# Weyl hydrodynamics in a strong magnetic field

Siyu Zhu  Physics Department, University of California, Santa Cruz, California
95064, USA  Grigory Bednik  Physics Department, University of California,
Santa Cruz, California 95064, USA  Sergey Syzranov  Physics Department,
University of California, Santa Cruz, California 95064, USA

###### Abstract

We study the hydrodynamic transport of electrons in a Weyl semimetal in a
strong magnetic field. Impurity scattering in a Weyl semimetal with two Weyl
nodes is strongly anisotropic as a function of the direction of the field and
is significantly suppressed if the field is perpendicular to the separation
between the nodes in momentum space. This allows for convenient access to the
hydrodynamic regime of transport, in which electron scattering is dominated by
interactions rather than by impurities. In a strong magnetic field, electrons
move predominantly parallel to the direction of the field, and the flow of the
electron liquid in a Weyl-semimetal junction resembles the Poiseuille flow of
a liquid in a pipe. We compute the viscosity of the Weyl liquid
microscopically and find that it weakly depends on the magnetic field and has
the temperature dependence $\eta(T)\propto T^{2}$. The hydrodynamic flow of
the Weyl liquid can be generated by a temperature gradient. The hydrodynamic
regime in a Weyl-semimetal junction can be probed via the thermal conductance
$G_{q}(B,T)\propto B^{2}T$ of the junction.

##  I Introducion

Hydrodynamics has recently been receiving attention as a paradigm for
describing transport in sufficiently clean materials with strong electron
correlations not amenable to exact microscopic treatment. Hydrodynamic
description deals with macroscopic degrees of freedom, such as the densities
of particles and their momenta.

The hydrodynamic regime requires that the electron-electron scattering rate
significantly exceed the electron-phonon and impurity scattering rates and is
predicted to lead to such uncovnentional phenomena as Gurzhi effect Gurzhi
(1963, 1968) (growing conductance with increasing temperature), current
vortices Levitov and Falkovich (2016); Bandurin  _et al._ (2016) and magnetic
dynamos in electron liquids Galitski  _et al._ (2018). Hydrodynamic transport
is also often discussed as a possible mechanism behind the linear-in-$T$
resistivity in high-temperature superconductors Davison  _et al._ (2014);
Hartnoll (2015).

Dirac materials in 2D (graphene Narozhny  _et al._ (2017); Narozhny (2019);
Lucas and Fong (2018)) and 3D (Weyl and Dirac semimetals Sukhachov  _et al._
(2018); Gorbar  _et al._ (2018a, b, c); Galitski  _et al._ (2018); Sukhachov
_et al._ (2021)) is another popular venue for theoretical studies of
hydrodynamic effects. Hydrodynamic flows in such systems simulate
ultrarelativistic interacting matter and, in the case of two dimensions, allow
for convenient visualisation (see, e.g., Ref. Ku  _et al._ (2020)).

Despite extensive theoretical studies, achieving the hydrodynamic regime is
rather challenging; materials that allow for conclusive experimental
observations of hydrodynamic transport are few and far between. Such
observations include manifestations of hydrodynamics in the nonlocal transport
in high-mobility $(Al,Ga)As$ heterostructures Gupta  _et al._ (2021); Braem
_et al._ (2018), magnetoresistive Bockhorn  _et al._ (2011); Alekseev (2016);
Mani  _et al._ (2013); Shi  _et al._ (2014) and Gurzhi effects de Jong and
Molenkamp (1995); Gusev  _et al._ (2018a, b) in $(Al,Ga)As$ heterostructures,
deviations from the Wiedemann-Franz law Jaoui  _et al._ (2018) in $WP_{2}$ and
a combination of magnetotransport phenomena in $PdCoO_{2}$ Moll  _et al._
(2016). Graphene provides another popular playground for observing
hydrodynamic phenomena Bandurin  _et al._ (2016, 2018); Sulpizio  _et al._
(2019); Gallagher  _et al._ (2019); Berdyugin  _et al._ (2019); Crossno  _et
al._ (2016); Ku  _et al._ (2020) (see Refs. Narozhny (2019); Lucas and Fong
(2018) for a comprehensive review).

Figure 1:  The flow of a Weyl liquid in a junction in a strong magnetic field.
The motion is effectively unidirectional because quasiparticles can move only
parallel or antiparallel to the magnetic field. The velocity of the liquid is
defined as the velocity of the reference frame (bath) in which thermalisation
of the liquid takes place. The dependence of the velocity on the transverse
coordinate creates shear stress. The corresponding dissipative forces are
determined by the Coulomb interaction between different layers of the liquid.

In this paper, we demonstrate that 3D Weyl semimetals (WSMs) is a readily
accessible platform for hydrodynamic transport and discuss manifestations of
such transport in them in strong magnetic fields. As demonstrated recently in
Ref. Bednik  _et al._ (2020), the impurity scattering time $\tau$ for
electrons in a Weyl semimetal with two nodes is strongly anisotropic as a
function of the direction of the magnetic field:

$\displaystyle\frac{1}{\tau}=\frac{1}{\tau_{0}}\cos^{2}\theta+\frac{1}{\tau_{1}},$
(1)

where $\theta$ is the angle between the field and the separation of the Weyl
nodes in momentum space and $1/\tau_{0}\gg 1/\tau_{1}$. The scattering rate is
strongly suppressed for $\theta$ close to $\pi/2$, i.e. for magnetic fields
perpendicular to the separation between the nodes, which makes the
hydrodynamic regime in a Weyl semimetal conveniently achievable by applying
the magnetic field in the respective direction (in addition to that, the
elastic scattering length in a strong magnetic field may be quite large even
away from $\theta=\frac{\pi}{2}$; see Appendix A).

In a strong magnetic field, Weyl electrons move predominantly (anti)parallel
to the direction of the field. As a result, the motion of the electron liquid
in a Weyl-semimetal junction in a magnetic field resembles the Poiseuille flow
Sutera and Skalak (1993); Landau and Lifshitz (1987) of a liquid in a pipe, as
shown in Fig. 1. The friction between the layers of the Weyl liquid moving
with different velocities leads to dissipation and viscosity. Due to the
effectively one-dimensional character of the flow, there is no transverse
transport of particles that leads to a common mechanism of viscosity in
liquids and gases Reif (2009). The viscosity comes, however, from the “Coulomb
drag” between layers of liquid moving with different velocities, a mechanism
introduced recently for conventional metals in Ref. Liao and Galitski (2020).

We derive the hydrodynamic equations describing the hydrodynamic motion of a
Weyl liquid in a strong magnetic field, compute microscopically the viscosity
of such a liquid and analyse the conduction of a Weyl-semimetal junction. We
find that the viscosity weakly depends on the magnetic field and strongly on
the temperature $T$ and, for realistic temperatures, is given by

$\displaystyle\eta=\frac{M}{12\pi v_{F}}T^{2},$ (2)

where the “mass” $M$ gives the inverse curvature of the quasiparticle
dispersion near the Weyl nodes and $v_{F}$ is the Fermi velocity. In the
hydrodynamic regime, the thermal conductance (the response of the energy flux
to a small temperature difference) of a Weyl-semimetal junction is given, up
to a non-universal coefficient of order unity which depends on the shape of
the junction, by

$\displaystyle G_{q}\sim\left(\frac{|e|B}{c}\right)^{2}\frac{S}{LMv_{F}}T,$
(3)

where $S$ is the cross-sectional area and $L$ is the length of the junction.

The paper is organised as follows. In Sec. II, we introduce the model of WSMs
in a strong magnetic field and discuss the approximations we use in this
paper. In Sec. III, we derive the hydrodynamic equations for the electron
liquid in such a semimetal. Sec. IV deals with the viscosity of such a liquid.
In Sec. V, we describe the hydrodynamic flow of the Weyl liquid generated by a
temperature gradient and the possibility of its experimental observation. We
conclude in Sec. VI.

##  II Model

We consider the model of a Weyl semimetal with two Weyl nodes, right (R) and
left (L) (shown in Fig. 2), and equal energies of the nodes. The Weyl
semimetal may have additional nodes that weakly affect the electron dynamics
between nodes L and R so long as the additional nodes are well separated in
momentum space from the nodes under consideration. The magnetic field ${\bf
B}$ is directed at angle $\theta$ with respect to the line connecting the two
nodes in momentum space. For simplicity, we assume that the quasiparticles
have no spin (apart from the pseudospin operator associated with the bands in
the Weyl semimetal) and have isotropic dispersions around each node. Our
quantitative results will hold, however, up to coefficients of order unity,
for an arbitrary type-I Weyl semimetal.

Figure 2:  Orientation of the magnetic field relative to the locations of the
Weyl nodes in momentum space.

We focus on the ultraquantum limit of the magnetic field

$\displaystyle
B>\left(\frac{2}{9}\right)^{\frac{1}{3}}\frac{\mu^{2}c}{|e|v_{F}^{2}},$ (4)

at which all electrons in equilibrium occupy the zeroth Landau level, where
$\mu$ is the chemical potential (measured from the energy of the Weyl nodes)
in the absence of the field; $v_{F}$ is the Fermi velocity; hereinafter we set
$\hbar=1$.

