Finally, note that if we are interested on the $\sigma_{00}$ term of the conductivity tensor, instead of Eq. (76), we have to use

$$ \sigma^{00}(q) = \frac{q_{\parallel}^2}{q_0^2} \left[ \sigma_L(q) - \lim_{q_0 \to 0} \sigma_L(q) \right], \quad (A14) $$

to avoid the appearance of any possible spurious term in the $\omega \to 0$ limit.

## Appendix B: Local Kubo conductivity expressions

There is a special case in the local limit (when we apply the $q_{\parallel} \to 0$ limit to the Kubo formula (69)) when results valid for all temperatures can be obtained. The local limit of the conductivities of a massive Dirac cone are

$$ \begin{aligned} \sigma_{xx}(\omega, \mathbf{0}, \mu, \Delta) &= i \frac{\sigma_0}{\pi} \left[ \frac{\mu^2 - \Delta^2}{|\mu|} \frac{1}{\Omega} \Theta(|\mu| - |\Delta|) + \frac{\Delta^2}{M\Omega} - \frac{\Omega^2 + 4\Delta^2}{2i\Omega^2} \tan^{-1} \left( \frac{i\Omega}{2M} \right) \right], \\ \sigma_{xy}(\omega, \mathbf{0}, \mu, \Delta) &= \frac{2\sigma_0}{\pi} \frac{\eta\Delta}{i\Omega} \tan^{-1} \left( \frac{i\Omega}{2M} \right), \end{aligned} \quad (B1) $$

where $\Omega = \hbar\omega + i\hbar\Gamma$, $\Gamma = \tau^{-1}$ and $M = \text{Max}[|\Delta|, |\mu|]$. These results are per Dirac cone and they are consistent with the ones found by other researchers [28][31][32][33][47][87][88][89]. The first term in $\sigma_{xx}$ corresponds to inter-band transitions, and the last two terms to inter-band transitions. Note that, in the local limit $q_{\parallel} \to 0$ one obtains $\sigma_{xx}(\omega, 0) = \sigma_{yy}(\omega, 0) = \sigma_L(\omega, 0) = \sigma_T(\omega, 0)$, and $\sigma_{xy}(\omega, 0) = -\sigma_{yx}(\omega, 0) = \sigma_H(\omega, 0)$.

From those results, we obtain the conductivity at zero temperature of a Dirac cone with $\Delta = 0$ mass gap as

$$ \begin{aligned} \sigma_{xx}(\omega, \mathbf{q}_{\parallel} = \mathbf{0}, T = 0, \Delta = 0) &= \sigma_{xx}^{\text{intra}}(\omega) + \sigma_{xx}^{\text{inter}}(\omega), \\ \sigma_{xx}^{\text{intra}}(\omega) &= \frac{\alpha c}{\pi} \frac{|\mu|}{-i\Omega}, \\ \sigma_{xx}^{\text{inter}}(\omega) &= \frac{\alpha c}{2\pi} \tan^{-1} \left( \frac{-i\Omega}{2|\mu|} \right), \\ \sigma_{xy}(\omega, \mathbf{q}_{\parallel} = \mathbf{0}, T = 0, \Delta = 0) &= 0. \end{aligned} \quad (B2) $$

From this result with $\Delta = 0$, in the non-dissipation limit for the interband term and for finite temperatures, by using the Maldague formula, the well-known result of Falkovsky is obtained as

$$ \begin{aligned} \sigma_{xx}^{F}(\omega, \Gamma, T, \mu) &= \sigma_{xx}^{F, \text{intra}}(\omega, \Gamma, T, \mu) + \sigma_{xx}^{F, \text{inter}}(\omega, 0, T, \mu), \\ \sigma_{xx}^{F, \text{intra}}(\omega, \Gamma, T, \mu) &= \frac{1}{\pi \hbar} \frac{2i \alpha c k_B T}{\omega + i \Gamma} \ln \left[ 2 \cosh \left( \frac{\mu}{2k_B T} \right) \right], \\ \sigma_{xx}^{F, \text{inter}}(\omega, \Gamma, T, \mu) &= \frac{\alpha c}{4} G \left( \left| \frac{\hbar \omega}{2} \right| \right) + i \frac{\alpha c}{4} \frac{4 \hbar \omega}{\pi} \int_0^{\infty} d\xi \frac{G(\xi) - G\left(\left| \frac{\hbar \omega}{2} \right|\right)}{(\hbar \omega)^2 - 4\xi^2}. \end{aligned} \quad (B3) $$

with

$$ G(\epsilon) = n_F(-\epsilon + \mu) - n_F(+\epsilon + \mu) = \frac{\sinh(\beta\epsilon)}{\cosh(\beta\mu) + \cosh(\beta\epsilon)}. \quad (B4) $$

The intraband term is obtained by the use of the Maldague formula (Eq. (92)) for Eq. (94), as

$$ \sigma_{xx}^{\text{intra}}(\omega, T) = \frac{\alpha c}{\pi} \frac{1}{-i\Omega} \frac{2}{\beta} \ln \left[ 2 \cosh \left( \frac{\beta\mu}{2} \right) \right] = \frac{\alpha c}{\pi \hbar} \frac{2i k_B T}{\omega + i\Gamma} \ln \left[ 2 \cosh \left( \frac{\mu}{2k_B T} \right) \right]. \quad (B5) $$

To derive the local interband conductivity at finite temperatures, we first need to apply the zeroth dissipation limit $\tau \to \infty$ ($\Gamma \to 0$) of the real part $\omega \in \mathbb{R}$ of the interband conductivity of graphene, as

$$ \lim_{\tau \to \infty} \mathrm{Re} [\sigma_{xx}^{\text{inter}}(\omega)] = \lim_{\tau \to \infty} \frac{\alpha c}{2\pi} \tan^{-1} \left( \frac{-i\Omega}{2|\mu|} \right) = \frac{\alpha c}{4} \left[ \Theta(\hbar\omega - 2|\mu|) + \Theta(-\hbar\omega - 2|\mu|) \right], \quad (B6) $$

which coincides with the result shown in [31] in the $T \to 0$ limit for $\omega > 0$. Note that $\mathrm{Re}[\sigma_{xx}^{\text{inter}}(-\omega)] = +\mathrm{Re}[\sigma_{xx}^{\text{inter}}(\omega)]$ is an even function in $\omega$.
Applying the Maldague formula (Eq. (92)) to Eq. (94), we obtain

$$ \mathrm{Re} [\sigma_{xx}^{\mathrm{inter}}(\omega, \mu, T)] = G \left( \left| \frac{\hbar\omega}{2} \right| \right) = \frac{\alpha c}{4} \left( \frac{1}{e^{\beta(\mu - \frac{\hbar\omega}{2})} + 1} - \frac{1}{e^{\beta(\mu + \frac{\hbar\omega}{2})} + 1} \right) = \frac{\alpha c}{4} \frac{\sinh\left(\beta\left|\frac{\hbar\omega}{2}\right|\right)}{\cosh\left(\beta\mu\right) + \cosh\left(\beta\left|\frac{\hbar\omega}{2}\right|\right)} \quad (\text{B7}) $$

We have the absolute value in the formula above because we are handling also negative real frequencies, while in [31] the result for only positive frequencies where derived. Applying a regularized version of the Kramers-Krönig relationships that avoids the use of the principal part of a function, that read as [100][101][102]:

$$ \begin{aligned} \sigma_R(x) &= \frac{2}{\pi} \int_0^\infty d\omega \frac{\omega \sigma_I(\omega) - x \sigma_I(x)}{\omega^2 - x^2}, \\ \sigma_I(x) &= \frac{2}{\pi} x \int_0^\infty d\omega \frac{\sigma_R(x) - \sigma_R(\omega)}{\omega^2 - x^2}, \end{aligned} \quad (\text{B8}) $$

it is immediate that

$$ \mathrm{Im} [\sigma^{\mathrm{inter}}(\omega, T)] = \frac{4\hbar\omega}{\pi} \int_{0}^{\infty} d\xi \frac{G(\xi) - G\left(\left|\frac{\hbar\omega}{2}\right|\right)}{(\hbar\omega)^2 - 4\xi^2}, \quad (\text{B9}) $$

which coincides with the imaginary part of the interband term of Falkovsky [103].

Finally, by using the Kramers-Krönig relation to find the real part of the conductivity at imaginary frequencies $\omega = i\xi$ [71]

$$ \mathrm{Re} [\sigma_{p,0}(i\xi)] = \frac{2}{\pi} \int_{0}^{\infty} d\omega \frac{\xi}{\omega^2 + \xi^2} \mathrm{Re} [\sigma_{p,0}(\omega)] = \frac{2}{\pi} \int_{0}^{\infty} d\omega \frac{\omega}{\omega^2 + \xi^2} \mathrm{Im} [\sigma_{p,0}(\omega)], \quad (\text{B10}) $$

we obtain the interband conductivity for imaginary frequencies as [15]

$$ \sigma_F^{\mathrm{inter}}(i\hbar\xi) = \frac{2}{\pi} \int_0^\infty d\omega \frac{\hbar\xi}{(\hbar\omega)^2 + (\hbar\xi)^2} G\left(\frac{\hbar\omega}{2}\right). \quad (\text{B11}) $$

Note that this result is completely equivalent to the use of the Maldague formula to $\sigma_{xx}^{\mathrm{inter}}$ given in Eq. (B2), and by making the substitution $\xi \to \xi + \Gamma$, we can automatically add the constant dissipation to this interband conductivity.

## Appendix C: Derivation of the Polarization and conductivity from the QFT-b model previously used in literature

In this appendix we will show how to derive the results for the polarization operator given in [8][21][9] presented in section VII from Eq. (102).

$$ \Pi^{\mu\nu}(q) = ig_s \frac{\alpha c}{\hbar} \int_k \frac{Z^{\mu\nu}(K_\alpha, q_\alpha)}{[K^\mu K_\mu - m^2] [S^\mu S_\mu - m^2]}, \quad (\text{C1}) $$

We are going to use the following notation: $S_\mu = K_\mu + q_\mu$, $K_\mu = (\tilde{k}_0 + \mu, \tilde{k}_\parallel) = (i\hbar\omega_n + \mu, \hbar v_F k_\parallel)$, $q_\mu = (\tilde{q}_0, \tilde{q}_\parallel) = (i\hbar\omega_m, \hbar v_F q_\parallel)$ (see tab. I), $Z^{\mu\nu}(K_\alpha, q_\alpha)$ is given in Eq. (104).

We apply the Matsubara formalism directly to the expression of Eq. (C1) to obtain the expression shown in [9]. Remembering that, for fermions we have $\tilde{k}_0 = i\hbar\omega_n = i\frac{2\pi}{\beta}(n + \frac{1}{2}) \forall n \in \mathbb{Z}$ and $\tilde{q}_0 = i\hbar\omega_m = i\frac{2\pi}{\beta}(m + \frac{1}{2}) \forall m \in \mathbb{Z}$, we
Applying the Maldague formula (Eq. (92)) to Eq. (94), we obtain

$$ \mathrm{Re} [\sigma_{xx}^{\mathrm{inter}}(\omega, \mu, T)] = G \left( \left| \frac{\hbar\omega}{2} \right| \right) = \frac{\alpha c}{4} \left( \frac{1}{e^{\beta(\mu - \frac{\hbar\omega}{2})} + 1} - \frac{1}{e^{\beta(\mu + \frac{\hbar\omega}{2})} + 1} \right) = \frac{\alpha c}{4} \frac{\sinh\left(\beta\left|\frac{\hbar\omega}{2}\right|\right)}{\cosh\left(\beta\mu\right) + \cosh\left(\beta\left|\frac{\hbar\omega}{2}\right|\right)} \quad (B7) $$

We have the absolute value in the formula above because we are handling also negative real frequencies, while in [31] the result for only positive frequencies where derived. Applying a regularized version of the Kramers-Krönig relationships that avoids the use of the principal part of a function, that read as [100][101][102]:

$$ \begin{aligned} \sigma_R(x) &= \frac{2}{\pi} \int_0^\infty d\omega \frac{\omega \sigma_I(\omega) - x \sigma_I(x)}{\omega^2 - x^2}, \\ \sigma_I(x) &= \frac{2}{\pi} x \int_0^\infty d\omega \frac{\sigma_R(x) - \sigma_R(\omega)}{\omega^2 - x^2}, \end{aligned} \quad (B8) $$

it is immediate that

$$ \mathrm{Im} [\sigma^{\mathrm{inter}}(\omega, T)] = \frac{4\hbar\omega}{\pi} \int_0^\infty d\xi \frac{G(\xi) - G\left(\left|\frac{\hbar\omega}{2}\right|\right)}{(\hbar\omega)^2 - 4\xi^2}, \quad (B9) $$

which coincides with the imaginary part of the interband term of Falkovsky [103].

Finally, by using the Kramers-Krönig relation to find the real part of the conductivity at imaginary frequencies $\omega = i\xi$ [71]

$$ \mathrm{Re} [\sigma_{p,0}(i\xi)] = \frac{2}{\pi} \int_0^\infty d\omega \frac{\xi}{\omega^2 + \xi^2} \mathrm{Re} [\sigma_{p,0}(\omega)] = \frac{2}{\pi} \int_0^\infty d\omega \frac{\omega}{\omega^2 + \xi^2} \mathrm{Im} [\sigma_{p,0}(\omega)], \quad (B10) $$

we obtain the interband conductivity for imaginary frequencies as [15]

$$ \sigma_F^{\mathrm{inter}}(i\hbar\xi) = \frac{2}{\pi} \int_0^\infty d\omega \frac{\hbar\xi}{(\hbar\omega)^2 + (\hbar\xi)^2} G\left(\frac{\hbar\omega}{2}\right). \quad (B11) $$

Note that this result is completely equivalent to the use of the Maldague formula to $\sigma_{xx}^{\mathrm{inter}}$ given in Eq. (B2), and by making the substitution $\xi \to \xi + \Gamma$, we can automatically add the constant dissipation to this interband conductivity.

## Appendix C: Derivation of the Polarization and conductivity from the QFT-b model previously used in literature

In this appendix we will show how to derive the results for the polarization operator given in [8][21][9] presented in section VII from Eq. (102).

$$ \Pi^{\mu\nu}(q) = ig_s \frac{\alpha c}{\hbar} \int_k \frac{Z^{\mu\nu}(K_\alpha, q_\alpha)}{[K^\mu K_\mu - m^2] [S^\mu S_\mu - m^2]}, \quad (C1) $$

We are going to use the following notation: $S_\mu = K_\mu + q_\mu$, $K_\mu = (\tilde{k}_0 + \mu, \tilde{k}_\parallel) = (i\hbar\omega_n + \mu, \hbar v_F k_\parallel)$, $q_\mu = (\tilde{q}_0, \tilde{q}_\parallel) = (i\hbar\omega_m, \hbar v_F q_\parallel)$ (see tab. I), $Z^{\mu\nu}(K_\alpha, q_\alpha)$ is given in Eq. (104).

