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OPEN Electric, thermal,

and thermoelectric
magnetoconductivity for Weyl/
multi‑Weyl semimetals
in planar Hall set‑ups induced
by the combined effects
of topology and strain
Leonardo Medel 1*, Rahul Ghosh 2, Alberto Martín‑Ruiz 1 & Ipsita Mandal 2,3

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 ∇r T ) 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.

There has been an incredible amount of research work focussing on the investigations of the transport proper-
ties 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

1
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, 04510 Ciudad de México,
México. 2Department of Physics, Shiv Nadar Institution of Eminence (SNIoE), Gautam Buddha Nagar, Uttar
Pradesh 201314, India. 3Freiburg Institute for Advanced Studies (FRIAS), University of Freiburg, D‑79104 Freiburg,
Germany. *email: leonardo.medel@correo.nucleares.unam.mx

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chiralities (i.e., χ = ±1). The values of J for Weyl (e.g., ­TaAs7–9 and HgTe-class ­materials10), double-Weyl (e.g.,
HgCr2 Se411 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/2 or 3π/2, the conven-
tional 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 a­ nomaly15–21. 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 quasiparti-
cles being proportional to J(E · B)22,23. 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 ∇ r T . In this scenario, too, a potential differ-
ence is induced along the y-axis due to the chiral a­ nomaly21,24 [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 t­ ensors25–40.
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., quasi-
particles) in such a way that they can be modelled as pseudogauge fields in the s­ emimetals37,41–47. The form of
these elastic gauge fields shows that they couple to the quasiparticles of the Weyl fermions with opposite chirali-
ties with opposite s­ igns37,44–46,48,49. 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 u ­ s38, 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 tempera-
ture 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 + χ B5 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 ∇r T ) applied along the x-axis. In other words, B = B(cos θ, sin θ , 0) and
B5 = B5 (cos θ5 , sin θ5 , 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, mag-
netothermal, and magnetothermoelectric response can be derived using the semiclassical Boltzmann formalism.

Figure 1.  Schematics showing the planar Hall (or planar thermal Hall) experimental set-up, where the sample
is subjected to an external electric field E x̂ (and/or a temperature gradient ∂x T x̂ ). 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.

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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 ten-
sors, 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 effec-
tive continuum Hamiltonian is given b ­ y3,4,14
 ky v⊥
Hχ (k) = dχ (k) · σ , k⊥ = kx2 + ky2 , φk = arctan( ) , αJ = J−1 ,
kx k0
  (1)
J J
dχ (k) = αJ k⊥ cos(Jφk ), αJ k⊥ sin(Jφk ), χ vz kz ,

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 by

2J
εχ,s (k) = (−1)s+1 ǫk , s ∈ {1, 2}, ǫk = αJ2 k⊥ + vz2 kz2 , (2)

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.
The band velocity of the chiral quasiparticles is given by
(−1)s  2 2J−2 2J−2

v (0)
χ,s (k) ≡ ∇k εχ ,s (k) = − J αJ k⊥ kx , J αJ2 k⊥ ky , vz2 kz . (3)
ǫk
The Berry curvature (BC) and the orbital magnetic moment (OMM), associated with the sth band, are expressed
­by51–54
(−1)s ǫ i jl  
χ,s (k) = i �∇k ψsχ (k)| × |∇k ψsχ (k)� ⇒ �iχ ,s (k) = dχ (k) · ∂kj dχ (k) × ∂kl dχ (k)
4 |dχ (k)|3
ie
(4)
  
and mχ,s (k) = − �∇ k ψs (k)| × H(k) − Eχ ,s (k) |∇ k ψs (k)�
2
e ǫ i jl  
⇒ miχ,s (k) = 2
dχ (k) · ∂kj dχ (k) × ∂kl dχ (k) ,
4 |dχ (k)|
respectively, where the indices i, j, and l ∈ {x, y, z}, and are used to denote the Cartesian components of the 3d
χ
vectors and tensors. The symbol |ψs (k)� denotes the normalized eigenvector corresponding to the band labelled
χ χ
by s, with {|ψ1 �, {|ψ2 �} forming an orthonornomal set for each node.
On evaluating the expressions in Eq. (4), using Eq. (1), we get

Figure 2.  Schematic dispersion of a single node in a (a) Weyl, (b) double-Weyl, and (c) triple-Weyl semimetal,
plotted against the kz kx-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.

