&+\mathcal{M}_{\bar{k}}\left({\mathbfcal{Z}}\times_{i\in\underline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}\times_{\widecheck{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{k}\times_{i\in\overline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}\right)\bigg{)}.\end{split}$
(32)

Notice that ${\rm
rank}\left(\mathcal{M}_{\bar{k}}\left({\mathbfcal{T}}\times_{i\in\underline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}\times_{\widecheck{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{k}\times_{i\in\overline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}\right)\right)\leq
r_{k}$, to apply Theorem 5 in [LZ20a], the key is to give an upper bound for

$\left\|\left(\mathcal{M}_{\bar{k}}\left({\mathbfcal{Z}}\times_{i\in\underline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}\times_{\widecheck{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{k}\times_{i\in\overline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}\right)\right)_{\max(r_{k})}\right\|_{q}.$

To simplify the notation, for $S_{j}\in\mathcal{S}_{j}^{(-\bar{k})}$, we let
$S_{j1}=S_{j}\bigcap\underline{\Omega}_{k},S_{j2}=S_{j}\bigcap(\widecheck{\Omega}_{k}\bigcup\overline{\Omega}_{k})$
and
$S_{j1}^{c}=S_{j}^{c}\bigcap\underline{\Omega}_{k},S_{j2}^{c}=S_{j}^{c}\bigcap(\widecheck{\Omega}_{k}\bigcup\overline{\Omega}_{k})$.
Then

$\begin{split}&\left\|\left(\mathcal{M}_{\bar{k}}\left({\mathbfcal{Z}}\times_{i\in\underline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}\times_{\widecheck{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{k}\times_{i\in\overline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}\right)\right)_{\max(r_{k})}\right\|_{q}\\\
=&\Big{\|}\Big{(}\mathcal{M}_{\bar{k}}\Big{(}{\mathbfcal{Z}}\times_{i\in\underline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}\left(P_{{\mathbf{U}}_{i^{\prime}}}+P_{{\mathbf{U}}_{i^{\prime}\perp}}\right)\times_{\widecheck{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{k}\left(P_{{\mathbf{U}}_{k}}+P_{{\mathbf{U}}_{k\perp}}\right)\\\
&\quad\quad\quad\quad\times_{i\in\overline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}\left(P_{{\mathbf{U}}_{i^{\prime}}}+P_{{\mathbf{U}}_{i^{\prime}\perp}}\right)\Big{)}\Big{)}_{\max(r_{k})}\Big{\|}_{q}\\\
\overset{(a)}{\leq}&\left\|\left(\mathcal{M}_{\bar{k}}\left({\mathbfcal{Z}}\times_{i\in\underline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}}}\times_{\widecheck{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{k}P_{{\mathbf{U}}_{k}}\times_{i\in\overline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}}}\right)\right)_{\max(r_{k})}\right\|_{q}\\\
&+\sum_{S_{1}\in\mathcal{S}_{1}^{(-\bar{k})}}\Big{\|}\Big{(}\mathcal{M}_{\bar{k}}\Big{(}{\mathbfcal{Z}}\times_{i\in
S_{11}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}\perp}}\times_{i\in
S_{11}^{c}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}}}\\\
&\quad\quad\quad\quad\times_{i\in
S_{12}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}\perp}}\times_{i\in
S^{c}_{12}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}}}\Big{)}\Big{)}_{\max(r_{k})}\Big{\|}_{q}+\cdots\\\
&+\sum_{S_{j}\in\mathcal{S}_{j}^{(-\bar{k})}}\Big{\|}\Big{(}\mathcal{M}_{\bar{k}}\Big{(}{\mathbfcal{Z}}\times_{i\in
S_{j1}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}\perp}}\times_{i\in
S_{j1}^{c}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}}}\\\
&\quad\quad\quad\quad\times_{i\in
S_{j2}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}\perp}}\times_{i\in
S^{c}_{j2}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}}}\Big{)}\Big{)}_{\max(r_{k})}\Big{\|}_{q}+\cdots\\\
&+\left\|\left(\mathcal{M}_{\bar{k}}\left({\mathbfcal{Z}}\times_{i\in\underline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}\perp}}\times_{\widecheck{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{k}P_{{\mathbf{U}}_{k\perp}}\times_{i\in\overline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}\perp}}\right)\right)_{\max(r_{k})}\right\|_{q}\\\
=&\left\|\left(\mathcal{M}_{\bar{k}}\left({\mathbfcal{Z}}\times_{i\in\underline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}}}\times_{\widecheck{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{k}P_{{\mathbf{U}}_{k}}\times_{i\in\overline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}}}\right)\right)_{\max(r_{k})}\right\|_{q}\\\
&+\sum_{j=1}^{d-1}\sum_{S_{j}\in\mathcal{S}_{j}^{(-\bar{k})}}\Big{\|}\Big{(}\mathcal{M}_{\bar{k}}\Big{(}{\mathbfcal{Z}}\times_{i\in
S_{j1}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}\perp}}\times_{i\in
S_{j1}^{c}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}}}\\\
&\quad\quad\quad\quad\times_{i\in
S_{j2}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}\perp}}\times_{i\in
S^{c}_{j2}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}}}\Big{)}\Big{)}_{\max(r_{k})}\Big{\|}_{q}.\end{split}$
(33)

Here $(a)$ is due to the triangle inequality for truncated Schatten-$q$ norm
given in Lemma 4 of [LZ20a]. The right hand side of (33) can be divided into
the sum of $d$ groups and the value of $j^{th}$ group is denoted as $G_{j}$
where

$G_{0}:=\left\|\left(\mathcal{M}_{\bar{k}}\left({\mathbfcal{Z}}\times_{i\in\underline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}}}\times_{\widecheck{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{k}P_{{\mathbf{U}}_{k}}\times_{i\in\overline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}}}\right)\right)_{\max(r_{k})}\right\|_{q}$

and for $1\leq j\leq d-1$, define $G_{j}$ to be,

$\begin{split}G_{j}=\sum_{S_{j}\in\mathcal{S}_{j}^{(-\bar{k})}}\Big{\|}\Big{(}\mathcal{M}_{\bar{k}}\Big{(}{\mathbfcal{Z}}\times_{i\in
S_{j1}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}\perp}}\times_{i\in
S_{j1}^{c}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}}}\\\
\times_{i\in
S_{j2}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}\perp}}\times_{i\in
S^{c}_{j2}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}}}\Big{)}\Big{)}_{\max(r_{k})}\Big{\|}_{q}.\end{split}$

Next we are going to upper bound $G_{j}\,(0\leq j\leq d-1)$.

  * •

Upper Bound of $G_{0}$.

$\begin{split}G_{0}=&\left\|\left(\mathcal{M}_{\bar{k}}\left({\mathbfcal{Z}}\times_{i\in\underline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}}}\times_{\widecheck{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{k}P_{{\mathbf{U}}_{k}}\times_{i\in\overline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}}}\right)\right)_{\max(r_{k})}\right\|_{q}\\\
=&\Big{\|}\Big{(}\mathcal{M}_{\bar{k}}\Big{(}\left({\mathbfcal{Z}}\times_{i\neq\bar{k}}{\mathbf{U}}_{i^{\prime}}^{\top}\right)\times_{i\in\underline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}{\mathbf{U}}_{i^{\prime}}\times_{\widecheck{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{k}{\mathbf{U}}_{k}\\\
&\quad\quad\quad\quad\quad\times_{i\in\overline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}{\mathbf{U}}_{i^{\prime}}\Big{)}\Big{)}_{\max(r_{k})}\Big{\|}_{q}\\\
\leq&\left\|\left(\mathcal{M}_{\bar{k}}\left({\mathbfcal{Z}}\times_{i\neq\bar{k}}{\mathbf{U}}_{i^{\prime}}^{\top}\right)\right)_{\max(r_{k})}\right\|_{q}=\tau_{1k}.\end{split}$

Here the inequality is due to the fact
$\|\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}{\mathbf{U}}_{i^{\prime}}\|\leq
1$ for any $t$ and the following fact: for any matrix ${\mathbf{A}}$ s.t.
$\|{\mathbf{A}}\|\leq 1$,

$\sigma_{k}\left(\mathcal{M}_{i}({\mathbfcal{Z}}\times_{j}{\mathbf{A}})\right)=\sigma_{k}\left(\mathcal{M}_{i}({\mathbfcal{Z}})\cdot{\mathbf{A}}^{\top}\right)\leq\sigma_{k}\left(\mathcal{M}_{i}({\mathbfcal{Z}})\right)\|{\mathbf{A}}\|\leq\sigma_{k}\left(\mathcal{M}_{i}({\mathbfcal{Z}})\right),$
(34)

where the first inequality is due to Lemma 1 of [LZ20a].

  * •

Upper Bound of $G_{j}\,(1\leq j\leq d-1)$.

$\begin{split}G_{j}&\leq{d-1\choose
j}\times\max_{S_{j}\in\mathcal{S}_{j}^{(-\bar{k})}}\\\
&\Big{\|}\Big{(}\mathcal{M}_{\bar{k}}\Big{(}{\mathbfcal{Z}}\times_{i\in
S_{j1}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}\perp}}\times_{i\in
S_{j1}^{c}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}}}\times_{i\in
S_{j2}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}\perp}}\\\
&\quad\quad\quad\times_{i\in
S^{c}_{j2}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}}}\Big{)}\Big{)}_{\max(r_{k})}\Big{\|}_{q}\\\
&\overset{(a)}{\leq}{d-1\choose
j}\max_{S_{j}\in\mathcal{S}_{j}^{(-\bar{k})}}\left(\prod_{i\in
S_{j1}}\left\|\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}{\mathbf{U}}_{i^{\prime}\perp}\right\|_{q}\prod_{i\in
S_{j2}}\left\|\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}{\mathbf{U}}_{i^{\prime}\perp}\right\|_{q}\right)\times\\\
&\left\|\left(\mathcal{M}_{\bar{k}}\left({\mathbfcal{Z}}\times_{i\in
S_{j1}}\frac{\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}\perp}}}{\left\|\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}\perp}}\right\|_{q}}\times_{i\in
S_{j2}}\frac{\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}\perp}}}{\left\|\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}P_{{\mathbf{U}}_{i^{\prime}\perp}}\right\|_{q}}\times_{i\in
S_{j}^{c}}{\mathbf{U}}_{i^{\prime}}^{\top}\right)\right)_{\max(r_{k})}\right\|_{q}.\end{split}$
(35)

