Universal superdiffusive modes in charged two dimensional liquids
UUniversal superdiffusive modes in charged two dimensional liquids
Egor I. Kiselev
Institut für Theorie der Kondensierten Materie, Karlsruher Institut für Technology, 76131 Karlsruhe, Germany
Using a hydrodynamic approach, we show that charge diffusion in two dimensional Coulombinteracting liquids with broken momentum conservation is intrinsically anomalous. The chargerelaxation is governed by an overdamped, superdiffusive plasmon mode. We demonstrate that thediffusing particles follow Lévy flight trajectories, and study the hydrodynamic collective modes underthe influence of magnetic fields. The latter are shown to slow down the superdiffusive process. Theresults are argued to be relevant to electron liquids in solids, as well as plasmas.
I. INTRODUCTION
Two dimensional electron systems are among themost studied in condensed matter science: ultra-cleangraphene sheets with impurity scattering lengths largerthan µ m [1] allow the observation of viscous electronflows [2–6], which were predicted almost 50 years ago[7, 8]. Twisted bilayer graphene is on its way to becomean important model system for strongly correlated elec-trons [9–11], and unconventional transport effects are ob-served in exceedingly pure delafossite metals [12–15].Hydrodynamic transport theories have been success-fully applied to predict the behavior of such systems [16–39]. A particularly intriguing trait of hydrodynamics isits universality. It can be derived from general symmetryprinciples without knowledge of the underlying micro-scopic theory. This makes the hydrodynamic approachparticularely interesting for the study of systems whereno microscopic picture has yet been established, such ase.g. strange metals [16, 29, 38].In this paper, the mode spectrum of a charged twodimensional liquid with weakly broken momentum con-servation is investigated within a hydrodynamic frame-work. We find that at large scales (or equivalently smallwavenumbers), the diffusion of charges is governed by asuperdiffusive mode, which was described by Dyakonovand Furman [40], and which we interpret as an over-damped plasmon. This mode is shown to be universalin the sense that it does not depend on any micorscopicdetails of the system and is determined by the rate ofmomentum relaxation, the charge density and the massdensity alone. We also elaborate on the how the modearises from the Lévy flight random walks of the individ-ual charged particles. Using coupled Langevin equations,we show that the particle motion is dominated by Lévyflights and obeys heavy-tailed Lévy stable statistics. Fur-thermore, we study the influence of magentic fields on themode spectrum and find that the superdiffusive motionis slowed down by magnetic fields.Anomalous diffusion has been discussed in the con-text of Yukawa liquids and dusty plasmas. Numericalresults implied that these systems are superdiffusive [41],however extensive simulations showed that the diffusionprocess is ultimately governed by Gaussian dynamics[42][43]. We reach a similar conclusion. In dusty plasmas,where the charged particles are screened by mobile back- ground charges and pair interactions are well describedby the Yukawa potential, ordinary diffusion prevails (seeSec. III B). However in one component Coulomb plas-mas [44] and certain colloidal suspensions [45], we expectsuperdiffusion as described in this paper. A. Main results
The diffusion of charges in two dimensional systemsis shown to be intrinsically anomalous due to Coulombinteractions and is described by the fractional differentialequation ∂ t ρ Q = 2 aτ | ∆ | ρ Q , (1)which is derived in Sec. III. Here, ρ Q is the chargedensity, τ is the momentum relaxation time (see Eq.(7)), a is a constant depending only on the homoge-neous background densities of charge and mass (see Eq.(15) and below), and | ∆ | is the fractional Laplacian[46, 47]. Solving Eq. (1) with the initial condition ρ Q ( t = 0 , r ) = Q δ ( r ) , we find (Eq. (21)) that the chargedensity follows a broadening Cauchy distribution: ρ Q ( t, r ) = Q aτ ∆ t π (cid:16) (2 aτ ∆ t ) + r (cid:17) / . (2)Eq. (2) was written down by Dyakonov and Furman inRef. [40]. Deriving the Eqs. (1), (2) from hydrodynam-ics, we show that superdiffusion prevails in virtually alltwo dimensional, Coulomb interacting systems. More-over, the superdiffusive behavior is universal in the sensethat the only parameters that enter Eq. (1) are a and τ ,whereas it is not influenced by the nature of the micro-scopic interactions. In particular, we demonstrate in Sec.III A that charge relaxation in quasi-relativistic Diracsystems such as charge neutral graphene, and twistedbilayer graphene is also superdiffusive. Here, τ has to bereplaced by τ c – the relaxation time of charge currents.The Cauchy distribution of Eq. (2) is a member ofthe family of Lévy stable distributions. The general the-ory of Lévy stability (see Refs. [48, 49]) implies that ifthe superdiffusive dynamics of Eq. (1) emerges from therandom motion of individual particles – a picture that is a r X i v : . [ c ond - m a t . s t r- e l ] F e b Figure 1. The Figure shows the step size distribution p (∆ r ) of a random walk as performed by Coulomb interacting, dif-fusing particles in two dimensions. At large step sizes, thedistribution clearly follows the p ∼ ∆ r − power-law whichleads to the superdiffusive dynamics described by Eq. (1).The data was obtained