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# Hydrodynamic theory of thermoelectric transport and negative magnetoresistance in Weyl semimetals

Contributed by Subir Sachdev, June 30, 2016 (sent for review June 3, 2016; reviewed by Anton Burkov and Kristan Jensen)

## Significance

Weyl semimetals are exotic materials with negative electrical magnetoresistance: when an electric and magnetic field are applied in parallel, the induced electrical current increases upon increasing magnetic field strength. This is due to an emergent axial quantum anomaly in Weyl semimetals. We present a universal description of thermoelectric transport in weakly disordered Weyl semimetals where electron–electron interactions are faster than electron–impurity scattering. We predict negative thermal magnetoresistance: upon applying a parallel temperature gradient and magnetic field, the induced heat current increases with increasing magnetic field strength. This is caused by a distinct emergent quantum anomaly—the axial–gravitational anomaly. Measuring this effect may be the most practical route to experimentally observing this anomaly in any branch of physics.

## Abstract

We present a theory of thermoelectric transport in weakly disordered Weyl semimetals where the electron–electron scattering time is faster than the electron–impurity scattering time. Our hydrodynamic theory consists of relativistic fluids at each Weyl node, coupled together by perturbatively small intervalley scattering, and long-range Coulomb interactions. The conductivity matrix of our theory is Onsager reciprocal and positive semidefinite. In addition to the usual axial anomaly, we account for the effects of a distinct, axial–gravitational anomaly expected to be present in Weyl semimetals. Negative thermal magnetoresistance is a sharp, experimentally accessible signature of this axial–gravitational anomaly, even beyond the hydrodynamic limit.

The recent theoretical predictions (1⇓–3) and experimental discoveries (4⇓–6) of Weyl semimetals open up an exciting new solid state playground for exploring the physics of anomalous quantum field theories. These anomalies can lead to striking signatures in simple transport measurements. Upon applying a magnetic field

So far, the theories of this negative magnetoresistance assume two facts about the dynamics of the quasiparticles of the Weyl semimetal. First, it is assumed that the quasiparticles are long lived, and that a kinetic description of their dynamics is valid. Second, it is assumed that the dominant scattering mechanism is between quasiparticles and impurities or phonons. In most simple crystals—including Weyl semimetals—it is likely that this description is reasonable.

However, there are exotic metals in which the quasiparticle–quasiparticle scattering time is much smaller than the quasiparticle–impurity/phonon scattering time. In such a finite temperature metal, the complicated quantum dynamics of quasiparticles reduces to classical hydrodynamics of long-lived quantities—charge, energy, and momentum—on long time and length scales. Most theoretical (19⇓⇓⇓⇓⇓–25) and experimental (26⇓–28) work on such electron fluids studies the dynamics of (weakly interacting) Fermi liquids in ultrapure crystals. As expected, the physics of a hydrodynamic electron fluid is qualitatively different from the kinetic regime where quasiparticle–impurity/phonon scattering dominates, and there are qualitatively distinct signatures to look for in experiments.

Experimental evidence for a strongly interacting quasirelativistic plasma of electrons and holes has recently emerged in graphene (29, 30). The relativistic hydrodynamic theories necessary to understand this plasma are different from ordinary Fermi liquid theory (31), and lead to qualitatively different transport phenomena (32, 33). The hydrodynamics necessary to describe an electron fluid in a Weyl material, when the Fermi energy is close to a Weyl node, is similar to the hydrodynamics of the graphene plasma, though with additional effects related to anomalies (34, 35). Such a quasirelativistic regime is where negative magnetoresistance is most pronounced (9), and also where interaction effects can be strongest, due to the lack of a large Fermi surface to provide effective screening.

In this paper, we develop a minimal hydrodynamic model for direct current (dc) thermoelectric transport in a disordered, interacting Weyl semimetal, where the Fermi energy is close to the Weyl nodes. The first hydrodynamic approach to transport in a Weyl semimetal may be found in ref. 36 (see also refs. 37, 38). In contrast to these, our approach ensures that the conductivity matrix is positive semidefinite and Onsager reciprocal. We apply an infinitesimal electric field

Although the qualitative form of our results (e.g.,

In this paper, we work in units where

## 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 *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 fermion*T*, and velocity *SI Appendix*. We will take the background electromagnetic field to be*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 *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 *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

The new coefficients **[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 *a*. We require that

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

## 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 *a*) obeys (42, 43)**[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:**[10]**. Hence, we demand that the matrix

Using standard arguments for Onsager reciprocity in statistical mechanics (44), one can show that *SI Appendix*, we will show using the memory matrix formalism (39, 45) that whenever the quantum mechanical operators **[8]**, and the symmetry and positive semidefiniteness of

## Equilibrium Fluid Flow

We now find an equilibrium solution to **[6]**. Beginning with the simple case of **[6]** neglecting the intervalley and anomalous terms. Using **[8]** it is straightforward to see that the intervalley terms also vanish on this solution. If

Following ref. 43, we can perturbatively construct a solution to the equations of motion when

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:

## Thermoelectric Conductivity

We now linearize the hydrodynamic equations around this equilibrium solution, applying infinitesimally small external electric fields

The hydrodynamic Eq. **6** must then be solved in this modified background. In linear response, the hydrodynamic variables become

We assume that the inhomogeneity in the chemical potential is small:*T*; *u* is our perturbative parameter, and we assume that *SI Appendix*, and we present highlights here. At leading order, the linearized hydrodynamic equations reduce to**[15]** in local thermal equilibrium.

Eq. **22** depends on

The total charge current is

The thermoelectric conductivity matrix is:*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:**[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 **[24]** and **[25]**, perpendicular to

The remaining nonvanishing transport coefficients are **[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 *B* and inversely proportional to **[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 **[27]**, and the Cauchy–Schwarz inequality

Our expression for the conductivity may seem ill posed—it explicitly depends on the matrix inverse **[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

## Violation of the Wiedemann–Franz Law

The thermal conductivity **[1]** by

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 *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 *B*, and saturates to a finite number as *B* becomes larger (but still

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 *SI Appendix*. We find that **[11]** makes the opposite assumption when

## 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

## Acknowledgments

R.A.D. is supported by the Gordon and Betty Moore Foundation EPiQS Initiative through Grant GBMF4306. A.L. and S.S. are supported by the National Science Foundation under Grant DMR-1360789 and Multi-University Research Initiative Grant W911NF-14-1-0003 from Army Research Office. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. S.S. also acknowledges support from Cenovus Energy at Perimeter Institute.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: sachdev{at}g.harvard.edu, lucas{at}fas.harvard.edu, or rdavison{at}physics.harvard.edu.

Author contributions: A.L., R.A.D., and S.S. designed research; A.L., R.A.D., and S.S. performed research; and A.L. and R.A.D. wrote the paper.

Reviewers: A.B., University of Waterloo; and K.J., San Francisco State University.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1608881113/-/DCSupplemental.

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