Collective excitations in Weyl semimetals in the hydrodynamic regime
P. O. Sukhachov E. V. Gorbar I. A. Shovkovy V. A. Miransky






Department of Applied Mathematics, Western University, London, Ontario, Canada N6A 5B7



Department of Physics, Taras Shevchenko National Kiev University, Kiev, 03680, Ukraine
Bogolyubov Institute for Theoretical Physics, Kiev, 03680, Ukraine



College of Integrative Sciences and Arts, Arizona State University, Mesa, Arizona 85212, USA
Department of Physics, Arizona State University, Tempe, Arizona 85287, USA



Department of Applied Mathematics, Western University, London, Ontario, Canada N6A 5B7


The spectrum of collective excitations in Weyl materials is studied by using consistent
hydrodynamics. The corresponding framework includes the vortical and chiral anomaly
effects, as well as the dependence on the separations between the Weyl nodes in energy b_0
and momentum 𝐛. The latter are introduced via the Chern–Simons contributions to the electric
current and charge densities in Maxwell's equations. It is found that, even in the absence of a
background magnetic field, certain collective excitations (e.g. the helicon-like modes and the anomalous Hall waves) are
strongly affected by the chiral shift 𝐛. In a background magnetic field,
the existence of the distinctive longitudinal and transverse anomalous Hall waves with a linear dispersion relation is predicted.
They originate from the oscillations of the electric charge density and electromagnetic fields,
in which different components of the fields are connected via the anomalous
Hall effect in Weyl semimetals.


Collective excitations in Weyl semimetals in the hydrodynamic regime
    V. A. Miransky
    June 11, 2018
====================================================================





§ INTRODUCTION




Collective excitations play an important role in all kinds of plasmas <cit.>,
ranging from the nonrelativistic electron-hole ones in solids <cit.> to various relativistic plasmas created in the
heavy-ion collisions <cit.>, the early Universe <cit.>,
and astrophysics <cit.>. With the discovery of graphene, whose quasiparticles
are described by the two-dimensional (2D) relativistic-like Dirac equation (for reviews, see, e.g. Refs. <cit.>),
it became clear that relativistic-like plasmas could be also realized and studied in condensed matter
systems. Furthermore, the discovery of Dirac and Weyl semimetals with three-dimensional (3D) chiral quasiparticles (for
recent reviews, see Refs. <cit.>)
made possible the study of relativistic phenomena in well-controlled table-top experiments.

The defining feature of Weyl semimetals is that the opposite chirality Weyl nodes are separated in energy and/or momentum.
This is in contrast to Dirac semimetals, where the location of such nodes
coincides.
The separation between the Weyl nodes is quantified by a pseudo-scalar
parameter b_0 and an axial vector 𝐛 (the latter is also known as the chiral shift) that break the
parity-inversion (PI) and time-reversal (TR) symmetries, respectively. In addition, an imbalance between the number densities of opposite
chirality carriers could be induced. These chiral asymmetries, in turn, lead to qualitatively new effects including unusual properties of collective excitations in the plasma.

The spectrum of collective excitations in chiral relativistic plasmas can be studied
by using any approach that takes into account the chiral nature of quasiparticles and accounts for
the chiral anomaly <cit.>. Among the most popular of them are the chiral kinetic theory (CKT)
<cit.> and the chiral hydrodynamics
<cit.>. The kinetic theory allows one to investigate a
wide range of collective excitations with wavelengths smaller or equal to the particle mean free path
ℓ_ mfp, provided the latter is much larger than the average interparticle distance. The
use of the chiral hydrodynamics, on the other hand, is limited to the length scales at which the
assumption of local equilibrium is satisfied, i.e. to collective excitations with wavelengths much
longer than ℓ_ mfp (or, equivalently, sufficiently small wave vectors, i.e. k ≪ 1/ℓ_ mfp).

The potential relevance of hydrodynamics for the description of the electron transport in solids
was understood long time ago by Gurzhi <cit.>.
The corresponding
framework is applicable only when the electron fluid is well defined and decoupled from the ion
lattice, i.e. when the rate of the electron-electron scattering is much higher than the rates of
electron scatterings on phonons and impurities. These conditions are usually met only in a finite
window of temperatures, limited by the growing scattering rates on impurities at small
and on phonons at high temperature. In the case of graphene, for example, such a hydrodynamic
window was discussed theoretically in Refs. <cit.> and observed
experimentally in Refs. <cit.>. It is interesting to mention that the viscous
hydrodynamic flow can provide a higher conduction than the ballistic transport <cit.>.
Recently, the hydrodynamic regime was also experimentally probed in Weyl semimetals <cit.>.
The hydrodynamical nature of the transport observations is supported by a characteristic dependence of the electrical resistivity on
the constriction channel width, as well as by a strong violation of the Wiedemann–Franz law with
the lowest value of the Lorenz number ever reported <cit.>.

Theoretically, the hydrodynamic approach was already used to describe the negative magnetoresistance
<cit.> and thermoelectric transport <cit.>
in Weyl semimetals. However, the corresponding framework lacked any information
on the separation between the Weyl nodes. It is worth reminding that such a critical information is also missing in the conventional
chiral kinetic theory <cit.>
and is known to cause a serious problem. Indeed,
this theory naively predicts an anomalous local nonconservation of the electric charge when both electromagnetic
and strain-induced pseudoelectromagnetic fields are applied to the system <cit.>. As we argued
in Ref. <cit.>, the problem is resolved in the framework of the consistent chiral kinetic
theory, in which the electric charge and current densities include the topological Chern–Simons contributions
<cit.> (also known as the Bardeen–Zumino terms <cit.>
in high energy physics). These contributions introduce the needed dependence on the energy separation b_0
and the chiral shift 𝐛 that are critical for
reproducing the correct chiral magnetic effect (CME) in equilibrium <cit.> and the anomalous Hall
effect (AHE) <cit.> in Weyl semimetals. By noting that the hydrodynamic equations
can be obtained by averaging the corresponding kinetic ones (see, e.g. Refs. <cit.>),
we recently argued <cit.> that, in the consistent hydrodynamics (CHD),
similar Chern–Simons contributions should be added to the electric current and charge densities in
Maxwell's equations. On the other hand, these terms are absent in the Euler equation and the energy conservation relation for the electron fluid.

Within the CKT approach, it was already demonstrated that the plasmons <cit.> and
helicons <cit.> are affected by the energy and momentum separations
between the Weyl nodes. At nonzero electric chemical
potential, for instance, the effect of the chiral shift is similar to that of a background magnetic field:
it lifts the degeneracy of plasmon frequencies and mixes the longitudinal modes with the transverse
ones. Also, as shown in Ref. <cit.>, the component of the chiral shift along the direction
of an external magnetic field modifies the effective helicon mass. It is reasonable to expect, therefore, that
the collective excitations in the hydrodynamic regime are also affected by b_0 and 𝐛.
The main goal of this paper is to investigate the corresponding spectrum of collective modes
in detail by using the CHD. (The case of transverse collective excitations was
briefly reported in Ref. <cit.>.)

The paper is organized as follows. In Sec. <ref>, we review the key features of the CHD
in Weyl materials <cit.>. The linearized hydrodynamic equations together
with Maxwell's equations are discussed in Sec. <ref>. The
collective excitations in the absence of a background magnetic field are studied in Sec. <ref>.
The longitudinal and transverse collective modes, defined with respect to the direction of the magnetic field, are
analyzed in Secs. <ref> and <ref>, respectively. In Sec. <ref>,
we discuss the range of validity of the obtained results and the experimental techniques that could be
used to test the predictions. The summary of our study is given in Sec. <ref>.
Technical details, including the explicit expressions for the transport coefficients and the frequencies of some collective modes are given in Appendix <ref>. Throughout this paper, we set the Boltzmann constant k_B=1.




§ CONSISTENT HYDRODYNAMIC THEORY




In this section, we review the CHD
in Weyl semimetals derived in Ref. <cit.>.
The corresponding equations take into account the explicit breaking of the Galilean invariance by the ion lattice
and include the effects of the vorticity and the chiral anomaly. (Note that the CHD also describes Dirac
semimetals if one sets b_0=0 and 𝐛=0.)

Let us begin with the electric and chiral charge continuity relations

    ∂_t ρ +(∇·𝐉)    =    0, 
    ∂_t ρ_5 +(∇·𝐉_5)    =    -e^3 (𝐄·𝐁)/2π^2 ħ^2 c,

where ρ and ρ_5 denote the total electric and chiral charge densities, respectively.
The total electric and chiral current densities are denoted as 𝐉 and 𝐉_5,
respectively. Their explicit expressions read <cit.>

    ρ   =    -e n +σ^(B)(𝐁·𝐮)/3v_F^2 + 5c σ^(ϵ, u)(𝐁·ω)/v_F^2
    +ρ_ CS,
    ρ_5    =    -en_5 +σ^(B)_5 (𝐁·𝐮) /3v_F^2,

and

    𝐉   =    -en𝐮 +σ^(V)ω +σ^(B)𝐁+cσ^(B)[𝐄×𝐮]/3 v_F^2  + 5c^2σ^(ϵ, u)[𝐄×ω]/v_F^2
    -[𝐮×∇]σ^(V)/3 +[∇×ω] σ^(ϵ, V)/2 +𝐉_ CS,
    𝐉_5    =    -en_5𝐮 +σ^(V)_5ω +σ^(B)_5𝐁+cσ^(B)_5[𝐄×𝐮]/3v_F^2 -[𝐮×∇]σ^(V)_5/3  +[∇×ω] σ^(ϵ, V)_5/2.

Here e is the absolute value of the electron charge, 𝐮 is the fluid velocity,
ω=[∇×𝐮]/2 is the vorticity, v_F is the Fermi velocity,
and c is the speed of light.
Here n and n_5 are the fermion and the chiral fermion number densities, respectively. The explicit expressions for the anomalous transport coefficients are given in Appendix <ref>.

In addition to the matter parts of the charge and current densities, there are also the
topological Chern–Simons contributions ρ_ CS and 𝐉_ CS
<cit.>
given by

    ρ_ CS   =   - e^3 (𝐛·𝐁)/2π^2ħ^2c^2,
    𝐉_ CS   =   -e^3b_0 𝐁/2π^2ħ^2c + e^3[𝐛×𝐄]/2π^2ħ^2c.

