The motion of electrons in conductors in the absence of external temperature and/or electro-chemical gradients is purely random thermal motion [1] . Once a spatial temperature gradient ∇T is introduced, the vector character of ∇ specifies a preferred direction, and therefore electrons flow along ∇T [2] . The purpose of this paper is to propose a class of materials where a temporal gradient ∂ t T alone (i.e. without any spatial gradient) is sufficient to generate electric and/or heat currents. Why is it important to be able to derive current by temporal gradient? In the so called hot deserts such as Sahara or Kavir and Lut deserts of Iran, due to clear sky, the heat loss in the night is substantial and the daily variation of temperature can be as great as 22 • C 1 . Therefore the hot deserts with their cold midnights and hot noons can serve as a free provider of ∂ t T through a half cycle of the spinning of the earth. Can this naturally available temporal gradient be employed in energy production?
To set the stage, suppose that for some reason, there is a preferred direction in the space determined by a vector ζ. Can the anisotropy arising from such a preferred direction organize the random motion of electrons into non-zero electric or heat currents? As we will discuss in this paper, there are certain materials where such a vector ζ not only does exist, but additionally, it enters the effective description of the motion of electrons in such a way that it mingles space and time coordinates, whereby creates a new spacetime structure. As a result, even a pure temporal gradient of a spatially uniform temperature T can drive a current via mixing of the space and time coordinates arising from ζ.
Unlike the electrons in the standard model of elementary particle physics [3] , the electrons in the solids are mounted on a lattice. Every periodic lattice structure belongs to one of the 230 possible space groups [4] . Some of these structures afford to give a low-energy effective theory for the elec- * ahmadreza.moradpour@gmail.com † mahdi.torabian@sharif.edu ‡ jafari@sharif.edu 1 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 theory as in certain structure of borophene [6, 7] or the organic compound α−(BEDT-TTF) 2 I 3 [8] [9] [10] [11] [12] [13] or certain deformations of graphene [14] . For the Dirac electrons in solids, the dispersion relation giving energy ε of the quasiparticles at every momentum k will be a Dirac cone, and enjoys an emergent Lorentz invariance at low energies corresponding to length scales much larger than the lattice spacing [5] . But in materials hosting a tilted Dirac cone, the cone-shaped dispersion relation is tilted in energy-momentum space [14, 15] . This tilt is characterized by set of parameters ζ. Presence of tilt in the Dirac cone, not only bestows the space with a preferred direction set by the vector ζ [14, 15] , but also gives rise to an emergent spacetime structure [16] [17] [18] . It turns out that, the tilt can be attributed to an underlying metric of the spacetime [16, 19, 20] . Therefore in the same way that Dirac materials enjoy an emergent Lorentz symmetry arising from an emergent Minkowski spacetime at long wave length for the lattice at hand, those materials that host a tilted Dirac cone can be assigned an emergent spacetime structure at long length scales which is controlled by ζ and is given by
where v F is the velocity scale that defines the emergent spacetime (which plays a role similar to the light speed c, but in real materials is two or three orders of magnitude smaller than c) and ζ encodes the tilt of the Dirac cone [20] . For the rest of the paper we will set v F = 1. The isometries of the spacetime (1) are appropriate deformations of the Lorentz group [16] . As such, the spacetime defined by Eq. (1) is a deformation of the Minkowski spacetime by a continuous parameter ζ. The presence of tilt ζ modifies many of the physical properties of the materials, in particular including their interfaces with superconductors [21, 22] . The above metric possesses a blackhole horizon [23] that stems from spatial variation of the Gallilean boost ζ in Eq. (1). Spatial variation of a parameter similar to ζ can emulate a black-hole horizion in atomic Bose-Einstein condensates [24] , polariton superfluids [25, 26] . The spin-polarized currents are also predicted to implement blackhole horizon for magnons [27] . Our proposal differs from the above systems in that (i) the tilted Dirac/Weyl systems required are at ambient conditions and (ii) the quasiparticles of the theory are fermions, namely electrons and holes which carry electric charges. As such, any effects arising from the emergent structure of the spacetime, will leave direct signature in almost any electron spectroscopy experiment, including of course the transport (conductivity) measurements.
