Distinguishing features of longitudinal magnetoconductivity for a Rarita-Schwinger-Weyl node

For an orthonormal set of eigenvectors, $\{\psi_{s}(\bm{k})\}$, the explicit forms of the positive-energy bands are given by

	$\displaystyle\psi_{1/2}(\bm{k})$	$\displaystyle=\begin{bmatrix}-\frac{\sqrt{3}\,e^{-3\,i\,\phi}\left(1-\cos^{2}\theta\right)\csc\left(\frac{\theta}{2}\right)}{4}&\frac{e^{-2\,i\,\phi}\cos\left(\frac{\theta}{2}\right)(3\cos\theta-1)}{2}&\frac{e^{-i\,\phi}\sin\left(\frac{\theta}{2}\right)\,(3\cos\theta+1)}{2}&\frac{\sqrt{3}\,\sin\left(\frac{\theta}{2}\right)\sin\theta}{2}\end{bmatrix}^{\rm T}$
	$\displaystyle\text{and }\psi_{3/2}(\bm{k})$	$\displaystyle=\begin{bmatrix}e^{-3\,i\,\phi}\cos^{3}\left(\frac{\theta}{2}\right)&\frac{\sqrt{3}\,e^{-2\,i\,\phi}\sin^{2}\theta\csc\left(\frac{\theta}{2}\right)}{4}&\frac{\sqrt{3}\,e^{-i\,\phi}\sin\left(\frac{\theta}{2}\right)\sin\theta}{2}&\sin^{3}\left(\frac{\theta}{2}\right)\end{bmatrix}^{\rm T}.$		(6)

This will be used to determine the strength of the scattering cross-sections and, also, an ansatz for the nonequilibirum quasiparticle-distributions.

Due to a nontrivial topology of the bandstructure, we need to compute the BC and the OMM, using the starting expressions of [5, 59]

$\displaystyle{\bm{\Omega}}_{s}(\bm{k})=i\left[\nabla_{\bm{k}}\psi_{s}({\bm{k}})\right]^{\dagger}\crossproduct\left[\nabla_{\bm{k}}\psi_{s}({\bm{k}})\right]\text{and }{\bm{m}}^{s}(\bm{k})=\frac{-\,i\,e}{2}\,\left[\nabla_{\bm{k}}\psi_{s}({\bm{k}})\right]^{\dagger}\crossproduct\Big{[}\left\{\mathcal{H}({\bm{k}})-\varepsilon^{s}({\bm{k}})\right\}\left\{\bm{\nabla}_{\bm{k}}\psi_{s}({\bm{k}})\right\}\Big{]},$

(7)

respectively. Evaluating these expressions for the RSW node described by $\mathcal{H}(\bm{k})$, we get [60, 33]

$\displaystyle{\bm{\Omega}}_{s}(\bm{k})$

$\displaystyle=\frac{-\,s\,\bm{k}}{k^{3}}\text{ and }\bm{m}_{s}(\bm{k})=\frac{-\,e\,v_{F}\,\mathcal{G}_{s}\,\bm{k}}{k^{2}}\,,\text{ where }\{\mathcal{G}_{\pm 1/2},\,\mathcal{G}_{\pm 3/2}\}=\left\{\frac{7}{4},\,\frac{3}{4}\right\}.$

(8)

Since $\bm{\Omega}_{s}(\bm{k})$ and $\bm{m}_{s}(\bm{k})$ are the intrinsic properties of the bandstructure, they depend only on the wavefunctions.

A nonzero BC manifests itself primarily via modifying the phase-space volume element, incorporated through the factor of [56, 6, 61, 57, 33, 15]

$\displaystyle{\mathcal{D}}_{s}(\bm{k})=\left[1+e\,\left\{{\bm{B}}\cdot\bm{\Omega}_{s}(\bm{k})\right\}\right]^{-1}.$

(9)

This, in turn, it affects the explicit forms of the Hamilton’s equations, which we will discuss In Sec. III. On the other hand, the OMM distorts the Fermi surfaces along the direction of $\bm{B}$, by adding the term

$\displaystyle\varepsilon_{s}^{(m)}(\bm{k})=-\,{\bm{B}}\cdot\bm{m}_{s}(\bm{k})\,,$

(10)

as schematically depicted in Fig. 1(b). Consequently, in addition to affecting the Fermi-Dirac distributions functions, it causes modifications of the group-velocities via adding

$\displaystyle{\bm{v}}_{s}^{(m)}({\bm{k}})\equiv\nabla_{\bm{k}}\varepsilon_{(\sigma)}(\bm{k})$

(11)

to $\bm{v}_{s}(\bm{k})$.