By simplifying (35), we get

$\begin{split}G_{j}&\leq{d-1\choose
j}\max_{S_{j}\in\mathcal{S}_{j}^{(-\bar{k})}}\left(\prod_{i\in
S_{j1}}\left\|\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}{\mathbf{U}}_{i^{\prime}\perp}\right\|_{q}\prod_{i\in
S_{j2}}\left\|\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}{\mathbf{U}}_{i^{\prime}\perp}\right\|_{q}\right)\\\
&\times\Big{\|}\Big{(}\mathcal{M}_{\bar{k}}\Big{(}{\mathbfcal{Z}}\times_{i\in
S_{j1}}\frac{\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}{\mathbf{U}}_{i^{\prime}\perp}}{\left\|\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}{\mathbf{U}}_{i^{\prime}\perp}\right\|_{q}}{\mathbf{U}}_{i^{\prime}\perp}^{\top}\\\
&\quad\quad\quad\quad\quad\times_{i\in
S_{j2}}\frac{\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}{\mathbf{U}}_{i^{\prime}\perp}}{\left\|\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}{\mathbf{U}}_{i^{\prime}\perp}\right\|_{q}}{\mathbf{U}}_{i^{\prime}\perp}^{\top}\times_{i\in
S_{j}^{c}}{\mathbf{U}}_{i^{\prime}}^{\top}\Big{)}\Big{)}_{\max(r_{k})}\Big{\|}_{q}\\\
&\overset{(b)}{\leq}{d-1\choose
j}\max_{S_{j}\in\mathcal{S}_{j}^{(-\bar{k})}}\left(\prod_{i\in
S_{j1}}\left\|\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}{\mathbf{U}}_{i^{\prime}\perp}\right\|_{q}\prod_{i\in
S_{j2}}\left\|\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}{\mathbf{U}}_{i^{\prime}\perp}\right\|_{q}\right)\times\tau_{j+1}\\\
&\leq{d-1\choose
j}\max_{S_{j}\in\mathcal{S}_{j}^{(-\bar{k})}}\left(e_{t}\right)^{|S_{j2}|}\left(e_{t+1}\right)^{|S_{j1}|}\tau_{j+1}\end{split}$
(36)

where (a) is due to the fact that
$\|\tilde{{\mathbf{U}}}_{i}^{(t)\top}{\mathbf{U}}_{i^{\prime}}\|\leq 1$ for
any $t$ and (34); (b) is due to the fact that
$\left\|\frac{\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}{\mathbf{U}}_{i^{\prime}\perp}}{\left\|\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}{\mathbf{U}}_{i^{\prime}\perp}\right\|_{q}}\right\|_{q}\leq
1$ and the definition of $\tau_{j+1}$.

So in summary, plug (36) into (33), now we have the following bound

$\begin{split}&\left\|\left(\mathcal{M}_{\bar{k}}\left({\mathbfcal{Z}}\times_{i\in\underline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t+1)\top}_{i^{\prime}}\times_{\widecheck{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{k}\times_{i\in\overline{\Omega}_{k}}\widetilde{{\mathbf{U}}}^{(t)\top}_{i^{\prime}}\right)\right)_{\max(r_{k})}\right\|_{q}\\\
\leq&\tau_{1k}+\sum_{j=1}^{d-1}{d-1\choose
j}\max_{S_{j}\in\mathcal{S}_{j}^{(-\bar{k})}}\left(e_{t}\right)^{|S_{j2}|}\left(e_{t+1}\right)^{|S_{j1}|}\tau_{j+1}.\end{split}$
(37)

Step 2. In this step we want to use induction to prove the following claim,

$e_{t}\leq
2^{\frac{d+3}{2}}\frac{\tau_{1}}{\lambda}+e_{0}/2^{t};t=0,1,\ldots.$ (38)

Claim (38) clearly holds if $t=0$. Assume (38) holds for $t$ and next we show
it also holds for $t+1$.

Let’s first show the upper bound of $e_{t}$ can be used to upper bound
$e_{t+1,1}$. Notice

$\begin{split}2\left\|\left(\mathcal{M}_{\bar{1}}\left({\mathbfcal{Z}}\times_{i\neq\bar{1}}\widetilde{{\mathbf{U}}}_{i^{\prime}}^{(t)\top}\right)\right)_{\max(r_{1})}\right\|_{q}\overset{(a)}{\geq}&\left\|\widetilde{{\mathbf{U}}}_{1\perp}^{(t+1)\top}{\mathbf{U}}_{1}\right\|\sigma_{r_{1}}\left(\mathcal{M}_{\bar{1}}\left({\mathbfcal{T}}\times_{i\neq\bar{1}}\widetilde{{\mathbf{U}}}_{i^{\prime}}^{(t)\top}\right)\right)\\\
\overset{(b)}{\geq}&\left(e_{t+1,1}\right)\sigma_{r_{1}}\left({\mathbf{T}}_{1}\left(\otimes_{i\neq\bar{1}}{\mathbf{U}}_{i^{\prime}}\right)\cdot\left(\otimes_{i\neq\bar{1}}{\mathbf{U}}_{i^{\prime}}^{\top}\widetilde{{\mathbf{U}}}_{i^{\prime}}^{(t)}\right)\right)\\\
\overset{(c)}{\geq}&\left(e_{t+1,1}\right)\lambda\left(\prod_{i\neq\bar{1}}\sigma_{\min}\left({\mathbf{U}}_{i^{\prime}}^{\top}\widetilde{{\mathbf{U}}}_{i^{\prime}}^{(t)}\right)\right)\\\
\overset{(d)}{\geq}&\left(e_{t+1,1}\right)\lambda(1-e_{t}^{2})^{\frac{d-1}{2}}.\end{split}$
(39)

Here (a) is due to Theorem 5 in [LZ20a] and the fact the left singular space
of
$\mathcal{M}_{\bar{1}}\left({\mathbfcal{T}}\times_{i\neq\bar{1}}\widetilde{{\mathbf{U}}}_{i^{\prime}}^{(t)\top}\right)$
is ${\mathbf{U}}_{1}$ and its rank is less than $r_{1}$, (b) is due to the
fact that the right singular space of ${\mathbf{T}}_{1}$ is
$\otimes_{i\neq\bar{1}}{\mathbf{U}}_{i^{\prime}}$, equation (5), (c) is by
Lemma 1 of [LZ20a] and (d) is due to the Lemma 1 of [CZ18].

Plugging the upper bound of
$\left\|\left(\mathcal{M}_{\bar{1}}\left({\mathbfcal{Z}}\times_{i\neq\bar{1}}\widetilde{{\mathbf{U}}}_{i^{\prime}}^{(t)\top}\right)\right)_{\max(r_{1})}\right\|_{q}$
in (37) into (39), we get

$e_{t+1,1}\leq 2\frac{\tau_{11}+\sum_{j=1}^{d-1}{d-1\choose
j}\left(e_{t}\right)^{j}\tau_{j+1}}{\lambda(1-e_{t}^{2})^{\frac{d-1}{2}}}.$
(40)

Next we show under condition (15), $e_{t+1,1}\leq
2^{\frac{d+3}{2}}\frac{\tau_{1}}{\lambda}+e_{0}/2^{t}$, i.e., the upper bound
of $e_{t}$ can be used to upper bound $e_{t+1,1}$. First, under (15), we have

$\lambda\geq
2^{\frac{d+6}{2}}\tau_{1}\lor\left(\sum_{j=1}^{d-1}2^{\frac{d+4-j}{2}}{d-1\choose
j}\tau_{j+1}\right),$ (41)

due to the fact that $\xi\geq\tau_{j}$ for $j=1,\ldots,d$. Since
$e_{0}\leq\frac{\sqrt{2}}{2}$ and (41), so $e_{t}\leq\frac{\sqrt{2}}{2}$ by
(38). So by (40), we have

$\begin{split}\left(\frac{1}{2}\right)^{\frac{d-1}{2}}e_{t+1,1}&\leq
2\frac{\tau_{11}+\sum_{j=1}^{d-1}{d-1\choose
j}\left(e_{t}\right)^{j}\tau_{j+1}}{\lambda}\\\ &\leq
2\left(\frac{\tau_{11}}{\lambda}+\frac{\sum_{j=1}^{d-1}{d-1\choose
j}(\frac{\sqrt{2}}{2})^{j-1}\tau_{j+1}}{\lambda}e_{t}\right)\\\
&\overset{\eqref{cond: lambda
condition}}{\leq}2\left(\frac{\tau_{11}}{\lambda}+2^{-\frac{d+3}{2}}e_{t}\right)\end{split}$
(42)

Since (38) holds for $t$ and plug in the upper bound of $e_{t}$ in (42),
multiply $2^{\frac{d-1}{2}}$ at both side of (42), we get

$e_{t+1,1}\leq
2^{\frac{d+3}{2}}\frac{\tau_{1}}{\lambda}+\frac{e_{0}}{2^{t+1}}\leq
2^{\frac{d+3}{2}}\frac{\tau_{1}}{\lambda}+\frac{e_{0}}{2^{t}}.$

So the upper bound of $e_{t}$ can be used to bound $e_{t+1,1}$ and by doing
similar analysis for all modes, we conclude that the upper bound for $e_{t}$
also holds for $e_{t+1}$, i.e., we have

$e_{t+1}\leq 2^{\frac{d+3}{2}}\frac{\tau_{1}}{\lambda}+\frac{e_{0}}{2^{t}}.$
(43)

Now we can show (38) also holds for $t+1$ given it holds for $t$. With (43),
when we do the similar analysis of (39) for other modes, we can use the same
upper bound of $e_{t}$ to bound $e_{t+1}$. So repeat (39), for any $1\leq
k\leq m$, we have

$e_{t+1,k}\left(1-e_{t}^{2}\right)^{\frac{d-1}{2}}\leq
2\frac{\tau_{1k}+\sum_{j=1}^{d-1}{d-1\choose
j}\left(e_{t}\right)^{j}\tau_{j+1}}{\lambda}.$ (44)

Since it is true for every $k$, we obtain the following recursive inequality
for $e_{t+1}$,

$e_{t+1}\left(1-e_{t}^{2}\right)^{\frac{d-1}{2}}\leq
2\frac{\tau_{1}+\sum_{j=1}^{d-1}{d-1\choose
j}\left(e_{t}\right)^{j}\tau_{j+1}}{\lambda}.$ (45)

By applying the same argument as in (42), we can show that

$e_{t+1}\leq 2^{\frac{d+3}{2}}\frac{\tau_{1}}{\lambda}+\frac{e_{0}}{2^{t+1}},$

given (38) holds for $e_{t}$.

