Electron currents from temporal gradients in tilted Dirac cone materials: Electric energy enabled by spacetime geometry
EElectron currents from temporal gradients in tilted Dirac cone materials:Electric energy enabled by pacetime geometry
A. Moradpouri, ∗ Mahdi Torabian, † and S.A. Jafari ‡ Department of Physics , Sharif University of Technology , Tehran 11155-9161 , Iran (Dated: July 8, 2020)Tilted Dirac/Weyl fermions admit a geometric description in terms of an effective spacetime metric. Using thismetric, we formulate the hydrodynamics theory for tilted Dirac/Weyl materials in d + 1 spacetime dimensions.We find that the mingling of spacetime through the off-diagonal components of the metric gives rise to: (i)heat and electric currents proportional to the temporal gradient of temperature, ∂ t T and (ii) a non-zero Hallcondductance σ ij ∝ ζ i ζ i where ζ j parametrizes the tilt in j ’th space direction. The finding (i) above suggeststhat naturally available sources of ∂ t T in hot deserts can serve as new concept for the extraction of electricityfrom the spacetime geometry. We find a further tilt-induced non-Drude contribution to conductivity which canbe experimentally disentangles from the usual Drude pole. I. INTRODUCTION
The motion of electrons in conductors in the absence of ex-ternal temperature and/or electro-chemical gradients is purelyrandom thermal motion [1]. Once a spatial temperature gra-dient ∇ T is introduced, the vector character of ∇ speci-fies a preferred direction, and therefore electrons flow along ∇ T [2]. The purpose of this paper is to propose a class ofmaterials where a temporal gradient ∂ t T alone (i.e. withoutany spatial gradient) is sufficient to generate electric and/orheat currents. Why is it important to be able to derive cur-rent by temporal gradient? In the so called hot deserts suchas Sahara or Kavir and Lut deserts of Iran, due to clear sky,the heat loss in the night is substantial and the daily variationof temperature can be as great as 22 ◦ C . Therefore the hotdeserts with their cold midnights and hot noons can serve asa free provider of ∂ t T through a half cycle of the spinning ofthe earth. Can this naturally available temporal gradient beemployed in energy production?To set the stage, suppose that for some reason, there is apreferred direction in the space determined by a vector ζ . Canthe anisotropy arising from such a preferred direction organizethe random motion of electrons into non-zero electric or heatcurrents? As we will discuss in this paper, there are certainmaterials where such a vector ζ not only does exist, but ad-ditionally, it enters the effective description of the motion ofelectrons in such a way that it mingles space and time coordi-nates, whereby creates a new spacetime structure. As a result,even a pure temporal gradient of a spatially uniform temper-ature T can drive a current via mixing of the space and timecoordinates arising from ζ .Unlike the electrons in the standard model of elementaryparticle physics [3], the electrons in the solids are mountedon a lattice. Every periodic lattice structure belongs to oneof the 230 possible space groups [4]. Some of these struc-tures afford to give a low-energy effective theory for the elec- ∗ [email protected] † [email protected] ‡ [email protected] See en.wikipedia.org/wiki/Desert trons that is mathematically equivalent to the Dirac theory(e.g. in graphene [5]) or deformations of the Dirac theoryas in certain structure of borophene [6, 7] or the organic com-pound α − (BEDT-TTF) I [8–13] or certain deformations ofgraphene [14]. For the Dirac electrons in solids, the disper-sion relation giving energy ε of the quasiparticles at everymomentum k will be a Dirac cone, and enjoys an emergentLorentz invariance at low energies corresponding to lengthscales much larger than the lattice spacing [5]. But in ma-terials hosting a tilted Dirac cone, the cone-shaped dispersionrelation is tilted in energy-momentum space [14, 15]. Thistilt is characterized by set of parameters ζ . Presence of tilt inthe Dirac cone, not only bestows the space with a preferreddirection set by the vector ζ [14, 15], but also gives rise toan emergent spacetime structure [16–18]. It turns out that,the tilt can be attributed to an underlying metric of the space-time [16, 19, 20]. Therefore in the same way that Dirac ma-terials enjoy an emergent Lorentz symmetry arising from anemergent Minkowski spacetime at long wave length for thelattice at hand, those materials that host a tilted Dirac cone canbe assigned an emergent spacetime structure at long lengthscales which is controlled by ζ and is given by ds = − v F dt + ( d r − ζ v F dt ) , (1)where v F is the velocity scale that defines the emergent space-time (which plays a role similar to the light speed c , but in realmaterials is two or three orders of magnitude smaller than c )and ζ encodes the tilt of the Dirac cone [20]. For the rest of thepaper we will set v F = 1 . The isometries of the spacetime (1)are appropriate deformations of the Lorentz group [16]. Assuch, the spacetime defined by Eq. (1) is a deformation of theMinkowski spacetime by a continuous parameter ζ . The pres-ence of tilt ζ modifies many of the physical properties of thematerials, in particular including their interfaces with super-conductors [21, 22]. The above metric possesses a blackholehorizon [23] that stems from spatial variation of the Gallileanboost ζ in Eq. (1). Spatial variation of a parameter simi-lar to ζ can emulate a black-hole horizion in atomic Bose-Einstein condensates [24], polariton superfluids [25, 26]. Thespin-polarized currents are also predicted to implement black-hole horizon for magnons [27]. Our proposal differs fromthe