In thermal equilibrium, the distribution of the quasiparticles follows the Fermi-Dirac distribution,

$\displaystyle f_{0}\big{(}\xi_{s}(\bm{k}),\mu,T\big{)}=\left[1+\exp\left\{\left(\xi_{s}(\bm{k})-\mu\right)/T\right\}\right]^{-1},\quad\xi_{s}({\bm{k}})={\varepsilon}_{s}({\bm{k}})+\varepsilon_{s}^{(m)}(\bm{k})\,,$

(12)

where $T$ is the temperature and $\xi_{s}({\bm{k}})$ is the effective energy (containing the OMM contribution). The quantity $f_{0}$ will appear in the equations that follow, where we will suppress showing its $\mu$- and $T$-dependence explicitly, just for the sake of uncluttering of notations. We note that, as $T\rightarrow 0$, $\partial_{u}f_{0}(u)\rightarrow-\delta(u-\mu)$, which serves to reduce the momentum-space integrals over a Fermi pocket to the respective Fermi surface.

The Hamilton’s equations of motion for the quasiparticles, occupying a given Bloch band, are described by [56, 6, 61, 57, 33, 15]

	$\displaystyle\dot{\bm{r}}$	$\displaystyle=\nabla_{\bm{k}}\,\xi_{s}-\dot{\bm{k}}\,\crossproduct\,\bm{\Omega}_{s}(\bm{k})\Rightarrow\,\dot{\bm{r}}=\mathcal{D}_{s}({\bm{k}})\left[\bm{w}_{s}({\bm{k}})+e\,{\bm{E}}\crossproduct\bm{\Omega}_{s}({\bm{k}})+e\left\{\bm{\Omega}_{s}(\bm{k})\cdot\bm{w}_{s}({\bm{k}})\right\}\bm{B}\right],$
		$\displaystyle\text{and }\dot{\bm{k}}=-\,e\left({\bm{E}}+\dot{\bm{r}}\,\crossproduct\,{\bm{B}}\right)\Rightarrow\dot{\bm{k}}=-\,e\,\mathcal{D}_{s}({\bm{k}})\left[{\bm{E}}+\bm{w}_{s}({\bm{k}})\crossproduct{\bm{B}}+e\left({\bm{E}}\cdot{\bm{B}}\right)\bm{\Omega}_{s}({\bm{k}})\right],$		(13)

where

$\displaystyle\bm{w}_{s}({\bm{k}})={\bm{v}}_{s}(\bm{k})+{\bm{v}}_{s}^{(m)}({\bm{k}})\,.$

(14)

The presence of various terms involving the BC and the OMM reflect the nontrivial role played by the topological properties, as compared to systems where such topology does not exist. For comparison, one can look at the structure of the phase-space equations in Refs. [62, 63], for example. We notice that the term appearing as $-\dot{\bm{k}}\,\crossproduct\,\bm{\Omega}_{s}$ represents an anomalous velocity, with the BC (existing in the momentum space) playing the counterpart of the magnetic field (existing in the real space).

The semiclassical Boltzmann’s transport formalism is implemented by starting with the basic kinetic equations, embodied by

$\displaystyle\left[\partial_{t}+{\bm{w}}_{s}({\bm{k}})\cdot\nabla_{\bm{r}}-e\left\{\bm{E}+{\bm{w}}_{s}({\bm{k}})\times{\bm{B}}\right\}\cdot\nabla_{\bm{k}}\right]f_{s}(\bm{r},\bm{k},t)=I_{\text{coll}}[f_{s}({\bm{k}})]\,.$

(15)