We apply the Matsubara formalism directly to the expression of Eq. (C1) to obtain the expression shown in [9]. Remembering that, for fermions we have $\tilde{k}_0 = i\hbar\omega_n = i\frac{2\pi}{\beta}(n + \frac{1}{2}) \forall n \in \mathbb{Z}$ and $\tilde{q}_0 = i\hbar\omega_m = i\frac{2\pi}{\beta}(m + \frac{1}{2}) \forall m \in \mathbb{Z}$, we
obtain

$$
\begin{align*}
\frac{\hbar \Pi^{\mu\nu}(q)}{ig_s \alpha c} &= \int_{-\infty}^{\infty} \frac{d\tilde{k}_0}{2\pi} \frac{Z^{\mu\nu}(\tilde{k}_0 + \mu, \tilde{k}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{[K^{\mu}K_{\mu} - m^2] [S^{\mu}S_{\mu} - m^2]} \\
&= \int_{-\infty}^{\infty} \frac{d\tilde{k}_0}{2\pi} \frac{Z^{\mu\nu}(\tilde{k}_0 + \mu, \tilde{k}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{[(\tilde{k}_0 + \mu)^2 - \epsilon_{\tilde{k}}^2] [(\tilde{k}_0 + \tilde{q}_0 + \mu)^2 - \epsilon_{\tilde{k}+\tilde{q}}^2]} \\
&= \frac{1}{\beta} \sum_{n \in \mathbb{Z}^{\text{Fermi}}} \frac{Z^{\mu\nu}(i\hbar\omega_n + \mu, \tilde{k}_{\parallel}, i\hbar\omega_m, \tilde{q}_{\parallel})}{[(i\hbar\omega_n + \mu)^2 - \epsilon_{\tilde{k}}^2] [(i\hbar\omega_n + i\hbar\omega_m + \mu)^2 - \epsilon_{\tilde{s}}^2]} \\
&= \left[ \frac{Z_{\mu\nu}(-i\hbar\omega_m - \epsilon_s, \tilde{k}_{\parallel}, i\hbar\omega_m, \tilde{q}_{\parallel})}{2\epsilon_s ((i\hbar\omega_m + \epsilon_s)^2 - \epsilon_{\tilde{k}}^2)} + \frac{Z_{\mu\nu}(-\epsilon_{\tilde{k}}, \tilde{k}_{\parallel}, i\hbar\omega_m, \tilde{q}_{\parallel})}{2\epsilon_{\tilde{k}} ((i\hbar\omega_m - \epsilon_{\tilde{k}})^2 - \epsilon_{\tilde{s}}^2)} \right] \\
&\quad - \sum_{\lambda=\pm} \frac{n_F(\epsilon_s - \lambda\mu)Z_{\mu\nu}(-i\hbar\omega_m + \epsilon_s, \tilde{k}_{\parallel}, i\hbar\omega_m, \tilde{q}_{\parallel})}{2\epsilon_s ((i\hbar\omega_m - \epsilon_s^\lambda)^2 - \epsilon_{\tilde{k}}^2)} - \sum_{\lambda=\pm} \frac{n_F(\epsilon_{\tilde{k}} - \lambda\mu)Z_{\mu\nu}(\epsilon_{\tilde{k}}^\lambda, \tilde{k}_{\parallel}, i\hbar\omega_m, \tilde{q}_{\parallel})}{2\epsilon_{\tilde{k}} ((i\hbar\omega_m + \epsilon_{\tilde{k}}^\lambda)^2 - \epsilon_{\tilde{s}}^2)}, \quad (C2)
\end{align*}
$$

where we have used that $n_F(-i\hbar\omega_m + \epsilon_k - \lambda\mu) = n_F(\epsilon_k - \lambda\mu)$ and $s_\parallel = k_\parallel + q_\parallel$ (see tab. I). Next, we analytically expand the Matsubara sum given in Eq. (C2) to the whole upper complex plane by applying the formal change $i\omega_m = \omega \in \mathbb{C}^+$ into the definition of the polarization operator given in Eq. (C1), using $\tilde{q}_0 = \hbar\omega$, we get

$$
\frac{\hbar \Pi^{\mu\nu}(q)}{ig_s \alpha c} = \left[ \frac{Z_{\mu\nu}(-\tilde{q}_0 - \epsilon_s, \tilde{k}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{2\epsilon_s ((\tilde{q}_0 + \epsilon_s)^2 - \epsilon_{\tilde{k}}^2)} + \frac{Z_{\mu\nu}(-\epsilon_{\tilde{k}}, \tilde{k}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{2\epsilon_{\tilde{k}} ((\tilde{q}_0 - \epsilon_{\tilde{k}})^2 - \epsilon_{\tilde{s}}^2)} \right] \\
- \sum_{\lambda=\pm} \frac{n_F(\epsilon_s - \lambda\mu)Z_{\mu\nu}(-\tilde{q}_0 + \epsilon_s^\lambda, \tilde{k}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{2\epsilon_s ((\tilde{q}_0 - \epsilon_s^\lambda)^2 - \epsilon_{\tilde{k}}^2)} - \sum_{\lambda=\pm} \frac{n_F(\epsilon_{\tilde{k}} - \lambda\mu)Z_{\mu\nu}(\epsilon_{\tilde{k}}^\lambda, \tilde{k}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{2\epsilon_{\tilde{k}} ((\tilde{q}_0 + \epsilon_{\tilde{k}}^\lambda)^2 - \epsilon_{\tilde{s}}^2)}. \quad (C3)
$$

The term in brackets corresponds to the integrand of the $T=0$ limit, while the second and third terms correspond to the correction due to the temperature. Therefore, following the notation of [9], the Polarization operator given in Eq. (C1) can be written as

$$
\Pi^{\mu\nu}(q) = \Pi_0^{\mu\nu}(q) + \Delta_T \Pi^{\mu\nu}(q), \quad (C4)
$$

by construction, $\Pi_0^{\mu\nu}(q)$ is independent of temperature and of the chemical potential $\mu$ [9], therefore, it corresponds to the interband conductivity with $\mu = k_B T = 0$ eV. On the other hand, to simplify $\Delta_T \Pi^{\mu\nu}(q)$, we apply the change of variables $\mathbf{k}_{\parallel} \rightarrow -(\mathbf{k}_{\parallel} + \mathbf{q}_{\parallel})$, we also make use of the symmetry of the relation of dispersion $\epsilon_{\mathbf{k}} = \epsilon_{-\mathbf{k}}$ and we transform the dummy variable $\lambda \rightarrow -\lambda$ to the first summand of $\Delta_T \Pi^{\mu\nu}(q)$ to obtain

$$
\begin{align*}
\int_{BZ} \frac{d^2 \mathbf{k}_{\parallel}}{(2\pi)^2} \sum_{\lambda=\pm} \frac{n_F(\epsilon_s - \lambda\mu) Z_{\mu\nu}(-\tilde{q}_0 + \epsilon_s^{\lambda}, \tilde{k}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{2\epsilon_s ((\tilde{q}_0 - \epsilon_s^{\lambda})^2 - \epsilon_{\tilde{k}}^2)} &= \int_{\mathbf{k}} \sum_{\lambda=\pm} \frac{n_F(\epsilon_{-\mathbf{k}} + \lambda\mu) Z_{\mu\nu}(-\tilde{q}_0 - \epsilon_{-\mathbf{k}}^{\lambda}, -\tilde{s}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{2\epsilon_{-\mathbf{k}} ((\tilde{q}_0 + \epsilon_{-\mathbf{k}}^{\lambda})^2 - \epsilon_{-\tilde{s}}^2)} \\
&= \int_{\mathbf{k}} \sum_{\lambda=\pm} \frac{n_F(\epsilon_{\mathbf{k}} + \lambda\mu) Z_{\mu\nu}(-\epsilon_{\mathbf{k}}^{\lambda}, -\tilde{s}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{2\epsilon_{\mathbf{k}} ((\tilde{q}_0 + \epsilon_{\mathbf{k}}^{\lambda})^2 - \epsilon_{\tilde{s}}^2)} \\
&= \int_{\mathbf{k}} \sum_{\lambda=\pm} \frac{n_F(\epsilon_{\mathbf{k}} + \lambda\mu) Z_{\mu\nu}(\epsilon_{\mathbf{k}}^{\lambda}, \tilde{k}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{2\epsilon_{\mathbf{k}} ((\tilde{q}_0 + \epsilon_{\mathbf{k}}^{\lambda})^2 - \epsilon_{\tilde{s}}^2)}. \tag{C5}
\end{align*}
$$

where we have used that

$$
Z_{\mu\nu}(-\epsilon_{\mathbf{k}}^{\lambda} - \tilde{q}_0, -\tilde{k}_{\parallel} - \tilde{q}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel}) = Z_{\mu\nu}(\epsilon_{\mathbf{k}}^{\lambda}, \tilde{k}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel}). \quad (C6)
$$

Joining all together, and using

$$
N_{\mu}(\epsilon) = \sum_{\eta=\pm} n_{F}(\epsilon + \eta\mu), \qquad (C7)
$$

which is Eq. (107) of the main text, we simplify $\Delta_T \Pi_0^{\mu\nu}(q)$ into

$$
\Delta_T \Pi_0^{\mu\nu}(q) = -ig_s \frac{\alpha c}{\hbar} \int_{\mathbf{k}} N_{\mu}(\epsilon_{\mathbf{k}}) \sum_{\lambda=\pm} \frac{Z_{\mu\nu}(\epsilon_{\mathbf{k}}^{\lambda}, \tilde{\mathbf{k}}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{2\epsilon_{\mathbf{k}} ((\tilde{q}_0 + \epsilon_{\mathbf{k}}^{\lambda})^2 - \epsilon_{\tilde{s}}^2)}. \quad (C8)
$$
obtain

$$
\begin{align*}
\frac{\hbar \Pi^{\mu\nu}(q)}{ig_s \alpha c} &= \int_{-\infty}^{\infty} \frac{d\tilde{k}_0}{2\pi} \frac{Z^{\mu\nu}(\tilde{k}_0 + \mu, \tilde{k}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{[K^{\mu}K_{\mu} - m^2] [S^{\mu}S_{\mu} - m^2]} \\
&= \int_{-\infty}^{\infty} \frac{d\tilde{k}_0}{2\pi} \frac{Z^{\mu\nu}(\tilde{k}_0 + \mu, \tilde{k}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{[(\tilde{k}_0 + \mu)^2 - \epsilon_{\tilde{k}}^2] [(\tilde{k}_0 + \tilde{q}_0 + \mu)^2 - \epsilon_{\tilde{k}+\tilde{q}}^2]} \\
&= \frac{1}{\beta} \sum_{n \in \mathbb{Z}^{\text{Fermi}}} \frac{Z^{\mu\nu}(i\hbar\omega_n + \mu, \tilde{k}_{\parallel}, i\hbar\omega_m, \tilde{q}_{\parallel})}{[(i\hbar\omega_n + \mu)^2 - \epsilon_{\tilde{k}}^2] [(i\hbar\omega_n + i\hbar\omega_m + \mu)^2 - \epsilon_{\tilde{s}}^2]} \\
&= \left[ \frac{Z_{\mu\nu}(-i\hbar\omega_m - \epsilon_s, \tilde{k}_{\parallel}, i\hbar\omega_m, \tilde{q}_{\parallel})}{2\epsilon_s ((i\hbar\omega_m + \epsilon_s)^2 - \epsilon_{\tilde{k}}^2)} + \frac{Z_{\mu\nu}(-\epsilon_k, \tilde{k}_{\parallel}, i\hbar\omega_m, \tilde{q}_{\parallel})}{2\epsilon_k ((i\hbar\omega_m - \epsilon_k)^2 - \epsilon_{\tilde{s}}^2)} \right] \\
&\quad - \sum_{\lambda=\pm} \frac{n_F(\epsilon_s - \lambda\mu)Z_{\mu\nu}(-i\hbar\omega_m + \epsilon_s, \tilde{k}_{\parallel}, i\hbar\omega_m, \tilde{q}_{\parallel})}{2\epsilon_s ((i\hbar\omega_m - \epsilon_s^\lambda)^2 - \epsilon_{\tilde{k}}^2)} - \sum_{\lambda=\pm} \frac{n_F(\epsilon_k - \lambda\mu)Z_{\mu\nu}(\epsilon_k^\lambda, \tilde{k}_{\parallel}, i\hbar\omega_m, \tilde{q}_{\parallel})}{2\epsilon_k ((i\hbar\omega_m + \epsilon_k^\lambda)^2 - \epsilon_{\tilde{s}}^2)}, \quad (C2)
\end{align*}
$$

where we have used that $n_F(-i\hbar\omega_m + \epsilon_k - \lambda\mu) = n_F(\epsilon_k - \lambda\mu)$ and $s_\parallel = k_\parallel + q_\parallel$ (see tab. I). Next, we analytically expand the Matsubara sum given in Eq. (C2) to the whole upper complex plane by applying the formal change $i\omega_m = \omega \in \mathbb{C}^+$ into the definition of the polarization operator given in Eq. (C1), using $\tilde{q}_0 = \hbar\omega$, we get

$$
\frac{\hbar \Pi^{\mu\nu}(q)}{ig_s \alpha c} = \left[ \frac{Z_{\mu\nu}(-\tilde{q}_0 - \epsilon_s, \tilde{k}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{2\epsilon_s ((\tilde{q}_0 + \epsilon_s)^2 - \epsilon_{\tilde{k}}^2)} + \frac{Z_{\mu\nu}(-\epsilon_k, \tilde{k}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{2\epsilon_k ((\tilde{q}_0 - \epsilon_k)^2 - \epsilon_{\tilde{s}}^2)} \right] \\
- \sum_{\lambda=\pm} \frac{n_F(\epsilon_s - \lambda\mu)Z_{\mu\nu}(-\tilde{q}_0 + \epsilon_s^\lambda, \tilde{k}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{2\epsilon_s ((\tilde{q}_0 - \epsilon_s^\lambda)^2 - \epsilon_{\tilde{k}}^2)} - \sum_{\lambda=\pm} \frac{n_F(\epsilon_k - \lambda\mu)Z_{\mu\nu}(\epsilon_k^\lambda, \tilde{k}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{2\epsilon_k ((\tilde{q}_0 + \epsilon_k^\lambda)^2 - \epsilon_{\tilde{s}}^2)}. \quad (C3)
$$

The term in brackets corresponds to the integrand of the $T=0$ limit, while the second and third terms correspond to the correction due to the temperature. Therefore, following the notation of [9], the Polarization operator given in Eq. (C1) can be written as

$$
\Pi^{\mu\nu}(q) = \Pi_0^{\mu\nu}(q) + \Delta_T \Pi^{\mu\nu}(q), \quad (C4)
$$

by construction, $\Pi_0^{\mu\nu}(q)$ is independent of temperature and of the chemical potential $\mu$ [9], therefore, it corresponds to the interband conductivity with $\mu = k_B T = 0$ eV. On the other hand, to simplify $\Delta_T \Pi^{\mu\nu}(q)$, we apply the change of variables $\mathbf{k}_{\parallel} \rightarrow -(\mathbf{k}_{\parallel} + \mathbf{q}_{\parallel})$, we also make use of the symmetry of the relation of dispersion $\epsilon_k = \epsilon_{-k}$ and we transform the dummy variable $\lambda \rightarrow -\lambda$ to the first summand of $\Delta_T \Pi^{\mu\nu}(q)$ to obtain

$$
\begin{align*}
\int_{BZ} \frac{d^2 \mathbf{k}_{\parallel}}{(2\pi)^2} \sum_{\lambda=\pm} \frac{n_F(\epsilon_s - \lambda\mu) Z_{\mu\nu}(-\tilde{q}_0 + \epsilon_s^{\lambda}, \tilde{k}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{2\epsilon_s ((\tilde{q}_0 - \epsilon_s^{\lambda})^2 - \epsilon_{\tilde{k}}^2)} &= \int_{\mathbf{k}} \sum_{\lambda=\pm} \frac{n_F(\epsilon_{-\mathbf{k}} + \lambda\mu) Z_{\mu\nu}(-\tilde{q}_0 - \epsilon_{-\mathbf{k}}^{\lambda}, -\tilde{s}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{2\epsilon_{-\mathbf{k}} ((\tilde{q}_0 + \epsilon_{-\mathbf{k}}^{\lambda})^2 - \epsilon_{-\tilde{s}}^2)} \\
&= \int_{\mathbf{k}} \sum_{\lambda=\pm} \frac{n_F(\epsilon_{\mathbf{k}} + \lambda\mu) Z_{\mu\nu}(-\epsilon_{\mathbf{k}}^{\lambda}, -\tilde{s}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{2\epsilon_{\mathbf{k}} ((\tilde{q}_0 + \epsilon_{\mathbf{k}}^{\lambda})^2 - \epsilon_{\tilde{s}}^2)} \\
&= \int_{\mathbf{k}} \sum_{\lambda=\pm} \frac{n_F(\epsilon_{\mathbf{k}} + \lambda\mu) Z_{\mu\nu}(\epsilon_{\mathbf{k}}^{\lambda}, \tilde{k}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{2\epsilon_{\mathbf{k}} ((\tilde{q}_0 + \epsilon_{\mathbf{k}}^{\lambda})^2 - \epsilon_{\tilde{s}}^2)}. \tag{C5}
\end{align*}
$$

where we have used that

$$
Z_{\mu\nu}(-\epsilon_{\mathbf{k}}^{\lambda} - \tilde{q}_0, -\tilde{k}_{\parallel} - \tilde{q}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel}) = Z_{\mu\nu}(\epsilon_{\mathbf{k}}^{\lambda}, \tilde{k}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel}). \quad (C6)
$$

Joining all together, and using

$$
N_{\mu}(\epsilon) = \sum_{\eta=\pm} n_{F}(\epsilon + \eta\mu), \qquad (C7)
$$

which is Eq. (107) of the main text, we simplify $\Delta_T \Pi_0^{\mu\nu}(q)$ into

$$
\Delta_T \Pi_0^{\mu\nu}(q) = -ig_s \frac{\alpha c}{\hbar} \int_{\mathbf{k}} N_{\mu}(\epsilon_{\mathbf{k}}) \sum_{\lambda=\pm} \frac{Z_{\mu\nu}(\epsilon_{\mathbf{k}}^{\lambda}, \tilde{\mathbf{k}}_{\parallel}, \tilde{q}_0, \tilde{q}_{\parallel})}{2\epsilon_{\mathbf{k}} ((\tilde{q}_0 + \epsilon_{\mathbf{k}}^{\lambda})^2 - \epsilon_{\tilde{s}}^2)}. \quad (C8)
$$
The same change of variables $k \to -(k+q)$ can be applied to the first summand of $\Pi_0^{\mu\nu}(q)$, obtaining

$$ \begin{aligned} \int_k \frac{Z_{\mu\nu}(-\tilde{q}_0 - \epsilon_s, \tilde{k}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_s ((\tilde{q}_0 + \epsilon_s)^2 - \epsilon_k^2)} &= \int_k \frac{Z_{\mu\nu}(-\tilde{q}_0 - \epsilon_{-k}, -\tilde{s}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_{-k} ((\tilde{q}_0 + \epsilon_{-k})^2 - \epsilon_s^2)} \\ &= \int_k \frac{Z_{\mu\nu}(-\epsilon_k - \tilde{q}_0, -\tilde{k}_{\|} - \tilde{q}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_k ((\tilde{q}_0 + \epsilon_k)^2 - \epsilon_s^2)} \\ &= \int_k \frac{Z_{\mu\nu}(\epsilon_k, \tilde{k}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_k ((\tilde{q}_0 + \epsilon_k)^2 - \epsilon_s^2)}, \end{aligned} \quad (C9) $$

where we have used the symmetry shown in Eq. (C6), then we have

$$ \begin{aligned} \Pi_0^{\mu\nu}(q) &= ig_s \frac{\alpha c}{\hbar} \int_k \left[ \frac{Z_{\mu\nu}(-\tilde{q}_0 - \epsilon_s, \tilde{k}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_s ((\tilde{q}_0 + \epsilon_s)^2 - \epsilon_k^2)} + \frac{Z_{\mu\nu}(-\epsilon_k, \tilde{k}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_k ((\tilde{q}_0 - \epsilon_k)^2 - \epsilon_s^2)} \right] \\ &= ig_s \frac{\alpha c}{\hbar} \int_k \left[ \frac{Z_{\mu\nu}(\epsilon_k, \tilde{k}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_k ((\tilde{q}_0 + \epsilon_k)^2 - \epsilon_s^2)} + \frac{Z_{\mu\nu}(-\epsilon_k, \tilde{k}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_k ((\tilde{q}_0 - \epsilon_k)^2 - \epsilon_s^2)} \right] \\ &= ig_s \frac{\alpha c}{\hbar} \int_k \sum_{\lambda=\pm} \frac{Z_{\mu\nu}(\epsilon_k^{\lambda}, \tilde{k}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_k ((\tilde{q}_0 + \epsilon_k^{\lambda})^2 - \epsilon_s^2)}. \end{aligned} \quad (C10) $$