Thus, (38) holds for all $t$. Given
$t_{\max}\geq\log(e_{0}\lambda/\tau_{1})\vee 1$, we get the upper bound for
$e_{t}$:

$e_{t}\leq\left(2^{\frac{d+3}{2}}+1\right)\frac{\tau_{1}}{\lambda}.$ (46)

Plug (45) and (41) into the upper bound (44), we get

$\begin{split}e_{t+1,k}&\leq
2\left(1-\frac{\left(2^{\frac{d+3}{2}}+1\right)^{2}\tau_{1}^{2}}{\lambda^{2}}\right)^{-\frac{d-1}{2}}\left(\frac{\tau_{1k}}{\lambda}+\sum_{j=1}^{d-1}\frac{{d-1\choose
j}\left(2^{\frac{d+3}{2}}+1\right)^{j}\tau_{1}^{j}\tau_{j+1}}{\lambda^{j+1}}\right),\end{split}$

for $k=1,\ldots,m$.

Finally the perturbation of signal subspaces follows by observing that
$e_{t,k}:=\left\|(\widetilde{{\mathbf{U}}}_{k\perp}^{(t)})^{\top}{\mathbf{U}}_{k}\right\|_{q}=\left\|\sin\Theta\left(\widetilde{{\mathbf{U}}}^{(t)}_{k},{\mathbf{U}}_{k}\right)\right\|_{q}$
due to Lemma 7 of [LZ20a].

Step 3. In this step, we are going to give an upper bound for the tensor
reconstruction error for
$\|\widetilde{{\mathbfcal{T}}}\times_{\Omega_{1}}P_{\widehat{{\mathbf{U}}}_{1}}\times\cdots\times_{\Omega_{m}}P_{\widehat{{\mathbf{U}}}_{m}}-{\mathbfcal{T}}\|_{{\rm
HS}}$.

For the rest of the proof, consider $q=2$ in the definition of
$\tau_{1},\ldots,\tau_{d}$. First, notice the following decomposition

$\begin{split}{\mathbfcal{T}}=&{\mathbfcal{T}}\times_{1}\left(P_{\widehat{{\mathbf{U}}}_{1^{\prime}}}+P_{\widehat{{\mathbf{U}}}_{1^{\prime}\perp}}\right)\times_{2}\left(P_{\widehat{{\mathbf{U}}}_{2^{\prime}}}+P_{\widehat{{\mathbf{U}}}_{2^{\prime}\perp}}\right)\times\cdots\times_{d}\left(P_{\widehat{{\mathbf{U}}}_{d^{\prime}}}+P_{\widehat{{\mathbf{U}}}_{d^{\prime}\perp}}\right)\\\
=&{\mathbfcal{T}}\times_{1}P_{\widehat{{\mathbf{U}}}_{1^{\prime}}}\times\cdots\times_{d}P_{\widehat{{\mathbf{U}}}_{d^{\prime}}}+{\mathbfcal{T}}\times_{1}P_{\widehat{{\mathbf{U}}}_{1^{\prime}\perp}}\times_{2}P_{\widehat{{\mathbf{U}}}_{2^{\prime}}}\times\cdots\times_{d}P_{\widehat{{\mathbf{U}}}_{d^{\prime}}}\\\
&+{\mathbfcal{T}}\times_{1}{\mathbf{I}}_{p_{1^{\prime}}}\times_{2}P_{\widehat{{\mathbf{U}}}_{2^{\prime}\perp}}\times\cdots\times_{d}P_{\widehat{{\mathbf{U}}}_{d^{\prime}}}+{\mathbfcal{T}}\times_{1}{\mathbf{I}}_{p_{1^{\prime}}}\times_{2}{\mathbf{I}}_{p_{2^{\prime}}}\times\cdots\times_{d}P_{\widehat{{\mathbf{U}}}_{d^{\prime}\perp}}\\\
&+\ldots+{\mathbfcal{T}}\times_{i\leq
d-1}{\mathbf{I}}_{p_{i^{\prime}}}\times_{d}P_{\widehat{{\mathbf{U}}}_{d^{\prime}\perp}}\\\
=&{\mathbfcal{T}}\times_{\Omega_{1}}P_{\widehat{{\mathbf{U}}}_{1}}\times\cdots\times_{\Omega_{m}}P_{\widehat{{\mathbf{U}}}_{m}}+\sum_{k=1}^{d}{\mathbfcal{T}}\times_{i<k}{\mathbf{I}}_{p_{i^{\prime}}}\times_{k}P_{\widehat{{\mathbf{U}}}_{k^{\prime}\perp}}\times_{i>k}P_{\widehat{{\mathbf{U}}}_{i^{\prime}}}.\end{split}$

Thus,

$\begin{split}&\left\|\widetilde{{\mathbfcal{T}}}\times_{\Omega_{1}}P_{\widehat{{\mathbf{U}}}_{1}}\times\cdots\times_{\Omega_{m}}P_{\widehat{{\mathbf{U}}}_{m}}-{\mathbfcal{T}}\right\|_{{\rm
HS}}\\\
\leq&\left\|(\widetilde{{\mathbfcal{T}}}-{\mathbfcal{T}})\times_{\Omega_{1}}P_{\widehat{{\mathbf{U}}}_{1}}\times\cdots\times_{\Omega_{m}}P_{\widehat{{\mathbf{U}}}_{m}}\right\|_{{\rm
HS}}+\sum_{k=1}^{d}\left\|{\mathbfcal{T}}\times_{i<k}{\mathbf{I}}_{p_{i^{\prime}}}\times_{k}P_{\widehat{{\mathbf{U}}}_{k^{\prime}\perp}}\times_{i>k}P_{\widehat{{\mathbf{U}}}_{i^{\prime}}}\right\|_{{\rm
HS}}\\\
\leq&\left\|{\mathbfcal{Z}}\times_{\Omega_{1}}P_{\widehat{{\mathbf{U}}}_{1}}\times\cdots\times_{\Omega_{m}}P_{\widehat{{\mathbf{U}}}_{m}}\right\|_{{\rm
HS}}+\sum_{k=1}^{d}\left\|\widehat{{\mathbf{U}}}_{k^{\prime}\perp}^{\top}\mathcal{M}_{k}({\mathbfcal{T}})\right\|_{F}.\end{split}$
(47)

Notice that for $k_{1},k_{2}\in\Omega_{k}$,
$\mathcal{M}_{k_{1}}({\mathbfcal{T}})=\mathcal{M}_{k_{2}}({\mathbfcal{T}})P_{1}$
where $P_{1}$ is a permutation matrix. So
$\sum_{k=1}^{d}\left\|\widehat{{\mathbf{U}}}_{k^{\prime}\perp}^{\top}\mathcal{M}_{k}({\mathbfcal{T}})\right\|_{F}=\sum_{k=1}^{m}|\Omega_{k}|\left\|\widehat{{\mathbf{U}}}_{k\perp}^{\top}{\mathbf{T}}_{k}\right\|_{F}$.

$\begin{split}&\left\|\widehat{{\mathbf{U}}}_{k\perp}^{\top}{\mathbf{T}}_{k}\right\|_{F}\left(1-\frac{\left(2^{\frac{d+3}{2}}+2\right)^{2}\tau_{1}^{2}}{\lambda^{2}}\right)^{-\frac{d-1}{2}}\\\
\overset{(a)}{\leq}&\left\|\widehat{{\mathbf{U}}}_{k\perp}^{\top}{\mathbf{T}}_{k}\right\|_{F}\sigma_{\min}\left(\left(\otimes_{i\in\underline{\Omega}_{k}}{\mathbf{U}}_{i^{\prime}}^{\top}\widetilde{{\mathbf{U}}}_{i}^{(t_{\max})}\otimes_{i\in\widecheck{\Omega}_{k}\bigcup\overline{\Omega}_{k}}{\mathbf{U}}_{i^{\prime}}^{\top}\widetilde{{\mathbf{U}}}_{i^{\prime}}^{(t_{\max}-1)}\right)\right)\\\
\overset{(b)}{=}&\left\|\widehat{{\mathbf{U}}}_{k\perp}^{\top}{\mathbf{T}}_{k}\otimes_{i\neq\bar{k}}{\mathbf{U}}_{i^{\prime}}\right\|_{F}\sigma_{\min}\left(\left(\otimes_{i\in\underline{\Omega}_{k}}{\mathbf{U}}_{i^{\prime}}^{\top}\widetilde{{\mathbf{U}}}_{i}^{(t_{\max})}\otimes_{i\in\widecheck{\Omega}_{k}\bigcup\overline{\Omega}_{k}}{\mathbf{U}}_{i^{\prime}}^{\top}\widetilde{{\mathbf{U}}}_{i^{\prime}}^{(t_{\max}-1)}\right)\right)\\\
\overset{(c)}{\leq}&\left\|\widehat{{\mathbf{U}}}_{k\perp}^{\top}{\mathbf{T}}_{k}\left(\otimes_{i\neq\bar{k}}{\mathbf{U}}_{i^{\prime}}\right)\left(\otimes_{i\neq\bar{k}}{\mathbf{U}}_{i^{\prime}}\right)^{\top}\left(\otimes_{i\in\underline{\Omega}_{k}}\widetilde{{\mathbf{U}}}_{i^{\prime}}^{(t_{\max})}\otimes_{i\in\widecheck{\Omega}_{k}\bigcup\overline{\Omega}_{k}}\widetilde{{\mathbf{U}}}_{i^{\prime}}^{(t_{\max}-1)}\right)\right\|_{F}\\\
\overset{(d)}{\leq}&2\left\|\left(\mathcal{M}_{\bar{k}}\left({\mathbfcal{Z}}\times_{i\in\underline{\Omega}_{k}}\widetilde{{\mathbf{U}}}_{i^{\prime}}^{(t_{\max})\top}\times_{i\in\widecheck{\Omega}_{k}\bigcup\overline{\Omega}_{k}}\widetilde{{\mathbf{U}}}_{i^{\prime}}^{(t_{\max}-1)\top}\right)\right)_{\max(r_{k})}\right\|_{F}\overset{(e)}{\leq}2\xi,\end{split}$
(48)

where $\xi:=\underset{\|{\mathbfcal{Y}}\|_{{\rm HS}}\leq 1,{\rm
rank}({\mathbfcal{Y}})\leq(r_{1^{\prime}},\ldots,r_{d^{\prime}})}{\sup}\langle{\mathbfcal{Z}},{\mathbfcal{Y}}\rangle$.
(a) is due to the fact
$\sigma_{\min}\left({\mathbf{U}}_{i^{\prime}}^{\top}\widetilde{{\mathbf{U}}}_{i}^{(t)}\right)=\sqrt{1-e_{t}^{2}}$
by Lemma 1 of [CZ18] and
$\max(e_{t_{\max}},e_{t_{\max}-1})\leq\frac{\left(2^{\frac{d+3}{2}}+2\right)\tau_{1}}{\lambda}$;
(b) is due to the fact that the right singular space of ${\mathbf{T}}_{k}$ is
$\otimes_{i\neq\bar{k}}{\mathbf{U}}_{i^{\prime}}$; (c) is due to Lemma 1 of
[LZ20a], equation (5) and properties of Kronecker product, (d) is due to
Theorem 1 in [LZ20a] and

