Quantum oscillations in acoustic phonons in Weyl semimetals
Song-Bo Zhang Jianhui Zhou




Institut für Theoretische Physik und Astrophysik, Universität
Würzburg, D-97074 Würzburg, Germany



Anhui Province Key Laboratory of Condensed Matter Physics at Extreme
Conditions, High Magnetic Field Laboratory,Chinese Academy of Sciences,
Hefei 230031, Anhui, China


We theoretically study the modification of the energy spectrum of long-wavelength acoustic phonons due to the electron-phonon interaction in a three-dimensional topological Weyl semimetal under the influence of quantizing magnetic fields. We find that the dispersion and attenuation of phonons show striking oscillatory behaviors when varying the magnetic field at low temperatures. These oscillations are distinct when the Fermi energy is in different energy regimes. Moreover, the van Hove singularity of the Weyl spectrum can manifest as a transition between different oscillation patterns when increasing the magnetic field. These phonon behaviors could provide testable fingerprints of the Fermi-surface morphology and relativistic feature of the Weyl semimetal.

Quantum oscillations in acoustic phonons in Weyl semimetals
    Jianhui Zhou
    December 30, 2023
===========================================================





§ INTRODUCTION


Three-dimensional (3D) Weyl semimetals are a novel topological phase
of matter in which two low-energy bands have isolated and linear
crossing points, termed as Weyl nodes, in momentum space <cit.>.
According to the fermion doubling theorem <cit.>,
these Weyl nodes appear in pairs with opposite chirality. They act as
sources of monopoles and anti-monopoles in momentum space <cit.>.
This nontrivial momentum-space topology results in various fascinating
phenomena, such as the chiral anomaly <cit.>,
chiral magnetic effect <cit.>,
anomalous magnetoresistance <cit.>
as well as exotic phonon dynamics <cit.>.
The Weyl semimetal breaks either time-reversal or inversion symmetry.
However, under the protection of some extra symmetries, the Dirac
semimetal, which may be simply regarded as a combination of Weyl semimetals
related by time-reversal or inversion symmetry, could exist and inherit
many interesting properties of Weyl semimetals. Very recently, Weyl and Dirac
semimetals have been predicted in a variety of materials <cit.>,
some of which have been confirmed experimentally <cit.>.

Applying a magnetic field in a 3D metal quantizes the continuous electronic
states to a series of discrete Landau bands. The corresponding electronic
orbit (also called cyclotron orbit) at a given energy ϵ is
closed. In momentum space, it encloses an area S(ϵ) normal
to the magnetic field. This area obeys the condition of Lifshitz-Onsager
quantization S(ϵ)=(2π eB/ħ)(ν+γ_L) <cit.>,
where B is the magnetic field and ν is a non-negative integer.
The phase mismatch 0⩽γ_L<1 consists of two
parts: γ_L=γ_M+γ_B. The
first part γ_M=1/2 is the Maslov index and the other part
is directly related to the Berry phase ϕ_B of the closed
orbit by γ_B=-ϕ_B/2π <cit.>.
As the result of the quantization, many measurable quantities oscillate
when varying the magnetic field at low temperatures, such as the magnetization
and longitudinal conductivity associated with the de Haas-van Alphen
and Shubnikov-de Haas effect, respectively. Several experiments on
the Dirac semimetals Cd_3As_2 <cit.>,
Na_3Bi <cit.> and the TaAs-class Weyl
semimetals <cit.>
have made efforts to detect the relativistic feature of Weyl fermions.
Very recently, people have commenced to experimentally explore the
behaviors of phonons in TaAs in the presence of magnetic fields <cit.>.
Therefore, theoretical investigations on the phonon physics in Weyl
semimetals under magnetic fields are highly desired.

In this paper, we study the modification of the energy spectrum
of long-wavelength acoustic phonons by the electron-phonon interaction
in doped Weyl semimetals under quantizing magnetic fields at low temperatures.
We find that when the Fermi energy lies in different energy regimes,
the modification of phonon dispersion ΔΩ_q shows distinct
oscillations when varying the magnetic field. When the Fermi energy
is close to the Weyl nodes, the oscillations correspond to a single
period which resembles that of the ideal Weyl fermions. However, due
to the higher-order momentum terms, the phase shift in the oscillations
differs from that of the ideal Weyl fermions by a value proportional
to the Fermi energy measured from the Weyl nodes. When the Fermi
energy lies above the Lifshitz transition energy, multiple periods
could coexist in ΔΩ_q, resulting in beating oscillations.
Moreover, a van Hove singularity could exist in the Weyl spectrum
and be signified as a transition between different patterns of oscillations.

At the transition point the background of the oscillations in ΔΩ_q
diverges, and the oscillations transmit between a beating pattern
with three periods and another with a single period.

 In addition, we show that the phonon attenuation exhibits sets of
periodic spike peaks at low temperatures the features of which are also very
sensitive to the position of the Fermi energy. These
findings could greatly enrich our understanding of Weyl semimetals
as well as their coupling to acoustic phonons.

The paper is organized as follows. In Sec. <ref>,
we introduce the minimal model for topological Weyl semimetals and
outline the approach to calculate the modification of the energy dispersion
and the attenuation of long-wavelength acoustic phonons in the Weyl
semimetal under magnetic fields. In Sec. <ref>,
we analyze the modification of phonon dispersion at zero temperature
in two complementary parameter regimes, respectively. In Sec. <ref>,
we calculate and discuss the phonon attenuation at low temperatures.
Section <ref> summarizes the results.



§ MODEL AND FORMALISM


The minimal model for topological Weyl semimetals is given by <cit.>

    ℋ( k)     =     v(k_xσ_x+k_yσ_y)+m(k_c^2-| k|^2)σ_z,

where σ_x,y,z are Pauli matrices acting on the two involved
orbits; v≡ v_Fħ is the Fermi velocity in the x-y
plane and k=(k_x,k_y,k_z) is the 3D momentum. The model
has two energy bands touching at two Weyl nodes at (0,0,± k_c),
respectively. Around the Weyl nodes, the model mimics two Weyl Hamiltonians
with opposite velocities ± v_z=±2mk_c in the z direction.
The sign ± indicates opposite chirality of the two Weyl nodes.
The momentum-quadratic term m| k|^2 is crucial to derive
the Weyl nodes as well as well localized surface states in the presence
of a surface. It breaks the Lorentz symmetry and modifies the linear
dispersion of the Weyl fermions away from the Weyl nodes. The model
 has a Lifshitz transition energy E_L≡ mk_c^2
at which the two separated Fermi surfaces merge to a single one; see Fig. <ref>(a). When
v_z>√(2)v, the model possesses also a van Hove singularity
at E_v≡ v√(v_z^2-v^2)/m where the density
of states (DOS) diverges. The model along with its time-reversal partner
has been used to well describe the low-energy physics of topological
Dirac semimetals such as Cd_3As_2 and Na_3Bi <cit.>.