In the absence of impurity scattering and interactions, the motion of
electrons is one-dimensional; quasiparticles can propagate only parallel or
antiparallel to the direction of the magnetic field ${\bf B}$ with the
velocity $v_{F}$. We assume that electrons move along the magnetic field at
node L and in the opposite direction at node R.

Impurities and screening of Coulomb interactions. The strength of Coulomb
interactions in the system is characterised by the dimensionless “fine
structure constant”

$\displaystyle\alpha=\frac{e^{2}}{\epsilon v_{F}},$ (5)

where $\epsilon$ is the dielectric constant. Most Weyl and Dirac materials
have sufficiently large values of the dielectric constant Jay-Gerin  _et al._
(1977); Madelung  _et al._ (1998); Jenkins  _et al._ (2016); Buckeridge  _et
al._ (2016) to ensure the condition $\alpha\ll 1$, which controls the
diagrammatic perturbation theory for the interactions used in this paper.

While we focus on the hydrodynamic regime of transport, the system may contain
a small amount of charged impurities. The hydrodynamic transport, studied in
this paper, will persist so long as the elastic scattering rate (1) is
significantly exceeded by the quasiparticle scattering rate due to electron-
electron interactions.

In the Thomas-Fermi approximation Abrikosov (1988), the screening radius of
static Coulomb interaction is given by Bednik  _et al._ (2020)

$\displaystyle\varkappa^{-1}=\sqrt{\frac{\pi\epsilon
v_{F}c}{2|e|^{3}B}}=l_{B}\sqrt{\frac{\pi}{2\alpha}}.$ (6)

The Thomas-Fermi approximation is justified in the limit Abrikosov (1988)
$|\mu|\gg\varkappa v_{F}$, which will be assumed throughout this paper.
However, our results still hold qualitatively for other values of $\mu$.

Spatial scales of the hydrodynamic flow. To apply the hydrodynamic
description, we assume that the macroscopic degrees of freedom of the electron
liquid, such as the velocity ${\bf u}$ of the liquid, vary smoothly in space,
on length scales $\lambda$ significantly exceeding the microscopic scales of
the system,

$\displaystyle\lambda\gg Q^{-1},l_{B},\varkappa^{-1},$ (7)

where $2Q$ is the separation between the Weyl nodes in momentum space; the
screening radius $\varkappa^{-1}$ is given by Eq. (6) and

$\displaystyle l_{B}=\sqrt{c/(|e|B)}$ (8)

is the magnetic length. The momentum separation $2Q$ between the Weyl nodes is
typically of the order of inverse atomic distances and is assumed to be the
largest momentum scale in the problem.

At low temperatures, the densities are correlated on length scales of the
order of the screening radius (6) $\varkappa^{-1}=l_{B}\sqrt{\pi/\alpha}$,
which exceeds the magnetic length $l_{B}$ due to the smallness of the coupling
constant $\alpha$. The correlations in the electron liquid may, therefore, be
assumed isotropic on lengthscales $\varkappa^{-1}\ll L\ll\lambda$ exceeding
the screening radius $\varkappa^{-1}$ but shorter than the characteristic
scales $\lambda$ of the variation of the macroscopic hydrodynamic parameters
such as the velocity of the liquid.

Energy scales. As we demonstrate in this paper, the viscosity of the Weyl
liquid strongly depends on its temperature. For magnetic fields on the order
of $1T$ or larger, the cyclotron frequency $v_{F}/l_{B}$ is on the order of
$50$meV or larger and, thus, significantly exceeds the temperatures $T$ used
in experiments on Weyl semimetals. Taking into account Eq. (6), realistic
energy scales may, therefore, be assumed to satisfy the conditions

$\displaystyle T\ll v_{F}\varkappa\ll v_{F}/l_{B}.$ (9)

##  III Hydrodynamic equations

###  III.1 Velocity of the hydrodynamic flow

In the hydrodynamic description, the flowing electron liquid may be considered
to be equilibrated in a moving reference frame. It is possible, therefore, to
introduce the velocity ${\bf u}$ of the electron liquid as the velocity of the
equilibrium reference frame. Considering the dispersion $\xi_{\bf k}=\pm
v_{F}k-u\left({\bf r}\right)k$ of the quasiparticles at the zeroth Landau
level near the left and the right nodes, the respective quasiparticle
occupation numbers are given by
$f_{L}\left(k,u,\mu_{L}\right)=f\left(k,u,\mu_{L}\right)$ and
$f_{R}\left(k,u,\mu_{R}\right)=f\left(-k,-u,\mu_{R}\right)$, where

$\displaystyle f\left(k,u,\mu\right)=\frac{1}{e^{\left[v_{F}k-u\left({\bf
r}\right)k-\mu\left({\bf r}\right)\right]/T}+1}$ (10)

and the momentum $k$ is measured from the respective node. Hereinafter, we
assume that the gradient of the velocity ${\bf u}$ is small and neglect the
effects of vorticity exemplified by the chiral vortical effect Vilenkin
(1980); Gorbar  _et al._ (2018a); Chen  _et al._ (2014).

The electron liquid may be thermalised by any bath of neutral excitations
(e.g. phonons) or the electrons themselves. The full hydrodynamic description
of a Weyl semimetal should include the hydrodynamic equations of motion of the
bath as well as those of the electron liquid. In this paper, we assume, for
simplicity, that the electron liquid acts as its own bath.

###  III.2 Hydrodynamic variables

We develop a hydrodynamic description of the electron liquid in terms of the
density of electrons near each node and the momentum density of the liquid.
The electron density $N_{L}$ near node L is measured relative to the
equilibrium states of an undoped Weyl semimetals in the absence of the flow
(i.e. for $u=0$):

$\displaystyle N_{L}=\frac{|e|B}{2\pi
c}\int\frac{dk}{2\pi}\left[f_{L}\left(k,u,\mu_{L}\right)-f_{L}\left(k,0,0\right)\right].$
(11)

Using the distribution functions
$f_{L}\left(k,u,\mu_{L}\right)=f\left(k,u,\mu_{L}\right)$ given by Eq. (10) we
obtain

$\displaystyle N_{L}=\frac{|e|B}{4\pi^{2}c}\frac{\mu_{L}}{v_{F}-u},$ (12)

where $\mu_{L}$ is the chemical potential at the left node.

Because each quasiparticle at the left node moves with the velocity $v_{F}$
along the magnetic field and carries a charge of $e=-|e|$, the electric
current $j_{L}$ carried by the electrons near this node (relative to the
equilibrium state in the absence of the flow) is given by

$\displaystyle j_{L}$ $\displaystyle=e\frac{|e|B}{2\pi
c}\int\frac{dk}{2\pi}v_{F}\left[f_{L}\left(k,u,\mu_{L}\right)-f_{L}\left(k,0,0\right)\right]$
$\displaystyle=ev_{F}N_{L}.$ (13)

Similarly, we compute the concentration $N_{R}$ of the electrons near the
right node:

$\displaystyle N_{R}=\frac{|e|B}{4\pi^{2}c}\frac{\mu_{R}}{v_{F}+u}$ (14)

and the current

$\displaystyle j_{R}=-ev_{F}N_{R}.$ (15)

###  III.3 Hydrodynamic equations

####  III.3.1 Continuity equations for densities

The continuity equation for the electron density $N_{L}$ reads

$\displaystyle\partial_{t}N_{L}+v_{F}\partial_{z}N_{L}=-\frac{e^{2}}{4\pi^{2}c}{\bf
E}\cdot{\bf B}-\frac{N_{L}-N_{R}}{\tau},$ (16)

where $v_{F}N_{L}$ is the flux of the density (measured relative to the
equilibrium state) along the magnetic field ($z$ axis); the first term in the
right-hand side (rhs) describes the change of the density $N_{L}$ due to the
chiral anomalyBurkov (2018); Son and Spivak (2013); Parameswaran  _et al._
(2014); Burkov (2014) in the presence of the electric field ${\bf E}$; the
second term in the rhs accounts for the elastic scattering of electrons
between the two nodes. In Eq. (16), $1/\tau$ is the rate of internodal elastic
scattering (due to collisions with impurities or other defects in the system).
Similarly, the continuity equation for the density $N_{R}$ is given by

$\displaystyle\partial_{t}N_{R}-v_{F}\partial_{z}N_{R}=\frac{e^{2}}{4\pi^{2}c}{\bf
E}\cdot{\bf B}-\frac{N_{R}-N_{L}}{\tau}.$ (17)

####  III.3.2 Navier-Stokes equation

In order to provide a complete hydrodynamic description of the electron liquid
in given electric and magnetic fields, the continuity equations (16) and (17)
have to be complemented by the Navier-Stokes equation for momentum density.
The momentum density near each individual Weyl node is not conserved due to
the interactions between electrons at different nodes.

The Navier-Stokes equation is given by

$\displaystyle\partial_{t}p+\partial_{z}J_{p}=F_{\text{E}}+F_{\text{scatt}}+F_{\text{visc}}-\partial_{z}P,$
(18)

where $p$ is the density of momentum along the $z$ axis; $J_{p}$ is the flux
of momentum; the force $F_{\text{E}}$ account for the change of the momentum
$p$ due to external electric and magnetic fields; $F_{\text{scatt}}$ describes
momentum relaxation due to impurity scattering; $F_{\text{visc}}$ is the force
that describes dissipative effects due to the viscosity of the electron liquid
and $P$ is the pressure of the electron liquid. In what immediately follows,
we compute these quantities microscopically in a weakly interacting Weyl
electron liquid.

