Weyl Hydrodynamics

We begin by developing our hydrodynamic treatment of the electron fluid, assuming the chemical potential lies close to the charge neutrality point for every node. For simplicity, we assume that the Weyl nodes are locally isotropic to reduce the number of effective parameters. It is likely straightforward, although tedious, to generalize and study anisotropic systems.

We will firstly review the hydrodynamic theory of a chiral fluid with an anomalous axial U(1) symmetry, derived in refs. 34, 35. Neglecting intervalley scattering, this theory describes the dynamics near one Weyl node. The equations of relativistic chiral hydrodynamics are the conservation laws for charge, energy, and momentum, modified by the external electromagnetic fields, which we denote with Fμν. On a curved space with Riemann tensor Rαβδγ, they read∇μTμν=FνμJμ−G16π2∇μ[ερσαβFρσRαβνμ] ,[2a]∇μJμ=−C8εμνρσFμνFρσ−G32π2εμνρσRβμναRαρσβ,[2b]where C is a coefficient related to the standard axial anomaly and G is a coefficient related to an axial–gravitational anomaly (40). For a Weyl fermionC=k4π2, G=k24,[3]with k∈ℤ the Berry flux associated with the Weyl node (41). Jμ and the energy–momentum tensor Tμν are related to the hydrodynamic variables of chemical potential μ, temperature T, and velocity uμ in a tightly constrained way (34, 35), which we review in the SI Appendix. We will take the background electromagnetic field to beF=Bdx∧dy+∂iμ0dxi∧dt,[4]with B as a constant. Constant B is required by Maxwell’s equations for the external electromagnetic field in equilibrium, at leading order.

A single chiral fluid cannot exist in a Weyl material. Instead, enough Weyl nodes must exist so that the “net” C for the material vanishes. This follows mathematically from the fact that the Brillouin zone of a crystal is necessarily a compact manifold and so the sum of the Berry fluxes associated with each node must vanish—this is the content of the Nielsen–Ninomiya theorem (7). Hence, we must consider the response of multiple chiral fluids when developing our theory of transport.

One might hope that so long as each chiral fluid has a well-behaved response, then the net conductivities are simply additive. This is not so: the transport problem is ill-posed for a single chiral fluid, once we apply a background magnetic field. To see this, suppose that we apply an electric field such that E⋅B≠0. Then, the total charge in the sample obeysdQtotdt=∫d3x ∂μJμ=CE⋅BV3,[5]with V3 the spatial volume of the metal. Even at the linear response level, we see that there is a necessary O(E) time dependence to any solution to the hydrodynamic equations (with spatial directions periodically identified). If there is no static solution to the equations of motion, then any dc conductivity is an ill-posed quantity to compute. There is also energy production in a uniform temperature gradient, proportional to G∇T⋅B, even when C=0 (SI Appendix).

The physically relevant solution to this issue is that multiple Weyl nodes exist in a real material, and this means that we must consider the coupled response of multiple chiral fluids. Rare intervalley processes mediated by phonons and/or impurities couple these chiral fluids together (8) and make the transport problem far richer for Weyl fluids than for simpler quantum critical fluids, including the Dirac fluid (32).

We label each valley fluid quantity with the labels ab…. For example, uaμ is the velocity of valley fluid a. To avoid being completely overwhelmed with free parameters, we only include coefficients at zeroth order in derivatives coupling distinct fluids together. In fact, this will be sufficient to capture the negative magnetoresistance, as we explain in the next section. Accounting for this coupling modifies the conservation equations to∇μJaμ=−Ca8εμνρσFμνFρσ−Ga32π2εμνρσRβμναRαρσβ−∑b[ℛabνb+Sabβb] [6a]∇μTaμν=FνμJμa−Ga16π2∇μ[ερσαβFρσRαβνμ]+uaν∑b[Uabνb+Vabβb], [6b]where we have defined βa≡1/Ta and νa≡βaμa. The transport problem is well-posed if∑aCa=∑aGa=0.[7]

The new coefficients ℛ,S,U, and V characterize the rate of the intervalley transfer of charge, energy, and momentum due to relative imbalances in chemical potential or temperature. In writing [6], we have chosen the intervalley scattering of energy and momentum to be relativistic. This makes the analysis easier as it preserves Lorentz covariance, but will not play an important role in our results. In particular, the intervalley momentum transfer processes are subleading effects in our theory of transport.

