Semiclassical Boltzmann transport theory for multi-Weyl semimetals
Sanghyun Park Seungchan Woo E. J. Mele Hongki Min






hmin@snu.ac.kr
^1 Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
^2 Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA




Multi-Weyl semimetals (m-WSMs) are a new type of Weyl semimetal that have linear dispersion along one symmetry direction but anisotropic nonlinear dispersion along the two transverse directions with a topological charge larger than one. Using the Boltzmann transport theory and fully incorporating the anisotropy of the system, we study the dc conductivity as a function of carrier density and temperature.  We find that the characteristic density and temperature dependence of the transport coefficients at the level of Boltzmann theory are controlled by the topological charge of the multi-Weyl point and distinguish m-WSMs from their linear Weyl counterparts.


Semiclassical Boltzmann transport theory for multi-Weyl semimetals
    Hongki Min^1,2
    December 30, 2023
==================================================================





Introduction. 
There has been a growing interest in three-dimensional (3D) analogs of graphene called Weyl semimetals (WSMs) where bands disperse linearly in all directions in momentum space around a twofold point degeneracy.  Most attention has been devoted to novel response functions in elementary WSMs which exhibit a linear dispersion; however, recently it has been realized that these are just the simplest members of a family of multi-Weyl semimetals (m-WSMs) <cit.> which are characterized instead by double (triple) Weyl-nodes with a linear dispersion along one symmetry direction but quadratic (cubic) dispersion along the remaining two directions. These multi-Weyl nodes have a topologically protected charge (also referred to as chirality) larger than one, a situation that can be stabilized by point group symmetries <cit.>.

 
 Noting that multilayer graphenes with certain stacking patterns support two-dimensional (2D) gapless low energy spectra with high chiralities, these m-WSMs can be regarded as the 3D version of multilayer graphenes.
 One can expect that their modified energy dispersion and spin- or pseudospin-momentum locking textures will have important consequences for various physical properties due both to an enhanced density of states (DOS) and the anisotropy in the energy dispersion, distinguishing m-WSMs from elementary WSMs. In this Rapid Communication, we demonstrate that this emerges already at the level of dc conductivity in the strong scattering limit described by semiclassical Boltzmann transport theory.  The transport properties of conventional linear WSMs have recently been explored theoretically by several authors <cit.>, and there have been theoretical works on the stability of charge-neutral double-Weyl nodes in the presence of Gaussian disorder <cit.> and the thermoelectric transport properties in double-Weyl semimetals <cit.>. However, as we show below, the density and temperature dependences of the dc conductivity for m-WSMs require an understanding of the effect of anisotropy in the nonlinear dispersion on the scattering. We develop this theory and find that it predicts characteristic power-law dependences of the conductivity on density and temperature that depend on the topological charge of the Weyl node and distinguish m-WSMs from their linear counterparts.




Model. 
The low-energy effective Hamiltonian for m-WSMs with chirality J near a single Weyl point is given by <cit.>

    H_J   =   ε_0 [ (k_- k_0)^J σ_+ + (k_+ k_0)^Jσ_-]+ħ v_z k_z σ_z,

where k_±=k_x± i k_y, σ_±=1 2(σ_x± iσ_y), σ are the Pauli matrices acting in the space of the two bands that make contact at the Weyl point,

and k_0 and ε_0 are the material-dependent parameters in units of momentum and energy, respectively. For simplicity, here we assumed an axial symmetry around the k_z axis. The eigenenergies of the Hamiltonian are given by ε_±=±ε_0√(k̃_∥^2J+c_z^2 k̃_z^2), where k̃=k/k_0, k̃_∥=√(k̃_x^2+k̃_y^2), and c_z=ħ v_z k_0/ε_0, thus the Hamiltonian H_J has a linear dispersion along the k_z direction for k_x=k_y=0, whereas a nonlinear dispersion ∼ k_∥^J along the in-plane direction for k_z=0. Note that the system described by the Hamiltonian in Eq. (<ref>) has a nontrivial topological charge characterized by the chirality index J <cit.>. [See Sec. <ref> in the Supplemental Material (SM) <cit.> for the eigenstates and DOS for m-WSMs.]


Boltzmann transport theory in anisotropic systems. 
We use semiclassical Boltzmann transport theory to calculate the density and temperature dependence of the dc conductivity, which is fundamental in understanding the transport properties of a system. Here we focus on the longitudinal part of the dc conductivity assuming time-reversal symmetry with vanishing Hall conductivities.


The Boltzmann transport theory is known to be valid in the high carrier density limit, and we assume that the Fermi energy is away from the Weyl node, as shown in experiments <cit.>. The limitation of the current approach will be discussed later.

For a d-dimensional isotropic system in which only a single band is involved in the scattering, it is well known that the momentum relaxation time at a wavevector k in the relaxation time approximation can be expressed as <cit.>

    1τ_ k=∫d^d k'  (2π)^d W_kk' (1-cosθ_kk'),

where W_kk'=2πħ n_ imp |V_kk'|^2 δ(ε_k-ε_k'), n_ imp is the impurity density, and V_kk' is the impurity potential describing a scattering from k to k'. The inverse relaxation time is a weighted average of the collision probability in which the forward scattering (θ_kk'=0) receives reduced weight.

For an anisotropic system,  the relaxation time approximation Eq. (<ref>) does not correctly describe the effects of the anisotropy on transport. Instead, coupled integral equations relating  the relaxation times at different angles need to be solved to treat the anisotropy in the nonequilibrium distribution <cit.>. The linearized Boltzmann transport equation for the distribution function f_k=f^(0)(ε)+δ f_k at energy ε=ε_k balances acceleration on the Fermi surface against the scattering rates 

    (-e)E·v_kS^(0)(ε)=∫d^d k' (2π)^d W_kk'(δ f_k-δ f_k'),

where

S^(0)(ε)=-∂ f^(0)(ε)∂ε, f^(0)(ε)=[e^β (ε-μ)+ 1]^-1 is the Fermi distribution function at equilibrium, and β=1 k_ B T.
We parametrize δ f_k in the form:

    δ f_k=(-e)(∑_i=1^d E^(i) v_k^(i)τ_k^(i))S^(0)(ε),

where E^(i), v_k^(i), and τ_k^(i) are the electric field, velocity, and relaxation time along the i-th direction, respectively.
After matching each coefficient in E^(i), we obtain an integral equation for the relaxation time,

    1=∫d^d k' (2π)^d W_kk'(τ_k^(i)-v_k'^(i) v_k^(i)τ_k'^(i)).

