We thank Han Hoe Yap for comments, and Jia-Huei Jiang for the codes. This work is supported by FRC grant R-144-000-402-114 and MOE grant MOE2018-T2-1-096.

Doppler shift
=============

In the appendices, we clarify some points made in the main texts and give some further details.

We first note that the function $\Pi^r_{\bf k}(\omega)$ is a description of the bosonic charge density plasmon. In particular, we consider a plasmon planewave $e^{i( {\bf k} \cdot {\bf r}-\omega t)}$ with frequency $\omega$ and wavevector ${\bf k}$. Let ${\bf v}_1$ describe the drift velocity of the electrons. If the wave ${\bf k}$ of the plasmon and the electron drift velocity ${\bf v}_1$ are in the same direction, electrons will “see” less vibrations, thus $\omega$ decreases. So the Doppler shift is, $\omega \rightarrow \omega + \delta \omega =  \omega - {\bf k} \cdot {\bf v}_1 = \omega - k_x v_1$.

Doppler shift or not for scalar photon self-energy
==================================================

The scalar photon self energies or polarization functions under the random phase approximation are, in time domain and real space, $$\begin{aligned}
\Pi_{jk}^>(t) &=& (-i\hbar)e^2 G_{jk}^>(t) G_{kj}^<(-t),\\
\Pi_{jk}^<(t) &=& (-i\hbar)e^2 G_{jk}^<(t) G_{kj}^>(-t),\\
\Pi_{jk}^r(t) &=& \theta(t) \bigl( \Pi_{jk}^>(t) - \Pi_{jk}^<(t)  \bigr).\end{aligned}$$ We obtain the frequency and wavevector domain 