The function $f_{s}(\bm{r},\bm{k},t)$ represents the nonequilibrium quasipaticle-distribution for the fermions populating band $s$, caused by a small deviation from $f_{0}(\xi_{s}(\bm{k}))$, as soon as probe fields (like electric field, temperature gradient [11, 33, 16], or chemical potential gradient [15]) are applied externally. On the right-hand side of Eq. (15), we have the collision integral, $I_{\text{coll}}[f_{s}({\bm{k}})]$. Obviously, its role is to account for the appropriate scattering processes relaxing $f_{s}({\bm{k}})$ towards the equilibrium value of $f_{0}(\xi_{s}({\bm{k}}))$. As seen from the equation itself, the probe field here is just an electric field ($\bm{E}$), and the deviation, $\delta f_{s}(\bm{r},\bm{k},t)\equiv f_{s}(\bm{r},\bm{k},t)-f_{0}(\xi_{s}(\bm{k}))$, is assumed to be of the same order of smallness (say, $\delta$) as $|\bm{E}|$. Since we restrict ourselves to spatially-uniform and time-independent electromagnetic fields, $f_{s}$ must must also be independent of position and time, so that $\delta f_{s}(\bm{r},\bm{k},t)=\delta f_{s}(\bm{k})$. Observing that $\delta f_{s}(\bm{k})\propto|\bm{E}|\propto\delta$, we expand the terms on both sides upto a given order in $\delta$. When we are interested in the linear response, we just retain the leading-order corrections, meaning terms which are linear-in-$\delta$.

In this paper, we will be focussing on the simple case of elastic point-scattering mechanisms, for which the collision integral takes the form of

$\displaystyle I_{\text{coll}}[f_{s}({\bm{k}})]=\sum\limits_{\tilde{s}}\int_{k^{\prime}}\mathcal{M}_{s,\tilde{s}}(\bm{k},\bm{k}^{\prime})\left[f_{\tilde{s}}(\bm{k}^{\prime}))-f_{s}(\bm{k})\right],\text{ where }\int_{k^{\prime}}\equiv\int\frac{d^{3}\bm{k}^{\prime}\,\mathcal{D}^{-1}_{\tilde{s}}({\bm{k}}^{\prime})}{(2\,\pi)^{3}}$

(16)

is the compact symbol for indicating a 3d momentum-space integral. Needless to say, it includes the modified phase-space factor caused by a nonzero BC. The scatterings being elastic, they do not involve any dissipation of energy. Using the Fermi’s golden rule, they can be expressed as

$\displaystyle\mathcal{M}_{s,\tilde{s}}(\bm{k},\bm{k}^{\prime})=\frac{2\,\pi\,\rho_{\rm imp}}{V}\,\Big{|}\left\{\psi_{\tilde{s}}({\bm{k}^{\prime}})\right\}^{\dagger}\;{\mathcal{V}}_{s,\tilde{s}}(\bm{k},\bm{k}^{\prime})\;\psi_{s}({\bm{k}})\Big{|}^{2}\,\delta\Big{(}\xi_{\tilde{s}}(\bm{k}^{\prime})-\xi_{s}(\bm{k})\Big{)}\,,$

(17)

where, $\rho_{\rm imp}$ denotes the impurity-concentration, $V$ stands for the system’s spatial volume, and ${\mathcal{V}}_{s,\tilde{s}}(\bm{k},\bm{k}^{\prime})$ represents the matrix-elements of the effective scattering-potential. The consideration of elastic and spinless scattering-centres reduces the potential to an identity matrix in the spinor space, i.e., ${\mathcal{V}}_{s,\tilde{s}}(\bm{k},\bm{k}^{\prime})=\mathbb{I}_{2\times 2}\,{\mathcal{V}}_{s,\tilde{s}}$. Thus, we can now use the version of

$\displaystyle\mathcal{M}_{s,\tilde{s}}(\bm{k},\bm{k}^{\prime})=\frac{2\,\pi\,\rho_{\rm imp}\,|{\mathcal{V}}_{s,\tilde{s}}|^{2}}{V}\,\Big{|}\left\{\psi_{\tilde{s}}({\bm{k}^{\prime}})\right\}^{\dagger}\;\psi_{s}({\bm{k}})\Big{|}^{2}\,\delta\Big{(}\xi_{\tilde{\chi},\tilde{s}}(\bm{k}^{\prime})-\xi_{\chi,s}(\bm{k})\Big{)}\,,$

(18)

which can be easily computed by using the spinorial forms shown in Eq. (II.1). Assuming a reciprocal interband-scattering function between the $s=1/2$ and $s=3/2$ bands, we set ${\mathcal{V}}_{1/2,3/2}={\mathcal{V}}_{3/2,1/2}$. For this system, altogether we need three parameters for the scattering strengths, leading to

