# Universal linear and nonlinear electrodynamics of a Dirac fluid

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Edited by Subir Sachdev, Harvard University, Cambridge, MA, and approved February 11, 2018 (received for review September 27, 2017)

## Significance

Electrons in pristine solids behave as a hydrodynamic fluid in a certain range of temperatures and frequencies. We show that the response of such a fluid to an electromagnetic field is different from what is predicted by the usual kinetic theory. Certain aspects of this response are universal, for example, a direct relation between the linear and second-order nonlinear optical conductivities. Discovery of this relation enriches our understanding of the light–matter interaction in diverse electron systems and new materials such as graphene.

## Abstract

A general relation is derived between the linear and second-order nonlinear ac conductivities of an electron system in the hydrodynamic regime of frequencies below the interparticle scattering rate. The magnitude and tensorial structure of the hydrodynamic nonlinear conductivity are shown to differ from their counterparts in the more familiar kinetic regime of higher frequencies. Due to universality of the hydrodynamic equations, the obtained formulas are valid for systems with an arbitrary Dirac-like dispersion, ranging from solid-state electron gases to free-space plasmas, either massive or massless, at any temperature, chemical potential, or space dimension. Predictions for photon drag and second-harmonic generation in graphene are presented as one application of this theory.

There has been a renewed interest in hydrodynamic phenomena in electron systems with a Dirac-like energy-momentum dispersion *Discussion*). Therefore, exploring electrodynamics of Dirac fluids may be practical.

In this work we focus on the second-order ac conductivity of a Dirac fluid, which controls nonlinear optical phenomena such as sum (difference) frequency generation and also photon drag. The hydrodynamic regime cannot be described by theories that neglect ee interactions. Indeed, applied to graphene, we find significant differences of our results from what one obtains at frequencies

## Second-Order Nonlinear ac Conductivity

Recall that the second-order conductivity is a third-rank tensor **2**) are the same in hydrodynamic (17), kinetic (30), and quantum (31) domains in this case. Conversely, if either *Lorentz-Invariant Dirac Fluid and Graphene*). While the interaction-induced mass renormalization in graphene at *A*). Similarly, *B*).

Let us now present a quick if nonrigorous derivation of Eq. **2**. Consider the expansion of a given Fourier harmonic of the electric current *Lorentz-Invariant Dirac Fluid and Graphene*):**7a** we get **2**. One can verify that for a nonrelativistic gas our formulas agree with those in the literature (17, 30).

## Lorentz-Invariant Dirac Fluid and Graphene

The case of a Lorentz-invariant Dirac fluid can be treated rigorously. Proposed solid-state examples of such fluids (2) actually lack true Lorentz invariance. Their matter and field components have different limiting velocities, v and c. However, if Coulomb interactions are weak, the approximate Lorentz invariance with velocity v holds. In graphene, this is so if the dielectric constant κ of the environment is large, so that the interaction constant **2**.

Let us introduce two additional quantities. One is the flow velocity 𝐮 that defines the electric current **7a** and **7b** are charge continuity equation and the energy conservation equation. Eq. **7c** is the relativistic Euler equation written in “covariant derivatives” **7d** for the Lorentz force

The linear response has already been treated at length (2, 4⇓⇓–7, 13, 14, 37). For example, for massless particles, **4**) decreases as *A*). The question of how the opposite trends of *Supporting Information, Linear ac Conductivity*), assuming the following form of the perturbed electron distribution function f with respect to its equilibrium value **8** with Eq. **5** at **8**, the effective Drude weight *A*. A quantitative theory of these crossover behaviors is a challenge for future work. Meanwhile, Fig. 3 indicates the existence of two separate frequency intervals where *Left Inset*.

Let us move on to the second-order conductivity, ignoring the momentum dissipation for now, **2**. The only unknown parameter is **3** applied to **7**, which is more tedious (*Supporting Information, Second-Order Conductivity in the Hydrodynamic Regime*) but gives the same result. This verifies the validity of our universal formula Eq. **2** for Dirac fluids.

Let us examine the T dependence of the spectral weight **10** predicts *Supporting Information, Second-Order Conductivity in the Kinetic Regime*), such calculations show that *B*).

## Second-Harmonic Generation and Photon Drag

A direct experimental probe of the second-order spectral weight is the second-harmonic generation (SHG), which corresponds to *A*). As explained above, hydrodynamics predicts a SHG signal that is two times larger at low T and much smaller at high T compared with the standard kinetic theory (21, 23) (Fig. 4*B*). The crossover from the kinetic regime to the hydrodynamic one would occur at temperature *B*.

Another effect controlled by *A*). [A recent work (40) studied a similar phenomenon for a surface plasmon playing the role of the incident beam.] To the second order in the in-plane field *p* polarization and *s* polarization, that is, polarizations in and orthogonal to the plane of incidence. For a beam with the in-plane momentum **2** that come from parameter **7c**. The resultant expression for **S33** in the limit *C* and *D*. The reason why in Eq. **17** **17** holds for scattering by short-range impurities. More general formulas can be found in ref. 41 and *Supporting Information, PD*.

## Discussion

Experimental investigation of solid-state hydrodynamics began with 2D electron gas in GaAs (3) and recently expanded to graphene (10, 11, 16) and the quasi-2D metal PdCoO_{2} (12). These dc electric transport measurements (3, 10, 12, 16) probed viscosity of the electron fluid and its dc thermal transport properties (11). Theoretical work (5) predicted the “perfect” Dirac fluid behavior of graphene, various attributes of linear response functions (4, 6, 7), and dc transport (14, 15) properties. In contrast, our work addresses the theory of nonlinear electron hydrodynamics at finite frequencies.

We have proved our key result Eq. **2** rigorously for a model of a Lorentz-invariant fluid, which is a reasonable approximation for materials with either a Dirac-like or a parabolic energy spectrum. For other systems some terms in Eq. **7** may be modified in a system-specific way. Our argument that Eq. **2** nevertheless applies universally is based on two additional general theorems: the Landau–Lifshitz–Pitaevskii formula for the ponderomotive force (35) and the triangular symmetry of the second-order conductivity demanded by energy conservation (Eq. **6** and the discussion below it). They ensure universal validity of Eq. **2** at least for small enough frequencies.

The following estimates suggest that the hydrodynamic regime

Finally, we reiterate that although we used graphene as the main example in this paper, our Eq. **3** relating linear and nonlinear ac conductivities is universal for Dirac fluids. This relation and also Eqs. **2** and **10** should apply as well to ultrapure parabolic band metals (12), semiconductors (3), surface states of topological insulators, Dirac/Weyl semimetals (50), and many other materials, as long as they are in the hydrodynamic regime.

## Acknowledgments

We thank G. Falkovich, M. Glazov, and G. Ni for discussions. Research on general aspects of the hydrodynamic regime is supported by the Department of Energy under Grant DE-SC0018218. Applications to graphene are supported by the Office of Naval Research under Grant N00014-15-1-2671 and those to semiconductors by the National Science Foundation under Grant ECCS-1640173 and also by the Semiconductor Research Corporation (SRC) through the Center for Excitonic Devices at University of California, San Diego, research 2701.002. D.N.B. is an investigator in Quantum Materials funded by the Gordon and Betty Moore Foundation’s Emergent Phenomena in Quantum Systems Initiative through Grant GBMF4533.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: sunzhiyuandt{at}gmail.com.

Author contributions: Z.S., D.N.B., and M.M.F. designed research; Z.S. and M.M.F. performed research; and Z.S. and M.M.F. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1717010115/-/DCSupplemental.

Published under the PNAS license.

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