For the isotropic case [τ_k^(i)=τ(ε) for a given energy ε=ε_ k], Eq. (<ref>) reduces to Eq. (<ref>). 






[See Sec. <ref> in SM <cit.> for applications of Eq. (<ref>) to m-WSMs.]
The current density J induced by an electric field E is then given by

    J^(i)=g∫d^d k (2π)^d (-e) v_k^(i)δ f_k≡σ_ijE^(j),

where g is the degeneracy factor and σ_ij is the conductivity tensor given by

    σ_ij=g e^2∫d^d k (2π)^d S^(0)(ε) v_k^(i) v_k^(j)τ_k^(j).

For the calculation, we set g=4 and v_z=v_0≡ε_0 ħ k_0.










Density dependence of dc conductivity. 
Consider the m-WSMs described by Eq. (<ref>) with chirality J and their dc conductivity as a function of carrier density at zero temperature. Due to the anisotropic energy dispersion with the axial symmetry, for J>1 the conductivity also will be anisotropic as σ_xx=σ_yy≠σ_zz. 

 We consider two types of impurity scattering: short-range impurities (e.g., lattice defects, vacancies, and dislocations) and charged impurities distributed randomly in the background. The impurity potential for short-range scatterers is given by a constant V_kk'=V_ short in momentum space (i.e., zero-range delta function in real space), whereas for charged Coulomb impurities in 3D it is given by


V_kk'=4π e^2 ϵ(q)|q|^2,

where ϵ(q) is the dielectric function for q=k-k'. Within the Thomas-Fermi approximation, the dielectric function can be approximated as ϵ(q)≈κ[1+(q_ TF^2/|q|^2)], where κ is the background dielectric constant, q_ TF=√(4π e^2 κ D(ε_ F)) is the Thomas-Fermi wave vector, and D(ε_ F) is the DOS at the Fermi energy ε_ F. The interaction strength for charged impurities can be characterized by an effective fine structure constant α=e^2κħ v_0. Note that q_ TF∝√(gα).





Figure <ref> shows the density dependence of the dc conductivity for charged impurity scattering at zero temperature. Because of the chirality J, m-WSMs have a characteristic density dependence in dc conductivity, which can be understood as follows. From Eq. (<ref>), we expect σ_ii∼ [v_ F^(i)]^2/V_ F^2, where v_ F^(i) is the Fermi velocity along the ith direction and V_ F^2 is the angle-averaged squared impurity potential at the Fermi energy ε_ F.








For m-WSMs, the in-plane component with k_z=0 and out-of-plane component with k_x=k_y=0 for the velocity at ε_ F are given by v_ F^(∥)=J v_0 r_ F^1-1 J and v_ F^(z)=v_0 c_z, respectively, where r_ F=ε_ F/ε_0. (See Sec. <ref> in SM <cit.>.)

For charged impurities, in the strong screening limit (gα≫ 1), V_ F∼ q_ TF^-2∼ D^-1(ε_ F)∼ε_ F^-2 J, thus we find


    σ_xx   ∼   ε_ F^2(1-1 J)ε_ F^4 J∼ n^2(J+1) J+2, 
    σ_zz   ∼   ε_ F^4 J∼ n^4 J+2.


Here, the DOS is D(ε)∼ε^2 J, thus ε_ F∼ n^J J+2.
In the weak screening limit (gα≪ 1), we expect V_ F∼ε_ F^-2ζ with 1 J≤ζ≤ 1, because the in-plane and out-of-plane components of the wavevector at ε_ F are k_ F^(∥)=k_0 r_ F^1 J and k_ F^(z)=k_0 r_ F/c_z, respectively. Thus, we find


    σ_xx   ∼   ε_ F^2(1-1 J)ε_ F^4ζ∼ n^2(J-1)+4Jζ J+2, 
    σ_zz   ∼   ε_ F^4ζ∼ n^4Jζ J+2.


(See Sec. <ref> in SM <cit.> for the analytic expressions of the dc conductivity for short-range impurities and for charged impurities in the strong screening limit, and a detailed discussion for charged impurities in the weak screening limit.)





Figure <ref> illustrates the evolution of the power-law density dependence of the dc conductivity as a function of the screening strength characterized by gα. Note that ζ=1 J in Eq. (<ref>) gives the same density exponent as in the strong screening limit in Eq. (<ref>).
Thus, as α increases, the density exponent evolves from that obtained in Eq. (<ref>) with decreasing ζ within the range 1 J≤ζ≤ 1. Here, nonmonotonic behavior in the density exponent  originates from the angle-dependent power law in the relaxation time, which manifests in the weak screening limit. (See Sec. <ref> in SM <cit.> for further discussion.)


Similarly, for short-ranged impurities, V_ F is a constant independent of density; in this case we find


    σ_xx   ∼   ε_ F^2(1-1 J)∼ n^2(J-1) J+2, 
    σ_zz   ∼   ε_ F^0 ∼ n^0.







The anisotropy in conductivity can be characterized by σ_xx/σ_zz. Figure <ref> shows σ_xx/σ_zz as a function of density for m-WSMs. Thus, as the carrier density increases, the anisotropy in conductivity increases. Interestingly, σ_xx/σ_zz for both short-range impurities and charged impurities in the strong screening limit is given by

    σ_xx/σ_zz∼ε_ F^2(1-1 J)∼ n^2(J-1) J+2.

Note that for arbitrary screening, ζs for σ_xx and σ_zz in Eq. (<ref>) are actually different, thus not cancelled in σ_xx/σ_zz and the power-law deviates from that in Eq. (<ref>). (See Sec. <ref> in SM <cit.> for the analytic/asymptotic expressions of the density dependence of σ_xx/σ_zz.)