$\displaystyle\big{|}{\mathcal{V}}_{1/2,1/2}\big{|}^{2}\equiv\frac{16\times 2\,\pi}{\rho_{\rm imp}}\,\beta_{\rm intra}^{1/2,1/2}\,,\quad\big{|}{\mathcal{V}}_{3/2,3/2}\big{|}^{2}\equiv\frac{16\times 2\,\pi}{\rho_{\rm imp}}\,\beta_{\rm intra}^{3/2,3/2}\,,\text{ and }\big{|}{\mathcal{V}}_{s,{\tilde{s}}}\big{|}^{2}\Big{|}_{s\neq\tilde{s}}\equiv\frac{16\times 2\,\pi}{\rho_{\rm imp}}\,\beta_{\rm inter}\,.$

(19)

We have added the self-explanatory subscripts, “intra” and “inter”, to clearly indicate the intraband- and interband-parts, respectively.

Gathering all the ingredients of the preceding subsections, we are now ready to solve the linearised Boltzmann equation, embodied by

$\displaystyle-e\,{\mathcal{D}}_{s}({\bm{k}})\left[\left\{{\bm{w}}_{s}({\bm{k}})+e\,\Big{(}{\bm{\Omega}}_{s}({\bm{k}})\cdot{\bm{w}}_{s}({\bm{k}})\Big{)}\bm{B}\right\}\cdot{\bm{E}}\;\;\frac{\partial f_{0}(\xi_{s}({\bm{k}}))}{\partial\xi_{s}({\bm{k}})}+\left\{{\bm{w}}_{s}({\bm{k}})\crossproduct{\bm{B}}\right\}\cdot\nabla_{\bm{k}}\,\delta f_{s}(\bm{k})\right]=I_{\text{coll}}[f_{s}({\bm{k}})]\,.$

(20)

Specifically, we choose the collinear electromagnetic fields to act along the $z$-axis, such that $\bm{B}=B\,\bm{\hat{z}}$ and $\bm{E}=E\,\bm{\hat{z}}$. To proceed further, we rewrite the deviation in the particles’ distribution-functions as

$\displaystyle\delta f_{s}(\bm{k})=-\,e\,\frac{\partial f_{0}(\xi_{s})}{\partial\xi_{s}}\,{\bm{E}}\cdot\bm{\Lambda}_{s}(\bm{k})=-\,e\,\frac{\partial f_{0}(\xi_{s}({\bm{k}}))}{\partial\xi_{s}({\bm{k}})}\,E\,{\Lambda}^{z}_{s}(\bm{k})\,,$

(21)

where $\bm{\Lambda}_{s}(\bm{k})$ is the vectorial mean-free path. Because of our specific choice of coordinates, only the $z$-component of $\bm{\Lambda}_{s}(\bm{k})$ [i.e., ${\Lambda}^{z}_{s}(\bm{k})$] emerges as the nontrivial component as a result of the probe electric field. Consequently, the only equation that needs to be solved turns out to be

$\displaystyle w^{z}_{s}({\bm{k}})+e\,B\left[{\bm{\Omega}}_{s}({\bm{k}})\cdot{\bm{w}}_{s}({\bm{k}})\right]-\,e\,B\left[{\bm{w}}_{s}({\bm{k}})\crossproduct\bm{\hat{z}}({\bm{k}})\right]\cdot\nabla_{\bm{k}}{\Lambda}^{z}_{s}(\bm{k})={\mathcal{D}}^{-1}_{s}({\bm{k}})\sum\limits_{\tilde{\chi},\tilde{s}}\int_{k^{\prime}}\mathcal{M}_{s,\tilde{s}}(\bm{k},\bm{k}^{\prime})\left[{\Lambda}^{z}_{\tilde{s}}(\bm{k}^{\prime})-{\Lambda}^{z}_{s}(\bm{k})\right].$

(22)