above systems in that (i) the tilted Dirac/Weyl systems a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l required are at ambient conditions and (ii) the quasiparticlesof the theory are fermions, namely electrons and holes whichcarry electric charges. As such, any effects arising from theemergent structure of the spacetime, will leave direct signa-ture in almost any electron spectroscopy experiment, includ-ing of course the transport (conductivity) measurements.In this paper, we set out to use the hydrodynamics as a tech-nical tool to study the electron fluid [28] in the tilted Diraccone mateirals where the spacetime is given by metric (1).Let us announce our result in advance: In any conductor, thespatial gradient of temperature can generate heat and electriccurrents. The mingling of the spacetime in Eq. (1) will allowthe materials with tilted Dirac cone to generate heat and elec-tric currents from pure temporal gradients. Our theoreticaltool to bring out this result is the hydrodynamics. Hydrody-namic is an effective long-time and long-distance descriptionof quantum many body systems that focuses on few conservedcollective variables rather than embarking on the formidbletask of addressing all the microscopic degrees of freedom.Hydrodynamics as a powerful universal approach, hasmany applications in various systems differing in microscopicdetails which are however, described by the same equations.Only gross symmetry properties has to be properly incor-porated into the formulation of hydrodynamic for a systemat hand. Applications of hydrodynamics approach in highenergy physics includes quark-qluen plasma [29, 30], par-ity violation [31], chiral anomaly [32] and dissipative su-perfluid [33]. Within the hydrodynamic approach, one cancome up with universal and model-independent predicationsuch as kinematic viscosity value [34, 35]. The hydrodynam-ics approach can also be applied to fluid-gravity correspon-dence [36] which relate the dynamics of the gravity side tothe hydrodynamic equations where the hydrodynamic fluctu-ation mode describes the fluctuations of black holes [37, 38].In this work, we will be interested in much simpler version ofhydrodynamics in a spacetime structure in d + 1 dimensionssubject to the metric (1) that describes electrons in sub-eV en-ergy scales in solids with tilted Dirac cone.The roadmap of the paper is as follows: In section II fol-lowing Lucas [39] we formulate the hydrodynamics for themetric (1) as a natural generalization of the hydrodynamicstheory of graphene. In section III, in addition to the usual tensor transport coefficients of standards solids, we introduce vector transport coefficients required for a consistence treat-ment of the hydrodynamics in spacetime (1) where we discussperfect fluid. It is followd by a treatment of the viscous fluidsin section IV in this new spacetime. We end the paper withdiscussion and summary of the paper in section V. II. HYDRODYNAMICS OF TILTED DIRAC FERMIONS
In this section we develop the hydrodynamics of electronsin a tilted Dirac cone material. For a planar material in d = 2 space dimensions, there is a two-parameter family of tilt de-formations given by (cid:126)ζ = ( ζ x , ζ y ) to the Dirac equation inthree dimensional spacetime. These deformations and the cor-responding dispersion relation can be obtained if instead of the conventional Lorentz metric η µν one applies the follow-ing metric tensor g µν = (cid:18) − ζ − ζ j − ζ i δ ij (cid:19) . (2)where ζ = ζ x + ζ y and δ is the × unit matrix [16, 20].In this parametrization of the tilt we adopt the normalization | ζ | < so that spacetime makes sense in admissible coordi-nates. For the sake of convenience in the following computa-tions, we introduce γ = (1 − ζ ) − / . The Greek indices runover 0,1 and 2 for spacetime coordinates and the Latin indicesrun over spatial coordinates.We assume that the electron-electron scattering rate τ − ee take over any other scattering rate such as electron-phonon( τ − − ph ) or electron-impurity scattering rate ( τ − − imp ). Thisregime is attainable in graphene that hosts upright Diraccone [40–43]. This has become possible by ability to tune thestrength of electron-electron interaction via gating that sets thescale of the Fermi surface. In such a regime, an effective de-scription at large distances ( (cid:29) v F τ ee ) over long time ( (cid:29) τ ee )is provided by the hydrodynamics equations given by conser-vation laws of Noether currents. Assuming translational in-variance and gauge invariance of the underlying microscopictheory, there are conserved energy-momentum tensor T µν andan Abelian current vector J µ . They constitute ten indepen-dent components subject to four constraints ∂ µ T µν = 0 and ∂ µ J µ = 0 .In order to find unique solutions to hydrodynamics equa-tion, it is assumed that the currents are determined throughfour auxiliary local thermodynamical quantities: the temper-ature T ( x ) , the chemical potential µ ( x ) , a normalized time-like velocity vector field u µ ( x ) ( i.e. u µ u µ = − ) and theirderivatives. The fluid observer moves along with the fluid andmeasures variables (local temperature, local mass density etc.)without ambiguities. The 3-velocity u µ is defined relative tothe Eulerian (arbitrary) observer. The fluid velocity v i is de-fined through v µ = u µ /u . The generalized Lorentz factoris defined as Γ ≡ − n µ u µ = u where n µ = ( − ,(cid:126) is thetime-like normal vector to the 2-space. An Eulerian observerattributes this factor to matter moving in the fluid frame. Forinstance, given the temperature measured by the fluid observer T , an Eulerian observer finds T E = Γ T Given an arbitrary vector, any tensor can be decomposed