We consider both the short-range and charged impurities by adding their scattering rates according to Matthiessen's rule assuming that each scattering mechanism is independent.  At low densities (but high enough to validate the Boltzmann theory) the charged impurity scattering always dominates the short-range scattering, while at high densities the short-range scattering dominates, irrespective of the chirality J and screening strength.



Temperature dependence of dc conductivity. 
In 3D materials, it is not easy to change the density of charge carriers by gating, because of screening in the bulk. However, the temperature dependence of dc conductivity can be used to understand the carrier dynamics of the system.
The effect of finite temperature arises from the energy averaging over the Fermi distribution function in Eq. (<ref>), and the temperature dependence of the screening of the impurity potential for charged impurities <cit.>.


From the invariance of carrier density with respect to temperature, we obtain the variation of the chemical potential μ(T) as a function of temperature T. Then the Thomas-Fermi wavevector q_ TF(T) in 3D at finite T can be expressed as q_ TF(T)=√(4π e^2κ∂ n ∂μ). 

In the low- and high-temperature limits, the chemical potential is given by

    μ/ε_ F   =   
    1- π^2/3J(T/T_ F)^2     (T≪ T_ F), 
    1 2η(2 J)Γ(2+2 J)(T/T_ F)^-2 J    (T≫ T_ F),

whereas the Thomas-Fermi wave vector is given by

    q_TF(T) /q_TF(0)    =    
    1- π^2/6J(T/T_ F)^2     (T≪ T_ F), 
    √(2η(2 J)Γ(1+2 J))(T/T_ F)^1 J   (T≫ T_ F),

where T_ F=ε_ F/k_ B is the Fermi temperature, and Γ and η are the gamma function and the Dirichlet eta function <cit.>, respectively.
(See Sec. <ref> in SM <cit.> for the temperature dependence of the chemical potential and Thomas-Fermi wave vector.)
In a single-band system, q_ TF(T) always decreases with T^-1 at high temperatures, whereas in m-WSMs, q_ TF(T) increases with T^1 J because of the thermal excitation of carriers that participate in the screening.





Figure <ref> shows the temperature dependence of dc conductivity for charged impurities. 
We find


    σ_xx(T)σ_xx(0)   =   
    1+C_xx(T T_ F)^2     (T≪ T_ F), 
    
    D_xx(T T_ F)^2+4ζ-2 J    (T≫ T_ F),
    
    σ_zz(T)σ_zz(0)   =   
    1+C_zz(T T_ F)^2     (T≪ T_ F), 
    
    D_zz(T T_ F)^4ζ    (T≫ T_ F).


As discussed, ζ varies within 1 J≤ζ≤ 1 and approaches 1 J in the strong screening limit (gα≫ 1). Here, the high-temperature coefficients D_ii>0, whereas the low-temperature coefficients C_ii change sign from negative to positive as α increases.
For short-range impurities, we find


    σ_xx(T)σ_xx(0)   =   
    1+C_xx^ short(T T_ F)^2     (T≪ T_ F), 
    
    D_xx^ short(T T_ F)^2(J-1) J    (T≫ T_ F),
    
    σ_zz(T)σ_zz(0)   =   
    1-e^-T_ F/T    (T≪ T_ F), 
    1 2+D_zz^ short(T T_ F)^-2+J J    (T≫ T_ F).


Here, C_xx^ short<0 and D_ii^ short>0. 
Note that for J=1, Eq. (<ref>a) becomes constant, and reduces to Eq. (<ref>b) if next order corrections are included. 

(See Sec. <ref> in SM <cit.> for the analytic/asymptotic expressions of the temperature coefficients, and the evolution of C_ii as a function of gα.)

To understand the temperature dependence, we can consider a situation where the thermally induced charge carriers participate in transport. Then the temperature dependence in the high-temperature limit can be obtained simply by replacing the ε_ F dependence with T in Eqs. (<ref>)-(<ref>), which describe the density dependence of dc conductivity. Similarly as in Fig. <ref>, σ_xx(T)/σ_zz(T) also increases with T at high temperatures.

For the charged impurities at high temperatures, and neglecting the effect of phonons, the conductivity increases with temperature, and mimics an insulating behavior.
By contrast, for short-range impurities at high temperatures, σ_zz(T) decreases with temperature and approaches 0.5 σ_zz(0), thus showing a metallic behavior. Interestingly, σ_xx(T)  shows contrasting behavior for J>1 and J=1, increasing (decreasing) with temperature for J>1 (J=1) showing insulating (metallic) behavior at high temperatures.





Discussion. 
We find that the dc conductivities in the Boltzmann limit show characteristic density and temperature dependences that depend strongly on the chirality of the system, revealing a signature of m-WSMs in transport measurements, which can be compared with experiments. In real materials with time reversal symmetry, multiple Weyl points with compensating chiralities will be present. The contributions from the individual nodes calculated by our  method are additive when the Weyl points are well separated and internode scattering is weak.
Our analysis is based on the semiclassical Boltzmann transport theory with the Thomas-Fermi approximation for screening and corrected for the anisotropy of the Fermi surface in m-WSMs. The Boltzmann transport theory is known to be valid in the high density limit. At low densities, inhomogeneous impurities induce a spatially varying local chemical potential, typically giving a minimum conductivity when the chemical potential is  at the Weyl node <cit.> and the problem is treated within the effective medium theory. Note that the Thomas-Fermi approximation used in this work is the long-wavelength limit of the random phase approximation (RPA), and neglects interband contributions to the polarization function <cit.>, thus deviating from the RPA result at low densities. Both simplifications become important in the low-density limit, which will be considered in our future work.  





The authors thank Shaffique Adam for helpful discussions and comments.
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education under Grant No. 2015R1D1A1A01058071.
E.J.M.'s work on this project was supported by the U.S. Department of Energy, Office of Basic Energy Sciences under Award No. DE-FG02-ER45118.
H.M. acknowledges travel support provided by the University Research Foundation at the University of Pennsylvania while this work was carried out.