The momentum density is given by

$\displaystyle p=$ $\displaystyle\frac{|e|B}{2\pi
c}\int\frac{dk}{2\pi}\left[f_{L}\left(k,u,\mu_{L}\right)-f_{L}\left(k,0,0\right)\right]\left(-Q\cos\theta+k\right)+\frac{|e|B}{2\pi
c}\int\frac{dk}{2\pi}\left[f_{R}\left(k,u,\mu_{R}\right)-f_{R}\left(k,0,0\right)\right]\left(Q\cos\theta+k\right)$
$\displaystyle=$ $\displaystyle
N_{L}\left[-Q\cos\theta+\frac{\mu_{L}}{2\left(v_{F}-u\right)}\right]+N_{R}\left[Q\cos\theta-\frac{\mu_{R}}{2\left(v_{F}+u\right)}\right]+\frac{|e|BT^{2}}{6c}\frac{v_{F}u}{(v_{F}^{2}-u^{2})^{2}},$
(19)

where $2Q$ is the separation between the Weyl nodes in momentum space. The
flux $J_{p}$ of momentum reads

$\displaystyle J_{p}=$ $\displaystyle v_{F}\frac{|e|B}{2\pi
c}\int\frac{dk}{2\pi}\left[f_{L}\left(k,u,\mu_{L}\right)-f_{L}\left(k,0,0\right)\right]\left(-Q\cos\theta+k\right)-v_{F}\frac{|e|B}{2\pi
c}\int\frac{dk}{2\pi}\left[f_{R}\left(k,u,\mu_{R}\right)-f_{R}\left(k,0,0\right)\right]\left(Q\cos\theta+k\right)$
$\displaystyle=$ $\displaystyle
v_{F}N_{L}\left[-Q\cos\theta+\frac{\mu_{L}}{2(v_{F}-u)}\right]-v_{F}N_{R}\left[Q\cos\theta-\frac{\mu_{R}}{2(v_{F}+u)}\right]+\frac{|e|BT^{2}}{12c}\frac{3v_{F}^{2}u^{2}-u^{4}}{\left(v_{F}^{2}-u^{2}\right)^{2}v_{F}}.$
(20)

Using Eq. (12) and (14), the divergence of the flux $J_{p}$ can be simplified
as

$\displaystyle\partial_{z}J_{p}=$
$\displaystyle\left\\{v_{F}\partial_{z}N_{L}\left[-Q\cos\theta+\frac{\mu_{L}}{2\left(v_{F}-u\right)}\right]-v_{F}\partial_{z}N_{R}\left[Q\cos\theta-\frac{\mu_{R}}{2\left(v_{F}+u\right)}\right]\right\\}$
$\displaystyle+\left[\frac{v_{F}N_{L}}{2}\partial_{z}\left(\frac{\mu_{L}}{v_{F}-u}\right)+\frac{v_{F}N_{R}}{2}\partial_{z}\left(\frac{\mu_{R}}{v_{F}+u}\right)\right]+v_{F}\frac{|e|B}{24c}\left[\partial_{z}\frac{T^{2}}{(v_{F}-u)^{2}}+\partial_{z}\frac{T^{2}}{(v_{F}+u)^{2}}\right]$
$\displaystyle=$ $\displaystyle
v_{F}\partial_{z}N_{L}\left(-Q\cos\theta+\frac{\mu_{L}}{v_{F}-u}\right)-v_{F}\partial_{z}N_{R}\left(Q\cos\theta-\frac{\mu_{R}}{v_{F}+u}\right)$
$\displaystyle+\frac{|e|BT^{2}}{6c}\frac{(3v_{F}^{2}u+u^{3})v_{F}}{\left(v_{F}^{2}-u^{2}\right)^{3}}\partial_{z}u+\frac{|e|BT}{6c}\frac{3v_{F}^{2}u^{2}-u^{4}}{\left(v_{F}^{2}-u^{2}\right)^{2}v_{F}}\partial_{z}T.$
(21)

The force $F_{E}$ is given by the change of the total momentum $p$ due to the
transfer of quasiparticles between the nodes because of the chiral anomaly:

$\displaystyle F_{E}=$ $\displaystyle\frac{|e|B}{2\pi
c}\partial_{t}\int\frac{dk}{2\pi}\left\\{f_{L}\left[k,u,\mu_{L}-|e|E_{z}t\left(v_{F}-u\right)\right]-f_{L}\left(k,u,\mu_{L}\right)\right\\}\left(-Q\cos\theta+k\right)\bigg{|}_{t=0}$
$\displaystyle+\frac{|e|B}{2\pi
c}\partial_{t}\int\frac{dk}{2\pi}\left\\{f_{R}\left[k,u,\mu_{R}+|e|E_{z}t\left(v_{F}+u\right)\right]-f_{R}\left(k,u,\mu_{R}\right)\right\\}\left(Q\cos\theta+k\right)\bigg{|}_{t=0}$
(22)

Using the distribution functions
$f_{L}\left(k,u,\mu_{L}\right)=f\left(k,u,\mu_{L}\right)$ given by Eq. (10) we
obtain

$\displaystyle F_{E}=$ $\displaystyle-\frac{e^{2}}{4\pi^{2}c}{\bf E}\cdot{\bf
B}\left(-Q\cos\theta+\frac{\mu_{L}}{v_{F}-u}\right)+\frac{e^{2}}{4\pi^{2}c}{\bf
E}\cdot{\bf B}\left(Q\cos\theta-\frac{\mu_{R}}{v_{F}+u}\right).$ (23)

In the limit of low temperatures $T$, Eq. (23) can be understood intuitively
as follows. The quantities $-Q\cos\theta+\frac{\mu_{L}}{v_{F}-u}$ and
$Q\cos\theta-\frac{\mu_{R}}{v_{F}+u}$ give the momenta of the quasiparticles
near the chemical potentials at the left and the right nodes and
$\frac{e^{2}}{4\pi^{2}c}{\bf E}\cdot{\bf B}$ is the rate of increase of the
quasiparticle density at the right node (or its decrease at the left node).
Multiplying these momenta by the corresponding rates of change of
quasiparticle densities gives the rate of change of the total momentum due to
an external electromagnetic field in the limit of zero temperature. We
emphasise, however, that the result (23) applies at all temperatures $T$.

The momentum relaxation rate due to impurity scattering is given by

$\displaystyle F_{scatt}=$ $\displaystyle\frac{|e|B}{2\pi
c}\frac{1}{\tau}\int\frac{dk}{2\pi}\left[f_{R}\left(-k,u,\mu_{R}\right)-f_{L}\left(k,u,\mu_{L}\right)\right]\left(-Q\cos\theta+k\right)$
$\displaystyle+\frac{|e|B}{2\pi
c}\frac{1}{\tau}\int\frac{dk}{2\pi}\left[f_{L}\left(-k,u,\mu_{L}\right)-f_{R}\left(k,u,\mu_{R}\right)\right]\left(Q\cos\theta-k\right)$
$\displaystyle=$
$\displaystyle\frac{N_{L}-N_{R}}{\tau}\left(2Q\cos\theta-\frac{\mu_{L}}{v_{F}-u}-\frac{\mu_{R}}{v_{F}+u}\right)-\frac{1}{\tau}\frac{|e|BT^{2}}{3c}\frac{uv_{F}}{\left(v_{F}^{2}-u^{2}\right)^{2}},$
(24)

where $1/\tau$ is the elastic internodal scattering rate introduced in Eqs.
(16) and (17). At $T=0$, Eq. (III.3.2) can be understood intuitively as
follows. At $T=0$, all the electron states with energies up to $\mu_{L}$ and
$\mu_{R}$ are filled at the left and right nodes, and it is possible to assume
that only electrons with energies
$\min(\mu_{L},\mu_{R})<\varepsilon<\max(\mu_{L},\mu_{R})$ get scattered
between the nodes. Then the quantities
$-Q\cos\theta+\frac{1}{2}\left(\frac{\mu_{L}}{v_{F}-u}+\frac{\mu_{R}}{v_{F}+u}\right)$
and
$Q\cos\theta-\frac{1}{2}\left(\frac{\mu_{L}}{v_{F}-u}+\frac{\mu_{R}}{v_{F}+u}\right)$
have the meaning of the average momenta of electrons at the left and the right
nodes that participate in these elastic scattering processes. Multiplying
these momenta by the rate $\frac{N_{L}-N_{R}}{\tau}$ of change of the
densities of electrons due to internodal scattering gives Eq. (III.3.2) at
$T=0$.