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2J−2 2 J−2
χ (−1)s J vz αJ2 k⊥ χ e J vz αJ2 k⊥
(5)
   
χ,s (k) = kx , ky , J kz , mχ ,s (k) = − kx , ky , J kz .
2 ǫk3 2
2 ǫk

From these expressions, we immediately observe the identity

mχ ,s (k) = −(−1)s e ǫk χ ,s (k) . (6)
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
(0)
with s = 1. Henceforth, we will use the notations εχ ,1 = εχ, v χ ,1 = v (0) (since it is independent of χ), �χ ,1 = �χ,
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.

Response tensors using the Boltzmann formalism

Using the semiclassical Boltzmann f­ ormalism57,58, 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, associ-
ated 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 as
 χ   χ χ 
 
Ji σij αij Ej
th,χ = χ χ
T αij ℓij −∂j T
. (7)
Ji
j
χ χ
Here, σij and αij represent the components of the magnetoelectric conductivity tensor (σ χ) and the magnetother-
moelectric conductivity tensor (α χ), respectively. While α χ determines the Peltier (�χ), 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 , �χ, and the magnetothermal conductivity tensor κ χ (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:
χ
  −1 χ χ
 χ  −1 χ χ
 χ  −1 χ
Sij = σ χ ii′ αi′ j , �ij = T αii′ σ χ i′ j , κij = ℓij − T αii′ σ χ i′ j′ αj′ j .
′ ′ ′ ′
(8)
a i i ,j

In order to include the effects from the OMM and the BC, we first define the quantitites

Eχ (k) = εχ (k) + εχ(m) (k) , εχ(m) (k) = − Bχ · mχ (k) ,

v χ (k) ≡ ∇k Eχ (k) = v (0) (k) + v (m)
χ (k) , (9)
(m) (m)
  −1
v χ (k) = ∇k εχ (k) , Dχ = 1 + e Bχ · χ (k) ,

where εχ(m) (k) is the Zeeman-like correction to the energy due to the OMM, v χ (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 ele-
ment 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.
Our weak-magnetic-field limit implies that
e Bχ · χ ≪ 1. (10)

Parameter SI units Natural units

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

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  = c = kB = 1, and 4 π ǫ0 = 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.

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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.

In our calculations, we keep terms upto O(|Bχ |2 ) and, thus, use

2
Dχ = 1 − e Bχ · χ + e2 Bχ · χ + O(|Bχ |3 ) . (11)
  

Also, the condition in Eq. (10) implies that |εχ(m) (k)| is small compared to εχ (k):

(12)
 
|Bχ · mχ | = εχ e Bχ · χ ≪ εχ .

This means that the Fermi-Dirac distribution can also be power expanded up to quadratic order in the
magnetic field, as follows:
1  m 2 ′′
f0 (Eχ ) = f0 (εχ ) + εχ(m) f0′ (εχ ) + ε f0 (εχ ) + O(|Bχ |3 ) , (13)
2 χ
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, con-
tributed by the conduction band, is given by
d3 k  ∂f0 (Eχ )

χ
σij = −e2 τ (14)
 
Dχ vχ i + e (v χ · χ ) Bχ i vχ j + e (v χ · χ ) Bχ j ,
(2 π)3 ∂ 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:

1. The intrinsic anomalous Hall term is given by

d3 k

AH ,χ AH(0) ,χ AH(1) ,χ AH(2) ,χ
σij = − e2 ǫijl �l (k) f0 (Eχ ) = σij + σij + σij + O(|Bχ |3 ) ,
(2 π)3 χ
d3 k d3 k
 
AH(0) ,χ AH(1) ,χ
σij = − e2 ǫijl � l
χ (k) f0 (εχ ) , σ ij = − e 2
ǫijl �l (k) εχ(m) f0′ (εχ ) , (15)
(2 π) 3 (2 π)3 χ
e2 ǫijl d3 k
  2
AH(2) ,χ
σij =− � l
χ (k) εχm f0′′ (εχ ) ,
2 (2 π)3
AH ,χ AH(0) ,χ
whose diagonal components (i.e., σii ) are automatically zero. The first term, σij , 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

σyxAH (1),χ = σxyAH (1),χ

∞ ∞ 4J−2 2 π
k⊥
f ′ (εχ ) (16)
 
∝ kz dkz dk⊥ dφ Bχ x cos φ + Bχ y sin φ = 0 ,
εχ5 0
−∞ 0 0
and σyxAH (2),χ = σyxAH (2),χ = 0 .

  Only the transverse out-of-plane components are nonzero, viz.