Finally, the full polarization operator can be written as

$$ \Pi^{\mu\nu}(q) = ig_s \frac{\alpha c}{\hbar} \int_k [1 - N_{\mu}(\epsilon_k)] \sum_{\lambda=\pm} \frac{Z_{\mu\nu}(\epsilon_k^{\lambda}, \tilde{k}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_k ((\tilde{q}_0 + \epsilon_k^{\lambda})^2 - \epsilon_s^2)}, \quad (C11) $$

which is the result shown in Eq. (106). In addition to that, from Eq. (104), it can be seen that the spatial part of the Polarization operator can be split as [6]

$$ Z_{ij}(k, q) = v_F^2 \left[ Z_L(k, q) \frac{\tilde{q}_i \tilde{q}_j}{\tilde{q}_{\|}^2} + Z_T(k, q) \left( \delta_{ij} - \frac{\tilde{q}_i \tilde{q}_j}{\tilde{q}_{\|}^2} \right) + S_{ij} Z_S(k, q) \right], \quad (C12) $$

with

$$ \begin{aligned} Z_{00}(k, q) &= 4 \left[ \tilde{k}_0 \tilde{s}_0 + \tilde{k}_{\|}^2 + \tilde{k}_{\|} \tilde{q}_{\|} \cos(\varphi) + m^2 \right], \\ Z_T(k, q) &= 4 \left[ \tilde{k}_0 \tilde{s}_0 - \left( \tilde{k}_{\|} \tilde{q}_{\|} \cos(\varphi) + \tilde{k}_{\|}^2 \cos(2\varphi) - m^2 \right) \right], \\ Z_L(k, q) &= 4 \left[ \tilde{k}_0 \tilde{s}_0 + \left( \tilde{k}_{\|} \tilde{q}_{\|} \cos(\varphi) + \tilde{k}_{\|}^2 \cos(2\varphi) - m^2 \right) \right], \\ Z_S(k, q) &= 4 \left[ \tilde{k}_{\|} \tilde{q}_{\|} \sin(\varphi) + \tilde{k}_{\|}^2 \sin(2\varphi) \right], \end{aligned} \quad (C13) $$

where $\tilde{k}_0 = \epsilon_k^{\lambda}$, $\tilde{s}_0 = \tilde{k}_0 + \tilde{q}_0$, $\tilde{k}_{\|} = \hbar v_F k_{\|}$ and $\tilde{q}_{\|} = \hbar v_F q_{\|}$ (see tab. I). Note that there is no Hall term because this model is topologically trivial ($C=0$). It is immediate to see that $\Pi_S(k, q) = 0$. Then, each term of the polarization operator can be written, using $p = \{00, L, T\}$ as

$$ \Pi_p(q) = -i2g_s \frac{\alpha c}{\hbar} \int_k \frac{1}{2\epsilon_k} [1 - N_{\mu}(\epsilon_k)] \sum_{\lambda=\pm} \left[ 1 + \frac{M_p(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|})}{Q(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|}) + 2\tilde{k}_{\|} \cdot \tilde{q}_{\|}} \right], \quad (C14) $$

where

$$ Q(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|}) = -\tilde{q}_z^2 - 2\tilde{q}_0 \epsilon_k^{\lambda}, \quad (C15) $$

$$ M_p(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|}) = \frac{1}{2} Z_p(\epsilon_k^{\lambda}, \tilde{k}_{\|}, \tilde{q}_0, \tilde{q}_{\|}) - Q(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|}) - 2\tilde{k}_{\|} \cdot \tilde{q}_{\|}, \quad (C16) $$

with $\tilde{q}_z^2 = \tilde{q}_0^2 - \tilde{q}_{\|}^2$ (see tab. I). In particular, we obtain

$$ M_{00}(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|}) = -\tilde{q}_z^2 + 4\tilde{q}_0\epsilon_k^{\lambda} + 4\epsilon_k^2, \quad (C17) $$

$$ M_L(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|}) = -\tilde{q}_z^2 + 4\tilde{q}_0\epsilon_k^{\lambda} + 4\tilde{k}_{\|}^2 \cos^2(\varphi), \quad (C18) $$

$$ M_T(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|}) = -\tilde{q}_z^2 + 4\tilde{q}_0\epsilon_k^{\lambda} + 4\tilde{k}_{\|}^2 \sin^2(\varphi) - 4\tilde{k}_{\|} \tilde{q}_{\|} \cos(\varphi), \quad (C19) $$

note that $M_{00}(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|})$ is not a function of $\varphi$, so we can write $M_{00}(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|})$ instead.
The same change of variables $k \to -(k+q)$ can be applied to the first summand of $\Pi_0^{\mu\nu}(q)$, obtaining

$$ \begin{aligned} \int_k \frac{Z_{\mu\nu}(-\tilde{q}_0 - \epsilon_s, \tilde{k}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_s ((\tilde{q}_0 + \epsilon_s)^2 - \epsilon_k^2)} &= \int_k \frac{Z_{\mu\nu}(-\tilde{q}_0 - \epsilon_{-k}, -\tilde{s}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_{-k} ((\tilde{q}_0 + \epsilon_{-k})^2 - \epsilon_s^2)} \\ &= \int_k \frac{Z_{\mu\nu}(-\epsilon_k - \tilde{q}_0, -\tilde{k}_{\|} - \tilde{q}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_k ((\tilde{q}_0 + \epsilon_k)^2 - \epsilon_s^2)} \\ &= \int_k \frac{Z_{\mu\nu}(\epsilon_k, \tilde{k}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_k ((\tilde{q}_0 + \epsilon_k)^2 - \epsilon_s^2)}, \end{aligned} \quad (C9) $$

where we have used the symmetry shown in Eq. (C6), then we have

$$ \begin{aligned} \Pi_0^{\mu\nu}(q) &= ig_s \frac{\alpha c}{\hbar} \int_k \left[ \frac{Z_{\mu\nu}(-\tilde{q}_0 - \epsilon_s, \tilde{k}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_s ((\tilde{q}_0 + \epsilon_s)^2 - \epsilon_k^2)} + \frac{Z_{\mu\nu}(-\epsilon_k, \tilde{k}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_k ((\tilde{q}_0 - \epsilon_k)^2 - \epsilon_s^2)} \right] \\ &= ig_s \frac{\alpha c}{\hbar} \int_k \left[ \frac{Z_{\mu\nu}(\epsilon_k, \tilde{k}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_k ((\tilde{q}_0 + \epsilon_k)^2 - \epsilon_s^2)} + \frac{Z_{\mu\nu}(-\epsilon_k, \tilde{k}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_k ((\tilde{q}_0 - \epsilon_k)^2 - \epsilon_s^2)} \right] \\ &= ig_s \frac{\alpha c}{\hbar} \int_k \sum_{\lambda=\pm} \frac{Z_{\mu\nu}(\epsilon_k^{\lambda}, \tilde{k}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_k ((\tilde{q}_0 + \epsilon_k^{\lambda})^2 - \epsilon_s^2)}. \end{aligned} \quad (C10) $$

Finally, the full polarization operator can be written as

$$ \Pi^{\mu\nu}(q) = ig_s \frac{\alpha c}{\hbar} \int_k [1 - N_{\mu}(\epsilon_k)] \sum_{\lambda=\pm} \frac{Z_{\mu\nu}(\epsilon_k^{\lambda}, \tilde{k}_{\|}, \tilde{q}_0, \tilde{q}_{\|})}{2\epsilon_k ((\tilde{q}_0 + \epsilon_k^{\lambda})^2 - \epsilon_s^2)}, \quad (C11) $$

which is the result shown in Eq. (106). In addition to that, from Eq. (104), it can be seen that the spatial part of the Polarization operator can be split as [6]

$$ Z_{ij}(k, q) = v_F^2 \left[ Z_L(k, q) \frac{\tilde{q}_i \tilde{q}_j}{\tilde{q}_{\|}^2} + Z_T(k, q) \left( \delta_{ij} - \frac{\tilde{q}_i \tilde{q}_j}{\tilde{q}_{\|}^2} \right) + S_{ij} Z_S(k, q) \right], \quad (C12) $$

with

$$ \begin{aligned} Z_{00}(k, q) &= 4 \left[ \tilde{k}_0 \tilde{s}_0 + \tilde{k}_{\|}^2 + \tilde{k}_{\|} \tilde{q}_{\|} \cos(\varphi) + m^2 \right], \\ Z_T(k, q) &= 4 \left[ \tilde{k}_0 \tilde{s}_0 - \left( \tilde{k}_{\|} \tilde{q}_{\|} \cos(\varphi) + \tilde{k}_{\|}^2 \cos(2\varphi) - m^2 \right) \right], \\ Z_L(k, q) &= 4 \left[ \tilde{k}_0 \tilde{s}_0 + \left( \tilde{k}_{\|} \tilde{q}_{\|} \cos(\varphi) + \tilde{k}_{\|}^2 \cos(2\varphi) - m^2 \right) \right], \\ Z_S(k, q) &= 4 \left[ \tilde{k}_{\|} \tilde{q}_{\|} \sin(\varphi) + \tilde{k}_{\|}^2 \sin(2\varphi) \right], \end{aligned} \quad (C13) $$

where $\tilde{k}_0 = \epsilon_k^{\lambda}$, $\tilde{s}_0 = \tilde{k}_0 + \tilde{q}_0$, $\tilde{k}_{\|} = \hbar v_F k_{\|}$ and $\tilde{q}_{\|} = \hbar v_F q_{\|}$ (see tab. I). Note that there is no Hall term because this model is topologically trivial ($C=0$). It is immediate to see that $\Pi_S(k, q) = 0$. Then, each term of the polarization operator can be written, using $p = \{00, L, T\}$ as

$$ \Pi_p(q) = -i2g_s \frac{\alpha c}{\hbar} \int_k \frac{1}{2\epsilon_k} [1 - N_{\mu}(\epsilon_k)] \sum_{\lambda=\pm} \left[ 1 + \frac{M_p(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|})}{Q(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|}) + 2\tilde{k}_{\|} \cdot \tilde{q}_{\|}} \right], \quad (C14) $$

where

$$ Q(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|}) = -\tilde{q}_z^2 - 2\tilde{q}_0 \epsilon_k^{\lambda}, \quad (C15) $$

$$ M_p(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|}) = \frac{1}{2} Z_p(\epsilon_k^{\lambda}, \tilde{k}_{\|}, \tilde{q}_0, \tilde{q}_{\|}) - Q(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|}) - 2\tilde{k}_{\|} \cdot \tilde{q}_{\|}, \quad (C16) $$

with $\tilde{q}_z^2 = \tilde{q}_0^2 - \tilde{q}_{\|}^2$ (see tab. I). In particular, we obtain

$$ M_{00}(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|}) = -\tilde{q}_z^2 + 4\tilde{q}_0\epsilon_k^{\lambda} + 4\epsilon_k^2, \quad (C17) $$

$$ M_L(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|}) = -\tilde{q}_z^2 + 4\tilde{q}_0\epsilon_k^{\lambda} + 4\tilde{k}_{\|}^2 \cos^2(\varphi), \quad (C18) $$

$$ M_T(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|}) = -\tilde{q}_z^2 + 4\tilde{q}_0\epsilon_k^{\lambda} + 4\tilde{k}_{\|}^2 \sin^2(\varphi) - 4\tilde{k}_{\|} \tilde{q}_{\|} \cos(\varphi), \quad (C19) $$

note that $M_{00}(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|})$ is not a function of $\varphi$, so we can write $M_{00}(\tilde{q}_0, \tilde{k}_{\|}, \tilde{q}_{\|})$ instead.
# 1. Temporal Polarization

For the case $\Pi_{\mu\nu}(q) = \Pi_{00}(q)$, and for imaginary frequencies $\tilde{q}_0 = i\Xi = i\hbar\xi$, we can transform Eq. (C14) into

$$ \Pi_{00}(q) = -2ig_s \frac{\alpha c}{\hbar} \int_0^{k_M} \frac{dk_\parallel}{2\pi} \frac{k_\parallel}{2\epsilon_k} [1 - N_\mu(\epsilon_k)] \sum_{\lambda=\pm} \int_0^{2\pi} \frac{d\varphi}{2\pi} \left[ 1 + \frac{M_{00}(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)}{Q(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) + 2\tilde{k}_\parallel \cdot \tilde{q}_\parallel} \right], \quad (C20) $$

where $\tilde{k}_\parallel = \hbar v_F k_\parallel$, $\tilde{q}_\parallel = \hbar v_F q_\parallel$ (see tab. I) and $k_M$ plays the role of an upper cut-off in frequencies, and

$$ \begin{aligned} Q(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) &= \Xi^2 + \tilde{q}_\parallel^2 - 2i\Xi\epsilon_k^\lambda, \\ M_{00}(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) &= -\Xi^2 - \tilde{q}_\parallel^2 + 4i\Xi\epsilon_k^\lambda + 4\epsilon_k^2. \end{aligned} \quad (C21) $$

By using $\tilde{k}_\parallel \cdot \tilde{q}_\parallel = \tilde{k}_\parallel \cdot \tilde{q}_\parallel \cos(\varphi)$ with $\tilde{k}_\parallel = \hbar v_F k_\parallel$ and $\tilde{q}_\parallel = \hbar v_F q_\parallel$, the angular integral can be carried out as

$$ \int_0^{2\pi} \frac{d\varphi}{2\pi} \frac{1}{Q + a \cos(\varphi)} = \frac{1}{\sqrt{Q^2 - a^2}}, \quad (C22) $$

Therefore, $\Pi_{00}(q)$ can be written as

$$ \Pi_{00}(q) = -2ig_s \frac{\alpha c}{\hbar} \int_0^{k_M} \frac{dk_\parallel}{2\pi} \frac{k_\parallel}{2\epsilon_k} [1 - N_\mu(\epsilon_k)] \left[ 2 + \sum_{\lambda=\pm} \frac{M_{00}(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)}{N(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)} \right]. \quad (C23) $$

with

$$ \begin{aligned} Q(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) &= \Xi^2 + \tilde{q}_\parallel^2 - 2i\lambda\Xi\epsilon_k, \\ M_{00}(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) &= -\Xi^2 - \tilde{q}_\parallel^2 + 4i\lambda\Xi\epsilon_k + 4\epsilon_k^2, \\ N(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) &= \sqrt{[Q(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)]^2 - (2\tilde{k}_\parallel\tilde{q}_\parallel)^2}. \end{aligned} \quad (C24) $$

Note that the ratio can be equivalently represented as

$$ \sum_{\lambda=\pm} \frac{M_{00}(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)}{N(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)} = 2\mathbb{R}\mathrm{e}\left[\frac{M_{00}(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)}{N(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)}\right], \quad (C25) $$

therefore, we obtain

$$ \Pi_{00}(q) = -2ig_s \frac{\alpha c}{\hbar} \int_0^{k_M} \frac{dk_\parallel}{2\pi} \frac{k_\parallel}{\epsilon_k} [1 - N_\mu(\epsilon_k)] \left[ 1 + \mathbb{R}\mathrm{e} \left[ \frac{M_{00}(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)}{N(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)} \right] \right], \quad (C26) $$

The next step is to change the integration variable from $k_\parallel$ to $\epsilon = \sqrt{(\hbar v_F k_\parallel)^2 + m^2}$, therefore, we get

$$ \Pi_{00}(q) = -2ig_s \frac{\alpha c}{\hbar} \int_m^{\epsilon_M} \frac{d\epsilon}{2\pi\hbar^2 v_F^2} [1 - N_\mu(\epsilon)] \left[ 1 + \mathbb{R}\mathrm{e} \left[ \frac{-\Xi^2 - \tilde{q}_\parallel^2 + 4i\Xi\epsilon + 4\epsilon^2}{\sqrt{[\Xi^2 + \tilde{q}_\parallel^2 - 2i\Xi\epsilon]^2 - (2\tilde{q}_\parallel)^2(\epsilon^2 - m^2)}} \right] \right], \quad (C27) $$