${\rm
rank}\left(\widehat{{\mathbf{U}}}_{k\perp}^{\top}{\mathbf{T}}_{k}\left(\otimes_{i\in\underline{\Omega}_{k}}\widetilde{{\mathbf{U}}}_{i^{\prime}}^{(t_{\max})}\otimes_{i\in\widecheck{\Omega}_{k}\bigcup\overline{\Omega}_{k}}\widetilde{{\mathbf{U}}}_{i^{\prime}}^{(t_{\max}-1)}\right)\right)\leq
r_{k};$

and the last inequality is due to the fact that

$\begin{split}&\left\|\left(\mathcal{M}_{\bar{k}}\left({\mathbfcal{Z}}\times_{i\in\underline{\Omega}_{k}}\widetilde{{\mathbf{U}}}_{i^{\prime}}^{(t_{\max})\top}\times_{i\in\widecheck{\Omega}_{k}\bigcup\overline{\Omega}_{k}}\widetilde{{\mathbf{U}}}_{i^{\prime}}^{(t_{\max}-1)\top}\right)\right)_{\max(r_{k})}\right\|_{F}\\\
=&\sup_{\|{\mathbf{X}}\|_{F}\leq 1,{\rm rank}({\mathbf{X}})\leq
r_{k}}\left\langle\mathcal{M}_{\bar{k}}\left({\mathbfcal{Z}}\times_{i\in\underline{\Omega}_{k}}\widetilde{{\mathbf{U}}}_{i^{\prime}}^{(t_{\max})\top}\times_{i\in\widecheck{\Omega}_{k}\bigcup\overline{\Omega}_{k}}\widetilde{{\mathbf{U}}}_{i^{\prime}}^{(t_{\max}-1)\top}\right),{\mathbf{X}}\right\rangle\leq\xi,\end{split}$

for $k=1,\ldots,d$ and the equality is due to Lemma 2 of [LZ20a].

Combined with the fact
$\left\|{\mathbfcal{Z}}\times_{\Omega_{1}}P_{\widehat{{\mathbf{U}}}_{1}}\times\cdots\times_{\Omega_{m}}P_{\widehat{{\mathbf{U}}}_{m}}\right\|_{{\rm
HS}}\leq\xi$ and (48) and plug them into (47), we have

$\begin{split}&\left\|\widetilde{{\mathbfcal{T}}}\times_{\Omega_{1}}P_{\widehat{{\mathbf{U}}}_{1}}\times\cdots\times_{\Omega_{m}}P_{\widehat{{\mathbf{U}}}_{m}}-{\mathbfcal{T}}\right\|_{{\rm
HS}}\\\
\leq&\left\|{\mathbfcal{Z}}\times_{\Omega_{1}}P_{\widehat{{\mathbf{U}}}_{1}}\times\cdots\times_{\Omega_{m}}P_{\widehat{{\mathbf{U}}}_{m}}\right\|_{{\rm
HS}}+\sum_{k=1}^{d}\left\|\widehat{{\mathbf{U}}}_{k^{\prime}\perp}^{\top}\mathcal{M}_{k}({\mathbfcal{T}})\right\|_{F}\\\
=&\left\|{\mathbfcal{Z}}\times_{\Omega_{1}}\widehat{{\mathbf{U}}}_{1}^{\top}\times\cdots\times_{\Omega_{m}}\widehat{{\mathbf{U}}}_{m}^{\top}\right\|_{{\rm
HS}}+\sum_{k=1}^{m}|\Omega_{k}|\left\|\widehat{{\mathbf{U}}}_{k\perp}^{\top}\mathcal{M}_{k}({\mathbfcal{T}})\right\|_{F}\\\
\leq&\left(1+2d\left(1-\frac{\left(2^{\frac{d+3}{2}}+2\right)^{2}\tau_{1}^{2}}{\lambda^{2}}\right)^{-\frac{d-1}{2}}\right)\xi.\end{split}$
(49)

Finally, notice that (10) follows exact the same proof as Step 3 except that
in (48), we need to replace
$\widetilde{{\mathbf{U}}}_{i}^{(t_{\max})},\widetilde{{\mathbf{U}}}_{i}^{(t_{\max}-1)}$
with $\widetilde{{\mathbf{U}}}_{i}^{(t)},\widetilde{{\mathbf{U}}}_{i}^{(t-1)}$
and use $\max(e_{t},e_{t-1})\leq
2^{\frac{d+3}{2}}\frac{\tau_{1}}{\lambda}+\frac{e_{0}}{2^{t-1}}$. Therefore,
we have finished the proof of this theorem. $\blacksquare$

###  B.5 Proof of Theorem 5

In this setting, we consider applying Corollary 1. To apply Corollary 1, we
only need to compute $\tau_{1i}$, $\xi$.

Consider the case $q=\infty$ and let’s first bound $\tau_{1k}$. Notice

$\mathcal{M}_{k}({\mathbfcal{Z}}\times_{i\neq
k}{\mathbf{U}}_{i}^{\top})=\mathcal{M}_{k}({\mathbfcal{Z}})\otimes_{i\neq
k}{\mathbf{U}}_{i}.$

Each row of $\mathcal{M}_{k}({\mathbfcal{Z}})\otimes_{i\neq
k}{\mathbf{U}}_{i}$ is independent multivariate Gaussian with covariance
matrix $\sigma{\mathbf{I}}_{r_{-k}}$ where $r_{-k}=\prod_{i\neq k}r_{i}$. By
random matrix theory [Ver10], we have

$\mathbb{P}\left(\|\mathcal{M}_{k}({\mathbfcal{Z}}\times_{i\neq
k}{\mathbf{U}}_{i}^{\top})\|/\sigma\leq\sqrt{p_{k}}+\sqrt{r_{-k}}+t\right)\geq
1-2\exp(-(p_{k}+r_{-k})/2).$

Since $r_{\max}\leq p_{\min}^{\frac{1}{d-1}}$, we have $\tau_{1k}\leq
C\sigma\sqrt{p_{k}}$ with probability at least $1-2\exp(-(p_{k}+r_{-k})/2)$.

Similarly by the proof of Lemma 5 of [ZX18], we have

$\xi\leq C\sigma\sqrt{\sum_{i=1}^{d}p_{i}r_{i}}$

w.p. at least $1-\exp(-cp_{\min})$.

As we mentioned in Remark 6, $\tau_{j}\leq\xi$. So the results follows by
plugging the bound of $\tau_{j},\xi$ into Corollary 1 and noticing that

$\frac{\sum_{j=1}^{d-1}{d-1\choose
j}\left(\left(2^{\frac{d+3}{2}}+1\right)\tau_{1}/\lambda\right)^{j}\tau_{j+1}}{\lambda}\leq
C\frac{\tau_{1}\max_{j}\tau_{j}}{\lambda^{2}},$

under the assumption of the signal to ratio $\lambda/\sigma$. $\blacksquare$

###  B.6 Proof of Lemma 3

The proof of this Lemma is straight forward.

$\begin{split}{\mathbfcal{T}}&={\mathbfcal{B}}\times_{1}{\mathbf{\Pi}}_{1}\times\cdots\times_{d}{\mathbf{\Pi}}_{d}\\\
&={\mathbfcal{B}}\times_{1}{\mathbf{\Pi}}_{1}({\mathbf{\Pi}}_{1}^{\top}{\mathbf{\Pi}}_{1})^{-\frac{1}{2}}({\mathbf{\Pi}}_{1}^{\top}{\mathbf{\Pi}}_{1})^{\frac{1}{2}}\times\cdots\times_{d}{\mathbf{\Pi}}_{d}({\mathbf{\Pi}}_{d}^{\top}{\mathbf{\Pi}}_{d})^{-\frac{1}{2}}({\mathbf{\Pi}}_{d}^{\top}{\mathbf{\Pi}}_{d})^{\frac{1}{2}}\\\
&=\left({\mathbfcal{B}}\times_{1}({\mathbf{\Pi}}_{1}^{\top}{\mathbf{\Pi}}_{1})^{\frac{1}{2}}\times\cdots\times_{d}({\mathbf{\Pi}}_{d}^{\top}{\mathbf{\Pi}}_{d})^{\frac{1}{2}}\right)\times_{1}{\mathbf{\Pi}}_{1}({\mathbf{\Pi}}_{1}^{\top}{\mathbf{\Pi}}_{1})^{-\frac{1}{2}}\times\cdots\times_{d}{\mathbf{\Pi}}_{d}({\mathbf{\Pi}}_{d}^{\top}{\mathbf{\Pi}}_{d})^{-\frac{1}{2}}\\\
&={\mathbfcal{S}}\times_{1}{\mathbf{\Pi}}_{1}({\mathbf{\Pi}}_{1}^{\top}{\mathbf{\Pi}}_{1})^{-\frac{1}{2}}{\mathbf{V}}_{1}\times\cdots\times_{d}{\mathbf{\Pi}}_{d}({\mathbf{\Pi}}_{d}^{\top}{\mathbf{\Pi}}_{d})^{-\frac{1}{2}}{\mathbf{V}}_{d},\end{split}$

where the last inequality comes from the assumption about the decomposition of
${\mathbfcal{B}}\times_{1}({\mathbf{\Pi}}_{1}^{\top}{\mathbf{\Pi}}_{1})^{\frac{1}{2}}\times\cdots\times_{d}({\mathbf{\Pi}}_{d}^{\top}{\mathbf{\Pi}}_{d})^{\frac{1}{2}}$.
$\blacksquare$

###  B.7 Proof of Theorem 6

First by the same argument in the proof of Theorem 5, with probability at
least $1-\exp(-cp_{\min})$, $\tau_{1k}\leq C\sigma\sqrt{p_{k}}$ and $\xi\leq
C\sigma\sqrt{\sum_{i=1}^{d}p_{i}r_{i}}$.

Under the assumption (A1), we have
$\lambda({\mathbfcal{S}}):=\min_{i}\sigma_{r_{i}}(\mathcal{M}_{i}({\mathbfcal{S}}))\geq
C\lambda\sqrt{\frac{\prod_{i=1}^{d}p_{i}}{\prod_{i=1}^{d}r_{i}}}$ by the
definition of ${\mathbfcal{S}}$.