Here it is important to emphasize that the above Chern–Simons contributions affect Maxwell's equations in Weyl semimetals and, in fact, are the only source of the dependence on the energy separation
b_0 and the chiral shift 𝐛 in the CHD.

A few comments regarding the charge and current densities are in order here. First, we note that the charge densities
are modified by the fluid velocity and vorticity. Next, as expected, the conventional CME and the chiral vortical effect (CVE)
<cit.>, as well as their chiral counterparts are reproduced in the current densities. Among the vorticity-related contributions,
there are terms proportional to the cross product of the fluid velocity and gradients of the thermodynamic variables,
as well as curls of vorticity. The terms ∝ (𝐁·𝐮) could be considered as anomalous charge
inflows. There is also a term ∝ (𝐁·ω) that is structurally similar, but its coefficient
does not depend on the chemical potentials. In the current densities, we also included the terms induced by the
simultaneous presence of the fluid velocity and the electric field. While they resemble the contributions obtained via the
usual Lorentz transformation of the terms with the magnetic field, their coefficients are different. This should not be
surprising, however, since there is no true Lorentz invariance in Weyl and Dirac semimetals.

As is well known, the hydrodynamic equations can be obtained from the corresponding kinetic ones
<cit.> by assuming that the deviations from the state of local equilibrium are small.
By making use of the consistent CKT <cit.>
in the relaxation-time approximation, we derived the
Euler equation and the energy conservation relation for the electron fluid in Weyl and Dirac semimetals in
Ref. <cit.> (for the details of the derivation, see the Supplemental Material in Ref. <cit.>).

The explicit form of the Euler equation reads

    1/v_F∂_t [ w  𝐮/v_F+σ^(ϵ,B)𝐁
    +ħω n_5/2] +∇_𝐫P + 4/15v_F[∑_j=1^3 𝐁_j ∇_𝐫𝐮_j + (𝐁·∇_𝐫)𝐮 +𝐁 (∇_𝐫·𝐮)] σ^(ϵ, B)
          + c [∇_𝐫×𝐄] σ^(ϵ, B)/3v_F
    +2/3v_F∑_j=1^3 𝐮_j∇_𝐫𝐁_j σ^(ϵ, B)
    -4σ^(ϵ, B)/15v_F[∑_j=1^3 𝐮_j∇_𝐫𝐁_j +(𝐮·∇_𝐫)𝐁]
    +5σ^(ϵ, u)∇_𝐫B^2/2
          + [(𝐁·∇_𝐫)ω + ∑_j=1^3 𝐁_j∇_𝐫ω_j  ] σ^(ϵ, V)/5c -∑_j=1^3 σ^(ϵ, V)ω_j ∇_𝐫𝐁_j/2c
    -σ^(ϵ, V)/5c[∑_j=1^3ω_j∇_𝐫𝐁_j + (ω·∇_𝐫)𝐁] 
           =-en𝐄 +1/c[𝐁×(en𝐮
    -σ^(V)ω/3)] +σ^(B)𝐮 (𝐄·𝐁)/3v_F^2 +5c σ^(ϵ, u) (𝐄·𝐁)ω/v_F
     - w  𝐮/v_F^2τ - ħω n_5/2 v_F τ,

where B≡|𝐁|, w=ϵ+P is the enthalpy density, ϵ is the energy density, and P is the pressure.
In this equation, we used the convention that the derivatives apply to all quantities standing to their right.
Further, τ denotes the relaxation time connected with the intravalley (chirality preserving) scattering,
which is the dominant dissipation mechanism. In this connection, we note that the relaxation time τ_5
for the chirality-flipping (intervalley) processes is usually much larger <cit.>
and, therefore, can be ignored.

A rather complicated form of the Euler equation (<ref>) for the electron fluid in Weyl and Dirac semimetals
is the result of including many anomalous terms and the vorticity effects. Before proceeding further,
therefore, it is instructive to discuss some of its most interesting features. Firstly, as we noted in Ref. <cit.>,
the Chern–Simons current density 𝐉_ CS does not contribute directly to the Euler
equation. Secondly, the dissipation effects are captured primarily by the term ∝𝐮/τ,
which is a distinctive feature of an electron fluid in solids <cit.>. It originates from the electron scattering
on phonons and/or impurities and breaks explicitly the Galilean symmetry. The breaking of the Galilean symmetry
should not be surprising because there is a preferred coordinate system in which the lattice ions are stationary.
It is worth noting that the term ∝𝐮/τ is responsible for the usual Ohm's
law in the steady state of the electron liquid. Indeed, at 𝐁= 0, the steady state is reached
when the right-hand side of the Euler equation (<ref>) vanishes, i.e. when
𝐮_ ave = - e n τ v_F^2𝐄/w. After taking into account the
hydrodynamic definition of the electric current density 𝐉 = -en 𝐮_ ave, the latter
reproduces the conventional Ohm's law. Thirdly, we note that the Lorentz force term on the right-hand
side of the Euler equation contains an extra factor of 1/3 in the CVE current. Lastly, all
terms ∝(𝐄·𝐁) on the right-hand side of Eq. (<ref>)
are related to the chiral anomaly.

In addition to the Euler equation, we also need the following energy conservation relation <cit.>:

    ∂_t ϵ
    +(∇_𝐫·𝐮)  w  +σ^(ϵ, u)[∑_i=1^3𝐁_i (𝐁·∇_𝐫)𝐮_i -2B^2(∇_𝐫·𝐮)] -2c(𝐄·[∇_𝐫×𝐮])σ^(ϵ,B)/3v_F
          -5cσ^(ϵ, u)(𝐄·[∇_𝐫×𝐁])
    +v_F(𝐁·∇_𝐫)σ^(ϵ, B)
    -2σ^(ϵ, u)[(𝐮·∇_𝐫)B^2 -3∑_i=1^3𝐮_i(𝐁·∇_𝐫)𝐁_i]
    
           +ħ v_F (∇_𝐫·ω)n_5/2 - (𝐄·[∇_𝐫×ω])σ^(ϵ, V)/2 =- 𝐄·(en𝐮-σ^(B)𝐁 -σ^(V)ω/3).

As in the Euler equation, the electrical force term on the right-hand side of Eq. (<ref>) includes
an extra factor of 1/3 in the CVE current. Also, there are no topological Chern–Simons contributions.

In order to obtain a self-consistent framework for the description of a chiral electron fluid in Weyl
semimetals, Eqs. (<ref>), (<ref>), (<ref>), and
(<ref>) should be supplemented by the standard Maxwell's equations. It is worthwhile to
note that Gauss's law includes the total electric charge density ρ due to electrons given
by Eq. (<ref>) and the background charge density ρ_ b (its value will be specified in the next section).
Similarly, Ampere's law contains the total electric current density given by Eq. (<ref>).




§ LINEARIZED HYDRODYNAMICS EQUATIONS




In the study of collective modes, it is justified to assume that the deviations of the local thermodynamic
parameters from their equilibrium values always remain small. Therefore, a simpler linearized form of
the hydrodynamic equations is sufficient. In this section, we present the corresponding linearized equations
of the CHD.

The state of global equilibrium in the chiral electron fluid is defined by the values of
the electric chemical potential μ_0, the chiral chemical potential μ_5,0, and temperature T_0.
The corresponding energy and electric charge densities are given by the following standard expressions:

    ϵ_0   =   μ^4_0+6μ^2_0μ_5,0^2+μ_5,0^4/4π^2ħ^3v_F^3 +T^2_0(μ^2_0+μ_5,0^2)/2ħ^3v_F^3 +7π^2T^4_0/60ħ^3v_F^3,
    ρ_0    =   
    -e n_0 = -eμ_0(μ^2_0+3μ^2_5,0+π^2T^2_0)/3π^2 ħ^3 v_F^3 .

In addition, the pressure and the enthalpy density are P_0=ϵ_0/3 and w_0=4ϵ_0/3, respectively.
The expression for the chiral charge density -e n_5,0 is similar to that in Eq. (<ref>), but with the electric and chiral chemical potentials interchanged, μ_0↔μ_5,0.

In an electrically neutral sample, the total equilibrium electron charge density ρ_0 defined with respect to the Weyl nodes
should be compensated by the background charge density due to ions in the material, i.e.
ρ_0 +ρ_ b =0. In the absence of a background magnetic field, this condition implies

    eμ_0(μ^2_0+3μ^2_5,0+π^2T^2_0)/3π^2 ħ^3 v_F^3 -ρ_ b=0,

where μ_0 is the equilibrium value of the electric chemical potential at B_0=0.

In the presence of the background magnetic field 𝐁_0∥𝐳̂, the electric current should be absent in
the global equilibrium state, i.e.

    𝐉_ eq = (σ^(B)_0  - e^3/2π^2ħ^2cb_0 )𝐁_0 = e^2 (μ_5,0 -eb_0 )𝐁_0/2π^2ħ^2c= 0.

This implies that μ_5,0=eb_0, i.e. the chiral chemical potential in the global equilibrium state
is unambiguously determined by the energy separation between the Weyl nodes. This also agrees with
the analysis within a band-theoretical model of Weyl semimetals <cit.>.

The inclusion of an external magnetic field does not affect the total electrical neutrality of the sample. This implies, therefore, that the global equilibrium value of the electric chemical potential μ_0,B in the
presence of the magnetic field should be self-consistently determined by the following neutrality condition:

    μ_0,B(μ_0,B^2+3μ^2_5,0+π^2T_0^2)/3π^2 ħ^3 v_F^3 +e^2 (𝐛·𝐁_0)/2π^2ħ^2c^2 -μ_0(μ^2_0+3μ^2_5,0+π^2T^2_0)/3π^2 ħ^3 v_F^3=0,

where we used Eqs. (<ref>), (<ref>), and the result in Eq. (<ref>).
(Here we assumed that temperature does not change when the magnetic field is included.) The
above equation defines the electric chemical potential μ_0,B as a function of 𝐁_0, 𝐛,
and T_0, as well as the reference value of the electric chemical potential μ_0. Therefore, when
𝐁_0 ≠0 one should replace μ_0 with μ_0,B in all thermodynamic functions, i.e.
w_0, ϵ_0, P_0, n_0, and n_5,0.

When a long-wavelength collective mode propagates through the chiral electron fluid, it perturbs the local
values of thermodynamic parameters and fields through the following deviations: δμ(x), δμ_5(x), δ T(x), and δ𝐁(x).
In general, the local fluid velocity δ𝐮(x) and the electric field δ𝐄(x) are also nonzero in the perturbed state.