In this paper, we set out to use the hydrodynamics as a technical tool to study the electron fluid [28] in the tilted Dirac cone mateirals where the spacetime is given by metric (1). Let us announce our result in advance: In any conductor, the spatial gradient of temperature can generate heat and electric currents. The mingling of the spacetime in Eq. (1) will allow the materials with tilted Dirac cone to generate heat and electric currents from pure temporal gradients. Our theoretical tool to bring out this result is the hydrodynamics. Hydrodynamic is an effective long-time and long-distance description of quantum many body systems that focuses on few conserved collective variables rather than embarking on the formidble task of addressing all the microscopic degrees of freedom.
Hydrodynamics as a powerful universal approach, has many applications in various systems differing in microscopic details which are however, described by the same equations. Only gross symmetry properties has to be properly incorporated into the formulation of hydrodynamic for a system at hand. Applications of hydrodynamics approach in high energy physics includes quark-qluen plasma [29, 30] , parity violation [31] , chiral anomaly [32] and dissipative superfluid [33] . Within the hydrodynamic approach, one can come up with universal and model-independent predication such as kinematic viscosity value [34, 35] . The hydrodynamics approach can also be applied to fluid-gravity correspondence [36] which relate the dynamics of the gravity side to the hydrodynamic equations where the hydrodynamic fluctuation mode describes the fluctuations of black holes [37, 38] . In this work, we will be interested in much simpler version of hydrodynamics in a spacetime structure in d + 1 dimensions subject to the metric (1) that describes electrons in sub-eV energy scales in solids with tilted Dirac cone.
The roadmap of the paper is as follows: In section II following Lucas [39] we formulate the hydrodynamics for the metric (1) as a natural generalization of the hydrodynamics theory 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 treatment of the hydrodynamics in spacetime (1) where we discuss perfect fluid. It is followd by a treatment of the viscous fluids in section IV in this new spacetime. We end the paper with discussion and summary of the paper in section V.
In this section we develop the hydrodynamics of electrons in a tilted Dirac cone material. For a planar material in d = 2 space dimensions, there is a two-parameter family of tilt deformations given by ζ = (ζ x , ζ y ) to the Dirac equation in three dimensional spacetime. These deformations and the corresponding dispersion relation can be obtained if instead of the conventional Lorentz metric η µν one applies the following metric tensor
where ζ 2 = ζ 2 x + ζ 2 y and δ is the 2 × 2 unit matrix [16, 20] . In this parametrization of the tilt we adopt the normalization |ζ|< 1 so that spacetime makes sense in admissible coordinates. For the sake of convenience in the following computations, we introduce γ = (1 − ζ 2 ) −1/2 . The Greek indices run over 0,1 and 2 for spacetime coordinates and the Latin indices run over spatial coordinates.
We assume that the electron-electron scattering rate τ −1 ee take over any other scattering rate such as electron-phonon (τ −1 e−ph ) or electron-impurity scattering rate (τ −1 e−imp ). This regime is attainable in graphene that hosts upright Dirac cone [40] [41] [42] [43] . This has become possible by ability to tune the strength of electron-electron interaction via gating that sets the scale of the Fermi surface. In such a regime, an effective description at large distances ( v F τ ee ) over long time ( τ ee ) is provided by the hydrodynamics equations given by conservation laws of Noether currents. Assuming translational invariance and gauge invariance of the underlying microscopic theory, there are conserved energy-momentum tensor T µν and an Abelian current vector J µ . They constitute ten independent components subject to four constraints ∂ µ T µν = 0 and ∂ µ J µ = 0.