In this paper, we consider a uniform magnetic field B applied in the z direction to the Weyl semimetal <cit.>. The electronic spectrum of the Weyl semimetal is split to a series
of Landau bands <cit.>

    ϵ_k_z^0    =    ω-ℳ_0(k_z),
    ϵ_k_z^ns    =    ω+s√(ℳ_n^2(k_z)+nη^2),  n⩾1

where η=√(2)v/ℓ_B, ω=m/ℓ_B^2, ℳ_n(k_z)=E_L-mk_z^2-2nω
with n∈{0,1,2,...} and s=±; see Fig. <ref>(b). ℓ_B=√(ħ/eB)
is the magnetic length. All the Landau bands have the degeneracy of
N_L=L_xL_y/2πℓ_B^2 with L_x,y,z the
lengths of the system in the x, y and z direction, respectively.
The ω term in the Landau bands stems from the quadratic term
in the model. It breaks the particle-hole symmetry between the n⩾1 Landau bands. This term plays an important role in the transition
of the patterns of quantum oscillations, as we show below. To facilitate
the derivation, we adopt the Landau gauge A=(-By,0,0). Hence,
the wavefunctions of the n⩾1 Landau bands can be written
as

    ⟨r|n,s,k_x,k_z⟩    =    ([ C_k_z↑^nsϕ_n-1k_xk_z(r);   C_k_z↓^nsϕ_nk_xk_z(r) ]),

where C_k_z↑^n+=-C_k_z↓^n-=cosθ_k_z^n,
C_k_z↓^n+=C_k_z↑^n-=sinθ_k_z^n,
2θ_k_z^n=arctan(√(n)η/ℳ_n) and

    ϕ_nk_xk_z(r)     =    e^ik_xx+ik_zze^-(y-y_0)^2/2ℓ_B^2√(2^nn!√(π)ℓ_BL_xL_z)ℋ_n(y-y_0/ℓ_B).

ℋ_n(x) is the n-th Hermite polynomial and y_0=k_xℓ_B^2
is the guiding center. Specially, the wavefunction of the chiral Landau
band with n=0 reads ⟨r|0,k_x,k_z⟩=(0,ϕ_0k_xk_z(r))^T.
Note that we focus on the orbital effect and neglect the Zeeman splitting
effect of the magnetic field throughout this paper.



In the presence of the magnetic field, the interaction between the
electrons and acoustic phonons can be written as 

    H_int    =    ∑_α,α'∑_qΓ_αα'^(0)(q)a_α^†a_α'(b_-q^†+b_q),

where the symbol α and α' are abbreviations for the
set of quantum numbers n, s, k_x and k_z; a_α^†
(a_α) is the creation (annihilation) operator for an electron
in the quantum state |α⟩ with energy ϵ_α;
b_q^† (b_q) is the creation (annihilation)
operator for an acoustic phonon with the momentum q=(q_x,q_y,q_z)
and energy Ω_q=v_pq; v_p/ħ is the phonon
velocity; Γ_αα'^(0)(q)=⟨α|U(q)e^-iq·r|α'⟩
are the matrix elements of electron-phonon interaction, U(q)
is the electron-phonon perturbation potential. For simplicity, we
consider U(q) to be independent of the electron state. For
longitudinal acoustic phonons |U(q)|^2=ħ^2D^2q/(2v_pρ),
where D is the deformation potential constant and ρ is the
mass density of the crystal <cit.>.
We utilize the simplified scalar form of the electron-phonon interaction and neglect the gauge-field interaction in Eq. (<ref>). It should be noted that the full electron-phonon interaction in Weyl semimetals is a 2×2 matrix in the pseudospin space, the non-diagonal elements of which could generate electromagnetic gauge fields <cit.>. The gauge field effect is interesting but goes beyond the scope of this paper.

The Dyson's equation for phonon Green's function is given by <cit.>

    D^-1(q,Ω)     =     D_0^-1(q,Ω)-Π(q,Ω),

where D_0(q,Ω)=2Ω_q/(Ω^2-Ω_q^2)
is the phonon Green's function in the absence of interaction. Π(q,Ω)
is the phonon self-energy. It can be calculated from a bubble diagram:

    Π(q,iω_m)     =    TV∑_iϵ_n∑_α,α'G(α,iϵ_n)Γ_αα'(q)
            × G(α',iϵ_n-iω_m)Γ_αα'^(0)*(q),

where G(α,iϵ_n) is the Green's function of an electron
in the state |α⟩. ϵ_n=(2n+1)π T with integers
n are the Matsubara frequencies for electrons. T is temperature
(the Boltzmann constant k_B=1 is taken) and V=L_xL_yL_z
is the volume of the system. Γ_αα'(q) represents
the electron-phonon interaction vertex. In the Weyl semimetal,
 Migdal's theorem is applicable with an accuracy up to the order
of v_p/v <cit.>. Note that the phonon velocity is
typically a few orders of magnitude smaller than the electron velocity.
One can thus replace Γ_αα'(q) and G(α,iϵ_n)
with Γ_αα'^(0)(q) and G^(0)(α,iϵ_n)=(iϵ_n-ϵ_α+μ)^-1,
respectively. Summing over the electron frequencies and then carrying
out the analytic continuation iω_n→Ω_q+iδ,
we can write the retarded phonon self-energy as

    Π^R(q,Ω_q)     =    4π^2χV∑ _α,α'|ℱ_αα'(q)|^2f(ϵ_α)-f(ϵ_α')ϵ_α-ϵ_α'-Ω_q-iδ,

where χ=ħ^2qD^2/(8π^2ρ v_p), f(ϵ_α)=[1+exp((ϵ_α-μ)/T)]^-1
is the Fermi function, δ is a positive infinitesimal and ℱ_αα'(q)≡⟨α|e^-iq·r|α'⟩
is the form factor. Using the wavefunctions stated in Eq. (<ref>),
the form factor is derived as

    ℱ_αα'(q)     =     D(q)[C_k_z↑^nsC_k_z'↑^n's'𝒥_n-1^n'-1(u)+C_k_z↓^nsC_k_z'↓^n's'𝒥_n^n'(u)]
          ×δ_k_x',k_x+q_xδ_k_z',k_z+q_z, 
    𝒥_n^n'(u)     =    ρ^|n-n'|L_min(n,n')^|n-n'|(u)√(min(n,n')!max(n,n')!),

where D(q)=e^iφ_q(n'-n)e^-iq_y(y_0'+y_0)/2e^-u/2,
u=ℓ_B^2(q_x^2+q_y^2)/2, φ_q=arctan(q_y/q_x)
and L_N^m(x) are the associated Laguerre polynomials. In particular,
for n=0, C_k_z↑^0=0 and C_k_z↓^0=1.