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e3 J vz Bχ y f0′ (εχ )
∞ e3 J vz Bχ y π2

σzxAH (1),χ (µχ ) = dεχ =− 1 + and
24 π 2 0 εχ 24 π 2 µχ 3 β 2 µ2χ
  (17)
AH (1),χ e3 J vz Bχ x π2
σzy (µχ ) = 1+ [if we have a nonzero y-component of E].
24 π 2 µχ 3 β 2 µ2χ
AH (2),χ AH (2),χ
  We note that σzx = σzy = 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.
2. 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 considera-
tions).
χ
The expression for σij , 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 ­expressions24
d3 k  Eχ − µ ∂f0 (Eχ )

χ
(18)
 
αij = e τ 3
Dχ vχ i + e (v χ · χ ) Bχ i vχ j + e (v χ · χ ) Bχ j ,
(2 π) T ∂ Eχ
and
d3 k  (Eχ − µ)2 ∂f0 (Eχ )

χ
(19)
 
ℓij = − τ 3
Dχ vχ i + e (v χ · χ ) Bχ i vχ j + e (v χ · χ ) Bχ j ,
(2 π) T ∂ Eχ
respectively. Since ℓχ determines the first term in the magnetothermal conductivity tensor κ χ , 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:
χ 0,χ �,χ m,χ
σij = σij + σij + σij . (20)

Here,
d 3 k (0) (0) ′

0,χ
σij = −e2 τ v v f (εχ ), (21)
(2 π)3 χi χj 0
is the conductivity surviving in the absence of a magnetic field (i.e., for Bχ = 0),
d3 k
  
�,χ ′
σij = −e4 τ Q χ i Qχ j f0 (εχ ), Q χ =  χ × v (0)
χ × Bχ , (22)
(2 π)3
is the contribution arising solely from the BC, and
 
d3k εχ(m)   εχ(m) 

m,χ (m)  ∂
σij = 2 e τ3
Qχ i vχj + ∇k · T χ ij + Bχ · V χ ij f ′ (εχ ) , (23)
(2 π)3 e 2 ∂εχ 0

(0) 1 (m) (0) (0) mχ (0)

where T χij = e χ Bχ i vχj + êi vχj and V χ ij = χ vχi vχj − ∂iv , (24)
2 2 e k χj
represents the contribution which goes to zero if OMM is set to zero. The symbol êi represents the unit vector
along the Cartesian coordinate axis labelled by i.
The longitudinal and transverse components of the magnetoelectric conductivity tensor σ χ (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 contribution
χ χ
Ji (µ) = σij (µ) Eχ j , (25)
χ
where σij 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 as

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


J(µ+ , µ− ) = J χ (µχ ) = J + (µ+ ) + J − (µ− )
χ =±1
 (26)
and J 5 (µ+ , µ− ) = χ J χ (µχ ) = J + (µ+ ) − J − (µ+ ) ,
χ =±1

respectively. These suggest to introduce analogous expressions for the conductivity tensors as
 χ  χ
σij (µ+ , µ− ) = σij (µχ ) and σ5ij = χ σij (µχ ) .
(27)
χ =±1 χ =±1

Using Eq. (27), we find that

Ji (µ+ , µ− ) = σij+ (µ+ ) E+j + σij− (µ− ) E−j = σij (µ+ , µ− ) Ej + σ5ij (µ+ , µ− ) E5j
(28)
and J5i (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 + χ E5, 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 define

�ij (Bχ ) = σij (µ+ , µ− ) − σij (µ+ , µ− )

Bχ =0
 (29)
and �5ij (Bχ ) = σ5ij (µ+ , µ− ) − σ5ij (µ+ , µ− )

Bχ =0

for the total and axial conductivity tensor components, respectively, after subtracting off the Bχ-independent
�,χ m,χ
parts. We denote the parts connected with σij and σij as (�ij� , �5ij
� ) and (� m , � m ), respectively. Therefore,
ij 5ij
if the OMM is not considered, each of (�ijm , �5ij
m ) goes to zero.