It is followed by the change of variable into a non-dimensional energy $u$, using $\tilde{\theta}_z = \sqrt{\Xi^2 + \tilde{q}_\parallel^2}$ (see tab. I), we apply the change of variable $\epsilon = \tilde{\theta}_z^{u/2}$, obtaining

$$ \Pi_{00}(q) = -i2g_s \frac{\alpha c}{\hbar} \int_{\frac{2m}{\tilde{\theta}_z}}^{\frac{2\epsilon_M}{\tilde{\theta}_z}} \frac{\tilde{\theta}_z du}{4\pi\hbar^2 v_F^2} \left[ 1 - N_\mu\left(\tilde{\theta}_z^{u/2}\right) \right] \left[ 1 + \mathbb{R}\mathrm{e} \left[ \frac{-\tilde{\theta}_z^2 + 2i\Xi\tilde{\theta}_z u + \tilde{\theta}_z^2 u^2}{\sqrt{[\tilde{\theta}_z^2 - i\Xi\tilde{\theta}_z u]^2 - (\tilde{\theta}_z^2 - \Xi^2)(\tilde{\theta}_z^2 u^2 - 4m^2)}} \right] \right], \quad (C28) $$
# 1. Temporal Polarization

For the case $\Pi_{\mu\nu}(q) = \Pi_{00}(q)$, and for imaginary frequencies $\tilde{q}_0 = i\Xi = i\hbar\xi$, we can transform Eq. (C14) into

$$ \Pi_{00}(q) = -2ig_s \frac{\alpha c}{\hbar} \int_0^{k_M} \frac{dk_\parallel}{2\pi} \frac{k_\parallel}{2\epsilon_k} [1 - N_\mu(\epsilon_k)] \sum_{\lambda=\pm} \int_0^{2\pi} \frac{d\varphi}{2\pi} \left[ 1 + \frac{M_{00}(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)}{Q(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) + 2\tilde{k}_\parallel \cdot \tilde{q}_\parallel} \right], \quad (C20) $$

where $\tilde{k}_\parallel = \hbar v_F k_\parallel$, $\tilde{q}_\parallel = \hbar v_F q_\parallel$ (see tab. I) and $k_M$ plays the role of an upper cut-off in frequencies, and

$$ \begin{aligned} Q(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) &= \Xi^2 + \tilde{q}_\parallel^2 - 2i\Xi\epsilon_k^\lambda, \\ M_{00}(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) &= -\Xi^2 - \tilde{q}_\parallel^2 + 4i\Xi\epsilon_k^\lambda + 4\epsilon_k^2. \end{aligned} \quad (C21) $$

By using $\tilde{k}_\parallel \cdot \tilde{q}_\parallel = \tilde{k}_\parallel \cdot \tilde{q}_\parallel \cos(\varphi)$ with $\tilde{k}_\parallel = \hbar v_F k_\parallel$ and $\tilde{q}_\parallel = \hbar v_F q_\parallel$, the angular integral can be carried out as

$$ \int_0^{2\pi} \frac{d\varphi}{2\pi} \frac{1}{Q + a \cos(\varphi)} = \frac{1}{\sqrt{Q^2 - a^2}}, \quad (C22) $$

Therefore, $\Pi_{00}(q)$ can be written as

$$ \Pi_{00}(q) = -2ig_s \frac{\alpha c}{\hbar} \int_0^{k_M} \frac{dk_\parallel}{2\pi} \frac{k_\parallel}{2\epsilon_k} [1 - N_\mu(\epsilon_k)] \left[ 2 + \sum_{\lambda=\pm} \frac{M_{00}(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)}{N(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)} \right]. \quad (C23) $$

with

$$ \begin{aligned} Q(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) &= \Xi^2 + \tilde{q}_\parallel^2 - 2i\lambda\Xi\epsilon_k, \\ M_{00}(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) &= -\Xi^2 - \tilde{q}_\parallel^2 + 4i\lambda\Xi\epsilon_k + 4\epsilon_k^2, \\ N(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) &= \sqrt{[Q(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)]^2 - (2\tilde{k}_\parallel\tilde{q}_\parallel)^2}. \end{aligned} \quad (C24) $$

Note that the ratio can be equivalently represented as

$$ \sum_{\lambda=\pm} \frac{M_{00}(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)}{N(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)} = 2\mathbb{R}\mathrm{e}\left[\frac{M_{00}(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)}{N(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)}\right], \quad (C25) $$

therefore, we obtain

$$ \Pi_{00}(q) = -2ig_s \frac{\alpha c}{\hbar} \int_0^{k_M} \frac{dk_\parallel}{2\pi} \frac{k_\parallel}{\epsilon_k} [1 - N_\mu(\epsilon_k)] \left[ 1 + \mathbb{R}\mathrm{e} \left[ \frac{M_{00}(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)}{N(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)} \right] \right], \quad (C26) $$

The next step is to change the integration variable from $k_\parallel$ to $\epsilon = \sqrt{(\hbar v_F k_\parallel)^2 + m^2}$, therefore, we get

$$ \Pi_{00}(q) = -2ig_s \frac{\alpha c}{\hbar} \int_m^{\epsilon_M} \frac{d\epsilon}{2\pi\hbar^2 v_F^2} [1 - N_\mu(\epsilon)] \left[ 1 + \mathbb{R}\mathrm{e} \left[ \frac{-\Xi^2 - \tilde{q}_\parallel^2 + 4i\Xi\epsilon + 4\epsilon^2}{\sqrt{[\Xi^2 + \tilde{q}_\parallel^2 - 2i\Xi\epsilon]^2 - (2\tilde{q}_\parallel)^2(\epsilon^2 - m^2)}} \right] \right], \quad (C27) $$

It is followed by the change of variable into a non-dimensional energy $u$, using $\tilde{\theta}_z = \sqrt{\Xi^2 + \tilde{q}_\parallel^2}$ (see tab. I), we apply the change of variable $\epsilon = \tilde{\theta}_z^{u/2}$, obtaining

$$ \Pi_{00}(q) = -i2g_s \frac{\alpha c}{\hbar} \int_{\frac{2m}{\tilde{\theta}_z}}^{\frac{2\epsilon_M}{\tilde{\theta}_z}} \frac{\tilde{\theta}_z du}{4\pi\hbar^2 v_F^2} \left[ 1 - N_\mu\left(\tilde{\theta}_z^{u/2}\right) \right] \left[ 1 + \mathbb{R}\mathrm{e} \left[ \frac{-\tilde{\theta}_z^2 + 2i\Xi\tilde{\theta}_z u + \tilde{\theta}_z^2 u^2}{\sqrt{[\tilde{\theta}_z^2 - i\Xi\tilde{\theta}_z u]^2 - (\tilde{\theta}_z^2 - \Xi^2)(\tilde{\theta}_z^2 u^2 - 4m^2)}} \right] \right], \quad (C28) $$
Taking a common factor of $\hat{\theta}_z > 0$, defining the nondimensional parameters $\gamma = \frac{\Xi}{\hat{\theta}_z}$ and $\delta = \frac{2m}{\hat{\theta}_z}$, and applying the limit $\epsilon_M \to \infty$ we obtain

$$ \Pi_{00}(q) = -i \frac{g_s \alpha c}{2\pi \hbar} \frac{\tilde{\theta}_z}{\hbar^2 v_F^2} \int_{\delta}^{\infty} du \left[ 1 - N_{\mu} \left( \tilde{\theta}_z \frac{u}{2} \right) \right] \left[ 1 - \mathrm{Re} \left[ \frac{1 - u^2 - 2i\gamma u}{\sqrt{1 - u^2 - 2i\gamma u + (1 - \gamma^2)\delta^2}} \right] \right], \quad (C29) $$

With this result, we obtain that $\Pi^{00}(q)$ obtained from the QFT-b model is equivalent to the polarization operator obtained from the non-local Kubo formula.

## 2. Longitudinal Polarization

Using the relation between the Longitudinal and Temporal terms of the Polarization operator derived from the transversality condition

$$ \Pi_L(q) = \frac{q_0^2}{q_{\parallel}^2} \Pi_{00}(q), \quad (C30) $$

the Longitudinal Polarization can be obtained in terms of $\Pi_{00}(q)$ as

$$ \Pi_L(q) = -2ig_s \frac{\alpha c}{\hbar} \frac{\tilde{q}_0^2}{\tilde{q}_{\parallel}^2} \int_{\mathbf{k}} \frac{1}{2\epsilon_{\mathbf{k}}} [1 - N_{\mu}(\epsilon_{\mathbf{k}})] \sum_{\lambda=\pm} \left[ 1 + \frac{M_{00}(\tilde{q}_0, \tilde{k}_{\parallel}, \tilde{q}_{\parallel})}{Q(\tilde{q}_0, \tilde{k}_{\parallel}, \tilde{q}_{\parallel}) + 2\tilde{k}_{\parallel} \cdot \tilde{q}_{\parallel}} \right], \quad (C31) $$

where we have used that $\tilde{q}_{\parallel} = \hbar v_F q_{\parallel}$ and $\tilde{q}_0 = \hbar q_0$ (see tab. I). This is Eq. (109) shown in Sect. VII. From this relation, by using the definitions of $M_{00}$ and $M_L$ given in Eq. (C17) and Eq. (C18) respectively, we obtain the following equality ($\tilde{q}_z = \sqrt{\tilde{q}_0^2 - \tilde{q}_{\parallel}^2}$)

$$ \begin{aligned} & \int_{\mathbf{k}} \frac{1}{2\epsilon_{\mathbf{k}}} [1 - N_{\mu}(\epsilon_{\mathbf{k}})] \sum_{\lambda=\pm} \frac{4\tilde{k}_{\parallel}^2 \cos^2(\varphi)}{Q(\tilde{q}_0, \tilde{k}_{\parallel}, \tilde{q}_{\parallel}) + 2\tilde{k}_{\parallel} \cdot \tilde{q}_{\parallel}} \\ & = \int_{\mathbf{k}} \frac{1}{2\epsilon_{\mathbf{k}}} [1 - N_{\mu}(\epsilon_{\mathbf{k}})] \sum_{\lambda=\pm} \left[ \frac{\tilde{q}_z^2}{\tilde{q}_{\parallel}^2} \left( 1 + \frac{-\tilde{q}_z^2 + 4\tilde{q}_0 \epsilon_{\mathbf{k}}^{\lambda}}{Q(\tilde{q}_0, \tilde{k}_{\parallel}, \tilde{q}_{\parallel}) + 2\tilde{k}_{\parallel} \cdot \tilde{q}_{\parallel}} \right) + \frac{\tilde{q}_0^2}{\tilde{q}_{\parallel}^2} \frac{4\epsilon_{\mathbf{k}}^2}{Q(\tilde{q}_0, \tilde{k}_{\parallel}, \tilde{q}_{\parallel}) + 2\tilde{k}_{\parallel} \cdot \tilde{q}_{\parallel}} \right], \end{aligned} \quad (C32) $$

that will be used to simplify other terms of the Polarization operator. Eq. (C32) is easily checked for imaginary frequencies after an integration over the angular variable $\varphi$ and, therefore, it is valid for all complex frequencies by analytical continuation.

For imaginary frequencies $\tilde{q}_0 = i\hbar\xi = i\Xi$, we have

$$ \Pi_L(q) = -g_s \frac{\alpha c}{2\pi \hbar} \tilde{\theta}_z \frac{\Xi^2}{\tilde{q}_{\parallel}^2} \int_{\delta}^{\infty} du \left[ 1 - N_{\mu} \left( \tilde{\theta}_z \frac{u}{2} \right) \right] \left[ 1 - \mathrm{Re} \left[ \frac{1 - u^2 - 2i\gamma u}{\sqrt{1 - u^2 - 2i\gamma u + (1 - \gamma^2)\delta^2}} \right] \right], \quad (C33) $$

Once $\Pi_L^{(0)}(q)$ is regularized following [9][21], we have

$$ \Pi_L(q) = \frac{g_s \alpha c \Xi^2}{8\pi \hbar \tilde{\theta}_z} \Psi(\delta) + g_s \frac{\alpha c \Xi^2}{2\pi \hbar \tilde{q}_{\parallel}^2} \tilde{\theta}_z \int_{\delta}^{\infty} du N_{\mu} \left( \tilde{\theta}_z \frac{u}{2} \right) \left[ 1 - \mathrm{Re} \left[ \frac{1 - u^2 - 2i\gamma u}{\sqrt{1 - u^2 - 2i\gamma u + (1 - \gamma^2)\delta^2}} \right] \right], \quad (C34) $$

Those are Eq. (116) and Eq. (117) in Sect. VII. The equivalent conductivity is $(\sigma_L^{QFT-b}(i\xi) = i\Pi_L(i\xi)/(i\xi) = \Pi_L(i\xi)/\xi)$

$$ \sigma_L^{QFT-b}(q) = \frac{g_s \alpha c}{8\pi} \gamma \Psi(\delta) + g_s \frac{\alpha c \Xi}{2\pi} \frac{\tilde{\theta}_z}{\tilde{q}_{\parallel}^2} \int_{\delta}^{\infty} du N_{\mu} \left( \tilde{\theta}_z \frac{u}{2} \right) \left[ 1 - \mathrm{Re} \left[ \frac{1 - u^2 - 2i\gamma u}{\sqrt{1 - u^2 - 2i\gamma u + (1 - \gamma^2)\delta^2}} \right] \right]. \quad (C35) $$
Taking a common factor of $\hat{\theta}_z > 0$, defining the nondimensional parameters $\gamma = \frac{\Xi}{\hat{\theta}_z}$ and $\delta = \frac{2m}{\hat{\theta}_z}$, and applying the limit $\epsilon_M \to \infty$ we obtain

$$ \Pi_{00}(q) = -i \frac{g_s \alpha c}{2\pi \hbar} \frac{\tilde{\theta}_z}{\hbar^2 v_F^2} \int_{\delta}^{\infty} du \left[ 1 - N_{\mu} \left( \tilde{\theta}_z \frac{u}{2} \right) \right] \left[ 1 - \text{Re} \left[ \frac{1 - u^2 - 2i\gamma u}{\sqrt{1 - u^2 - 2i\gamma u + (1 - \gamma^2)\delta^2}} \right] \right], \quad (C29) $$

With this result, we obtain that $\Pi^{00}(q)$ obtained from the QFT-b model is equivalent to the polarization operator obtained from the non-local Kubo formula.

## 2. Longitudinal Polarization

Using the relation between the Longitudinal and Temporal terms of the Polarization operator derived from the transversality condition

$$ \Pi_L(q) = \frac{q_0^2}{q_{\parallel}^2} \Pi_{00}(q), \quad (C30) $$

the Longitudinal Polarization can be obtained in terms of $\Pi_{00}(q)$ as

$$ \Pi_L(q) = -2ig_s \frac{\alpha c}{\hbar} \frac{\tilde{q}_0^2}{\tilde{q}_{\parallel}^2} \int_{\mathbf{k}} \frac{1}{2\epsilon_{\mathbf{k}}} [1 - N_{\mu}(\epsilon_{\mathbf{k}})] \sum_{\lambda=\pm} \left[ 1 + \frac{M_{00}(\tilde{q}_0, \tilde{k}_{\parallel}, \tilde{q}_{\parallel})}{Q(\tilde{q}_0, \tilde{k}_{\parallel}, \tilde{q}_{\parallel}) + 2\tilde{k}_{\parallel} \cdot \tilde{q}_{\parallel}} \right], \quad (C31) $$

where we have used that $\tilde{q}_{\parallel} = \hbar v_F q_{\parallel}$ and $\tilde{q}_0 = \hbar q_0$ (see tab. I). This is Eq. (109) shown in Sect. VII. From this relation, by using the definitions of $M_{00}$ and $M_L$ given in Eq. (C17) and Eq. (C18) respectively, we obtain the following equality ($\tilde{q}_z = \sqrt{\tilde{q}_0^2 - \tilde{q}_{\parallel}^2}$)

$$ \int_{\mathbf{k}} \frac{1}{2\epsilon_{\mathbf{k}}} [1 - N_{\mu}(\epsilon_{\mathbf{k}})] \sum_{\lambda=\pm} \frac{4\tilde{k}_{\parallel}^2 \cos^2(\varphi)}{Q(\tilde{q}_0, \tilde{k}_{\parallel}, \tilde{q}_{\parallel}) + 2\tilde{k}_{\parallel} \cdot \tilde{q}_{\parallel}} \\ = \int_{\mathbf{k}} \frac{1}{2\epsilon_{\mathbf{k}}} [1 - N_{\mu}(\epsilon_{\mathbf{k}})] \sum_{\lambda=\pm} \left[ \frac{\tilde{q}_z^2}{\tilde{q}_{\parallel}^2} \left( 1 + \frac{-\tilde{q}_z^2 + 4\tilde{q}_0 \epsilon_{\mathbf{k}}^{\lambda}}{Q(\tilde{q}_0, \tilde{k}_{\parallel}, \tilde{q}_{\parallel}) + 2\tilde{k}_{\parallel} \cdot \tilde{q}_{\parallel}} \right) + \frac{\tilde{q}_0^2}{\tilde{q}_{\parallel}^2} \frac{4\epsilon_{\mathbf{k}}^2}{Q(\tilde{q}_0, \tilde{k}_{\parallel}, \tilde{q}_{\parallel}) + 2\tilde{k}_{\parallel} \cdot \tilde{q}_{\parallel}} \right], \quad (C32) $$

that will be used to simplify other terms of the Polarization operator. Eq. (C32) is easily checked for imaginary frequencies after an integration over the angular variable $\varphi$ and, therefore, it is valid for all complex frequencies by analytical continuation.