As mentioned before, we are interested in the case where the wavelength
of the acoustic phonon is sufficiently large so that qℓ_B≪1
and q≪ k_c. In this case, 𝒥_n^n'(ρ)≈δ_n,n'
and C_k_z+q^ns≈ C_k_z^ns can be taken. These,
along with the energy denominator ϵ_α-ϵ_α'-Ω_q
in Eq. (<ref>), make only the terms with s=s'
important. Under these circumstances, the interband transitions are
forbidden, and the form factor simplifies to

    |ℱ_αα'(q)|     =    δ_k_x',k_xδ_k_z',k_z+q_zδ_n,n'δ_s,s'.

With Eq. (<ref>) in Eq. (<ref>)
and considering small phonon self-energy for reality, the modification
of phonon dispersion ΔΩ_q=Re[Π^R(q,Ω_q)]
and the attenuation of phonons Γ_q=-Im[Π^R(q,Ω_q)]
can be written as

    ΔΩ_q    =    χℓ_B^2∑_n,s∫ dk_zf(ϵ_k_z^ns)-f(ϵ_k_z+q_z^ns)ϵ_k_z^ns-ϵ_k_z+q_z^ns-Ω_q,
    Γ_q    =    πχℓ_B^2∑_n,s∫ dk_z[f(ϵ_k_z+q_z^ns)-f(ϵ_k_z^ns)]
                          ×δ(ϵ_k_z^ns-ϵ_k_z+q_z^ns-Ω_q),

respectively, where the sum runs over all Landau bands indexed by
{n,s}. We have considered a large Weyl system L_x,y,z→∞
and transformed the summation over k_z to an integral. The factor ℓ_B^-2 accounts for the Landau degeneracy which stems from the summation over k_x. According
to Eqs. (<ref>) and (<ref>), ΔΩ_q and Γ_q are non-zero only when
q_z is finite, due to the conservation of momentum and energy.
For a given phonon momentum q, both ΔΩ_q and
Γ_q are most pronounced when q∥B whereas
suppressed when q⊥B. At zero temperature T=0, we can
further approximate

    f(ϵ_k_z^ns)-f(ϵ_k_z+q_z^ns)≈-qcosθ v_k_z^nsδ(ϵ_k_z^ns-μ)

by considering q≪ k_z, where v_k_z^ns=∂ϵ_k_z^ns/∂ k_z
and θ is the angle between the directions of the magnetic
field and the phonon propagation. Then, it follows from Eqs. (<ref>)
and (<ref>) that

    ΔΩ_q    =     -χℓ_B^2∑_n,s∫ dk_zδ(ϵ_k_z^ns-μ)v_k_z^nscosθv_k_z^nscosθ+v_p,
    Γ_q    =    πχ v_pℓ_B^2∑_n,s∫ dk_zδ(ϵ_k_z^ns-μ)δ(v_k_z^nscosθ+v_p).

Note that the phonon attenuation can also be obtained
from ΔΩ_q by applying the Kramers-Krönig
relation. Equations (<ref>) and (<ref>)
indicate that only the electrons around the Fermi surface participate
in the renormalization of phonon dispersion ΔΩ_q as well as the phonon
attenuation Γ_q. Both ΔΩ_q and Γ_q are proportional to the momentum of the long-wave phonon, similar to the undressed phonon spectrum Ω_q=v_pq. Thus, these modifications can be directly related to the modification of phonon velocity.  Moreover, they show singularities periodically when the condition v_k_z^nscosθ+v_p=0
is met <cit.> In the following, we focus on the case of parallel propagation with
θ=0, unless specified otherwise. We emphasize
that the main results remain valid when θ deviates slightly
from zero.



§ PHONON DISPERSION MODIFICATION AT ZERO TEMPERATURE


In this section, we show that at zero temperature, ΔΩ_q
exhibits singularity and oscillatory behaviors when varying the magnetic
field. As indicated in Eq. (<ref>), the singularities
occur when the component of the electron velocity in the field direction
coincides with that of the phonons. Away from the singularities, the phonon
velocity is much smaller than the electron velocity. Thus, ΔΩ_q
can be rewritten as

    ΔΩ_q    =     -χℓ_B^2∫ dk_z𝒟(k_z),

where

    𝒟(k_z)     =    δ(ϵ_k_z^0-μ)+∑_n=1^∞∑_s=±δ(ϵ_k_z^ns-μ).

This indicates that ΔΩ_q is directly related to the
electron DOS at the Fermi energy. It could therefore provide an alternative
way to probe the information encoded in the electron DOS.

Next, we calculate the DOS function 𝒟(k_z) explicitly
with the Landau bands stated in Eq. (<ref>). The oscillation
behavior occurs in the semiclassical limit, i.e., |μ|≫η,ω,
where many Landau bands cross the Fermi energy. In this limit, 𝒟(k_z)
can be evaluated by using Poisson's summation rule:

    𝒟(k_z)     =     2|μ|Re[∑_j=0^∞∫_0^∞dnδ(ε_nk_z^2-μ^2)exp(i2π jn)]
            +12[δ(ε_0k_z-μ)-δ(ε_0k_z+μ)],

where ε_0k_z=m(k_z^2-k_c^2), ε_nk_z=√(ℳ_n^2(k_z)+nη^2)
with n∈{1,2,⋯}, and for convenience we have defined μ≡μ-ω.
Eventually, 𝒟(k_z) can be rewritten as

    𝒟(k_z)     =    ℓ_B^2|μ_-|ℱ_k_z∑_l=±Θ(n_l,k_z)[1+2∑_j=1^∞cos(2π jn_l,k_z)]
              +12[δ(ε_0k_z-μ)-δ(ε_0k_z+μ)],

where Θ(x) is the Heaviside step function and the index functions
read

    n_±,k_z    =    ℓ_B^22m^2(k_c^2-k_z^2)-v^2±ℱ_k_z4m^2,
    ℱ_k_z    =    √(v^4-4v^2m^2(k_c^2-k_z^2)+4m^2μ^2).