Longitudinal magnetoconductivity
Using the explicit expressions derived in Appendix B, we have
0,χ m,χ
χ
σxx (µχ ) = σxx �,χ
(µχ ) + σxx (µχ ) + σxx (µχ ) , (30)
where

0,χ e2 τ J
σxx (µχ ) = �2 (µχ ) ,
6 π 2 vz
2
e4 τ vz αJJ �− 2 (µχ ) Ŵ 2 − 1 
 

J
�,χ
σxx (µχ ) = J
 9 1  gxbc (J) Bχ2 x + gybc (J) Bχ2 y ,
3
128 π 2 Ŵ 2−J (31)
2
e4 τ vz αJJ �− 2 (µχ ) Ŵ 2 − 1J 
 

m,χ
σxx (µχ ) = 3
J
9 1 gxm (J) Bχ2 x + gym (J) Bχ2 y ,
128 π 2 Ŵ 2 − J

and

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Figure 5.  Comparison of the values of the functions defined in Eqs. (32) and (35) for J = 1, 2, 3.

gxbc (J) = J 32 J 2 − 19 J + 3 , gybc (J) = J(3 J − 1)(2 J − 1),

 

37 J 4 − 100 J 3 + 74 J 2 − 21 J + 2 3 J 4 − 12 J 3 − 4 J 2 + 9 J − 2 (32)
gxm (J) = , gym (J) = .
J J
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 Bχ x-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,
χ
for a WSM, the OMM part always reduces the value of σxx by adding a negative contribution.
In order to estimate the effects of the OMM (which was neglected in many earlier works), we plot the behav-
iour of xx ,  m ,   , and  m in Figs. 6 and 7. The interplay of the BC-only and the OMM-induced parts
xx 5xx 5xx
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-B cos θ and/or linear-in-B sin θ terms to appear, in addition to the quadratic-in-Bi dependence
of the untilted WSMs/mWSMs.

Planar Hall conductivity

Using the explicit expressions derived in Appendix C, we have
χ χ 0,χ �,χ m,χ
σyx (µχ ) = σxy (µχ ) = σxy (µχ ) + σxy (µχ ) + σxy (µχ ) , (33)
where
2
e4 τ vz αJJ �− 2 (µχ ) Ŵ 2 − 1
 
0,χ J
σxy (µχ ) = 0 , �,χ
σxy (µχ ) = J
 f bc (J) Bχ x Bχ y ,
Ŵ 92 − 1J
3

64 π 2
2 (34)
e4 τ vz αJJ �− 2 (µχ ) Ŵ 2 − 1
 
m,χ J
σxy (µχ ) = J
 f m (J) Bχ x Bχ y ,
Ŵ 92 − 1J
3

64 π 2
and
2
f bc (J) = J 13 J 2 − 7 J + 1 , f m (J) = 17 J 3 − 44 J 2 + 39 J − 15 + (35)
 
.
J
From Fig. 5, we find that (1) f bc (J) is positive for all J-values; (2) f m (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 xy  ,  m ,   , and  m in Figs. 8 and
xy 5xy 5xy
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

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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 xx  (  ) represents the part of  ( ) originating purely from the
5xx xx 5xx
BC-contributions (i.e., with no OMM), xx m ( m ) is the contribution which vanishes if the OMM is not at all
5xx
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.

from π to 2π , which results from the emergence of terms linearly proportional to the components of B, rather
than just the quadratic ones.

Magnetothermoelectric conductivity
Defining
χ   
Fij = Dχ vχ i + e (v χ · χ ) Bχ i vχ j + e (v χ · χ ) Bχ j , (36)
we expand it as
χ 0,χ 1,�,χ 1,m,χ 2,�,χ 2,m,χ 2,(�,m),χ
Fij = Fij + Fij + Fij + Fij + Fij + Fij + O(|Bχ |3 ) , (37)

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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 xx  (  ) represents the part of  ( ) originating purely from the
5xx xx 5xx
BC-contributions (i.e., with no OMM), xx m ( m ) is the contribution which vanishes if the OMM is neglected.
5xx
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.

where
0,χ (0) (0)
Fij = vχ i vχ j ,
  
1,�,χ (0) (0)  (0) (0)
= e v (0)

Fij χ · χ vχi Bχ j + Bχ i vχj − e Bχ · χ vχi vχj ,
 
1,m,χ (0) (m) (m) (0)
Fij = vχi vχj + vχi vχj ,
2,�,χ 2,m,χ (m) (m) (38)
Fij = e2 Qχ i Qχ j , Fij = vχi vχj ,
     
2,(�,m),χ (m) (m) (0) (0)
Fij = e v (0)
χ · χ vχi Bχ j + Bχ i vχj + e v (m)
χ · χ vχi Bχ j + Bχ i vχj
  (0) (m) (m) (0)

− e Bχ · χ vχi vχj + vχi vχj .

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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 xy  (  ) represents the part of  ( ) originating purely from the
5xy xy 5xy
BC-contributions (i.e., with no OMM), xy m ( m ) is the contribution which vanishes if the OMM is not at all
5xy
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.