As usual in the study of collective excitations, we look for the solutions in the form of
plain waves with the frequency ω and the wave vector 𝐤, i.e. δμ(x) =
δμ  e^-iω t+i𝐤𝐫 and the similar expressions for other perturbed
quantities. To the linear order in perturbations, the electric and chiral charge continuity relations
(<ref>) and (<ref>) are given by

    ωδρ-(𝐤·δ𝐉)=0,
          ωδρ_5 -(𝐤·δ𝐉_5)
    =- i e^3/2π^2ħ^2c(𝐁_0·δ𝐄),

where

    δρ   =    -eδ n +(𝐁_0·δ𝐮) σ^(B)/3v_F^2
    +i 5 c^2 σ^(ϵ, u) (𝐁_0· [𝐤×δ𝐮])/2v_F^2
    -e^3(𝐛·δ𝐁)/2π^2ħ^2c^2,
    δρ_5   =    -eδ n_5 +(𝐁_0·δ𝐮) σ^(B)_5/3v_F^2 ,

and

    δ𝐉   =    -en_0δ𝐮 + 𝐁_0δσ^(B)
    + e^3[𝐛×δ𝐄]/2π^2ħ^2c
    +i/2σ^(V)[𝐤×δ𝐮]
    -1/4σ^(ϵ, V)[𝐤×[𝐤×δ𝐮]],
    δ𝐉_5    =    -en_5,0δ𝐮 + σ^(B)_5 δ𝐁
    +𝐁_0δσ^(B)_5
    +i/2σ^(V)_5[𝐤×δ𝐮]
    -1/4σ^(ϵ, V)_5[𝐤×[𝐤×δ𝐮]].

Here we assumed that all coefficients are evaluated for the global equilibrium values
of the thermodynamic parameters, as well as used condition (<ref>)
and the expression for the vorticity deviation δω = i[𝐤×δ𝐮]/2.

The linearized forms of the Euler equation (<ref>) and the energy conservation
relation (<ref>) are given by

    ω/v_F{ w_0  δ𝐮/v_F+𝐁_0δσ^(ϵ,B) + σ^(ϵ,B)δ𝐁 +iħ n_5,0[𝐤×δ𝐮]/4} -𝐤δ P -4 σ^(ϵ,B)/15v_F[ 𝐤(𝐁_0·δ𝐮) +δ𝐮(𝐁_0·𝐤) +𝐁_0(𝐤·δ𝐮)] 
           -c σ^(ϵ, B)/3v_F[𝐤×δ𝐄]
    -5σ^(ϵ, u)𝐤(𝐁_0·δ𝐁)
    -iσ^(ϵ, V)/10c{(𝐁_0·𝐤)[𝐤×δ𝐮] +𝐤(𝐁_0·[𝐤×δ𝐮])}
          =-en_0i δ𝐄 +i/c[𝐁_0×(en_0δ𝐮
    - iσ^(V) [𝐤×δ𝐮]/6)] -i w_0  δ𝐮/v_F^2τ +ħ n_5,0 [𝐤×δ𝐮]/4v_Fτ

and

    ωδϵ
    - w_0 (𝐤·δ𝐮)  -σ^(ϵ, u)[
    (𝐁_0·δ𝐮)(𝐁_0·𝐤) - 2B_0^2(𝐤·δ𝐮)]
    -v_F(𝐁_0·𝐤)δσ^(ϵ,B) =iσ^(B)(δ𝐄·𝐁_0),

respectively. Note that the last equation is not modified by the effects of the vorticity.

Finally, after taking into account Faraday's law δ𝐁=(c/ω) [𝐤×δ𝐄],
the remaining Maxwell's equation for δ𝐄 is given by

    (ω^2 -c^2k^2/ε_eμ_m)δ𝐄 +𝐤(𝐤·δ𝐄)
    =4π i ω/ε_e(en_0δ𝐮 - 𝐁_0δσ^(B)
    - e^3[𝐛×δ𝐄]/2π^2ħ^2c
    -i σ^(V)[𝐤×δ𝐮]/2
    +σ^(ϵ, V)[𝐤×[𝐤×δ𝐮]]/4).

In general, the chiral shift 𝐛 can be given in components as follows:
𝐛=(b_⊥, b̃_⊥, b_∥), where b_⊥ and b̃_⊥
are the two independent components perpendicular to 𝐁_0 and b_∥ is
the component parallel to 𝐁_0. In the case of the longitudinal (with respect to the
magnetic field) propagation of collective modes, the rotational symmetry can be used to eliminate one of the
perpendicular components and, consequently, we can set b̃_⊥=0 without loss
of generality. For transverse modes, however, both b_⊥ and b̃_⊥
are relevant. By definition, we set b_⊥ to be the component of the chiral shift that
is perpendicular to 𝐁_0, but parallel to 𝐤, while b̃_⊥ is
perpendicular to both 𝐤 and 𝐁_0.

In the following sections, we will analyze the collective excitations in Dirac and Weyl materials
by using the linearized form of the CHD equations obtained above.




§ COLLECTIVE EXCITATIONS IN THE ABSENCE OF EXTERNAL MAGNETIC FIELD




In this section, we study the collective excitations at 𝐁_0=0. Even in this rather simple case, the spectrum of
collective excitations is affected by the chiral shift and contains qualitatively new modes.




 §.§ Dirac semimetals and PI symmetry broken Weyl semimetals




In order to test the CHD, let us start from the simplest case of Dirac semimetals.
Therefore, we set |𝐛|=b_0=0, as well as μ_5,0=0 in accordance with
Eq. (<ref>). We also note that, according to
Eq. (<ref>), we have μ_0,B=μ_0 in the absence of a background magnetic field.

When the electric and chiral charge densities vanish (i.e. at μ_0=μ_5,0=0),
we obtain the following roots of the characteristic equation [which is the determinant of the system (<ref>), (<ref>), and (<ref>)–(<ref>)]:

    ω_ d   =    -i/τ,
    ω_ s, ±   =    -i/2τ± i√(3-4τ^2v_F^2|𝐤|^2)/2√(3)τ,

and the roots corresponding to the standard in-medium electromagnetic (light) waves, i.e. ω_ light, ± =± c|𝐤|/√(ε_e μ_m). Here ε_e is the electric permittivity and μ_m is the magnetic permeability. As is easy to check, the latter roots are associated with oscillations
of electromagnetic fields, but induce no hydrodynamic motion of the electron fluid. In a strict sense,
therefore, they are not hydrodynamic modes. As we will see later, however, under certain conditions
the electromagnetic waves could hybridize with the electron fluid oscillations and produce new types of collective
excitations.

The first solution, ω_ d, corresponds to a doubly degenerate diffusive mode. The next
pair of solutions in Eq. (<ref>) describes damped sound waves.
As is easy to check, in the limit of large τ, the corresponding frequencies ω_ s, ±
approach the standard dispersion relation for sound waves in plasma, ω_ s, ±≃± v_F|𝐤|/√(3). By analyzing the eigenstates of these sound waves, one can verify
that, as expected, they are sustained by oscillations of the fluid velocity and do not induce any local
electromagnetic field.

The spectrum changes in the case of a nonzero electric charge density ρ_0 (i.e. at μ_0≠ 0,
but still μ_5,0=0). In the long-wavelength limit (|𝐤|→ 0), the corresponding dispersion relations
of collective modes in Dirac semimetals read

    ω_ d, ±   ≈   
    -i/2τ±i/2τ√(1-16π v_F^2 ρ_0^2 τ^2/ε_e  w_0 )
    +O(|𝐤|),
    ω_ s, ±   =    -i/2τ±i/2τ√(1-4τ^2 v_F^2 (12πρ_0^2 +w_0|𝐤|^2)/3ε_e  w_0 ),
    ω_ d,1   =   ω_ d,2≈ -ic^2|𝐤|^2  w_0 /4πμ_m v_F^2 ρ_0^2 τ+O(|𝐤|^3).

First, as is clear from the expressions for ω_ d, ± and ω_ s, ± in Eqs. (<ref>) and (<ref>),
a nonzero electric charge density makes the dissipative and sound waves gapped. [Note that each of the former modes is doubly degenerate.]
In addition to these gapped modes, there are also gapless
doubly degenerate diffusive waves with ω = ω_ d,1 and ω = ω_ d,2,
which appear to be hybridized electromagnetic waves.

The results here can be easily generalized to the case of Weyl semimetals with a broken PI (i.e. with b_0≠ 0),
but intact TR  symmetry (i.e. |𝐛|=0).
In such a case, in accordance with Eq. (<ref>),
the chiral chemical potential is nonzero in equilibrium, i.e. μ_5,0=eb_0≠0. When the electric charge density
is also nonzero, we find that the frequencies of the collective modes in Eqs. (<ref>),
(<ref>), and (<ref>) remain unchanged to the leading
order in the wave vector. However, in the limit of vanishing electric charge density (i.e. μ_0=0, but μ_5,0≠ 0),
the long-wavelength spectrum of the diffusive waves is modified by a nonzero n_5,0. In particular,
the corresponding frequencies are given by

    ω_ d,± = -i/τ4 w_0  ∓ħ n_5,0 v_F |𝐤|/4 w_0
    ±ħ n_5,0 v_F |𝐤|.

As for the sound and light waves, they remain the same, see Eq. (<ref>) and the text after it.




 §.§ Weyl semimetals with broken TR symmetry




In this subsection, we consider the case of Weyl semimetals with a broken TR symmetry, which is characterized
by |𝐛|≠0. In the simplest case of vanishing chemical potentials μ_0=μ_5,0=0, we find
that the frequencies of both diffusive and sound waves coincide with those in a Dirac semimetal, see
Eqs. (<ref>) and (<ref>).
In addition, there are also collective modes that strongly depend on the chiral shift. Their dispersion relations read

    ω_ AHW, ±   =   ±√(2e^6μ_m |𝐛|^2 + π^2 c^4ε_e ħ^4 |𝐤|^2 - 2e^3√(μ_m(e^6 μ_m |𝐛|^4 +π^2 c^4ε_e ħ^4 (𝐤·𝐛)^2)))/π c ε_e ħ^2 √(μ_m),
    ω_ gAHW, ±   =   ±√(2e^6μ_m |𝐛|^2 + π^2 c^4ε_e ħ^4 |𝐤|^2 + 2e^3√(μ_m (e^6 μ_m |𝐛|^4 +π^2 c^4ε_e ħ^4 (𝐤·𝐛)^2)))/π c ε_e ħ^2 √(μ_m).