In order to find unique solutions to hydrodynamics equation, it is assumed that the currents are determined through four auxiliary local thermodynamical quantities: the temperature T (x), the chemical potential µ(x), a normalized timelike velocity vector field u µ (x) (i.e. u µ u µ = −1) and their derivatives. The fluid observer moves along with the fluid and measures variables (local temperature, local mass density etc.) without ambiguities. The 3-velocity u µ is defined relative to the Eulerian (arbitrary) observer. The fluid velocity v i is defined through v µ = u µ /u 0 . The generalized Lorentz factor is defined as Γ ≡ −n µ u µ = u 0 where n µ = (−1, 0) is the time-like normal vector to the 2-space. An Eulerian observer attributes this factor to matter moving in the fluid frame. For instance, given the temperature measured by the fluid observer T , an Eulerian observer finds T E = ΓT Given an arbitrary vector, any tensor can be decomposed to its transverse and longitudinal components with respect to that vector. We have a freedom to identify u µ with the velocity of energy flow in the so-called Landau frame u µ ∼ T µν u ν . Moreover, T and µ can be defined so that
where n is the number density of charge carriers and is the energy density. In this frame, the energy-momentum tensor and particle current can be decomposed respectively as follows [44, 45] 
where P is a scalar, j µ , P µν and t µν are transverse tensors that satisfy u µ j µ = u µ P µν = u µ t µν . Tensor t is symmetric traceless t µ µ = 0. Tensor P is defined as
which is called the projection tensor; it is symmetric and in general has non-vanishing trace. The inverse metric is
The remaining elements P, j µ and t µν are determined in terms of derivatives of hydrodynamic variables and yield the constituent equations at desired order. At first order hydrodynamics we find
where p is pressure in the local rest frame, ξ B is the bulk viscosity, σ Q is the intrinsic conductivity and η is the shear viscosity. Eq (8) is written in the presence of an external electromagnetic field determined by F νρ . In passing, we recall that coefficients in zero-order hydrodynamics , p and n are fixed by T , µ and the equation of state in equilibrium thermodynamics [46] [47] [48] . Moreover, the non-negative parameters ζ B , σ Q and η (the Wilsonian coefficients of the effective hydrodynamic theory) are either measured in experiments or determined from an underlying microscopic (quantum) theory.
We define the response of the electric current J and the heat current Q to an external electric field E, spatial gradient ∇T and possibly temporal variation ∂ 0 T of temperature as follows
where σ ij , α ij ,ᾱ ij andκ ij are the usual tensor response coefficients relating the electric and heat currents to spatial gradients of electrochemical poential or temperature [2] . Anticipating electric/heat currents in response to temporal gradient ∂ 0 T , we have additionally introduced the vector transport coefficents β i and γ i . All the above coefficients are functions of t − t by time-translational invariance. We compute the above transport coefficients within hydrodynamics theory. We imagine that fluid is perturbed around its equilibrium state (specified by µ 0 , T 0 and u µ 0 = (1 − ζ 2 ) −1/2 (1, 0) in the rest frame of the fluid exposed to E 0 = 0) by a slight amount parametrized as follows
The first order perturbation in the electric current is given by
(12) Moreover, the leading order perturbation in momentum is
with which we compute thermal conductivity through
An interesting feature of Eq. (8) in the tilted Dirac/Weyl materials is a genuine effect where temporal variations generate currents: We note that in a linear hydrodynamic theory the term in the brackets in this equation are already first order, and therefore at this order P µν → g µν . Therefore the spatial component J i of the current (in addition to the first term nu i ) will acquire a contribution proportional to ∂ 0 µ and ∂ 0 T which is accompanied by the tilt parameters g i0 = −ζ i . This effect is solely dependent on the tilt parameters ζ i and vanishes for upright Dirac/Weyl systems where ζ i = 0. Therefore, we extend the conventional thermoelectric coefficients to account for this through additional transport coefficients β i and γ i in Eqns. (9) and (10).