0,χ 1,�,χ 1,m,χ

Here, Fij is Bχ-independent, Fij is linear in the components of the BC, Fij is linear in the OMM-
2,�,χ 1,m,χ
contributions, Fij is quadratic in the components of the BC, Fij is quadratic in the OMM-contributions,
2,(�,m),χ
and Fij is a mixed term which contains products of the BC-components and the OMM-contributions.
χ χ Eχ −µ
Using the above expressions for the function defined as Gij = Fij T , the weak-field expansion gives us
 (m)
χ χ εχ − µ  0,χ 1,�,χ 1,m,χ εχ
Gij = Fij + Fij + Fij + Fij + O(|Bχ |3 )
T T (39)
0,χ 1,�,χ 1,m,χ 2,�,χ 2,m,χ 2,(�,m),χ
= Gij + Gij + Gij + Gij + Gij + Gij + O(|Bχ |3 ) .

Here,

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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 xy  (  ) represents the part of  ( ) originating purely from the
5xy xy 5xy
BC-contributions (i.e., with no OMM), xy m ( m ) is the contribution which vanishes if the OMM is neglected.
5xy
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.

0,χ   1,�,χ  
0,χ Fij εχ − µ 1,�,χ Fij εχ − µ
Gij = , Gij = ,
T T
0,χ 1,m,χ  2,�,χ 
Fij εχ(m)
 
Fij εχ −µ −µ
Fij εχ
1,m,χ
Gij = +
2,�,χ
, Gij = , (40)
T T T
1,m,χ 2,m,χ 1,�,χ (m) 2,(�,m),χ 
εχ(m)
  
2,m,χ Fij Fij εχ − µ 2,(�,m),χ Fij εχ Fij εχ − µ
Gij = + , Gij = + ,
T T T T
χ
with the parts named in a similar spirit as done for the case of Fij .
χ
Defining a third function Hij = Gij f0′ (Eχ ), and using Eq. (13), its expansion turns out to be

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 2
0,χ
  Gij εχ(m) f0′′′ (εχ )
χ χ 0,χ 1,�,χ 1,m,χ
Hij = Gij f0′ (εχ ) + Gij + Gij + Gij εχ(m) f0′′ (εχ ) + + O(|Bχ |3 )
2 (41)
0,χ 1,�,χ 1,m,χ 2,�,χ 2,m,χ 2,(�,m),χ
= Gij + Gij + Gij + Gij + Gij + Gij + O(|Bχ |3 ) .

Here,
0,χ 0,χ 1,�,χ 1,�,χ ′ 1,m,χ 0,χ 1,m,χ
Hij = Gij f0′ (εχ ) , Hij = Gij f0 (εχ ) , Hij = Gij εχ(m) f0′′ (εχ ) + Gij f0′ (εχ ) ,
 2
0,χ
Gij εχ(m) f0′′′ (εχ )
2,�,χ 2,�,χ ′ 2,m,χ 1,m,χ 2,m,χ
Hij = Gij f0 (εχ ) , Hij = + Gij εχ(m) f0′′ (εχ ) + Gij f0′ (εχ ) ,
2
2,(�,m),χ 1,�,χ (m) ′′ 2,(�,m),χ ′
Hij = Gij εχ f0 (εχ ) + Gij f0 (εχ ) .
(42)
χ χ
The nomenclature of these parts follow the same scheme as that for Fij and Gij .
Starting from Eq. (18), applying the weak-in-magnetic-field limit using the expansions outlined above, the
expression for the magnetoelectric conductivity tensor α χ is dissociated into
χ 0,χ �,χ m,χ �,χ 1,�,χ 2,�,χ 1,m,χ 2,m,χ 2,(�,m),χ
αij = αij + +αij + αij , where αij = αij + αij , α m,χ = αij + αij + αij .
(43)

0,χ 1,�,χ 2,�,χ 1,m,χ 2,m,χ 2,(�,m),χ 0,χ 1,�,χ 2,�,χ 1,m,χ
The terms αij , αij , αij , αij , αij , and αij consist of the integrands Hij , Hij , Hij , Hij ,
2,m,χ 2,(�,m),χ χ m,χ
Hij , and Hij , respectively. Analogous to the case of σij , the term αij goes to zero if the OMM is set
to zero.
The longitudinal and transverse components of the magnetothermoelectric conductivity tensor α χ (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 as
 χ  χ
αij = αij and α5ij = χ αij ,
(44)
χ χ

respectively. Subtracting off the Bχ-independent parts, we define


Aij (Bχ ) = αij (µ+ , µ− ) − αij (µ+ , µ− )

Bχ =0
 (45)
and A5ij (Bχ ) = α5ij (µ+ , µ− ) − α5ij (µ+ , µ− ) .