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Supplemental Material:

Semiclassical Boltzmann transport theory for multi-Weyl semimetals

















§ EIGENSTATES AND DENSITY OF STATES FOR MULTI-WEYL SEMIMETALS


Let us consider the eigenstates and density of states (DOS) for the low-energy effective Hamiltonian of m-WSMs described by Eq. (<ref>) in the main text:

    H_J
       =   ε_0(
    [  c_z k̃_z    k̃_-^J;    k̃_+^J -c_z k̃_z;           ]),

where k̃=k/k_0 and c_z=ħ v_z k_0/ε_0. To avoid difficulties associated with anisotropic dispersions, we consider the following coordinate transformation <cit.>

    k_x   → k_0(rsinθ)^1 Jcosϕ,
    
    k_y   → k_0(rsinθ)^1 Jsinϕ,
    
    k_z   →k_0  c_z rcosθ,

which transforms the Hamiltonian into the following form:

    H=ε_0 r (
    [        cosθ sinθ e^-iJϕ;  sinθ e^iJϕ       -cosθ;             ]).

In the transformed coordinates, the energy dispersion is given by ε_±(r)=±ε_0 r and the corresponding eigenstate is given by


    |+⟩   =   (
    [        cosθ 2; sinθ 2 e^i Jϕ;               ]), 
    
    |-⟩   =   (
    [       -sinθ 2; cosθ 2 e^i Jϕ ]).



The Jacobian 𝒥 corresponding to this transformation is given by

    𝒥=
    |
    [ ∂ k_x/∂ r  ∂ k_x/∂θ  ∂ k_x/∂ϕ; ∂ k_y/∂ r  ∂ k_y/∂θ  ∂ k_y/∂ϕ; ∂ k_z/∂ r  ∂ k_z/∂θ  ∂ k_z/∂ϕ;           ]|=k_0^3  c_z J r^2 Jsin^2 J-1θ≡𝒥(r,θ).


Note that for the + band, the band velocity v_ k^(i)=1ħε_+,k∂  k_i can be expressed as


    v_ k^(x)   =   J v_0 r^1-1 Jsin^2-1 Jθcosϕ, 
    
    v_ k^(y)   =   J v_0 r^1-1 Jsin^2-1 Jθsinϕ, 
    
    v_ k^(z)   =   c_z v_0 cosθ,


where v_0=ε_0ħ k_0.






The DOS at energy ε>0 can be obtained as

    D(ε)   =   g∫d^3k (2π)^3δ(ε-ε_+,k) 
       =   g ∫_0^∞dr ∫_0^πdθ∫_0^2πdϕ𝒥(r,θ) (2π)^3δ(ε-ε_0 r) 
       =   g B(1 2,1 J) 4π^2 c_z Jk_0^3ε_0(εε_0)^2 J,

where g is the number of degenerate Weyl nodes.
Here, we used the relation ∫_0^π/2 dθcos^mθsin^nθ=1 2 B(m+1 2,n+1 2), where B(m,n)=Γ(m)Γ(n) Γ(m+n) is the beta function and Γ(x)=∫_0^∞ dt  t^x-1e^-t is the gamma function <cit.>.
Note that the Thomas-Fermi wavevector is determined by the DOS at the Fermi energy ε_ F given by

    q_ TF=√(4π e^2 κ D(ε_ F))=k_0 √(g α B(1 2,1 J)π c_z J)(ε_ Fε_0)^1 J,

where α=e^2κħ v_0 is the effective fine structure constant.

The carrier density is then given by

    n=∫_0^ε_ Fdε D(ε)=n_0 g B(1 2,1 J) 4π^2 c_z (J+2)(ε_ Fε_0)^2 J+1,

where n_0=k_0^3.
Note that ε_ F∼ n^J  J+2 and D(ε_ F)∼ n^2 J+2.





§ DENSITY DEPENDENCE OF DC CONDUCTIVITY IN MULTI-WEYL SEMIMETALS AT ZERO TEMPERATURE



In this section, we derive the dc conductivity at zero temperature for 3D anisotropic systems with an anisotropic energy dispersion which has an axial symmetry around the k_z-axis (i.e. independent of ϕ), as in the m-WSMs described by Eq. (<ref>) in the main text. To take into account the anisotropy of the energy dispersion, we express the anisotropic Boltzmann equation in Eq. (<ref>) in the main text using the transformed coordinates in Eq. (<ref>) assuming an axial symmetry around the k_z-axis:

    1   =   ∫_0^∞dr'∫_0^πdθ'∫_0^2πdϕ'𝒥(r',θ') (2π)^3 W_kk'(τ_k^(i)-v_k'^(i) v_k^(i)τ_k'^(i)) 
       =   ∫_0^∞dr'∫_0^πdθ'∫_0^2πdϕ'
    k_0^3 r'^2 Jsin^2 J-1θ' (2π)^3 c_z J[2πħ n_ imp |V_kk'|^2 F_kk'δ(ε_0 r-ε_0 r')](τ_k^(i)-d_kk'^(i)τ_k'^(i)) 
       =   2πħ n_ impk_0^3 r^2 J (2π)^2 c_z J ε_0∫_-1^1dcosθ' (1-cos^2θ')^1 J-1∫_0^2πdϕ' 2π |V_kk'|^2 F_kk'(τ_k^(i)-d_kk'^(i)τ_k'^(i)),

where d_kk'^(i)=v_k'^(i)/v_k^(i) and F_kk' =  1/2[1+cosθcosθ' +sinθsinθ' cos J(ϕ - ϕ')] is the square of the wavefunction overlap between k and k' states in the same band.
Let us define ρ_0=k_0^3  (2π)^2  c_z ε_0, V_0=ε_0 k_0^3, and 1τ_0(r)=2πħ n_ imp V_0^2 ρ_0. Then with μ=cosθ, we have

    1=r^2 J J∫_-1^1dμ'(1-μ'^2)^1 J-1∫_0^2πdϕ' 2π |Ṽ_kk'|^2 F_kk'(τ̃_k^(i)-d_kk'^(i)τ̃_k'^(i)),

where Ṽ_kk'=V_kk'/V_0 and τ̃_k^(i)=τ_k^(i)/τ_0.