These are the electromagnetic (light) modes modified by the Chern–Simons contributions in Maxwell's
equations. (In the limit |𝐛|→0, both modes reduce to the usual in-medium light waves.) As is
easy to check, the propagation of these modes is affected by a nonzero chiral shift
𝐛 via the anomalous Hall effect currents ∝ [𝐛×𝐄]. Such currents
mix the longitudinal and transverse (with respect to the direction of the wave vector) components of the
oscillating electromagnetic fields. While one of the modes, namely the anomalous Hall wave (AHW) with the
frequency ω_ AHW, ± given by Eq. (<ref>), remains gappless, the other one,
namely the gapped anomalous Hall wave (gAHW) with the frequency ω_ gAHW, ± given by Eq. (<ref>)
acquires a gap 2e^3|𝐛|/(π c ε_e ħ^2).
As expected, both types of the anomalous Hall waves
are nondissipative. Also, strictly speaking, these waves are nonhydrodynamic modes since they are not accompanied by fluid oscillations.

It should be pointed that the dispersion relations of both anomalous Hall waves depend on the direction of their
propagation with respect to the orientation of the chiral shift. This dependence is particularly strong in
the case of the gapless mode. Indeed, when the AHW propagates perpendicularly to 𝐛
(i.e. 𝐤⊥𝐛), it reduces to the regular in-medium light wave, i.e.

    lim_k_∥→0ω_ AHW, ± = ω_ light, ±.

However, for 𝐤∥𝐛, the AHW
turns into the anomalous helicon (AH) with a quadratic dispersion relation, i.e.

    lim_k_⊥→0ω_ AHW, ± = ω_ AH, ±≈±π c^3ħ^2 k_∥^2/2e^3μ_m |𝐛| +O(|𝐤|^3).

It should be emphasized that the limit |𝐛|→ 0 cannot be taken in the last expression,
which is valid only at long wavelengths. However, such a limit is well defined in the exact
expression (<ref>) and, as expected, reproduces the dispersion
relation of the usual light wave.

The result in Eq. (<ref>) is quite remarkable. It shows that the helicon-type
excitations can exist in Weyl semimetals without any background magnetic field and at vanishing electric charge
density. This is in drastic contrast to usual plasmas, in which helicons exist only at nonzero magnetic field and electric
charge density. In connection to this AH mode, we should remark that a similar dispersion relation can be formally
obtained by taking the limit of a vanishing background magnetic field in the result found in Ref. <cit.>, where the CKT approach was used. However,
we note that the corresponding conclusion is not reliable
because the analysis in Ref. <cit.>
heavily relies on the presence of a cyclotron resonance and, consequently, the presence of a magnetic field.

It is also interesting to consider the case of Weyl semimetals with nonzero electric charge density (i.e. μ_0≠0).
By setting μ_5,0=0, we find the following frequencies of the diffusive wave and the gAHW in the long-wavelength limit:

    ω_ d, ±   ≈    -i/2τ±e^3 |𝐛|/cπε_e ħ^2 - i √( w_0 (π c ε_e ħ^2 ∓ 2iτ e^3 |𝐛| )^2 -16π^3ε_e c^2v_F^2 ħ^4 ρ_0^2 τ^2)/2π c ε_e ħ^2 τ√( w_0 )
    +O(|𝐤|),
    ω_ gAHW, ±   ≈    -i/2τ±e^3 |𝐛|/cπε_e ħ^2 + i √( w_0 (π c ε_e ħ^2 ∓ 2iτ e^3 |𝐛| )^2 -16π^3ε_e c^2v_F^2 ħ^4 ρ_0^2 τ^2)/2π c ε_e ħ^2 τ√( w_0 ) +O(|𝐤|).

In the same limit, the frequencies of the AHW are given by Eq. (<ref>) in Appendix <ref> [we will discuss a few limiting cases below].
In addition, while the gap in the sound frequencies ω_ s, ± is still given by Eq. (<ref>), the dependency on the wave vector is different.
Note that all these modes are the generalized versions of those at zero charge density.

In order to clarify the nature of the AHW, it is instructive to consider the two limiting cases: 𝐤⊥𝐛 (b_∥→0) and 𝐤∥𝐛 (b_⊥→0).
In the limit b_∥→0, the AHW becomes completely
diffusive with the following frequencies:

    lim_b_∥→0ω_ AHW, +   =   ω_ d,1≈
    -ic^2|𝐤|^2  w_0 /4πμ_m v_F^2 ρ_0^2 τ+O(|𝐤|^3),
    lim_b_∥→0ω_ AHW, -   =   ω_ d,2≈
    -iv_F^2|𝐤|^2  w_0  τ(3π^3 c^4 ħ^4 ρ_0^2 +e^6μ_m w_0  b_⊥^2 )/3μ_m (4π^4 c^2v_F^4 ħ^4 ρ_0^4τ^2 +e^6  w_0 ^2 b_⊥^2)+O(|𝐤|^3).

The degeneracy of these diffusive waves is lifted by a nonzero component of the chiral shift b_⊥.

The same modes in the limit b_⊥→0 are qualitatively different. In this case, instead of the completely diffusive waves,
we obtain the AH modes with frequencies

    lim_b_⊥→0ω_ AHW, ±=ω_ AH, ±≈
    - π c^3 ħ^2  w_0  |𝐤|^2 (2 iπ^2 c  ħ^2 v_F^2 ρ_0^2 τ∓ w_0 e^3b_∥)/2 μ_m(4π^4 c^2 v_F^4ħ^4 ρ_0^4 τ^2 +e^6  w_0 ^2 b_∥^2 )  + O(|𝐤|^3).

By comparing the above result with that in Eq. (<ref>), we find that the AH
becomes dissipative, with the imaginary part determined by the electric charge density.

For completeness, let us also briefly mention what happens in the case of Weyl semimetals where both TR and PI
symmetries are broken. Such Weyl semimetals are characterized not only by |𝐛|≠ 0, but also a nonzero
chiral chemical potential μ_5,0=eb_0≠0. At nonzero electric charge density, we find that the expressions
for ω_ AHW, ±, ω_ d, ±, and ω_ gAHW, ±  [see Eqs. (<ref>), (<ref>), and (<ref>)] remain unchanged to the leading order in the wave vector.
As for the case of the vanishing electric charge density
(i.e. μ_0=0), the only mode affected by a nonzero μ_5,0 will be the diffusive wave given in
Eq. (<ref>).

The real and imaginary parts of the collective modes frequencies for several choices of 𝐛
and μ_0 are shown in Figs. <ref>(a) and <ref>(b), respectively.
For our numerical estimates, we set the electric permittivity ε_e=1 and the magnetic permeability μ_m = 1, as well as used the following representative set of model parameters:

    v_F≈1.49×10^8 ,     τ =10^-12 ,      a ≈ 25.5×10^-8 ,     
    b= ħ c/e b_ latt ,     
    b_ latt≈ 0.3π/a,

which are close to the parameters that describe the low-energy spectrum of electron quasiparticles
in Cd_3As_2 <cit.>. Note also that, in general, the relaxation time is a function of chemical potentials and temperature. However, henceforth, we will neglect such a dependence in order to present our results as clearly as possible.

The values of the wave vectors and frequencies in Figs. <ref>(a) and <ref>(b) are given in units of
K_0=10^-6π/a ≈12.3 ^-1 and Ω_0=v_F K_0≈1.84,
which can be viewed as the characteristic values in the problem at hand. In order to avoid the clutter,
we did not show the dispersion relation of the gAHW in the figure. Numerically, the corresponding mode
is weakly dispersing with (ω_ gAHW)≈ 2e^3|𝐛|/(π c ε_e ħ^2) and has a relatively small imaginary part at μ_0≠0.





Let us briefly discuss the dispersion relations of the collective modes presented in Figs. <ref>(a) and <ref>(b).
At μ_0=0, the AHWs are dissipationless and have linear dispersion laws when
𝐤⊥𝐛 and quadratic ones when 𝐤∥𝐛. While
the former case describes nothing else but the usual in-medium light waves, the latter is the AH mode given by Eq. (<ref>), whose existence
is a distinctive feature of Weyl semimetals with a broken TR symmetry. When
μ_0≠0 and b=b_⊥, the AHWs are completely diffusive gapless waves, with
frequencies ω_ AHW,+=ω_d,1 and ω_ AHW,-=ω_d,2
given by Eqs. (<ref>) and (<ref>), respectively.
In addition, while the real part of the AH frequency given by Eq. (<ref>)
is almost unaffected by a nonzero μ_0, a small imaginary part develops and makes the wave slightly
damped.
Last but not least, we checked that a nonzero μ_5,0, even as large as μ_0, leads to rather small quantitative changes in the spectrum of the collective excitations.




§ LONGITUDINAL PROPAGATION OF COLLECTIVE MODES




In this section, we consider the case of the longitudinal propagation of collective modes with respect to
the external magnetic field 𝐁_0 (i.e. 𝐤∥𝐁_0).




 §.§ Dirac semimetals and PI symmetry broken Weyl semimetals




It is instructive to start our analysis from the
simplest case of Dirac semimetals, i.e. |𝐛|=b_0=0. At μ_0=μ_5,0=0, the spectrum of the longitudinal collective modes can be obtained analytically.
We find, in particular, that the frequencies of the diffusive and electromagnetic
(light) waves are unaffected by 𝐁_0, see Eq. (<ref>) and the text after it.
On the other hand, the dispersion relation of the
sound wave changes and is now given by

    ω_ s, ± = -i/2τ±i/2τ√(1 -4τ^2 v_F^2|𝐤|^2 w_0
    -σ^(ϵ, u)[2|𝐤|^2B_0^2 -(𝐤·𝐁_0)]/3 w_0 ),

which is applicable for both longitudinal and transverse propagation of the sound waves.
In addition, there is the gapped chiral magnetic wave (gCMW) with the frequency

    ω_ gCMW,± = ±eB_0 √(3v_F^3 (4π e^2T_0^2 +3ε_e ħ^3 v_F^3 k_∥^2))/2π^2T_0^2 c √(ε_e ħ).