Bχ =0
�,χ m,χ
We denote the parts connected with αij and αij as (A� ij , 5ij ) and (Aij ,
A� m
5ij ), respectively. Therefore, if the
Am
OMM is not considered, each of (Aij , A5ij ) goes to zero.
m m

Longitudinal magnetothermoelectric coefficient

Using the explicit expressions derived in Appendix D, we have
0,χ m,χ
χ
αxx (µχ ) = αxx �,χ
(µχ ) + αxx (µχ ) + αxx (µχ ) , (46)
where
2
√
π e3 τ vz αJJ Ŵ 2 − 1J gxbc Bχ2 x + gybc Bχ2 y
 
0,χ e τ J µχ �,χ
αxx (µχ ) =− , αxx (µχ ) = ,
9 vz β 1+ 2  J
192 β µχ J Ŵ 29 − 1J


2 (47)
√
π e3 τ vz αJJ Ŵ 2 − 1J gxm Bχ2 x + gym Bχ2 y
 
m,χ
αxx (µχ ) = .
1+ 2 J
192 µχ J β Ŵ 92 − 1J
 

Transverse magnetothermoelectric coefficient

Using the explicit expressions derived in Appendix E, we have
χ χ 0,χ �,χ m,χ
αyx (µχ ) = αxy (µχ ) = αxy (µχ ) + αxy (µχ ) + αxy (µχ ) , (48)
where

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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 A xx ( A5xx ) represents the part of Axx ( A5xx ) originating purely from the

BC-contributions (i.e., with no OMM), Axx ( Amm
5xx ) 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.

2
√ 3
π e τ vz αJJ Ŵ 2 − 1J f bc (J)
 
0,χ �,χ
αxy (µχ ) = 0, αxy (µχ ) = Bχ x Bχ y ,
1+ 2  J
96 β µχ J Ŵ 92 − 1J


2 (49)
√
π e3 τ vz αJJ Ŵ 2 − 1J f m (J)
 
m,χ
αxy (µχ ) = Bχ x Bχ y .
1+ 2  J
96 β µχ J Ŵ 92 − 1J


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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 A xx ( A5xx ) represents the part of Axx ( A5xx ) originating purely from the

BC-contributions (i.e., with no OMM), Am xx ( Am ) is the contribution which vanishes if the OMM is neglected.
5xx
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.

Mott relation
From the explicit expressions of the in-plane longitudinal and transverse components of σ χ and α χ, which we
have derived [cf. Eqs. (31), (34), (47), and (49)], we can immediately infer that the relation
χ 3eβ χ
∂µχ σij (µχ ) = − α (µχ ) + O(β −2 ) (50)
π 2 ij
χ π2 χ
is satisfied. This is equivalent to satisfying the Mott relation αij (µχ ) = − 3 e β ∂µχ σij (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 A ij and Aij can be readily inferred from that of ij and ij , which we
m  m

have already discussed in the preceding section. Nevertheless, we provide here some representative plots of the
(1) longitudinal planar components A xx , Axx , A5xx , and A5xx in Figs. 10 and 11; (2) transverse planar components
m  m

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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 A xy ( A5xy ) represents the part of Axy ( A5xy ) originating purely from the


BC-contributions (i.e., with no OMM), Am xy ( A5xy ) is the contribution which vanishes if OMM is not at all
m

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.

xy , Axy , A5xy , and A5xy in Figs. 12 and 13. We choose a different set of values for µ+ and µ−, compared to those
A m  m

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.

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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 A xy ( A5xy ) represents the part of Axy ( A5xy ) originating purely from the


BC-contributions (i.e., with no OMM), Axy ( Am

m
5xy ) 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
2  2
χ χ χ
G̃ij = Fij Eχ − µ / T = Fij εχ + εχ(m) − µ / T . Its weak-field expansion is given by
 2
 2   (m)
ε (m)
χ χ εχ − µ

0,χ 1,�,χ 1,m,χ
 εχ − µ εχ 0,χ χ
G̃ij = Fij + 2 Fij + Fij + Fij + Fij + O(|Bχ |3 )
T T T (51)
0,χ 1,�,χ 1,m,χ 2,�,χ 2,m,χ 2,(�,m),χ
= G̃ij + G̃ij + Gij + G̃ij + G̃ij + G̃ij + O(|Bχ |3 ) .