The force $F_{visc}=\partial_{x}T_{xz}+\partial_{y}T_{yz}+\partial_{z}T_{zz}$
describes the dissipative effects due to the viscosity of the liquid, where
$T_{xz}$, $T_{yz}$ and $T_{zz}$ are the components of the stress tensor. While
the liquid can move only along the direction of the magnetic field (the $z$
axis, see Fig. 1) in the strong magnetic field under consideration, the
velocity $u$ of this motion is different for different transverse coordinates
$x$ and $y$ for the same $z$, which creates shear stress. The total viscous
force is given by

$\displaystyle
F_{visc}=\eta\left(\partial_{x}^{2}+\partial_{y}^{2}\right)u+\kappa\partial_{z}^{2}u,$
(25)

where $u$ is the velocity of the electron liquid defined in Sec. (III.1);
$\eta$ is the shear viscosity, the response of the stress forces between
layers of the electron liquid flowing along the $z$ axis to the transverse
gradient of the velocity $u$; the coefficient $\kappa$ characterises the
response of the strain to the longitudinal spatial change of the velocity $u$.
We microscopically demonstrate in Sec. (IV) that $\eta\gg\kappa$.

The pressure $P$ of the Weyl liquid, computed in Appendix B, is given by

$\displaystyle P=P_{0}+\frac{|e|B}{12cv_{F}}T^{2},$ (26)

where $P_{0}$ is a temperature-independent contribution that depends on the
details of the quasiparticle dispersion away from the Weyl nodes.

Combining Eqs. (18), (III.3.2), (23), (III.3.2) and (25), we arrive at the
Navier-Stokes equation (18) in the form

$\displaystyle\partial_{t}p+v_{F}\partial_{z}N_{L}\left(-Q\cos\theta+\frac{\mu_{L}}{v_{F}-u}\right)-v_{F}\partial_{z}N_{R}\left(Q\cos\theta-\frac{\mu_{R}}{v_{F}+u}\right)$
$\displaystyle+\frac{|e|BT^{2}}{6c}\frac{(3v_{F}^{2}u+u^{3})v_{F}}{\left(v_{F}^{2}-u^{2}\right)^{3}}\partial_{z}u+\frac{|e|BT}{6c}\frac{3v_{F}^{2}u^{2}-u^{4}}{\left(v_{F}^{2}-u^{2}\right)^{2}v_{F}}\partial_{z}T=$
$\displaystyle-\frac{e^{2}}{4\pi^{2}c}{\bf E}\cdot{\bf
B}\left(-Q\cos\theta+\frac{\mu_{L}}{v_{F}-u}\right)+\frac{e^{2}}{4\pi^{2}c}{\bf
E}\cdot{\bf B}\left(Q\cos\theta-\frac{\mu_{R}}{v_{F}+u}\right)$
$\displaystyle+\frac{N_{L}-N_{R}}{\tau}\left(2Q\cos\theta-\frac{\mu_{L}}{v_{F}-u}-\frac{\mu_{R}}{v_{F}+u}\right)-\frac{1}{\tau}\frac{|e|BT^{2}}{3c}\frac{uv_{F}}{\left(v_{F}^{2}-u^{2}\right)^{2}}+\eta\left(\partial_{x}^{2}+\partial_{y}^{2}\right)u+\kappa\partial_{z}^{2}u-\partial_{z}P.$
(27)

Equations (16), (17) and (III.3.2) provide a complete hydrodynamic description
of a Weyl liquid in a strong magnetic field.

##  IV Viscosity

In this section, we compute microscopically the viscosity of a Weyl liquid in
a strong magnetic field. The viscosity tensor is determined by the correlator
of the corresponding components of the stress tensor (see, for example, Ref.
Bradlyn  _et al._ (2012)) and can be represented in the form

$\displaystyle\eta_{ijkl}(\omega)=\frac{1}{\Omega}\left[{\cal
B}_{ijkl}(\omega)-{\cal B}_{ijkl}(0)\right],$ (28)

where

$\displaystyle{\cal B}_{ijkl}(\omega)=$ $\displaystyle\frac{1}{2}\int d{\bf
r}\int_{-\beta}^{\beta}d\tau\left<T_{\tau}{\hat{T}}_{ij}({\bf
r},\tau){\hat{T}}_{kl}({\bf
0},0)\right>e^{i\Omega\tau}\bigg{|}_{i\Omega\rightarrow\omega+i0}$ (29)

is the retarded correlator of the components ${\hat{T}}_{ij}$ and
${\hat{T}}_{kl}$ of the stress tensor operator and
$i\Omega\rightarrow\omega+i0$ is our convention for the analytic continuation
from positive Matsubara frequencies $\Omega$ to the real frequency $\omega$
Mahan (2008); Abrikosov  _et al._ (1975).

Strictly speaking, the viscosity of the electron liquid depends on the
velocity ${\bf u}$ of the liquid at a given location, and the averaging
$\left<\ldots\right>$ in Eq. (29) should be carried out with respect to the
equilibrium Fermi-Dirac distribution (10) in the reference frame of the moving
liquid. However, because realistic velocities ${\bf u}$ are significantly
exceeded by the Fermi velocity $v_{F}$, the dependence of the viscosity on the
velocity ${\bf u}$ may be neglected and averaging over the equilibrium state
of a stationary liquid may be used when computing the viscosity tensor (28).
In what follows, we evaluate explicitly the Matsubara correlator in Eq. (29).

The stress tensor ${\hat{T}}_{ij}$ includes two qualitatively distinct
components Martin and Schwinger (1959). The first, kinetic, component is
independent of the interaction in the system and for a Weyl semimetal with two
nodes, is given by

$\displaystyle{\hat{T}}_{ij}^{(0)}({\bf
r})=\sum_{\chi=L,R}{\hat{\psi}}^{\dagger}_{\chi}({\bf
r}){\hat{v}}_{i\chi}{\hat{k}}_{j}{\hat{\psi}}_{\chi}({\bf r}),$ (30)

where the summation is carried out over the nodes $\chi=L,R$;
${\hat{\psi}}^{\dagger}_{\chi}({\bf r})$ and ${\hat{\psi}}_{\chi}({\bf r})$
are the creation and annihilation operators of the electrons at node $\chi$;
${\hat{v}}_{i\chi}$ is the $i$-th component of the velocity operator at node
$\chi$ and ${\hat{k}}_{j}=-i\frac{\partial}{\partial r_{j}}$ is the $j$-th
momentum component. The second contribution to the stress tensor
${\hat{T}}_{ij}$ is determined by the electron-electron interactions Martin
and Schwinger (1959) (see also Refs. Link  _et al._ (2018) and Liao and
Galitski (2020)) and, in the limit of smooth variations of the gradients of
the macroscopic parameters of the liquid [cf. the condition (7)] (“local
uniformity approximation” of Ref. Martin and Schwinger (1959)), is given by

$\displaystyle{\hat{T}}_{ij}^{\text{(int)}}({\bf
r})=-\frac{1}{2}\sum_{\chi,\chi^{\prime}=L,R}\int
d{\bm{\rho}}\,\,{\hat{\psi}}^{\dagger}_{\chi}\left({\bf
r}+\frac{{\bm{\rho}}}{2}\right){\hat{\psi}}^{\dagger}_{\chi^{\prime}}\left({\bf
r}-\frac{{\bm{\rho}}}{2}\right)$ $\displaystyle\rho_{i}\frac{\partial
V({\bm{\rho}})}{\partial\rho_{j}}{\hat{\psi}}_{\chi^{\prime}}\left({\bf
r}-\frac{{\bm{\rho}}}{2}\right){\hat{\psi}}_{\chi}\left({\bf
r}+\frac{{\bm{\rho}}}{2}\right),$ (31)

where $V({\bm{\rho}})=\frac{e^{2}}{\varepsilon\rho}$ is the Coulomb
interaction potential.

###  IV.1 Shear viscosity

In what immediately follows, we compute the viscosity
$\eta=\eta_{xzxz}=\eta_{yzyz}$ that describes the response of the shear stress
Landau and Lifshitz (1970) $T_{xz}$ and $T_{yz}$ of the liquid flowing along
the $z$ axis, the direction of the magnetic field (see Fig. 1), to the
transverse gradients $\frac{\partial u}{\partial x}$ and $\frac{\partial
u}{\partial y}$ of the velocity. Because the quasiparticles at both nodes can
move only (anti-)parallel to the magnetic field ($v_{x\chi}=v_{y\chi}=0$,
$v_{z\chi}\neq 0$), there is no kinetic contribution (30) to the components
$T_{xz}$ and $T_{yz}$ of the stress tensor, which determine the viscosity
$\eta$. The absence of such a contribution is due to the effectively one-
dimensional character of the motion of the liquid. Indeed, in conventional
liquids and gases, in which interactions between particles may be considered
to be contact, the viscosity comes from the transverse transport of particles
between parallel moving layers of liquid and the transfer of the longitudinal
momentum of those particles Reif (2009). Because transverse transport is
negligible in a Weyl semimetal in a strong magnetic field, the viscosity is
dominated by the Coulomb drag between parallel moving layers of liquid, which
comes from the long-range character of (screened) Coulomb interactions. In
what follows, we compute, therefore, the Matsubara correlator [cf. Eq. (29)]
of the interaction contributions (IV) to the stress tensor.

The electron liquids may relax momentum via processes of quasiparticle
scattering between the nodes. Due to the long-range nature of Coulomb
interactions, with the characteristic momentum scale $\varkappa$ given by Eq.
(6), such processes have a rate suppressed by the small parameter
$\varkappa/Q\ll 1$ and will not be considered here.

