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Scientific Reports
2045-2322
Nature Publishing Group UK London

68615
10.1038/s41598-024-68615-0
Article
Electric, thermal, and thermoelectric magnetoconductivity for Weyl/multi-Weyl semimetals in planar Hall set-ups induced by the combined effects of topology and strain
Medel Leonardo leonardo.medel@correo.nucleares.unam.mx

1
Ghosh Rahul 2
Martín-Ruiz Alberto 1
Mandal Ipsita 23
1 https://ror.org/01tmp8f25 grid.9486.3 0000 0001 2159 0001 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, 04510 Ciudad de México, México
2 Department of Physics, Shiv Nadar Institution of Eminence (SNIoE), Gautam Buddha Nagar, Uttar Pradesh 201314 India
3 https://ror.org/0245cg223 grid.5963.9 0000 0004 0491 7203 Freiburg Institute for Advanced Studies (FRIAS), University of Freiburg, D-79104 Freiburg, Germany
13 9 2024
13 9 2024
2024
14 213904 6 2024
25 7 2024
© The Author(s) 2024
2024
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We continue our investigation of the response tensors in planar Hall (or planar thermal Hall) configurations where a three-dimensional Weyl/multi-Weyl semimetal is subjected to the combined influence of an electric field E (and/or temperature gradient ∇rT) and an effective magnetic field Bχ, generalizing the considerations of Phys. Rev. B 108 (2023) 155132 and Physica E 159 (2024) 115914. The electromagnetic fields are oriented at a generic angle with respect to each other, thus leading to the possibility of having collinear components, which do not arise in a Hall set-up. The net effective magnetic field Bχ consists of two parts—(a) an actual/physical magnetic field B applied externally; and (b) an emergent magnetic field B5 which quantifies the elastic deformations of the sample. B5 is an axial pseudomagnetic field because it couples to conjugate nodal points with opposite chiralities with opposite signs. Using a semiclassical Boltzmann formalism, we derive the generic expressions for the response tensors, including the effects of the Berry curvature (BC) and the orbital magnetic moment (OMM), which arise due to a nontrivial topology of the bandstructures. We elucidate the interplay of the BC-only and the OMM-dependent parts in the longitudinal and transverse (or Hall) components of the electric, thermal, and thermoelectric response tensors. Especially, for the co-planar transverse components of the response tensors, the OMM part acts exclusively in opposition (sync) with the BC-only part for the Weyl (multi-Weyl) semimetals.

Subject terms

Electronic properties and materials
Topological matter
CONACyT (México)CF- 428214 DGAPA- UNAMAG100224 European Union’s Horizon 2020 research and innovation programmeMarie Sklodowska-Curie grant agreement number 754340 Mandal Ipsita issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

There has been an incredible amount of research work focussing on the investigations of the transport properties of semimetals, which are systems harbouring band-crossing points in the Brillouin zone (BZ). Two or more bands cross at the nodal points where the densities of states go to zero. Among the three-dimensional (3d) semimetals with twofold nodal points, the well-known examples include the Weyl semimetals (WSMs)1,2 and the multi-Weyl semimetals (mWSMs)3–5, whose bandstructures exhibit nontrivial topology quantified by the Berry phase. The nodal points for both the WSMs and the mWSMs are protected by the point-group symmetries of the crystal lattice4. In the language of the Berry curvature (BC) flux, each nodal point acts as a source or sink in the momentum space and, thus, acts as an analogue of the elusive magnetic monopole. The value of the monopole charge is equal to the Chern number arising from the Berry connection. A consequence of the Nielson-Ninomiya theorem6, applicable for systems in an odd number of spatial dimensions, the nodal points exist in pairs, with each pair carrying Chern numbers ±J. Thus, the pair acts as a source and a sink of the BC flux. Since the sign of the monopole charge is called the chirality χ of the assoiciated node, the two nodes in a pair are of opposite chiralities (i.e., χ=±1). The values of J for Weyl (e.g., TaAs7–9 and HgTe-class materials10), double-Weyl (e.g., HgCr2Se411 and SrSi212,13), and triple-Weyl nodes (e.g., transition-metal monochalcogenides14) are equal to one, two, and three, respectively.

Suppose we consider an experimental set-up with a WSM/mWSM semimetal subjected to an external uniform electric field E along the x-axis and a uniform external magnetic field B along the y-axis. Since B is perpendicular to E, a potential difference (known as the Hall voltage) will be generated along the z-axis. This phenomenon is the well-known Hall effect. However, if we apply B making an angle θ with E, where θ≠π/2or3π/2, the conventional Hall voltage induced from the Lorentz force is zero along the y-axis. Nonetheless, due to nontrivial Chern numbers, a voltage difference VPH appears along this direction, as shown in Fig. 1. This is known as the planar Hall effect (PHE), arising due to the chiral anomaly15–21. The chiral anomaly refers to the phenomenon of charge pumping from one node to its conjugate when E·B≠0, originating from a local non-conservation of electric charge in the vicinity of an individual node, with the rate of change of the number density of chiral quasiparticles being proportional to JE·B22,23. The associated transport coefficients, related to this set-up, are referred to as the longitudinal magnetoconductivity (LMC) and the planar Hall conductivity (PHC), which depend on the value of θ. In an analogous set-up, we observe the planar thermal Hall effect (PTHE), where we replace the external electric field by (or add) an external temperature gradient ∇rT. In this scenario, too, a potential difference is induced along the y-axis due to the chiral anomaly21,24 [cf. Fig. 1], with the response coefficients known as the longitudinal thermoelectric coefficient (LTEC) and transverse thermoelectric coefficient (TTEC). There has been a tremendous amount of efforts to determine the behaviour of these response tensors25–40.Figure 1 Schematics showing the planar Hall (or planar thermal Hall) experimental set-up, where the sample is subjected to an external electric field Ex^ (and/or a temperature gradient ∂xTx^). An external magnetic field B is applied such that it makes an angle θ with the existing electric field (and/or the temperature gradient). In addition, the sample is considered to be under the influence of a mechanical strain, whose effect is incorporated via an artificial chiral gauge field B5, making an angle θ5 with the x-axis. The resulting planar Hall (or planar thermal Hall) voltage, generated along the y-axis, is indicated by the symbol VPH.

In a planar Hall (or thermal Hall) set-up, if a semimetal is subjected to mechanical strain, it induces elastic deformations of the material. The elastic deformations couple to the electronic degrees of freedom (i.e., quasiparticles) in such a way that they can be modelled as pseudogauge fields in the semimetals37,41–47. The form of these elastic gauge fields shows that they couple to the quasiparticles of the Weyl fermions with opposite chiralities with opposite signs37,44–46,48,49. Due to the chiral nature of the coupling between the emergent vector fields and the itinerant fermionic carriers, this provides an example of axial gauge fields in three dimensions. This is to be contrasted with the actual electromagnetic fields, which couple to all the nodes with the same sign. While a uniform pseudomagnetic field B5 can be generated when a WSM/mWSM nanowire is put under torsion, a pseudoelectric field E5 appears on dynamically stretching and compressing the crystal along an axis (which can be achieved, for example, by driving longitudinal sound waves)46. Direct evidence of the generation of such pseudoelectromagnetic fields in doped semimetals has been obtained in experiments50. In an earlier work by the two of us38, the response tensors have been computed for WSMs and mWSMs, but neglecting the orbital magnetic moment (OMM)51,52, which is another artifact of a nontrivial topology in the bandstructures. Although the effects of OMM were included in the computations of Ref.37, only the electric conductivity at zero temperature was studied. Hence, in this paper, we derive all the relevant electric, thermal, and thermoelectric response tensors, associated with the planar Hall and planar thermal set-ups, which constitute a complete description incorporating both the BC and the OMM.