Here,

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0,χ  2
0,χ Fij εχ − µ
G̃ij = ,
T
1,�,χ   2
1,�,χ Fij εχ − µ
G̃ij = ,
T
0,χ   (m) 1,m,χ  2
1,m,χ 2 Fij εχ − µ εχ Fij εχ − µ
G̃ij = + ,
T T
2,�,χ  2 (52)
2,�,χ Fij εχ −µ
G̃ij = ,
T
 2
1,m,χ  2,m,χ  2 0,χ
εχ − µ εχ(m) Fij εχ(m)

2,m,χ 2 Fij Fij εχ − µ
G̃ij = + + ,
T T T
1,�,χ  2,(�,m),χ  2
− µ εχ(m)

2,(�,m),χ 2 Fij εχ Fij εχ −µ
G̃ij = + .
T T
χ
We need to define one last function H̃ij = G̃ij f0′ (Eχ ). Using Eq. (13), its expansion turns out to be
 2
0,χ
  G̃ij εχ(m) f0′′′ (εχ )
χ χ 0,χ 1,�,χ 1,m,χ
H̃ij = G̃ij f0′ (εχ ) + G̃ij + G̃ij + G̃ij εχ(m) f0′′ (εχ ) + + O(|Bχ |3 )
2 (53)
0,χ 1,�,χ 1,m,χ 2,�,χ 2,m,χ 2,(�,m),χ
= G̃ij + G̃ij + G̃ij + G̃ij + G̃ij + G̃ij + O(|Bχ |3 ) ,

where
0,χ 0,χ 1,�,χ 1,�,χ
H̃ij = G̃ij f0′ (εχ ) , H̃ij = G̃ij f0′ (εχ ) ,
1,m,χ 0,χ 1,m,χ 2,�,χ 2,�,χ
H̃ij = G̃ij εχ(m) f0′′ (εχ ) + G̃ij f0′ (εχ ) , H̃ij = G̃ij f0′ (εχ ) ,
2
(54)

0,χ
G̃ij εχ(m) f0′′′ (εχ )
2,m,χ 1,m,χ (m) ′′ 2,m,χ ′
H̃ij = + G̃ij εχ f0 (εχ ) + G̃ij f0 (εχ ) ,
2
2,(�,m),χ 1,�,χ (m) ′′ 2,(�,m),χ ′
H̃ij = G̃ij εχ f0 (εχ ) + G̃ij f0 (εχ ) .

Using the expressions shown above, the integrand in Eq. (19) is expanded in the weak-in-magnetic-field
limit, leading to
χ 0,χ �,χ m,χ �,χ 1,�,χ 2,�,χ m,χ 1,m,χ 2,m,χ 2,(�,m),χ
ℓij = ℓij + +ℓij + ℓij , where ℓij = ℓij + ℓij , ℓij = ℓij + ℓij + ℓij . (55)
χ 0,χ 1,�,χ 2,�,χ 1,m,χ 2,m,χ 2,(�,m),χ
Analogous to αij , the terms ℓij , ℓij , ℓij , ℓij , ℓij , and ℓij have been labelled such that they
0,χ 1,�,χ 2,�,χ 1,m,χ 2,m,χ 2,(�,m),χ
are defined by the integrands H̃ij , H̃ij , H̃ij , H̃ij , H̃ij , and H̃ij , respectively. Analogous to the
χ χ m,χ
cases of σij and αij , the term ℓij 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 define
2,m,χ ℓ,2 ℓ,2 ℓ,2
ℓij = I1,ij + I2,ij + I3,ij , (56)

where
 2
εχ(m)  εχ − µ ε (m)
 
d3 k
  
ℓ,2 (0) (0) (0) (m) (m) (0) χ
I1,ij = −τ v v + 2 vχi vχj + vχi vχj
(2 π)3 χi χj T T
 2 
(m) (m) εχ − µ
+ vχi vχj f0′ (εχ ) ,
T
 εχ − µ 2 ε(m)
 
d 3 k  (0) (m)
 
ℓ,2 (m) (0) χ
I2,ij = −τ v v
χi χj + v v
χi χj (57)
(2 π)3 T
 2
εχ − µ εχ(m) 
 