We plug Eq. (<ref>) into Eq. (<ref>)
and perform the integration over k_z. Due to the large value
of ℓ_B, the factor cos(2π jn_l,k_z) is a rapidly
oscillating function of k_z. The other factors in Eq. (<ref>)
are smooth in k_z. We can employ the saddle point approximation
to the integration by expanding n_l,k_z around its extremal
points k_i over which the integration is important <cit.>.
According to Eq. (<ref>), n_+,k_z has three
possible extremal points:

    k_i^+    =     0, ±√(k_c^2-μ^2/v^2),

while n_-,k_z has only one at k_i^-=0. Thus, the integration
over k_z can be recast as

    ∫ dk_zΘ(n_l,k_z)ℱ_k_zcos(2π jn_l,k_z)
    
    =    ∑_k_i^lΘ(n_l,k_i^l)ℱ_k_i^l√(jn_l,k_i^l”)|cos(2π jn_l,k_i^l±π4),

where n_l,k_z”≡∂^2n_k_z^l/∂ k_z^2
and the + (-) sign is taken if n_l,k_i^l”>0 (<0).
Finally, ΔΩ_q can be written as three parts:

    ΔΩ_q    =ΔΩ_q^osc+    ΔΩ_q^b+ΔΩ_q',

where

    ΔΩ_q^osc    =-2χ|μ|∑_l=±∑_k_i^l∑_j=1^∞
       ×Θ(n_l,k_i^l)ℱ_k_i^l√(jn_l,k_i^l”)cos(2π jn_l,k_i^l±π4).
    ΔΩ_q^b    =-χ|μ|∑_l=±∫ dk_zΘ(n_k_z^l)ℱ_k_z,
    ΔΩ_q'     =χ2ℓ_B^2[Θ(E_L-μ)√(m(E_L-μ))-Θ(E_L+μ)√(m(E_L+μ))].

The first part ΔΩ_q^osc, consisting of several
sets of harmonic terms, describes the quantum oscillations of the
phonon spectrum. Since the Landau index n_k_z is related to
the cross section of the Fermi surface at k_z by n_k_z=ℓ_B^2S(k_z,μ)/2π,
these oscillations are essentially governed by the extremal cross sections
of the Fermi surfaces normal to the magnetic field. Therefore, the
oscillations of Ω_q are of the same type of the de Haas-van
Alphen oscillations and can be used to detect the Fermi surface morphology
of electrons. In reality, a finite broadening of Landau bands due to, e.g., impurity
scattering, electron-phonon interaction or thermal excitation at finite temperatures, smears out
higher harmonics and make only the first harmonic important <cit.>.
Therefore, the quantum oscillations are customarily described by only
the first (j=1) harmonic term in experiments. In the following,
we use the first harmonic term to represent a set of harmonics for
convenience unless specified otherwise. The second and last parts
ΔΩ_q^b and ΔΩ_q' give the background
of the oscillations. In general, ΔΩ_q^b also
depends on the magnetic field via the modified Fermi energy μ=μ-ω.
This dependence results from the quadratic term in the model for topological
Weyl semimetals. In weak fields ω≪|μ|, ΔΩ_q^b
becomes weakly dependent of the magnetic field. ΔΩ_q'
refers to the special contribution from the chiral Landau band. In
the semiclassical limit, this part is much smaller compared to the
other two parts. Finally, we note that ΔΩ_q is linear
in χ=ħ^2qD^2/(8π^2ρ v_p), hence the modified
phonon velocity v_p+Δ v_p=v_p(1+ΔΩ_q/v_pq)
remains a constant with respect to q. In the weak field limit ℓ_B→∞,
both ΔΩ_q^osc and ΔΩ_q' vanish,
whereas ΔΩ_q^b remains finite and reproduces
the results in the absence of the magnetic field.

Now, we are ready to analyze ΔΩ_q concretely. In order
to illustrate the results clearly, we consider two complementary cases:
v_z<√(2)v and v_z>√(2)v, respectively.






 §.§ The case with v_z<√(2)v


If v_z<√(2)v, the Weyl semimetal possesses three different
energy regimes, namely, the low- (|μ|<E_L), moderate-
(E_L<|μ|<vk_c) and high-energy (|μ|>vk_c) regimes.
In these regimes, the Fermi surfaces take distinct geometries. While
E_L refers to the transition energy at which the two Fermi
surfaces begin to merge to a single one, vk_c is the energy at
which the Fermi surface changes from a dumbbell shape to a convex
sphere. As a result, the phonon dispersion shows different patterns
of oscillations in the magnetic field when μ is in different
regimes, which we discuss in detail below. For the moment, we consider
μ to be away from the boundaries between the different energy
regimes. When μ is close to the boundaries, the variation of
the magnetic field could lead to transitions between different patterns
of oscillations. This is more evident for the opposite case v_z>√(2)v
and will discussed later. Figure <ref>
displays typical ΔΩ_q as a function of ℓ_B^2.
In the plots, we use the typical parameters, e.g., for the Dirac semimetal
Cd_3As_2, v=6 eV·A, eV·A, k_c=0.15
A^-1, m=15 eV·A^2 and ρ=7×10^3 kg/m^3,
and for long wavelength acoustic phonons, D=20 eV, v_p=0.015
eV·A and q=10^-4 A^-1<cit.>. The blue dots are the analytic results of Eq. (<ref>)
with the first 10^4 harmonics, while the pink dots are the numerical
integration for Eq. (<ref>). They are in perfect agreement
with each other.

We first examine the low-energy limit {η,ω}≪|μ|≪ E_L,
where the semimetal possesses two separated pockets of ideal Weyl
fermions located at k=±(0,0,k_c), respectively. According
to Eq. (<ref>), the phonon dispersion modification
ΔΩ_q can be written as

    ΔΩ_q    =-    4μ^2χv^2v_z-4|μ|χvv_zℓ_B∑_j=1^∞1√(j)cos(2jπ𝒫_1ℓ_B^2-π4).