toits transverse and longitudinal components with respect to thatvector. We have a freedom to identify u µ with the velocityof energy flow in the so-called Landau frame u µ ∼ T µν u ν .Moreover, T and µ can be defined so that u µ J µ = − n and u µ T µν = − (cid:15)u ν , (3)where n is the number density of charge carriers and (cid:15) is theenergy density. In this frame, the energy-momentum tensorand particle current can be decomposed respectively as fol-lows [44, 45] J µ = nu µ + j µ , (4) T µν = (cid:15)u µ u ν + P P µν + t µν , (5)where P is a scalar, j µ , P µν and t µν are transverse tensorsthat satisfy u µ j µ = u µ P µν = u µ t µν . Tensor t is symmetrictraceless t µµ = 0 . Tensor P is defined as P µν = g µν + u µ u ν , (6)which is called the projection tensor; it is symmetric and ingeneral has non-vanishing trace. The inverse metric is g µν = (cid:18) − − ζ i − ζ j δ ij − ζ i ζ j (cid:19) . (7)The remaining elements P, j µ and t µν are determined interms of derivatives of hydrodynamic variables and yield theconstituent equations at desired order. At first order hydrody-namics we find P = p − ζ B ∂ µ u µ ,j µ = − σ Q T P µν ∂ ν ( µ/T ) + σ Q P µν F νρ u ρ ,t µν η P µρ P νσ (( ∂ ρ u σ + ∂ σ u ρ − g ρσ ∂ α u α )+ ξ B g ρσ ∂ α u α ) , (8)where p is pressure in the local rest frame, ξ B is the bulk vis-cosity, σ Q is the intrinsic conductivity and η is the shear vis-cosity. Eq (8) is written in the presence of an external electro-magnetic field determined by F νρ . In passing, we recall thatcoefficients in zero-order hydrodynamics (cid:15) , p and n are fixedby T , µ and the equation of state in equilibrium thermody-namics [46–48]. Moreover, the non-negative parameters ζ B , σ Q and η (the Wilsonian coefficients of the effective hydro-dynamic theory) are either measured in experiments or deter-mined from an underlying microscopic (quantum) theory. III. EMERGENT VECTOR TRANSPORT COEFFICIENTSIN TILTED DIRAC SYSTEM:
We define the response of the electric current (cid:126)J and the heatcurrent (cid:126)Q to an external electric field (cid:126)E , spatial gradient (cid:126) ∇ T and possibly temporal variation ∂ T of temperature as follows J i ( t ) = (cid:90) d t (cid:48) (cid:20) σ ij E j ( t (cid:48) ) − α ij ∂ j T ( t (cid:48) ) − β i T ( t (cid:48) ) ∂ µ ( t (cid:48) ) T ( t (cid:48) ) (cid:21) , (9) Q i ( t ) = (cid:90) d t (cid:48) (cid:20) T ¯ α ij E j ( t (cid:48) ) − ¯ κ ij ∂ j T ( t (cid:48) ) − µγ i T ( t (cid:48) ) ∂ µ ( t (cid:48) ) T ( t (cid:48) ) (cid:21) , (10)where σ ij , α ij , ¯ α ij and ¯ κ ij are the usual tensor response co-efficients relating the electric and heat currents to spatial gra-dients of electrochemical poential or temperature [2]. Antic-ipating electric/heat currents in response to temporal gradient ∂ T , we have additionally introduced the vector transport co-efficents β i and γ i . All the above coefficients are functions of t − t (cid:48) by time-translational invariance.We compute the above transport coefficients within hydro-dynamics theory. We imagine that fluid is perturbed aroundits equilibrium state (specified by µ , T and u µ = (1 − ζ ) − / (1 ,(cid:126) in the rest frame of the fluid exposed to (cid:126)E = (cid:126) )by a slight amount parametrized as follows δT ( t, (cid:126)x ) , δu µ ( t, (cid:126)x ) = γ ( γ (cid:126)ζ · δ(cid:126)v, δ(cid:126)v ) , δ (cid:126)E ( t, (cid:126)x ) . (11) The first order perturbation in the electric current is given by δJ i = nγδv i − σ Q ζ i µ T ∂ δT − σ Q g ij (cid:16) γδE j − µ T ∂ j δT (cid:17) . (12)Moreover, the leading order perturbation in momentum is δT i = γ ( (cid:15) + p ) δv i − δP ζ i (13) + ηγ ζ ζ i (2 ∂ δu − g ∂ α δu α ) − η ( ζ j ζ i + γ ζ g ij )( ∂ δu j + ∂ j δu − g j ∂ α δu α )+ ηζ j g ik ( ∂ j δu k + ∂ k δu j − g jk ∂ α δu α ) + ξζ i ∂ α δu α . with which we compute thermal conductivity through δQ i = δT i − δ ( µJ i ) . (14)An interesting feature of Eq. (8) in the tilted Dirac/Weylmaterials is a genuine effect where temporal variations gen-erate currents: We note that in a linear hydrodynamic theorythe term in the brackets in this equation are already first order,and therefore at this order P µν → g µν . Therefore the spatialcomponent J i of the current (in addition to the first term nu i )will acquire a contribution proportional to ∂ µ and ∂ T whichis accompanied by the tilt parameters g i = − ζ i . This effectis solely dependent on the tilt parameters ζ i and vanishes forupright Dirac/Weyl systems where ζ i = 0 . Therefore, we ex-tend the conventional thermoelectric coefficients to accountfor this through additional transport coefficients β i and γ i inEqns. (9) and (10).We solve the hydrodynamics equations to evaluate the elec-tric current and thermal current in the presence of backgroundelectric field and temperature spatiotemporal gradients. Atthis leading order, the charge conservation ∂ µ J µ = 0 implies ∂ t (cid:104) γδn − nγ ζ i δv i + µ T σ Q ( γ ζ ∂ − ζ i ∂ i ) δT + σ Q ζ i γδE i (cid:105) + ∂ i ( δJ i ) = 0 . (15)Similarly, the energy conservation ∂ µ T µ = 0 implies ∂ t (cid:104) γ δ(cid:15) + γ ζ δp − γ ( (cid:15) + p ) ζ i δv i − ηγ ζ (2 ∂ δu − g ∂ α δu α )+2 ηζ γ ζ i ( ∂ i δu + ∂ δu i − g i ∂ α δu α ) − ηζ i ζ j ( ∂ i δu j + ∂ j δu i − g ij ∂ α δu α ) − ξγ ζ ∂ α u α (cid:105) + ∂ i δT i = 0 . (16)Finally, the momentum conservation ∂ µ T µi = 0 gives − (cid:15) + p τ imp δu i = ∂ ( δT i )+ ∂ i (cid:104) δP g ij − ηζ i ζ j (2 ∂ δu − g ∂ α δu α ) (17) + η ( ζ i g jk + ζ j g ik )( ∂ δu k + ∂ k δu − g k ∂ α δv α ) − ηg il g kj ( ∂ l δu k + ∂ k δu l − g lk ∂ α u α ) − ξg ij ∂ α δv α (cid:105) , where the parameter τ imp is the relaxation time due to scat-tering from impurities. The above expressions in the squarebracket are leading order perturbations in number density,energy, density and the stress tensor. We apply equations(15), (16) and (17) to extract transport coefficients. In pass-ing we note that, in the fluid rest frame the electric field, thetemperature gradient and the pressure gradient are balanced as − ∂ i P = nE i + s∂ i T . Thus, we apply the following constraint − ∂ i δP = n δE i + s ∂ i δT, (18)in the leading order in perturbations. A. Perfect fluid: Non-Drude features