We solve the hydrodynamics equations to evaluate the electric current and thermal current in the presence of background electric field and temperature spatiotemporal gradients. At this leading order, the charge conservation ∂ µ J µ = 0 implies
Similarly, the energy conservation ∂ µ T µ0 = 0 implies
Finally, the momentum conservation ∂ µ T µi = 0 gives
where the parameter τ imp is the relaxation time due to scattering from impurities. The above expressions in the square bracket 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 passing we note that, in the fluid rest frame the electric field, the temperature gradient and the pressure gradient are balanced as −∂ i P = nE i +s∂ i T . Thus, we apply the following constraint
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 uniform solutions with ∂ j δv i = 0. With these assumptions, the energy-momentum conservation implies
We can solve the above equations for the velocity perturbation
where C −1 is the inverse matrix of
which is explicitly computed in the appendix. Then, we compute the electric current (12) as a response to spatiotemporal variation of the electric field and temperature. Finally, by applying equations (18) and (21) we can read the coefficients in (9) as follows
The pole structure of the electric and heat conductivity tensors are the same. This is because they are both related to the determinant of the matrix C. Therefore, we focus on the poles of the conductivity. We find that the conductivity tensor has the following two poles
In the upright Dirac cone with Minkowski spacetime structure, ζ → 0, both poles ω 1 and ω 2 of Eqns. (26) and (27) reduce to the Drude result. To understand these poles, we have defined a redshifted relaxation time τ lab imp = γτ imp . In this definition τ imp can be interpreted as the microscopic relaxation time experienced by electrons in the spacetime with a given ζ, while τ lab imp can be interpreted as the same time measured in the laboratory by the experimentalist sitting in the laboratory spacetime having ζ = 0. As such, the pole at ω 1 is a natural extension of the Drude pole to the geometry (1). However, the new pole ω 2 arises from the new spacetime structure. Although at ζ → 0 limit it becomes degenerate with the Drude pole, but at ζ → 1 it can become up to 3 times larger than ω 1 . Both poles are on the imaginary axis, and their real part is zero, as they are caused by very low-energy (Drude) excitations across the Fermi level. Then the question will be, is there a way to distinguish the contributions from the Drude pole ω 1 and the spacetime pole ω 2 ? To answer this question, we need to look at the residues at the two poles that determine the spectral intensity associated with each pole.
Let us start by looking at the residue of the absorptive part of the conductivity, namely the longitudinal conductivity σ xx . The residues at the above poles are computed as follows 2
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 information in residues of the poles of σ yy . As pointed out, both ω 1 and ω 2 poles are on the imaginary axis and their real part is zero. To disentangle their contribution, not that the meaning of the longitudinal conductivity σ xx is the current along the applied electric field (both assumed along the x direction). But then the x axis can subtend an angle θ with the tilt vector ζ. This can be an interesting variable. Therefore, in Fig. 1 we have plotted the dependence of the residudes of the longitudinal part of the conductivity on the polar angle θ of the tilt vector. The solid (dashed) lines correspond to the Drude-like pole ω 1 (spacetime pole ω 2 ). Various colors correspond to different tilt magnitudes. Both poles have a dipolar pattern. But their nodal structure is different which helps to identify which pole is contributing the spectral weight. When the electric field is applied along the tilt direction (θ = 0), the residue of the Drude-like pole vanishes and the absorption is contributed by ω 2 pole. By rotating the applied electric field away from the tilt direction, the Drude-like pole ω 1 takes over the spacetime pole ω 2 . When the applied electric field is completely perpendicular to ζ, only ω 1 contributes to the conductivity. The common feature of solid and dashed curves in Fig. 1 is that the spectral weight of both ω 1 and ω 2 poles decreases by increasing the tilt ζ. 