For imaginary frequencies $\tilde{q}_0 = i\hbar\xi = i\Xi$, we have

$$ \Pi_L(q) = -g_s \frac{\alpha c}{2\pi \hbar} \tilde{\theta}_z \frac{\Xi^2}{\tilde{q}_{\parallel}^2} \int_{\delta}^{\infty} du \left[ 1 - N_{\mu} \left( \tilde{\theta}_z \frac{u}{2} \right) \right] \left[ 1 - \text{Re} \left[ \frac{1 - u^2 - 2i\gamma u}{\sqrt{1 - u^2 - 2i\gamma u + (1 - \gamma^2)\delta^2}} \right] \right], \quad (C33) $$

Once $\Pi_L^{(0)}(q)$ is regularized following [9][21], we have

$$ \Pi_L(q) = \frac{g_s \alpha c \Xi^2}{8\pi \hbar \tilde{\theta}_z} \Psi(\delta) + g_s \frac{\alpha c \Xi^2}{2\pi \hbar \tilde{q}_{\parallel}^2} \tilde{\theta}_z \int_{\delta}^{\infty} du N_{\mu} \left( \tilde{\theta}_z \frac{u}{2} \right) \left[ 1 - \text{Re} \left[ \frac{1 - u^2 - 2i\gamma u}{\sqrt{1 - u^2 - 2i\gamma u + (1 - \gamma^2)\delta^2}} \right] \right], \quad (C34) $$

Those are Eq. (116) and Eq. (117) in Sect. VII. The equivalent conductivity is $(\sigma_L^{QFT-b}(i\xi) = i\Pi_L(i\xi)/(i\xi) = \Pi_L(i\xi)/\xi)$

$$ \sigma_L^{QFT-b}(q) = \frac{g_s \alpha c}{8\pi} \gamma \Psi(\delta) + g_s \frac{\alpha c \Xi}{2\pi} \frac{\tilde{\theta}_z}{\tilde{q}_{\parallel}^2} \int_{\delta}^{\infty} du N_{\mu} \left( \tilde{\theta}_z \frac{u}{2} \right) \left[ 1 - \text{Re} \left[ \frac{1 - u^2 - 2i\gamma u}{\sqrt{1 - u^2 - 2i\gamma u + (1 - \gamma^2)\delta^2}} \right] \right]. \quad (C35) $$
### 3. Transversal Polarization

By using Eq. (C32) in Eq. (C14) with Eq. (C19), the Transversal Polarization term can be simplified as

$$ \Pi_T(q) = 2ig_s \frac{\alpha c}{\hbar} \frac{\tilde{q}_0^2}{\tilde{q}_\parallel^2} \int_{\mathbf{k}} \frac{1}{2\epsilon_{\mathbf{k}}} [1 - N_\mu(\epsilon_{\mathbf{k}})] \sum_{\lambda=\pm} \left[ 1 + \frac{M_{00}(\tilde{q}_0, \tilde{k}_\parallel, \tilde{q}_\parallel) - 4 \frac{\tilde{q}_0^2}{\tilde{q}_0^2} (\tilde{k}_\parallel^2 + \tilde{q}_0 \epsilon_{\mathbf{k}})}{Q(\tilde{q}_0, \tilde{k}_\parallel, \tilde{q}_\parallel) + 2 \tilde{k}_\parallel \cdot \tilde{q}_\parallel} \right]. \quad (C36) $$

Here we study the special case of imaginary frequencies $\tilde{q}_0 = i\hbar\xi = i\Xi$, in polar coordinates

$$ \Pi_T(q) = g_s \frac{\alpha c}{\hbar} \frac{-\Xi^2}{\tilde{q}_\parallel^2} \int_0^{k_M} \frac{dk_\parallel}{2\pi} \frac{k_\parallel}{\epsilon_{\mathbf{k}}} [1 - N_\mu(\epsilon_{\mathbf{k}})] \int_0^{2\pi} \frac{d\varphi}{2\pi} \sum_{\lambda=\pm} \left[ 1 + \frac{M_{00}(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) + 4 \frac{\tilde{q}_\parallel^2}{\Xi^2} (\tilde{k}_\parallel^2 + i\lambda\Xi\epsilon_{\mathbf{k}})}{Q(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) + 2\tilde{k}_\parallel\tilde{q}_\parallel \cos(\varphi)} \right]. \quad (C37) $$

The integral in $\varphi$ can be carried out using Eq. (C22)

$$ \int_0^{2\pi} \frac{d\varphi}{2\pi} \frac{1}{Q + a \cos(\varphi)} = \frac{1}{\sqrt{Q^2 - a^2}}. \quad (C38) $$

Therefore, $\Pi_T(q)$ can be written as

$$ \Pi_T(q) = g_s \frac{\alpha c}{\hbar} \frac{-\Xi^2}{\tilde{q}_\parallel^2} \int_0^{k_M} \frac{dk_\parallel}{2\pi} \frac{k_\parallel}{\epsilon_{\mathbf{k}}} [1 - N_\mu(\epsilon_{\mathbf{k}})] \sum_{\lambda=\pm} \left[ 1 + \frac{M_{00}(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) + 4 \frac{\tilde{q}_\parallel^2}{\Xi^2} (\tilde{k}_\parallel^2 + i\lambda\Xi\epsilon_{\mathbf{k}})}{\sqrt{[Q(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)]^2 - (2\tilde{k}_\parallel\tilde{q}_\parallel)^2}} \right]. \quad (C39) $$

Similarly to what was done in Eq. (C25), using $t = \lambda^2 = 1$, the ratio can be equivalently represented as

$$ \sum_{\lambda=\pm} \frac{M_T(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)}{N(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)} = 2\operatorname{Re} \left[ \frac{M_T(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)}{N(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)} \right], \quad (C40) $$

therefore, we obtain

$$ \Pi_T(q) = g_s \frac{\alpha c}{\hbar} \frac{-\Xi^2}{\tilde{q}_\parallel^2} \int_0^{k_M} \frac{dk_\parallel}{2\pi} \frac{k_\parallel}{\epsilon_{\mathbf{k}}} [1 - N_\mu(\epsilon_{\mathbf{k}})] \left[ 2 + 2\operatorname{Re} \left[ \frac{M_{00}(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) + 4 \frac{\tilde{q}_\parallel^2}{\Xi^2} (\tilde{k}_\parallel^2 + i\Xi\epsilon_{\mathbf{k}})}{\sqrt{[Q(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)]^2 - (2\tilde{k}_\parallel\tilde{q}_\parallel)^2}} \right] \right]. \quad (C41) $$

Now we apply the change of coordinates $\epsilon = \sqrt{(\hbar v_F k_\parallel)^2 + m^2}$

$$ \Pi_T(q) = 2g_s \frac{\alpha c}{\hbar} \frac{-\Xi^2}{\tilde{q}_\parallel^2} \int_m^{\epsilon_M} \frac{d\epsilon}{2\pi} [1 - N_\mu(\epsilon)] \left[ 1 + \operatorname{Re} \left[ \frac{-\Xi^2 - \tilde{q}_\parallel^2 + 4i\Xi\epsilon + 4\epsilon^2 + 4 \frac{\tilde{q}_\parallel^2}{\Xi^2} (\epsilon^2 - m^2 + i\Xi\epsilon)}{\sqrt{(\Xi^2 + \tilde{q}_\parallel^2 - 2i\Xi\epsilon)^2 - 4(\epsilon^2 - m^2)\tilde{q}_\parallel^2}} \right] \right]. \quad (C42) $$

It is followed by the change of variable into a non-dimensional energy, using $\tilde{\theta}_z = \sqrt{\Xi^2 + \tilde{q}_\parallel^2}$ (and, therefore, $\tilde{q}_\parallel^2 = \tilde{\theta}_z^2 - \Xi^2$), we apply the change of variable $\epsilon = \tilde{\theta}_z^{u/2}$, obtaining

$$ \Pi_T(q) = 2g_s \frac{\alpha c}{\hbar} \frac{-\Xi^2}{\tilde{q}_\parallel^2} \int_{\frac{2m}{\tilde{q}_\parallel}}^{\frac{2\epsilon_M}{\tilde{q}_\parallel}} \frac{du}{2\pi} \frac{\tilde{\theta}_z}{2} [1 - N_\mu(\tilde{\theta}_z^{u/2})] \left[ 1 + \operatorname{Re} \left[ \frac{-\tilde{\theta}_z^2 + 2i\Xi\tilde{\theta}_z u + \tilde{\theta}_z^2 u^2 + 4 \left( \frac{\tilde{\theta}_z^2}{\Xi^2} - 1 \right) \left( \tilde{\theta}_z^2 \frac{u^2}{4} - m^2 + i\Xi\tilde{\theta}_z^{u/2} \right)}{\sqrt{\left( \tilde{\theta}_z^2 - i\Xi\tilde{\theta}_z u \right)^2 - \left( \tilde{\theta}_z^2 u^2 - 4m^2 \right) \left( \tilde{\theta}_z^2 - \Xi^2 \right)}} \right] \right] \quad (C43) $$

Using the definitions $\delta = \frac{2m}{\tilde{\theta}_z}$ and $\gamma = \frac{\Xi}{\tilde{\theta}_z}$, we can simplify this integral into

$$ \Pi_T(q) = 2g_s \frac{\alpha c}{\hbar} \frac{-\Xi^2}{\tilde{q}_\parallel^2} \int_\delta^{\frac{2\epsilon_M}{\tilde{q}_\parallel}} \frac{du}{2\pi} \frac{\tilde{\theta}_z}{2} [1 - N_\mu(\tilde{\theta}_z^{u/2})] \left[ 1 + \operatorname{Re} \left[ \frac{-1 + 2i\gamma u + u^2 + (\gamma^2 - 1)(u^2 - \delta^2 + 2i\gamma u)}{\sqrt{(1 - i\gamma u)^2 - (u^2 - \delta^2)(1 - \gamma^2)}} \right] \right]. \quad (C44) $$
### 3. Transversal Polarization

By using Eq. (C32) in Eq. (C14) with Eq. (C19), the Transversal Polarization term can be simplified as

$$ \Pi_T(q) = 2ig_s \frac{\alpha c}{\hbar} \frac{\tilde{q}_0^2}{\tilde{q}_\parallel^2} \int_{\mathbf{k}} \frac{1}{2\epsilon_{\mathbf{k}}} [1 - N_\mu(\epsilon_{\mathbf{k}})] \sum_{\lambda=\pm} \left[ 1 + \frac{M_{00}(\tilde{q}_0, \tilde{k}_\parallel, \tilde{q}_\parallel) - 4 \frac{\tilde{q}_0^2}{\tilde{q}_0^2} (\tilde{k}_\parallel^2 + \tilde{q}_0 \epsilon_{\mathbf{k}})}{Q(\tilde{q}_0, \tilde{k}_\parallel, \tilde{q}_\parallel) + 2 \tilde{k}_\parallel \cdot \tilde{q}_\parallel} \right]. \quad (C36) $$

Here we study the special case of imaginary frequencies $\tilde{q}_0 = i\hbar\xi = i\Xi$, in polar coordinates

$$ \Pi_T(q) = g_s \frac{\alpha c}{\hbar} \frac{-\Xi^2}{\tilde{q}_\parallel^2} \int_0^{k_M} \frac{dk_\parallel}{2\pi} \frac{k_\parallel}{\epsilon_{\mathbf{k}}} [1 - N_\mu(\epsilon_{\mathbf{k}})] \int_0^{2\pi} \frac{d\varphi}{2\pi} \sum_{\lambda=\pm} \left[ 1 + \frac{M_{00}(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) + 4 \frac{\tilde{q}_\parallel^2}{\Xi^2} (\tilde{k}_\parallel^2 + i\lambda\Xi\epsilon_{\mathbf{k}})}{Q(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) + 2\tilde{k}_\parallel\tilde{q}_\parallel \cos(\varphi)} \right]. \quad (C37) $$

The integral in $\varphi$ can be carried out using Eq. (C22)

$$ \int_0^{2\pi} \frac{d\varphi}{2\pi} \frac{1}{Q + a \cos(\varphi)} = \frac{1}{\sqrt{Q^2 - a^2}}. \quad (C38) $$

Therefore, $\Pi_T(q)$ can be written as

$$ \Pi_T(q) = g_s \frac{\alpha c}{\hbar} \frac{-\Xi^2}{\tilde{q}_\parallel^2} \int_0^{k_M} \frac{dk_\parallel}{2\pi} \frac{k_\parallel}{\epsilon_{\mathbf{k}}} [1 - N_\mu(\epsilon_{\mathbf{k}})] \sum_{\lambda=\pm} \left[ 1 + \frac{M_{00}(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) + 4 \frac{\tilde{q}_\parallel^2}{\Xi^2} (\tilde{k}_\parallel^2 + i\lambda\Xi\epsilon_{\mathbf{k}})}{\sqrt{[Q(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)]^2 - (2\tilde{k}_\parallel\tilde{q}_\parallel)^2}} \right]. \quad (C39) $$

Similarly to what was done in Eq. (C25), using $t = \lambda^2 = 1$, the ratio can be equivalently represented as

$$ \sum_{\lambda=\pm} \frac{M_T(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)}{N(i\lambda\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)} = 2\operatorname{Re} \left[ \frac{M_T(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)}{N(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)} \right], \quad (C40) $$

therefore, we obtain

$$ \Pi_T(q) = g_s \frac{\alpha c}{\hbar} \frac{-\Xi^2}{\tilde{q}_\parallel^2} \int_0^{k_M} \frac{dk_\parallel}{2\pi} \frac{k_\parallel}{\epsilon_{\mathbf{k}}} [1 - N_\mu(\epsilon_{\mathbf{k}})] \left[ 2 + 2\operatorname{Re} \left[ \frac{M_{00}(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel) + 4 \frac{\tilde{q}_\parallel^2}{\Xi^2} (\tilde{k}_\parallel^2 + i\Xi\epsilon_{\mathbf{k}})}{\sqrt{[Q(i\Xi, \tilde{k}_\parallel, \tilde{q}_\parallel)]^2 - (2\tilde{k}_\parallel\tilde{q}_\parallel)^2}} \right] \right]. \quad (C41) $$

Now we apply the change of coordinates $\epsilon = \sqrt{(\hbar v_F k_\parallel)^2 + m^2}$

$$ \Pi_T(q) = 2g_s \frac{\alpha c}{\hbar} \frac{-\Xi^2}{\tilde{q}_\parallel^2} \int_m^{\epsilon_M} \frac{d\epsilon}{2\pi} [1 - N_\mu(\epsilon)] \left[ 1 + \operatorname{Re} \left[ \frac{-\Xi^2 - \tilde{q}_\parallel^2 + 4i\Xi\epsilon + 4\epsilon^2 + 4 \frac{\tilde{q}_\parallel^2}{\Xi^2} (\epsilon^2 - m^2 + i\Xi\epsilon)}{\sqrt{(\Xi^2 + \tilde{q}_\parallel^2 - 2i\Xi\epsilon)^2 - 4(\epsilon^2 - m^2)\tilde{q}_\parallel^2}} \right] \right]. \quad (C42) $$

It is followed by the change of variable into a non-dimensional energy, using $\tilde{\theta}_z = \sqrt{\Xi^2 + \tilde{q}_\parallel^2}$ (and, therefore, $\tilde{q}_\parallel^2 = \tilde{\theta}_z^2 - \Xi^2$), we apply the change of variable $\epsilon = \tilde{\theta}_z^{u/2}$, obtaining

$$ \Pi_T(q) = 2g_s \frac{\alpha c}{\hbar} \frac{-\Xi^2}{\tilde{q}_\parallel^2} \int_{\frac{2m}{\tilde{q}_\parallel}}^{\frac{2\epsilon_M}{\tilde{q}_\parallel}} \frac{du}{2\pi} \frac{\tilde{\theta}_z}{2} [1 - N_\mu(\tilde{\theta}_z^{u/2})] \left[ 1 + \operatorname{Re} \left[ \frac{-\tilde{\theta}_z^2 + 2i\Xi\tilde{\theta}_z u + \tilde{\theta}_z^2 u^2 + 4 \left( \frac{\tilde{\theta}_z^2}{\Xi^2} - 1 \right) \left( \tilde{\theta}_z^2 \frac{u^2}{4} - m^2 + i\Xi\tilde{\theta}_z^{u/2} \right)}{\sqrt{\left( \tilde{\theta}_z^2 - i\Xi\tilde{\theta}_z u \right)^2 - \left( \tilde{\theta}_z^2 u^2 - 4m^2 \right) \left( \tilde{\theta}_z^2 - \Xi^2 \right)}} \right] \right]. \quad (C43) $$

Using the definitions $\delta = \frac{2m}{\tilde{\theta}_z}$ and $\gamma = \frac{\Xi}{\tilde{\theta}_z}$, we can simplify this integral into

$$ \Pi_T(q) = 2g_s \frac{\alpha c}{\hbar} \frac{-\Xi^2}{\tilde{q}_\parallel^2} \int_{\delta}^{\frac{2\epsilon_M}{\tilde{q}_\parallel}} \frac{du}{2\pi} \frac{\tilde{\theta}_z}{2} [1 - N_\mu(\tilde{\theta}_z^{u/2})] \left[ 1 + \operatorname{Re} \left[ \frac{-1 + 2i\gamma u + u^2 + (\gamma^2 - 1)(u^2 - \delta^2 + 2i\gamma u)}{\sqrt{(1 - i\gamma u)^2 - (u^2 - \delta^2)(1 - \gamma^2)}} \right] \right]. \quad (C44) $$
This integral can be further simplified, and the cut-off can be eliminated to obtain

$$ \Pi_T(q) = \frac{g_s \alpha c \Xi^2}{2\pi\hbar \tilde{q}_\parallel^2} \tilde{\theta}_z \int_{\delta}^{\infty} du \left[ 1 - N_{\mu}(\tilde{\theta}_z \frac{u}{2}) \right] \left[ 1 + \mathrm{Re} \left[ \frac{(1 + i\gamma^{-1}u)^2 + (\gamma^{-2} - 1)\delta^2}{\sqrt{1 - 2i\gamma u - u^2 + (1 - \gamma^2)\delta^2}} \right] \right]. \quad (C45) $$