As shown in Fig. 1, the co-planar E and Bχ=B+χB5 set-ups considered here consist of a nonzero B5 and a nonzero B in the xy-plane. These two parts of the effective magnetic field Bχ are oriented at the angles θ and θ5, respectively, with respect to E (or ∇rT) applied along the x-axis. In other words, B=Bcosθ,sinθ,0 and B5=B5cosθ5,sinθ5,0, where B≡|B| and B5≡|B5|. We consider the weak magnetic field limit with low values of B and B5, such that the formation of the Landau levels can be ignored, and the magnetoelectric, magnetothermal, and magnetothermoelectric response can be derived using the semiclassical Boltzmann formalism. The paper is organized as follows: In Sect. "Model", we describe the low-energy effective Hamiltonians for the WSMs and mWSMs. In Sects. "Magnetoelectric conductivity", "Magnetothermoelectric conductivity", and "Magnetothermal coefficient", we show the explicit expressions of the in-plane components of the response tensors, and discuss their behaviour in some relevant parameter regimes. Finally, we conclude with a summary and outlook in Sect. "Summary and future perspectives". The appendices are devoted to explaining the details of the intermediate steps used to derive the final expressions in the main text.

Model

In the vicinity of a nodal point with chirality χ and Berry monopole charge of magnitude J, the low-energy effective continuum Hamiltonian is given by3,4,141 Hχ(k)=dχ(k)·σ,k⊥=kx2+ky2,ϕk=arctan(kykx),αJ=v⊥k0J-1,dχ(k)=αJk⊥Jcos(Jϕk),αJk⊥Jsin(Jϕk),χvzkz,

where σ={σx,σy,σz} is the vector operator consisting of the three Pauli matrices, σ0 is the 2×2 identity matrix, χ∈{1,-1} denotes the chirality of the node, and vz (v⊥) is the Fermi velocity along the z-direction (xy-plane). The parameter k0 has the dimension of momentum, whose value depends on the microscopic details of the material in consideration. The eigenvalues of the Hamiltonian are given by2 εχ,s(k)=(-1)s+1ϵk,s∈{1,2},ϵk=αJ2k⊥2J+vz2kz2,

where the value 1 (2) for s represents the conduction (valence) band [cf. Fig. 2]. We note that we recover the linear and isotropic nature of a WSM by setting J=1 and α1=vz.Figure 2 Schematic dispersion of a single node in a (a) Weyl, (b) double-Weyl, and (c) triple-Weyl semimetal, plotted against the kzkx-plane. The double(triple)-Weyl node shows an anisotropic hybrid dispersion with a quadratic(cubic)-in-momentum dependence along the kx-direction. In order to pinpoint the direction-dependent features, the projections of the dispersion along the respective momentum axes are also shown.

The band velocity of the chiral quasiparticles is given by3 vχ,s(0)(k)≡∇kεχ,s(k)=-(-1)sϵkJαJ2k⊥2J-2kx,JαJ2k⊥2J-2ky,vz2kz.

The Berry curvature (BC) and the orbital magnetic moment (OMM), associated with the sth band, are expressed by51–544 Ωχ,s(k)=i⟨∇kψsχ(k)|×|∇kψsχ(k)⟩⇒Ωχ,si(k)=(-1)sϵjli4|dχ(k)|3dχ(k)·∂kjdχ(k)×∂kldχ(k)andmχ,s(k)=-ie2⟨∇kψs(k)|×H(k)-Eχ,s(k)|∇kψs(k)⟩⇒mχ,si(k)=eϵjli4|dχ(k)|2dχ(k)·∂kjdχ(k)×∂kldχ(k),

respectively, where the indices i, j, and l ∈{x,y,z}, and are used to denote the Cartesian components of the 3d vectors and tensors. The symbol |ψsχ(k)⟩ denotes the normalized eigenvector corresponding to the band labelled by s, with {|ψ1χ⟩,{|ψ2χ⟩} forming an orthonornomal set for each node.

On evaluating the expressions in Eq. (4), using Eq. (1), we get5 Ωχ,s(k)=χ(-1)sJvzαJ2k⊥2J-22ϵk3kx,ky,Jkz,mχ,s(k)=-χeJvzαJ2k⊥2J-22ϵk2kx,ky,Jkz.

From these expressions, we immediately observe the identity6 mχ,s(k)=-(-1)seϵkΩχ,s(k).

While the BC changes sign with s, the OMM does not.

In this paper, we will take a positive value of the chemical potential μ, such that it cuts the conduction bands with s=1. Henceforth, we will use the notations εχ,1=εχ, vχ,1(0)=v(0) (since it is independent of χ), Ωχ,1=Ωχ, and mχ,1=mχ, in order to avoid cluttering. The ranges of the values of the parameters that we will use in our computations are shown in Table 1. Table 1 The values of the various parameters which we have used in plotting the transport coefficients are tabulated here. Since αJ=v⊥/k0J-1, we get α1=vz, α2=v⊥/k0, and α3=v⊥/k02=α22/α1. In terms of natural units, we need to set ħ=c=kB=1, and 4πϵ0=137. In our plots, we have used v⊥=vz (from the table entry), leading to α2=3.9×10-5 eV-1 and α3=2.298×10-6 eV-2. For J=2 and J=3, v⊥ has been set equal to vz for the sake of simplicity, while the isotropic dispersion for J=1 has v⊥=vz automatically.

Parameter	SI units	Natural units	
vz from Ref.55	15×105 m s-1	0.005	
τ from Ref.56	10-13s	152 eV-1	
T from Ref.20	10-100K	8.617×10-4-8.617×10-3 eV	
B and B5 from Ref.48	0-10 Tesla	0-2000eV2	
μ from Refs.20,55	1.6×10-21-1.6×10-20 J	0.01-0.1 eV	

Response tensors using the Boltzmann formalism

Using the semiclassical Boltzmann formalism57,58, the transport coefficients can be determined in the weak |Bχ| limit, which applies to the regime of small cyclotron frequency, implying that the Landau-level quantization can be ignored. The detailed steps can be found in the Appendix A of Ref.38, which we do not repeat here for the sake of brevity. Moreover, we work with the relation-time approximation for the collision integral, taking a simplistic momentum-independent relaxation time τ. Furthermore, we assume that intranode scatterings are dominant over the internode scattering processes, such that τ corresponds only to the former.

Let the contributions to the average electric and thermal current densities from the quasiparticles, associated with the node of chirality χ, be Jχ and Jth,χ, respectively. The response matrix, which relates the resulting generalized currents to the driving electric potential gradient and/or temperature gradient, can be expressed as7 JiχJith,χ=∑jσijχαijχTαijχℓijχEj-∂jT.