The first term is independent of the magnetic field. It is quadratic
in μ and corresponds to the result in the absence of the magnetic
field. The second term is periodic in ℓ_B^2 with the period
𝒫_1=μ^2/2v^2 quadratic in μ/v and the amplitude
4|μ|/(vv_∥ℓ_B). Unlike the 3D conventional electron
gas, the phase shift (i.e., -π/4) in the oscillations is always
negative, regardless of the type of carriers. From this phase shift,
a Berry phase of π can be deduced, as expected for ideal Weyl
fermions.

Next, we look at the low-energy regime |μ|<E_L,
in which the semimetal still has two separated and equivalent
pockets of electrons. In this regime, ΔΩ_q is formulated
as

    ΔΩ_q    =     -χ|μ|mvlnℱ_+ℱ_--χ𝒜_1ℓ_B∑_j=1^∞1√(j)
          ×cos[2jπ(𝒫_1ℓ_B^2-γ_1)-π4],

where 𝒫_1=μ^2/2v^2, ℱ_±=v^2±2m|μ|+2v√(m(E_L±|μ|)),
γ_1=mμ/v^2, 𝒜_1=4|μ|/vv_z' and v_z'=v_z√(1-(μ/vk_c)^2).
The background of the oscillations is written in a logarithmic form.
It depends weakly on the magnetic field, see the black dashed line
in Fig. <ref>(a). The period 𝒫_1
of the oscillations is the same as that of ideal Weyl fermions. This
indicates the relativistic Weyl properties of the two equivalent pockets.
However, compared to ideal Weyl fermions, the Weyl fermions in this
topological semimetal are modified in two aspects. First, the Fermi
velocity v_z' is reduced by the Fermi energy as v_z√(1-(μ/vk_c)^2).
As a result, the amplitude of the harmonic χ 𝒜_1/ℓ_B=4|μ|/vv_z'
is enhanced. Second, the oscillations acquire an additional phase
shift γ_1=mμ/v^2 proportional to μ. Thus, the Berry
phase of the electrons is modified as ϕ_B=π(1-2γ_1)=π(1-2mμ/v^2).
It is no longer quantized in π. It decreases for the n-type doping
μ>0 whereas increases for the p-type doping μ<0.


In experiments, the Berry phase can be measured using the approach
of Landau index analysis. In explicit, we first assigns the ordered indices ν={1,2,⋯}
to the locations B_ν of the maxima of the oscillations, next
plots ν as a function of 1/B_ν and then extrapolates the
plot to 1/B_ν→0 to get the intercept of ν, say
ν_0.  Then, the Berry phase is found via the relation ϕ_B=(2πν_0+5π/4) mod 2π.
As mentioned before, the Berry phase for ideal Weyl fermions is π.
The presence of the quadratic term in the topological semimetal gives
rise to a correction of the Berry phase 2πγ_1=-2π mμ/v^2.
For Cd_3As_2 with typical parameters: k_c≃0.15
A^-1, v≃510 eV·A (i.e., v/ħ≃0.751.5×10^6
m/s) <cit.>,
m≃15 eV·A^2 <cit.>, and in the
low-energy regime, the Fermi energy μ can be as large as the
Lifshitz transition energy ≃0.3 eV. Thus, the correction can
be as large as ≃0.10.4π . For Na_3Bi
with typical parameters: k_c≃0.1 A^-1, v≃2.5
eV·A <cit.>, m≃10 eV·A^2 <cit.>
and μ≃0.1 eV, we can obtain a correction up to ≃0.3π.
This correction may account for the discrepancy between the observations
of the Berry phase in the n-type doped Dirac semimetals and the ideal
value of π <cit.>.


In the moderate-energy regime E_L<|μ|<vk_c,
there is only a single Fermi surface in the semimetal. It takes a
dumbbell shape with two equivalent maximum cross sections at around
k_z=± k_c as well as a minimum cross section at k_z=0,
see the inset of Fig. <ref>(b). Thus, ΔΩ_q
has two harmonics and exhibits a beating pattern of oscillations,
as shown in Fig. <ref>(b). The first harmonic
is associated with the two maximum cross sections, similar to that
in the low-energy regime, whereas the second one is associated with the minimum
cross section. The corresponding ΔΩ_q reads

    ΔΩ_q    =     -χ|μ|mvlnℱ_+ℱ_0-χℓ_B∑_κ∈{1,2}∑_j=1^∞𝒜_κ√(j)
          ×cos[2jπ(𝒫_κℓ_B^2-γ_κ)+(-1)^κπ4],

where ℱ_0=√(v^4-4v^2mE_L+4m^2μ^2).
The first harmonic has the same forms of the period 𝒫_1,
phase shift γ_1 and amplitude 𝒜_1 as those
in the low-energy regime. In contrast, the period, phase shift and
amplitude of the second harmonic behave more complicatedly. They can
be written as 𝒫_2=(2mE_L-v^2+ℱ_0)/4m^2,
γ_2=mμ/ℱ_0 and 𝒜_2=2|μ|/√(ℱ_0|v^2-ℱ_0|),
respectively. Figure <ref> displays the periods 𝒫_1,2,
phase shifts γ_1,2 and amplitudes 𝒜_1,2 of
the two harmonics as functions of μ. Unlike 𝒫_1∝μ^2,
𝒫_2 deviates from the quadratic dependence on μ,
which signifies the non-relativistic nature of the associated electrons
centered at k_z=0. Similar to γ_1, γ_2 increases
as increasing |μ|. However, it is always larger than γ_1.
This indicates a larger deviation of the Berry phase from π and
hence provides another signature of the non-relativistic property
of the second harmonic. A similar beating oscillation occurs in the
Shubnikov-de Haas oscillation due to the topological band inversion
even in the absence of Zeeman splitting <cit.>. The
double-period Shubnikov-de Haas oscillation has been experimentally
observed on Cd_3As_2 <cit.>.


Finally, we analyze the high-energy regime |μ|>vk_c,
where the Fermi surface becomes a convex spheroid centered at k=0.
ΔΩ_q again oscillates with a single period and can
be formulated as

    ΔΩ_q    =     -χ|μ|mvlnℱ_+ℱ_0-χ𝒜_2ℓ_B∑_j=1^∞1√(j)
          ×cos[2π j(𝒫_2ℓ_B^2-γ_2)-π4].

The oscillations are contributed purely by the second harmonic. This
implies that the carriers in the electronic system behave more like
a non-relativistic electron gas. When increasing |μ|, the period 𝒫_2
increases, while the phase shift γ_2 approaches the constant
sgn(μ)π/2 and the amplitude 𝒜_2 to 1/m.
In the extreme limit |μ|≫ vk_c, 𝒜_2≈1/m,
ℱ_0=2m|μ|, 𝒫_2=|μ|/2m and γ_2=sgn(μ)/2.
Equation (<ref>) resembles the results for a
3D conventional electron gas with parabolic dispersion.