The gap of this excitation is not identical to, but shares some similarity with the cyclotron frequency Ω_c∼ B_0/T_0,
which is also proportional to the magnetic field and inversely proportional to temperature. While, naively, this mode can be
identified with the cyclotron wave in the chiral electron plasma, it could be also related to the chiral magnetic wave (CMW) <cit.>. Indeed, the gCMW is connected with the coupled oscillations of the electric and chiral chemical potentials amended by the oscillating electric field along 𝐁_0. However, unlike the conventional CMW, the gCMW includes the effects of the dynamical electromagnetism.

The dispersion relations of the collective modes become rather complicated at nonzero electric charge density,
which is quantified by μ_0≠0. In total, we found 8 nontrivial solutions of the characteristic equation.
Here, for the sake of clarity and brevity, we analyze only the most interesting of them, i.e.

    ω_ d,±   ≈    -i/2τ∓v_F^2 ρ_0 B_0/2c w_0 - i √(ε_e [c  w_0 ∓  iτ v_F^2 ρ_0 B_0]^2 -16π v_F^2c^2ρ_0^2  w_0 τ^2 )/2c√(ε_e) w_0 τ +O(k_∥),
    ω_ gCMW, ±   ≈    -i/2τ∓v_F^2 ρ_0 B_0/2c w_0 + i √(ε_e [c w_0 ∓  iτ v_F^2 ρ_0 B_0]^2 -16π v_F^2c^2ρ_0^2 w_0 τ^2 )/2c√(ε_e) w_0 τ +O(k_∥),
    ω_ h, ±   ≈   ∓ck_∥^2B_0/4πμ_m ρ_0 -i/τc^2k_∥^2 w_0 /4πμ_m v_F^2 ρ_0^2 +O(k_∥^3).

By comparing the dispersion relations ω_ gAHW,± in Eq. (<ref>) and ω_ gCMW, ± in Eq. (<ref>),
we see that the chiral shift at B_0=0 in the former
plays a similar role to that of the magnetic field 𝐁_0 at |𝐛|=0 in the latter.
This indicates that the gAHW and gCMW are closely related collective modes.

These results can be easily generalized to the case of Weyl semimetals with a broken PI symmetry. This can be done
by including a nonzero chiral chemical potential, i.e. μ_5,0=eb_0≠0. As is easy to check, in
this case, the diffusive and sound waves are modified. The frequency of the former is given in
Eq. (<ref>). The dispersion relations of the sound waves, on the other
hand, are

    ω_ s, +   ≈    -iv_F^2k_∥^2 τ (15μ_5,0^2 +7π^2T_0^2) ( w_0 -μ_5,0n_5,0-σ^(ϵ, u)B_0^2)/3 w_0
    (5μ_5,0^2 +7π^2T_0^2)
     +O(k_∥^3),
    ω_ s, -   ≈    -i2c w_0 (3μ_5,0^2+π^2T_0^2)/τ[2c w_0 (3μ_5,0^2+π^2T_0^2) -v_F ħμ_5,0σ^(B) B_0^2 ] +O(k_∥).

In addition to the conventional electromagnetic (light) wave, there is also the gCMW with the frequency

    ω_ gCMW,± = ±e^2B_0√(5μ_5,0^2+7π^2T^2_0)/cπħ^2√(ε_e g_0) +O(k_∥),

where used the following shorthand notation:

    g_0= 15μ_5,0^4+6π^2μ_5,0^2T_0^2+7π^4T_0^4/3π v_F^3 ħ^3.

In general, however, we find that all modifications in the spectra of collective modes are quantitative,
rather than qualitative.

When both electric and chiral charge densities are nonzero, we checked that the frequencies
ω_ d, ±, ω_ gCMW,±, and ω_ h, ± remain unchanged to the leading order in
small |𝐤|. For their explicit expressions, see  Eqs. (<ref>), (<ref>), and (<ref>), respectively.




 §.§ Weyl semimetals with 𝐛⊥𝐁_0




Let us now consider the case of the longitudinal waves (i.e. 𝐤∥𝐁_0) in
Weyl semimetals with 𝐛⊥𝐁_0 and |𝐛|=b_⊥.
The analysis is relatively simple for zero electric and chiral charge densities.
We find that the frequencies of the
diffusive, sound, and light modes are given by the same expressions as in the case of Dirac semimetals,
see Eqs. (<ref>), (<ref>), as well as
the text after Eq. (<ref>).
On the other hand, the chiral shift changes
the dispersion relations of the gCMW by turning it into the gAHW with the following frequency:

    ω_ gAHW,±≈±e^2 √(3ε_e v_F^3 ħ^3 B_0^2 +4π e^2 T^2_0 b_⊥^2)/cε_e ħ^2 T_0 √(π^3)
    +O(k_∥^2).

The corresponding exact expression is given by Eq. (<ref>) in Appendix <ref>.
We note that both B_0 and b_⊥ play a similar role in producing a nonzero gap for the electromagnetic (light)
mode and, thus, transforming it into the gAHW.

Interestingly, b_⊥ also gives rise to a qualitatively different type of the AHW, which we call the
longitudinal anomalous Hall wave (lAHW).
(The transverse AHW, which was predicted
in Ref. <cit.>, will be analyzed in Sec. <ref>.) While the exact frequency is given by Eq. (<ref>) in Appendix <ref>, the corresponding result in the long-wavelength limit reads

    ω_ lAHW,±≈±ħ B_0 k_∥√(3v_F^3 (π^3 c^4 ħ T^2_0 +3e^4μ_m v_F^3 b_⊥^2))/cT_0√(π^3μ_m (3ε_e v_F^3 ħ^3 B_0^2 +4π e^2T^2_0b_⊥^2 )) +O(k_∥^3).

Some features of the lAHW resemble those of a general AHW, but the roles that the chiral shift component b_⊥ and the magnetic
field strength B_0 play are different. Furthermore, unlike the AHW, the lAHW has the characteristic linear dispersion law.

In order to clarify the origin of the lAHW, let us present the corresponding nontrivial equations
that describe this mode within the CHD. In the case of μ_0=μ_5,0=0
and 𝐛⊥𝐁_0, they read

    T^2ω/3 v_F^3 ħδμ+  eB_0 k_∥/2π^2cδμ_5 =0,
          eB_0 k_∥/2π^2 cδμ + T^2 ω/3v_F^3ħδμ_5 -i e^2B_0/2π^2cδ E_∥ =0,
          (ω^2  -c^2k^2_∥/ε_e μ_m)δẼ_⊥ - i2e^3ω b_⊥/π cε_e ħ^2δ E_∥=0,
          ε_eωδ E_∥ + i2e^2/π c ħ^2(B_0 δμ_5 +eb_⊥δẼ_⊥)=0,

where δẼ_⊥ denotes the component of the oscillating electric field parallel to [𝐁_0×𝐛].
The first two equations are the continuity relations for the electric and chiral charges, respectively. The other two
come from Maxwell's equations. We also used Faraday's law δ𝐁=(c/ω) [𝐤×δ𝐄], as well as took into account that δ T=0 and δ𝐮=0
for this particular AHW mode.

From the continuity equations (<ref>) and (<ref>),
we see that the oscillations of the electric and chiral chemical potentials are coupled to each other.
However, because of the chiral anomaly, they also drive the oscillations of δ E_∥.
Then, due to the AHE, represented by the terms proportional to b_⊥ in Maxwell's equations (<ref>) and (<ref>), the lAHW solution becomes self-consistent.
Indeed, the oscillations of δ E_∥
drive the topological current 𝐉_ AHE∝[𝐛×δ𝐄], which then induces an oscillating electric field δẼ_⊥.
Finally, δẼ_⊥ allows for the component of 𝐉_ AHE parallel to 𝐁_0,
which together with δ E_∥ leads to the oscillating CME current,
𝐉_ CME∝𝐁_0 δμ_5, thus closing the cycle.

Despite some similarities with the conventional CMW, which is also
driven by the coupled oscillations of the electric and chiral chemical potentials, the lAHW is a
profoundly different mode that relies on the dynamical electromagnetism and the Chern–Simons
currents in the chiral electron fluid. In this connection, we may add that the CMW is affected by the
dynamical electromagnetism too. As shown in Refs. <cit.>
within the CKT, the CMW turns into a chiral plasmon with a nonzero gap.

It is also interesting to briefly discuss the case of Weyl semimetals with broken PI and TR symmetries.
This is covered by including a nonzero μ_5,0. At μ_0=0, the frequencies of collective modes
affected by μ_5,0 are

    ω_ gAHW,±   ≈   ±e^2√(4e^2b_⊥^2g_0 +ε_e B_0^2 (5μ_5,0^2+7π^2T_0^2))/πε_e c ħ^2 √(g_0) +O(k_∥),
    ω_ lAHW,±   ≈   ±B_0 k_∥√((5μ_5,0^2+7π^2T_0^2) [π c^4 ħ (3μ_5,0^2+π^2T_0^2) +3e^4 μ_m v_F^3 b_⊥^2])/c√(πμ_m ħ (3μ_5,0^2+π^2T_0^2)[ε_e B_0^2(5μ_5,0^2+7π^2T_0^2) +4e^2b_⊥^2 g_0]) +O(k_∥^2).

At the same time, the frequencies of the sound waves ω_ s, + and ω_ s, - are given by
Eqs. (<ref>) and (<ref>), respectively.
Similarly to the case 𝐁_0=0, the effect of the chiral chemical potential is quantitative. In the most general case μ_0≠0 and μ_5,0≠0,
the analytical expressions for the frequencies are too cumbersome to be presented here. They can be easily analyzed,
however, by using numerical methods.




 §.§ Weyl semimetals with 𝐛∥𝐁_0




Further, we turn to the case of Weyl semimetals with the chiral shift parallel to the magnetic field,
i.e. 𝐛∥𝐁_0. Because of the topological Chern–Simons contribution
in the electric charge density, ρ_ CS∝𝐛·𝐁_0, the
global equilibrium electric chemical potential μ_0,B is no longer equal to its reference value
at B_0=0, i.e. μ_0, see Eq. (<ref>).