The gradient expansion may be different for each fluid, but we will assume that Jaμ and Taμν depend only on fluid a. We require that∑a or bℛab=∑a or bSab=∑a or bUab=∑a or bVab=0.[8]This ensures that globally charge and energy are conserved, as well as that uniform shifts in the background chemical potential and/or temperature, for all fluids simultaneously, are exact zero modes of the equations of motion.

For simplicity in [6], we have implicitly assumed that the Weyl nodes are all at the same chemical potential in equilibrium. This is generally not true for realistic Weyl materials. As nontrivial issues in hydrodynamics already arise without making this generalization, we will stick to the case where all Weyl nodes are at the same chemical potential in equilibrium in this paper.

For the remainder of this paper, we will be interested in transport in flat spacetimes where Rμναβ=0. Except where otherwise stated, we will assume Minkowski space from now on. Hence, for most purposes, we write partial derivatives ∂μ rather than covariant derivatives ∇μ. However, we will continue to use the covariant derivative ∇μ at intermediate steps of the calculations where it is necessary.

Thermodynamic Constraints

We will now derive the constraints on our hydrodynamic parameters which are imposed by demanding that the second law of thermodynamics is obeyed locally. Without intervalley coupling processes, and at the ideal fluid level (derivative corrections, including Fμν, are neglected), the second law of thermodynamics implies that the total entropy current sμ (where sa is the entropy density of fluid a) obeys (42, 43)∂μsμ=∂μ(∑asauaμ)=0.[9]In the more generic, nonideal, case the right-hand side of [9] must be nonnegative. In our theory of coupled chiral fluids, the right-hand side of [9] does not vanish already at the ideal fluid level:∂μsμ=∑ab(βa[Uabνb+Vabβb]+νa[ℛabνb+Sabβb])≥0.[10]There is no possible change we can make to the entropy current that is local and that can subtract off the right-hand side of [10]. Hence, we demand that the matrixA≡(ℛ −S−U V)[11]is positive semidefinite.

Using standard arguments for Onsager reciprocity in statistical mechanics (44), one can show that A=AT. In the SI Appendix, we will show using the memory matrix formalism (39, 45) that whenever the quantum mechanical operators na and ϵa are naturally defined:AIJ=Tlimω→0Im(Gx˙Ix˙JR(ω))ω,[12]where xI denotes (na,ϵa) and dots denote time derivatives. We also prove [8], and the symmetry and positive semidefiniteness of A through the memory matrix formalism, at the quantum mechanical level.

Equilibrium Fluid Flow

We now find an equilibrium solution to [6]. Beginning with the simple case of B=0, it is straightforward to see following ref. 46 that an equilibrium solution isμa=μ0(x),[13a]Ta=T0=constant,[13b]uaμ=(1,0).[13c]Indeed as pointed out in refs. 32, 46, this exactly satisfies [6] neglecting the intervalley and anomalous terms. Using [8] it is straightforward to see that the intervalley terms also vanish on this solution. If B=0 then CεμνρσFμνFρσ=0, and hence this is an exact solution to the hydrodynamic equations. We define the parameter ξ as the typical correlation length of μ(x): roughly speaking, ξ∼|μ0|/|∂xμ0|.

Following ref. 43, we can perturbatively construct a solution to the equations of motion when B≠0, assuming thatB≪T2 and 1≪ξT.[14]Both of these assumptions are necessary for our hydrodynamic formalism to be physically sensible. Using these assumptions, it is consistent at leading order to only change via≠0, but to keep μa and Ta the same:vza=Caμ02B2(ϵa+Pa)+GaT02Bϵa+Pa≡v(μ(x),T,Bi).[15]

It may seem surprising that in a single chiral fluid, there would be a nonvanishing charge current. This is a well-known phenomenon called the chiral magnetic effect (for a recent review, see ref. 47). In our model, the net current flow is the sum of the valley contributions:Jz=∑aJaz=∑aCaμ0B=0,[16]and so indeed, this complies with the expectation that the net current in a solid-state system will vanish in equilibrium, as discussed (in more generality) in refs. 41, 48.