by integrating the system of coupledLangevin equations of Eq. (54). certainly true for classical particles – the step size distri-bution p (∆ r ) characterizing the particles’ random walksmust decay as a ∆ r − power-law: p (∆ r ) ∼ ∆ r − , ∆ r → ∞ . (3)Such a slow power-law decay invalidates the central limittheorem, so that in the limit of many random steps, theparticle distribution does not converge to a gaussian, butto the heavy-tailed Cauchy distribution of Eq. (2). Suchrandom walks are known as Lévy flights. To demonstratethe Lévy flight nature of the charge relaxation process,we performed a computational experiment (see Sec. VI).The step size distributions of diffusing Coulomb inter-acting particles was studied using the coupled Langevinequations (54). As shown in Fig. 1, the numerical stepsize distribution indeed obeys the power-law (3), demon-strating that the particles are travelling on Lévy flighttrajectories and their dynamics is governed by Eq. (1)[50, 51] at large scales. The distance travelled by theparticles scales as r ( t ) ∼ aτ t, which is much faster then the r ( t ) ∼ √ Dt law of normaldiffusion: the full width at half maximum of the Cauchydistribution (2) broadens with a constant velocity v =2 aτ .Studying the collective mode spectrum of the chargedtwo dimensional liquid, we show that the superdiffusivemode can be interpreted as an overdamped plasmon. Inthe presence of momentum relaxation, the conventionalplasmon mode ω pl = √ aq becomes purely imaginaryfor small q (see Fig. 2). The superdiffusive mode then emerges as an imaginary branch of the plasmon disper-sion relation: ω + = − iaτ | q | (see Eqs. (16), (17) and(31)). We also study the collective modes in the pres-ence of magnetic fields (Sec. V), and find that superdif-fusion is slowed down by a factor of (cid:0) − ω c τ (cid:1) , where ω c is the (small) cyclotron frequency (see Eq. (53)). Therelaxation of charges is then governed by the equation ∂ t ρ Q = 2 aτ (cid:0) − ω c τ (cid:1) | ∆ | ρ Q . Apart from the superdiffusive mode, we derive the mag-netoplasmon dispersion at finite τ (see Eqs. (49)-(52)and Figs. 3, 4). It is noteworthy that the limits ω c → and q → are not interchangeable and result in differ-ent dispersion relations. This behavior is discussed belowEq. (46).In Sec. IV, we discuss the connection between the Ein-stein relation and the superdiffusive behavior. We derivethe Einstein relation D = σ (cid:107) ( ω → , χ ρ Q ρ Q (0 , q → , where χ ρ Q ρ Q is the charge susceptibility, from hydrody-namics, showing that the diffusion constant D appearsin a diffusive pole of the nonlocal longitudinal conduc-tivity σ (cid:107) . Despite the presence of the diffusive pole ω D = − iDq , the equilibration of inhomogeneous chargeor current distributions is superdiffusive, and is not de-scribed by an ordinary diffusion equation. This changeswhen a gate is located in the vicinity of the two dimen-sional system. If the distance between the gate and the2D system is sufficiently small, the long range Coulombpotential becomes subleading to the capacitance of thegate. In this case, the relaxation of charges indeed fol-lows a diffusion equation, and the Einstein relation givesthe diffusion constant (see Sec. IV C).Finally, the contribution of the superdiffusive mode thespecific heat of a two dimensional liquid was calculated(Sec. VII). At low temperatures we find c V = c T − c T + c T − O (cid:0) T (cid:1) , (4)where all higher order terms are of odd powers in T . Thecoefficients are given by c = q ∗ aτ , c = ζ (3)8 πa τ , where q ∗ is a momentum cut-off. The coefficent in front of the T term does not depend on q ∗ . This term is due to thenon-analyticity of the superdiffusive mode at q = 0 (seeEq. (58)). The result (4) is very different from the ∼ T specific heat of the undamped plasmon mode ω pl = √ aq [52]. For a normal diffusive mode in two dimension wefind c V,g = d T + d (cid:48) T log (cid:18) T (cid:19) + d T + O (cid:0) T (cid:1) . (5)The T log (1 /T ) contribution, which is dominant at lowtemperatures is not uncommon for two dimensional sys-tems. In Sr Ru O , the T log (1 /T ) contribution hasbeen observed experimentally [53]. Other mechanismsleading to a T log (1 /T ) dependence of the specific heatare quantum critical fluctuations of overdamped bosonicmodes with a dynamical exponent z = 2 [54] and scat-tering between hot Fermi pocket and cold Fermi surfaceelectrons in Sr Ru O [55].The remainder of this paper is organized as follows: inSec. II we introduce the hydrodynamic framework that isused throughout the paper. Sec. III presents a derivationof the Eqs. (1) and (2). Yukawa liquids and two dimen-sional Dirac systems are discussed. Sec. IV deals withthe Einstein relation and with gated 2D systems. Theinfluence of magnetic fields on the spectrum of collectivemodes is investigated in Sec. V. Sec. VI presents thenumerical results on the Langevin dynamics of chargedparticles. Finally, Sec. VII deals with the contributionsof collectives modes to the specific heat. II. HYDRODYNAMICS