Here, σijχ and αijχ represent the components of the magnetoelectric conductivity tensor (σχ) and the magnetothermoelectric conductivity tensor (αχ), respectively. While αχ determines the Peltier (Πχ), Seebeck (S), and Nernst coefficients, ℓχ is the linear response tensor relating the heat current density to the temperature gradient, at a vanishing electric field. Sχ , Πχ, and the magnetothermal conductivity tensor κχ (which provides the coefficients between the heat current density and the temperature gradient at vanishing electric current) can be extracted from the coefficients on the right-hand-side of Eq. (7), via the following relations57,58:8 Sijχ=∑a′σχii′-1αi′jχ,Πijχ=T∑i′αii′χσχi′j-1,κijχ=ℓijχ-T∑i′,j′αii′χσχi′j′-1αj′jχ.

In order to include the effects from the OMM and the BC, we first define the quantitites9 Eχ(k)=εχ(k)+εχ(m)(k),εχ(m)(k)=-Bχ·mχ(k),vχ(k)≡∇kEχ(k)=v(0)(k)+vχ(m)(k),vχ(m)(k)=∇kεχ(m)(k),Dχ=1+eBχ·Ωχ(k)-1,

where εχ(m)(k) is the Zeeman-like correction to the energy due to the OMM, vχ(k) is the modified band velocity of the Bloch electrons after including εχ(m)(k), and Dχ is the modification factor of the phase space volume element due to a nonzero BC. The modification of the effective Fermi surface, on including the correction εχ(m)(k), is shown schematically in Fig. 3.Figure 3 Schematics of the Fermi surfaces for one node of a (a) WSM and (b) double-Weyl semimetal, without and with the OMM-correction for the effective energy dispersion. Here we have taken the effective magnetic field to be directed purely along the x-axis.

Our weak-magnetic-field limit implies that10 eBχ·Ωχ≪1.

In our calculations, we keep terms upto O(|Bχ|2) and, thus, use11 Dχ=1-eBχ·Ωχ+e2Bχ·Ωχ2+O(|Bχ|3).

Also, the condition in Eq. (10) implies that |εχ(m)(k)| is small compared to εχ(k):12 |Bχ·mχ|=εχeBχ·Ωχ≪εχ.

This means that the Fermi-Dirac distribution can also be power expanded up to quadratic order in the magnetic field, as follows:13 f0(Eχ)=f0(εχ)+εχ(m)f0′(εχ)+12εχm2f0″(εχ)+O(|Bχ|3),

where the prime indicates derivative with respect to the energy argument of f0.

The general expression for the magnetoelectric conductivity tensor for an isolated node of chirality χ, contributed by the conduction band, is given by14 σijχ=-e2τ∫d3k(2π)3Dχvχi+e(vχ·Ωχ)Bχivχj+e(vχ·Ωχ)Bχj∂f0(Eχ)∂Eχ,

where we do not include the parts coming from the “intrinsic anomalous Hall” effect and the so-called Lorentz-force contribution. This is because of the following reasons: The intrinsic anomalous Hall term is given by 15 σijAH,χ=-e2ϵijl∫d3k(2π)3Ωχl(k)f0(Eχ)=σijAH(0),χ+σijAH(1),χ+σijAH(2),χ+O(|Bχ|3),σijAH(0),χ=-e2ϵijl∫d3k(2π)3Ωχl(k)f0(εχ),σijAH(1),χ=-e2ϵijl∫d3k(2π)3Ωχl(k)εχ(m)f0′(εχ),σijAH(2),χ=-e2ϵijl2∫d3k(2π)3Ωχl(k)εχm2f0″(εχ),

whose diagonal components (i.e., σiiAH,χ) are automatically zero. The first term, σijAH(0),χ, is Bχ-independent and vanishes identically. The nonzero OMM generates Bχ-dependent terms. However, for our configuration consisting of E- and Bχ-components lying in the xy-plane, we have 16 σyxAH(1),χ=σxyAH(1),χ∝∫-∞∞kzdkz∫0∞dk⊥k⊥4J-2εχ5f0′(εχ)∫02πdϕBχxcosϕ+Bχysinϕ=0,andσyxAH(2),χ=σyxAH(2),χ=0.

Only the transverse out-of-plane components are nonzero, viz. 17 σzxAH(1),χ(μχ)=e3JvzBχy24π2∫0∞dεχf0′(εχ)εχ=-e3JvzBχy24π2μχ1+π23β2μχ2andσzyAH(1),χ(μχ)=e3JvzBχx24π2μχ1+π23β2μχ2[if we have a nonzeroy-component ofE].

We note that σzxAH(2),χ=σzyAH(2),χ=0. Since we are focussing on the in-plane components of the response tensors, we will not discuss further the behaviour of these nonzero out-of-plane components.

The Lorentz-force part shows a behaviour analogous to the intrinsic anomalous Hall part described above, with vanishing in-plane components (see Ref.40 for generic arguments stemming from symmetry considerations).

The expression for σijχ, shown above, includes the effects of the BC and the OMM. Here, f0 is the equilibrium Fermi-Dirac distribution at temperature T=1/β and chemical potential μχ. As discussed in Refs.37,38, while a purely physical magnetic field B gives a quadratic-dependence of the response on the overall magnetic field, inclusion of a nonzero axial part B5 opens up the possibility of generating linear and parabolic behaviour of the response tensors.

Analogous to Eq. (14), we have the general expressions2418 αijχ=eτ∫d3k(2π)3Dχvχi+e(vχ·Ωχ)Bχivχj+e(vχ·Ωχ)BχjEχ-μT∂f0(Eχ)∂Eχ,

and19 ℓijχ=-τ∫d3k(2π)3Dχvχi+e(vχ·Ωχ)Bχivχj+e(vχ·Ωχ)Bχj(Eχ-μ)2T∂f0(Eχ)∂Eχ,

respectively. Since ℓχ determines the first term in the magnetothermal conductivity tensor κχ, we will often loosely refer to ℓχ itself as the magnetothermal coefficient.

Magnetoelectric conductivity

Working in the weak-in-magnetic-field limit, which allows expansions in the components of Bχ, the terms in Eq. (14) are disentangled into a sum of three terms, having distinct origins, as follows:20 σijχ=σij0,χ+σijΩ,χ+σijm,χ.

Here,21 σij0,χ=-e2τ∫d3k(2π)3vχi(0)vχj(0)f0′(εχ),

is the conductivity surviving in the absence of a magnetic field (i.e., for Bχ=0),22 σijΩ,χ=-e4τ∫d3k(2π)3QχiQχjf0′(εχ),Qχ=Ωχ×vχ(0)×Bχ,

is the contribution arising solely from the BC, and23 σijm,χ=2e3τ∫d3k(2π)3Qχivχj(m)+εχ(m)e∇k·Tχij+εχ(m)2Bχ·Vχij∂∂εχf0′(εχ),

24 whereTχij=eΩχBχivχj(0)+12e^ivχj(m)andVχij=Ωχvχi(0)vχj(0)-mχ2e∂kivχj(0),

represents the contribution which goes to zero if OMM is set to zero. The symbol e^i represents the unit vector along the Cartesian coordinate axis labelled by i.