Another possible mechanism of viscosity comes from the Coulomb drag
Pogrebinskii (1977); Price (1983); Narozhny and Levchenko (2016) between
layers of the electron liquid moving parallel to each other, as shown in Fig.
1. In the presence of the transverse gradient of the velocity ${\bf u}$,
different layers of the electron liquid move with different velocities, with
Coulomb interactions resulting in effective friction forces between the
layers. This mechanism of viscosity has been pioneered in Ref. Liao and
Galitski (2020) for a conventional Fermi liquid. Under the made
approximations, it also dominates the viscosity of Weyl fermions in a strong
magnetic field considered here.

Figure 3:  Diagrams for computing the viscosity of the electron liquid. (a)
The vertex corresponding to the interaction contribution to the stress tensor.
(b) The diagram for the drag contribution to the viscosity. (c) A block in
diagram (b) which takes into account the screening of the interactions in the
RPA.

The drag contribution to the Matsubara correlator ${\cal B}_{ijkl}(i\Omega)$
in Eq. 29 corresponds to the diagram in Fig. 3b, where the interaction
contribution to the stress tensor $T_{ij}$ corresponds to the vertex shown in
Fig. 3a. To describe the screening of the interactions, we use the random
phase approximation (RPA) Mahan (2008); Abrikosov (1988), as shown in Fig. 3c.
A prefactor of $2$ in diagram 3b comes from two possible pairings of the ends
of the stress-tensor vertex 3a in the correlator that this diagram describes.

The correlator ${\cal B}_{ijkl}(i\Omega)$, corresponding to the diagram 3b,
can be evaluated in the momentum representation as

$\displaystyle{\cal B}_{ijkl}(i\Omega)=$
$\displaystyle\frac{T}{2}\sum_{i\nu}\int\frac{d^{3}{\bf
q}}{(2\pi)^{3}}\frac{\partial}{\partial q_{i}}\left[q_{j}V({\bf
q})\right]\frac{\partial}{\partial q_{k}}\left[q_{l}V({\bf q})\right]$
$\displaystyle\frac{\Pi\left(i\Omega+i\nu,{\bf q}\right)}{1-V({\bf
q})\Pi\left(i\Omega+i\nu,{\bf q}\right)}\frac{\Pi\left(i\nu,{\bf
q}\right)}{1-V({\bf q})\Pi\left(i\nu,{\bf q}\right)},$ (32)

where $V({\bf q})=\frac{4\pi e^{2}}{\epsilon q^{2}}$ is the bare propagator of
Coulomb interactions and $\Pi(i\nu,q)$ is the polarisation operator, which
corresponds to a simple fermionic bubble in the diagrams in Fig. 3c in the
limit of a small coupling $\alpha\ll 1$. A microscopic calculation of the
polarisation operator are presented in Appendix C.

Equation (32) contains a Matsubara sum of the form
$I(i\Omega)=T\sum_{i\nu}D(i\nu+i\Omega)D(i\nu)$, which can be conveniently
computed by contour integration in the complex $\nu$ plane that gives

$\displaystyle I(i\Omega)=$ $\displaystyle\frac{1}{4\pi
i}\int_{-\infty}^{+\infty}\coth\frac{\varepsilon}{2T}\left[D_{A}(\varepsilon+i\Omega)D_{R}(\varepsilon)-D_{R}(\varepsilon+i\Omega)D_{A}(\varepsilon)\right]d\varepsilon$
$\displaystyle+\frac{1}{4\pi
i}\int_{-\infty}^{+\infty}\coth\frac{\varepsilon}{2T}\left[D_{R}(\varepsilon)D_{R}(\varepsilon-i\Omega)-D_{A}(\varepsilon)D_{A}(\varepsilon-i\Omega)\right]d\varepsilon,$

where $D_{A}(\varepsilon)$ and $D_{R}(\varepsilon)\equiv
D_{A}^{*}(\varepsilon)$ are the advanced and retarded versions of the
correlator $D(i\nu)$, i.e. obtained from it by analytic continuation from,
respectively, the lower and the upper half-planes (see, for example, Refs.
Perel and Eliashberg (1962) and Kamenev and Oreg (1995) for the details of the
contour integration). Performing such contour integration and the analytic
continuation $i\Omega\rightarrow\omega+i0$ and utilising Eq. (28) gives, in
the limit of low frequencies $\omega$,

$\displaystyle\eta(\omega\rightarrow 0)=$
$\displaystyle\frac{1}{T}\int\frac{d\varepsilon}{2\pi}\frac{d^{3}{\bf
q}}{(2\pi)^{3}}\left[\frac{q_{x}q_{z}V^{\prime}(q)}{2q\sinh\frac{\varepsilon}{2T}}\right]^{2}$
$\displaystyle\frac{\left[\mathrm{Im}\,\Pi_{R}(\varepsilon,{\bf
q})\right]^{2}}{\left\\{\left[1-V({\bf
q})\,\mathrm{Re}\,\Pi_{R}(\varepsilon,{\bf q})\right]^{2}+\left[V({\bf
q})\,\mathrm{Im}\,\Pi_{R}(\varepsilon,{\bf q})\right]^{2}\right\\}^{2}},$ (33)

where $\Pi_{R}(\varepsilon,{\bf q})$ is the retarded polarisation operator
obtained by analytic continuation from the Matsubara polarisation operator
$\Pi(i\nu,{\bf q})$ given by Eq. (34). Equation (33) has been obtained in Ref.
Liao and Galitski (2020) using Keldysh technique. In what immediately follows,
we evaluate explicitly the polarisation operator for experimentally important
frequency and momentum scales.

#### Polarisation operator

The polarisation operator

$\displaystyle\Pi({\bf r};{\bf
r}^{\prime};i\Omega)=-\frac{1}{2}\int_{-\beta}^{\beta}\left<{\hat{n}}({\bf
r},\tau){\hat{n}}({\bf r}^{\prime},0)\right>e^{i\Omega\tau}d\tau,$ (34)

where ${\hat{n}}({\bf r},\tau)$ is the electron density, is evaluated
explicitly in Appendix C. In the limit $\varkappa^{-1}\gg l_{B}$ under
consideration and at momenta $|{\bf q}|\ll l_{B}^{-1}$, the Fourier-transform
of the retarded polarisation operator is given by

$\displaystyle\Pi_{R}(\varepsilon,{\bf q})=$ $\displaystyle\frac{|e|B}{2\pi
c}\sum_{\chi=L,R}\int\frac{dk}{2\pi}\frac{f_{\chi}(k,0,\mu_{\chi})-f_{\chi}(k+q_{z},0,\mu_{\chi})}{E_{\chi
k}-E_{\chi(k+q_{z})}+\varepsilon+i0},$ (35)

where the integration is carried out over the momentum $k$ along the direction
of the magnetic field; $f_{\chi}(k,0,\mu_{\chi})$ is the distribution function
of the electrons at node $\chi$ [cf. Eq. (10)] and $E_{\chi k}$ is the
corresponding electron dispersion (as we clarify below, the deviation of the
dispersion from the linear dependence $\pm v_{F}k$ needs to be taken into
account for evaluating the viscosity of the system).

The real and imaginary parts of the retarded polarisation operator
$\Pi_{R}(\varepsilon,{\bf q})$ describe, respectively, the screening of
Coulomb interactions and the decay of the density waves in the Weyl liquid
(Landau damping). In what immediately follows, we evaluate these contributions
explicitly.

#### Screening

The main contribution to the static viscosity (33) comes from the energies
$\varepsilon$ on the order of the temperature $T$, which is significantly
exceeded by the cyclotron frequency [see Eq. (9)] and the bandwidth of the
quasiparticle dispersion. This allow us to neglect the
$\varepsilon$-dependence of the real part $\text{Re}\Pi_{R}$ of the retarded
polarisation operator.

Similarly, we neglect the dependence of $\text{Re}\Pi_{R}$ on the momentum
${\bf q}$ whose characteristic values are on the order of the inverse
screening radius $\varkappa$ given by Eq. (6) and significantly exceeded by
the inverse magnetic length $l_{B}^{-1}$ [cf. the condition (9)] and the
momentum scales of the quasiparticle band. Below, we will show that the
typical scale of the momentum component $q_{z}$ that contributes to the
viscosity is even smaller and is on the order of $T/v_{F}$.

The real part of the polarisation operator is, therefore, given by the density
of the electron states at the Fermi level (with the minus sign):

$\displaystyle\text{Re}\,\Pi_{R}(\varepsilon,{\bf
q})=-\frac{|e|B}{2\pi^{2}cv_{F}}=-\frac{\epsilon\varkappa^{2}}{4\pi e^{2}},$
(36)

where $\varkappa$ is the inverse screening radius of Coulomb interactions
given by Eq. (6).

#### Landau damping

For the existence of a finite damping (to the leading order in interactions),
it is necessary to take into account the curvature of the electron dispersion
near the nodes. Indeed, for linearly dispersive quasiparticles, density waves
composed of electrons near one node propagate with the velocity $\pm v_{F}$
and lack dispersion. The conservation of momentum in any process involving
only electrons near one node also enforces energy conservation, which is why
all momentum conserving processes contribute to the damping and lead to a
singular $\propto\sum_{\pm}\delta(\varepsilon\pm v_{F}q_{z})$ imaginary part
of the lowest-order polarisation operator (35).