The motion of a charged two dimensional liquid aregoverned by the laws of momentum and charge conser-vation and the corresponding continuity equations. In-troducing the flow velocity u , the charge denstiy ρ Q andmass density ρ M , we can write the charge current as j Q ,i = ρ Q u i , and the momentum density as g i = ρ M u i .In the case of a Galilean invariant system, we have ρ M = mρ , ρ Q = eρ , where ρ is the particle numberdensity and m , e are the mass and charge of the parti-cles that constitute the liquid. If the Galilean invarianceis broken, u can be introduced as a field sourcing theconserved crystal momentum, and the densities ρ Q and ρ M can be defined using the memory matrix formalism[21, 38] (see Appendix A for details).The hydrodynamic equations that we will use in thefollowing are continuity equation for the charge density ρ Q : ∂ t ρ Q + ∂ i ( ρ Q u i ) = 0 (6)and the Navier-Stokes equation, which is the continuityequation for the momentum density: ∂ t ( ρ M u i ) + ∂ j Π ij = − τ ρ M u i − ρ Q ∇ φ. (7)Here, Π ij is the momentum current tensor and φ is theelectrostatic potential. τ is the relaxation time of themomentum density and accounts for momentum dissipa-tion, e.g. due to impurities. The above equations arevery similar to the equations of classical hydrodynamics[56, 57]. However, since charged liquids are considered,we need to take care of electrostatic forces induced by aninhomogeneous charge density. The electrostatic poten-tial φ in Eq. (7) therefore depends not only on externallyapplied fields but also on the charge density: φ = φ [ ρ Q ] .In general one can write Π ij = ρ M u j u i + pδ ij − τ ij , (8) Here, ρ M is the mass density, p is the fluid’s pressure.The viscous stress tensor τ ij can be written as τ ij = η ijkl ∂ k u l using the viscosity tensor η ijkl . We limit ourselfto isotropic systems where the viscosous stress tensor canbe written in terms of the shear viscosity η and bulkviscosity ζ : τ ij = η∂ j ∂ j u i + ζ∂ i ∂ j u j . (9)The hydrodynamic equations (6), (7) describe the macro-scopic dynamics of translation invariant fluids with orwithout Galilean invariance without making assumptionson the nature of microscopic interactions. Therefore,they are useful tools to study the dynamics of strangemetals and other materials where a microscopic theory iscurrently out of reach [16, 19, 38]. III. SUPERDIFFUSION
In the following it will be useful to separate the densi-ties ρ Q , ρ M into a homogeneous background and a smallfluctuating term: ρ Q / M = ρ (0) Q / M + ρ (1) Q / M ( t, r ) . (10)In the absence of external fields, the electrostatic po-tential is determined by the inhomogeneous part of thecharge density: φ ( r ) = 1 ε ˆ d x (cid:48) ρ (1) Q ( t, r ) | r − r (cid:48) | , (11)where ε is the dielectric constant of the substrate. Thepotential φ in Eqs. (7) and (11) is the hydrodynamic ana-logue of the self-consistent potentials of the Landau-Silin[58] and Vlasov [59] theories. After a Fourier transformthe above equation reads φ ( q ) = V ( q ) ρ (1) Q ( ω, q ) with V ( q ) = 2 π/q . We will be interested in the system’sresponse to small inhomogeneities at small q . Let ustherefore sort out the higher order terms. The pressureterm in Eq. (7) can be written as ∇ p = (cid:16) K/ρ (0) Q (cid:17) ∇ ρ (1) Q with the bulk modulus K = ρ Q ( ∂p/∂ρ Q ) . Using thecontinuity equation for ρ Q , we find that the pressure termis of order q : ∇ i p ∝ q i q j u j . The viscous terms ∂ j τ ij alsoare of order q . On the other hand, using Eq. (6) onefinds ρ (1) Q ( ω, q ) = q i ω ρ (0) Q u i ( ω, q ) . (12)Therefore, linearizing Eq. (7) in u i and ρ (1) Q / M and per-forming a Fourier transform we obtain, to first order in q , (cid:0) − iω + τ − (cid:1) u i = − (cid:16) ρ (0) Q (cid:17) ρ (0) M ε ( iq i ) V ( q ) q j ω u j . (13)Being interested in the longitudinal solutions to Eq. (13),we set u ∝ q . The above equation then reduces to ω (cid:0) iω − τ − (cid:1) = i (cid:16) ρ (0) Q (cid:17) ρ (0) M ε q V ( q ) . (14)It follows ω ± = − i τ ± (cid:114) aq − τ (15)with a = π (cid:16) ρ (0) Q (cid:17) / (cid:16) ερ (0) M (cid:17) . In the absence of momen-tum dissipation, i.e. in the limit τ → ∞ , Eq. (15) re-duces to the well known 2D plasmon dispersion ω = √ aq [60]. Eq. (15) describes a damped out plasmon mode,which is purely imaginary below a threshold wavevector q ∗ = 1 / (cid:0) τ a (cid:1) . This purely imaginary branch of the dis-persion corresponds to a superdiffusive mode, as we willshortly see. For Fermi liquids, Eq. (14), which describesthe plasmon pole in the presence of disorder, has beenderived diagrammatically in Ref. [61]. As shown here, itcan be justified on much more general grounds. Expand-ing Eq. (15) for small q we find ω − ≈ − iτ + 2 iaτ | q | (16) ω + ≈ − iaτ | q | . (17)Finally, there exists a transverse mode with u · q = 0 which is given by ω ⊥ = − i/τ + O (cid:0) q (cid:1) . (18)The dispersion relation of Eq. (17) describes a su-perdiffusive mode for the charge density ρ Q . In con-trast to the | q | dependence of Eq. (17), simple diffusivemodes are governed by a dispersion relation ω = − iDq ,where D is the diffusion constant. In space time coor-dinates this translates to the well known diffusion equa-tion ∂ t ρ Q = D ∇ ρ Q . On the other hand, Eq. (17), viaEq. (13), leads to a fractional diffusion equation for thecharge density: ∂ t ρ Q = 2 aτ | ∆ | ρ Q . (19)The fractional laplace operator | ∆ | α is defined via it’sproperties under the Fourier transform: F (cid:104) | ∆ | α f (cid:105) ( q ) = − | q | α F [ f ] ( q ) [46, 47]. The special case α = 1 is used inEq. (19).Fractional diffusion equations [51] are used to describesuperdiffusion in systems as different as random media[62] and financial markets [63]. Indeed, the fractionaldiffusion equation that we arrived at can be interpreted asthe continuous time limit