The longitudinal and transverse components of the magnetoelectric conductivity tensor σχ (i.e., the LMC and the PHC) are computed from the starting expression shown in Eq. (14). The details of the intermediate steps are given in Appendices B and C. Before delving into the investigation of the behaviour of these response coefficients in the subsequent subsections, first let us analyze the relation between the current and the axial electromagnetic fields.

For the quasiparticles with chirality χ, we have the electric current contribution25 Jiχ(μ)=σijχ(μ)Eχj,

where σijχ is given by Eq. (14). For two conjugate nodes with chemical potential values μ+ and μ- (as shown schematically in Fig. 4 for WSMs), we define the total and axial currents as26 J(μ+,μ-)=∑χ=±1Jχ(μχ)=J+(μ+)+J-(μ-)andJ5(μ+,μ-)=∑χ=±1χJχ(μχ)=J+(μ+)-J-(μ+),

respectively. These suggest to introduce analogous expressions for the conductivity tensors as27 σij(μ+,μ-)=∑χ=±1σijχ(μχ)andσ5ij=∑χ=±1χσijχ(μχ).

Figure 4 A pair of conjugate Weyl nodes with the corresponding chemical potentials tuned to two different values, μ+ and μ-.

Using Eq. (27), we find that28 Ji(μ+,μ-)=σij+(μ+)E+j+σij-(μ-)E-j=σij(μ+,μ-)Ej+σ5ij(μ+,μ-)E5jandJ5i(μ+,μ-)=σij+(μ+)E+j-σij-(μ-)E-j=σ5ij(μ+,μ-)Ej+σij(μ+,μ-)E5j,

where E5 is an axial pseudoelectric field, which can be generated artificially (as explained in the introduction). For the node with chirality χ, analogous to Bχ, the physical E and the axial E5 add up to give the effective electric field Eχ=E+χE5, reflecting the dependence on χ. In this paper, we deal with the case where E5=0.

From the above expressions for the total and axial currents, we now discuss the behaviour of the total and axial LMC and PHC as functions of θ, which is the angle between E and B, with the former chosen to be directed along the x-axis [cf. Fig. 1]. For the illustration of the behaviour of the response, we define29 Σij(Bχ)=σij(μ+,μ-)-σij(μ+,μ-)|Bχ=0andΣ5ij(Bχ)=σ5ij(μ+,μ-)-σ5ij(μ+,μ-)|Bχ=0

for the total and axial conductivity tensor components, respectively, after subtracting off the Bχ-independent parts. We denote the parts connected with σijΩ,χ and σijm,χ as (ΣijΩ,Σ5ijΩ) and (Σijm,Σ5ijm), respectively. Therefore, if the OMM is not considered, each of (Σijm,Σ5ijm) goes to zero.

Longitudinal magnetoconductivity

Using the explicit expressions derived in Appendix B, we have30 σxxχ(μχ)=σxx0,χ(μχ)+σxxΩ,χ(μχ)+σxxm,χ(μχ),

where31 σxx0,χ(μχ)=e2τJ6π2vzΛ2(μχ),σxxΩ,χ(μχ)=e4τvzαJ2JΛ-2J(μχ)128π32Γ(2-1J)Γ(92-1J)gxbc(J)Bχx2+gybc(J)Bχy2,σxxm,χ(μχ)=e4τvzαJ2JΛ-2J(μχ)Γ(2-1J)128π32Γ(92-1J)gxm(J)Bχx2+gym(J)Bχy2,

and32 gxbc(J)=J32J2-19J+3,gybc(J)=J3J-12J-1,gxm(J)=37J4-100J3+74J2-21J+2J,gym(J)=3J4-12J3-4J2+9J-2J.

From Fig. 5, we find that (1) gxbc(J) and gybc(J) are positive for all J-values; (2) gxm(J) is negative for J=1 and positive for J=2,3; (3) gym(J) is negative for all J-values. For J=1, the OMM acts in opposition to the BC-only term for the Bχx-part, and reduces the overall response. On the other hand, for the mWSMs, the OMM adds up to the BC-only term for the Bχx-part, thus increasing the overall response. For the Bχy-part, the OMM and the BC-only parts always have opposite signs and, thus, tend to reduce the overall response’s magnitude. Therefore, for a WSM, the OMM part always reduces the value of σxxχ by adding a negative contribution.Figure 5 Comparison of the values of the functions defined in Eqs. (32) and (35) for J=1,2,3.

In order to estimate the effects of the OMM (which was neglected in many earlier works), we plot the behaviour of ΣxxΩ, Σxxm, Σ5xxΩ, and Σ5xxm in Figs. 6 and 7. The interplay of the BC-only and the OMM-induced parts are illustrated via some representative parameter values. In agreement with our comparison of the g-values, we find that for a WSM, the OMM can even change the sign of the response, depending on the net magnetic field. However, for J=2,3, the gxm and the gym-parts have opposite signs — hence, their combined effects may increase or reduce the overall response. From Fig. 6, we find that, turning on a nonzero B5-part changes the periodicity, with respect to θ, from π to 2π. This is completely expected because the axial pseudomagnetic field causes linear-in-Bcosθ and/or linear-in-Bsinθ terms to appear, in addition to the quadratic-in-Bi dependence of the untilted WSMs/mWSMs. Figure 6 The total and axial combinations of the LMC (in units of eV) for the two conjugate nodes, defined in Eq. (29), as functions of θ, using various values of B (in units of eV2), B5 (in units of eV2), and θ5 (as indicated in the plotlabels). We have set vz=0.005, τ=151 eV-1, β=1160 eV-1, μ+=0.4 eV, and μ-=0.2 eV. As explained below Eq. (29), while ΣxxΩ (Σ5xxΩ) represents the part of Σxx (Σ5xx) originating purely from the BC-contributions (i.e., with no OMM), Σxxm (Σ5xxm) is the contribution which vanishes if the OMM is not at all considered. We have used the superscript ξ to indicate that, along the vertical axis, we have plotted the BC-only, OMM, and total parts, with the colour-coding shown in the plotlegends. The values of the maxima and minima of the curves are strongly dependent on the values of J.

Figure 7 The total and axial combinations of the LMC (in units of eV) for the two conjugate nodes, defined in Eq. (29), as functions of B (in units of eV2) and B5 (in units of eV2), using various values of θ and θ5 (as indicated in the plotlabels). We have set vz=0.005, τ=151 eV-1, β=1160 eV-1, μ+=0.4 eV, and μ-=0.2 eV. As explained below Eq. (29), while ΣxxΩ (Σ5xxΩ) represents the part of Σxx (Σ5xx) originating purely from the BC-contributions (i.e., with no OMM), Σxxm (Σ5xxm) is the contribution which vanishes if the OMM is neglected. We have used the superscript ξ to indicate that, along the vertical axis, we have plotted the BC-only, OMM, and total parts, with the colour-coding shown in the plotlegends.