Once $\Pi_T^{(0)}(q)$ is regularized following [9][21], we have

$$ \Pi_T(q) = \frac{g_s \alpha c}{8\pi \hbar} \tilde{\theta}_z \Psi(\delta) - g_s \frac{\alpha c}{2\pi \hbar} \frac{\Xi^2}{\tilde{q}_\parallel^2} \tilde{\theta}_z \int_{\delta}^{\infty} du N_{\mu}(\tilde{\theta}_z \frac{u}{2}) \left[ 1 + \mathrm{Re} \left[ \frac{(1 + i\gamma^{-1}u)^2 + (\gamma^{-2} - 1)\delta^2}{\sqrt{1 - 2i\gamma u - u^2 + (1 - \gamma^2)\delta^2}} \right] \right]. \quad (C46) $$

Those are Eq. (118) and Eq. (119) in Sect. VII. The equivalent transversal conductivity is $(\sigma_T^{QFT-b}(i\xi) = i\Pi_T(i\xi)/(i\xi) = \Pi_T(i\xi)/\xi)$

$$ \sigma_T(q) = \frac{g_s \alpha c}{8\pi} \gamma^{-1} \Psi(\delta) - g_s \frac{\alpha c \Xi}{2\pi \tilde{q}_\parallel^2} \tilde{\theta}_z \int_{\delta}^{\infty} du N_{\mu}(\tilde{\theta}_z \frac{u}{2}) \left[ 1 + \mathrm{Re} \left[ \frac{(1 + i\gamma^{-1}u)^2 + (\gamma^{-2} - 1)\delta^2}{\sqrt{1 - 2i\gamma u - u^2 + (1 - \gamma^2)\delta^2}} \right] \right]. \quad (C47) $$

## 4. Polarization Trace

The trace of the polarization operator can be written in terms of the longitudinal and transversal polarization terms (see Eq. (77)) as

$$ \Pi_{\text{tr}}(q) = -\Pi_T(q) - \frac{q_z^2}{q_{\parallel}^2} \Pi_{00}(q), \quad (C48) $$

a result derived from the transversality condition. Using Eqs. (C31) and (C36), we obtain

$$ \Pi_{\text{tr}}(q) = -ig_s \frac{\alpha c}{\hbar} \int_{\mathbf{k}} \frac{1}{2\epsilon_{\mathbf{k}}} [1 - N_{\mu}(\epsilon_{\mathbf{k}})] \sum_{\lambda=\pm} \left[ 1 + \frac{\tilde{q}_0^2 - \tilde{q}_{\parallel}^2 + 4\Delta^2}{Q(\tilde{q}_0, \tilde{k}_{\parallel}, \tilde{q}_{\parallel}) + 2\tilde{k}_{\parallel} \cdot \tilde{q}_{\parallel}} \right]. \quad (C49) $$

In imaginary frequencies, after with the same procedure as the one used to obtain $\Pi_{00}$, we get

$$ \Pi_{\text{tr}}(q) = -ig_s \frac{\alpha c}{\hbar} \frac{\tilde{\theta}_z}{2\pi\hbar^2 v_F^2} \int_{\delta}^{\infty} du \left[ 1 - N_{\mu}\left(\tilde{\theta}_z \frac{u}{2}\right) \right] \sum_{\lambda=\pm} \left[ 1 + \mathrm{Re} \left[ \frac{\delta^2 - 1}{\sqrt{1 - u^2 - 2i\lambda\gamma u + (1 - \gamma^2)\delta^2}} \right] \right]. \quad (C50) $$

# Appendix D: Derivation of the conductivity from the Kubo formula

In this appendix, starting from the Ohm Law $J_\mu = \sigma_{\mu\nu}E^\nu$ (compare with Eq. (51)), we are going to derive the expression of the conductivity from the Kubo formula [5], where the switching starts at $t \to -\infty$ to avoid the transitory to the new (non-equilibrium) steady state

$$ \sigma_{\mu\nu}(t, \mathbf{q}) = \int_{-\infty}^{t} d\tau \Pi_{\mu\nu}(\tau, \mathbf{q}) = \int_{-\infty}^{t} d\tau \mathrm{Tr} (\hat{\rho}_{\beta}[J_{\mu}(\tau, \mathbf{q}), J_{\mu}^{*}(0, \mathbf{q})]), \quad (D1) $$

where the Polarization operator, instead of be given by Eq. (53), is equivalently defined in real time as ($[A, B]$ is the commutator between A and B)

$$ \Pi_{\mu\nu}(\tau, \mathbf{q}) = \mathrm{Tr} (\hat{\rho}_{\beta}[J_{\mu}(\tau, \mathbf{q}), J_{\mu}^{*}(0, \mathbf{q})]), \quad (D2) $$

and as a consequence we have

$$ \partial_t \sigma_{\mu\nu}(t, \mathbf{q}) = \Pi_{\mu\nu}(t, \mathbf{q}). \quad (D3) $$

We start from the (causal) microscopic Ohm law in the position space.

$$ J_{\mu}(t, \mathbf{x}) = \int_{-\infty}^{t} d\tau \int d^d\mathbf{y} \sigma_{\mu\nu}(t - \tau, \mathbf{x} - \mathbf{y}) E^{\nu}(\tau, \mathbf{y}). \quad (D4) $$
This integral can be further simplified, and the cut-off can be eliminated to obtain

$$ \Pi_T(q) = \frac{g_s \alpha c \Xi^2}{2\pi\hbar \tilde{q}_{\parallel}^2} \tilde{\theta}_z \int_{\delta}^{\infty} du \left[ 1 - N_{\mu}(\tilde{\theta}_z \frac{u}{2}) \right] \left[ 1 + \mathrm{Re} \left[ \frac{(1 + i\gamma^{-1}u)^2 + (\gamma^{-2} - 1)\delta^2}{\sqrt{1 - 2i\gamma u - u^2 + (1 - \gamma^2)\delta^2}} \right] \right]. \quad (C45) $$

Once $\Pi_T^{(0)}(q)$ is regularized following [9][21], we have

$$ \Pi_T(q) = \frac{g_s \alpha c}{8\pi \hbar} \tilde{\theta}_z \Psi(\delta) - g_s \frac{\alpha c}{2\pi \hbar} \frac{\Xi^2}{\tilde{q}_{\parallel}^2} \tilde{\theta}_z \int_{\delta}^{\infty} du N_{\mu}(\tilde{\theta}_z \frac{u}{2}) \left[ 1 + \mathrm{Re} \left[ \frac{(1 + i\gamma^{-1}u)^2 + (\gamma^{-2} - 1)\delta^2}{\sqrt{1 - 2i\gamma u - u^2 + (1 - \gamma^2)\delta^2}} \right] \right]. \quad (C46) $$

Those are Eq. (118) and Eq. (119) in Sect. VII. The equivalent transversal conductivity is $(\sigma_T^{QFT-b}(i\xi) = i\Pi_T(i\xi)/(i\xi) = \Pi_T(i\xi)/\xi)$

$$ \sigma_T(q) = \frac{g_s \alpha c}{8\pi} \gamma^{-1} \Psi(\delta) - g_s \frac{\alpha c \Xi}{2\pi \tilde{q}_{\parallel}^2} \tilde{\theta}_z \int_{\delta}^{\infty} du N_{\mu}(\tilde{\theta}_z \frac{u}{2}) \left[ 1 + \mathrm{Re} \left[ \frac{(1 + i\gamma^{-1}u)^2 + (\gamma^{-2} - 1)\delta^2}{\sqrt{1 - 2i\gamma u - u^2 + (1 - \gamma^2)\delta^2}} \right] \right]. \quad (C47) $$

## 4. Polarization Trace

The trace of the polarization operator can be written in terms of the longitudinal and transversal polarization terms (see Eq. (77)) as

$$ \Pi_{\mathrm{tr}}(q) = -\Pi_T(q) - \frac{q_z^2}{q_{\parallel}^2} \Pi_{00}(q), \quad (C48) $$

a result derived from the transversality condition. Using Eqs. (C31) and (C36), we obtain

$$ \Pi_{\mathrm{tr}}(q) = -ig_s \frac{\alpha c}{\hbar} \int_{\mathbf{k}} \frac{1}{2\epsilon_{\mathbf{k}}} [1 - N_{\mu}(\epsilon_{\mathbf{k}})] \sum_{\lambda=\pm} \left[ 1 + \frac{\tilde{q}_0^2 - \tilde{q}_{\parallel}^2 + 4\Delta^2}{Q(\tilde{q}_0, \tilde{k}_{\parallel}, \tilde{q}_{\parallel}) + 2\tilde{k}_{\parallel} \cdot \tilde{q}_{\parallel}} \right]. \quad (C49) $$

In imaginary frequencies, after with the same procedure as the one used to obtain $\Pi_{00}$, we get

$$ \Pi_{\mathrm{tr}}(q) = -ig_s \frac{\alpha c}{\hbar} \frac{\tilde{\theta}_z}{2\pi \hbar^2 v_F^2} \int_{\delta}^{\infty} du \left[ 1 - N_{\mu}\left(\tilde{\theta}_z \frac{u}{2}\right) \right] \sum_{\lambda=\pm} \left[ 1 + \mathrm{Re} \left[ \frac{\delta^2 - 1}{\sqrt{1 - u^2 - 2i\lambda\gamma u + (1 - \gamma^2)\delta^2}} \right] \right]. \quad (C50) $$

## Appendix D: Derivation of the conductivity from the Kubo formula

In this appendix, starting from the Ohm Law $J_{\mu} = \sigma_{\mu\nu}E^{\nu}$ (compare with Eq. (51)), we are going to derive the expression of the conductivity from the Kubo formula [5], where the switching starts at $t \to -\infty$ to avoid the transitory to the new (non-equilibrium) steady state

$$ \sigma_{\mu\nu}(t, \mathbf{q}) = \int_{-\infty}^{t} d\tau \Pi_{\mu\nu}(\tau, \mathbf{q}) = \int_{-\infty}^{t} d\tau \mathrm{Tr} (\hat{\rho}_{\beta}[J_{\mu}(\tau, \mathbf{q}), J_{\mu}^{*}(0, \mathbf{q})]), \quad (D1) $$

where the Polarization operator, instead of be given by Eq. (53), is equivalently defined in real time as ($[A, B]$ is the commutator between A and B)

$$ \Pi_{\mu\nu}(\tau, \mathbf{q}) = \mathrm{Tr} (\hat{\rho}_{\beta}[J_{\mu}(\tau, \mathbf{q}), J_{\mu}^{*}(0, \mathbf{q})]), \quad (D2) $$

and as a consequence we have

$$ \partial_t \sigma_{\mu\nu}(t, \mathbf{q}) = \Pi_{\mu\nu}(t, \mathbf{q}). \quad (D3) $$

We start from the (causal) microscopic Ohm law in the position space.

$$ J_{\mu}(t, \mathbf{x}) = \int_{-\infty}^{t} d\tau \int d^{d}\mathbf{y} \sigma_{\mu\nu}(t - \tau, \mathbf{x} - \mathbf{y}) E^{\nu}(\tau, \mathbf{y}). \quad (D4) $$
Applying a Fourier transform in the position's coordinates, using the convolution theorem, and substituting $E^\nu(\tau, \mathbf{q}) = -\partial_\tau A^\nu(\tau, \mathbf{q})$ (we are using the Coulomb Gauge in these calculations, as usual) we get

$$J_{\mu}(t, \mathbf{q}) = - \int_{-\infty}^{t} d\tau \sigma_{\mu\nu}(t - \tau, \mathbf{q}) \partial_{\tau} A^{\nu}(\tau, \mathbf{q}). \qquad (D5)$$

Here we apply an integration by parts in time

$$J_{\mu}(t, \mathbf{q}) = - \left[ \sigma_{\mu\nu}(t - \tau, \mathbf{q}) A^{\nu}(\tau, \mathbf{q}) \right]_{-\infty}^{t} + (-1)^2 \int_{-\infty}^{t} d\tau \left[ \partial_{\tau} \sigma_{\mu\nu}(t - \tau, \mathbf{q}) \right] A^{\nu}(\tau, \mathbf{q}). \qquad (D6)$$

Using Eq. (D3) and that the integral of the derivative is the initial function, we obtain

$$J_{\mu}(t, \mathbf{q}) = - \left[ A^{\nu}(\tau, \mathbf{q}) \int_{-\infty}^{\tau} d\tau_1 \partial_{\tau_1} \sigma_{\mu\nu}(t - \tau_1, \mathbf{q}) \right]_{-\infty}^{t} + \int_{-\infty}^{t} d\tau \Pi_{\mu\nu}(t - \tau, \mathbf{q}) A^{\nu}(\tau, \mathbf{q}). \qquad (D7)$$

We can cancel out the boundary $t \to -\infty$ integral term

$$J_{\mu}(t, \mathbf{q}) = - \left[ A^{\nu}(t, \mathbf{q}) \int_{-\infty}^{t} d\tau_1 \Pi_{\mu\nu}(t - \tau_1, \mathbf{q}) - 0 \right] + \int_{-\infty}^{t} d\tau \Pi_{\mu\nu}(t - \tau, \mathbf{q}) A^{\nu}(\tau, \mathbf{q}), \qquad (D8)$$

and this result can be simplified into

$$J_{\mu}(t, \mathbf{q}) = - \int_{-\infty}^{t} d\tau \Pi_{\mu\nu}(t - \tau, \mathbf{q}) A^{\nu}(t, \mathbf{q}) + \int_{-\infty}^{t} d\tau \Pi_{\mu\nu}(t - \tau, \mathbf{q}) A^{\nu}(\tau, \mathbf{q}), \qquad (D9)$$

$$J_{\mu}(t, \mathbf{q}) = \int_{-\infty}^{t} d\tau \Pi_{\mu\nu}(t - \tau, \mathbf{q}) [A^{\nu}(\tau, \mathbf{q}) - A^{\nu}(t, \mathbf{q})]. \qquad (D10)$$

Note that this result is already different from the linear relation between the electric conductivity and the potential vector proposed in Eq. (56) that can be written as a function of time as

$$J_{\mu}(t, \mathbf{q}) = \int_{-\infty}^{t} d\tau \Pi_{\mu\nu}(t - \tau) A^{\nu}(\tau, \mathbf{q}). \qquad (D11)$$

We can write the integral as a kernel proportional to the retarded $A^\nu(\tau, \mathbf{q})$ by using

$$\begin{aligned} \int_{-\infty}^{t} d\tau \Pi_{\mu\nu}(t - \tau, \mathbf{q}) A^{\nu}(t, \mathbf{q}) &= \int_{-\infty}^{t} d\tau_1 \Pi_{\mu\nu}(t - \tau_1, \mathbf{q}) A^{\nu}(t, \mathbf{q}) \\ &= \int_{-\infty}^{t} d\tau_1 \Pi_{\mu\nu}(t - \tau_1, \mathbf{q}) \int_{-\infty}^{t} d\tau A^{\nu}(t, \mathbf{q}) \delta(\tau - t) \\ &= \int_{-\infty}^{t} d\tau \int_{-\infty}^{t} d\tau_1 \Pi_{\mu\nu}(t - \tau_1, \mathbf{q}) A^{\nu}(t, \mathbf{q}) \delta(\tau - t) \\ &= \int_{-\infty}^{t} d\tau \delta(\tau - t) \left( \int_{-\infty}^{\tau} d\tau_1 \Pi_{\mu\nu}(\tau - \tau_1, \mathbf{q}) \right) A^{\nu}(\tau, \mathbf{q}). \end{aligned} \qquad (D12)$$

Introducing Eq. (D12) in Eq. (D10), we have

$$J_{\mu}(t, \mathbf{q}) = \int_{-\infty}^{t} d\tau \left[ \Pi_{\mu\nu}(t - \tau, \mathbf{q}) - \delta(\tau - t) \left( \int_{-\infty}^{\tau} d\tau_1 \Pi_{\mu\nu}(\tau - \tau_1, \mathbf{q}) \right) \right] A^{\nu}(\tau, \mathbf{q}). \qquad (D13)$$

In temporal frequency space, using

$$\Pi_{\mu\nu}(\omega, \mathbf{q}) = \int_{-\infty}^{\infty} d\tau_1 e^{-i\omega\tau_1} \Theta(\tau - \tau_1) \Pi_{\mu\nu}(\tau - \tau_1, \mathbf{q}) = \int_{-\infty}^{\tau} d\tau_1 e^{-i\omega\tau_1} \Pi_{\mu\nu}(\tau - \tau_1, \mathbf{q}), \qquad (D14)$$
Applying a Fourier transform in the position's coordinates, using the convolution theorem, and substituting $E^\nu(\tau, \mathbf{q}) = -\partial_\tau A^\nu(\tau, \mathbf{q})$ (we are using the Coulomb Gauge in these calculations, as usual) we get

$$J_{\mu}(t, \mathbf{q}) = - \int_{-\infty}^{t} d\tau \sigma_{\mu\nu}(t - \tau, \mathbf{q}) \partial_{\tau} A^{\nu}(\tau, \mathbf{q}). \qquad (D5)$$