of a stochastic process involvingLévy flights. To see this, let us solve Eq. (19). Taking theFourier transform of the spatial portion of the equationand using the initial condition ρ Q ( t , r ) = Q δ ( r ) where Q is the charge, we find ρ Q (∆ t, q ) = Q e − iaτ | q | ∆ t . (20) Here, we have abbreviated ∆ t = t − t . Taking the inverseFourier transform one obtains ρ Q (∆ t, r ) = Q aτ ∆ t π (cid:16) (2 aτ ∆ t ) + r (cid:17) / . (21)This function is interpreted as the probability distribu-tion for the distances a particle travels in a period of time ∆ t starting at r = 0 , i.e. the step size distribution of arandom walk. While it’s mean value vanishes by symme-try and it’s variance is infinite: (cid:10) r (cid:11) = ∞ , it is easy to seethat that its width grows linearly with t . The distance ∆ r that a particle travels therefore scales as ∆ r ∼ t, which is much faster than in ordinary diffusive processes,where the distance scales as ∆ x ∼ √ t .In general, one dimensional Lévy flights are charac-terized by heavy-tailed power-law step size distributionswhich scale as p ( x ) ∼ x − ( α +1) for large step sizes x [48, 49]. Here, < α < is an exponent which fully char-acterizes the Lévy stable distribution function [49]. Thetraveled distance of a random walker scales as ∆ x ∼ t α .In d dimensions the step sizes are distributed accordingto p ( r ) ∼ r − ( α + d ) (22)[64]. Thus Eqs. (17) and (19) indeed describe a Lévyflight with exponent α = 1 . Another way to see thatEq. (19) describes a Lévy flight is to remember thatthe characteristic function of a (symmetric) Lévy stabledistribution is (cid:10) e − i q · r (cid:11) = e − γ | q | α , where γ characterizesthe width of the distribution. For α = 1 this indeedcorresponds to the solution of Eq. (19) in Fourier spaceEq. (20). A. Dirac liquids
The prime example of a Dirac liquid is graphene at thecharge neutrality point. At finite temperatures, equalnumbers of particles and holes are excited, such that thesystem remains charge neutral. Thus ρ (0) Q = 0 holds.Homogeneous electric currents consist of equal numbersof electrons and holes. However at finite wavevectors, ρ (1) Q = q i ω j Q ,i holds, such that the self-consistent potential(11) must be included [65]. In charge neutral graphene,electric currents are relaxed by interaction effects sincethey are not protected by momentum conservation [66].The corresponding relaxation time τ c damps out the plas-mon mode just as τ does in Eq. (15). The collectivemode structure of this system was studied in Ref. [65].The damped plasmon mode is given by ω ± = − i τ c ± (cid:115) vqτ V − τ c . (23)Here, v is the electron group velocity and τ V = πk B T (cid:126) αN log(2) with the fine structure constant α = e εv (cid:126) . τ V character-izes the strength of the electrostatic repulsion. For small q we find a superdiffusive mode ω + = − i vτ c τ V q. (24)Similar physics will prevail in other charge neutral sys-tems such as twisted bilayer graphene, since at smallwavenumbers the electric current will always follow thedynamics ( ∂ t + τ ) j (1) Q ( t, r ) = −∇ ´ d x (cid:48) ρ (1) Q ( t, r (cid:48) ) ε | r − r (cid:48) | , which,together with the continuity equation will result in a su-perdiffusive mode. Interestingly, the phase space behav-ior of Dirac liquids is also superdiffusive [67]. B. Yukawa liquids
In a Yukawa liquid charges interact with a Yukawapair-potential V Y ( q ) = 2 πq + κ , (25)where κ is the inverse screening length. Such a screenedinteraction potential arises, when the considered chargesare screened by mobile background charges, as for ex-ample in dusty plasmas [68]. Two dimensional Yukawaliquids are widely studied (see e.g. [44, 69, 70]). In par-ticular, superdiffusion has been discussed [41], but ulti-mately ruled out in favor of normal diffusion [42].Within our hydrodynamic model, it is readily shownthat diffusion in a 2D Yukawa liquid is indeed Gaussian.To this end we replace V ( q ) in Eq. (14) by the Yukawapotential (25). Instead of Eq. (15) we then obtain ω Y ± = − i τ ± (cid:115) aq q + κ − τ . (26) For small q , Eq. (26) reduces to ω Y − ≈ − iτ + i aτκ q ω Y + ≈ − i aτκ q . (27)The mode ω + describes normal diffusion where the dif-fusion constant is given by D Y = aτκ . If κ is large, thebulk and shear viscosities, which also give a contributionof order O (cid:0) q (cid:1) will enter the expression for the diffusionconstant. Eq. (27) is therefore a good approximation,if the screening is weak, i.e. κ → . In this case, thedynamics of a weakly inhomogeneous charge distributionis described by the diffusion equation ∂ t ρ Q = D ∇ ρ Q . (28) IV. BEHAVIOR AT LARGER WAVENUMBERS,EINSTEIN RELATION, GATINGA. Behavior at O (cid:0) q (cid:1) The full dispersion relations ω ± ( q ) which are also validat larger q can be efficiently obtained from the well knowncondition for collective excitations ε ( ω, q ) = 0 , where ε ( ω, q ) = φ ext ( ω, q ) /φ ( ω, q ) is the dielectric function ofthe 2D material. Here φ = φ ext + φ ind is the total electricpotential, where φ ext is due to external sources and φ ind is sourced by the inhomogenious charge carrier density ρ (1) Q . From the definition of ε we find ε ( ω, q ) = 1 − χ ρ Q ρ Q ( ω, q ) V ( q ) . (29)Here χ ρ Q ρ Q is the charge susceptibility which is definedvia the relation ρ (1) Q ( ω, q ) = χ ρ Q ρ Q ( ω, q ) φ ( ω, q ) . (30)The condition ε ( ω, q ) = 0 together with the Eqs. (33)and (35) then gives ω ± ( q ) = − i τ − iq ζ + η ρ (0) M τ ± (cid:118)(cid:117)(cid:117)(cid:116) aq (cid:32) Kq aρ (0) M (cid:33) − τ (cid:32) q ( ζ + η ) ρ (0) M + 1 (cid:33) . (31)For small q , Eq. (31) reduces to the expression given inEq. (15). A third mode ω ⊥ is easily found by setting u · q = 0 in the Navier-Stokes equation (7). This transversemode obeys the dispersion relation ω ⊥ = − iτ − iηq . (32)The modes of Eqs. (31), (32) are shown in Fig. 2. B. Einstein relation