In order to describe a finite dispersion of the charge density waves, we take
into account the non-linearity of the quasiparticle dispersion near the Weyl
nodes:

$\displaystyle E_{k}=\pm v_{F}k+\frac{k^{2}}{2M},$ (37)

where “$+$” and “$-$” correspond, respectively, to the left and the right
nodes. The dispersion (37) and the momentum $k$ are measured, respectively,
from the Fermi level and Fermi momentum. The energy scale $Mv_{F}^{2}$ is the
largest energy scale in the problems and, in the case it is determined by the
band structure of the Weyl semimetal, may be assumed to be on the order of
several electronvolt.

Utilising Eqs. (35) and (37) gives

$\displaystyle\text{Im}\,\Pi_{R}(\varepsilon,{\bf q})=\frac{|e|B}{4\pi
c}\frac{M}{|q_{z}|}\sum_{\pm}$
$\displaystyle\frac{\pm\sinh\frac{\varepsilon}{2T}}{\cosh\left[\frac{Mv_{F}^{2}(\varepsilon\pm
v_{F}q_{z})}{2\varepsilon
T}+\frac{\varepsilon}{4T}\right]\cosh\left[\frac{Mv_{F}^{2}(\varepsilon\pm
v_{F}q_{z})}{2\varepsilon T}-\frac{\varepsilon}{4T}\right]}.$ (38)

The terms with “+” and “-” correspond to functions sharply peaked at
$\varepsilon\pm v_{F}q_{z}=0$ and account, respectively, for the contribution
of the left and right nodes.

The characteristic values $\varepsilon$ and $v_{F}q_{z}$ that contribute to
the viscosity are on the order of the temperature $T$. When deriving Eq. (38)
we neglected, therefore, the effect of the small energy
$\frac{q_{z}^{2}}{2M}\sim\frac{T^{2}}{Mv_{F}^{2}}\ll 1$ on the distribution
functions $f_{\chi}(k,0,\mu_{\chi})$ and $f_{\chi}(k+q_{z},0,\mu_{\chi})$ in
the polarisation operator (35).

#### The value of the shear viscosity

Utilising Eqs. (33) and (36) and the smallness of the Landau damping, the
viscosity can be rewritten in the form

$\displaystyle\eta=\frac{1}{T}\left(\frac{4\pi
e^{2}}{\epsilon}\right)^{2}\int\frac{d\varepsilon}{2\pi}\frac{d^{3}{\bf
q}}{(2\pi)^{3}}\frac{q_{x}^{2}q_{z}^{2}}{\sinh^{2}\frac{\varepsilon}{2T}}\frac{\left[\mathrm{Im}\,\Pi_{R}(\varepsilon,{\bf
q})\right]^{2}}{\left(q^{2}+\varkappa^{2}\right)^{4}}.$ (39)

Because the imaginary part $\text{Im}\,\Pi_{R}(\varepsilon,{\bf q})$ of the
retarded polarisation operator is sharply peaked at $\varepsilon\pm vq_{z}=0$
only momenta $q_{z}$ on the order of $T/v_{F}$ contribute to the viscosity. By
contrast, the transverse momenta $q_{x}$ and $q_{y}$ have characteristic
values on the order of $\varkappa$, which significantly exceed $T/v_{F}$ [see
the condition (9)]. This allows us to neglect the dependence of the
denominator in Eq. (39) on the momentum $q_{z}$. Integrating out the
transverse momenta $q_{x}$ and $q_{y}$ gives

$\displaystyle\eta=\frac{1}{3\pi T\varkappa^{4}}\left(\frac{\pi
e^{2}}{\epsilon}\right)^{2}\int\frac{d\varepsilon}{2\pi}\frac{dq_{z}}{2\pi}\frac{\left[\mathrm{Im}\,\Pi_{R}(\varepsilon,{\bf
q})\right]^{2}}{\sinh^{2}\frac{\varepsilon}{2T}}q_{z}^{2}.$ (40)

Using Eq. (38) and introducing variables $s=\frac{\varepsilon}{4T}$ and
$t=\frac{Mv_{F}^{2}(v_{F}q_{z}-\varepsilon)}{2\varepsilon T}$, Eq. (40) can be
represented in the form

$\displaystyle\eta=$
$\displaystyle\frac{1}{3\pi^{3}T\varkappa^{4}}\left(\frac{\pi
e^{2}}{\epsilon}\right)^{2}\left(\frac{|e|B}{4\pi c}\right)^{2}$
$\displaystyle\frac{16T^{3}M}{v_{F}^{3}}\int\frac{|s|dtds}{\cosh^{2}(s+t)\cosh^{2}(s-t)},$
(41)

which gives the viscosity

$\displaystyle\eta=\frac{M}{12\pi v_{F}}T^{2}.$ (42)

Equation (42) is our main result for the viscosity of a Weyl liquid in a
strong magnetic field.

Due to the drag character of the analyzed mechanism of viscosity, it has the
same temperature dependence, $\propto T^{2}$, as the drag resistivity between
two parallel conductive layers Narozhny and Levchenko (2016) and can be
understood from phase-space considerations similar to those explaining the
$\propto T^{2}$ dependence of the electron-electron scattering rate in a
conventional metal Abrikosov (1988); Gantmakher (2005). Indeed, each electron
can collide with electrons in a parallel layer of the liquid with energies in
a window of order $T$ near the Fermi energy. The characteristic width of the
layer of momenta into which the electron can get scattered is also
proportional to the temperature $T$. In the presence of a finite curvature
$M^{-1}$ of the dispersion, the final momentum of the other electron
participating in the collision is fixed by the energy and momentum
conservation laws. This results, therefore, in the $\propto T^{2}$ scattering
rate that manifests itself in the viscosity (42).

###  IV.2 Longitudinal response

In this subsection, we evaluate the coefficient $\kappa$ that characterises
the response of stress component $T_{zz}$ to the gradient $\frac{\partial
u}{\partial z}$ [cf. Eq. (25)]. This coefficient has both drag and kinetic
contributions.

$\displaystyle\kappa^{\text{drag}}=\frac{1}{T}\left(\frac{4\pi
e^{2}}{\epsilon}\right)^{2}\int\frac{d\varepsilon}{2\pi}\frac{d^{3}{\bf
q}}{(2\pi)^{3}}\frac{q_{z}^{4}}{\sinh^{2}\frac{\varepsilon}{2T}}\frac{\left[\mathrm{Im}\,\Pi_{R}(\varepsilon,{\bf
q})\right]^{2}}{\left(q^{2}+\varkappa^{2}\right)^{4}}.$ (43)

Using Eq. (38) and introducing variables $s=\frac{\varepsilon}{4T}$,
$t=\frac{Mv_{F}^{2}(v_{F}q_{z}-\varepsilon)}{2\varepsilon T}$ and
$b=\frac{4T}{v_{F}}$, Eq. (43) can be represented in the form

$\displaystyle\kappa^{\text{drag}}=\frac{M}{12\pi
v_{F}}T^{2}\cdot\frac{4}{\varkappa^{2}}\int\frac{\left(bs\right)^{2}|s|dsdt}{\cosh^{2}\left(s-t\right)\cosh^{2}\left(s+t\right)}.$
(44)

By introducing Eq. (42), Eq. (44) can be expressed as

$\displaystyle\kappa^{\text{drag}}$
$\displaystyle=\frac{4}{\varkappa^{2}}\int\frac{\left(bs\right)^{2}|s|dsdt}{\cosh^{2}\left(s-t\right)\cosh^{2}\left(s+t\right)}\eta$
$\displaystyle=\frac{3\zeta\left(3\right)b^{2}}{\varkappa^{2}}\eta\sim\frac{M}{v_{F}^{3}\varkappa^{2}}T^{4}.$
(45)

For the low temperatures $T\ll\varkappa v_{F}$ under consideration [cf. Eq.
(9)], the contribution (45) is significantly exceeded by the shear viscosity
(42). The contribution (45) is proportional to the typical value of
$q_{z}^{4}\propto T^{4}$ and, as a result, contains an extra power of $T^{2}$
relative to the shear viscosity (42). It is possible to show that the kinetic
contribution [i.e. containing the kinetic vertices (30)] has the same
dependence on the typical value of $q_{z}$ and the $\propto T^{4}$ temperature
dependence and is also significantly suppressed compared to the shear
viscosity $\eta$. We leave, however, a rigorous calculation of this
contribution for future studies.

##  V Temperature-generated flow and potential for experimental observation

In this section, we address the possibility of experimental observation of the
discussed hydrodynamic flow of a Weyl electron liquid in a strong magnetic
field. In a sufficiently long Weyl-semimetal junction, whose length exceeds
the elastic scattering length $\tau v_{F}$, the conductance is independent of
the viscosity $\eta$. Indeed, according to Eqs. (16) and (17), a longitudinal
electric field $E$ results in a stationary imbalance of the electron densities
$N_{L}-N_{R}=-\frac{e^{2}}{4\pi^{2}c}{\bf B}\cdot{\bf E}\tau$, which leads to
a finite conductivity $\sigma=\frac{|e|^{3}v_{F}}{4\pi^{2}c}B\tau$ matching
the conductivity in a system in the non-hydrodynamic (diffusive) regime Son
and Spivak (2013); Burkov (2018).