Here we apply an integration by parts in time

$$J_{\mu}(t, \mathbf{q}) = - \left[ \sigma_{\mu\nu}(t - \tau, \mathbf{q}) A^{\nu}(\tau, \mathbf{q}) \right]_{-\infty}^{t} + (-1)^2 \int_{-\infty}^{t} d\tau \left[ \partial_{\tau} \sigma_{\mu\nu}(t - \tau, \mathbf{q}) \right] A^{\nu}(\tau, \mathbf{q}). \qquad (D6)$$

Using Eq. (D3) and that the integral of the derivative is the initial function, we obtain

$$J_{\mu}(t, \mathbf{q}) = - \left[ A^{\nu}(\tau, \mathbf{q}) \int_{-\infty}^{\tau} d\tau_{1} \partial_{\tau_{1}} \sigma_{\mu\nu}(t - \tau_{1}, \mathbf{q}) \right]_{-\infty}^{t} + \int_{-\infty}^{t} d\tau \Pi_{\mu\nu}(t - \tau, \mathbf{q}) A^{\nu}(\tau, \mathbf{q}). \qquad (D7)$$

We can cancel out the boundary $t \to -\infty$ integral term

$$J_{\mu}(t, \mathbf{q}) = - \left[ A^{\nu}(t, \mathbf{q}) \int_{-\infty}^{t} d\tau_{1} \Pi_{\mu\nu}(t - \tau_{1}, \mathbf{q}) - 0 \right] + \int_{-\infty}^{t} d\tau \Pi_{\mu\nu}(t - \tau, \mathbf{q}) A^{\nu}(\tau, \mathbf{q}), \qquad (D8)$$

and this result can be simplified into

$$J_{\mu}(t, \mathbf{q}) = - \int_{-\infty}^{t} d\tau \Pi_{\mu\nu}(t - \tau, \mathbf{q}) A^{\nu}(t, \mathbf{q}) + \int_{-\infty}^{t} d\tau \Pi_{\mu\nu}(t - \tau, \mathbf{q}) A^{\nu}(\tau, \mathbf{q}), \qquad (D9)$$

$$J_{\mu}(t, \mathbf{q}) = \int_{-\infty}^{t} d\tau \Pi_{\mu\nu}(t - \tau, \mathbf{q}) [A^{\nu}(\tau, \mathbf{q}) - A^{\nu}(t, \mathbf{q})]. \qquad (D10)$$

Note that this result is already different from the linear relation between the electric conductivity and the potential vector proposed in Eq. (56) that can be written as a function of time as

$$J_{\mu}(t, \mathbf{q}) = \int_{-\infty}^{t} d\tau \Pi_{\mu\nu}(t - \tau) A^{\nu}(\tau, \mathbf{q}). \qquad (D11)$$

We can write the integral as a kernel proportional to the retarded $A^\nu(\tau, \mathbf{q})$ by using

$$\begin{aligned} \int_{-\infty}^{t} d\tau \Pi_{\mu\nu}(t - \tau, \mathbf{q}) A^{\nu}(t, \mathbf{q}) &= \int_{-\infty}^{t} d\tau_{1} \Pi_{\mu\nu}(t - \tau_{1}, \mathbf{q}) A^{\nu}(t, \mathbf{q}) \\ &= \int_{-\infty}^{t} d\tau_{1} \Pi_{\mu\nu}(t - \tau_{1}, \mathbf{q}) \int_{-\infty}^{t} d\tau A^{\nu}(t, \mathbf{q}) \delta(\tau - t) \\ &= \int_{-\infty}^{t} d\tau \int_{-\infty}^{t} d\tau_{1} \Pi_{\mu\nu}(t - \tau_{1}, \mathbf{q}) A^{\nu}(t, \mathbf{q}) \delta(\tau - t) \\ &= \int_{-\infty}^{t} d\tau \delta(\tau - t) \left( \int_{-\infty}^{\tau} d\tau_{1} \Pi_{\mu\nu}(\tau - \tau_{1}, \mathbf{q}) \right) A^{\nu}(\tau, \mathbf{q}). \end{aligned} \qquad (D12)$$

Introducing Eq. (D12) in Eq. (D10), we have

$$J_{\mu}(t, \mathbf{q}) = \int_{-\infty}^{t} d\tau \left[ \Pi_{\mu\nu}(t - \tau, \mathbf{q}) - \delta(\tau - t) \left( \int_{-\infty}^{\tau} d\tau_{1} \Pi_{\mu\nu}(\tau - \tau_{1}, \mathbf{q}) \right) \right] A^{\nu}(\tau, \mathbf{q}). \qquad (D13)$$

In temporal frequency space, using

$$\Pi_{\mu\nu}(\omega, \mathbf{q}) = \int_{-\infty}^{\infty} d\tau_{1} e^{-i\omega\tau_{1}} \Theta(\tau - \tau_{1}) \Pi_{\mu\nu}(\tau - \tau_{1}, \mathbf{q}) = \int_{-\infty}^{\tau} d\tau_{1} e^{-i\omega\tau_{1}} \Pi_{\mu\nu}(\tau - \tau_{1}, \mathbf{q}), \qquad (D14)$$
we obtain the relation

$$J_{\mu}(\omega, \mathbf{q}) = \left[ \Pi_{\mu\nu}(\omega, \mathbf{q}) - \left( \int_{-\infty}^{\tau} d\tau_1 \Pi_{\mu\nu}(\tau - \tau_1, \mathbf{q}) \right) \right] A^{\nu}(\omega, \mathbf{q}), \qquad (D15)$$

$$J_{\mu}(\omega, \mathbf{q}) = \left[ \Pi_{\mu\nu}(\omega, \mathbf{q}) - \lim_{\omega \to 0} \Pi_{\mu\nu}(\omega, \mathbf{q}) \right] A^{\nu}(\omega, \mathbf{q}) \neq \Pi_{\mu\nu}(\mathbf{q}) A^{\nu}(\omega, \mathbf{q}). \qquad (D16)$$

This is a result in contradiction with the usual assumed relation provided in Eq. (56). Finally, using the microscopic Ohm law in temporal frequency space, we have from Eq. (D5)

$$J_{\mu}(\omega, \mathbf{q}) = \sigma_{\mu\nu}(\omega, \mathbf{q})(-i\omega)A^{\nu}(\omega, \mathbf{q}). \qquad (D17)$$

Then we can compare Eq. (D16) and Eq. (D17) to obtain the so-called Luttinger formula [62][63]

$$\sigma_{\mu\nu}(\omega, \mathbf{q}) = \frac{\Pi_{\mu\nu}(\omega, \mathbf{q}) - \lim_{\omega \to 0} \Pi_{\mu\nu}(\omega, \mathbf{q})}{-i\omega}. \qquad (D18)$$

So we conclude that, assuming the microscopic Ohm law given in Eq. (D4), and the relation between the conductivity $\sigma_{\mu\nu}$ and the Polarization operator $\Pi_{\mu\nu}$ derived from the Kubo formula given in Eq. (D1), the Fourier transform of the conductivity tensor is given by Eq. (D18). Therefore, we conclude that the correct conductivity should be obtained from Eq. (D18). Note that this subtraction, that we derived here simply from Ohm law and time causality, naturally implies that it cannot exist stationary current in absence of electric field, i.e. that always $J_{\mu}(\omega = 0, \mathbf{q}) = 0$ even when $A_{\mu}(\omega = 0, \mathbf{q}) \neq 0$. This is a strong physical requirement, already discussed by A.A. Abrikosov in 1963 [64] and more recently in [28].

[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004), https://www.science.org/doi/pdf/10.1126/science.1102896.

[2] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).

[3] L. M. Woods, D. A. R. Dalvit, A. Tkatchenko, P. Rodriguez-Lopez, A. W. Rodriguez, and R. Podgornik, Rev. Mod. Phys. 88, 045003 (2016).

[4] S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Rev. Mod. Phys. 83, 407 (2011).

[5] R. Kubo, Journal of the Physical Society of Japan 12, 570 (1957), https://doi.org/10.1143/JPSJ.12.570.

[6] P. Rodriguez-Lopez, W. J. M. Kort-Kamp, D. A. R. Dalvit, and L. M. Woods, Phys. Rev. Mater. 2, 014003 (2018).

[7] G. W. Hanson, IEEE Transactions on Antennas and Propagation 56, 747 (2008).

[8] M. Liu, Y. Zhang, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, Phys. Rev. Lett. 126, 206802 (2021).

[9] M. Bordag, G. L. Klimchitskaya, V. M. Mostepanenko, and V. M. Petrov, Phys. Rev. D 91, 045037 (2015).

[10] P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin, Phys. Rev. B 74, 235443 (2006).

[11] A. B. Kashuba, Phys. Rev. B 78, 085415 (2008).

[12] L. Fritz, J. Schmalian, M. Müller, and S. Sachdev, Phys. Rev. B 78, 085416 (2008).

[13] Y. Jeyar, M. Antezza, and B. Guizal, Phys. Rev. E 107, 025306 (2023).

[14] P. Rodriguez-Lopez, D.-N. Le, I. V. Bondarev, M. Antezza, and L. M. Woods, Phys. Rev. B 109, 035422 (2024).

[15] C. Abbas, B. Guizal, and M. Antezza, Phys. Rev. Lett. 118, 126101 (2017).

[16] Y. Jeyar, M. Luo, K. Austry, B. Guizal, Y. Zheng, H. B. Chan, and M. Antezza, Phys. Rev. A 108, 062811 (2023).

[17] Y.-M. Zhang, M. Antezza, and J.-S. Wang, International Journal of Heat and Mass Transfer 194, 123076 (2022).

[18] G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. D 107, 105007 (2023).

[19] G. L. Klimchitskaya and V. M. Mostepanenko, Physics 5, 1013 (2023).

[20] V. R. Khalilov and I. V. Mamsurov, The European Physical Journal C 75, 167 (2015).

[21] G. Bimonte, G. L. Klimchitskaya, and V. M. Mostepanenko, Phys. Rev. B 96, 115430 (2017).

[22] J. F. Annett, Superconductivity, superfluids and condensates, Oxford master series in condensed matter physics (Oxford Univ. Press, Oxford, 2004).

[23] F. London, Phys. Rev. 74, 562 (1948).

[24] J. Bardeen, Phys. Rev. 81, 469 (1951).

[25] P. Coleman, Introduction to Many-Body Physics (Cambridge University Press, 2015).

[26] G. Gómez-Santos, Phys. Rev. B 80, 245424 (2009).

[27] N. M. R. Peres, Rev. Mod. Phys. 82, 2673 (2010).

[28] L. A. Falkovsky and A. A. Varlamov, The European
we obtain the relation

$$J_{\mu}(\omega, \mathbf{q}) = \left[ \Pi_{\mu\nu}(\omega, \mathbf{q}) - \left( \int_{-\infty}^{\tau} d\tau_1 \Pi_{\mu\nu}(\tau - \tau_1, \mathbf{q}) \right) \right] A^{\nu}(\omega, \mathbf{q}), \quad (D15)$$

$$J_{\mu}(\omega, \mathbf{q}) = \left[ \Pi_{\mu\nu}(\omega, \mathbf{q}) - \lim_{\omega \to 0} \Pi_{\mu\nu}(\omega, \mathbf{q}) \right] A^{\nu}(\omega, \mathbf{q}) \neq \Pi_{\mu\nu}(\mathbf{q}) A^{\nu}(\omega, \mathbf{q}). \quad (D16)$$

This is a result in contradiction with the usual assumed relation provided in Eq. (56). Finally, using the microscopic Ohm law in temporal frequency space, we have from Eq. (D5)

$$J_{\mu}(\omega, \mathbf{q}) = \sigma_{\mu\nu}(\omega, \mathbf{q})(-i\omega)A^{\nu}(\omega, \mathbf{q}). \quad (D17)$$

Then we can compare Eq. (D16) and Eq. (D17) to obtain the so-called Luttinger formula [62][63]

$$\sigma_{\mu\nu}(\omega, \mathbf{q}) = \frac{\Pi_{\mu\nu}(\omega, \mathbf{q}) - \lim_{\omega \to 0} \Pi_{\mu\nu}(\omega, \mathbf{q})}{-i\omega}. \quad (D18)$$

So we conclude that, assuming the microscopic Ohm law given in Eq. (D4), and the relation between the conductivity $\sigma_{\mu\nu}$ and the Polarization operator $\Pi_{\mu\nu}$ derived from the Kubo formula given in Eq. (D1), the Fourier transform of the conductivity tensor is given by Eq. (D18). Therefore, we conclude that the correct conductivity should be obtained from Eq. (D18). Note that this subtraction, that we derived here simply from Ohm law and time causality, naturally implies that it cannot exist stationary current in absence of electric field, i.e. that always $J_{\mu}(\omega = 0, \mathbf{q}) = 0$ even when $A_{\mu}(\omega = 0, \mathbf{q}) \neq 0$. This is a strong physical requirement, already discussed by A.A. Abrikosov in 1963 [64] and more recently in [28].

[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004), https://www.science.org/doi/pdf/10.1126/science.1102896.

[2] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).

[3] L. M. Woods, D. A. R. Dalvit, A. Tkatchenko, P. Rodriguez-Lopez, A. W. Rodriguez, and R. Podgornik, Rev. Mod. Phys. 88, 045003 (2016).

[4] S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Rev. Mod. Phys. 83, 407 (2011).

[5] R. Kubo, Journal of the Physical Society of Japan 12, 570 (1957), https://doi.org/10.1143/JPSJ.12.570.

[6] P. Rodriguez-Lopez, W. J. M. Kort-Kamp, D. A. R. Dalvit, and L. M. Woods, Phys. Rev. Mater. 2, 014003 (2018).

[7] G. W. Hanson, IEEE Transactions on Antennas and Propagation 56, 747 (2008).

[8] M. Liu, Y. Zhang, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, Phys. Rev. Lett. 126, 206802 (2021).

[9] M. Bordag, G. L. Klimchitskaya, V. M. Mostepanenko, and V. M. Petrov, Phys. Rev. D 91, 045037 (2015).

[10] P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin, Phys. Rev. B 74, 235443 (2006).

[11] A. B. Kashuba, Phys. Rev. B 78, 085415 (2008).

[12] L. Fritz, J. Schmalian, M. Müller, and S. Sachdev, Phys. Rev. B 78, 085416 (2008).

[13] Y. Jeyar, M. Antezza, and B. Guizal, Phys. Rev. E 107, 025306 (2023).

[14] P. Rodriguez-Lopez, D.-N. Le, I. V. Bondarev, M. Antezza, and L. M. Woods, Phys. Rev. B 109, 035422 (2024).

[15] C. Abbas, B. Guizal, and M. Antezza, Phys. Rev. Lett. 118, 126101 (2017).

[16] Y. Jeyar, M. Luo, K. Austry, B. Guizal, Y. Zheng, H. B. Chan, and M. Antezza, Phys. Rev. A 108, 062811 (2023).

[17] Y.-M. Zhang, M. Antezza, and J.-S. Wang, International Journal of Heat and Mass Transfer 194, 123076 (2022).

[18] G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. D 107, 105007 (2023).

[19] G. L. Klimchitskaya and V. M. Mostepanenko, Physics 5, 1013 (2023).

[20] V. R. Khalilov and I. V. Mamsurov, The European Physical Journal C 75, 167 (2015).

[21] G. Bimonte, G. L. Klimchitskaya, and V. M. Mostepanenko, Phys. Rev. B 96, 115430 (2017).

[22] J. F. Annett, Superconductivity, superfluids and condensates, Oxford master series in condensed matter physics (Oxford Univ. Press, Oxford, 2004).

[23] F. London, Phys. Rev. 74, 562 (1948).

[24] J. Bardeen, Phys. Rev. 81, 469 (1951).

[25] P. Coleman, Introduction to Many-Body Physics (Cambridge University Press, 2015).

[26] G. Gómez-Santos, Phys. Rev. B 80, 245424 (2009).

[27] N. M. R. Peres, Rev. Mod. Phys. 82, 2673 (2010).

[28] L. A. Falkovsky and A. A. Varlamov, The European
Physical Journal B **56**, 281 (2007).

[29] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, Phys. Rev. Lett. **96**, 256802 (2006).

[30] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, New Journal of Physics **11**, 095013 (2009).

[31] L. A. Falkovsky, Journal of Experimental and Theoretical Physics **106**, 575 (2008).

[32] I. V. Fialkovsky, V. N. Marachevsky, and D. V. Vassilevich, Phys. Rev. B **84**, 035446 (2011).

[33] I. V. Fialkovsky and D. V. Vassilevich, International Journal of Modern Physics A **27**, 1260007 (2012), https://doi.org/10.1142/S0217751X1260007X.

[34] P. Allen, in *Conceptual Foundations of Materials*, Contemporary Concepts of Condensed Matter Science, Vol. 2, edited by S. G. Louie and M. L. Cohen (Elsevier, 2006) pp. 165–218.

[35] K. W. K. Shung, Phys. Rev. B **34**, 979 (1986).

[36] J. González, F. Guinea, and M. A. H. Vozmediano, Modern Physics Letters B **07**, 1593 (1993), https://doi.org/10.1142/S0217984993001612.

[37] B. Wunsch, T. Stauber, F. Sols, and F. Guinea, New Journal of Physics **8**, 318 (2006).

[38] J. González, F. Guinea, and M. Vozmediano, Nuclear Physics B **424**, 595 (1994).

[39] J. González, F. Guinea, and M. A. H. Vozmediano, Phys. Rev. B **59**, R2474 (1999).