Assuming τ̃_k^(i)=τ̃^(i)(μ) from the axial symmetry,

    1=w̃^(i)(μ)τ̃^(i)(μ)-∫_-1^1dμ' w̃^(i)(μ,μ')τ̃^(i)(μ'),

where



    w̃^(i)(μ)   =   r^2 J J∫_-1^1dμ'(1-μ'^2)^1 J-1∫_0^2πdϕ' 2π |Ṽ_kk'|^2 F_kk', 
    w̃^(i)(μ,μ')   =   r^2 J J(1-μ'^2)^1 J-1∫_0^2πdϕ' 2π |Ṽ_kk'|^2 F_kk' d_kk'^(i).



Now let us discretize θ or equivalently μ=cosθ to μ_n (n=1,2,⋯,N) with an interval Δμ=2/N. Then for τ̃_n^(i)=τ̃^(i)(μ_n), we have

    1=P_n^(i)τ̃_n^(i) - ∑_n' P_nn'^(i)τ̃_n'^(i),

where P_n^(i)=w̃^(i)(μ_n) is an N-vector and P_nn'^(i)=w̃^(i)(μ_n,μ_n')Δμ is an N× N matrix which relate the θ-dependent relaxation times. Note that Eq. (<ref>) has a similar structure for the multiband scattering <cit.> in which the relaxation time can be obtained by solving coupled equations, which relate the relaxation times for different energy bands involved in the scattering.

Then the dc conductivity at zero temperature is given by

    σ_ij   =   g e^2∫d^3 k (2π)^3δ(ε_ k-ε_ F) v_k^(i) v_k^(j)τ_k^(j)
       =   g e^2 ∫_0^∞dr ∫_0^πdθ∫_0^2πdϕk_0^3 r^2 Jsin^2 J-1θ (2π)^3 c_z Jδ(ε_0 r-ε_ F) v_k^(i) v_k^(j)τ_k^(j)
       =   σ_0  J∫_0^∞dr r^2 J∫_-1^1dμ(1-μ^2)^1 J-1∫_0^2πdϕ 2πδ(r-r_ F) ṽ_k^(i)ṽ_k^(j)τ̃_k^(j),

where σ_0=g e^2 ρ_0 v_0^2 τ_0, r_ F=ε_ F/ε_0,  and ṽ_k^(i)=v_k^(i)/v_0. Thus, from Eq. (<ref>), we have


    σ_xxσ_0   =   J r_ F^2 2∫_-1^1dμ(1-μ^2)τ̃^(x)(μ), 
    σ_zzσ_0   =   c_z^2 r_ F^2 J J∫_-1^1dμ(1-μ^2)^1 J-1μ^2 τ̃^(z)(μ).


Note that τ_0, v_0, ρ_0 and σ_0 are the density independent normalization constants in units of time, velocity, DOS, and conductivity, respectively. In addition, from the axial symmetry around the k_z-axis, σ_xx=σ_yy.

For the short-range impurities, V_kk' is independent of density. Thus from Eq. (<ref>), ω̃^(i)(μ)∼ε_ F^2 J and τ̃^(i)(μ)∼ε_ F^-2 J at the Fermi energy ε_ F. Note that ε_ F∼ n^J J+2. Therefore we have


    σ_xx   ∼   ε_ F^2-2 J∼ n^2(J-1) J+2, 
    σ_zz   ∼   ε_ F^0 ∼ n^0.



For charged impurities in the strong screening limit, V_kk'∼ q_ TF^-2∼ D^-1(ε_ F)∼ε_ F^-2 J, thus ω̃^(i)(μ)∼ε_ F^2 J-4 J and τ̃^(i)(μ)∼ε_ F^2 J at ε_ F. Therefore we have


    σ_xx   ∼   ε_ F^2+2 J∼ n^2(J+1) J+2, 
    σ_zz   ∼   ε_ F^4 J∼ n^4 J+2.



For charged impurities in the weak screening limit, from V_kk'∼ |k-k'|^-2 and Eq. (<ref>), we expect the potential average on the Fermi surface as V_ F∼ε_ F^-2ζ with 1 J≤ζ≤ 1 (assuming no logarithmic correction), thus ω̃^(i)(μ)∼ε_ F^2 J-4ζ and τ̃^(i)(μ)∼ε_ F^4ζ-2 J at ε_ F. Therefore, we have


    σ_xx   ∼   ε_ F^2+4ζ-2 J∼ n^2(J-1)+4Jζ J+2, 
    σ_zz   ∼   ε_ F^4ζ∼ n^4Jζ J+2.


Here, ζs in σ_xx and σ_zz do not need to be the same, as explained later in this section. Note that ζ=1 J gives the same density exponent corresponding to the strong screening limit.  

For the short-range impurities, it turns out that the relaxation time is independent of polar angles θ. Assuming τ^(i)(μ)=τ^(i) from the beginning, for short-range impurity potential V_kk'=V_ short, Eq. (<ref>) reduces to

    1τ̃^(i)=r^2 J JṼ_ short^2  ∫_-1^1dμ'(1-μ'^2)^1 J-1∫_0^2πdϕ' 2π  F_kk'(1-d_kk'^(i)),

where Ṽ_ short=V_ short/V_0.
Then we find that the relaxation time τ^(i)(ε) at at energy ε=rε_0 is


    1τ̃^(x)(ε)   =   r^2 J 2JṼ_ short^2 B(1 2,1 J)-δ_J1r^2 3Ṽ_ short^2, 
    1τ̃^(z)(ε)   =   r^2 J 2JṼ_ short^2 B(1 2,1 J+1).


From Eq. (<ref>), finally we obtain


    σ_xxσ_0   =   2J r_ F^2  3τ̃^(x)(ε_ F), 
    σ_zzσ_0   =   c_z^2 r_ F^2 J J B(3 2,1 J)  τ̃^(z)(ε_ F)=J c_z^2 Ṽ_ short^2.


Note that σ_xx/σ_0=Ṽ_ short^-2, 16  3π r_ FṼ_ short^-2, 12  B(1 2,1 3) r_ F^4 3Ṽ_ short^-2 for J=1,2,3, respectively, and the obtained analytic expressions are consistent with the density dependence in Eq. (<ref>). From Eq. (<ref>), we find 
σ_xx/σ_zz=1 c_z^2, 8 r_ F 3π c_z^2, 4 r_ F^4 3 B(1 2,1 3) c_z^2 for J=1,2,3, respectively, and the anisotropy between σ_xx and σ_zz increases as the Fermi energy or the carrier density increases.