Another unusual effect arising from the tilt in ideal fluid is the presence of a non-zero Hall and thermal Hall coefficeitns in the absence of external magnetic field. Due to offdiagonal metric elements of the metric g ij ∝ ζ i ζ i , a non-zero Hall and thermal Hall coefficient proportional to ζ i ζ j appear in Eqs. (23) and (24) . This effect also solely depends on the presence of the tilt ζ i . Mathematically, in the absence of ζ i (isotropic space), the conductivity tensor σ ij (as in the case of graphene) will become proportional to δ ij . But in the present case, the anisotropy of the space will be reflected in a ζ i ζ j dependence in all tensorial quantities including the conductivity and heat conductivity tensors. As such, the non-zero Hall and thermal Hall coefficients in the tilted Dirac cone can be directly attributed to the structure of the spacetime. Now let us consider the off-diagonal (transverse) component of the conductivity, namely σ xy . It turns out that there will be a tilt-induced Hall response σ xy which is given by
Res(σ xy ω2 ) =
Mathematically, σ ij (and also α ij ) being tensors are naturally expected to have a term proportional to g ij ∼ ζ i ζ j . But what is the physical meaning of a non-zero Hall coefficient in the absence of a background B z in the new spacetime geometry? Indeed in the small tilt limit, the metric (1) can be viewed as a superposition of an additional "Lorentz" boost with the boost parameter given by ζ [49] . As such the applied electric field can act like an effective B-field, generating the Hall response. But note that our finding is not limited to small ζ and is valid for any finite ζ. The residues of the Hall conductivity arising from ω 1 and ω 2 poles are plotted in Fig. 2 . As can be seen both poles display quadrapular pattern. Also their behavior with ζ is similar. Both intensities decrease upon decreasing the tilt magnitude ζ.
The conclusion is that, the longitudinal conductivity is a suitable measure to disentangle the role of Drude-like pole ω 1 and spactime pole ω 2 in the conductivity of tilted Dirac material sheets.
Now let us turn our attention to the heat (energy) transport coefficients. The thermal current in a non-viscous homogeneous fluid is given by
where δJ i is given in (12) . We use the energy conservation equation (19) and equation of state = 2 · P to compute the pressure perturbation δP as follows
Ignoring the last term in the above equation, we compute the coefficients in thermal current (10) by applying equation (18), (21) , (32) and (33) as follows
where I i j is defined as
The last term in (33) gives an additional contribution toᾱ ij and κ ij . To see how the effect of the last term can be measured, let us consider the dc limit ω = 0. In this limit the above equations are regular and well behaved. This last term in the absence of dependence on the time argument t will give a term in the heat current which has the following form,
The above equation has directly measurable consequence: Irrespective of whether the tilted Dirac cone system is stimulated by external electric field or external temperature gradient, an accumulative heat current will be generated in the system which is proportional to the time t during which the external field is applied. This heat current is (i) in the direction of ζ 3 and (ii) is controlled by the component of δ E (or ∇δT ) along the ζ. The resulting diploar angular profile can be used to immediately map the direction of the vector ζ. Put it another way, the later means that the accumulative heat current vanishes if the externally applied field or temperature gradient is transverse to the tilt vector ζ. The accumulative nature (proportionality to time t during which the external field is applied) of this heat current means that if the experiment is perfomed for a long enough time t, it can take over any other terms in the heat current given in Eqns. (34, 35) . This increase can not be continued indefinitely, as beyond certain point, nonlinear effects take over, and the linear reaponse ceased to be valid. This effect solely depends on the presence of a tilt vector ζ, and as such has no analog in other solid-state systems. The peculiar ζ dependence along with a t-linear depencendence of this particular form of heat current can be employed to separate it from the other terms in the heat current. The anomalous heat transport in 8P mmn borophene studied in Ref. [50] can be re-examined in the light of this new term.