[40] P. K. Pyatkovskiy, Journal of Physics: Condensed Matter **21**, 025506 (2008).

[41] G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. D **102**, 016006 (2020).

[42] G. L. Klimchitskaya and V. M. Mostepanenko, Universe **6**, 10.3390/universe06090150 (2020).

[43] G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. D **97**, 085001 (2018).

[44] G. L. Klimchitskaya, V. M. Mostepanenko, and B. E. Sernelius, Phys. Rev. B **89**, 125407 (2014).

[45] B. E. Sernelius, Phys. Rev. B **85**, 195427 (2012).

[46] I. V. Fialkovsky and D. V. Vassilevich, Modern Physics Letters A **31**, 1630047 (2016), https://doi.org/10.1142/S0217732316300470.

[47] M. Bordag, I. V. Fialkovsky, D. M. Gitman, and D. V. Vassilevich, Phys. Rev. B **80**, 245406 (2009).

[48] A. H. C. Neto, Selected topics in graphene physics (2018).

[49] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, International Journal of Modern Physics B **21**, 4611 (2007), https://doi.org/10.1142/S0217979207038022.

[50] M. Ezawa, Journal of the Physical Society of Japan **84**, 121003 (2015), https://doi.org/10.7566/JPSJ.84.121003.

[51] M. Ezawa, Phys. Rev. B **87**, 155415 (2013).

[52] P. R. Wallace, Phys. Rev. **71**, 622 (1947).

[53] M. Jablan, H. Buljan, and M. Soljačić, Phys. Rev. B **80**, 245435 (2009).

[54] S. A. Hartnoll and A. P. Mackenzie, Rev. Mod. Phys. **94**, 041002 (2022).

[55] A. A. Abrikosov, I. Dzyaloshinskii, L. P. Gorkov, and R. A. Silverman, *Methods of quantum field theory in statistical physics* (Dover, New York, NY, 1975).

[56] S. Adam, E. H. Hwang, V. M. Galitski, and S. D. Sarma, Proceedings of the National Academy of Sciences **104**, 18392 (2007), https://www.pnas.org/doi/pdf/10.1073/pnas.0704772104.

[57] L. Szunyogh and P. Weinberger, Journal of Physics: Condensed Matter **11**, 10451 (1999).

[58] M. Cao, Y. Luo, Y. Xie, Z. Tan, G. Fan, Q. Guo, Y. Su, Z. Li, and D.-B. Xiong, Advanced Materials Interfaces **6**, 1900468 (2019), https://onlinelibrary.wiley.com/doi/pdf/10.1002/admi.201900468.

[59] J. M. Link, P. P. Orth, D. E. Sheehy, and J. Schmalian, Phys. Rev. B **93**, 235447 (2016).

[60] S. Teber, *Field theoretic study of electron-electron interaction effects in Dirac liquids*, Phd thesis, Sorbonne Université, Paris, France (2018), arXiv:1810.08428 [cond-mat.mes-hall].

[61] A. J. Millis, Optical conductivity and correlated electron physics, in *Strong interactions in low dimensions*, edited by D. Baeriswyl and L. Degiorgi (Springer Netherlands, Dordrecht, 2004) pp. 195–235.

[62] J. M. Luttinger, Transport theory, in *Mathematical Methods in Solid State and Superfluid Theory: Scottish Universities' Summer School*, edited by R. C. Clark and G. H. Derrick (Springer US, Boston, MA, 1968) pp. 157–193.

[63] L. S. A. Vernes and P. Weinberger, Phase Transitions **75**, 167 (2002), https://doi.org/10.1080/01411590290023067.

[64] A. A. Abrikosov, Zh. Eksp. Teor. Fiz. **44**, 2039 (1963), [Sov. Phys. JETP **17**, 6, p. 2039 (1963)].

[65] M. Bordag and I. G. Pirozhenko, Phys. Rev. D **91**, 085038 (2015).

[66] W.-K. Tse and A. H. MacDonald, Phys. Rev. Lett. **105**, 057401 (2010).

[67] A. Watson, D. Margetis, and M. Luskin, arXiv.org (2023).

[68] A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein, Phys. Rev. B **50**, 7526 (1994).

[69] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, Journal of Physics: Condensed Matter **19**, 026222 (2006).

[70] G. D. Mahan, *Many Particle Physics, Third Edition* (Plenum, New York, 2000).

[71] D. Drosdoff and L. M. Woods, Phys. Rev. B **82**, 155459 (2010).

[72] D. Drosdoff, A. Phan, L. Woods, I. Bondarev, and J. Dobson, The European Physical Journal B **85**, 1434 (2012).

[73] A. Watson, D. Margetis, and M. Luskin, Japan Journal of Industrial and Applied Mathematics **40**, 1765 (2023).

[74] V. Zeitlin, Physics Letters B **352**, 422 (1995).

[75] N. Dorey and N. Mavromatos, Nuclear Physics B **386**, 614 (1992).

[76] M. Ezawa, Phys. Rev. Lett. **109**, 055502 (2012).

[77] M. Ezawa, Phys. Rev. B **86**, 161407(R) (2012).

[78] M. Ezawa, Phys. Rev. Lett. **110**, 026603 (2013).

[79] P. Rodriguez-Lopez, W. J. M. Kort-Kamp, D. A. R. Dalvit, and L. M. Woods, Nature Communications **8**, 14699 (2017).

[80] G. Giuliani and G. Vignale, *Quantum Theory of the Electron Liquid* (Cambridge University Press, 2005).

[81] P. F. Maldague, Surface Science **73**, 296 (1978).

[82] D. Drosdoff and L. M. Woods, Phys. Rev. B **82**, 155459 (2010).

[83] T. Zhu, M. Antezza, and J.-S. Wang, Phys. Rev. B **103**, 125421 (2021).

[84] J.-S. Wang and M. Antezza, Phys. Rev. B **109**, 125105 (2024).

[85] N. A. Sinitsyn, J. E. Hill, H. Min, J. Sinova, and A. H. MacDonald, Phys. Rev. Lett. **97**, 106804 (2006).
Physical Journal B **56**, 281 (2007).

[29] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, Phys. Rev. Lett. **96**, 256802 (2006).

[30] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, New Journal of Physics **11**, 095013 (2009).

[31] L. A. Falkovsky, Journal of Experimental and Theoretical Physics **106**, 575 (2008).

[32] I. V. Fialkovsky, V. N. Marachevsky, and D. V. Vassilevich, Phys. Rev. B **84**, 035446 (2011).

[33] I. V. Fialkovsky and D. V. Vassilevich, International Journal of Modern Physics A **27**, 126007 (2012), https://doi.org/10.1142/S0217751X1260007X.

[34] P. Allen, in *Conceptual Foundations of Materials*, Contemporary Concepts of Condensed Matter Science, Vol. 2, edited by S. G. Louie and M. L. Cohen (Elsevier, 2006) pp. 165–218.

[35] K. W. K. Shung, Phys. Rev. B **34**, 979 (1986).

[36] J. González, F. Guinea, and M. A. H. Vozmediano, Modern Physics Letters B **07**, 1593 (1993), https://doi.org/10.1142/S0217984993001612.

[37] B. Wunsch, T. Stauber, F. Sols, and F. Guinea, New Journal of Physics **8**, 318 (2006).

[38] J. González, F. Guinea, and M. Vozmediano, Nuclear Physics B **424**, 595 (1994).

[39] J. González, F. Guinea, and M. A. H. Vozmediano, Phys. Rev. B **59**, R2474 (1999).

[40] P. K. Pyatkovskiy, Journal of Physics: Condensed Matter **21**, 025506 (2008).

[41] G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. D **102**, 016006 (2020).

[42] G. L. Klimchitskaya and V. M. Mostepanenko, Universe **6**, 10.3390/universe6090150 (2020).

[43] G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. D **97**, 085001 (2018).

[44] G. L. Klimchitskaya, V. M. Mostepanenko, and B. E. Sernelius, Phys. Rev. B **89**, 125407 (2014).

[45] B. E. Sernelius, Phys. Rev. B **85**, 195427 (2012).

[46] I. V. Fialkovsky and D. V. Vassilevich, Modern Physics Letters A **31**, 1630047 (2016), https://doi.org/10.1142/S0217732316300470.

[47] M. Bordag, I. V. Fialkovsky, D. M. Gitman, and D. V. Vassilevich, Phys. Rev. B **80**, 245406 (2009).

[48] A. H. C. Neto, Selected topics in graphene physics (2018).

[49] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, International Journal of Modern Physics B **21**, 4611 (2007), https://doi.org/10.1142/S0217979207038022.

[50] M. Ezawa, Journal of the Physical Society of Japan **84**, 121003 (2015), https://doi.org/10.7566/JPSJ.84.121003.

[51] M. Ezawa, Phys. Rev. B **87**, 155415 (2013).

[52] P. R. Wallace, Phys. Rev. **71**, 622 (1947).

[53] M. Jablan, H. Buljan, and M. Soljačić, Phys. Rev. B **80**, 245435 (2009).

[54] S. A. Hartnoll and A. P. Mackenzie, Rev. Mod. Phys. **94**, 041002 (2022).

[55] A. A. Abrikosov, I. Dzyaloshinskii, L. P. Gorkov, and R. A. Silverman, *Methods of quantum field theory in statistical physics* (Dover, New York, NY, 1975).

[56] S. Adam, E. H. Hwang, V. M. Galitski, and S. D. Sarma, Proceedings of the National Academy of Sciences **104**, 18392 (2007), https://www.pnas.org/doi/pdf/10.1073/pnas.0704772104.

[57] L. Szunyogh and P. Weinberger, Journal of Physics: Condensed Matter **11**, 10451 (1999).

[58] M. Cao, Y. Luo, Y. Xie, Z. Tan, G. Fan, Q. Guo, Y. Su, Z. Li, and D.-B. Xiong, Advanced Materials Interfaces **6**, 1900468 (2019), https://onlinelibrary.wiley.com/doi/pdf/10.1002/admi.201900468.

[59] J. M. Link, P. P. Orth, D. E. Sheehy, and J. Schmalian, Phys. Rev. B **93**, 235447 (2016).

[60] S. Teber, Field theoretic study of electron-electron interaction effects in Dirac liquids, Phd thesis, Sorbonne Universite, Paris, France (2018), arXiv:1810.08428 [cond-mat.mes-hall].

[61] A. J. Millis, Optical conductivity and correlated electron physics, in *Strong interactions in low dimensions*, edited by D. Baeriswyl and L. Degiorgi (Springer Netherlands, Dordrecht, 2004) pp. 195–235.

[62] J. M. Luttinger, Transport theory, in *Mathematical Methods in Solid State and Superfluid Theory: Scottish Universities' Summer School*, edited by R. C. Clark and G. H. Derrick (Springer US, Boston, MA, 1968) pp. 157–193.

[63] L. S. A. Vernes and P. Weinberger, Phase Transitions **75**, 167 (2002), https://doi.org/10.1080/01411590290023067.

[64] A. A. Abrikosov, Zh. Eksp. Teor. Fiz. **44**, 2039 (1963), [Sov. Phys. JETP **17**, 6, p. 2039 (1963)].

[65] M. Bordag and I. G. Pirozhenko, Phys. Rev. D **91**, 085038 (2015).

[66] W.-K. Tse and A. H. MacDonald, Phys. Rev. Lett. **105**, 057401 (2010).

[67] A. Watson, D. Margetis, and M. Luskin, arXiv.org (2023).

[68] A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein, Phys. Rev. B **50**, 7526 (1994).

[69] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, Journal of Physics: Condensed Matter **19**, 026222 (2006).

[70] G. D. Mahan, *Many Particle Physics, Third Edition* (Plenum, New York, 2000).

[71] D. Drosdoff and L. M. Woods, Phys. Rev. B **82**, 155459 (2010).

[72] D. Drosdoff, A. Phan, L. Woods, I. Bondarev, and J. Dobson, The European Physical Journal B **85**, 1434 (2012).

[73] A. Watson, D. Margetis, and M. Luskin, Japan Journal of Industrial and Applied Mathematics **40**, 1765 (2023).

[74] V. Zeitlin, Physics Letters B **352**, 422 (1995).

[75] N. Dorey and N. Mavromatos, Nuclear Physics B **386**, 614 (1992).

[76] M. Ezawa, Phys. Rev. Lett. **109**, 055502 (2012).

[77] M. Ezawa, Phys. Rev. B **86**, 161407(R) (2012).

[78] M. Ezawa, Phys. Rev. Lett. **110**, 026603 (2013).

[79] P. Rodriguez-Lopez, W. J. M. Kort-Kamp, D. A. R. Dalvit, and L. M. Woods, Nature Communications **8**, 14699 (2017).

[80] G. Giuliani and G. Vignale, *Quantum Theory of the Electron Liquid* (Cambridge University Press, 2005).

[81] P. F. Maldague, Surface Science **73**, 296 (1978).

[82] D. Drosdoff and L. M. Woods, Phys. Rev. B **82**, 155459 (2010).

[83] T. Zhu, M. Antezza, and J.-S. Wang, Phys. Rev. B **103**, 125421 (2021).

[84] J.-S. Wang and M. Antezza, Phys. Rev. B **109**, 125105 (2024).

[85] N. A. Sinitsyn, J. E. Hill, H. Min, J. Sinova, and A. H. MacDonald, Phys. Rev. Lett. **97**, 106804 (2006).
[86] L. A. Falkovsky and S. S. Pershoguba, Phys. Rev. B **76**, 153410 (2007).

[87] X. Xiao and W. Wen, Phys. Rev. B **88**, 045442 (2013).

[88] W.-K. Tse and A. H. MacDonald, Phys. Rev. Lett. **105**, 057401 (2010).

[89] W.-K. Tse and A. H. MacDonald, Phys. Rev. B **84**, 205327 (2011).

[90] A. G. Grushin, P. Rodriguez-Lopez, and A. Cortijo, Phys. Rev. B **84**, 045119 (2011).

[91] P. Rodriguez-Lopez and A. G. Grushin, Phys. Rev. Lett. **112**, 056804 (2014).

[92] G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. B **93**, 245419 (2016).

[93] G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. B **94**, 195405 (2016).

[94] G. L. Klimchitskaya, V. M. Mostepanenko, and V. M. Petrov, Phys. Rev. B **96**, 235432 (2017).

[95] M. Bordag, I. Fialkovsky, N. Khusnutdinov, and D. Vasilevich, Phys. Rev. B **104**, 195431 (2021).

[96] G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Phys. Rev. B **89**, 115419 (2014).

[97] R. D. Pisarski, Phys. Rev. D **29**, 2423 (1984).

[98] N. R. Poniatowski, American Journal of Physics **87**, 436 (2019), https://pubs.aip.org/aapt/ajp/article-pdf/87/6/436/13079072/436_1_online.pdf.

[99] P. Rodriguez-Lopez and M. Antezza, Future Publication.

[100] M. Dresselhaus, G. Dresselhaus, S. Cronin, and A. Filho, *Solid State Properties: From Bulk to Nano*, Graduate Texts in Physics (Springer Berlin Heidelberg, 2018).

[101] J. D. Jackson, *Classical electrodynamics*, 3rd ed. (Wiley, New York, NY, 1999).

[102] W. Jones and N. March, *Theoretical Solid State Physics*, Dover books on physics and chemistry No. v. 1 (Wiley-Interscience, 1973).

[103] B. Guizal and M. Antezza, Phys. Rev. B **93**, 115427 (2016).
[86] L. A. Falkovsky and S. S. Pershoguba, Phys. Rev. B **76**, 153410 (2007).

[87] X. Xiao and W. Wen, Phys. Rev. B **88**, 045442 (2013).

[88] W.-K. Tse and A. H. MacDonald, Phys. Rev. Lett. **105**, 057401 (2010).

[89] W.-K. Tse and A. H. MacDonald, Phys. Rev. B **84**, 205327 (2011).

[90] A. G. Grushin, P. Rodriguez-Lopez, and A. Cortijo, Phys. Rev. B **84**, 045119 (2011).

[91] P. Rodriguez-Lopez and A. G. Grushin, Phys. Rev. Lett. **112**, 056804 (2014).

[92] G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. B **93**, 245419 (2016).

[93] G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. B **94**, 195405 (2016).

[94] G. L. Klimchitskaya, V. M. Mostepanenko, and V. M. Petrov, Phys. Rev. B **96**, 235432 (2017).

[95] M. Bordag, I. Fialkovsky, N. Khusnutdinov, and D. Vasilevich, Phys. Rev. B **104**, 195431 (2021).

[96] G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Phys. Rev. B **89**, 115419 (2014).

[97] R. D. Pisarski, Phys. Rev. D **29**, 2423 (1984).

[98] N. R. Poniatowski, American Journal of Physics **87**, 436 (2019), https://pubs.aip.org/aapt/ajp/article-pdf/87/6/436/13079072/436_1_online.pdf.

[99] P. Rodriguez-Lopez and M. Antezza, Future Publication.

[100] M. Dresselhaus, G. Dresselhaus, S. Cronin, and A. Filho, *Solid State Properties: From Bulk to Nano*, Graduate Texts in Physics (Springer Berlin Heidelberg, 2018).

[101] J. D. Jackson, *Classical electrodynamics*, 3rd ed. (Wiley, New York, NY, 1999).

[102] W. Jones and N. March, *Theoretical Solid State Physics*, Dover books on physics and chemistry No. v. 1 (Wiley-Interscience, 1973).

[103] B. Guizal and M. Antezza, Phys. Rev. B **93**, 115427 (2016).