Before closing this subsection, let us discuss the amplitude
of the oscillations. The phonon dispersion modification could be found
by measuring the change of the phonon velocity compared to the unperturbed
one, ΔΩ_q/Ω_q=Δ v_p/v_p. The ratio
is determined by χ𝒜_1,2/(Ω_qℓ_B).
For the typical parameters for the acoustic phonons in the Dirac semimetal
Cd_3As_2, v_p=0.015 eV·A, ρ=7000 kg/m^3 <cit.>,
D=20 eV <cit.> and q=10^-4 A^-1, we obtain
χ≈3.3×10^-5 eV^2·A^3. Hence, we estimate
the phonon dispersion modification as 10^-410^-3
for μ≈0.1 eV and ℓ_B≈10^3 A. This is experimentally
detectable, for example, by the pulse-echo technique <cit.>.
A finite scattering lifetime (or broadening) is usually present in the electronic bands, due to impurity scattering or electron-phonon interaction <cit.>.
In the semiclassical regime, we can assume a constant lifetime τ_q for the Landau bands. Then, we can derive a Dingle damping factor e^-j π m^*/(eBτ_q) for the j-th harmonic, where m^* is the effective mass. The Dingle factors make the first harmonic dominant whereas higher harmonics become less important. However, in a clean Weyl metal, the scattering lifetime τ_q is very long while the mass m^* is small. Therefore, we could expect to observe a large number of harmonics at low temperatures.






 §.§ The case with v_z>√(2)v




  §.§.§ Oscillations in four energy regimes


Now we turn to the complementary case with v_z>√(2)v. The
band structure of the Weyl semimetal in this case is shown in Fig. <ref>(a).
In addition to E_L and vk_c, there is another characteristic
energy, the van Hove singularity at E_v=v√(v_z^2-v^2)/2m,
where the DOS diverges; see Fig. <ref>(b). Therefore,
we can define four different energy regimes separated by E_v, E_L,
and vk_c. Suppose μ is away from the boundaries. ΔΩ_q
shows distinct patterns of oscillations in these four regimes, similar
to the case with v_z<√(2)v. The low-energy regime |μ|<E_v
is exactly the same as that for the case with v_z<√(2)v.
There are two separated and equivalent convex Fermi surfaces at (0,0,± k'_c)
where k'_c=±√(k_c^2-μ^2/v^2). Thus, ΔΩ_q
oscillates with a single period and can be formulated by Eq. (<ref>).
In the second energy regime E_v<|μ|<min(E_L,vk_c),
the two separated Fermi surfaces merge to form a dumbbell-shaped Fermi
surface with a sphere inside; see the insert in Fig. <ref>(b).
Note that this is different from the case with v_z<√(2)v
where the two Fermi surfaces merge at E_L rather than
E_v and no sphere appears inside. Thus, the dumbbell-shaped
Fermi surface has an additional maximal cross section for the inside
sphere. As a result, ΔΩ_q displays beating oscillations
with three periods, see the small field region in Fig. <ref>(a).
In the third energy regime min(E_L,vk_c)<|μ|<max(E_L,vk_c),
ΔΩ_q also shows a beating pattern of oscillations but
with only two periods. If E_L<|μ|<vk_c, then the inside
Fermi surface disappears and the outside Fermi surface remains dumbbell
shaped. The formula for ΔΩ_q is the same as Eq. (<ref>)
with the periods given by 𝒫_1 and 𝒫_2.
On the other hand, if vk_c<|μ|<E_L, then the outside
Fermi surface becomes convex but it coexists with the inside Fermi
surface. Thus, the two periods of the oscillations are instead given
by 𝒫_2 and 𝒫_3, where 𝒫_3=(2mE_L-v^2-ℱ_0)/4m^2.
The typical oscillations in this case are shown in Fig. <ref>.
Finally, in the high-energy regime |μ|>max(E_L,vk_c),
the Fermi surface becomes a single convex sphere. We have again simple
oscillations with only one period and formulated by Eq. (<ref>).







  §.§.§ Van Hove singularity and transitions of oscillation patterns




When the Fermi energy μ is close to the van Hove singularity,
we can observe a transition of different patterns of oscillations
when increasing the magnetic field. To show this, we consider E_vμ<min(E_L,vk_c)
and display the typical oscillations in Fig <ref>(a).
Before the transition, the oscillations have three periods and can
be described by

    ΔΩ_q^osc    =     -χℓ_B∑_κ∈{1,2,3}∑_j=1^∞𝒜_κ√(j)
          ×cos[2jπ(𝒫_κℓ_B^2-γ_κ)+(-1)^κπ4],

where γ_3=-γ_2 and 𝒜_3=2|μ|/√(ℱ_0|ℱ_0+v^2|).
After the transition, the oscillations have only one period and are
described instead by

    ΔΩ_q^osc    =     -χ𝒜_1ℓ_B∑_j=1^∞1√(j)cos[2jπ(𝒫_1ℓ_B^2-γ_1)-π4].

When the transition happens, the background of the oscillation also
diverges. According to Eq. (<ref>), the background
ΔΩ_q^b of oscillations at μ≃ E_v
can be written as

    ΔΩ_q^b    =     -χ|μ|vmlnℱ'_+ℱ'_-4m^2|μ^2-E_v^2|,

where ℱ'_±=± v^2+2m|μ|+2v√(m(E_L±|μ|)).
It takes the same form of the DOS but with the Fermi energy modified by
the magnetic field μ=μ-ω. The divergence of
ΔΩ_q^b happens at |μ|=E_v.
Thus, the critical magnetic field is given by B_c=(ħ/me)|E_v-|μ||.
The replacement μ=μ-ω in Eq. (<ref>)
indicates that the transition induced by the magnetic field stems
from the quadratic term in Eq. (<ref>).
A similar transition happens for min(E_L,vk_c)<μ-E_v
but with the two patterns exchanging their sides, see Fig. <ref>(b).
We note finally that the transition of oscillation patterns is a manifestation
of the van Hove singularity which is absent in the case with v_z<√(2)v.