One striking aspect of the transport of heat and electric currents in tilted Dirac cone systems is the appearance of new transport coefficients, β i and γ i which are given by equations (25) and (36) , respectively. These vector transport coefficients (as opposed to tensor transport coefficients) relate the temporal gradients of electrochemical potential and temperature to electric and heat currents. As can be seen from Eqns. (25) and (36) , at the present order of calculations, these are frequency independent and are furthermore proportional to the only available vector, namely ζ i . The fact that these vector transport coefficients are given by product of σ Q and ζ i , means that these coefficients quantifying the conversion of temporal gradients to electric and heat currents are essentially determined by the ability σ Q of the material to conduct (electricity). In the absence of tilt in normal conductors, since the vector ζ i is zero, there will be no preferred dirction in the space, and therefore the vector transport coefficients γ i and β i remain inert. In tilted Dirac materials, these vector transport coefficients find a unique opportunity to become active and play a significant role. Generally, there can be materials with odd number of Dirac valleys [51] . But quite often, the Dirac cones come in pairs. This has to do with Fermion doubling problem according to which, putting a Dirac theory on a (hypercubic) 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 are already active. But the later case corresponding to lattices on which Fermion doubling occures, requires some discussion: If the material is inversion symmetric, its two valleys come with tilt parameters ζ ± = ±ζ [20] . Since the number of Dirac valleys is even, the currents from the two valleys cancel each other. Therefore, despite that the β i and γ i transport coefficients for a single valley are active, for the whole mateirial hosting even number of valleys they cancel each other's effect in an infinite system. However, if the materials lacks an inversion center such that ζ + + ζ − becomes non-zero, then a net electric (heat) current enabled by the coefficient β i (γ i ) can flow in response to temporal gradient of temperature (or electrochemical potential if one wishes to make it depend on time). Therefore one possible rout to realization of the present effect is to search for a tilted Dirac cone in a material without inversion center.
The second rout can be built around the ideas of "valleytronics" [53] that are popular in planar (2D) mateirals. The essential idea is that imposing appropriate boundary condition by cutting appropriate edges, one can create valley valves [54] at the other end of which the "population" of the two valleys is imbalanced. In this approach, although the material is inversion symmetric and has even number of valleys with opposite tilt parameres ±ζ, that cancell out for infinite system, in finite systems with appropriate boundaries, the population imbalance between the valleys gives rise to a net electric current ∼ β i (n + − n − ) or heat current ∼ γ i (n + − n − ) which is driven by the imbalanced population of the symmetric valleys.
The surface of crystalline topological insulators is also predicted to host tilted Dirc cone [55] . To exploite only one Dirac cone, one needs to emply the sem-infinte geometry which is not useful for practical uses. In practice the systems have finite thickness, and tehrefore the other tilted Dirac cone at the other surface will nearly cancel the effect of the first.
So far we have brought up essential physics of TDFs for ideal fluid of electrons. It is now expedient to discuss the effect of viscosity in such systems. In this case, the energy and momentum conservation equations in Fourier space are respectively given as follows
Solving for the perturbation in the velocity field we find
where F −1 is the inverse matrix of
and C is the same as Eq. (22) for dissipationless fluid. The explicit 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
where just the matrix F encompassing viscosity parameters replaces matrix C of non-viscose case.
The bulk viscosity at this order can be ignored [39] and shear viscosity can be controled by temperature and its dependence for graphene is as follow [40] :
where α = cα QED /(ε r v F ) and ε r is the relative permeability of the material with respect to vacuum and α QED is the fine structure constant. Essential scales hare are temperature and Since v F in TDFs is comparable to graphene, we expect this formula to give reasonable estimate of η in TDFs. Moreover, we find the coefficients in thermal current (10) by using equation (18), (41) , (32) and (33) as
where I is defined through (37) for dissipationless fluid and V i j encodes the viscosity information as follows:
As can be seen, viscosity are given by zeros of determinant of matrix F which is now a fourth degree polynomial and it must have two extra poles. It turns out that only two of these poles are causal and reduce to the ω 1 and ω 2 poles of the ideal fluid in Eq. (26) and (27) . As can be seen in Fig. 3 the poles do not alter much by viscosity (which is controlled by temperatue). In Fig. 4 we have polar plotted the residue of the σ xx at the ω 1 (solid lines) and ω 2 (dashed lines) poles. Similarly in Fig. 5 we have plotted the same information for the Hall conductance σ xy . As can be seen the two figures 4 and 5 parallel their ideal counteparts in Figs. 1 and 2 , respectively. As can be seen their qualitative features are identical. So we do not expect the viscosity to heavily affect the conductivity of the TDFs.