Looking at the structure of the terms on both the sides, we infer that ${\Lambda}^{z}_{s}\equiv{\Lambda}^{z}_{s}(\mu,\theta)$ at an energy $\mu$, which is independent of $\phi$. The fact that $\delta f_{s}(\bm{k})$ can depend only $\theta$ and $\mu$ follows from the constraints imposed by the delta functions, reducing an integral over the full momentum space to one over the Fermi-surface contour with energy-profile $\xi_{s}(\bm{k})=\mu$. Essentially, the dependence on the azimuthal angle $\phi$ drops out because of the residual rotational symmetry of the system about the $z$-axis. The equations of the Fermi surfaces are derived by solving $\xi_{s}(k_{F}^{s},\theta)=\mu$, with the $\theta$-dependent radii of $\{k_{F}^{s}(\theta)\}$ parametrising the local Fermi momenta. Overall, the justification of assuming the $\phi$-independent forms of $\{{\Lambda}^{z}_{s}\equiv{\Lambda}^{z}_{s}(\mu,\theta)\}$ is evident from the self-consistency — $\left[{\bm{\Omega}}_{s}(\bm{k})\cdot{\bm{w}}_{s}(\bm{k})\right]$ is $\phi$-independent and $\left[{\bm{w}}_{s}(\bm{k})\crossproduct\bm{\hat{z}}\right]\cdot\nabla_{\bm{k}}{\Lambda}^{z}_{s}(\mu,\theta)$ reduces to zero.

Making use of the above arguments, we can now compute the spinor overlaps, $\big{|}\left\{\psi_{\tilde{s}}({\bm{k}^{\prime}})\right\}^{\dagger}\;\psi_{s}({\bm{k}})\big{|}^{2}$, with the help of Eq. (II.1). Knowing that the dependence of the azimuthal angles, $\phi$ and $\phi^{\prime}$, will disappear once we perform the azimuthal-angle integrations, we list the overlap-functions as

	$\displaystyle{\mathcal{T}}_{s,\tilde{s}}(\theta,\theta^{\prime})$	$\displaystyle=\left[5-3\cos^{2}\theta^{\prime}+\cos\theta\left(17\cos\theta^{\prime}-27\cos^{3}\theta^{\prime}\right)+\cos^{2}\theta\left(9\cos^{2}\theta^{\prime}-3\right)+9\cos^{3}\theta\cos\theta^{\prime}\left(5\cos^{2}\theta^{\prime}-3\right)\right]\beta_{\rm intra}^{1,1}$
		$\displaystyle\quad+\left[5-3\cos^{2}\theta^{\prime}+\cos\theta\left(9\cos\theta^{\prime}-3\cos^{3}\theta^{\prime}\right)+\cos^{2}\theta\left(9\cos^{2}\theta^{\prime}-3\right)+\cos^{3}\theta\left(5\cos^{3}\theta^{\prime}-3\cos\theta^{\prime}\right)\right]\beta_{\rm intra}^{3,3}$
		$\displaystyle\quad+\left[3+3\cos^{2}\theta^{\prime}+\cos^{3}\theta\left(9\cos\theta^{\prime}-15\cos^{3}\theta^{\prime}\right)+\cos\theta\left(9\cos^{3}\theta^{\prime}-3\cos\theta^{\prime}\right)+\cos^{2}\theta\left(3-9\cos^{2}\theta^{\prime}\right)\right]\beta_{\rm inter}\,.$		(23)

Here, the answer contains the results obtained after already performing the trivial $\phi$-integrals. We are now ready to solve Eq. (22), starting from its simplified version of

$\displaystyle h_{s}(\mu,\theta)=\sum_{\tilde{s}}V\int_{k^{\prime}}\,\mathcal{M}_{s,\tilde{s}}(\bm{k},\bm{k}^{\prime})\,{\Lambda}^{z}_{\tilde{s}}(\mu,\theta^{\prime})-\frac{{\Lambda}^{z}_{s}(\mu,\theta)}{\tau_{s}(\mu,\theta)}\,,$

(24)

where

$\displaystyle\tau^{-1}_{s}(\mu,\theta)=\sum_{\tilde{s}}V\int_{k^{\prime}}\mathcal{M}_{s,\tilde{s}}(\bm{k},\bm{k}^{\prime})\,,\quad h_{s}(\mu,\theta)={\mathcal{D}}_{s}({\bm{k}})\left[w^{z}_{s}({\bm{k}})+e\,B\left\{{\bm{\Omega}}_{s}({\bm{k}})\cdot{\bm{w}}_{s}({\bm{k}})\right\}\right].$

(25)

Incorporating the energy-conservation restrictions, which reduce the integrals to the respective Fermi surfaces at energy $\mu$, we simply it even further to