Although the charge dynamics in a two dimensionalliquid is superdiffusive, transport coefficients obey theEinstein relation: The longitudinal conductivity σ (cid:107) is de-fined via Ohm’s law: j Q ,i ( ω, q ) = σ (cid:107) ( ω, q ) E i ( ω, q ) with E ∝ q , where the electric field is determined by the gra-dient of the total electrostatic potential: E = −∇ φ . In Figure 2. The dispersion relation of the damped plasmonmode is shown. In the presence of momentum relaxation,the well known √ q -plasmon mode is damped out, such thatbelow a certain threshold wavenumber Re ( ω ± ( q )) = 0 holds.Interestingly, the plasmon is also damped out at large q , dueto both viscosity and momentum relaxation. The blue linecorresponds to the superdiffusive mode ω + ( q ) (see Eq. (17)). the regime of linear response, we find from Eq. (7) σ (cid:107) ( ω, q ) = τ (cid:16) ρ (0) Q (cid:17) /ρ (0) M − iωτ + iτ q ω Kρ (0) M + q η + ζρ (0) M . (33)Here we have again used the relation ∇ p = (cid:16) K/ρ (0) Q (cid:17) ∇ ρ (1) Q where K = ρ Q ( ∂p/∂ρ Q ) is the bulkmodulus. We also assumed an isotropic system where theviscous stress tensor reduces to τ ij = η∂ j ∂ j u i + ζ∂ i ∂ j u j ,with the shear and bulk viscosities η , ζ . Let us relate thebulk modulus to the charge susceptibility χ ρ Q ρ Q . In thestatic case ω = 0 , forces stemming from fluctuations of φ ( x ) are balanced by pressure changes: − i q ρ (0) Q φ (0 , q ) = (cid:16) K/ρ (0) Q (cid:17) i q ρ (1) Q (0 , q ) . It follows χ ρ Q ρ Q (0 , q →
0) = − (cid:16) ρ (0) Q (cid:17) K − . (34)This is a special case of the relation σ (cid:107) ( ω, q ) = iωq χ ρ Q ρ Q ( ω, q ) , (35)which can be obtained from the Kubo expression for σ and the continuity equation [71]. The conductivity (33) has a diffusive pole ω D ( q ) = − iDq (36)for small q (see Eq. (38) for the full expression). Here, D is the diffusion constant D = Kτ /ρ (0) M . However, thepole (36) does not coincide with the superdiffusive mode ω + ( q ) of Eq. (17), because the electric conductivity char-acterizes the system’s response to the total electric field E = −∇ φ . For a given electrostatic potential φ , thecharge density is fixed through Eq. (30). Indeed, a chargedistribution evolving according to Eq. (36) would notsolve the Eqs. (7), (6) (see also Ref. [72] for a similardiscussion). Combining (33), (34) we obtain the Einsteinrelation D = σ (cid:107) ( ω → , χ ρ Q ρ Q (0 , q → . (37)The order of limits for ω and q is essential, since at finite ω , q , Eq. (35) determines the ration σ (cid:107) /χ ρ Q ρ Q . Finallywe note, that even though diffusion is normal for Yukawainteracting charges (see Eq. (28)), the diffusion constant D Y is not equal to the D of Eq. (37). C. Gated systems
Two dimensional solid-state systems can be manipu-lated by gates [73, 74]. In particular, if the distance be-tween the 2D channel and the gate is smaller than thelength scales of the charge inhomogeneities inside the 2Dlayer, the Poisson term (11) on the right of Eq. (7) canbe replaced by a capacitive term φ C = ρ (1) Q /C , where C is the gate capacitance per unit area. This is the local ca-pacitance approximation, which is appropriate for manygated devices [75–78]. The force −∇ φ C stemming fromthe capacitive term can be absorbed into the pressureterm of the Navier-Stokes equation with the substitution K → ˜ K = K + ρ (0) Q C .
The absence of the Poisson term (11) changes the dis-persion relations of the hydrodynamic modes. Instead ofEq. (14), the hydrodynamic modes are now determinedby ω g ± = − i + iνq τ ± (cid:118)(cid:117)(cid:117)(cid:116) ˜ Kρ (0) M q − (1 + νq ) τ . (38)The subscript g indicates, that we are considering a gatedsystem with a uniform electrostatic potential. For small q we have ω g − = − iτ − iq (cid:32) ντ − ˜ Kτρ ( ) M (cid:33) (39) ω g + = − i ˜ Kq τρ (0) M . (40)While ω g − is gapped, ω g + is an ordinary diffusive modewith a diffusion constant ˜ D = ˜ Kτ /ρ (0) M . For a gatedstructure the diffusion constant governing the diffusionof charges is indeed equal to the one obtained from theEinstein relation (37). V. MAGNETIC FIELDS
Magnetic fields qualitatively change the spectrum ofcollective excitations of a charged liquid. Under the in-fluence of a magnetic field B charges oscillate at the cy-clotron frequency ω c = Bρ (0) Q /ρ (0) M c. (41)At finite wavevectors, the cyclotron resonance mergeswith the plasmon and gives rise to the magnetoplasmonmode. We are interested in how the dispersion relations of collectives modes change when both a momentum re-laxation time τ and a uniform magnetic field are added.This can be studied by adding a uniform magnetic fieldoriented perpendicular to the fluid plane to the Navier-Stokes equation: ∂ t ( ρ M u i ) + ∂ j Π ij = − τ ρ M u i − ρ Q ∇ φ + Bc ε ij j Q ,j . (42)The last term in Eq. (42) describes the Lorentz force ex-erted on the fluid by the magnetic field. We assume thatthe magnetic field is weak, such that Landau quantizu-ation effects, as well as the localization of electrons