We start by ignoring dissipation η = ξ = 0 , Furthermore,we are interested in a homogeneous flow; i.e. spatially uni-form solutions with ∂ j δv i = 0 . With these assumptions, theenergy-momentum conservation implies ∂ t (cid:104) γ ( δ(cid:15) + ζ δP ) − γ ( (cid:15) + p ) ζ j δv j (cid:105) − ζ i ∂ i δP = 0 , (19) ∂ t (cid:104) γ ( (cid:15) + p ) δv i − ζ i δP (cid:105) + g ij ∂ j δP = − (cid:15) + p τ imp γδv i . (20)We can solve the above equations for the velocity perturbation δv i = ( C − ) ij (cid:104) g kj − ζ j ζ k γ (2 + ζ ) (cid:105) ( − ∂ k δP ) , (21)where C − is the inverse matrix of C ij = (cid:104) γ (cid:15) + p τ imp (1 − iγωτ imp ) (cid:105) δ ij + (cid:104) iωγ (cid:15) + p ζ (cid:105) ζ i ζ j , (22)which is explicitly computed in the appendix. Then, we com-pute the electric current (12) as a response to spatiotemporalvariation of the electric field and temperature. Finally, by ap-plying equations (18) and (21) we can read the coefficients in(9) as follows σ ij = γn ( C − ) ik (cid:104) g kj − ζ j ζ k γ (2 + ζ ) (cid:105) − g ij γσ Q , (23) α ij = − γn s ( C − ) ik (cid:104) g kj − ζ j ζ k γ (2 + ζ ) (cid:105) − µ T σ Q g ij , (24) β i = − σ Q ζ i . (25)The pole structure of the electric and heat conductivity tensorsare the same. This is because they are both related to the de-terminant of the matrix C . Therefore, we focus on the poles ofthe conductivity. We find that the conductivity tensor has thefollowing two poles ω = − iγτ imp ≡ − iτ labimp , (26) ω = 2 + ζ − ζ ω . (27)In the upright Dirac cone with Minkowski spacetime struc-ture, ζ → , both poles ω and ω of Eqns. (26) and (27) re-duce to the Drude result. To understand these poles, we havedefined a redshifted relaxation time τ labimp = γτ imp . In this def-inition τ imp can be interpreted as the microscopic relaxation time experienced by electrons in the spacetime with a given ζ , while τ labimp can be interpreted as the same time measured inthe laboratory by the experimentalist sitting in the laboratoryspacetime having ζ = 0 . As such, the pole at ω is a naturalextension of the Drude pole to the geometry (1). However,the new pole ω arises from the new spacetime structure. Al-though at ζ → limit it becomes degenerate with the Drudepole, but at ζ → it can become up to times larger than ω . Both poles are on the imaginary axis, and their real partis zero, as they are caused by very low-energy (Drude) exci-tations across the Fermi level. Then the question will be, isthere a way to distinguish the contributions from the Drudepole ω and the spacetime pole ω ? To answer this question,we need to look at the residues at the two poles that determinethe spectral intensity associated with each pole.Let us start by looking at the residue of the absorptive partof the conductivity, namely the longitudinal conductivity σ xx .The residues at the above poles are computed as follows Res( σ xxω ) = n v (cid:15) + p ζ y γζ , (28) Res( σ xxω ) = 2 n v (cid:15) + p ζ x γ (2 − ζ ) ζ . (29)By symmertries of space time σ yy can be extracted from σ xx only by exchanging ζ x ↔ ζ y . So there is no new informationin residues of the poles of σ yy .As pointed out, both ω and ω poles are on the imaginaryaxis and their real part is zero. To disentangle their contribu-tion, not that the meaning of the longitudinal conductivity σ xx is the current along the applied electric field (both assumedalong the x direction). But then the x axis can subtend an an-gle θ with the tilt vector ζ . This can be an interesting variable.Therefore, in Fig. 1 we have plotted the dependence of theresidudes of the longitudinal part of the conductivity on thepolar angle θ of the tilt vector. The solid (dashed) lines cor-respond to the Drude-like pole ω (spacetime pole ω ). Vari-ous colors correspond to different tilt magnitudes. Both poleshave a dipolar pattern. But their nodal structure is differentwhich helps to identify which pole is contributing the spectralweight. When the electric field is applied along the tilt direc-tion ( θ = 0 ), the residue of the Drude-like pole vanishes andthe absorption is contributed by ω pole. By rotating the ap-plied electric field away from the tilt direction, the Drude-likepole ω takes over the spacetime pole ω . When the appliedelectric field is completely perpendicular to ζ , only ω con-tributes to the conductivity. The common feature of solid anddashed curves in Fig. 1 is that the spectral weight of both ω and ω poles decreases by increasing the tilt ζ . In all plots of this paper, the Fermi velocity v F has been explicitely in-cluded and expressed in units of [ rτ imp ] where r is a length scale corre-sponding to average one electron and is related to the density by nr = 1 .For typical n ∼ cm − , we get r = 10 − cm. Furthermore a typicalvalue of τ imp is equal to − s . FIG. 1. Polar dependence of residue of the longitudinal conduc-tivity σ xx for Drude like pole ω (solid lines) and spacetime pole ω (dashed lines) in units of [ mev.τ ] . The polar angle is mea-sured from the direction of the tilt vector ζ . B. Tilt-induced Hall response
Another unusual effect arising from the tilt in ideal fluidis the presence of a non-zero Hall and thermal Hall coeffi-ceitns in the absence of external magnetic field.