§ PHONON ATTENUATION


In this section, we discuss the phonon attenuation. Using Migdal's
theorem, the phonon attenuation Γ_q is given by Eq. (<ref>).
At low temperatures T≪{η,ω}, and for small broadening
of Landau bands, Γ_q exhibits an oscillatory and striking
spike-like pattern, called giant quantum oscillations <cit.>, as
the magnetic field varies. These peaks stem from the resonant absorption
of the acoustic phonons. The spikes occur periodically when the Fermi
energy μ falls near the edge of one Landau band and the small
component of the Fermi velocity in the field direction equals the
phonon velocity.

To facilitate the analysis of the phonon attenuation, we first analyze
the zero-temperature limit and then consider the effect of a low temperature.
The phonon attenuation at zero temperature is given by Eq. (<ref>).
In the semiclassical limit, we apply Poisson's summation rule to
Eq. (<ref>) and rewrite it as

    Γ_q^0=    πχ v_pℓ_B^2   ∑_s∫_0^∞dn∫ dk_zδ(ϵ_k_z^ns-μ)
       ×   δ(v_k_z^ns+v_p)[1+2∑_j=1^∞cos(2π jn_k_z)].

Similar to the previous section for the modification of phonon dispersion
ΔΩ_q, we have focused on the case with θ=0.
Integrating over n and k_z, this yields

    Γ_q^0    =    πχ v_p|μ|∑_s,l∑_k_i^l{Θ(n_l,k_z)ℱ_k_z|∂^2ϵ_k_z^ns/∂ k_z^2|_n_l
                     ×[1+2∑_j=1^∞cos(2π jn_l,k_z)]}_k_i^l,

where n_l,k_z with l=± is given by Eq. (<ref>);
the sum runs over all possible solutions k_i^l to v_k_z^n_ls=0.
The phonon attenuation Γ_q^0 is linear in the momentum
of phonons q, but independent of the velocity v_p. The oscillations
Γ_q^0 are also determined essentially by the extremal
cross sections of the Fermi surface and have the same periods in the
oscillations of ΔΩ_q. Therefore, the measurement of
phonon attenuation provides an alternative way to detect the morphology
of the Fermi surface. In contrast to ΔΩ_q, there is no
suppressing factor of 1/√(j) accompanying each term. Thus,
the higher-order harmonics also make important contributions and give
rise to a series of giant peaks at low temperatures. The positions
of the peaks in the phonon attenuation coincide with that of the singularities
in ΔΩ_q. However, Γ_q^0 takes the opposite
sign of ΔΩ_q, which indicates that they are in opposite
phases. It is also noteworthy that, unlike ΔΩ_q, two
δ functions cancel out in the integrals over n and k_z.
As a result, there is no phase shift of ±π/4 arising from
the saddle point approximation. This may make it clearer to extrapolate
the Berry phase by the index plotting.



The effect of a low temperature on Γ_q can be taken into
account via the convolution <cit.>

    Γ_q    =     -∫_-∞^∞dξ∂ f(ξ)∂ξΓ_q^0(ξ),

where the derivative of the Fermi function ∂ f(ξ)/∂ξ=-cosh^-2[ξ-μ)/2T]/4T
is strongly peaked at ξ=μ with a width determined by the temperature.
Consider a general integral of the form

    Γ̃    =    14T∫_-∞^∞dξcosh^-2(ξ-μ/2T)K(ξ)cos[ϕ(ξ)]

where K(ξ)≡Θ(n_l,k_z)/(ℱ_k_z|∂^2ϵ_k_z^ns/∂ k_z^2|_n_l)
is a smooth function of ξ and ϕ(ξ)≡2π jn_l,k_z
is also smooth but oscillatory in ξ. At low temperatures, the
most important part of the integral arises from the neighborhood of
ξ=μ. Due to the large value ℓ_B^2 in n_l,k_z,
the term cos[ϕ(ξ)] oscillates rapidly with varying ξ.
We expand ϕ(ξ) to the first order in (ξ-μ) and replace
the function K(ξ) by its value at ξ=μ in Eq. (<ref>).
Then, the integral can be derived to

    Γ̃    =     -λsinh(λ)K(μ)cos[ϕ(μ)],

where λ=π T[∂ϕ(ξ)/∂ξ]_ξ=μ. Applying
this approach to all terms with a cosine function in Eq. (<ref>),
we obtain

    Γ_q    =    πχ v_p|μ|∑_s,l∑_k_i^lΘ(n_l,k_i^l)ℱ_k_z|∂^2ϵ_k_z^ns/∂ k_z^2|_n_l,k_i^l
           ×[1+2∑_j=1^∞λ_j^i,lcos(2π jn_l,k_i^l)sinh(λ_j^i,l)],

where λ_j^i,l=2π^2jT[∂ n_l,k_i^l/∂ξ]_ξ=μ.

Now we consider the case with v_z<√(2)v and analyze the features
of Γ_q in the three different energy regimes. Similar analysis
and main results apply to the other case with v_z>√(2)v.
We first look at the low-energy regime μ<E_L. The phonon
attenuation Γ_q reads

    Γ_q    =    Γ_I{1+2∑_j=1^∞λ_jcos[2π j(𝒫_1ℓ_B^2-γ_1)]sinh(λ_j)},

where λ_j=2π^2jℓ_B^2T|μ|/v^2; Γ_I=χ v_p𝒜_1^2/8
is the magnitude of the phonon attenuation in the absence of the magnetic
field. Γ_q has the same period 𝒫_1 as that
stated for ΔΩ_q. In Fig. <ref>(a),
we plot typical Γ_q as functions of ℓ_B^2 at a
small μ(<E_L) and at three different low temperatures,
respectively. The spike peaks appear periodically and their magnitudes
decay when decreasing the magnetic field or increasing the temperature.
According to Eq. (<ref>), the spike peaks are given
by

    Γ_q^P1    =    Γ_I[1+2∑_j=1^∞λ_jsinh(λ_j)].

At a low temperature T≪ v^2/2π^2ℓ_B^2|μ|, we can
replace the sum over j by an integral and obtain

    Γ_q^P1   ≈   v^24ℓ_B^2T|μ|Γ_I.

The magnitude of a peak is proportional to the magnetic field at which
the peak takes place, see the dashed line in Fig. <ref>(a).
Since Γ_I is essentially independent of T, the
peak is inversely proportional to the temperature, consistent with
the numerical results presented in Fig. <ref>.



To study the shape of the peaks, we turn to Eq. (<ref>)
and write approximately the phonon attenuation as

    Γ_q    =    πχ v_p/4l_B^2T∑_n√(n)η/k_c^2-n/ℓ_B^2cosh^-2(√(n)η-ℓ_n^2/2T).