For charged impurities in the strong screening limit, the impurity potential becomes V_kk'≈ V^ strong_ screen≡4π e^2 κ q_ TF^2, having the same feature of the short-range impurity potential. Thus, the relaxation time is also independent of polar angles and similar analytic expressions can be obtained by replacing Ṽ_ short by Ṽ^ strong_ screen in Eqs. (<ref>) and (<ref>), where Ṽ^ strong_ screen=V^ strong_ screen/V_0=4πα k_0^2/q_ TF^2. Then the relaxation time is given by


    1τ̃^(x)(ε)   =   8π^4 c_z^2 J  g^2 B(1 2,1 J) r^2 J r_ F^-4 J-δ_J14π^4 c_z^2  3 g^2 r^2 r_ F^-4, 
    1τ̃^(z)(ε)   =   16π^4 c_z^2 J  (J+2) g^2 B(1 2,1 J) r^2 J r_ F^-4 J,


thus, in the strong screening limit, we obtain


    σ_xxσ_0   =   2J r_ F^2  3τ̃^(x)(ε_ F), 
    σ_zzσ_0   =   c_z^2 r_ F^2 J J B(3 2,1 J)  τ̃^(z)(ε_ F)=g^2 B^2(1 2,1 J)  16π^4 J r_ F^4 J.


Note that σ_xx/σ_0=g^2 4π^4 c_z^2 r_ F^4, g^2  12π^3 c_z^2 r_ F^3, g^2 B(1 2,1 3) 12π^4 c_z^2 r_ F^8 3 for J=1,2,3, respectively, and the obtained analytic expressions are consistent with the density dependence in Eq. (<ref>). Also note that σ_xx/σ_zz has the same form with that obtained for short-range impurities.





For charged impurities at arbitrary screening, the relaxation time in general depends on polar angles for J>1. In addition, as seen in Fig. <ref> in the main text, the density exponent shows non-monotonic behavior as a function of gα. From Eq. (<ref>), for a given wavevector k=(k_0(r_ Fsinθ)^1 J,0,k_0  c_z r_ Fcosθ) at the Fermi energy, the average of the squared Coulomb potential on the Fermi surface is given by

    <V^2(θ)>_ F=1 2∫_-1^1dcosθ' (1-cos^2θ')^1 J-1∫_0^2πdϕ' 2π |V_kk'|^2.

Then assuming <V^2(θ)>_ F∼ r_ F^-4ζ(θ), we can obtain the angle dependent exponent ζ(θ) with 1 J≤ζ(θ)≤ 1. Figure <ref> shows ζ(θ) for several values of θ=0, π/6, π/2. This angle-dependent power-law gives rise to a significant non-monotonic behavior of τ_z and σ_zz in gα, which originates from the competition between two inverse length scales, q_ TF∼ r_ F^1 J and k_ F^(z)∼ r_ F. Note that the in-plane component of the wavevector k_ F^(∥)∼ r_ F^1 J at the Fermi energy has the same Fermi energy dependence with q_ TF, showing a monotonic-like behavior of τ_x and σ_xx in gα. As gα increases, ζ(θ) eventually  approaches 1/J irrespective of θ, obtained in the strong screening limit.




§ TEMPERATURE DEPENDENCE OF CHEMICAL POTENTIAL AND THOMAS-FERMI WAVEVECTOR IN MULTI-WEYL SEMIMETALS



In this section, we derive the temperature dependent chemical potential and Thomas-Fermi wavevector in a general gapless electron-hole system, and apply the results to m-WSMs. Suppose that a gapless electron-hole system has a DOS given by D(ε) = C_α |ε|^α-1(ε), where C_α is a constant and (ε) is a step function. For a d-dimensional electron gas with an isotropic energy dispersion ε∼ k^J, α=d/J, whereas for m-WSMs, D(ε) ∝ε^2  J from Eq. (<ref>), thus α =2 J+1.

When the temperature is finite, the chemical potential μ deviates from the Fermi energy ε_ F due to the broadening of the Fermi distribution function f^(0)(ε,μ)=[e^β (ε-μ)+ 1]^-1 where β=1 k_ BT. Since the charge carrier density n does not vary under the temperature change, we have

    n=∫_-∞^∞dε D(ε)f^(0)(ε ,μ)
    =∫_0^∞dε D(ε)[f^(0)(ε ,μ)+f^(0)(-ε ,μ)] ≡∫_-∞^ε_ Fdε D(ε).

Then the carrier density measured from the charge neutral point, Δ n≡.n|_μ-.n|_μ=0, is given by

    Δ n= ∫_0^∞dε D(ε)[f^(0)(ε ,μ)-f^(0)(ε ,-μ)] ≡∫_0^ε_ Fdε D(ε).

Here, we used f(-ε,μ)=1-f(ε,-μ).

Before proceeding further, let us consider the following integral:

    ∫_0^∞dx x^α-1 z^-1 e^x+1   =   ∫_0^∞dx x^α-1 z e^-x 1+ze^-x =-∫_0^∞dx x^α-1∑_n=1^∞ (-z)^n e^-nx
       =^t=nx   [∫_0^∞ dt  t^n-1e^-t][-∑_n=1^∞(-z)^n  n^α]=Γ(α) F_α(z),

where Γ(α)=∫_0^∞ dt  t^α-1e^-t is the gamma function and F_α(z)=-∑_n=1^∞(-z)^n  n^α. Note that Γ(α)=(α-1)Γ(α-1) with Γ(1)=1 and Γ(1/2)=√(π), and F_α(z)=z∂∂ zF_α+1(z).

Using the above result, we obtain

    Δ n=C_α (k_ B T)^αΓ(α) [F_α(z)-F_α(z^-1)]
    =C_α/αε_ F^α,

where z=e^βμ, which is called the fugacity. Thus, finally we have

    F_α(z)-F_α(z^-1) = (βε_ F)^α/Γ(α+1).