Notice that under the signal strength in the Theorem, the first order
perturbation error in
$\|\sin\Theta(\widehat{{\mathbf{U}}}_{i},{\mathbf{U}}_{i})\|$ dominates and
the error bounds for
$\|\sin\Theta(\widehat{{\mathbf{U}}}_{i},{\mathbf{U}}_{i})\|$ and
$\|\widehat{{\mathbfcal{T}}}-{\mathbfcal{T}}\|_{{\rm HS}}$ are directly
consequences of Corollary 1 by using the signal strength
$\lambda({\mathbfcal{S}})$.

For the cocluster membership recovery, since

$\|\sin\Theta(\widehat{{\mathbf{U}}}_{i},{\mathbf{U}}_{i})\|_{F}\leq\sqrt{r_{i}}\|\sin\Theta(\widehat{{\mathbf{U}}}_{i},{\mathbf{U}}_{i})\|\leq
C\frac{\sqrt{p_{i}r_{i}}}{\lambda/\sigma}\sqrt{\frac{\prod_{i=1}^{d}r_{i}}{\prod_{i=1}^{d}p_{i}}}.$

Then by the proof of Corollary 3.2 of [LR15], the error bounds for
$err(\widehat{{\mathbf{\Pi}}}_{i},{\mathbf{\Pi}}_{i})$ and
$\widetilde{err}(\widehat{{\mathbf{\Pi}}}_{i},{\mathbf{\Pi}}_{i})$ follow.
$\blacksquare$


# Optical effects induced in unmagnetized chiral cold plasma

Filipe S. Ribeiroa<EMAIL_ADDRESS><EMAIL_ADDRESS>Pedro D. S. Silvaa<EMAIL_ADDRESS><EMAIL_ADDRESS>Manoel M. Ferreira Jr.a,b<EMAIL_ADDRESS><EMAIL_ADDRESS>aPrograma de Pós-graduação em Física, Universidade
Federal do Maranhão, Campus Universitário do Bacanga, São Luís, Maranhão
65080-805, Brazil  bDepartamento de Física, Universidade Federal do Maranhão,
Campus Universitário do Bacanga, São Luís, Maranhão 65080-805, Brazil

###### Abstract

Unmagnetized cold plasma modes are investigated in the context of the chiral
Maxwell-Carroll-Field-Jackiw (MCJF) electrodynamics, where the axion chiral
factor acts retrieving some typical properties of magnetized plasmas. The
Maxwell equations are rewritten for a cold, uniform, and collisionless fluid
plasma model, allowing us to determine the dispersion relation, new refractive
indices and propagating modes. We find four distinct refractive indices
modified by the purely timelike CFJ background that plays the magnetic
conductivity chiral parameter role, associated with right-circularly polarized
[RCP] and left-circularly polarized [LCP] waves. For each refractive index,
the propagation and absorption zones are determined and illustrated for some
specific parameter values. Modified RCP and LCP helicons are found in the low-
frequency regime. The optical behavior is investigated, revealing that the
chiral factor induces birefringence, measured in terms of the rotatory power
(RP). The dichroism coefficient is carried out for the absorbing zones. The
negative refraction zones may enhance the involved rotatory power, yielding RP
sign reversion, a feature of rotating plasmas and MCFJ chiral plasmas.

###### pacs:

11.30.Cp, 41.20.Jb, 41.90.+e, 42.25.Lc

##  I Introduction

The properties of electromagnetic modes in a cold magnetized plasma are
described by the standard Maxwell equations written in continuous media [1, 2,
3, 4, 5, 6, 7, 8, 9], which provides the basis for the radio wave propagation
in the ionosphere, where the earth magnetic field plays a key role [10, 11,
12]. Although most plasma investigations occur in the presence of an averaged
magnetic field, non-magnetized plasmas have also been explored in some
extension, including topics such as vortex beams propagation [13, 14, 15, 16],
dusty plasma systems [17, 18, 19, 20, 21, 22], nonlinear propagating waves
[23, 24, 25, 27, 26] and self-focusing beams [28, 29]. In the astrophysical
context, electrodynamics of unmagnetized plasma was investigated in an
expanding universe scenario [30]. Recently, unmagnetized plasmas have been
considered around rotating black holes to study the properties of shadows [31,
32, 33, 34, 35].

Cold plasma theory is usually addressed in terms of the differential
equations,

$\displaystyle\frac{\partial n}{\partial
t}+\mathbf{\nabla}\cdot\left(n\mathbf{u}\right)=0,$ (1)
$\displaystyle\frac{\partial\mathbf{u}}{\partial
t}+\mathbf{u}\cdot\mathbf{\nabla
u}=\frac{q}{m}\left(\mathbf{E}+\mathbf{u}\times\mathbf{B}_{0}\right),$ (2)

where $n$ is the electron number density, $\mathbf{u}$ is the electron fluid
velocity field, $q$, and $m$ are the electron charge and mass, respectively.
For unmagnetized plasmas, defined in the absence of an external magnetic
field, $\mathbf{B}_{0}=0$, the term $(\mathbf{u}\times\mathbf{B}_{0})$ on the
right-hand side of Eq. (2) should be omitted. When one considers that the ions
are infinitely massive and thermal/collisional effects can be neglected, the
cold plasmas [8] are well described by a linearized system of equations
written in terms of the fluctuations of relevant physical quantities around
average values (constant in time and space),

$\displaystyle n$ $\displaystyle=n_{0}+\delta n,$ (3a)
$\displaystyle\mathbf{u}$ $\displaystyle=\delta\mathbf{u},$ (3b)
$\displaystyle\mathbf{E}$ $\displaystyle=\delta\mathbf{E}$ (3c)
$\displaystyle\mathbf{B}$ $\displaystyle=\delta\mathbf{B},$ (3d)

with $\delta n$, $\delta\mathbf{u}$, $\delta\mathbf{E}$, and
$\delta\mathbf{B}$ being first-order plane wave magnitude perturbations, and
the quantity $n_{0}$ is the average density value. Equation (3d) is the
unmagnetized simple version of the linearized full expression,
$\mathbf{B}=\mathbf{B}_{0}+\delta\mathbf{B}$, that holds for magnetized
plasmas. Performing the usual procedure, we write the corresponding diagonal
dielectric tensor,

$\varepsilon_{ij}(\omega)=\varepsilon_{0}\left(1-\omega_{p}^{2}/\omega^{2}\right)\delta_{ij},$
(4)

where $\omega_{p}^{2}=n_{0}q^{2}/{m\epsilon_{0}}$ is the plasma frequency.

The usual investigation for wave propagation in plasmas yields a well-known
longitudinal plasma oscillation, at the frequency $\omega=\omega_{p}$, and a
linearly polarized mode for the transversal propagating wave, associated with
the following refractive index:

$n=\sqrt{1-{\omega_{p}^{2}}/{\omega^{2}}}.$ (5)

The expression (5) also entails the existence of a cutoff at the plasma
frequency $\omega_{p}$, below which the transversal mode can not propagate.

The optical activity, manifested as the rotation of the polarization vector of
the propagating wave (birefringence), is one optical signature of magnetized
plasmas where the external field implies anisotropy, inducing the so-called
Faraday effect [36]. In the absence of an external magnetic field,
conventional plasmas do not exhibit optical activity, which is an important
difference between these two cases. Optical activity has been an important
topic in condensed matter chiral systems [37], being a feature of bi-isotropic
[38] and bi-anisotropic electrodynamics [39, 40, 41, 42, 43, 44], in which
birefringence and optical rotation [45] occur.

As well-known, the MCFJ theory is endowed with CPT and Lorentz violation (LV),
representing the CPT-odd piece of the U(1) gauge sector of the Standart Model
Extension (SME) [46, 47]. The SME has been widely examined in many scenarios,
such as in radiative evaluations [48, 49], topological defects solutions [50],
supersymmetry [51], and classical and quantum aspects [52].

The MCFJ electrodynamics has also a relevant connection with the axion
Lagrangian [53, 54],

$\displaystyle\mathrm{{\mathcal{L}}}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\theta(\mathbf{E}\cdot\mathbf{B)},$
(6)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field
strength tensor and $\theta$ is the scalar axion field. In the case the axion
derivative is a constant vector, $\partial_{\mu}\theta=(k_{AF})_{\mu}$,
Lagrangian (6) yields the MCFJ term,

$\displaystyle\mathrm{{\mathcal{L}}}=-\frac{1}{4}G^{\mu\nu}F_{\mu\nu}+\frac{1}{4}\epsilon^{\mu\nu\alpha\beta}\left(k_{AF}\right)_{\mu}A_{\nu}F_{\alpha\beta},$
(7)

where $\left(k_{AF}\right)_{\mu}$ is the 4-vector CFJ background and
$G^{\mu\nu}=\frac{1}{2}\chi^{\mu\nu\alpha\beta}F_{\alpha\beta}$ is the
constitutive field strength tensor that defines the continuous substrate where
this theory is set111The 4-rank tensor, $\chi^{\mu\nu\alpha\beta}$ is the
medium constitutive tensor [55], which provides the electric and magnetic
responses of the medium. The electric permittivity and magnetic permeability
tensors are $\epsilon_{ij}\equiv\chi^{0ij0}$ and
$\mu^{-1}_{lk}\equiv\frac{1}{4}\epsilon_{ijl}\chi^{ijmn}\epsilon_{mnk}$,
respectively. The usual isotropic constitutive relations are provided from
isotropic polarization and magnetization. Then, it holds
$\epsilon_{ij}=\epsilon\delta_{ij}$ and $\mu^{-1}_{ij}=\mu^{-1}\delta_{ij}$,
which yield the relations $\mathbf{D}=\epsilon\mathbf{E}$,
$\mathbf{H}=\mu^{-1}\mathbf{B}$..

Lagrangian (7) describes the MFCJ electrodynamics in matter and yields the
modified inhomogeneous Maxwell equations,

$\displaystyle\nabla\cdot\mathbf{D}$
$\displaystyle=-\mathbf{k}_{AF}\cdot\mathbf{B},$ (8)
$\displaystyle\nabla\times\mathbf{H}-\frac{\partial\mathbf{D}}{\partial t}$
$\displaystyle=-k_{AF}^{0}\mathbf{B}+\mathbf{k}_{AF}\times\mathbf{E},$ (9)

where $G^{i0}=D^{i}$, $G^{ij}=-\epsilon_{ijk}H^{k}$, and
$\quad(k_{AF})^{\mu}=(k_{AF}^{0},\mathbf{k}_{AF})$. The homogeneous Maxwell
equations are obtained from the Bianchi identity
$\partial_{\mu}\tilde{F}^{\mu\nu}=0$.