oncyclotron orbits can be neglected. Linearizing Eq. (42)in u i , ρ (1) M and ρ (1) Q , performing a Fourier transform andwriting the equation in terms of matrices, we find D ( ω, q ) u ( ω, q ) = 0 , (43)with D = (cid:0) − iω + τ − (cid:1) + (cid:18) aiωq + iKρ (0) M ω + ζρ (0) M (cid:19) q + νq (cid:18) aiωq + iKρ (0) M ω + ζρ (0) M (cid:19) q q − ω c (cid:18) aiωq + iKρ (0) M ω + ζρ (0) M (cid:19) q q + ω c (cid:0) − iω + τ − (cid:1) + (cid:18) aiωq + iKρ (0) M ω + ζρ (0) M (cid:19) q + ηρ (0) M q . (44)Here ν = η/ρ (0) M is the kinematic viscosity. The dispersion relations of collective modes can be found by setting det ( A ) = 0 . (45)In the limit τ → ∞ , Eq. (45) gives the magnetoplasmon dispersion [79, 80] ω mp, ± = ± (cid:112) aq + ω c , (46)where the conventional square-root plasmon spectrum is gapped out by the magnetic field. It is interesting to notethat in Eq. (46) the two limits q → and ω c → are not interchageable, yielding ω mp, + ≈ (cid:112) aq + ω c √ aq , ω c (cid:28) aq (47)and ω mp, + ≈ ω c + aq/ω c , aq (cid:28) ω c . (48)Either the cyclotron motion or the plasmon waves dominate the collective behavior. This behavior is even morestriking at finite τ . The modes ω − , ω ⊥ of the Eqs. (15) and (18) the become ω mag − ≈ − iτ − ic + 2 iaτ (cid:0) − ω c τ (cid:1) q − ic q − i c q , ω c (cid:28) aq (49) ω mag − ≈ − ω c (cid:0) aτ q (cid:1) − iτ + iaτ (cid:0) − ω c τ (cid:1) q , aq (cid:28) ω c (50) ω mag ⊥ ≈ − iτ + ic + ic q + i c q , ω c (cid:28) aq (51) ω mag ⊥ ≈ ω c (cid:0) aτ q (cid:1) − iτ + iaτ (cid:0) − ω c τ (cid:1) q , aq (cid:28) ω c , (52)where we have used the abbreviations c = ω c ζ − Kτ a τ ρ (0) M , c = ω c a τ νρ (0) M + τ K + ζ − ζKτ a τ (cid:16) ρ (0) M (cid:17) and c = ω c aτ . In the limit ofsmall but finite magnetic fields, the modes ω mag − , ω mag ⊥ acquire a dispersive real part of ± ω c (cid:0) aτ q (cid:1) and becomewavelike, albeit heavily damped. The mode spectrum for larger q is quite complicated and is depicted in Figs. 3 and4. For the superdiffusive mode (17) the two limits q → and ω c → are interchageable. In both cases the superdiffusivedispersion reads ω mag+ = − iaτ (cid:0) − ω c τ (cid:1) q. (53)The superdiffusion is thus slower by a factor of − ω c τ . Figure 3. Collective modes of a charged two dimensional liq-uid in the presence of momentum relaxation and a perpendic-ular magnetic field. The two damped magnetoplasmon modes ω mag ⊥ and ω mag − and the superdiffusive mode ω mag+ are shown.The colored dashed lines correspond to the approximations ofEqs. (50), (52) and (53). The blue dashed line depicts thedamped plasmon dispersion in the absence of magnetic fields ω ± given in Eq. (31), which is a reasonable approximationto the magnetoplasmon dispersion at larger q . At sufficientlylarge magnetic fields, the cyclotron resonance and the dampedplasmon mode merge. Here ω c = 0 . τ − was chosen, where ω c is the cyclotron frequency and τ − is the rate of momentumrelaxation. VI. LANGEVIN EQUATIONS AND LÉVYFLIGHTS
Diffusion processes can be modeled with Langevin-typestochastic equations. Here, we demonstrate that chargedparticles interacting via the Coulomb potential, while un-dergoing diffusion, indeed follow Lévy flight trajectoriesthat produce the superdiffusive dynamics of Eq. (17).Our starting point are coupled Langevin equations for
Figure 4. Collective modes of a charged two dimensional liq-uid in the presence of momentum relaxation and a perpendic-ular magnetic field. In contrast to Fig. 3, a smaller magneticfield ( ω c = 0 . τ − ) was chosen. The two damped magneto-plasmon modes ω mag ⊥ and ω mag − and the superdiffusive mode ω mag+ are shown. The colored dashed lines correspond to theapproximations of Eqs. (50), (52) and (53). The blue dashedline depicts the damped plasmon dispersion in the absence ofmagnetic fields ω ± given in Eq. (31), which is a reasonableapproximation to the magnetoplasmon dispersion at larger q .The cyclotron resonance and the damped plasmon mode arewell separated, whereas at larger fields strengths, these modesmerge (see Fig. 3). the particle coordinates r i ( t ) [42, 44, 81]: ¨ r ( i ) = − Q m N (cid:88) j (cid:54) = i (cid:12)(cid:12) r ( j ) − r ( i ) (cid:12)(cid:12) − τ ˙ r + 1 m η ( i ) . (54) Q is the particle charge and η i is an uncorrelatedstochastic force for which holds (cid:68) η ( i ) k ( t ) η ( j ) l ( t (cid:48) ) (cid:69) = κδ ij δ kl δ ( t − t (cid:48) ) . The Eq. (54) could describe a one-component Coulomb plasma [44] (as e.g. realized bymacroions in colloidal suspensions [45, 82]). Here κ = mk B T τ − holds due to the Einstein relation.It is a textbook result that without the Coulomb termin Eq. (54), the particles will undergo Brownian motion.Indeed, for Q = 0 the response to the stochastic force isgiven by r ( i ) ( t ) = τm ˆ t dt (cid:48) (cid:18) − e − ( t − t (cid:48) ) τ (cid:19) η ( i ) ( t (cid:48) ) where we assumed that η was switched on at t = 0 . Forthe variance r ( i ) ( t ) follows (cid:68) r ( i ) ( t ) · r ( j ) ( t ) (cid:69) ≈ δ ij τ κm t, which corresponds to an ordinary Gaussian diffusion pro-cess. To demonstrate, how the non-Gaussian superdiffu-sive dynamics of Eq. (17) emerges once the Coulombinteractions are turned on, we integrate Eq. (54) nu-merically. In the simulations, we used periodic bound-ary conditions and have chosen t c = (cid:112) mr c / Q as ourunit of time. The length r c is chosen arbitrarily (boxsize L = 25 r c ) but can be