Due to off-diagonal metric elements of the metric g ij ∝ ζ i ζ i , a non-zeroHall and thermal Hall coefficient proportional to ζ i ζ j appearin Eqs. (23) and (24). This effect also solely depends on thepresence of the tilt ζ i . Mathematically, in the absence of ζ i (isotropic space), the conductivity tensor σ ij (as in the case ofgraphene) will become proportional to δ ij . But in the presentcase, the anisotropy of the space will be reflected in a ζ i ζ j dependence in all tensorial quantities including the conduc-tivity and heat conductivity tensors. As such, the non-zeroHall and thermal Hall coefficients in the tilted Dirac cone canbe directly attributed to the structure of the spacetime.Now let us consider the off-diagonal (transverse) compo-nent of the conductivity, namely σ xy . It turns out that therewill be a tilt-induced Hall response σ xy which is given by Res( σ xyω ) = − n v (cid:15) + p ζ x ζ y γζ . (30) Res( σ xyω ) = 2 n v (cid:15) + p ζ x ζ y γ (2 − ζ ) ζ . (31)Mathematically, σ ij (and also α ij ) being tensors are naturallyexpected to have a term proportional to g ij ∼ ζ i ζ j . But whatis the physical meaning of a non-zero Hall coefficient in theabsence of a background B z in the new spacetime geometry?Indeed in the small tilt limit, the metric (1) can be viewed as asuperposition of an additional ”Lorentz” boost with the boostparameter given by ζ [49]. As such the applied electric field FIG. 2. Polar dependence of residue of σ xy for ω (solid lines) and ω (dashed lines) in units of [ mev.τ ] . The polar angle denotes theanble between the applied electric field and the tilt vector ζ . can act like an effective B-field, generating the Hall response.But note that our finding is not limited to small ζ and is validfor any finite ζ . The residues of the Hall conductivity arisingfrom ω and ω poles are plotted in Fig. 2. As can be seenboth poles display quadrapular pattern. Also their behaviorwith ζ is similar. Both intensities decrease upon decreasingthe tilt magnitude ζ .The conclusion is that, the longitudinal conductivity is asuitable measure to disentangle the role of Drude-like pole ω and spactime pole ω in the conductivity of tilted Diracmaterial sheets. C. Accumulative heat current
Now let us turn our attention to the heat (energy) transportcoefficients. The thermal current in a non-viscous homoge-neous fluid is given by Q i = γ ( (cid:15) + p ) δv i − ζ i δP − µ δJ i , (32)where δJ i is given in (12). We use the energy conservationequation (19) and equation of state (cid:15) = 2 · P to compute thepressure perturbation δP as follows δP ( t ) = γ ζ (cid:104) (cid:15) + p ) ζ j δv j + γ − ζ j (cid:90) t d t (cid:48) ∂ j δP ( t (cid:48) ) (cid:105) . (33)Ignoring the last term in the above equation, we compute thecoefficients in thermal current (10) by applying equation (18),(21), (32) and (33) as follows ¯ α ij = n T I im ( C − ) mk (cid:104) g kj − ζ j ζ k γ (2 + ζ ) (cid:105) + γ µ T σ Q g ij , (34) ¯ κ ij = s I im ( C − ) mk (cid:104) g kj − ζ j ζ k γ (2 + ζ ) (cid:105) − µ T σ Q g ij , (35) γ i = − σ Q ζ i , (36)where I ij is defined as I ij = (cid:104) γ ( (cid:15) + p ) − γµ n ) (cid:105) δ ij − (cid:104) γ (cid:15) + p ζ (cid:105) ζ i ζ j . (37)The last term in (33) gives an additional contribution to ¯ α ij and κ ij . To see how the effect of the last term can be mea-sured, let us consider the dc limit ω = 0 . In this limit theabove equations are regular and well behaved. This last termin the absence of dependence on the time argument t (cid:48) will givea term in the heat current which has the following form, Q i ( t ) ⊃ t γ − ζ ζ i ζ j ( nδE j + s∂ j δT ) . (38)The above equation has directly measurable consequence: Ir-respective of whether the tilted Dirac cone system is stimu-lated by external electric field or external temperature gra-dient, an accumulative heat current will be generated in thesystem which is proportional to the time t during which theexternal field is applied. This heat current is (i) in the di-rection of (cid:126)ζ and (ii) is controlled by the component of δ (cid:126)E (or (cid:126) ∇ δT ) along the (cid:126)ζ . The resulting diploar angular profilecan be used to immediately map the direction of the vector (cid:126)ζ . Put it another way, the later means that the accumulativeheat current vanishes if the externally applied field or temper-ature gradient is transverse to the tilt vector (cid:126)ζ . The accumula-tive nature (proportionality to time t during which the externalfield is applied) of this heat current means that if the experi-ment is perfomed for a long enough time t , it can take overany other terms in the heat current given in Eqns. (34, 35).This increase can not be continued indefinitely, as beyond cer-tain point, nonlinear effects take over, and the linear reaponseceased to be valid.This effect solely depends on the presence of a tilt vector (cid:126)ζ , and as such has no analog in other solid-state systems. Thepeculiar (cid:126)ζ dependence along with a t -linear depencendence ofthis particular form of heat current can be employed to sepa-rate it from the other terms in the heat current. The anomalousheat transport in P mmn borophene studied in Ref. [50] canbe re-examined in the light of this new term. In the absence of a vector (cid:126)ζ in the spacetime, all directions are equivalent.For the tilted Dirac materials, besides modifying the metric of the space-time, the vector (cid:126)ζ specifies a preferred direction in the space. Therefore theproportionality of (cid:126)Q ∝ (cid:126)ζ is intuitively appealing. D. Vector transport coefficients
One striking aspect of the transport of heat and electric cur-rents in tilted Dirac cone systems is the appearance of newtransport coefficients, β i and γ i which are given by equa-tions (25) and (36), respectively. These vector transport co-efficients (as opposed to tensor transport coefficients) relatethe temporal gradients of electrochemical potential and tem-perature to electric and heat currents. As can be seen fromEqns. (25) and (36), at the present order of calculations, theseare frequency independent and are furthermore proportionalto the only available vector, namely ζ i . The fact that thesevector transport coefficients are given by product of σ Q and ζ i , means that these coefficients quantifying the conversionof temporal gradients to electric and heat currents are essen-tially determined by the ability σ Q of the material to conduct(electricity). In the absence of tilt in normal conductors, sincethe vector ζ i is zero, there will be no preferred dirction in thespace, and therefore the vector transport coefficients γ i and β i remain inert. In tilted Dirac materials, these vector transportcoefficients find a unique opportunity to become active andplay a significant role. Generally, there can be materials withodd number of Dirac valleys [51]. But quite often, the Diraccones come in pairs. This has to do with Fermion doublingproblem according to which, putting a Dirac theory on a (hy-percubic) lattice doubles the number of Dirac fermions [52].In the former case where the number of Dirac nodes is odd,there is no challenge and the vector transport coefficients arealready active. But the later case corresponding to lattices onwhich Fermion doubling occures, requires some discussion: Ifthe material is inversion symmetric, its two valleys come withtilt parameters ζ ± = ± ζ [20]. Since the number of Diracvalleys is even, the currents from the two valleys cancel eachother. Therefore, despite that the β i and γ i transport coeffi-cients for a single valley are active, for the whole mateirialhosting even number of valleys they cancel each other’s ef-fect in an infinite system. However, if the materials lacks aninversion center such that ζ + + ζ − becomes non-zero, thena net electric (heat) current enabled by the coefficient β i ( γ i )can flow in response to temporal gradient of temperature (orelectrochemical potential if one wishes to make it depend ontime). Therefore one possible rout to realization of the presenteffect is to search for a tilted Dirac cone in a material withoutinversion center.The second rout can be built around the ideas of ”val-leytronics” [53] that are popular in planar (2D) mateirals. Theessential idea is that imposing appropriate boundary conditionby cutting appropriate edges, one can create valley valves [54]at the other end of which the ”population” of the two valleysis imbalanced. In this approach, although the material is in-version symmetric and has even number of valleys with op-posite tilt parameres ± ζ , that cancell out for infinite system,in finite systems with appropriate boundaries, the populationimbalance between the valleys gives rise to a net electric cur-rent ∼ β i ( n + − n − ) or heat current ∼ γ i ( n + − n − ) which isdriven by the imbalanced population of the symmetric valleys.The surface of crystalline topological insulators is also pre-dicted to host tilted Dirc cone [55]. To exploite only one Diraccone, one needs to emply the sem-infinte geometry which isnot useful for practical uses. In practice the systems have fi-nite thickness, and tehrefore the other tilted Dirac cone at theother surface will nearly cancel the effect of the first. IV. VISCOUS FLOW OF TILTED DIRAC FERMIONS
So far we have brought up essential physics of TDFs forideal fluid of electrons. It is now expedient to discuss theeffect of viscosity in such systems. In this case, the energyand momentum conservation equations in Fourier space arerespectively given as follows − iωδP = γ ζ (cid:104) − iω ( (cid:15) + p ) + ζ γω ( η + ξ ) (cid:105) ζ j δv j + 1 γ (2 + ζ ) ζ j ∂ j δP, (39) (cid:104) g kj − ζ j ζ k γ (2 + ζ ) (cid:105) ( − ∂ j δP ) = (40) (cid:104) γ (cid:15) + p τ imp (1 − iωγτ imp ) δ ij + 2 iωγ (cid:15) + p ζ ζ i ζ j − ηω γ (cid:104) ζ ζ ζ i ζ j + ζ δ ij (cid:105) − ξω γ
22 + ζ ζ i ζ j (cid:105) δv j . Solving for the perturbation in the velocity field we find δv i = ( F − ) ik (cid:104) g kj − ζ j ζ k γ (2 + ζ ) (cid:105) ( − ∂ j δP ) , (41)where F − is the inverse matrix of F ij = C ij − ηω γ (cid:104) ζ ζ ζ i ζ j + ζ δ ij (cid:105) − ξω γ
22 + ζ ζ i ζ j , (42)and C is the same as Eq. (22) for dissipationless fluid. Theexplicit form of inverse matrix is presented in the appendix.Then similar to the no-viscous fluid, by applying equations(12), (18) and (41) we can read the coefficients in (9) as σ ij = γn ( F − ) ik (cid:104) g kj − ζ j ζ k γ (2 + ζ ) (cid:105) − g ij γσ Q , (43) α ij = − γn s ( F − ) ik (cid:104) g kj − ζ j ζ k γ (2 + ζ ) (cid:105) − µ T σ Q g ij = − sn σ ij − g ij σ Q (cid:16) µ T + γ sn (cid:17) (44)where just the matrix F encompassing viscosity parametersreplaces matrix C of non-viscose case.The bulk viscosity at this order can be ignored [39] andshear viscosity can be controled by temperature and its de-pendence for graphene is as follow[40]: η = 0 .
45 ( k B T ) (cid:125) ( v F α ) (45)where α = cα QED / ( ε r v F ) and ε r is the relative permeabilityof the material with respect to vacuum and α QED is the finestructure constant. Essential scales hare are temperature and
FIG. 3. Temperature dependence of roots at fixed ζ x = ζ y = 0 . the kinetic energy of the Dirac electrons set by v F . Since v F in TDFs is comparable to graphene, we expect this formula togive reasonable estimate of η in TDFs.Moreover, we find the coefficients in thermal current (10)by using equation (18), (41), (32) and (33) as ¯ α ij = n T ( I im + iω V im )( F − ) mk (cid:104) g kj − ζ j ζ k γ (2 + ζ ) (cid:105) + γ µ T σ Q g ij , (46) ¯ κ ij = s ( I im + iω V im )( F − ) mk (cid:104) g kj − ζ j ζ k γ (2 + ζ ) (cid:105) − µ T σ Q g ij = s T n ¯ α ij − g ij σ Q µ T ( µ + γs ) , (47)where I is defined through (37) for dissipationless fluid and V ij encodes the viscosity information as follows: V ij = ηγ ζ (cid:16) δ ij −
12 + ζ ζ i ζ j (cid:17) + ξγ
12 + ζ ζ i ζ j . (48)As can be seen, viscosity are given by zeros of determinantof matrix F which is now a fourth degree polynomial and itmust have two extra poles. It turns out that only two of thesepoles are causal and reduce to the ω and ω poles of the idealfluid in Eq. (26) and (27). As can be seen in Fig. 3 the polesdo not alter much by viscosity (which is controlled by temper-atue). In Fig. 4 we have polar plotted the residue of the σ xx atthe ω (solid lines) and ω (dashed lines) poles. Similarly inFig. 5 we have plotted the same information for the Hall con-ductance σ xy . As can be seen the two figures 4 and 5 paralleltheir ideal counteparts in Figs. 1 and 2, respectively. As canbe seen their qualitative features are identical. So we do notexpect the viscosity to heavily affect the conductivity of theTDFs.To see this in a more quantitative setting, in Fig. 6, we haveploar ploted the residues of the longitudinal conductivity σ xx for the viscous flow (solid line) and ideal flow (dashed lines).As before, the vertitcal lobe (blue) corresponds to the Drude-like pole ω , while the horizontal lobe (red) corresponds tothe spacetime pole ω . Fig. 6 nicely shows how the viscousflow is continuously connected to the ideal flow. One furtherinformation that can be extracted from this figure is that the FIG. 4. Polar dependence of residue of σ xx for ω (solid lines) and ω (dashed lines) in units of [ mev.τ ] for viscous fluid at T =150 K. spectral intensity in ideal flow is larger than the viscous flow.Although the quantitative difference is small, but it shows thatthe light can be better absorbed by the viscous TDFs than theideal TDFs.Now that we have given a comprehensive comparison be-tween the conductivity tensor for the ideal and viscous flow ofTDFs, we are ready to present the actual experimentally ex-pected conductivity lineshapes. As noted in Eq. (26) and (27),and their viscous extensions in Fig. 3, both poles are purely FIG. 5. Residues at the poles ω (solid) and ω (dashed) of the σ xy for viscous flow at T = 150 K. FIG. 6. Comparison between the angular dependence of the residueof σ xx at fixed ζ = 0 . for viscous fluid (solid-lines) and ideal fluid(dashed lines). The poles ω and ω are shown by blue and red color.Spectral weights are larger for ideal fluid of TDFs. imaginary and their real part is zero. This comforms to theintution, as in a Fermi surface built on a tilted Dirac cone dis-persion, one does not expect higher energy absorptions to takeplace. One still has the low-energy particle-hole excitationsacross the Fermi surface.Fig. 7 shows the real (solid line) and imaginary (dashedline) parts of the longitudinal conductivity σ xx for the fluidof TDFs. The viscosity corresponds to T = 150 K. The angleis chosen to be θ = π/ because as we learned from Figs. 1and 4, along this direction both ω and ω poles comparablycontribute. As can be seen, larger tilt values reduce the heightof the Drude peak. Figure 8 shows the same information asFig. 