Thermoelectric Conductivity

We now linearize the hydrodynamic equations around this equilibrium solution, applying infinitesimally small external electric fields E˜i, and temperature gradients ζ˜i=−∂i⁡log⁡T to the fluid. Although we have placed an equal sign in this equation, we stress that we will apply ζ˜i in such a way we may apply a constant temperature gradient on a compact space (with periodic boundary conditions). Applying a constant E˜i is simple, and corresponds to turning on an external electric field in Fμν. Applying a constant ζ˜i is more subtle, and can be done by changing the spacetime metric to (49):ds2=ημνdxμdxν−2e−iωt−iωζ˜idxidt,[17]ω is a regulator, which we take to 0 at the end of the calculation. This spacetime is flat (Rαβγδ=0). To account for both E˜i and ζ˜i, the external gauge field is modified to A+A˜, whereA˜i=−(E˜i−μ0(x)ζ˜i)e−iωt−iω.[18]

The hydrodynamic Eq. 6 must then be solved in this modified background. In linear response, the hydrodynamic variables becomeμa=μ0(x)+μ˜a(x),[19a]Ta=T0+T˜a(x),[19b]uaμ≈(1+viaζ˜ie−iωtiω,via+v˜ia).[19c]Note that tilded variables represent objects that are first order in linear response. The correction to uat is necessary to ensure that uμuμ=−1 is maintained. In general we cannot solve the linearized equations analytically, except in the limit of perturbatively weak disorder and magnetic field strength.

We assume that the inhomogeneity in the chemical potential is small:μ0=μ¯0+uμ^0(x),[20]with u≪μ¯0 and T; u is our perturbative parameter, and we assume that μ^0 is a zero-mean random function with unit variance. We assume the scalingsB∼u2  and   ℛ,S,U,V∼u6.[21]The hydrodynamic equations can be solved perturbatively, and the charge and heat currents may be spatially averaged on this perturbative solution. The computation is presented in the SI Appendix, and we present highlights here. At leading order, the linearized hydrodynamic equations reduce to∂i[naw˜ia+σQa(E˜i−∂iμ˜a−μ0T(Tζ˜i−∂iT˜a))]=CaE˜iBi−∑b[ℛabν˜b+Sabβ˜b][22a]∂i[Tsaw˜ia−μ0σQa(E˜i−∂iμ˜a−μ0T(Tζ˜i−∂iT˜a))]=2GaT02ζ˜iBi+∑b[(ℛabμ0+Uab)ν˜b+(Sabμ0+Vab)β˜b][22b]na(∂iμ˜a−E˜i)+sa(∂iT˜a−Tζ˜i)=εijkw˜janaBk.[22c]We have definedv˜ia=w˜ia+∂vi∂μμ˜a+∂vi∂TT˜a.[23]Here, w˜ia represents the fluid velocity after subtracting the contribution coming from [15] in local thermal equilibrium.

Eq. 22 depends on Ga, despite the fact that our spacetime is flat. This follows from the subtle fact that thermodynamic consistency of the anomalous quantum field theory on curved spacetimes requires that the axial–gravitational anomaly alters the thermodynamics of fluids on flat spacetimes (40).

The total charge current is J˜i=∑aJ˜ai, and the total heat current is Q˜i=∑aT˜ati−μ0J˜i. At leading order in perturbation theory, we find that the charge current in each valley fluid may be written asJ˜ai=naV˜ia+CaM˜aBi,[24]and the heat current per valley, T˜ati−μ0J˜ai, may be written asQ˜ai=TsaV˜ia+2GaT0T˜aBi.[25]In the above expressions V˜ia is a homogeneous O(u−2) contribution to w˜ia, and M˜a and T˜a are O(u−4) homogeneous contributions to μ˜a and T˜a, respectively.