related to the Wigner-Seitzradius a : a = √ πn ≈ . r c . At t = 0 , the system con-sists of a uniform background distribution of particles ρ (0) Q = 0 . /r c and a small number of non-equlibriumparticles ( n = 20 ) localized completely within the unitsquare Θ (1 − | x | ) Θ (1 − | y | ) . The coupling parame-ter Γ is small Γ = Q / ( ak B T ) ≈ . and the dampingis substantial: τ = 0 . t c .The movements of the particles at t > can be in-terpreted as random walks. From the discussion of Sec.III, one expects that the distances travelled by the par-ticles during an interval ∆ t are distributed according tothe Cauchy distribution Eq. (21). I.e., if the diffusion isindeed anomalous with a coefficient α = 1 , the step sizes ∆ r of the random walk will follow a fat-tail powerlawdistribution decaying as p (∆ r ) ∼ r . (55)The superdiffusive behavior manifests itself at smallwavevectors, i.e. large distances, therefore the behav-ior for small step sizes will deviate from the ∆ r − law.Since the variance of the Cauchy distribution is not de-fined, we cannot identify the superdiffusive dynamics bymeasuring the correlation function (cid:10) r ( i ) ( t ) · r ( j ) ( t ) (cid:11) . In-stead, the step size distribution of Eq. (55) can be usedto study the Lévy flight nature of the diffusion process.Fig. 1 shows the step size distribution obtained inour computational experiment. The power-law decay ofEqs. (21), (55) for large ∆ r is clearly observed. Thus,the diffusive random motion of Coulomb interacting twodimensional particles is a non-Gaussian, Lévy stable ran-dom walk. From the general properties of such randomwalks, we know that the mean travelled distance of a par-ticles grows as t , in contrast to the √ t scaling of normaldiffusion [48, 49]. VII. SPECIFIC HEAT
Finally, we want to discuss the contributions of the (su-per)diffusive modes to the specific heat C of a chargedtwo dimensional liquid. The thermodynamics of collec-tive excitation has been ithe subject of many studies (seee.g. [52, 83–85]). In Appendix B, we show that themodes’ contribution to internal energy density E is givenby E = ˆ ∞ dε εν ( ε ) e βε − , where β = 1 /k B T and ν ( ε ) is the density of states of themodes: ν ( ε ) = − π ˆ q ∗ qdq π Im [ G ( ε, q )] . (56) G ( ε, q ) is the Green’s function of the diffusion equation. q ∗ serves as a momentum cut-off. The specific heat isdefined as c V = ∂E∂T . To simplify the analysis, we will focus on low tempera-tures, where the relevant modes will be the superdiffusivemode of Eq. (17) while the gapped mode (16) will onlygain importance at higher temperatures. We begin withthe superdiffusive mode ω + = − ia | q | and obtain ν ( ε ) = 2 aq ∗ τ − ε tan − (2 aq ∗ τ /ε )8 a τ π . (57)The superdiffusive mode contribution to the heat capac-ity for small temperatures is then given by c V = c T − c T + c T − O (cid:0) T (cid:1) , where all higher orders are of odd powers in T . For thecoefficient of the linear term we obtain c = q ∗ aτ . Sur-prisingly, c = ζ (3)8 πa τ does not depend on the momentumcut-off q ∗ and has a negative sign (although C is alwayspositive). This is due to the fact that the T -dependencecan be traced back to the nonanalyticity of the superdif-fusive mode at small q . We observe that the integrandof Eq. (56) depends on ε , yet the density of states ν ( ε ) has a linear term in ε . We can extract this term from theintegral of Eq. (56): lim ε → ν ( ε ) − ν (0) ε = lim ε → πaτ ˆ q ∗ dq π ε (cid:18) a τ q ε + 4 a τ q − q (cid:19) = − πa τ ˆ ∞−∞ dq δ ( q ) . (58)For small energies, the DOS is ν ( ε ) ≈ q ∗ π aτ − ε πa τ , (59)0giving the above values of c , c . It is well known thatto leading order at low temperatures, the specific heat C of Galilei invariant Fermi liquids is linear in the tem-perature, just as for free Fermions. However the analogydoes not hold beyond the leading order term. Nonana-lytic terms in the fermion self energy of two dimensionalFermi liquids result in corrections to the specific heat δC which behave as δC ∼ T [86–88]. This result is true forboth Coulomb and short range interactions [89]. Herewe show that in the presence of momentum relaxation,the nonanalytic superdiffusive mode of Eq. (17) as wellcontributes a ∼ T correction to the specific heat, how-ever with an opposite sign. This is in contrast to theplasmon resonance of a two dimensional charged systemwhere the plasmon dispersion is given by ω = √ aq andonly contributes a sub-subleading ∼ T term [52].For the specific heat contribution of the diffusive mode ω g + of gated 2D systems we obtain c V,g = d T + d (cid:48) T log (cid:18) T (cid:19) + d T + O (cid:0) T (cid:1) . (60)Gating thus qualitatively changes the specific heat of acharged two dimensional system. The low temperaturebehavior will be dominated by the T log (1 /T ) term. In-terestingly, several mechanisms have been discussed thatlead to a T log (1 /T ) temperature dependence of the spe-cific heat in two dimensional systems, such as quantumcritical fluctuations of overdamped bosonic modes with adynamical exponent z = 2 [54] and scattering betweenhot Fermi pocket and cold Fermi surface electrons inSr Ru O [55]. For Sr Ru O , the T log (1 /T ) contri-bution has been observed experimentally [53]. VIII. ACKNOWEDGEMENTS
I am greatful to Michael Bonitz, Igor Gornyi and JörgSchmalian for helpful comments and discussions. I alsowant thank Bhilahari Jeevanesan and Jonas Karcher whohelped to improve this manuscript.