(0) (0)
+ 2 vχi vχj f0′′ (εχ ) ,
T
(0) (0)
 2  (m) 2
d 3 k vχi vχj εχ − µ εχ

ℓ,2
I3,ij = −τ f0′′′ (εχ ) .
(2 π)3 2 T
In a similar spirit, we define

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2,(m,�),χ ℓ,3 ℓ,3 ℓ,3

ℓij = I1,ij + I2,ij + I3,ij , (58)

with
εχ − µ εχ(m)
 
d3 k ′

ℓ,3
I1,ij = −2eτ f (εχ )
(2 π)3 0 T
     (0) (0) 
(0) (0) (0)
× v χ · χ vχi Bχ j + Bχ i vχj − Bχ · χ vχi vχj ,
d 3 k  (0)
     
ℓ,3 (m) (m) (0) (0)
I2,ij = −eτ 3
v χ · χ vχi Bχ j + Bχ i vχj + v (m)
χ · χ vχi Bχ j + Bχ i vχj
(2 π)
 (0) (m)  ε − µ2
(m) (0) χ
f0′ (εχ ) ,

− Bχ · χ vχi vχj + vχi vχj
T
2
 (0) (0) εχ − µ εχ(m) ′′

d 3 k  (0)
    
ℓ,3 (0) (0)
I3,ij = −eτ v χ ·  χ v B
χi χ j + B v
χ i χj − B χ ·  χ v v
χi χj f0 (εχ ) .
(2 π)3 T
(59)
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 σ χ and α χ. 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 have

ℓχxx (µχ ) = ℓ0,χ �,χ m,χ

xx (µχ ) + ℓxx (µχ ) + ℓxx (µχ ) , (60)
where
J τ µ2χ T
ℓ0,χ
xx (µχ ) = ,
18 vz
2
√ 2
T Ŵ 2 − 1J  bc
J
 
π e τ v z αJ

�,χ
ℓxx (µχ ) =  9 1  gx (J) Bχ2 x + gybc (J) Bχ2 y ,
2
Ŵ 2−J (61)
3 × 128 µχJ
2
√ 2
π e τ vz αJJ T Ŵ 2 − 1J  m
 

m,χ
ℓxx (µχ ) = 2
 9 1  gx (J) Bχ2 x + gym (J) Bχ2 y .
Ŵ 2−J
3 × 128 µχJ

Transverse magnetothermal coefficient

Using the explicit expressions derived in Appendix G, we have

ℓχyx (µχ ) = ℓχxy (µχ ) = ℓ0,χ �,χ m,χ

xy (µχ ) + ℓxy (µχ ) + ℓxy (µχ ) , (62)
where

ℓ0,χ
xy (µχ ) = 0 ,
2
√ 2
π e τ vz αJJ T Ŵ 2 − 1J
 
ℓ�,χ
xy (µχ ) =  f bc (J) Bχ x Bχ y ,
Ŵ 92 − 1J
2

3 × 64 µχJ
(63)
2
√ 2
π e τ vz αJJ T Ŵ 2 − 1J
 
m,χ
ℓxy (µχ ) =  f m (J) Bχ x Bχ y .
Ŵ 92 − 1J
2

J
3 × 64 µχ

Wiedemann‑Franz law
From the explicit expressions of the in-plane longitudinal and transverse components of σ χ and ℓχ, which we
have derived [cf. Eqs. (31), (34), (61), and (62)], we immediately find that the relation
3 e2 χ
χ
σij = ℓ + O(β −2 ) (64)
π 2 T ij
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.

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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 ∇r T ) 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 ∇r T ) 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 + χ B5. 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 possibili-
ties 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 Co3Sn 2S 262, 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
0,χ 0,χ 0,χ
J-values, σxx (µχ ) and ℓxx (µχ ) are proportional to µ2χ, while αxx (µχ ) ∝ µχ. For the magnetic-field-dependent
−2 χ −1− 2J χ 2 χ
parts, the leading-order µχ-dependence goes as (i) µχ J for σij , (ii) µχ for αij , and (iii) µ J for ℓij . 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 apply-
ing 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 �µ ∝ (E · B)τv (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 s­ emimetals65, 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 n­ odes27,28,36,39, 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 a­ pproximation36. 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 cir-
cularly polarized light.

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

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Received: 4 June 2024; Accepted: 25 July 2024

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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 pre-
pared the figures. R.G. helped in some of the calculations. A.M.-R. reviewed the write-up. All authors reviewed
the manuscript.

Competing interests
The authors declare no competing interests.

Additional information
Supplementary Information The online version contains supplementary material available at https://​doi.​org/​
10.​1038/​s41598-​024-​68615-0.
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