The modified Ampère’s law (9) contains a chiral magnetic current density

$\mathbf{J}_{B}=k_{AF}^{0}\mathbf{B},$ (10)

where $k_{AF}^{0}$ plays the role of magnetic conductivity. The other
additional term that appears in Ampère’s law is the anomalous Hall current
density,

$\mathbf{J}_{AH}=\mathbf{k}_{AF}\times\mathbf{E},$ (11)

where the chiral vector $\mathbf{k}_{AF}$ represents the anomalous Hall
conductivity. These currents have been used to investigate electromagnetic
properties of matter endowed with the CME and AHE [56], in connection with the
description of Weyl semimetals.

The chiral magnetic effect (CME) [57, 58, 59, 60] is a macroscopic quantum
effect and consists of a linear magnetic current law, ${\bf
J}={\sigma}_{B}{\bf B}$, arising from an asymmetry (imbalance) between the
number density of left- and right-handed chiral fermions, being largely
addressed in many distinct contexts [63, 62, 61, 64, 65, 68, 66, 67, 68]. Non-
linear CME has also been considered in Weyl semimetals (WSMs) [69], where
electric and magnetic fields applied yield a current effectively proportional
to $B^{2}$, that is, ${\bf J}={\sigma}({\bf E}\cdot{\bf B}){\bf B}$ [72, 71,
70]. The anomalous Hall effect (AHE) is the macroscopic generation of an
electric current in the presence of an electric field, due to the separation
between the energy-crossing points in momentum space for right-handed and
left-handed fermions [74, 75, 76, 73]. Such effect has been investigated in
different contexts, such as in non-collinear antiferromagnets [77, 78], chiral
spin-liquids [79], and WSMs [80]. It has also been considered in the
propagation of surface plasmon-polaritons in WSMs [81]. Optical effects in
systems endowed with AHE were also addressed, such as WSM with broken time-
reversal and inversion symmetries [80, 82], as well as in the context of the
MCFJ electrodynamics in continuous media [83].

Cold magnetized plasma has been examined in the context of chiral bi-isotropic
extended constitutive relations, with the attainment of the negative
refraction associated with the chirality parameter [84, 85]. Chiral cold
plasmas were also investigated in the scenario of the axion-like Maxwell-
Carroll-Field-Jackiw (MCFJ) electrodynamics [86, 87], considering the
influence of the chiral magnetic current and anomalous Hall current. The
properties of the propagating modes have been discussed, such as
birefringence, absorption, and optical rotation, highlighting the role played
by the chiral scalar factor, $k_{AF}^{0}$, and the chiral vector
$\mathbf{k}_{AF}$, both engendering RP reversion.

In this work, we study the propagation of electromagnetic waves in an
unmagnetized chiral (parity-odd) plasma ruled by the MCFJ electrodynamics
endowed with the purely timelike chiral factor,
$\left(k_{AF}\right)^{\mu}=\left(k_{AF}^{0},\textbf{0}\right)$, which also
plays the role of chiral magnetic conductivity. This restriction is also very
usual in cold dark matter research, where the space variation of the axion
field can be neglected, with $\mathbf{\nabla}\theta=\mathbf{k}_{AF}=0$. The
main purpose of this work is to investigate propagation and absorption zones,
as well as the conditions of such behaviors. Thus, the dispersion relations
and refractive indices are determined. Optical effects, such as birefringence
and dichroism, are discussed and used to compare different cold chiral plasma
scenarios with the magnetized conventional behavior. The chiral factor appears
bringing about new properties to the unmagnetized panorama, as optical
activity, negative refraction and RP sign reversion.

This paper is outlined as follows. In Sec. II, we obtain the dispersion
relation, refractive indices, and propagating modes for the chiral cold
unmagnetized plasma under consideration. The profile of the refractive indices
is examined for two possible configurations. The dispersion relations,
$\omega\times k$, are also displayed and discussed. In Sec. III, optical
effects (birefringence and dichroism) are addressed in the cases examined. In
Sec. IV, we summarize the results and perspectives.

##  II Wave propagation in chiral plasma

In this section, we derive the collective electromagnetic modes for a cold
unmagnetized chiral plasma, whose modified Maxwell equations yield

$\left[n^{2}\delta_{ij}-n^{i}n^{j}-\frac{{\varepsilon}_{ij}}{\varepsilon_{0}}-i\frac{V_{0}}{\omega}\epsilon_{ikj}n^{k}\right]{E}^{j}=0,$
(12)

with the eletrical permittivity $\varepsilon_{ij}$ given in (4) and
$V_{0}=k_{AF}^{0}/\varepsilon_{0}$, the redefined timelike component of the
chiral background, while $\mathbf{n}=c\mathbf{k}/\omega$ is the refractive
index. In matrix form, Eq. (12) reads

$\begin{bmatrix}n^{2}-n_{x}^{2}-P&-n_{x}n_{y}+i\left(V_{0}/\omega\right)n_{z}&-n_{x}n_{z}-i\left(V_{0}/\omega\right)n_{y}\\\
-n_{x}n_{y}-i\left(V_{0}/\omega\right)n_{z}&n^{2}-n_{y}^{2}-P&-n_{y}n_{z}+i\left(V_{0}/\omega\right)n_{x}\\\
-n_{x}n_{z}+i\left(V_{0}/\omega\right)n_{y}&-n_{y}n_{z}-i\left(V_{0}/\omega\right)n_{x}&n^{2}-n_{z}^{2}-P\end{bmatrix}\begin{bmatrix}\delta
E_{x}\\\ \delta E_{y}\\\ \delta E_{z}\end{bmatrix}=0,$ (13)

where

$P=1-\frac{\omega_{p}^{2}}{\omega^{2}}.$ (14)

For simplicity, lets us consider the refractive index parallel to the magnetic
field, $\mathbf{n}=n\hat{z}$, such that the latter system becomes

$\begin{bmatrix}n^{2}-P&i\left(V_{0}/\omega\right)n&0\\\
-i\left(V_{0}/\omega\right)n&n^{2}-P&0\\\
0&0&-P\end{bmatrix}\begin{bmatrix}\delta E_{x}\\\ \delta E_{y}\\\ \delta
E_{z}\end{bmatrix}=0,$ (15)

whose solution stems from the null determinant condition,
$\mathrm{det}[M_{ij}]=0$,

$P\left[\omega^{2}\left(n^{2}-P\right)^{2}-n^{2}V_{0}^{2}\right]=0,$ (16)

the dispersion equation of the system. For longitudinal wave configuration,
$\delta\mathbf{E}=(0,0,\delta E_{z})$, a non propagating mode appears at the
plasma frequency, $\omega=\omega_{p}$, associated with $P=0$. Such a
longitudinal oscillation is the same one existing for a usual unmagnetized
plasma, being called plasmon. Plasmon is also known in solid-state context,
representing the collective mode of electrons vibrating (longitudinally) under
the action of the electromagnetic field.

For transverse waves, $\mathbf{n}\perp\delta\mathbf{E}$ or
$\delta\mathbf{E}=(\delta E_{x},\delta E_{y},0)$, the dispersion relation (16)
yields

$\left(n^{2}-P\right)^{2}-n^{2}\left(V_{0}/\omega\right)^{2}=0,$ (17)

also written as,

$n^{2}\pm\frac{V_{0}}{\omega}n-P=0,$ (18)

which provides, using the relation (14), the following refractive indices:

$\displaystyle n_{R,M}$
$\displaystyle=-\frac{V_{0}}{2\omega}\pm\sqrt{1+\left(\frac{V_{0}}{2\omega}\right)^{2}-\frac{\omega_{p}^{2}}{\omega^{2}}},$
(19) $\displaystyle n_{L,E}$
$\displaystyle=\frac{V_{0}}{2\omega}\pm\sqrt{1+\left(\frac{V_{0}}{2\omega}\right)^{2}-\frac{\omega_{p}^{2}}{\omega^{2}}}.$
(20)

The refractive indices above may be real, imaginary, or complex (presenting
both pieces) at some frequency ranges, where real and imaginary parts are
associated with the propagation and absorption of the electromagnetic mode,
respectively.

The propagating modes associated with the refractive indices in (19) and (20)
are the eigenvectors (with a null eigenvalue) obtained from the system (15),
in terms of circularly polarized waves. One obtains a LCP mode, associated
with the indices $n_{L}$ and $n_{E}$,

$n_{L},n_{E}\ \mapsto\
\mathbf{{E}}_{LCP}=\frac{i}{\sqrt{2}}\begin{bmatrix}1\\\ i\\\ 0\end{bmatrix},$
(21)

and a RCP mode, connected with the indices $n_{R}$ and $n_{M}$,

$n_{R},n_{M}\ \mapsto\
\mathbf{{E}}_{RCP}=\frac{i}{\sqrt{2}}\begin{bmatrix}1\\\ -i\\\
0\end{bmatrix}.$ (22)

It is easy to notice that the indices $n_{R}$, $n_{E}$, given by Eqs. (19) and
(20) share the same standard cutoff frequency, $\omega_{p}$, while the
refractive indices $n_{L}$ and $n_{M}$ have no real roots. It is important to
notice that the radicand,

$R\left(\omega\right)=1+\left(\frac{V_{0}}{2\omega}\right)^{2}-\frac{\omega_{p}^{2}}{\omega^{2}},$
(23)

in Eqs. (19) and (20), has a single positive root, given by

$\omega_{\mathit{rad}}=\sqrt{\omega_{p}^{2}-V_{0}^{2}/4}.$ (24)

Such a root, when real or imaginary, is useful to define two main scenarios,
introduced below.

  * •

Scenario 1: for

$V_{0}^{2}/4<\omega_{p}^{2},$ (25)

the root $\omega_{\mathit{rad}}$ is real, and the factor
$R\left(\omega\right)$ can be negative for $\omega<\omega_{\mathit{rad}}$ or
positive for $\omega>\omega_{\mathit{rad}}$. In this case, the indices (19)
and (20) are complex for $\omega<\omega_{\mathit{rad}}$ and real for
$\omega>\omega_{\mathit{rad}}$.

  * •

Scenario 2: for

$V_{0}^{2}/4>\omega_{p}^{2}.$ (26)

the root $\omega_{\mathit{rad}}$ is imaginary and the factor
$R\left(\omega\right)$ is always positive, implying real indices for any
frequency.

In the following, the behavior of the refractive indices (19) and (20) is
examined and illustrated, considering the conditions (25) and (26).

In the absence of the chiral factor, $V_{0}=0$, the indices (19) and (20)
reduce to the unmagnetized conventional expressions,

$\tilde{n}_{\pm}=\pm\sqrt{1-\frac{\omega_{p}^{2}}{\omega^{2}}},\\\ $ (27)

which will be used as comparison standard for indicating the chiral factor
effects.