The thermoelectric conductivity matrix is:σxx=σyy=∑ana2ΓaΓa2+B2na2,[26a]σxy=∑aBna3Γa2+B2na2,[26b]σzz=∑ana2Γa+sB2,[26c]κ¯xx=κ¯yy=∑aTsa2ΓaΓa2+B2na2,[26d]κ¯xy=∑aBTnasa2Γa2+B2na2,[26e]κ¯zz=∑aTsa2Γa+hB2,[26f]αxx=αyy=∑anasaΓaΓa2+B2na2,[26g]αxy=−αyx=∑aBna2saΓa2+B2na2,[26h]αzz=∑anasaΓa+aB2,[26i]where we have defined the four parameterss≡T(Ca Caμ)(ℛab −Sab−Uab Vab)−1(CbCbμ),[27a]h≡4T4(0 Ga)(ℛab −Sab−Uab Vab)−1(0Gb),[27b]a≡2T2(0 Ga)(ℛab −Sab−Uab Vab)−1(CbCbμ),[27c]Γa≡T02(sa(∂na/∂μ)−na(∂sa/∂μ))23σQa(ϵa+Pa)2u2.[27d]In these expressions, sums over valley indices are implicit. Coefficients odd under z→−z (such as σxz) vanish. Note that all of the contributions to the conductivities listed above are of the same order O(u−2) in our perturbative expansion, explaining the particular scaling limit (21) in u that was taken.

We have not listed the full set of transport coefficients. The unlisted transport coefficients are related to those in ref. 26 by Onsager reciprocity:σij(B)=σji(−B),[28a]κ¯ij(B)=κ¯ji(−B),[28b]αij(B)=α¯ji(−B).[28c]The symmetry of A is crucial in order for the final conductivity matrix to obey [28].

Evidently, the conductivities perpendicular to the magnetic field are Drude-like. This follows from principles, which are by now very well understood (31, 39). In these weakly disordered fluids, the transport coefficients are only limited by the rate at which momentum relaxes due to the disordered chemical potential Γa/(ϵa+Pa), and/or the rate at which the magnetic field relaxes momentum (by “rotating” it in the xy plane), Bna/(ϵa+Pa). This latter energy scale is the hydrodynamic cyclotron frequency (31). In our hydrodynamic theory, we can see this momentum “bottleneck” through the fact that the components of the charge and heat currents in [24] and [25], perpendicular to Bi, are proportional to the same fluid velocity V˜ia∼u−2 at leading order. The transport of the fluid is dominated by the slow rate at which this large velocity can relax. Because the heat current and charge current are proportional to this velocity field, the contribution of each valley fluid to σ, α, and κ¯ are all proportional to one another in the xy plane.

The remaining nonvanishing transport coefficients are σzz, αzz, and κ¯zz. From [26], we see that these conductivities are a sum of a Drude-like contribution (because this is transport parallel to the magnetic field, there is no magnetic momentum relaxation) from each valley, as before, and a new “anomalous” contribution that couples the valley fluids together. This anomalous contribution has a qualitatively similar origin as that discovered in refs. 8, 36. It can crudely be understood as follows: the chemical potential and temperature imbalances M˜ and T˜ are proportional to B and inversely proportional to A, as the homogeneous contributions to the right-hand side of [22] cancel. Such thermodynamic imbalances lead to corrections to valley fluid charge and heat currents, analogous to the chiral magnetic effect—these are the linear in B terms in [24] and [25]. Combining these scalings together immediately gives us the qualitative form of the anomalous contributions to the conductivity matrix.

The positive semidefiniteness of the thermoelectric conductivity matrix is guaranteed. Thinking of the conductivity matrices as a sum of the anomalous contribution and Drude contributions for each valley, it suffices to show each piece is positive-definite individually. The Drude pieces are manifestly positive definite, as is well known (it is an elementary exercise in linear algebra to confirm). To show the anomalous pieces are positive semidefinite, it suffices to show that sh≥Ta2. This follows from [27], and the Cauchy–Schwarz inequality (v1TAv1)(v2TAv2)≥(v1TAv2)2 for any vectors v1,2, and a symmetric, positive-semidefinite matrix A. These arguments also guarantee s,h>0.

Our expression for the conductivity may seem ill posed—it explicitly depends on the matrix inverse A−1, but A is not invertible. In fact, the kernel of A has two linearly independent vectors: (1a,1a) and (1a,−1a), with 1a denoting a vector with valley indices with each entry equal to 1. However, in the final formula for the thermoelectric conductivities, which sums over all valley fluids, we see that A−1 is contracted with vectors that are orthogonal to the kernel of A due to [7]. The expression for the conductivities is therefore finite and unique.