To see this in a more quantitative setting, in Fig. 6 , we have ploar 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 Drudelike pole ω 1 , while the horizontal lobe (red) corresponds to the spacetime pole ω 2 . Fig. 6 nicely shows how the viscous flow is continuously connected to the ideal flow. One further information that can be extracted from this figure is that the spectral intensity in ideal flow is larger than the viscous flow. Although the quantitative difference is small, but it shows that the light can be better absorbed by the viscous TDFs than the ideal TDFs. Now that we have given a comprehensive comparison between the conductivity tensor for the ideal and viscous flow of TDFs, we are ready to present the actual experimentally expected conductivity lineshapes. As noted in Eq. (26) and (27) imaginary and their real part is zero. This comforms to the intution, as in a Fermi surface built on a tilted Dirac cone dispersion, one does not expect higher energy absorptions to take place. One still has the low-energy particle-hole excitations across the Fermi surface. Fig. 7 shows the real (solid line) and imaginary (dashed line) parts of the longitudinal conductivity σ xx for the fluid of TDFs. The viscosity corresponds to T = 150K. The angle is chosen to be θ = π/4 because as we learned from Figs. 1 and 4, along this direction both ω 1 and ω 2 poles comparably contribute. As can be seen, larger tilt values reduce the height of the Drude peak. Figure 8 shows the same information as Fig. 7 for the Hall conductivity σ xy . Again we have plotted this curve for θ = π/4 at which both poles have compara- Now that we are done with the viscosity dependence of the tensor conductivity σ ij , let us comment about the effect of viscosity on vector transport coefficients. Since the shear viscosity η does not enter the definition of the charge current, the presence of viscosity does not change the vector transport coefficients β i and γ i . Note further that in the presence of viscosity, the transport coefficients α andᾱ will not be the same. The explicit form of the transport coefficients are given in the appendix.
Finally, we turn to the accumulative current in non-ideal fluid: As long as the velocity field is homogeneous, the shear viscosity η does not modify the energy equation (16) . Therefore when combined with the heat current equation, it will not lead to any modification in the accumulative current. The conclusion is that, the accumulative current is not affected by viscous forces.
In this paper, we have investigated the hydrodynamics of tilted Dirac fermions that live in the tilted Dirac materials. The essential feature of these materials is that the "tilt" deformation 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 geometry which distinguishes them from the rest of solid state systems. What our hydrodynamics theory in this paper predicts is that the mixing of space and time coordinates in such solids gives rise to transport properties which have no analogue in other solid state systems where there is no mixing between space and time coordinates.
The first non-trivial consequence of the spacetime structure determined by tilt parameters ζ i is that it gives rise to offdiagonal (Hall) transport coefficients in the absence of magnetic field. These Hall coefficients appear in both charge and heat transport. By generic symmetry structure of the spacetime in such solids, the conductivity tensors will be proportional to g ij = δ ij − ζ i ζ j . This is the root cause of Hall coeffient without a magnetic field. The intuitive understanding of the anomalous Hall response σ xy ∼ ζ 1 ζ 2 come from the small tilt limit where the tilt can be viewed as a boost parameter that converts the electric field into a magnetic field, whereby a Hall response can be generated. But our theory is valid for arbitrary tilt value.
The second non-trivial consequence of the structure of spacetime in these solids is appearance of an additional contribution to the heat current that depends on the duration of the exposure to driving electrochemical poential gradients or thermal gradients. We dub accumulative currents to describe these type of heat current.