###  II.1 About the index $n_{R}$

We now discuss some aspects of the index $n_{R}$, which has a root at
$\omega=\omega_{p}$. The behavior of $n_{R}$ in terms of the dimensionless
parameter $\omega/\omega_{c}$ is illustrated in Figs. 1 and 2, which show the
real and imaginary pieces of the refractive index $n_{R}$ under the condition
(25) and (26), respectively.

Figure 1: Index of refraction $n_{R}$ under the condition (25). The dashed red
(black) line corresponds to the imaginary piece of $n_{R}$ ($\tilde{n}_{+}$),
while the solid red (black) line represents the real piece of $n_{R}$
($\tilde{n}_{+}$), where the index $\tilde{n}_{+}$ is given in Eq. (27). The
scalar chiral parameter induces a new propagation window of negative
refraction near the plasma frequency. Here, $V_{0}=(3/2)\omega_{p}$, and
$\omega_{p}=1$ $\mathrm{rad}$ $s^{-1}$. Figure 2: Index of refraction $n_{R}$
under the condition (26). The dashed red (black) line corresponds to the
imaginary piece of $n_{R}$ ($\tilde{n}_{+}$), while the solid red (black) line
represents the real piece of $n_{R}$ ($\tilde{n}_{+}$). The scalar chiral
parameter induces a propagation window of negative refraction for
$0<\omega<\omega_{p}$. Here, $V_{0}=(5/2)\omega_{p}$, and $\omega_{p}=1$
$\mathrm{rad}$ $s^{-1}$.

For the condition (25), we point out:

  1. (i)

For $\omega\rightarrow 0$, the real piece of $n_{R}$ tends to infinity
(negatively), $\mathrm{Re}[n_{R}]\rightarrow-\infty$, while its imaginary part
tends to infinity (positively), $\mathrm{Im}[n_{R}]\rightarrow\infty$. This
behavior differs from the usual case, where $\mathrm{Re}[\tilde{n}_{+}]=0$ and
$\mathrm{Im}[\tilde{n}_{+}]\rightarrow\infty$ as $\omega\rightarrow 0$. See
black lines in Fig. 1.

  2. (ii)

For $0<\omega<\omega_{\mathit{rad}}$, it occurs a negative refractive index
zone with absorption, where $\mathrm{Re}[n_{R}]<0$ and $\mathrm{Im}[n_{R}]\neq
0$, as shown in Fig. 1. In the usual case, this negative refraction region
does not exist.

  3. (iii)

For $\omega_{\mathit{rad}}<\omega<\omega_{p}$, an attenuation-free propagation
zone with negative refraction is opened, that is, $\mathrm{Re}[n_{R}]<0$ and
$\mathrm{Im}[n_{R}]=0$.

  4. (iv)

For $\omega>\omega_{p}$, $n_{R}$ is always real and positive,
$\mathrm{Re}[n_{R}]>0$ and $\mathrm{Im}[n_{R}]=0$, corresponding to a
propagation zone, with $n_{R}\rightarrow 1$ in the high-frequency limit.

On the other hand, under the condition (26), only propagation zones are
reported. The first one occurs for the range $0<\omega<\omega_{p}$ and is
marked with negative refraction ($\mathrm{Re}[n_{R}]<0$ and
$\mathrm{Im}[n_{R}]=0$). The second one is defined for $\omega>\omega_{p}$,
where $\mathrm{Re}[n_{R}]$ increases monotonically in the high-frequency
limit, as illustrated in Fig. 2.

###  II.2 About the index $n_{M}$

The index $n_{M}$, given in Eq. (19) and plotted in Fig. 3, is always
negative, having no real root, as it happens in the magnetized chiral plasma
as well (see Ref. [86]). Under the condition (25), it holds

Figure 3: Index of refraction $n_{M}$ under the condition (25). The dashed red
(black) line corresponds to the imaginary piece of $n_{M}$ ($\tilde{n}_{-}$),
while the solid red (black) line represents the real piece of $n_{M}$
($\tilde{n}_{-}$). Here, $V_{0}=(3/2)\omega_{p}$, and $\omega_{p}=1$
$\mathrm{rad}$ $s^{-1}$. Figure 4: Index of refraction $n_{M}$ under the
condition (26). The dashed red (black) line corresponds to the imaginary piece
of $n_{M}$ ($\tilde{n}_{-}$), while the solid red (black) line represents the
real piece of $n_{M}$ ($\tilde{n}_{-}$). Here, $V_{0}=(5/2)\omega_{p}$, and
$\omega_{p}=1$ $\mathrm{rad}$ $s^{-1}$.

  1. (i)

For $\omega\rightarrow 0$, the real and imaginary pieces of $n_{M}$ diverges
(negatively), $\mathrm{Re}[n_{M}],\mathrm{Im}[n_{M}]\rightarrow-\infty$. For
$0<\omega<\omega_{\mathit{rad}}$, the index $n_{M}$ presents a negative
refraction zone with absorption, that is, $\mathrm{Re}[n_{M}]<0$ and
$\mathrm{Im}[n_{M}]\neq 0$, as shown in Fig. 3. This behavior contrasts with
the one of the unmagnetized conventional index, $\tilde{n}_{-}$, whose real
piece is null (it does not propagate). See the dashed black line in Fig. 3.

  2. (ii)

For $\omega>\omega_{\mathit{rad}}$, it occurs a propagation zone with negative
refraction, $\mathrm{Re}[n_{M}]<0$ and $\mathrm{Im}[n_{M}]=0$, expanding the
full propagation zone that, in the usual case, is defined for
$\omega>\omega_{p}$.

  3. (iii)

For $\omega\rightarrow\infty$, $n_{M}\rightarrow-1$, the same behavior of the
usual case in the high-frequency limit.

Considering the condition (26), the index $n_{M}$ is always real and negative
for $\omega>0$, with a monotonically decreasing magnitude, as shown in Fig. 4,
which represents a propagation zone with negative refraction for all
frequencies.

###  II.3 About the index $n_{L}$

The index $n_{L}$ is always positive and has no real root. It is the mirror
image of the index $n_{M}$ in relation to the abscissas axis. Under the
condition (25), its general behavior is represented in Fig. 5 and detailed
below.

  1. (i)

For $\omega\rightarrow 0$,
$\mathrm{Re}[n_{L}],\mathrm{Im}[n_{L}]\rightarrow+\infty$. For
$0<\omega<\omega_{\mathit{rad}}$, the index is complex, $\mathrm{Re}[n_{L}]>0$
and $\mathrm{Im}[n_{L}]\neq 0$, as highlighted in Fig. 5. This partial
propagating behavior differs from the standard unmagnetized case, where there
is only absorption in this range.

  2. (ii)

For $\omega>\omega_{\mathit{rad}}$, there appears a full propagation zone,
with $\mathrm{Re}[n_{L}]>0$ and $\mathrm{Im}[n_{L}]=0$, enlarged in comparison
with the usual case, which begins for $\omega>\omega_{p}$.

  3. (iii)

For $\omega\rightarrow\infty$, $n_{L}\rightarrow 1$, a similar behavior to the
usual case in this limit.

Figure 5: Index of refraction $n_{L}$ under the condition (25). The dashed red
(black) line corresponds to the imaginary piece of $n_{L}$ ($\tilde{n}_{+}$),
while the solid red (black) line represents the real piece of $n_{L}$
($\tilde{n}_{+}$). Here, $V_{0}=(3/2)\omega_{p}$, and $\omega_{p}=1$
$\mathrm{rad}$ $s^{-1}$. Figure 6: Index of refraction $n_{L}$ under the
condition (26). The dashed red (black) line corresponds to the imaginary piece
of $n_{L}$ ($\tilde{n}_{+}$), while the solid red (black) line represents the
real piece of $n_{L}$ ($\tilde{n}_{+}$). Here, $V_{0}=(5/2)\omega_{p}$, and
$\omega_{p}=1$ $\mathrm{rad}$ $s^{-1}$.

Under the condition (26) the refractive index $n_{L}$ is depicted in Fig. 6,
being always positive for $\omega>0$, with $\mathrm{Re}[n_{L}]>0$ and
$\mathrm{Im}[n_{L}]=0$. It is the mirror image of the index $n_{M}$ in
relation to the frequency axis, see Fig. 4, corresponding to a propagating
mode for all frequencies.

###  II.4 About the index $n_{E}$

The index $n_{E}$, given in Eq. (20), has root at $\omega=\omega_{p}$. Under
the condition (26), it presents the following features:

  1. (i)

For $\omega\rightarrow 0$, one has $\mathrm{Re}[n_{E}]\rightarrow\infty$ and
$\mathrm{Im}[n_{E}]\rightarrow-\infty$, which differs from the usual case,
$\mathrm{Re}[\tilde{n}_{-}]=0$ as $\omega\rightarrow 0$. For
$0<\omega<\omega_{\mathit{rad}}$, the index $n_{E}$ is complex, as highlighted
in Fig. 7.

  2. (ii)

For $\omega_{\mathit{rad}}<\omega<\omega_{p}$, the index is real, positive,
and decreasing with $\omega$, representing a propagation zone that does not
exist in standard case. See Fig. 7

  3. (iii)

For $\omega>\omega_{p}$, the index $n_{E}$ becomes negative, defining a
propagation zone with negative refraction, where its magnitude progressively
increases to the asymptotic value, $n_{E}\rightarrow-1$, in a very similar way
of the usual case.

On the other hand, for the condition (26), there is no absorption, since
$n_{E}$ is real for any frequency, see Fig. 8. In this case,
$\mathrm{Re}[n_{E}]>0$ for $\omega<\omega_{p}$, corresponding to a propagation
zone that appears in the place of the full usual absorbing zone. Furthermore,
for $\omega>\omega_{p}$, $\mathrm{Re}[n_{E}]<0$, and there is a propagation
zone with negative refraction, presenting a similar behavior to the usual
case.

Figure 7: Index of refraction $n_{E}$ under the condition (25). The dashed red
(black) line corresponds to the imaginary piece of $n_{E}$ ($\tilde{n}_{-}$),
while the solid red (black) line represents the real piece of $n_{E}$
($\tilde{n}_{-}$). Here, $V_{0}=(3/2)\omega_{p}$, and $\omega_{p}=1$
$\mathrm{rad}$ $s^{-1}$. Figure 8: Index of refraction $n_{E}$ under the
condition (26). The dashed red (black) line corresponds to the imaginary piece
of $n_{E}$ ($\tilde{n}_{-}$), while the solid red (black) line represents the
real piece of $n_{E}$ ($\tilde{n}_{-}$). Here, $V_{0}=(5/2)\omega_{p}$, and
$\omega_{p}=1$ $\mathrm{rad}$ $s^{-1}$.