The hydrodynamic properties of the systems, however, manifest themselves in
heat transport. The hydrodynamic flow can be generated by a temperature
gradient and detected through the dependence of the heat flux on the
temperature and magnetic field.

For a stationary flow, the momentum flux and electron densities at nodes $L$
and $R$ do not change, $\partial_{t}p=\partial_{t}N_{L}=\partial_{t}N_{R}=0$.
Multiplying the continuity equations (16) and (17) by, respectively,
$-Q\cos\theta+\frac{\mu_{L}}{v_{F}-u}$ and
$Q\cos\theta-\frac{\mu_{R}}{\left(v_{F}+u\right)}$ and subtracting from the
Navier-Stokes equation (III.3.2) gives

$\displaystyle\eta\left(\partial_{x}^{2}+\partial_{y}^{2}\right)u+\kappa\partial_{z}^{2}u-\partial_{z}P$
$\displaystyle-\frac{|e|B}{6c}\frac{v_{F}u}{\left(v_{F}^{2}-u^{2}\right)^{2}}\left(\frac{2}{\tau}+\frac{3v_{F}^{2}+u^{2}}{v_{F}^{2}-u^{2}}\partial_{z}u\right.$
$\displaystyle\left.+\frac{3v_{F}^{2}u-u^{3}}{v_{F}^{2}}\frac{\partial_{z}T}{T}\right)T^{2}=0.$
(46)

At small velocities $u$ and temperatures $T$, the term
$\kappa\partial_{z}^{2}u$ and contributions in the last two lines of Eq. (V),
of the order of $uT^{2}$ in temperature and velocity, can be neglected.
Equation (V) then matches the equation for the flow of a conventional liquid
in a pipe Landau and Lifshitz (1987); Sutera and Skalak (1993).

In accordance with the Hagen–Poiseuille equation Landau and Lifshitz (1987);
Sutera and Skalak (1993), the hydrodynamic velocity $u$ of such a liquid in
the middle of the junction is given by

$\displaystyle u=\frac{\zeta S}{\eta L}\Delta P,$ (47)

where $\zeta$ is a coefficient of order unity that depends on the transverse
shape of the junction; $L$ is the length of the junction; $S$ is its cross-
sectional area and $\Delta P$ is the pressure difference between the two ends
of the junction.

The pressure difference $\Delta P$ may be generated by different temperatures
at the ends of the junction. Utilising Eq. (26), we estimate the flow velocity
of the Weyl liquid as

$\displaystyle u\sim\frac{|e|B}{c}\frac{S}{M}\frac{\partial_{z}T}{T},$ (48)

where the “mass” $M$ describes the inverse curvature of the quasiparticle
dispersion and is introduced in Eq. (37). Using Eq. (48) and assuming that the
energy scale $Mv_{F}^{2}$ is given by the quasiparticle bandwidth and is of
the order of $1eV$, we estimate that velocities $u$ of the order of $v_{F}\sim
10^{8}\frac{cm}{s}$ can be achieved in a junction of size $\sqrt{S}\sim L\sim
100nm$ (in all dimensions) in a magnetic field $B\sim 1T$ and for temperature
gradients $\partial_{z}T\sim T/L$. The hydrodynamic regime is further favoured
by larger system sizes and magnetic fields.

The flow of the electron liquid is associated with the heat flux (energy
current) in the system, given by

$\displaystyle q$ $\displaystyle=\frac{|e|B}{2\pi
c}v_{F}\int\frac{dk}{2\pi}v_{F}k\left[f_{L}(k,u,\mu_{L})-f_{L}(k,0,0)\right]$
$\displaystyle-\frac{|e|B}{2\pi
c}v_{F}\int\frac{dk}{2\pi}(-v_{F}k)\left[f_{R}(k,u,\mu_{R})-f_{R}(k,0,0)\right]$
$\displaystyle=$
$\displaystyle\frac{|e|Bv_{F}^{2}}{8\pi^{2}c}\left[\frac{\mu_{L}^{2}}{(v_{F}-u)^{2}}-\frac{\mu_{R}^{2}}{(v_{F}+u)^{2}}\right]+\frac{|e|B}{6c}\frac{v_{F}^{3}u}{(v_{F}^{2}-u^{2})^{2}}T^{2},$
(49)

which can be used to detect the hydrodynamic flow and measure the average
velocity $u$ of the flow.

In the absence of the electric field ${\bf E}$, there is no electric current
flowing through the system, as follows from Eqs. 16 and 17 and the charge
neutrality condition $N_{L}+N_{R}=const$, which require
$\frac{\mu_{L}}{v_{F}-u}=\frac{\mu_{R}}{v_{F}+u}$. According to Eq. (49), the
energy current in the absence of the charge current is, therefore,
proportional to the hydrodynamic velocity $u$ of the current:

$\displaystyle
q\approx\frac{|e|B}{6c}\frac{u}{v_{F}}T^{2}\sim\left(\frac{|e|B}{c}\right)^{2}\frac{S}{Mv_{F}}T\partial_{z}T.$
(50)

The hydrodynamic flow can, thus, be generated by a temperature difference
$\Delta T$ at the ends of the junction and detected through the temperature-
and magnetic-field dependence of the heat conductance $G_{q}(T,B)$, the
response of the total energy flux to $\Delta T$. Estimating the gradient
$\partial_{z}T$ of the temperature as $\partial_{z}T\sim\Delta T/L$, where $L$
is the length of the junction, gives

$\displaystyle G_{q}\sim\left(\frac{|e|B}{c}\right)^{2}\frac{S}{LMv_{F}}T.$
(51)

##  VI Conclusion

In conclusion, we have studied the hydrodynamic motion of the electron liquid
in a Weyl semimetal with two Weyl nodes in a strong magnetic field. Such
systems provide a conveniently accessible platform for achieving the
hydrodynamic regime of transport because the impurity scattering rate of Weyl
fermions is strongly suppressed for certain directions of the magnetic field,
perpendicular to the separation of Weyl nodes in momentum space.

Because Weyl fermions in a quantising magnetic field move parallel or
antiparallel to the field, the motion of the liquid resembles Poiseuille flow
of a conventional liquid in a pipe (see Fig. 1). The viscosity of such a
liquid is dominated by the interactions between parallel layers of the liquid
moving with different velocities. We have derived the hydrodynamic equations
of motion of such a liquid for a Weyl semimetal with two Weyl nodes and
computed microscopically its viscosity. For realistic temperatures, the
temperature dependence of the viscosity is given by $\eta(T)\propto T^{2}$.
The hydrodynamic flow of the electron liquid in a Weyl-semimetal junction can
be generated by a temperature gradient and probed via the heat conductance
$G_{q}\propto B^{2}T$ of the junction.

We are grateful to I. Shovkovy for numerous insightful discussions. Part of
this work was performed at the Aspen Center for Physics, which is supported by
the National Science Foundation grant PHY-1607611.

##  Appendix A Estimates of scattering rates in a strong magnetic field

In this section, we provide estimates of the internodal elastic scattering
rate Bednik  _et al._ (2020)

$\displaystyle\frac{1}{\tau}$ $\displaystyle\approx\frac{2\pi
n_{\text{imp}}|e|Bv_{F}}{c\hbar}\left(\frac{e^{2}}{\hbar\epsilon
v_{F}}\frac{\cos\theta}{2Q^{2}}\right)^{2}$ (52)

in a Weyl semimetal in the ultraquantum regime, where $2Q$ is the momentum
separation between the nodes; $n_{\text{imp}}$ is the concentration of
impurities; $\epsilon$ is the dielectric constant and $\theta$ is the angle
between the direction of the field and the separation of the nodes. Because
the scattering rate strongly depends on the node separation $2Q$,
$1/\tau\propto Q^{-4}$, it is rather sensitive to the details of the band
structure and may differ by orders of magnitude even in Weyl semimetals with
close parameters.

Compound | $v_{F}\times 10^{7}cm/s$ | $Q\times 10^{6}cm^{-1}$ | Source for $Q$ and $v_{F}$ | $\frac{1}{\tau}\times 10^{7}\cos^{2}{\theta}s^{-1}$
---|---|---|---|---
TaAs | 3.2 | 1.425 |  Grassano  _et al._ (2018) | 9.71
NbAs | 3.0 | 0.473 |  | 853
WTeS | $4.0$ | 37.8 |  Meng  _et al._ (2019)111To our knowledge, there is no available data for the Fermi velocity in these compounds, which is why we use the value $v_{F}=4.0\cdot 10^{7}cm/s$. | $1.57\times 10^{-5}$
WTeSe | $4.0$ | 26.2 |  | $6.80\times 10^{-5}$
Table 1:  The Fermi velocities, node separation and scattering rates (52) for
four type-I Weyl semimetals in the magnetic field $B=1T$. For the estimates,
we used $n=10^{15}cm^{-3}$ and $\epsilon=30$.

In Table. 1, we provide scattering rates for several of the Weyl semimetals.
These estimates show that even in materials with the strongest scattering, the
elastic scattering length in the ultraquantum limit may be on the order of
$10^{-2}cm$ and may exceed or be comparable to the size of the sample even for
angles $\theta$ away from $\pi/2$, which makes the hydrodynamic regime
conveniently accessible.