Planar Hall conductivity

Using the explicit expressions derived in Appendix C, we have33 σyxχ(μχ)=σxyχ(μχ)=σxy0,χ(μχ)+σxyΩ,χ(μχ)+σxym,χ(μχ),

where34 σxy0,χ(μχ)=0,σxyΩ,χ(μχ)=e4τvzαJ2JΛ-2J(μχ)64π32Γ(2-1J)Γ(92-1J)fbc(J)BχxBχy,σxym,χ(μχ)=e4τvzαJ2JΛ-2J(μχ)64π32Γ(2-1J)Γ(92-1J)fm(J)BχxBχy,

and35 fbc(J)=J13J2-7J+1,fm(J)=17J3-44J2+39J-15+2J.

From Fig. 5, we find that (1) fbc(J) is positive for all J-values; (2) fm(J) is negative for J=1 and positive for J=2,3. For J=1, the OMM thus acts in opposition to the BC-only term, and reduces the magnitude of the overall PHC. In contrast, for the mWSMs, the OMM adds up to the BC-only term, thus increasing the overall response.

In order to estimate the effects of the OMM, we plot the behaviour of ΣxyΩ, Σxym, Σ5xyΩ, and Σ5xym in Figs. 8 and 9. The interplay of the BC-only and the OMM-induced parts are illustrated via the same parameter values as considered for the LMC. In agreement with our comparison of the f-values, we find that for a WSM, the OMM always reduces the response. On the other hand, for J=2,3, the effect of OMM is to enhance the overall response. Analogous to the LMC, Fig. 8 shows that a nonzero B5-part changes the periodicity with respect to θ from π to 2π, which results from the emergence of terms linearly proportional to the components of B, rather than just the quadratic ones. Figure 8 The total and axial combinations of the PHC (in units of eV) for the two conjugate nodes, defined in Eq. (29), as functions of θ, using various values of B (in units of eV2), B5 (in units of eV2), and θ5 (as indicated in the plotlabels). We have set vz=0.005, τ=151 eV-1, β=1160 eV-1, μ+=0.4 eV, and μ-=0.2 eV. As explained below Eq. (29), while ΣxyΩ (Σ5xyΩ) represents the part of Σxy (Σ5xy) originating purely from the BC-contributions (i.e., with no OMM), Σxym (Σ5xym) is the contribution which vanishes if the OMM is not at all considered. We have used the superscript ξ to indicate that, along the vertical axis, we have plotted the BC-only, OMM, and (BC + OMM) parts, with the colour-coding shown in the plotlegends. The values of the maxima and minima of the curves are strongly dependent on the values of J.

Figure 9 The total and axial combinations of the PHC (in units of eV) for the two conjugate nodes, defined in Eq. (29), as functions of B (in units of eV2) and B5 (in units of eV2), using various values of θ and θ5 (as indicated in the plotlabels). We have set vz=0.005, τ=151 eV-1, β=1160 eV-1, μ+=0.4 eV, and μ-=0.2 eV. As explained below Eq. (29), while ΣxyΩ (Σ5xyΩ) represents the part of Σxy (Σ5xy) originating purely from the BC-contributions (i.e., with no OMM), Σxym (Σ5xym) is the contribution which vanishes if the OMM is neglected. We have used the superscript ξ to indicate that, along the vertical axis, we have plotted the BC-only, OMM, and (BC + OMM) parts, with the colour-coding shown in the plotlegends.

Magnetothermoelectric conductivity

Defining36 Fijχ=Dχvχi+e(vχ·Ωχ)Bχivχj+e(vχ·Ωχ)Bχj,

we expand it as37 Fijχ=Fij0,χ+Fij1,Ω,χ+Fij1,m,χ+Fij2,Ω,χ+Fij2,m,χ+Fij2,(Ω,m),χ+O(|Bχ|3),

where38 Fij0,χ=vχi(0)vχj(0),Fij1,Ω,χ=evχ(0)·Ωχvχi(0)Bχj+Bχivχj(0)-eBχ·Ωχvχi(0)vχj(0),Fij1,m,χ=vχi(0)vχj(m)+vχi(m)vχj(0),Fij2,Ω,χ=e2QχiQχj,Fij2,m,χ=vχi(m)vχj(m),Fij2,(Ω,m),χ=evχ(0)·Ωχvχi(m)Bχj+Bχivχj(m)+evχ(m)·Ωχvχi(0)Bχj+Bχivχj(0)-eBχ·Ωχvχi(0)vχj(m)+vχi(m)vχj(0).

Here, Fij0,χ is Bχ-independent, Fij1,Ω,χ is linear in the components of the BC, Fij1,m,χ is linear in the OMM-contributions, Fij2,Ω,χ is quadratic in the components of the BC, Fij1,m,χ is quadratic in the OMM-contributions, and Fij2,(Ω,m),χ is a mixed term which contains products of the BC-components and the OMM-contributions.

Using the above expressions for the function defined as Gijχ=FijχEχ-μT, the weak-field expansion gives us39 Gijχ=Fijχεχ-μT+Fij0,χ+Fij1,Ω,χ+Fij1,m,χεχ(m)T+O(|Bχ|3)=Gij0,χ+Gij1,Ω,χ+Gij1,m,χ+Gij2,Ω,χ+Gij2,m,χ+Gij2,(Ω,m),χ+O(|Bχ|3).

Here,40 Gij0,χ=Fij0,χεχ-μT,Gij1,Ω,χ=Fij1,Ω,χεχ-μT,Gij1,m,χ=Fij0,χεχ(m)T+Fij1,m,χεχ-μT,Gij2,Ω,χ=Fij2,Ω,χεχ-μT,Gij2,m,χ=Fij1,m,χεχ(m)T+Fij2,m,χεχ-μT,Gij2,(Ω,m),χ=Fij1,Ω,χεχ(m)T+Fij2,(Ω,m),χεχ-μT,

with the parts named in a similar spirit as done for the case of Fijχ.

Defining a third function Hijχ=Gijf0′(Eχ), and using Eq. (13), its expansion turns out to be41 Hijχ=Gijχf0′(εχ)+Gij0,χ+Gij1,Ω,χ+Gij1,m,χεχ(m)f0″(εχ)+Gij0,χεχ(m)2f0″′(εχ)2+O(|Bχ|3)=Gij0,χ+Gij1,Ω,χ+Gij1,m,χ+Gij2,Ω,χ+Gij2,m,χ+Gij2,(Ω,m),χ+O(|Bχ|3).

Here,42 Hij0,χ=Gij0,χf0′(εχ),Hij1,Ω,χ=Gij1,Ω,χf0′(εχ),Hij1,m,χ=Gij0,χεχ(m)f0″(εχ)+Gij1,m,χf0′(εχ),Hij2,Ω,χ=Gij2,Ω,χf0′(εχ),Hij2,m,χ=Gij0,χεχ(m)2f0″′(εχ)2+Gij1,m,χεχ(m)f0″(εχ)+Gij2,m,χf0′(εχ),Hij2,(Ω,m),χ=Gij1,Ω,χεχ(m)f0″(εχ)+Gij2,(Ω,m),χf0′(εχ).

The nomenclature of these parts follow the same scheme as that for Fijχ and Gijχ.