By solving the above equation with respect to z for a given T, we can obtain the chemical potential μ=k_ B T ln z.

At low temperatures, βμ→∞ thus z→∞. Note that from the Sommerfeld expansion <cit.>

    lim_z→∞∫_0^∞dxH(x)/z^-1e^x+1≈∫_0^βμdxH(x) + π^2/6∂ H(βμ)/∂ x ,

where H(x) is a function which diverges no more rapidly than a polynomial as x→∞.

Then for H(x) = x^α-1 and using Eq. (<ref>), Eq. (<ref>) becomes

    lim_z→∞ F_α(z)≈(βμ)^αΓ(α+1)[1+π^2 6α(α-1)(βμ)^2],

whereas F_α(z^-1)=z^-1-z^-2 2^α+⋯ vanishes as z→∞.

Thus, we can obtain the low-temperature correction as

    μ/ε_ F≈ 1- π^2/6(α-1) (T/T_ F)^2,

where T_ F=ε_ F/k_ B is the Fermi temperature.

At high temperatures, βμ→ 0 due to the finite carrier densities, thus z→ 1. From z≈ 1+βμ+1 2(βμ)^2 for |βμ|≪ 1,

    lim_z→ 1 F_α(z)≈η(α)+η(α-1)βμ + 1/2η(α-2)(βμ)^2,

where η(α)=F_α(1) is the Dirichlet eta function <cit.>. 

Thus, we have F_α(z) - F_α(z^-1) ≈ 2η(α-1)βμ, and obtain the following high-temperature asymptotic form:

    μ/ε_ F≈1/2η(α-1)Γ(α+1)(T_ F/T)^α-1 .


For m-WSMs, α =2 J+1 and we obtain

    μ/ε_ F   =   
    1- π^2/3J(T/T_ F)^2     (T≪ T_ F), 
    1 2η(2 J)Γ(2+2 J)(T/T_ F)^-2 J    (T≫ T_ F).


Next, consider the temperature dependent Thomas-Fermi wavevector q_ TF(T). Note that in 3D, q_ TF^2(0)=4π e^2κ D(ε_ F) and at finite T, q_ TF^2(T)=4π e^2κ∂ n ∂μ. Thus we have

    q_TF^2(T) /q_TF^2(0)  = ∂ε_ F/∂μ =  Γ(α)/(βε_ F)^α-1[F_α-1(z) +F_α-1(z^-1)].

For a given T, the chemical potential (or equivalently fugacity z) is calculated using the density invariance in Eq. (<ref>), and then q_TF(T) is obtained from the above relation.

At low temperatures, μ(T) is given by Eq. (<ref>), thus

    q_TF^2(T) /q_TF^2(0)    ≈   Γ(α)/(βε_ F)^α-1[(βμ)^α-1/Γ(α)(1+π^2/6(α-1)(α-2)/(βμ)^2) + 0(z^-1-z^-2/2^α-1)] 
       ≈   μ/ε_ F+π^2/6(α-1)(α-2)/(βε_ F)^2≈ 1-π^2 6(α-1)(T/T_ F)^2.

At high temperatures, μ(T) is given by Eq. (<ref>), thus

    q_TF^2(T) /q_TF^2(0)    ≈   Γ(α)/(βε_ F)^α-1[2η(α-1)+η(α-3)(βμ)^2] 
       ≈    2η(α-1) Γ(α)(T/T_ F)^α-1.



For m-WSMs, we find

    q_TF(T) /q_TF(0)    =    
    1- π^2/6J(T/T_ F)^2      (T≪ T_ F), 
    √(2η(2 J)Γ(1+2 J))(T/T_ F)^1 J    (T≫ T_ F),

where q_ TF(0)=q_ TF is given by Eq. (<ref>).

Figure <ref> shows the temperature dependence of the chemical potential and Thomas-Fermi wavevector in m-WSMs.









§ TEMPERATURE DEPENDENCE OF DC CONDUCTIVITY IN MULTI-WEYL SEMIMETALS



From Eq. (<ref>) in the main text, we can easily generalize the conductivity tensor at zero temperature to that at finite temperature. For f^(0)(ε)=[z^-1e^βε+ 1]^-1, S^(0)(ε)=-∂ f^(0)(ε)∂ε=β f^(0)(ε) (1-f^(0)(ε))=β z^-1 e^βε (z^-1 e^βε+ 1)^2.
Then the conductivity tensor at finite temperature is given by

    σ_ij(T)   =   g e^2∫d^3 k (2π)^3(-∂ f^(0)(ε_k)∂ε) v_k^(i) v_k^(j)τ_k^(j)
       =   g e^2 ∫_0^∞dr ∫_0^πdθ∫_0^2πdϕk_0^3 r^2 Jsin^2 J-1θ (2π)^3 c_z Jβ z^-1 e^βε_0 r (z^-1 e^βε_0 r+ 1)^2 v_k^(i) v_k^(j)τ_k^(j)
       =   σ_0  J∫_0^∞dr r^2 J∫_-1^1dμ(1-μ^2)^1 J-1∫_0^2πdϕ 2πβε_0 z^-1 e^βε_0 r (z^-1 e^βε_0 r+ 1)^2ṽ_k^(i)ṽ_k^(j)τ̃_k^(j).


Thus from Eq. (<ref>), we have


    σ_xx(T)   =   σ_0 J 2∫_0^∞ dr  r^2 βε_0 z^-1 e^βε_0 r (z^-1 e^βε_0 r+ 1)^2∫_-1^1dμ(1-μ^2)τ̃^(x)(μ), 
    σ_zz(T)   =   σ_0 c_z^2  J∫_0^∞ dr  r^2 Jβε_0 z^-1 e^βε_0 r (z^-1 e^βε_0 r+ 1)^2∫_-1^1dμ(1-μ^2)^1 J-1μ^2 τ̃^(z)(μ).