We present a simple example of our theory for a fluid with two identical Weyl nodes of opposite chirality in the SI Appendix, along with a demonstration that the equations of motion are unchanged when we account for long-range Coulomb interactions, or impose electric fields and temperature gradients through boundary conditions in a finite domain. Hence, the transport coefficients we have computed above are in fact those which will be measured in experiment.

In this paper, we used inhomogeneity in the chemical potential to relax momentum when B=0. By following the hydrodynamic derivation in ref. 46, other mechanisms for disorder likely lead to the same thermoelectric conductivities as reported in ref. 26, but with a different formula for Γa.

Violation of the Wiedemann–Franz Law

The thermal conductivity κij usually measured in experiments is defined with the boundary conditions J˜i=0 (as opposed to κ¯ij, which is defined with E˜i=0). This thermal conductivity is related to the elements of the transport matrix [1] byκij=κ¯ij−Tα¯ikσkl−1αlj.[29]In an ordinary metal, the Wiedemann–Franz (WF) law states that (50)ℒij≡κijTσij=π23.[30]The numerical constant of π2/3 comes from the assumption that the quasiparticles are fermions, and that the dominant interactions are between quasiparticles and phonons or impurities, but otherwise is robust to microscopic details.

In general, our model will violate the WF law. Details of this computation are provided in the SI Appendix. In general, the WF law is violated by an O(1) constant, which depends on the magnetic field B. However, in the special case where we have valley fluids of opposite chirality but otherwise identical equations of state, we find that the transverse Lorenz ratios ℒxx, ℒxy, ℒyy are all parametrically smaller than 1 (in fact, they vanish at leading order in perturbation theory). In contrast, we find that ℒzz∼B2 at small B, and saturates to a finite number as B becomes larger (but still ≪T2). This dramatic angular dependence of the WF law would be a sharp experimental test of our formalism in a strongly correlated Weyl material.

If the intervalley scattering rate is almost vanishing, the anomalous conductivities of a weakly interacting Weyl gas are still computable with our formalism. Weak intravalley scattering processes bring the “Fermi liquid” at each Weyl node to thermal equilibrium, and A may be computed via semiclassical kinetic theory. Assuming that intervalley scattering occurs elastically off of pointlike impurities, we compute s, a, and h in the SI Appendix. We find that ℒzzanom<π2/3, asymptotically approaching the WF law when μ≫T. This is in contrast to the nonanomalous conductivities of a semiconductor, where under similar assumptions ℒzz>π2/3 (51). The Mott relation between αzzanom and σzzanom differs by an overall sign from the standard relation. These discrepancies occur because we have assumed that elastic intervalley scattering is weaker than intravalley thermalization. In contrast, [11] makes the opposite assumption when μ≫T, and so recovers all ordinary metallic phenomenology. Even in this limit, negative thermal magnetoresistance is a consequence of nonvanishing Ga.

Outlook

In this paper, we have systematically developed a hydrodynamic theory of thermoelectric transport in a Weyl semimetal where quasiparticle–quasiparticle scattering is faster than quasiparticle–impurity and/or quasiparticle–phonon scattering. We have demonstrated the presence of longitudinal negative magnetoresistance in all thermoelectric conductivities. New phenomenological parameters introduced in our classical model may be directly computed using the memory matrix formalism given a microscopic quantum mechanical model of a Weyl semimetal. Our formalism is directly applicable to microscopic models of interacting Weyl semimetals where all relevant nodes are at the same Fermi energy. Our model should be generalized to the case where different nodes are at different Fermi energies, though our main results about the nature of negative magnetoresistance likely do not change qualitatively.

Previously, exotic proposals had been put forth to measure the axial–gravitational anomaly in an experiment. Measurements involving rotating cylinders of a Weyl semimetal have been proposed in refs. 41, 52, and it is possible that the rotational speed of neutron stars is related to this anomaly (53). A nonvanishing negative magnetoresistance in either αzz or κ¯zz is a direct experimental signature of the axial–gravitational anomaly. It is exciting that a relatively mundane transport experiment on a Weyl semimetal is capable of detecting this novel anomaly.