Starting from Eq. (18), applying the weak-in-magnetic-field limit using the expansions outlined above, the expression for the magnetoelectric conductivity tensor αχ is dissociated into43 αijχ=αij0,χ++αijΩ,χ+αijm,χ,whereαijΩ,χ=αij1,Ω,χ+αij2,Ω,χ,αm,χ=αij1,m,χ+αij2,m,χ+αij2,(Ω,m),χ.

The terms αij0,χ, αij1,Ω,χ, αij2,Ω,χ, αij1,m,χ, αij2,m,χ, and αij2,(Ω,m),χ consist of the integrands Hij0,χ, Hij1,Ω,χ, Hij2,Ω,χ, Hij1,m,χ, Hij2,m,χ, and Hij2,(Ω,m),χ, respectively. Analogous to the case of σijχ, the term αijm,χ goes to zero if the OMM is set to zero.

The longitudinal and transverse components of the magnetothermoelectric conductivity tensor αχ (i.e., the LTEC and the TTEC) are computed from the expressions shown above. The details of the intermediate steps have been relegated to Appendices D and E.

Akin to the magnetoconductivity tensors, we define the total and axial magnetothermoelectric tensors as44 αij=∑χαijχandα5ij=∑χχαijχ,

respectively. Subtracting off the Bχ-independent parts, we define45 Aij(Bχ)=αij(μ+,μ-)-αij(μ+,μ-)|Bχ=0andA5ij(Bχ)=α5ij(μ+,μ-)-α5ij(μ+,μ-)|Bχ=0.

We denote the parts connected with αijΩ,χ and αijm,χ as (AijΩ,A5ijΩ) and (Aijm,A5ijm), respectively. Therefore, if the OMM is not considered, each of (Aijm,A5ijm) goes to zero.

Longitudinal magnetothermoelectric coefficient

Using the explicit expressions derived in Appendix D, we have46 αxxχ(μχ)=αxx0,χ(μχ)+αxxΩ,χ(μχ)+αxxm,χ(μχ),

where47 αxx0,χ(μχ)=-eτJμχ9vzβ,αxxΩ,χ(μχ)=πe3τvzαJ2JΓ(2-1J)192βμχ1+2JΓ(92-1J)gxbcBχx2+gybcBχy2J,αxxm,χ(μχ)=πe3τvzαJ2JΓ(2-1J)192μχ1+2JβΓ(92-1J)gxmBχx2+gymBχy2J.

Transverse magnetothermoelectric coefficient

Using the explicit expressions derived in Appendix E, we have48 αyxχ(μχ)=αxyχ(μχ)=αxy0,χ(μχ)+αxyΩ,χ(μχ)+αxym,χ(μχ),

where49 αxy0,χ(μχ)=0,αxyΩ,χ(μχ)=πe3τvzαJ2JΓ(2-1J)96βμχ1+2JΓ(92-1J)fbc(J)JBχxBχy,αxym,χ(μχ)=πe3τvzαJ2JΓ(2-1J)96βμχ1+2JΓ(92-1J)fm(J)JBχxBχy.

Mott relation

From the explicit expressions of the in-plane longitudinal and transverse components of σχ and αχ, which we have derived [cf. Eqs. (31), (34), (47), and (49)], we can immediately infer that the relation50 ∂μχσijχ(μχ)=-3eβπ2αijχ(μχ)+O(β-2)

is satisfied. This is equivalent to satisfying the Mott relation αijχ(μχ)=-π23eβ∂μχσijχ(μχ), which holds in the limit β→∞57, and can be derived mathematically through the Sommerfeld expansion. Hence, we find that the Mott relation continues to hold even in the presence of OMM, agreeing with the results in Ref.59, where generic settings have been considered.

Due to the Mott relation, the nature of AijΩ and Aijm can be readily inferred from that of ΣijΩ and Σijm, which we have already discussed in the preceding section. Nevertheless, we provide here some representative plots of the (1) longitudinal planar components AxxΩ, Axxm, A5xxΩ, and A5xxm in Figs. 10 and 11; (2) transverse planar components AxyΩ, Axym, A5xyΩ, and A5xym in Figs. 12 and 13. We choose a different set of values for μ+ and μ-, compared to those chosen for Figs. 6, 7, 8, 9, so as to cover a somewhat different parameter range for illustrative purposes. In fact, we have chosen here μ+-μ-<0, compared to the chosen value of μ+-μ->0 for the earlier section, which leads to a sign-flip of the cyan, orange, and light-green curves representing the axial combinations of the response tensors. Figure 10 The total and axial combinations of the LTEC (in units of eV) for the two conjugate nodes, defined in Eq. (45), as functions of θ, using various values of B (in units of eV2), B5 (in units of eV2), and θ5 (as indicated in the plotlabels). We have set vz=0.005, τ=151 eV-1, β=1160 eV-1, μ+=0.2 eV, and μ-=0.35 eV. As explained below Eq. (45), while AxxΩ (A5xxΩ) represents the part of Axx (A5xx) originating purely from the BC-contributions (i.e., with no OMM), Axxm (A5xxm) is the contribution which vanishes if OMM is not at all considered. We have used the superscript ξ to indicate that, along the vertical axis, we have plotted the BC-only, OMM, and (BC + OMM) parts, with the colour-coding shown in the plotlegends. The values of the maxima and minima of the curves are strongly dependent on the values of J.

Figure 11 The total and axial combinations of the LTEC (in units of eV) for the two conjugate nodes, defined in Eq. (45), as functions of B (in units of eV2) and B5 (in units of eV2), using various values of θ and θ5 (as indicated in the plotlabels). We have set vz=0.005, τ=151 eV-1, β=1160 eV-1, μ+=0.2 eV, and μ-=0.35 eV. As explained below Eq. (45), while AxxΩ (A5xxΩ) represents the part of Axx (A5xx) originating purely from the BC-contributions (i.e., with no OMM), Axxm (A5xxm) is the contribution which vanishes if the OMM is neglected. We have used the superscript ξ to indicate that, along the vertical axis, we have plotted the BC-only, OMM, and (BC + OMM) parts, with the colour-coding shown in the plotlegends.

Figure 12 The total and axial combinations of the TTEC (in units of eV) for the two conjugate nodes, defined in Eq. (45), as functions of θ, using various values of B (in units of eV2), B5 (in units of eV2), and θ5 (as indicated in the plotlabels). We have set vz=0.005, τ=151 eV-1, β=1160 eV-1, μ+=0.2 eV, and μ-=0.35 eV. As explained below Eq. (45), while AxyΩ (A5xyΩ) represents the part of Axy (A5xy) originating purely from the BC-contributions (i.e., with no OMM), Axym (A5xym) is the contribution which vanishes if OMM is not at all considered. We have used the superscript ξ to indicate that, along the vertical axis, we have plotted the BC-only, OMM, and (BC + OMM) parts, with the colour-coding shown in the plotlegends. The values of the maxima and minima of the curves are strongly dependent on the values of J.

