound materials, where charge carriers move in layers -or even edges, in some cases. That is, while their positioning is confined to a lower number of dimensions, the charge carriers are, in essence, still subject to interactions determined by the '1/r' Coulomb potential in three spatial dimensions -in contrast to, e.g., the 'ln r' potential in two dimensions. In a proper description of such a material, this feature must therefore also be incorporated into an effective model. Meaning that while the system one wishes to study is effectively (2 + 1)-dimensional, one has still to keep in mind that it is composed of particles whose electromagnetic interaction is determined by Maxwell's equations in (3 + 1)-dimensions.
From a holographic perspective, this requires to incorporate a mechanism that keeps particles 'in place'. From a top-down perspective, this role is expected to be played by an object like a D-brane, as illustrated in figure 1, which is also how aspects of graphene and related systems were holographically modeled from a top-down perspective previously [8][9][10][11][12][13]. This description, however, will create some difficulties from a practical standpoint. As pointed out in previous work [2], gravitational back-reaction seems crucial to determine material properties like plasmon excitations. For D-branes, however, going beyond a probe limit -in which back-reaction is neglected -is a rather difficult task where, in general, no efficient computational methods are at hand.
Nevertheless, to demonstrate that the interplay of confinement to a lower-dimensional subspace and gravitational back-reaction in a holographic model will combine to provide physically realistic results, we will present a toy-model that incorporates all the necessary features. The important point to keep in mind is that it is not so much the details of the gravitational interaction with the brane that is relevant, rather that there is some mediating interaction that couples the dynamics of the brane and the background. Thus, as a proxy, consider the gravitational interaction 'projected' onto the brane -which is sufficient, since we are only interested in studying phenomena due to particles that are, ultimately, constrained to not being able to move beyond the dimensions in which the brane extends. The crucial ingredient will be the appropriate choice of the function p L in (3.3) for the condition on the boundary at asymptotic infinity, such that it properly reflects the feature of a codimension one system, where the gauge potential can permeate one dimension more than the induced current.
To derive this condition, consider first an ordinary (3 + 1)-dimensional system. The longitudinal boundary condition can be derived rather straightforwardly from the Coulomb Figure 1. Schematics of how a holographic setup for a physically realistic model for graphene would have to be conceived. While charge carriers, and thus the induced electric current δJ, on the boundary are confined to 2 + 1 dimensions, the boundary potential δA stems from an interaction defined in 3 + 1 dimensions. Therefore, a realistic holographic setup would require a brane-construction that keeps particles in place and electromagnetic phenomena are described by the bulk physics projected onto the brane.
potential. For simplicity, we work in Coulomb gauge A t = φ and A x = 0. In the absence of external sources, a perturbation of the internal charge density δρ must be related to a change in the potential, Before proceeding, we emphasize that even though this seem like an instantaneous Coulomb interaction, this is the fully relativistic result following from a retarded interaction. The apparent conundrum is simply due to the fact that the choice of Coulomb gauge makes the interaction just look instantaneous, while it, of course, still preserves causality -see e.g. [14]. At any rate, after Fourier-transforming, In terms of bulk fields this corresponds to Though, when working in a holographic description at fixed chemical potential, it is more convenient to gauge transform these conditions into And this is exactly the plasmon condition (3.1) for codimension zero. However, if the charges are confined to a plane z = 0, the integral (4.1) changes to Thus, in Fourier-space, Comparing to (4.2), this leads to the boundary conditions being adjusted with a factor k/2. That is, in the (2 + 1)-dimensional boundary, the boundary conditions are instead Thus, the foremost effect of restricting the currents to one dimension less than the gauge potential is a factor of |k|/2. This motivates us to construct a holographic toy model for a codimension one system as follows. To avoid the difficulties of the dynamics when it comes to gravitational back-reaction to the brane embedding, we will, as a proxy, just take a simple lower-dimensional charged system coupled directly to gravity, like a planar Reissner-Nordström black hole, because the exact details of the back-reaction are far less important than the proper boundary conditions to identify the physical modes. And for the latter, we have to remedy the fact that we actually would have one additional dimension to include. This, we achieve by including the factor |k|/2 into the boundary condition (3.3), and we will show below that this is indeed the crucial ingredient to find a physically accurate response.