$\displaystyle h_{s}(\mu,\theta)+\frac{{\Lambda}^{z}_{s}(\mu,\theta)}{\tau_{s}(\mu,\theta)}=\sum_{\tilde{s}}\frac{\rho_{\rm imp}\,|{\mathcal{V}}_{s,\tilde{s}}|^{2}}{16\times 4\,\pi}\int d\theta^{\prime}\,\frac{\sin\theta^{\prime}\left(k^{\prime}\right)^{3}\,{\mathcal{D}}^{-1}_{\tilde{s}}({\bm{k}}^{\prime})}{|\bm{k}^{\prime}\cdot{\bm{w}}_{s}(\bm{k}^{\prime})|}\,{\mathcal{T}}_{s,\tilde{s}}(\theta,\theta^{\prime})\,{\Lambda}^{z}_{\tilde{s}}(\mu,\theta^{\prime})\Big{|}_{k^{\prime}=k_{F}^{\tilde{s}}}\,.$

(26)

We note that the factor $\left(k^{\prime}\right)^{3}\sin\theta^{\prime}$ arises as the Jacobian for switching to the spherical polar coordinates. Tackling the factors of $\delta\Big{(}\xi_{\tilde{s}}(\bm{k}^{\prime})-\mu\Big{)}$, arising from Eq. (18), needs re-expressing them in the language of $\delta(k^{\prime}-k_{F}^{\tilde{s}})$, which sources the appearance of the other multiplicative factor of

$$\big{|}\bm{\hat{k}^{\prime}}\cdot\nabla_{\bm{k}^{\prime}}\xi_{s}({\bm{k}^{\prime}})\big{|}^{-1}=k^{\prime}/|\bm{k}^{\prime}\cdot{\bm{w}}_{s}(\bm{k}^{\prime})|\,.$$

As a final step, observing the powers of $\cos\theta$ in (III.2)], we set forth an ansatz of

$\displaystyle{\Lambda}^{z}_{s}(\mu,\theta)$

$\displaystyle=\tau_{s}(\mu,\theta)\left[\lambda_{s}-h_{s}+a_{s}\cos\theta+b_{s}\cos^{2}\theta+c_{s}\cos^{3}\theta\right].$

(27)

In total we have 8 undetermined coefficients so determine, denoted as the set $\{\lambda_{s},\,a_{s},\,b_{s},\,c_{s}\}$. We have 8 linear equations at our disposal, extracted from Eq. (26) by equating the coefficients of $1$, $\cos\theta$, $\cos^{2}\theta$, and $\cos^{3}\theta$ on both the sides for each value of $s$. This system of linear equations can be expressed compactly as a matrix equation, captured by

$\displaystyle\mathcal{A}\,\mathcal{N}=\Upsilon\,,\text{ where }\mathcal{N}=\begin{bmatrix}\lambda_{1/2}&a_{1/2}&b_{1/2}&c_{1/2}&\lambda_{3/2}&a_{3/2}&b_{3/2}&c_{3/2}\end{bmatrix}^{\rm T}\,.$

(28)

The explicit forms of the square and row matrices, $\mathcal{A}$ and $\Upsilon$, are demonstrated in the appendix. Now there is a catch —- the eight equations represented above are not linearly independent. This is because $\mathcal{A}$’s rank is lower than 8. It is actually well and good since it prevents from the system from becoming overdetermined, as we must also employ the constraint of the electron-number conservation, embodied by

$\displaystyle\sum\limits_{s}\int_{k}\delta f_{s}(\bm{k})=0\,.$

(29)

Plugging in the solutions obtained, the charge-current density along the $z$-direction is computed as

$\displaystyle J_{z}^{\rm tot}=-\,\frac{e}{V}\sum_{s}\int_{k}(\dot{\bm{r}}\cdot\bm{\hat{z}})\;\delta f_{s}(\bm{k})\,,$

(30)

finally leading to the desired longitudinal magnetoconductivity,

$\displaystyle\sigma_{zz}^{\rm tot}=\sum_{s}\sigma_{zz}^{s}\,,\quad\sigma_{zz}^{s}=-\,\frac{e^{2}}{V}\int\frac{d^{3}{\bm{k}}}{(2\,\pi)^{3}}\left[w^{z}_{s}(\bm{k})+e\,B\left\{\bm{\Omega}_{s}(\bm{k})\cdot\bm{w}_{s}(\bm{k})\right\}\right]\delta\big{(}\xi_{s}(\bm{k})-\mu\big{)}\,{\Lambda}^{z}_{s}(\mu,\theta)\,.$

(31)