7 for the Hall conductivity σ xy . Again we have plottedthis curve for θ = π/ at which both poles have compara- FIG. 7. Real part (solid-lines) and imaginary part (dashed-lines) of σ xx for viscous fluid of TDFs at T=150 K in units of [ mev.τ imp ] .The hidden structure inside the broad Drude peak can be revealed bypolar dependence in Figs. 1 and 4. FIG. 8. Real part (solid-line) and imaginary part (dashed-lines) of σ xy for viscous fluid of TDFs at T=150 K in units of [ mev.τ imp ] . ble contributions as demonstrated in Figs. 2 and 5. The Hallconductivity in contrast to the longitudinal conductivity showsenhancement of the central peak upon increasing the tilt ζ .Now that we are done with the viscosity dependence of thetensor conductivity σ ij , let us comment about the effect ofviscosity on vector transport coefficients. Since the shear vis-cosity η does not enter the definition of the charge current,the presence of viscosity does not change the vector transportcoefficients β i and γ i . Note further that in the presence of vis-cosity, the transport coefficients α and ¯ α will not be the same.The explicit form of the transport coefficients are given in theappendix.Finally, we turn to the accumulative current in non-idealfluid: As long as the velocity field is homogeneous, the shearviscosity η does not modify the energy equation (16). There-fore when combined with the heat current equation, it willnot lead to any modification in the accumulative current. Theconclusion is that, the accumulative current is not affected byviscous forces. V. SUMMARY AND OUTLOOK
In this paper, we have investigated the hydrodynamics oftilted Dirac fermions that live in the tilted Dirac materials. Theessential feature of these materials is that the ”tilt” deforma-tion of the Dirac (or Weyl if we are in three space dimensions)can be neatly encoded into the spacetime metric. As such,these materials are bestowed with an emergent spacetime ge-ometry which distinguishes them from the rest of solid statesystems. What our hydrodynamics theory in this paper pre-dicts is that the mixing of space and time coordinates in suchsolids gives rise to transport properties which have no ana-logue in other solid state systems where there is no mixingbetween space and time coordinates.The first non-trivial consequence of the spacetime structuredetermined by tilt parameters ζ i is that it gives rise to off-diagonal (Hall) transport coefficients in the absence of mag-netic field . These Hall coefficients appear in both charge and heat transport. By generic symmetry structure of the space-time in such solids, the conductivity tensors will be propor-tional to g ij = δ ij − ζ i ζ j . This is the root cause of Hallcoeffient without a magnetic field. The intuitive understand-ing of the anomalous Hall response σ xy ∼ ζ ζ come fromthe small tilt limit where the tilt can be viewed as a boost pa-rameter that converts the electric field into a magnetic field,whereby a Hall response can be generated. But our theory isvalid for arbitrary tilt value.The second non-trivial consequence of the structure ofspacetime in these solids is appearance of an additional con-tribution to the heat current that depends on the duration ofthe exposure to driving electrochemical poential gradients orthermal gradients. We dub accumulative currents to describethese type of heat current.The third non-trivial and perhaps the most important impor-tant aspect of transport in tilted Dirac materials which mighthave far reaching technological consequences is that a tem-poral gradient of temperature or electrochemical potential canbe converted into electric or heat currents. The fact that thenature provides free ∂ t T in hot deserts from mid-night to mid-day might transfrom this unique capability of TDFs into atechnological revolution. This property arises, because thestrcture of spacetime in tilted Dirac fermion solids is suchthat it allows for a mixing between space and time coordi-nates. Therefore it is not surprising that temporal gradientscan drive currents, pretty much the same way spatial gradi-ents can drive currents. We have introduced the notion of vector transport coefficients (as opposed to commonly used, tensor ones), to formalize and quantify these effects. The im-portant technological advantage of such effects is that, via thevector transport coefficient β i , a gradual heating of these ma-terials ( ∂ t T ) generates electric currents. Furthermore we findthat this transport coefficient is given by β i = σ Q ζ i which isnothing but the ability σ Q of the material to conduct and thetilt vector ζ .The challenge of usign this source of electricity is that sincethe transport coefficent β i is proportional to ζ i , in those ma-terials where the number of Dirac nodes is even and are in-version symmetric, such that for every tilted Dirac cone, thereis another tilted Dirac cone with opposite ζ i , these transportcoefficients cancel each other’s effect. To remedy this, eitherone has to search for tilted Dirac cone materials with odd num-ber of Dirac cones [51], or if they have even number of Diracnodes, one has to search in lattices with broken inversion sym-metry that violates ζ + = − ζ − . Still there is a hope even ifnone of these systems are available: Building on the ideas ofvalleytronics [53, 54], populating a system of TDFs that pos-sesses even number of Dirac nodes with ζ + = − ζ − , appro-priately chosen boundaries in nano-electronic devices can stillgive rise to out-of-equilibirum valley polarization effects thatwill eventually generate a non-zero total current from tem-poral gradients of T (and/or µ ). In fact in such valleytron-ics setup, the valley polarization translates into an effective (cid:126)ζ polarization which therefore activates a net non-zero vectortransport coefficient. The corresponding heat transport coef-ficient γ i makes materilas with TDFs a suitable candidate inapplications that require to direct the heat current into a given0direction. This direction in TDFs is set by ζ . VI. ACKNOWLEDGEMENTS
S. A. J. was supported by grant No. G960214, researchdeputy of Sharif University of Technology and Iran ScienceElites Federation (ISEF). He thanks M. Mohajerani for pro- viding an inspiring working environment during the COVID-19 outbreak. M. T. is supported by the research deputy ofSharif University of Technology.