To start with, let us consider the case where μ_0=0 but a nonzero electric charge density
is topologically induced by the external magnetic field, i.e. ρ_0=e^3 b_∥B_0/(2π^2 c^2ħ^2).
As in the case 𝐛⊥𝐁_0 and μ_0≠0, there are eight nontrivial solutions
with rather cumbersome expressions for the dispersion relations.
It is possible to identify among them
the diffusive waves, the gAHW, and the AH mode. The corresponding frequencies read

    ω_ d, ±   ≈    -i/2τ±e^3b_∥(4π c^2  w_0  +ε_e v_F^2 B_0^2)/4π^2 c^3 ε_e ħ^2  w_0 
       -   i √((2π^2 c^3 ε_e ħ^2  w_0  ±4iπ c^2 e^3  w_0 τ b_∥± ie^3ε_e v_F^2 τ b_∥ B_0^2 )^2 ∓ 32 i π^3 τ c^5 e^3 ε_e ħ^2 b_∥  w_0 ^2)/4π^2 c^3ε_e ħ^2 w_0  τ +O(k_∥),
    ω_ gAHW, ±   ≈    -i/2τ±e^3b_∥(4π c^2  w_0  +ε_e v_F^2 B_0^2)/4π^2 c^3 ε_e ħ^2  w_0 
       +   i √((2π^2 c^3 ε_e ħ^2  w_0  ±4iπ c^2 e^3  w_0 τ b_∥± ie^3ε_e v_F^2 τ b_∥ B_0^2 )^2 ∓ 32 i π^3 τ c^5 e^3 ε_e ħ^2 b_∥  w_0 ^2)/4π^2 c^3ε_e ħ^2 w_0  τ +O(k_∥),

and

    ω_ AH, ±≈±π c^3 ħ^2k_∥^2 /2μ_m e^3b_∥ - iτv_F^2B_0^2k_∥^2 /4πμ_m  w_0  +O(k_∥^3),

respectively.
Note that the chiral shift has a profound effect on the value of the gap in the gAHW.
In fact, the background magnetic field does not produce any gap by itself when |𝐛|=0. Note also that lim_b_∥→0ω_ d, ±=-i/τ, which coincides with that in Eq. (<ref>) obtained for B_0=0.
As for the AH modes, their frequencies in Eq. (<ref>) are somewhat
similar to those in Eqs. (<ref>) and (<ref>).

In the case of μ_0≠0, for the sake of simplicity, we present only the frequencies of the
AH excitations, i.e.

    ω_ AH, ±≈ - π c^3ħ^2 [ 2iπ^2τ c^3 ħ^2 v_F^2 ρ_0^2 w_0
    ∓| e^3c^2 b_∥ w_0 ^2 +τ^2 B_0 v_F^4 ρ_0^2(2π^2 c^2 ħ^2 ρ_0 + e^3 b_∥B_0)| ] k_∥^2/2μ_m [e^6c^2 b_∥^2  w_0 ^2+τ^2v_F^4ρ_0^2(2π^2c^2ħ^2 ρ_0 + e^3b_∥ B_0)^2] +O(k_∥^3).

Note that at B_0≠0 the AH modes in Weyl semimetals contain a nonzero imaginary part, which, in principle, could make them strongly dissipative. For example, by using Eq. (<ref>), we can estimate the characteristic value of τ, at which the real and imaginary parts are equal, as

    τ^* = 2π^2c^3ħ^2  w_0 /e^3v_F^2 b_∥B_0^2.

For the numerical parameters used previously, see Figs. <ref>(a) and <ref>(b), as well as Eq. (<ref>),
we find that
τ^*≈ 10^-12 and is comparable to typical values of τ in Weyl materials.

If, in addition to the TR, the PI symmetry is also broken, we find that the frequencies ω_ d, ± and
ω_ gAHW, ± [see Eqs. (<ref>) and (<ref>), respectively] remain unchanged at a nonzero
μ_5,0. There are also three gapped solutions and three other solutions with the frequencies quadratic in k_∥.
Their expressions are rather cumbersome and will not be presented here.

The whole spectrum of collective modes can be straightforwardly analyzed by using numerical methods.
The frequencies of the most interesting AHW and AH modes are presented in Figs. <ref>(a) and <ref>(b).
As in the B_0=0 case, in order to avoid the unnecessary clutter, the results for the gAHW are omitted.
The corresponding frequency of the gAHW is determined almost exclusively by the chiral shift, i.e.
ω_ gAHW≈
2e^3|𝐛|/(π c ε_e ħ^2), and depends very weakly
on the wave vector. A small nonzero imaginary part of ω_ gAHW appears at
𝐛∥𝐁_0 and/or at μ_0≠0. By noting that realistic values of
μ_5,0 produce very small changes in the spectra, we set μ_5,0=0 for all modes shown
in Fig. <ref>.

As we see from Figs. <ref>(a) and <ref>(b), the lAHW is rather strongly affected by the chemical
potential μ_0 and is transformed into a weakly dissipative AHW with a linear dispersion law.
In the case 𝐛∥𝐁_0, there are the AH modes with frequencies ω_ AH≡ω_ AH, +, which are quadratic in k_∥. Unlike the AHW, they are weakly affected by a
nonzero electric chemical potential μ_0.
By comparing Figs. <ref>(a) and <ref>(b), we find that the imaginary parts of the AH frequencies are almost of the same order as the real ones.
We conclude, therefore, that such waves are
likely to be highly dissipative or even overdamped in the hydrodynamic regime in Weyl semimetals.








§ TRANSVERSE PROPAGATION OF COLLECTIVE MODES




In this section, we study the transverse collective modes (i.e. 𝐤⊥𝐁_0)
by using the CHD.




 §.§ Dirac semimetals and PI symmetry broken Weyl semimetals




Once again, we start from the simplest case of Dirac semimetals, i.e. |𝐛|=b_0=0.
When the chemical potentials vanish, μ_0=μ_5,0=0, we obtain the usual
types of modes: the diffusive and sound waves, as well as the electromagnetic (light) ones. The
diffusive and light waves have the same frequencies as at B_0=0, see
Eq. (<ref>) and the text after it.
There is also the gCMW with the frequency similar to that in Eq. (<ref>),
albeit with k_∥ replaced by k_⊥.
As one can see from Eq. (<ref>), the only mode that is truly different is the sound wave.
At μ_0≠0 (but still at μ_5,0=0), the gaps of the diffusive waves and the gCMW
are given by Eqs. (<ref>) and (<ref>), respectively.
The helicons similar to those in Eq. (<ref>) are absent, however, when
𝐤⊥𝐁_0.

By including a nonzero chiral chemical potential, it is also straightforward to generalize the above results to the case of Weyl
semimetals with a broken PI symmetry. When
the electric charge density vanishes, we find in the spectrum the diffusive and sound modes,
the gCMW, and the usual in-medium electromagnetic (light) waves. The gaps for most collective
modes, i.e. the diffusive modes, one of the sound waves, and the gCMW, are given by the same
expressions as in the case of the longitudinal propagation, i.e. Eqs. (<ref>), (<ref>)
and (<ref>), respectively.
The other sound wave, i.e.
ω_ s, +, is different and its frequency reads

    ω_ s, +≈ -iτ  v_F^2 k_⊥^2 (15μ_5,0^2+7π^2T_0^2) ( w_0  -μ_5,0n_5,0 -2σ^(ϵ, u)B_0^2)/3 w_0  (5μ_5,0^2+7π^2T_0^2) + O(k_⊥^3).

In the case with μ_0≠0 and μ_5,0≠0, we find that the frequencies of the diffusive
modes and the gCMW are given by the same expressions as at μ_5=0
[see Eqs. (<ref>) and (<ref>), respectively].
There are also three gapped modes with rather
cumbersome dispersion relations.




 §.§ Weyl semimetals with 𝐛⊥𝐁_0




In this subsection, we study the collective excitations in a Weyl semimetal with 𝐛⊥𝐁_0.
The results at μ_0=μ_5,0=0 read

    ω_ AHW,±   =   ±1/cε_eħ^2 T_0√(2π^3 μ_m){2π^3ε_e ħ^4 T_0^2 c^4 k_⊥^2
    +e^4μ_m(4π e^2 T_0^2 b_⊥,  tot^2 +3ε_e v_F^3ħ^3B_0^2 ) 
       -   e^3√(μ_m)√(e^2μ_m(4π e^2 T_0^2 b_⊥,  tot^2 +3ε_e v_F^3ħ^3B_0^2 )^2 +16π^4ε_e c^4 ħ^4 T_0^4b_⊥^2 k_⊥^2)}^1/2,
    
    
    ω_ gAHW,±   =   ±1/cε_eħ^2 T_0√(2π^3 μ_m){2π^3ε_e ħ^4 T_0^2 c^4 k_⊥^2 +e^4μ_m(4π e^2 T_0^2 b_⊥,  tot^2 +3ε_e v_F^3ħ^3B_0^2 ) 
       +   e^3√(μ_m)√(e^2μ_m(4π e^2 T_0^2 b_⊥,  tot^2 +3ε_e v_F^3ħ^3B_0^2 )^2 +16π^4ε_e c^4 ħ^4 T_0^4b_⊥^2 k_⊥^2)}^1/2,

where b_⊥,  tot^2=b_⊥^2+b̃_⊥^2 (recall that b_⊥ and b̃_⊥ correspond to the components of the chiral shift parallel and perpendicular to 𝐤, respectively). The spectrum of collective
modes in this regime also contains the diffusive and sound waves with the same frequencies as in
Eqs. (<ref>) and (<ref>),
respectively.

Even at zero charge density, the results for ω_ AHW,± and ω_ gAHW,±
are rather complicated. In order to get a deeper insight
into the properties of these collective modes, it is useful to consider the two limiting cases:
b̃_⊥→0 and b_⊥→0. In the first case, to the leading order in the wave
vector, we obtain

    lim_b̃_⊥→0ω_ AHW,±≈±ck_⊥√(3v_F^3ħ^3)B_0/√(μ_m(4π e^2T_0^2 b_⊥^2 +3ε_e v_F^3ħ^3 B_0^2)) +O(k_⊥^2)

and the gap lim_b̃_⊥→0ω_ gAHW,± is given by Eq. (<ref>).
Therefore, when 𝐛∥𝐤, the AHW with 𝐤⊥𝐁_0 is transformed
into the transverse anomalous Hall wave (tAHW), which was first obtained in Ref. <cit.>.
By following the discussion in Ref. <cit.>, it is instructive to illustrate the origin of the
tAHW. The corresponding set of nontrivial equations reads

    4ϵ_0 ω/T_0δ T - k_⊥( w_0 -2B_0^2σ^(ϵ, u)) δ u_⊥ =0,
          k_⊥/T_0δ T - i+ωτ/v_F^2τδ u_⊥
    +5ck_⊥^2B_0 σ^(ϵ, u)/ω  w_0 δẼ_⊥ =0,
          T^2 ω/3v_F^3ħδμ_5 -i e^2B_0/2π^2cδ E_∥ =0,
          (ω^2  -c^2k^2_⊥/ε_eμ_m)δẼ_⊥
    - 2i e^3ω b_⊥/π cε_e ħ^2δ E_∥=0,
          (ω^2  -c^2k^2_⊥/ε_eμ_m)δ E_∥
    +2ie^2ω/π cε_e ħ^2(B_0δμ_5 +eb_⊥δẼ_⊥) =0.