Results
As mentioned above, we simulate a holographic codimension one system by taking a planar Reissner-Nordström model with a (2 + 1)-dimensional boundary. And to counteract that test-charges in this description would, implicitly, tend to have the wrong potential, we include the corrective factor |k|/2 in the plasmon condition (3.3), which would be there if interactions in the system would, correctly, be described by electromagnetism in (3 + 1)dimensions. Up to this change in the boundary condition, conventions and setup are identical to [2]. The longitudinal result is presented in figures 2 and 3. As expected, for a non-zero charge, the dispersion relation is ω ∝ √ k for small k, and ω ≈ k, i.e. the dispersion of a free mode, for large k. The small k behavior can in fact be derived analytically to be Furthermore, the consistency of the model can be checked by considering that it must be subject to certain sum rules. And, within numerical precision, we indeed can verify that lim k→0 ∞ 0 dω Im [ω (ω, k)] −1 = −π/2 , as required, c.f. [15] for details. Beyond that, we can reach regions of parameter space which has only recently become accessible to experiments, such as µ ≈ T relevant for the Dirac fluid in graphene [16][17][18]. The extreme case µ/T = 0 is possible to observe for truly neutral media (such as He-3), and our result for this case can be seen in figure 2 and 3. Here, we can note that for small k there is the zero sound mode ω ≈ k/ √ 2 and the linear ω ≈ k mode for large k. This is  in accordance with the expectation that the plasmon mode of charged systems turns into the zero sound mode for neutral ones. Since we can access the finite µ T region, we can also predict how this happens, with the result being that there are three regions. For small k, there is the plasmon ω ∝ √ k, for intermediate k, there is zero sound ω ≈ k/ √ 2 and for large k, there is the free linear mode ω ≈ k. One way to present these different regions is by plotting the derivative of the logarithms, shown in figure 4, with the regions mentioned above as I, II and III in the figure.
In particular, note the significant difference in physics between the two linear modes in figure 2 and 3. The zero sound mode depends on changes in the Fermi surface, and is thus dimension dependent (hence the factor 1/ √ 2) and increasingly unstable when increasing k. The free mode, being dimension independent, has speed 1, in units where c = 1, and an imaginary part largely unaffected by increasing k.
It is also worth stressing that the dispersion we compute holographically agrees with the results from conventional condensed matter approaches for the regions of configuration space where comparisons can be made. In particular, in the small k region where ω ∝ √ k, in addition to matching the real part of the dispersion also the linear dependence of the imaginary part of ω on k, c.f. figure 4, matches recent computations for this"collisionless plasmon" part of the configuration space [19]. Also, for small k the results match hydrodynamic 2 results [20], but note however that since there is no mechanism for collisions in our model, a proper hydrodynamic region is not to be expected.

Conclusion
In this paper we analyze the conditions all physical modes in holography must satisfy, building on the results of [2], when the boundary theory includes electromagnetism satisfying Maxwell's equations. We find that for isotropic systems, there are appropriate mixed boundary conditions that translate the necessary features from the boundary theory into the bulk, (3.3) and (3.4) for longitudinal and transverse modes respectively. We also point out that, in the case of holographic Maxwell theories, poles of the screened correlator χ sc , corresponding to QNMs, do not yield physical modes.
We then demonstrate that this method of relating the physical set-up in the boundary theory to the boundary conditions on the equations of motion in the bulk is not only instructive and intuitive, but also a powerful tool when regarding other types of Maxwell theories, such as codimension one materials. For such, we obtain the very characteristic ω ∝ √ k dispersion for small k, using the simple planar AdS 4 -RN model. This has previously been impossible both in bottom-up systems, since the mode is heavily dimension dependent, and in top-down systems, since the mode also requires a dynamical back-reacted metric.
The use of mixed boundary conditions that are tailored to the boundary theory opens up a wide range of possible follow-up studies to previous work, as they can be applied to most previously studied bulk theories. Especially interesting are models that include a physically more realistic mechanism for dynamical polarization, and back-reacted top-down models that have a bulk theory that better aligns with the boundary theory in codimension one systems while still keeping backreacted gravity effects in the bulk.