To derive the asymptotic behaviors of σ_ii(T)/σ_ii(0) at low and high temperatures, let us rewrite Eq. (<ref>) in the main text, in the following energy integral form:

    σ_ii(T)   =   g e^2 I ∫_0^∞ dε(-∂ f^(0)(ε)∂ε) D(ε) [v^(i)(ε)]^2 τ^(i)(ε,T),

where I is a factor from the angular integration. Note that the factor I will be canceled by σ_ii(0) later. Assuming that τ^(i)(ε, T) can be decomposed as

    τ^(i)(ε, T) = τ^(i)(ε) g^(i)(T/T_ F),

where g^(i)(T/T_ F) is the energy-independent correction term from the screening effect with g^(i)(0)≡1, we can separate the contributions from the energy averaging over the Fermi distribution and the temperature dependent screening. Suppose D(ε) ∝ε^α-1, v^(i)(ε) ∝ε^ν, and τ^(i)(ε) ∝ε^γ. Then we can express σ_ii(T) as

    σ_ii(T)   =   C ∫_0^∞ dε(-∂ f^(0)(ε)∂ε) ε^α-1+2ν+γ g^(i)(T/T_ F) 
       =    C (k_ BT)^δΓ(δ+1)F_δ(z)g (T/T_ F),

where C is a constant and δ≡α-1+2ν+γ. Note that Eq. (<ref>) reduces to σ_ii(0)= C ε_ F^δ at zero temperature.
Therefore, after eliminating C, we have

    σ_ii(T)σ_ii(0)= Γ(δ+1)F_δ(z)/(βε_ F)^δ g^(i)(T/T_ F).

For short-range impurities, g^(i)(T T_ F)=1.
For charged impurities at low temperatures, from the form of the low-temperature correction for the Thomas-Fermi wavevector in Eq. (<ref>), we expect

    g^(i)(T T_ F)≈ 1-A^(i)(T T_ F)^2.

Note that A^(i) depends on the screening strength, and in the strong screening limit, from Eq. (<ref>) we have A^(i)=2π^2 3J.
At high temperatures, however, τ^(i)(ε,T) cannot be simply decomposed as Eq. (<ref>). The energy averaging typically dominates over the screening contribution <cit.>, and the screening correction g(T T_ F) only gives a constant factor without changing the temperature power. Assuming g^(i)(T T_ F)≈ 1 at high temperatures, then in the low and high temperature limits, we have

    σ_ii(T)σ_ii(0)   =   
    1+[π^2/6(δ-α)δ-A^(i)] (T/T_ F)^2      (T≪ T_ F), 
    Γ(δ+1) η(δ)(T/T_ F)^δ    (T≫ T_ F).







Now, consider m-WSM with α=2/J+1.
For short-range impurities, g^(i)(T/T_ F)=1, and from the energy dependence of the relaxation time in Eq. (<ref>), γ=-2/J. Thus, we find


    σ_xx(T)σ_xx(0)   =   
    1+ π^2/3(J-1/J)(J-4/J)(T T_ F)^2      (T≪ T_ F), 
    Γ(3-2/J)η(2-2/J)(T T_ F)^2-2/J    (T≫ T_ F),
    
    σ_zz(T)σ_zz(0)   =   
    1-e^-T_ F/T    (T≪ T_ F), 
    1/2+ 1/8η(2 J)Γ(2 J+2)(T/T_ F)^-2+J J    (T≫ T_ F).



For charged impurities in the strong screening limit, from Eq. (<ref>), τ^(i)(ε)∼ε^-2 J thus γ=-2/J at low temperatures, whereas at high temperatures τ^(i)(ε)∼ε^-2 J+4 J because thermally induced charge carriers participate in transport giving γ=2/J. Combining the temperature dependent screening correction with A^(i)=2π^2 3J at low temperatures, we find


    σ_xx(T)σ_xx(0)   =   
    1+ π^2/3(J^2-7J+4/J^2)(T T_ F)^2       (T≪ T_ F), 
    Γ(3+2/J)η(2+2/J)(T T_ F)^2+2/J    (T≫ T_ F),
    
    σ_zz(T)σ_zz(0)   =   
    1-2π^2/3J(T T_ F)^2     (T≪ T_ F), 
    Γ(1+4/J)η(4/J)(T T_ F)^4/J    (T≫ T_ F).



For charged impurities at arbitrary screening, from the Fermi energy dependence of the relaxation time discussed in Sec. <ref>, γ=4ζ-2/J with 1 J≤ζ≤ 1 at high temperatures. Thus, we can express the low and high temperature asymptotic forms as


    σ_xx(T)σ_xx(0)   =   
    1+ C_xx(T T_ F)^2     (T≪ T_ F), 
    Γ(3+4ζ-2 J)ζ(2+4ζ-2 J)(T T_ F)^2+4ζ-2 J    (T≫ T_ F),
    
    σ_zz(T)σ_zz(0)   =   
    1+ C_zz(T T_ F)^2     (T≪ T_ F), 
    Γ(1+4ζ)η(4ζ)(T T_ F)^4ζ    (T≫ T_ F).


As explained in Sec. <ref>, ζs in σ_xx and σ_zz do not need to be the same.
Note that ζ=1 J in Eq. (<ref>) gives the same high-temperature exponent corresponding to the strong screening limit in Eq. (<ref>), and the temperature dependent conductivity has the high-temperature asymptotic form given by Eq. (<ref>) with ζ which varies within 1 J≤ζ≤ 1 and approaches 1 J in the strong screening limit.

Figure <ref> shows the evolution of the low-temperature coefficients C_xx and C_zz in Eq. (<ref>) for charged impurities as a function of the screening strength gα.
Above a critical gα, C_xx and C_zz become negative, thus the conductivity decreases with temperature, showing a metallic behavior. As gα increases further, the low-temperature coefficients eventually approach C_xx=π^2/3(J^2-7J+4/J^2) and C_zz=-2π^2/3J, as obtained in Eq. (<ref>). The non-monotonic behavior in the low-temperature coefficients C_zz as a function of gα for J>1 originates from the angle-dependent power-law in the relaxation time, similarly as shown in Fig. <ref> in the main text.



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DasSarma2015_SM
S. Das Sarma and E. H. Hwang,
Charge transport in gapless electron-hole systems with arbitrary band dispersion,
Phys. Rev. B 91, 195104 (2015).