A specific peak is described by a corresponding term in the summation
of Eq. (<ref>). In general, the shape of a peak can be
described by

    Γ_n    =     A_ncosh^-2[|μ|(ℓ_B^2-ℓ_n^2)/(2ℓ_n^2T)],

where ℓ_n^2 corresponds to the magnetic field where the
center of the peak locates; the magnitude A_n can be taken as
a constant with respect to ℓ_B around the peak. From Eq. (<ref>),
the half width of the peak is found as

    (δℓ_B^2)_n    =     3.53ℓ_n^2T/|μ|=1.76ℓ_n^2Tγ_2/(m𝒫_2).

It is linear in T and ℓ_B^2, as shown in Fig. <ref>.

In the moderate-energy regime, the phonon attenuation consists
of two different periods:

    Γ_q    =    Γ_II{ζ_1[1+2∑_j=1^∞λ_jcos[2π j(𝒫_1ℓ_B^2-γ_1)]sinh(λ_j)]
           +ζ_2[1+2∑_j=1^∞λ_j'cos[2π j(𝒫_2ℓ_B^2-γ_2)]sinh(λ_j')]},

where λ_j'=2π^2jℓ_B^2T|μ|/ℱ_0,
Γ_II=χ v_p(𝒜_1^2+2𝒜_2^2)/8
with ζ_1=𝒜_1^2/(𝒜_1^2+2𝒜_2^2)
and ζ_2=2𝒜_2^2/(𝒜_1^2+2𝒜_2^2)
is the phonon attenuation in the absence of magnetic field. These
two periods, analogous to the previous argument for ΔΩ_q,
can be attributed to the Weyl and non-relativistic electrons, respectively.
Equipped with Eq. (<ref>), we can identify the complicated
pattern of the phonon attenuation in Fig. <ref>(b)
as a combination of two simple sets of spike peaks with different
periods and magnitudes, as shown in Fig. <ref>.
One set of peaks associating with the Weyl electrons is still described
by Eq. (<ref>). The other set of peaks is related
to non-relativistic electrons and can be instead depicted by

    Γ_q^P2    =    χ v_p𝒜_2^24|μ|ℱ_04ℓ_B^2T.

Similar to Γ_q^P1, Γ_q^P2 are linear in the
magnetic field and inversely proportional to the temperature. The
half width for these peaks can also be written in the form

    (δℓ_B^2)_n    =     1.76ℓ_n^2Tγ_2/(m𝒫_2).

Note that in this energy regime, the peaks are not always well separated by broad minima. Two peaks from different
sets may merge to be a single enhanced peak; see the peak around ℓ_B^2∼0.7×10^4A^2
in Fig. <ref>.

Finally, we look at the high-energy regime. The phonon attenuation
oscillates with only the second period and reads

    Γ_q    =    Γ_III[1+2∑_j=1^∞λ_j'cos(2π j(𝒫_2ℓ_B^2-γ_2))sinh(λ_j')].

where Γ_III=χ v_p𝒜_2^2/4. It associates
with the non-relativistic electron pocket and the corresponding set
of peaks is given by Eq. (<ref>). In the extreme limit
|μ|≫ vk_c, the peaks become

    Γ_q^P2(μ,T) = χ v_p/(8mℓ_B^2T),

which is independent of the Fermi energy μ.

Let us end this section by discussing some experimental relevance
for the phonon attenuation. For the Dirac semimetal Cd_3As_2
with typical parameters v≃510 eV·A,
and for μ≃0.1 eV and B≃1 T, a comparable ratio between
the half width and the period (δℓ_B^2)_n∼1/𝒫_1
yields the temperature T∼0.57v^2/ℓ_B^2|μ|∼29×10^-3
eV (i.e., 25-100 K). Thus, the spike peaks could be observed at temperatures
on the order of 1K. The electron scattering would broaden the electron
bands, and obscure the k_z of absorbing electrons at the band
edges. This would further broaden the spike peaks. Thus, to observe
the giant oscillations, the broadening of peaks needs to be much
smaller than the separation between the peaks. This requirement turns out as ql ≫ 1, where l = vτ_q is the mean
free length and τ_q the quantum scattering lifetime
of electrons. These conditions could be simultaneously
satisfied in the Dirac semimetals Cd_3As_2, Na_3Bi and the Weyl semimetals TaAs
in which the Shubinkov-de Haas oscillations have been observed <cit.>.
Similar to the modification of phonon dispersion, the phonon attenuation
can also be measured in the pulse-echo set-up including the phase-sensitive-detection
technique <cit.>. Moreover, the spike-like character
of the phonon attenuation at low temperatures allows us to determine
more accurately the field location of the peaks for the index plot.
This constitutes an advantage of the quantum oscillations of phonon
attenuation over other de Haas-van Alphen oscillations.



§ SUMMARY 

In summary, we have showed that the energy spectrum of long-wavelength acoustic phonons propagating in a Weyl semimetal exhibits exotic singularity and oscillatory behaviors as the magnetic field varies at low temperatures. These behaviors are very sensitive to the Fermi energy of the Weyl semimetal. When the Fermi energy lies above the Lifshitz transition energy, multiple periods appear in the oscillations, leading to beating behaviors. The van Hove singularity of the Weyl spectrum could manifest as a transition between different patterns of oscillations. We have revealed that the phonon attenuation displays similar oscillatory but spike-peaked behaviors. We have also briefly discussed the experimental relevance of these phonon behaviors in the candidate systems Cd_2As_3 and Na_3Bi.





We thank Shun-Qing Shen and Hai-Zhou Lu for useful discussions. S.B.Z. was supported by the DFG (SPP1666, SFB1170 “ToCoTronics”),
the Würzburg-Dresden Cluster of Excellence ct.qmat, EXC2147, project-id
39085490, and the Elitenetzwerk Bayern Graduate School on “Topological insulators”.
J.Z. was supported by the 100 Talents Program of Chinese Academy of Sciences (CAS) and by the High Magnetic Field Laboratory of Anhui Province.











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[footNote2()]footNote2
  note When the Fermi energy is close to the Weyl nodes, the Fermi surfaces are almost isotropic. Thus, the results in this low-energy regime should be independent of the direction of the magnetic field. However, when the Fermi energy is far away from the Weyl nodes, then the anisotropy of the electronic structure becomes pronounced. In this case, our theory is not applicable when the direction of magnetic field is deviated far from the z direction.Stop
  
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[Note1()]Note1
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