The solutions of Eq. (26) are obtained numerically for a given set of parameter-values. In our illustrations, we resort to plotting

$$\delta\sigma_{zz}(s)\equiv\sigma^{s}_{zz}(B)/\sigma^{s}_{zz}(B=0)-1,$$

where the residual conductivity at $B=0$ (i.e., the Drude part) is subtracted off. The parameter values have been taken from Ref. [58]. There, $\hbar v_{F}=1.23$ eV Å, which translates into

$\displaystyle v_{F}=\frac{1.23\times 10^{-10}\text{ m eV}}{6.58\times 10^{-16}\text{ eV s}}=0.1869\times 10^{6}\text{ m/s}\simeq 0.06\times 10^{-2}\,c\,,$

where $c$ is the speed of light. Therefore, in natural units, we set the value of $v_{F}$ to $6\times 10^{-4}$.

To start with, we attempt to make sense of the response-characteristics for the scenarios when there is no interband scattering, i.e., $\beta_{\rm inter}$ is set to zero. This leads to the reduction of $\mathcal{A}$ into the direct sum of the two matrices, $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$, each of them being a $4\times 4$ matrix. In every case, we find that they have rank $3$ (instead of $4$), with $\int_{k}\delta f_{s}(\bm{k})=0$ providing the necessary extra equation. Fig. 2 captures such a scenario. Curiously, the parabolic-shaped response-profiles curve in opposite ways for $s=1/2$ and $s=3/2$. In order to gauge the nature of the OMM-contributions, the lower panel represent the behaviour bbtained by switching off the OMM. It leads to the conclusion that (I) the OMM-parts come with a positive sign and are strong enough to flip the BC-only response into the positive domain for $s=1/2$; (II) the OMM-contributions have a negative sign which assist in increasing the magnitude of the already negative-valued response (from the BC-only parts) for $s=3/2$.

Next, we consider the generic situations when the quasiparticles populating both the bands can scatter with each other. The $8\times 8$ matrix $\mathcal{A}$ is found to have rank $7$, ensuring the internal consistency of the available equations. As discussed earlier, this means that the charge-conservation provides the remaining linearly independent equation [viz., Eq. (29)]. Fig. 3 provides some representative curves. From the data presented here, as well as our scans of the parameter space (not shown here), we find that for $\beta_{\rm inter}/\beta_{\rm intra}^{s,s}$ greater than some small threshold value, the curves for $s=3/2$ are turned upside down (into the negative domain), while the $s=1/2$ continuing to curve upwards (remaining positive). To probe into the role of the OMM-sourced parts, we plot the nature of the curves by switching off the OMM (lower panel of Fig. 3). Intriguingly, the curves for $s=1/2$ and $s=3/2$ behave in an opposite nature, the former always remaining negatve and curving downward, irrespective of the values of $\beta_{\rm inter}$. Overall, comparing the upper and the lower panels, we conclude that, while the OMM-parts have a positive sign and are strong enough to flip the BC-only response into the positive domain for $s=1/2$, the OMM-contributions have a negative sign and try to pull the positive-valued response (from the BC-only parts) towards lower values for $s=3/2$.

Overall, $\sigma^{s}_{zz}(B)$ comprises only even powers of $B$, which is in agreement with the Onsager-Casimir reciprocity relation, $\sigma^{\rm intra(inter)}_{zz}(\mathbf{B})=\sigma^{\rm intra(inter)}_{zz}(-\mathbf{B})$ [64, 65, 66]. From analysing the generic features, we conclude that there is little wisdom to be gained from the $\beta_{\rm inter}=0$ to infer anything regarding the $\beta_{\rm inter}\neq 0$ case. The only thing that matches between the two separate analyses is regarding the signs and strengths of the OMM-sourced parts. Albeit, the rationale behind not to try to extract similarities between the two cases is that the charge conservation equation(s) that come into picture are distinctly different — (I) for a single band with $\beta_{\rm inter}=0$, the charge conservation must be satisfied individually for that particular band; (II) a nonzero interband-coupling demands that the net charge, from summing over the two bands, must be conserved. The observations serve to reinforce the recurrent phenomenon that the response from a single isolated band is fundamentally different from the case when the quasiparticles are free to scatter with those populating other bands. In fact, the same thing has been onserved for the two bands of an isolated Kramers-Weyl node [35], where downward-curved parabolas morph into upward-curved ones for each of the bands, as soon as $\beta_{\rm inter}$ is turned on.