They are the energy conservation, Euler, chiral charge continuity, and two Maxwell's equations.
In the derivation, we also used δ𝐁=(c/ω) [𝐤×δ𝐄].

Just like in the case of the lAHW, the propagation of the tAHW is sustained by a dynamical version
of the AHE. However, there is an important difference between these two forms of the collective excitations. First, we note that there are oscillations of the electric chemical potential in the former, which are absent in the latter.
Most importantly, while the lAHW is a purely electromagnetic wave, in which the hydrodynamic sector is decoupled, the
tAHW involves both sectors coupled via the magnetic field 𝐁_0. Indeed, in the
electromagnetic sector, owing to the chiral anomaly, the oscillating chemical potential δμ_5
leads to the electric field δ E_∥. Then, the AHE allows for the oscillations of δẼ_⊥
that, in turn, change the local momentum density of the electron fluid via the last term on the left-hand side of
the Euler equation (<ref>). In the end, therefore, the tAHW
is a hybrid collective excitation of the electromagnetic field and the chiral electron fluid that is made possible by the
topological Chern–Simons current.
It is interesting to note that while the tAHW drives oscillations of the fluid velocity, surprisingly, it
remains nondissipative. We can speculate that the presence of an intrinsic viscosity of the electron fluid
might change this nondissipative character.

In the case of 𝐛⊥𝐤, the AHW at 𝐤⊥𝐁_0 is transformed
into the electromagnetic (light) wave lim_b_⊥→0ω_ AHW, ±=ω_ light,±.
As for the gapped mode, i.e. the gAHW, its frequency becomes

    lim_b_⊥→0ω_ gAHW, ± = ±√(π^3 c^4ε_e ħ^4 T_0^2k_⊥^2 + 3v_F^3 ε_e e^4 ħ^3 B_0^2 +4π e^6 μ_m T_0^2b̃_⊥^2)/cε_e ħ^2 T_0√(π^3μ_m).


Before concluding this subsection, let us also discuss Weyl semimetals in which the PI symmetry
is also broken. In this case b_0≠ 0 and the chiral chemical potential is nonzero. There are two
qualitatively different cases with 𝐛 parallel and perpendicular to 𝐤⊥𝐁_0,
respectively.

When 𝐛 is parallel to 𝐤 (and ρ_0=0), we find that the frequencies of the diffusive modes, one of the sound modes, and the gAHW
are the same as for the case of the longitudinal propagation of the collective
modes. They are given by Eqs. (<ref>), (<ref>), and (<ref>). respectively. The frequency of the other sound node, ω_ s, +, is given by
Eq. (<ref>). The dispersion relation of the tAHW is

    ω_ tAHW,±≈±ck_⊥B_0 √(5μ_5,0^2 +7π^2T_0^2)/√(μ_m[ε_e B_0^2 (5μ_5,0^2+7π^2T_0^2) +4e^2b_⊥^2 g_0]) +O(k_⊥^2).

The same results are also valid for the diffusive modes, the gAHW, and one of the sound modes when 𝐛∥[𝐁_0×𝐤] (and ρ_0=0), albeit in the gAHW, one should replace b_⊥→b̃_⊥ and the tAHW is replaced by the usual
electromagnetic (light) wave.
The second completely diffusive sound wave has the following frequency:

    ω_ s, +≈ -iτv_F^2k_⊥^2 [ε_e B_0^2(15μ_5,0^2+7π^2T_0^2) ( w_0  -μ_5,0n_5,0-2σ^(ϵ, u)B_0^2) +4e^2b̃_⊥^2 g_0 ( w_0 -2σ^(ϵ, u)B_0^2)] /3 w_0  [ε_e B_0^2 (5μ_5,0^2+7π^2T_0^2) +4e^2b̃_⊥^2 g_0]
    +O(k_⊥^3).

In both cases (with 𝐛⊥𝐁_0), at nonzero electric and chiral charge densities, we find that there
is the gapless diffusive wave
with the same frequency as in Eq. (<ref>).
In addition, there also are seven gapped solutions with rather complicated dispersion relations.

The numerical results for the tAHW frequency, as well as its counterparts at μ_0=10
are presented in Figs. <ref>(a) and <ref>(b). As in the previous cases, we
do not show the frequency of the gAHW, whose real part is approximately
given by ω_ gAHW≈2e^3|𝐛|/(π c ε_e ħ^2) and is weakly
dispersive.
It also has a small imaginary part that is present at nonzero ρ_0 (induced by either μ_0
or b_∥).





As we can see from Figs. <ref>(a) and <ref>(b), while the frequency of the tAHW is linear in
k_⊥ and does not contain any imaginary part at μ_0=0, the situation changes
qualitatively at μ_0≠0.
Indeed, the tAHWs transform into completely diffusive waves at μ_0≠0 and |k_⊥|≲ K_0/2, see Fig. <ref>(b).




 §.§ Weyl semimetals with 𝐛∥𝐁_0




Finally, we consider the special case of the transverse collective modes (with 𝐤⊥𝐁_0)
when 𝐛∥𝐁_0. In the absence of the reference chemical potential, i.e. μ_0=0, the
electric charge density is still induced via the topological contribution, which according to Eq. (<ref>),  reads ρ_0 = e^3(𝐛·𝐁_0)/(2π^2 c^2ħ^2).
By solving the characteristic
equation, we find that, to the leading order in |𝐤|, the frequencies of the diffusive waves and the gAHW are given by Eqs. (<ref>) and (<ref>), respectively.
The analytical expressions for the frequencies of other collective modes are more complicated.

If both TR and PI symmetries are broken in a Weyl semimetal, we find seven gapped modes at μ_0=0, as well as the following gapless
diffusive one:

    ω_ s, +   =    -iτv_F^2k_⊥^2/12πμ_m  w_0  (5μ_5,0^2+7π^2T_0^2){4πμ_m (15μ_5,0^2+7π^2T_0^2) ( w_0  -μ_5,0n_5,0) 
       +   B_0^2 [15μ_5,0^2 +21π^2T_0^2 +4πμ_m(45μ_5,0^2 +91π^2T_0^2)σ^(ϵ, u)]}.

The dispersion relations of all modes become rather complicated at μ_0≠0 and μ_5,0≠0
and, consequently, we will not present the corresponding analytical solutions.




§ DISCUSSIONS




In this section, we discuss the range of validity of the present study, as well as its possible experimental
implications. One of the simplifications that we relied on was the use of a Weyl semimetal model with only two Weyl
nodes. The majority of the experimentally discovered (see, e.g. Ref. <cit.>) Weyl semimetals, however, have multiple pairs of Weyl nodes separated by
2𝐛^(n), where index n labels different pairs of the nodes.
By taking into account that the chiral
shift enters the CHD equations only linearly via the Chern–Simons terms (i.e. the topological charge
and current densities) in Maxwell's equations, the effects of multiple pairs simply add up and can be rendered via 𝐛→𝐛_ eff≡∑_n𝐛^(n).
In other words, when the TR symmetry is broken, the spectrum of collective modes is affected  in the same way as predicted
by the model with a single pair of Weyl nodes, albeit with 𝐛 replaced by 𝐛_ eff≠ 0.

The situation is different in the special case of Weyl semimetals with an intact TR symmetry, in which
the total number of Weyl nodes is a multiple of four. This is due to the fact that the TR symmetry
maps each pair of the opposite-chirality Weyl nodes separated by 2𝐛^(n) to the other pair
separated by -2𝐛^(n). Consequently, the effective chiral shift 𝐛_ eff
should vanish and play no role in the hydrodynamic regime of the TR symmetric Weyl semimetals.
In addition, some observational signatures due to a nonzero chiral chemical potential
μ_5,0=eb_0≠ 0 should be still expected if the PI symmetry is broken.

Another simplification of the present study was the treatment of the dissipation effects.
In particular, we used the relaxation-time approximation in order to estimate the chirality-preserving (intravalley)
scattering processes (with the relaxation time τ), but completely ignored the chirality-flipping
(intervalley) processes (with the relaxation time τ_5). Such an approximation can be
justified by noting that usually τ_5≫τ (see, e.g. Ref. <cit.>).
In addition, the dependence of the relaxation time on the chemical potentials and temperature was also ignored. While
such a dependence leads only to quantitative corrections to the quasiparticles dispersion relations,
it could provide qualitative effects in the dependence of ω on μ_0 and/or T_0.

We would like to argue that many predicted distinctive features of the collective modes spectrum
that stem from the chiral shift
can be observed in realistic materials. Among the most promising candidates,
we could suggest the antiferromagnetic half Heusler compounds, such as GdPtBi and NdPtBi. The latter
were predicted to be TR symmetry breaking Weyl semimetals <cit.>
in an applied magnetic field. Another type of suitable Weyl semimetals are the theoretically proposed magnetic
Heusler compounds such as XCo_2Z (where X=V,Zr,Ti,Nb,Hf and Z=Si,Ge,Sn), VCo_2Al, and VCo_2Ga
<cit.>. As an added benefit, the latter have only two Weyl nodes near
the Fermi level and the separation between the nodes is comparable to the size of the Brillouin zone.

In order to experimentally test the unusual features of the collective excitations in Weyl semimetals,
one could use a similar experimental setup as in usual metals (see, e.g. Ref. <cit.>). This
requires, in particular, measuring the transmission amplitude of electromagnetic waves through a Weyl
crystal as a function of the frequency at a fixed external magnetic field or as a function of the field at
a fixed frequency. The response will be a series of resonances originating from the interference of
standing waves, from which the parameters of the dispersion laws could be extracted. Depending
on an experimental technique, the effects of various directions of the chiral shift can be studied
by changing the orientation of the magnetic field and/or crystal.