Appendix A
In this appendix we give explicit expressions for the transport coefficients of TDFs. The inverse matrix F − used in (42) iscomputed as follows F − = 1 f + f ω + f ω + f ω + f ω (cid:18) a + a ω + a ω b ω + b ω b ω + b ω c + c ω + c ω (cid:19) , (A1)where the above coefficients are given by a = γ (cid:15) + p τ imp , a = iγ (cid:15) + p ζ ( ζ y − ζ x − , a = 2 γ ζ y ζ ξ + γ ζ (2 + ζ )(1 − ζ y ) − ζ y ζ η,b = − iγ (cid:15) + p ζ ζ x ζ y , b = γ ζ (2 ζ + 5) η − ξ ζ ζ x ζ y ,c = γ (cid:15) + p τ imp , c = iγ (cid:15) + p ζ ( ζ x − ζ y − , c = 2 γ ζ x ζ ξ + γ ζ (2 + ζ )(1 − ζ x ) − ζ x ζ η, and f = a c , f = a c + a c , f = a c + a c + a c − b , f = a c + a c − b b , f = a c − b . Then, the electric conductivity is explicitly given by σ xx = n [ a (2 + ζ y − ζ x )] + [ a (2 + ζ y − ζ x ) − ζ x ζ y b ] ω + [ a (2 + ζ y − ζ x ) − ζ x ζ y b ] ω (2 + ζ )(1 − ζ ) ( f + f ω + f ω + f ω + f ω ) − σ Q (1 − ζ x )(1 − ζ ) , (A2) σ yy = σ xx ( ζ x ↔ ζ y ) , (A3) σ xy = n − ζ x ζ y a + [ b (2 + ζ x − ζ y ) − ζ x ζ y a ] ω + [ b (2 + ζ x − ζ y ) − ζ x ζ y a ] ω (2 + ζ )(1 − ζ ) ( f + f ω + f ω + f ω + f ω ) + σ Q ζ x ζ y (1 − ζ ) , (A4) σ yx = σ xy ( ζ x ↔ ζ y ) . (A5)Moreover, the transport coefficients ¯ α ij are computed as follows ¯ α xx = n T (cid:16) [ I xx a (2 + ζ y − ζ x ) − ζ x ζ y c I xy ] + [( I xx a + I xy b + V xx a )(2 + ζ y − ζ x ) − ζ x ζ y ( I xx b + I xy c + c e xy )] ω (2 + ζ )( f + f ω + f ω + f ω + f ω )+ [( I xx a + I xy b + V xx a + V xy b )(2 + ζ y − ζ x ) − ζ x ζ y ( I xx b + I xy c + V xx b + V xy c )] ω (2 + ζ )( f + f ω + f ω + f ω + f ω )+ [( V xx a + V xy b )(2 + ζ y − ζ x ) − ζ x ζ y ( V xx b + V xy c )] ω (2 + ζ )( f + f ω + f ω + f ω + f ω ) (cid:17) + µ T − ζ x (1 − ζ ) , (A6) ¯ α yy = ¯ α xx ( ζ x ↔ ζ y ) , (A7) ¯ α xy = n T (cid:16) [ I xy c (2 + ζ x − ζ y ) − ζ x ζ y a I xx ] + [( I xx b + I xy c + e xy c )(2 + ζ x − ζ y ) − ζ x ζ y ( I xx a + I xy b + a e xx )] ω (2 + ζ )( f + f ω + f ω + f ω + f ω )+ [( I xx b + I xy c + e xx b + V xy c )(2 + ζ x − ζ y ) − ζ x ζ y ( I xx a + I xy b + V xx a + V xy b )] ω (2 + ζ )( f + f ω + f ω + f ω + f ω )+ [( V xx b + V xy c )(2 + ζ x − ζ y ) − ζ x ζ y ( V xx a + V xy b )] ω (2 + ζ )( f + f ω + f ω + f ω + f ω ) (cid:17) − µ T ζ x ζ y (1 − ζ ) , (A8) ¯ α yx = ¯ α xy , (A9)where I xx = γ ( (cid:15) + p ) − γµ n − γ (cid:15) + p ζ ζ x ζ x , I yy = I xx ( ζ x ↔ ζ y ) , I xy = − γ (cid:15) + p ζ ζ x ζ y = I yx , V xx = 2 iηζ x [ ζ + ζ −
1] + iξζ x (1 − ζ )(2 + ζ )(1 − ζ ) , V yy = V xx ( ζ x ↔ ζ y ) , V xy = 2 iηζ x ζ y [ ζ + ζ −
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Modern Condensed Matter Physics (Cambridge University Press, New York, 2019).[3] M. D. Schwartz,
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