##  Appendix B Hydrodynamic pressure of the Weyl liquid

The Sommerfeld expansion of the grand potential of the electron liquid in
volume $V$

$\displaystyle\Omega(T)=-TV\int d\varepsilon\,{\cal
N}_{0}(\varepsilon)\ln\left(1+e^{\frac{\mu-\varepsilon}{T}}\right)=-\int
d\varepsilon\frac{N(\varepsilon)}{e^{\frac{\varepsilon-\mu}{T}}+1},$ (53)

gives

$\displaystyle\Omega(T)=\Omega(0)-\frac{\pi^{2}}{6}V{\cal N}_{0}(\mu)T^{2},$
(54)

where ${\cal N}_{0}(\varepsilon)$ is the density of states and
$N(\varepsilon)$ is the number of electron states in the system with energies
smaller than $\varepsilon$. Near the nodes of a Weyl semimetal in a strong
magnetic field, the density of states per node is given by ${\cal
N}_{0}(\varepsilon)=\frac{|e|B}{4\pi^{2}cv_{F}}$.

Using that $\Omega=-PV$, we obtain the pressure of the equilibrium electron
gas in a two-node Weyl semimetal in the form

$\displaystyle P=P_{0}+\frac{|e|B}{12cv_{F}}T^{2},$ (55)

where $P_{0}$ is a temperature-independent contribution which depends on the
details of the quasiparticle dispersion away from the Weyl nodes.

##  Appendix C Details of the calculation of the polarisation operator

In this section, we provide the details of the calculation of the polarisation
operator in a Weyl liquid in a magnetic field in the ultraquantum limit, in
which only the zeroth Landau level contributes. In what follows, we use the
Landau gauge

$\displaystyle{\bf A}=\left(-By,0,0\right)$ (56)

for the vector potential of the magnetic field. For this gauge, the momentum
${\bf k}_{xz}$ in the $xz$ plane is a good quantum number.

To the lowest order in interactions, the Matsubara polarisation operator is
given by

$\displaystyle\Pi({\bf r},{\bf r}^{\prime},i\Omega)=2T\sum_{i\omega}\sum_{{\bf
k}_{xz},{\bf k}_{xz}^{\prime}}\frac{\Psi_{{\bf k}_{xz}}^{*}({\bf r})\Psi_{{\bf
k}_{xz}}({\bf r}^{\prime})\Psi_{{\bf k}_{xz}^{\prime}}^{*}({\bf
r}^{\prime})\Psi_{{\bf k}_{xz}^{\prime}}({\bf r})}{\left(i\omega+i\Omega-
E_{{\bf k}_{xz}}\right)\left(i\omega-E_{{\bf k}_{xz}^{\prime}}\right)},$ (57)

where a prefactor of $2$ accounts from the presence of two nodes in the Weyl
liquid, which contribute equally to the polarisation operator; $E_{{\bf
k}_{xz}}$ is the quasiparticle dispersion at the zeroth Landau level with the
momentum ${\bf k}_{xz}$ in the $xz$ plane and

$\displaystyle\Psi_{{\bf k}_{xz}}({\bf r})=H_{{\bf
k}_{xz}}(y)\exp\left[{i(k_{x}x+k_{z}z)}\right]$ (58)

is the orbital part of the corresponding wavefunction, where

$\displaystyle H_{{\bf k}_{xz}}(y)={\left(\pi
l_{B}^{2}S_{xz}^{2}\right)^{-\frac{1}{4}}}\exp\left[-\frac{1}{2}\left(y/l_{B}-k_{x}l_{B}\right)^{2}\right];$
(59)

$S_{xz}$ is the area of the $xz$ cross-section of the system, which, for
simplicity, is assumed to be constant along the $z$ axis; $l_{B}$ is the
magnetic length given by Eq. (8).

For the evaluation of the viscosity in this paper, we focus on the
correlations of electron densities on distances $|{\bf r}-{\bf r}^{\prime}|\gg
l_{B}$ significantly exceeding the magnetic length $l_{B}$. To evaluate the
polarisation operator (57) at these scales, we first Fourier-transform it with
respect to the coordinate differences $x-x^{\prime}$ and $z-z^{\prime}$, using
the exact translational invariance along the $x$ and $z$ axes:

$\displaystyle\Pi({\bf
q}_{xz},y,y^{\prime},i\Omega)=2T\sum_{i\omega}\sum_{{\bf
k}_{xz}}\frac{H_{({\bf k}+{\bf q})_{xz}}^{*}(y)H_{({\bf k}+{\bf
q})_{xz}}(y^{\prime})H_{{\bf k}_{xz}}^{*}(y^{\prime})H_{{\bf
k}_{xz}}(y)}{\left[i\omega+i\Omega-E_{({\bf k}+{\bf
q})_{xz}}\right]\left(i\omega-E_{{\bf k}_{xz}}\right)}.$ (60)

For long distances, $|{\bf r}-{\bf r}^{\prime}|\gg l_{B}$, it is sufficient to
consider only small momenta $|{\bf q}_{xz}|\ll l_{B}^{-1}$. Because the
characteristic lengthscale of the function $H_{\bf k}(y)$, given by Eq. (59)
is $l_{B}$, it allows us to neglect the momentum ${\bf q}_{xz}$ in Eq. (60)
and make the approximations $H_{({\bf k}+{\bf q})_{xz}}^{*}(y)\approx H_{{\bf
k}_{xz}}^{*}(y)$, $H_{({\bf k}+{\bf q})_{xz}}(y^{\prime})\approx H_{{\bf
k}_{xz}}(y^{\prime})$.

The summand in Eq. (60) is peaked at the values of $y$ and $y^{\prime}$ given
by $k_{x}l_{b}^{2}$ and has a characteristic width of $l_{B}$ with respect to
both of these coordinates. Taking into account the summation with respect to
all values of $k_{x}$, the operator may be considered, at distances
$|y-y^{\prime}|\gg l_{B}$, as a sharply peaked function of $y-y^{\prime}$ and
approximated as

$\displaystyle\Pi({\bf q}_{xz},y,y^{\prime},i\Omega)=$
$\displaystyle\delta(y-y^{\prime})\int\left\\{2T\sum_{i\omega}\sum_{{\bf
k}_{xz}}\frac{H_{{\bf k}_{xz}}^{*}(y)H_{{\bf k}_{xz}}(y^{\prime})H_{{\bf
k}_{xz}}^{*}(y^{\prime})H_{{\bf k}_{xz}}(y)}{\left[i\omega+i\Omega-E_{({\bf
k}+{\bf q})_{xz}}\right]\left(i\omega-E_{{\bf
k}_{xz}}\right)}\right\\}dy^{\prime}$ $\displaystyle=$ $\displaystyle
2\delta(y-y^{\prime})T\sum_{i\omega}\sum_{{\bf k}_{xz}}\frac{H_{{\bf
k}_{xz}}^{*}(y)H_{{\bf k}_{xz}}(y)}{\left[i\omega+i\Omega-E_{({\bf k}+{\bf
q})_{xz}}\right]\left(i\omega-E_{{\bf k}_{xz}}\right)}$ $\displaystyle=$
$\displaystyle
2\delta(y-y^{\prime})T\sum_{i\omega}\int\frac{dk_{z}}{2\pi}\frac{dk_{x}}{2\pi}\frac{1}{l_{B}\sqrt{\pi}}\frac{\exp\left[-(l_{B}k_{x}-y/l_{B})^{2}\right]}{\left[i\omega+i\Omega-
E_{({\bf k}+{\bf q})_{xz}}\right]\left(i\omega-E_{{\bf k}_{xz}}\right)}$
$\displaystyle=$ $\displaystyle\frac{\delta(y-y^{\prime})}{2\pi
l_{B}^{2}}\sum_{i\omega}\int\frac{dk_{z}}{2\pi}\frac{1}{\left[i\omega+i\Omega-
E_{({\bf k}+{\bf q})_{xz}}\right]\left(i\omega-E_{{\bf k}_{xz}}\right)},$ (61)

where we have taken into account the dispersion $E_{\bf k}$ depends only on
the momentum component $k_{z}$ along the magnetic field and is independent of
the component $k_{x}$.

Fourier-transforming Eq. (61) gives

$\displaystyle\Pi({\bf q},i\Omega)=\frac{|e|B}{2\pi
c}\sum_{i\omega}\int\frac{dk_{z}}{2\pi}\frac{1}{\left[i\omega+i\Omega-E_{({\bf
k}+{\bf q})_{z}}\right]\left(i\omega-E_{k_{z}}\right)},$ (62)

which matches the polarisation operator of an effectively one-dimensional
systems with the dispersion $E_{k_{z}}$ and a degeneracy of $\frac{|e|B}{2\pi
c}$ per transverse area. The analytic continuation of Eq. (62) from the upper
half-plane of Matsubara frequencies, $i\Omega\rightarrow\varepsilon+i0$, to
real frequencies $\varepsilon$ gives the retarded polarisation operator (35).

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  * Note (1) To our knowledge, there is no available data for the Fermi velocity in these compounds, which is why we use the value $v_{F}=4.0\cdot 10^{7}cm/s$.