§ SUMMARY




In this paper, by making use of the consistent hydrodynamic framework proposed in Ref. <cit.>,
we analyzed the spectrum of collective excitations in Weyl and Dirac materials
with and without a background magnetic field. The underlying framework includes the effects of the chiral
anomaly and the vorticity, as well as the topological Chern–Simons (or Bardeen–Zumino) contributions to the
electric current and charge densities.
While the charge and current densities in Maxwell's equations include the topological Chern–Simons terms,
no such terms appear in the Euler equation and the energy conservation
relation. This can be explained by the fact that the hydrodynamic
sector of the theory is not
sensitive to the filled electron states deep below the Fermi surface, which are primarily responsible for the
Chern–Simons terms. At the same time, Maxwell's equations are affected by the total charge
and current densities, including those from the filled electron states.
As a result, the properties of collective modes are profoundly modified by b_0 and 𝐛,
which determine the separations between the Weyl nodes in energy and momentum, respectively.

By considering the limit of vanishing magnetic field and electric charge density,
we found that the spectrum of collective excitations in Dirac semimetals
includes a diffusive wave, hydrodynamic sound modes, and in-medium electromagnetic
(light) waves. Generically, the presence of an electric charge density ρ_0 modifies these waves
by generating nonzero gaps. When the external magnetic field 𝐁_0 is present, the gaps are
induced for the electromagnetic waves even in the absence of an electric charge density. In addition,
we also found that a nonzero 𝐁_0 changes the dispersion relation of the sound wave.
Further, in agreement with the findings in Ref. <cit.>, the external magnetic field gives rise to helicons
propagating along the direction of 𝐁_0.
In the presence of the electron-phonon interaction and/or the electron scattering on impurities, most of the collective modes,
including the helicons, become dissipative.
This is not true, however, for the modes that are primarily electromagnetic in nature.

In Weyl semimetals with a broken TR symmetry, we found that the topological contributions associated with
the chiral shift parameter qualitatively change the properties of some collective modes and, in some cases,
give rise to novel types of excitations. For example, in the case of vanishing background magnetic field
and zero electric charge density, the presence of the chiral shift 𝐛 produces a gap proportional to
|𝐛| for one of the anomalous Hall waves. As for the other (gapless) anomalous Hall wave, depending on the
orientation of the wave vector 𝐤 with respect to 𝐛, it interpolates between the usual
in-medium electromagnetic (light) wave (at 𝐤⊥𝐛) and the anomalous helicon wave
(at 𝐤∥𝐛). The latter is a unique helicon with a quadratic dispersion relation that
appears in Weyl semimetals in the absence of background magnetic fields and at vanishing electric charge density.

By studying the spectra of longitudinal and transverse (with respect to the direction of 𝐁_0) collective waves,
it is found that 𝐁_0 strongly affects the corresponding dispersion relations.
At vanishing electric charge density and 𝐛⊥𝐁_0, the longitudinal modes include diffusive waves, sound modes, and the anomalous Hall waves.
One of these Hall waves is gapped and the value of its gap is determined by the
magnetic field and the chiral shift. Another closely related excitation is a special type of the anomalous Hall waves called the
longitudinal anomalous Hall wave. This dissipationless wave
involves the oscillations of the electric and chiral charge densities as well as the electric field perpendicular
to the chiral shift. A dynamical version of the anomalous Hall effect plays an essential role in sustaining
the propagation of such a wave. In the case of 𝐛∥𝐁_0, we found modified
gapped anomalous Hall and diffusive waves, as well as the anomalous helicons. While the latter are somewhat
similar to those at 𝐁_0=0, they are dissipative. Our numerical estimates suggest that,
for realistic values of model parameters, the dissipation may be sufficiently strong to completely damp the
helicon waves.

Generically, the transverse collective modes at vanishing electric charge density are of the
same type: diffusive waves, sound modes, and the anomalous Hall waves. When the wave vector is parallel
to the chiral shift, one of anomalous Hall waves turns into the transverse anomalous Hall wave that was previously
discussed in Ref. <cit.>. When ρ_0=0, its frequency is real, linear in the wave vector, as well as
proportional to the magnetic field and inversely proportional to the chiral shift. While this mode looks similar to
the longitudinal anomalous Hall wave, it is quite different in nature. Indeed, this is a hybridized
version of electromagnetic and hydrodynamic waves. At ρ_0≠ 0, we found that the transverse anomalous
Hall wave is transformed into a dissipative wave with a quadratic dispersion law. At sufficiently small values of
the wave vector, this wave becomes completely diffusive and resembles the behavior of an overdamped
sound mode.


The work of E.V.G. was partially supported by the Program of Fundamental Research of the
Physics and Astronomy Division of the National Academy of Sciences of Ukraine.
The work of V.A.M. and P.O.S. was supported by the Natural Sciences and Engineering Research Council of Canada.
The work of I.A.S. was supported by the U.S. National Science Foundation under Grants PHY-1404232
and PHY-1713950.








§ ADDITIONAL FORMULAS




In this section, we present the explicit expressions for the anomalous transport coefficients and some anomalous Hall waves (AHW) frequencies.




 §.§ Anomalous transport coefficients




The anomalous transport coefficients used in the consistent hydrodynamics equations are given by

    σ^(B)   =   e^2μ_5/2π^2ħ^2c,     σ_5^(B) = e^2μ/2π^2ħ^2c, 
    σ^(ϵ, B)   =    -eμμ_5/2π^2ħ^2v_F c,     σ^(ϵ, u) = e^2v_F/120π^2ħ c^2,
    σ^(V)   =   -eμμ_5/π^2v_F^2ħ^2,     σ_5^(V) = -e/2π^2ħ^2v_F^2(μ^2+μ_5^2 + π^2T^2/3),
    σ^(ϵ, V)   =    -eμ/6π^2ħ v_F,     σ^(ϵ, V)_5 = -eμ_5/6π^2ħ v_F,

where e is the absolute value of the electron charge, μ denotes the electric chemical potential, μ_5 is the chiral chemical potential, T is temperature, v_F is the Fermi velocity, and c in the speed of light.
Note that the coefficients in Eqs. (<ref>)–(<ref>) agree with those obtained
in Refs. <cit.> in the “no-drag" frame
<cit.>.




 §.§ Anomalous Hall waves frequencies




Next, we present the frequencies of some AHW. Let us start with the case of the Weyl semimetal with a broken time-reversal symmetry in the absence of the external magnetic field 𝐁_0. The frequencies of the AHWs in the leading order in the wave vector 𝐤 are

    ω_ AHW, ±   ≈    -i w_0 |𝐤|^2/24πμ_m v_F^2ρ_0^2τ(4π^4 c^2v_F^4 ħ^4 ρ_0^4 τ^2 +e^6 w_0^2|𝐛|^2){
    e^6b_⊥^2 w_0(3c^2w_0+4πμ_m v_F^4ρ_0^2τ^2) +24 π^4 v_F^4c^4 ħ^4 ρ_0^4τ^2
       ±    e^3 w_0 √(e^6b_⊥^4(3c^2 w_0  -4πμ_m v_F^4 ρ_0^2 τ^2)^2 -48π c^2v_F^4 ρ_0^2 τ^2 b_∥^2 (3π^3c^4ħ^4 ρ_0^2 +e^6μ_m b_⊥^2 w_0 ))} +O(|𝐤|^3).

Here w_0=ϵ_0+P_0 is the equilibrium enthalpy density, ϵ_0 is the equilibrium energy density, P_0 is the equilibrium pressure, ρ_0 is the equilibrium electric charge density, μ_m is the magnetic permeability, ε_e is the electric permittivity, and τ is the relaxation time connected with the intravalley (chirality preserving) scattering. [See also Eqs. (<ref>)–(<ref>) and the notations after them in the main text of the paper.] In addition, we used b_∥ and b_⊥ for the parallel and perpendicular components of the chiral shift defined with respect to the direction of the wave vector 𝐤.

Further, we consider the case of the longitudinal, with respect to the external magnetic field 𝐁_0, propagation of the collective waves.
In addition, we assume that the chiral shift is perpendicular to the magnetic field, i.e. 𝐛⊥𝐁_0 and |𝐛|=b_⊥.
The frequencies of the gapped anomalous Hall wave (gAHW) and longitudinal anomalous Hall wave (lAHW), which are considered in Sec. <ref>, read

    ω_ gAHW,±   =   ±1/2π^2 ε_e cħ^2T_0^2√(2μ_m){4πμ_m e^4 T_0^2 (4π e^2 T_0^2 b_⊥^2 +3ε_e v_F^3 ħ^3 B_0^2) +ε_e ħ^4 k_∥^2(4π^4c^4T_0^4 +9μ_m ε_e ħ^2 v_F^6 e^2B_0^2) 
       +   {ε_e^2ħ^6[4π^4 c^4 ħ T_0^4 k_∥^2 -3e^2B_0^2 μ_m v_F^3 (4π e^2T_0^2+3ε_e ħ^3v_F^3 k_∥^2)]^2
    +32π^2ε_e μ_m ħ^3e^6 T_0^4 b_⊥^2 
       ×   [4π^4c^4ħ T_0^4 k_∥^2 +3μ_mv_F^3 e^2B_0^2(4π e^2T_0^2-3ε_e ħ^3v_F^3 k_∥^2) ] +256π^4μ_m^2 e^12 T_0^8 b_⊥^4 }^1/2}^1/2

and

    ω_ lAHW,±   =   ±1/2π^2 ε_e c ħ^2T_0^2√(2μ_m){4πμ_m e^4 T_0^2 (4π e^2 T_0^2 b_⊥^2 +3ε_e v_F^3 ħ^3 B_0^2) +ε_e ħ^4 k_∥^2 (4π^4c^4T_0^4 +9μ_m ε_e ħ^2 v_F^6 e^2B_0^2) 
       -   {ε_e^2ħ^6[4π^4 c^4 ħ T_0^4 k_∥^2 -3e^2B_0^2 μ_m v_F^3 (4π e^2T_0^2+3ε_e ħ^3v_F^3 k_∥^2)]^2 +32π^2ε_e μ_m ħ^3e^6 T_0^4 b_⊥^2 
       ×   [4π^4c^4ħ T_0^4 k_∥^2 +3μ_mv_F^3 e^2B_0^2(4π e^2T_0^2-3ε_e ħ^3v_F^3 k_∥^2) ] +256π^4 μ_m^2 e^12 T_0^8 b_⊥^4 }^1/2}^1/2,

respectively. Their counterparts in the long-wavelength limit are given by Eqs. (<ref>) and (<ref>), respectively, in the main text.


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