The third non-trivial and perhaps the most important important aspect of transport in tilted Dirac materials which might have far reaching technological consequences is that a temporal gradient of temperature or electrochemical potential can be converted into electric or heat currents. The fact that the nature provides free ∂ t T in hot deserts from mid-night to midday might transfrom this unique capability of TDFs into a technological revolution. This property arises, because the strcture of spacetime in tilted Dirac fermion solids is such that it allows for a mixing between space and time coordinates. Therefore it is not surprising that temporal gradients can drive currents, pretty much the same way spatial gradients 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 important technological advantage of such effects is that, via the vector transport coefficient β i , a gradual heating of these materials (∂ t T ) generates electric currents. Furthermore we find that this transport coefficient is given by β i = σ Q ζ i which is nothing but the ability σ Q of the material to conduct and the tilt vector ζ.
The challenge of usign this source of electricity is that since the transport coefficent β i is proportional to ζ i , in those materials where the number of Dirac nodes is even and are inversion symmetric, such that for every tilted Dirac cone, there is another tilted Dirac cone with opposite ζ i , these transport coefficients cancel each other's effect. To remedy this, either one has to search for tilted Dirac cone materials with odd number of Dirac cones [51] , or if they have even number of Dirac nodes, one has to search in lattices with broken inversion symmetry that violates ζ + = −ζ − . Still there is a hope even if none of these systems are available: Building on the ideas of valleytronics [53, 54] , populating a system of TDFs that possesses even number of Dirac nodes with ζ + = −ζ − , appropriately chosen boundaries in nano-electronic devices can still give rise to out-of-equilibirum valley polarization effects that will eventually generate a non-zero total current from temporal gradients of T (and/or µ). In fact in such valleytronics setup, the valley polarization translates into an effective ζ polarization which therefore activates a net non-zero vector transport coefficient. The corresponding heat transport coefficient γ i makes materilas with TDFs a suitable candidate in applications that require to direct the heat current into a given direction. This direction in TDFs is set by ζ. In this appendix we give explicit expressions for the transport coefficients of TDFs. The inverse matrix F −1 used in (42) is computed as follows
where the above coefficients are given by a 0 = γ 0 + p 0 τ imp , a 1 = iγ 2 0 + p 0 2 + ζ 2 (ζ 2 y − ζ 2 x − 2), a 2 = 2γ 3 ζ 2 y 2 + ζ 2 ξ + γ 3 ζ 2 (2 + ζ 2 )(1 − 2ζ 2 y ) − ζ 2 y 2 + ζ 2 η, b 1 = −2iγ 2 0 + p 0 2 + ζ 2 ζ x ζ y , b 2 = γ 3 ζ 2 (2ζ 2 + 5)η − 2ξ 2 + ζ 2 ζ x ζ y ,
Then, the electric conductivity is explicitly given by
σ xy = n 2 0 −3ζ x ζ y a 0 + [b 1 (2 + ζ 2 x − 2ζ 2 y ) − 3ζ x ζ y a 1 ]ω + [b 2 (2 + ζ 2 x − 2ζ 2 y ) − 3ζ x ζ y a 2 ]ω 2 (2 + ζ 2 )(1 − ζ 2 ) 1 2 (f 0 + f 1 ω + f 2 ω 2 + f 3 ω 3 + f 4 ω 4 )
σ yx = σ xy (ζ x ↔ ζ y ).
Moreover, the transport coefficientsᾱ ij are computed as follows α xx = n 0 T 0
[I x x a 0 (2 + ζ 2 y − 2ζ 2 x ) − 3ζ x ζ y c 0 I x y ] + [(I x x a 1 + I x y b 1 + V x x a 0 )(2 + ζ 2 y − 2ζ 2 x ) − 3ζ x ζ y (I x x b 1 + I x y c 1 + c 0 e x y )]ω (2 + ζ 2 )(f 0 + f 1 ω + f 2 ω 2 + f 3 ω 3 + f 4 ω 4 )
where I x x = γ 2 ( 0 + p 0 ) − γµ 0 n 0 − 2γ 2 0 + p 0 2 + ζ 2 ζ x ζ x , I y y = I x x (ζ x ↔ ζ y ), I x y = −2γ 2 0 + p 0 2 + ζ 2 ζ x ζ y = I y x ,