Figure 13 The total and axial combinations of the TTEC (in units of eV) for the two conjugate nodes, defined in Eq. (45), as functions of B (in units of eV2) and B5 (in units of eV2), using various values of θ and θ5 (as indicated in the plotlabels). We have set vz=0.005, τ=151 eV-1, β=1160 eV-1, μ+=0.2 eV, and μ-=0.35 eV. As explained below Eq. (45), while AxyΩ (A5xyΩ) represents the part of Axy (A5xy) originating purely from the BC-contributions (i.e., with no OMM), Axym (A5xym) is the contribution which vanishes if the OMM is neglected. We have used the superscript ξ to indicate that, along the vertical axis, we have plotted the BC-only, OMM, and (BC + OMM) parts, with the colour-coding shown in the plotlegends.

Magnetothermal coefficient

Using the function Fij, whose Taylor-expanded form has been shown in Eq. (37), we now define the function G~ijχ=FijχEχ-μ2/T=Fijχεχ+εχ(m)-μ2/T. Its weak-field expansion is given by51 G~ijχ=Fijχεχ-μ2T+2Fij0,χ+Fij1,Ω,χ+Fij1,m,χεχ-μεχ(m)T+Fij0,χεχ(m)2T+O(|Bχ|3)=G~ij0,χ+G~ij1,Ω,χ+Gij1,m,χ+G~ij2,Ω,χ+G~ij2,m,χ+G~ij2,(Ω,m),χ+O(|Bχ|3).

Here,52 G~ij0,χ=Fij0,χεχ-μ2T,G~ij1,Ω,χ=Fij1,Ω,χεχ-μ2T,G~ij1,m,χ=2Fij0,χεχ-μεχ(m)T+Fij1,m,χεχ-μ2T,G~ij2,Ω,χ=Fij2,Ω,χεχ-μ2T,G~ij2,m,χ=2Fij1,m,χεχ-μεχ(m)T+Fij2,m,χεχ-μ2T+Fij0,χεχ(m)2T,G~ij2,(Ω,m),χ=2Fij1,Ω,χεχ-μεχ(m)T+Fij2,(Ω,m),χεχ-μ2T.

We need to define one last function H~ijχ=G~ijf0′(Eχ). Using Eq. (13), its expansion turns out to be53 H~ijχ=G~ijχf0′(εχ)+G~ij0,χ+G~ij1,Ω,χ+G~ij1,m,χεχ(m)f0″(εχ)+G~ij0,χεχ(m)2f0″′(εχ)2+O(|Bχ|3)=G~ij0,χ+G~ij1,Ω,χ+G~ij1,m,χ+G~ij2,Ω,χ+G~ij2,m,χ+G~ij2,(Ω,m),χ+O(|Bχ|3),

where54 H~ij0,χ=G~ij0,χf0′(εχ),H~ij1,Ω,χ=G~ij1,Ω,χf0′(εχ),H~ij1,m,χ=G~ij0,χεχ(m)f0″(εχ)+G~ij1,m,χf0′(εχ),H~ij2,Ω,χ=G~ij2,Ω,χf0′(εχ),H~ij2,m,χ=G~ij0,χεχ(m)2f0″′(εχ)2+G~ij1,m,χεχ(m)f0″(εχ)+G~ij2,m,χf0′(εχ),H~ij2,(Ω,m),χ=G~ij1,Ω,χεχ(m)f0″(εχ)+G~ij2,(Ω,m),χf0′(εχ).

Using the expressions shown above, the integrand in Eq. (19) is expanded in the weak-in-magnetic-field limit, leading to55 ℓijχ=ℓij0,χ++ℓijΩ,χ+ℓijm,χ,whereℓijΩ,χ=ℓij1,Ω,χ+ℓij2,Ω,χ,ℓijm,χ=ℓij1,m,χ+ℓij2,m,χ+ℓij2,(Ω,m),χ.

Analogous to αijχ, the terms ℓij0,χ, ℓij1,Ω,χ, ℓij2,Ω,χ, ℓij1,m,χ, ℓij2,m,χ, and ℓij2,(Ω,m),χ have been labelled such that they are defined by the integrands H~ij0,χ, H~ij1,Ω,χ, H~ij2,Ω,χ, H~ij1,m,χ, H~ij2,m,χ, and H~ij2,(Ω,m),χ, respectively. Analogous to the cases of σijχ and αijχ, the term ℓijm,χ goes to zero if the OMM is set to zero.

The expressions are cumbersome and, hence, it is useful to break them up into smaller bits. In particular, we define56 ℓij2,m,χ=I1,ijℓ,2+I2,ijℓ,2+I3,ijℓ,2,

where57 I1,ijℓ,2=-τ∫d3k2π3[vχi0vχj0εχm2T+2vχi(0)vχj(m)+vχi(m)vχj(0)εχ-μεχ(m)T+vχi(m)vχj(m)εχ-μ2T]f0′(εχ),I2,ijℓ,2=-τ∫d3k2π3[vχi(0)vχj(m)+vχi(m)vχj(0)εχ-μ2εχ(m)T+2vχi(0)vχj(0)εχ-μεχ(m)2T]f0″(εχ),I3,ijℓ,2=-τ∫d3k2π3vχi(0)vχj(0)2εχ-μ2εχ(m)2Tf0″′(εχ).

In a similar spirit, we define58 ℓij2,(m,Ω),χ=I1,ijℓ,3+I2,ijℓ,3+I3,ijℓ,3,

with59 I1,ijℓ,3=-2eτ∫d3k(2π)3f0′(εχ)εχ-μεχ(m)T×vχ(0)·Ωχvχi(0)Bχj+Bχivχj(0)-Bχ·Ωχvχi(0)vχj(0),I2,ijℓ,3=-eτ∫d3k(2π)3[vχ(0)·Ωχvχi(m)Bχj+Bχivχj(m)+vχ(m)·Ωχvχi(0)Bχj+Bχivχj(0)-Bχ·Ωχvχi(0)vχj(m)+vχi(m)vχj(0)]εχ-μ2Tf0′(εχ),I3,ijℓ,3=-eτ∫d3k(2π)3[vχ(0)·Ωχvχi(0)Bχj+Bχivχj(0)-Bχ·Ωχvχi(0)vχj(0)]εχ-μ2εχ(m)Tf0″(εχ).

The longitudinal and transverse components of the tensor ℓχ are computed from the expressions shown above, in the same way as we have done for σχ and αχ. The details of the intermediate steps have been relegated to Appendices F and G.

Longitudinal magnetothermal coefficient

Using the explicit expressions derived in Appendix F, we have60 ℓxxχ(μχ)=ℓxx0,χ(μχ)+ℓxxΩ,χ(μχ)+ℓxxm,χ(μχ),

where61 ℓxx0,χ(μχ)=Jτμχ2T18vz,ℓxxΩ,χ(μχ)=πe2τvzαJ2JT3×128μχ2JΓ(2-1J)Γ(92-1J)gxbc(J)Bχx2+gybc(J)Bχy2,ℓxxm,χ(μχ)=πe2τvzαJ2JT3×128μχ2JΓ(2-1J)Γ(92-1J)gxm(J)Bχx2+gym(J)Bχy2.