Appendix A: Charge and mass densities in systemswithout Galilean invariance
In Galilean invariant systems the notions of mass andcharge densities are straightforward. If ρ is the particlenumber density, the mass density is given by ρ M = mρ and the charge density by ρ Q = eρ , where m and e are themass and charge of a particle. However many solid statesystems do not exhibit Galilean invariance, and it is use-full to extend the definitions of mass and charge densitiesto non-Galilean invariant, yet translation invariant sys-tems, where momentum conservation ensures the validityof hydrodynamics. Here the velocity u i ( x ) is defined asa source of the conserved crystal momentum [38]. With the shift H → H − ˆ d x u i ( t, x ) P i ( x ) , (A1)where H is the full Hamiltonian of the system and P i ( x ) is the momentum operator, the densities ρ M ( t, x ) , ρ Q ( t, x ) can be defined as response functions and calcu-lated using the memory matrix formalism [21, 38]. Mem-ory matrices allow to construct a hydrodynamic approx-imation to a quantum system by restricting the infinite-dimensional space of possible observables to a few con-served quantities and quantities which decay at very longtime scales. Sticking to the notation of Ref. [38], wewill call these quantities X A . Their thermodynamic con-jugate shall be called U B . An important object is thegeneralized conductivity σ AB . The memory matrix for-malism provides efficient means for its calculation. Thegeneralized conductivity relates the quantities X A to thefields ˙ U B : (cid:104) X A (cid:105) = − σ AB ˙ U B , (A2)where a summation over the index B labelling the(quasi)conserved quantities is implied. σ AB can be ex-pressed in terms of retarded Green’s functions σ AB ( z, q ) = 1 iz (cid:0) G RAB ( z, q ) − G RAB ( i , q ) (cid:1) (A3)with G RAB ( t, x ) = − i Θ ( t ) (cid:104) [ X A ( t, x ) , X B (0 , )] (cid:105) . As is customary in memory matrix literature, we usedthe Laplace transform G RAB ( z, q ) = ˆ ∞ dt e izt G RAB ( t, q ) . In our case the quantities X A include the momentum P i , which, following Eq. (A1), is sourced by the velocity u i . Using Eq. (A2), we write (cid:104) P i (cid:105) = − izσ P i ,u i u i . (A4)The above equation suggests that the mass densityshould be defined as ρ M = − izσ P i ,u i . It follows (cid:104) P i (cid:105) ( t, x ) = ˆ dt (cid:48) d x (cid:48) ρ M ( t − t (cid:48) , x − x (cid:48) ) u i ( t (cid:48) , x (cid:48) ) . (A5)Finally, we should keep in mind that the scales of hydro-dynamic temporal and spatial inhomogeneities t h and l h are much smaller than any time or length scale ˜ t , ˜ l char-acterizing the Hamiltonian H . Since the Green’s func-tions in the Eq. (A3) are calculated at vanishing flowvelocities, they will decay on scales given by | t − t (cid:48) | ≈ ˜ t , | x − x (cid:48) | ≈ ˜ l . On the other hand, the flow velocity u i varies on scales t h , l h . Thus, in the hydrodynamic limit1 ˜ t (cid:28) t h , ˜ l (cid:28) l h , Eq. (A5) can be approximated by thelocal relation g i ( t, x ) ≈ ρ M ( t, x ) u i ( t, x ) . (A6) ρ M ( t, x ) is the mass density used throughout the text.Similarely, we arrive at ρ Q = − izσ J Q ,i ,u i , where J Q ,i isthe electric current operator, and finally j Q ,i ≈ ρ Q ( t, x ) u i ( t, x ) . (A7) Appendix B: Specific heat contribution of collectivemodes
We begin with the partition function Z = ˆ D [ φ q ] e − (cid:80) q ,n φ q ,iωn G − ( iω n , q ) φ q ,iωn . (B1)Here, G ( iω n , q ) is the Green’s function of the dampedbosonic plasmon mode G ( iω n , q ) = 1 − iω n + ω + ( q ) , (B2) where ω + ( q ) was introduced in Eq. (17). φ q is repre-senting the bosonic plasmon fields. The heat capacitycan be calculated from the internal energy E , which isgiven by [90, 91] E = − ∂∂β ˆ d q (2 π ) (cid:88) n ln (cid:2) βG − ( iω n , q ) (cid:3) . (B3)First, we evaluate the Matsubara sum over bosonic fre-quencies ω n = 2 πn/β by rewriting it as a contour integralaround the imaginary axis of a variable ε : E = − ∂∂β ˆ d q (2 π ) ˆ C dε (2 πi ) βe βε − β ( − ε + ω + ( q ))] . In the following, it will be convenient to use the abbre-viation ω + ( q ) ≈ − iaτ q ≡ ξ. (B4)The integrand has a branch cut at Re ε > and Im ε = ξ .Correspondingly, the contour can be deformed such thatit encircles the line Im ε = − iaτ q running from toinfinity and back. The internal energy is then given by E = − ∂∂β ˆ d q (2 π ) ˆ ∞ dε (2 πi ) (cid:26) βe β ( ε + ξ ) − (cid:2) β (cid:0) − (cid:0) ε + ξ + i + (cid:1) + ξ (cid:1)(cid:3) − βe β ( ε + ξ ) − (cid:2) β (cid:0) − (cid:0) ε + ξ − i + (cid:1) + ξ (cid:1)(cid:3)(cid:27) (B5)or E = − ∂∂β ˆ d q (2 π ) ˆ ∞ dεπ βe βε − β ( − ε + ξ )]) . (B6)Simplifying the expression we obtain E = − ∂∂β ˆ d q (2 π ) ˆ ∞ dεπ (cid:26) ∂∂ε (cid:0) ln (cid:0) − e βε (cid:1) − βε (cid:1) × Im (ln [ β ( − ε + ξ )]) } = ∂∂β ˆ d q (2 π ) ˆ ∞ dεπ (cid:8)(cid:0) ln (cid:0) − e βε (cid:1) − βε (cid:1) × Im (cid:18) − ε + ξ (cid:19)(cid:27) = − ˆ d q (2 π ) ˆ ∞ dεπ εe βε − G ( ε, q ) . (B7)Keeping in mind that the imaginary part of the Green’sfunction determines the spectral function A ( ε, q ) via − π Im G ( ε, q ) = A ( ε, q ) , (B8)and the density of states ν ( ε ) is given by ν ( ε ) = ˆ d q (2 π ) A ( ε, q ) , (B9)formula (B7) can be interpreted as an energy average overthe Bose-Einstein distribution weightened by the densityof states: E = ˆ ∞ dε εν ( ε ) e βε − . (B10) [1] L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran,T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller,J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, One-dimensional electrical contact to a two-dimensionalmaterial , Science , 614 (2013). [2] M. J. M. de Jong and L. W. Molenkamp, Hydrodynamicelectron flow in high-mobility wires , Phys. Rev. B ,13389 (1995).[3] J. A. Sulpizio, L. Ella, A. Rozen, J. Birkbeck, D. J.Perello, D. Dutta, M. Ben-Shalom, T. Taniguchi,K. Watanabe, T. Holder, R. Queiroz, A. Principi,A. Stern, T. Scaffidi, A. K. Geim, and S. Ilani, Visu-alizing poiseuille flow of hydrodynamic electrons , Nature , 75 (2019).[4] G. Gusev, A. Jaroshevich, A. Levin, Z. Kvon, andA. Bakarov,
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