Transverse magnetothermal coefficient

Using the explicit expressions derived in Appendix G, we have62 ℓyxχ(μχ)=ℓxyχ(μχ)=ℓxy0,χ(μχ)+ℓxyΩ,χ(μχ)+ℓxym,χ(μχ),

where63 ℓxy0,χ(μχ)=0,ℓxyΩ,χ(μχ)=πe2τvzαJ2JT3×64μχ2JΓ(2-1J)Γ(92-1J)fbc(J)BχxBχy,ℓxym,χ(μχ)=πe2τvzαJ2JT3×64μχ2JΓ(2-1J)Γ(92-1J)fm(J)BχxBχy.

Wiedemann-Franz law

From the explicit expressions of the in-plane longitudinal and transverse components of σχ and ℓχ, which we have derived [cf. Eqs. (31), (34), (61), and (62)], we immediately find that the relation64 σijχ=3e2π2Tℓijχ+O(β-2)

is satisfied. This is equivalent to satisfying the Wiedemann-Franz law, which holds in the limit β→∞57. Hence, we have demonstrated that the Wiedemann-Franz law continues to hold even in the presence of OMM. This relation tells us that, knowing the nature of the magnetoelectric conductivity, we can infer the behaviour of the magnetothermal coefficient. Therefore, it is not necessary to provide any separate plots for this response.

Summary and future perspectives

In this paper, we have considered planar Hall (or planar thermal Hall) configurations such that a 3d Weyl or multi-Weyl semimetal is subjected to a conjunction of an electric field E (and/or temperature gradient ∇rT) and an effective magnetic field Bχ, oriented at a generic angle with respect to each other. The z-axis is chosen to be along the direction along which the mWSM shows a linear-in-momentum dispersion, and is perpendicular to the plane of E (or ∇rT) and Bχ. The effective magnetic field consists of two parts — (a) an actual/physical magnetic field B, and (b) an emergent magnetic field B5 which arises if the sample is subjected to elastic deformations (strain tensor field). Since B5 exhibits a chiral nature, because it couples to conjugate nodal points of opposite chiralities with opposite signs, Bχ is given by B+χB5. The relative orientations of these two constituents of Bχ, with respect to the direction of the electric field (or temperature gradient), give rise to a rich variety of possibilities in the characteristics of the electric, thermal, and thermoelectric response tensors. We have derived explicit expressions for these response coefficients, which have helped us to identify unambiguously the interplay of the BC- and OMM-contributions. In addition, we have illustrated the overall behaviour of the response in some realistic parameter regimes. We have found that the total (i.e., sum) and the axial (i.e., difference) combinations of the response, from the two conjugate nodes, depend strongly on the specific value of J. In particular, for the planar transverse components of the response tensors, while the OMM part acts exclusively in opposition with the BC-only part for the Weyl semimetals, the former syncs with the latter for J>1, thereby enhancing the overall response. The strain-induced B5 provides a way to have linear-in-B terms in the response coefficients for untilted nodes, in addition to the quadratic-in-B dependence.

The B2-dependence and the π-periodic behaviour (with respect to θ) of the magnetoelectric conductivity in the absence of B5 have been observed in numerous experiments, which involve materials like ZrTe560, TaAs61, NbP and NbAs17, and Co3Sn2S262, known to host Weyl nodes. Furthermore, the magnetothermal coefficient has also been measured in materials such as NdAlSi63, which again shows the expected B2-dependence. It is possible to modify these experimental set-ups to devise the mechanism for applying strain gradients to the samples, thus leading to the realization of B564 in addition.

For the case of a negative chemical potential, we need to focus on the valence band at the corresponding node. The energy, band velocity, and Berry curvature will have opposite signs compared to the case of the positive chemical potential. For the magnetic-field-independent “Drude” parts, we have found that, independent of the J-values, σxx0,χ(μχ) and ℓxx0,χ(μχ) are proportional to μχ2, while αxx0,χ(μχ)∝μχ. For the magnetic-field-dependent parts, the leading-order μχ-dependence goes as (i) μχ-2J for σijχ, (ii) μχ-1-2J for αijχ, and (iii) μ2J for ℓijχ. Hence, for J=3, the non-Drude parts depend on a fractional power of μχ, which physically makes no sense for μχ<0. Now we must remember that, in our formalism, we have not included μχ in the starting Hamiltonian Hχ, but have included it in the Fermi distribution function. This has allowed us to derive analytical expressions by applying the Sommerfeld expansion, and the form of the final expressions (summarized above) are an artifact of our specific procedure. Therefore, we conclude that it is possible to derive the results for μχ<0 by implementing the same procedure, but considering the transport contributed by a valence band (with the sign changes quoted above), for J=1,2. However, it will not work for J=3 due to the presence of fractional powers of μχ. The results for J=1,2 are thus expected to have a dependence on the various parameters similar to the μχ>0 case that we have considered here, modulo possible minus signs for the various contributing parts. As for the J=3 case, we can include the negative chemical potential at the Hamiltonian level, such that it directly enters into the energy eigenvalue expressions, and then numerically evaluate the behaviour of the various response tensors.

In Refs.27,28, the authors have included a momentum-independent internode scattering time τv, in addition to the intranode scattering time τ. The inclusion of the internode processes results in stabilizing the chiral anomaly, as it inherently leads to different chemical potentials in the two conjugate WSM/mWSM nodes. By plugging in τv as a phenomenological constant (because of ignoring its momentum dependence), Ref.27 has considered the resulting semiclassical Boltzmann equations for the two nodes in an untilted WSM. The authors have shown that a finite τv is tied to a difference in the chemical potential between the two nodes, given by Δμ∝E·Bτv (in the absence of a pseudomagnetic field). They have inferred that this results in the magnetoelectric conductivity tensor components acquiring extra contributions ∝τv and normal physical conditions dictate that τv≫τ. The same behaviour is found for the case of the thermoelectric conductivity tensor. A more complete treatment can be found in36, where the authors do not assume a momentum-independent internode-scattering time,  and they go beyond the relaxation-time approximation. It remains to be seen how the above calculations pan out for the mWSMs and multifold semimetals65, and what are the forms of the resulting final expressions.

In the future, it will be worthwhile to perform the same calculations by including tilted nodes27,28,36,39, because tilting is applicable for generic scenarios. A tilt can induce terms which are linearly-dependent on B, even in the absence of a strain-induced B5-part. Furthermore, a more realistic calculational set-up should include internode scatterings28 and going beyond the relaxation-time approximation36. Although we have considered the weak-magnetic-field scenario in this paper, under the influence of a strong quantizing magnetic field, we have to incorporate the effects of the discrete Landau levels while computing the linear response33,66–68. Other auxiliary directions include the study of linear and nonlinear response in the presence of disorder and/or strong correlations69–76. One could also explore the effects of a time-periodic drive21,77,78, for instance, by shining circularly polarized light.

Supplementary Information

Supplementary Information.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-024-68615-0.

Acknowledgements

LM and AMR acknowledge support from CONACyT (México) under project number CF- 428214, and DGAPA-UNAM under project number AG100224. IM’s research has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement number 754340.

Author contributions

L.M. did the calculations. I.M. perceived the idea, wrote the main manuscript text, analyzed the results, and prepared the figures. R.G. helped in some of the calculations. A.M.-R. reviewed the write-up. All authors reviewed the manuscript.

Data availibility

All data generated or analysed during this study are included in this published article.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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