###  II.5 Low-frequency modes

Low-frequency modes are usually defined in magnetized plasmas, considering the
regime

$\displaystyle\omega\ll\omega_{p},\quad\omega_{c}\ll\omega_{p},\quad\omega\ll\omega_{c},$
(28)

which provides a well-known RCP helicon mode, described by the following
refractive index:

$\displaystyle n_{-}$
$\displaystyle=\omega_{p}\sqrt{\frac{1}{\omega\omega_{c}}},$ (29)

where $\omega_{c}$ is the cyclotron frequency for a charged particle in a
magnetic field. Helicons are RCP modes that propagate at very low frequencies
and along the magnetic field axis. See Ref. [5] (Chapter 9), and Ref. [6]
(Chapter 8) for basic details. In a non-magnetized plasma, the low-frequency
limit is defined by only one condition,

$\displaystyle\omega\ll\omega_{p},$ (30)

which implemented on the refractive indices (19) and (20), yields

$\displaystyle\bar{n}_{R,M}$
$\displaystyle=-\frac{V_{0}}{2\omega}\pm\frac{1}{2\omega}\sqrt{V_{0}^{2}-4\omega_{p}^{2}},$
(31) $\displaystyle\bar{n}_{L,E}$
$\displaystyle=\frac{V_{0}}{2\omega}\pm\frac{1}{2\omega}\sqrt{V_{0}^{2}-4\omega_{p}^{2}},$
(32)

where we have used the “bar” notation to indicate the helicons quantities. The
indices (31) and (32) indicate the existence of both RCP and LCP helicons,
which propagate in the low-energy regime, due to the presence of the chiral
timelike component, $V_{0}$. These helicon modes are dispersive, with
magnitude going with the inverse of the frequency,

  * •

Under the condition (25), $V_{0}^{2}/4<\omega_{p}^{2}$, the indices (31) and
(32) become complex implying lossy behavior.

  * •

Under the condition (26), $V_{0}^{2}/4>\omega_{p}^{2}$, the indices
$\bar{n}_{R,M}$ and $\bar{n}_{L,E}$ are real, being associated with
propagating dispersive helicon modes. Moreover, there occurs negative
refraction, associated with $\bar{n}_{R}$, $\bar{n}_{M}$, and $\bar{n}_{E}$.

In the context of the magnetized plasmas in chiral MCFJ electrodynamics,
addressed in Ref. [86], the helicons indices $\bar{n}_{R,E}$ display a
constant and nondispersive behavior, differently of the indices (31) and (32).
Another important point is that the indices (31) and (32) represent low-
frequency propagating modes (RCP and LCP) with different refractive indices.
This scenario differs from the magnetized chiral plasmas theory, where
different low-frequency modes associated with the same index $\bar{n}_{R,E}$
occur, as seen in Ref. [86] (Section IV.E).

###  II.6 Dispersion relations behavior

In this section, we present a graph representation of the dispersion relations
associated with the obtained circular modes in dimensionless plots of
frequency $\times$ wave vector, that is,
$\left(\omega/\omega_{p}\right)\times\left(k/\omega_{p}\right)$. Starting from
the indices $n_{R}$ and $n_{M}$, under the condition (25), we illustrate the
dispersion relations in Fig. 9. The propagation occurs for
$\omega>\omega_{\mathit{rad}}$, while absorption takes place for
$\omega<\omega_{\mathit{rad}}$. Since $\omega_{\mathit{rad}}<\omega_{p}$, we
observe a reduced absorption zone for $n_{R}$ and $n_{M}$ (the pale pinked
zone) in relation to the one of the usual indices ($\tilde{n}_{\pm}$). For
$\omega_{\mathit{rad}}<\omega<\omega_{p}$, the unusual negative refraction
($k<0$) region of $n_{R}$ becomes evident. Under the condition (26), the
indices $n_{R}$ and $n_{M}$ are entirely real and the associated modes
propagate for any frequencies, as shown in Fig. 10. As clearly shown, the
index $n_{R}$ (blue line) becomes negative for $0<\omega<\omega_{p}$.

Figure 9: Dispersion relations related to refractive indices $n_{R}$ (solid
blue line) and $n_{M}$ (solid black line), under the condition (25). The
dashed black line represents the indices of the standard case,
$\tilde{n}_{\pm}$. The highlighted area in $n_{R,M}$ red (gray plus red)
indicates the absorption zone for $n_{R,M}$ ($\tilde{n}_{\pm}$). Here, we have
used $V_{0}=(3/2)\omega_{p}$, with $\omega_{p}=1$ $\mathrm{rad}$ $s^{-1}$.
Figure 10: Dispersion relations related to refractive indices $n_{R}$ (solid
blue line) and $n_{M}$ (solid black line), under the condition (26). The
dashed black line represents the indices of the standard case,
$\tilde{n}_{\pm}$. The highlighted area in gray indicates the absorption zone
for $\tilde{n}_{\pm}$. The plot also reveals the absence of lossy region for
$n_{R,M}$. Here, we have used $V_{0}=(5/2)\omega_{p}$, with $\omega_{p}=1$
$\mathrm{rad}$ $s^{-1}$. Figure 11: Dispersion relations related to refractive
indices $n_{L}$ (solid blue line) and $n_{E}$ (solid black line) under the
condition (25). The dashed black line represents the indices of the standard
case, $\tilde{n}_{\pm}$. The highlighted area in red (gray and red) indicates
the absorption zone for $n_{R,M}$ ($\tilde{n}_{\pm}$). Here, we have used
$V_{0}=(3/2)\omega_{p}$, with $\omega_{p}=1$ $\mathrm{rad}$ $s^{-1}$. Figure
12: Dispersion relations related to refractive indices $n_{L}$ (solid blue
line) and $n_{E}$ (solid black line) under the condition (26). The dashed
black line represents the indices of the standard case, $\tilde{n}_{\pm}$. The
highlighted gray area represents the absorption zone for$\tilde{n}_{\pm}$. The
plot also reveals the absence of a lossy region for $n_{L,E}$. Here, we have
used $V_{0}=(5/2)\omega_{p}$, with $\omega_{p}=1$ $\mathrm{rad}$ $s^{-1}$.

Negative refraction also arises in the chiral timelike MCFJ magnetized plasma
[86], while is not observed in the spacelike MCFJ magnetized case [87]. Thus,
such a negative dispersion is a feature of timelike MCFJ electrodynamics.

The dispersion relations associated with $n_{L}$ and $n_{E}$, under the
condition (25), are depicted in Fig. 11. Absortion and propagation zones
occurs for $\omega<\omega_{\mathit{rad}}$ and $\omega>\omega_{\mathit{rad}}$,
respectively. The scenario of shortened absorption zone is similar to the one
of Fig. 9, but with the index $n_{E}$ exhibiting negative refraction in the
place of $n_{R}$. On the other hand, under the condition (26), there is no
absorption and the propagation occurs for $\omega>0$, as illustrated in Fig.
12, which finds similarity with the plot of Fig. 10.

##  III Birefringence, rotatory power and dichroism

Optical activity usually manifests as a rotation of the polarization vector of
the propagating wave. Such a phenomenon, in the case of circularly polarized
modes, is called circular birefringence. It occurs when the RCP and LCP modes
propagate at different phase velocities. For the indices (19) and (20), the
corresponding phase velocities, $v_{R}=1/n_{R}$, $v_{L}=1/n_{L}$,
$v_{E}=1/n_{E}$ and $v_{M}=1/n_{M}$, are distinct, and engender birefringence
that is measure in terms of the rotatory power,

$\delta=-\frac{\omega}{2}\left(\mathrm{Re}\left[n_{LCP}\right]-\mathrm{Re}\left[n_{RCP}\right]\right),$
(33)

where $n_{LCP}$ and $n_{RCP}$ are the refractive indices for different
circular polarizations. The rotatory power (RP) is useful to optical
characterization of matter, including crystals [88, 89], organic compounds
[90, 91], graphene phenomena at terahertz band [92], gas of fast-spinning
molecules [93], chiral metamaterials [94, 95, 96], chiral semimetals [97, 98],
and in the determination of the rotation direction of pulsars [99].

###  III.1 Rotatory power

We write the rotatory power for the present chiral plasma considering the
refractive indices $n_{L}$, $n_{E}$, associated with the LCP wave, and the
indices $n_{R}$, $n_{M}$, associated to the RCP wave. Starting from the
indices $n_{L}$ and $n_{R}$, the corresponding RP,

$\delta_{LR}=-\frac{\omega}{2}\left(\mathrm{Re}[n_{L}]-\mathrm{Re}[n_{R}]\right),$
(34)

yields simply a constant (non-dispersive) RP, namely

$\delta_{LR}=-\frac{\omega}{2}\mathrm{Re}\left[V_{0}/\omega\right]=-\frac{V_{0}}{2},$
(35)

induced by the presence of the chiral parameter $V_{0}$.

Considering now the refractive indices $n_{E}$ and $n_{R}$, the rotatory power
is written as

$\delta_{ER}=-\frac{\omega}{2}\left(\mathrm{Re}[n_{E}]-\mathrm{Re}[n_{R}]\right),$
(36)

or,

$\delta_{ER}=-\frac{\omega}{2}\mathrm{Re}\left[V_{0}/\omega-2\sqrt{R\left(\omega\right)}\right],$
(37)

which is depicted in Fig. 13. Under the condition (25), one has
$R\left(\omega\right)<0$ for $\omega<\omega_{\mathit{rad}}$, so that the term
$\sqrt{R\left(\omega\right)}$ is pure imaginary in this range. The RP (37)
then coincides with the one (35) for $\omega<\omega_{\mathit{rad}}$. For
$\omega>\omega_{\mathit{rad}}$ the term $\sqrt{R\left(\omega\right)}$ becomes
real and starts to contribute (positively) to the RP, as seen in Fig. 13. It
is also worth mentioning that the RP (37) undergoes a sign reversal at the
plasma frequency $\omega_{p}$. Moreover, notice that such an increasing RP is
a consequence of the $n_{E}$ negative refraction zone.

The behavior of the RP (37) under the condition (26) is illustrated by the
brown line in Fig. 13. In this case, one has $R\left(\omega\right)>0$ for any
frequency, so the root $\sqrt{R\left(\omega\right)}$ contributes to the RP for
$\omega>0$, yielding the softly increasing profile of Fig. 13. The sign
reversal behavior at $\omega_{p}$ is not altered, since it occurs for
$\omega=\omega_{p}$.

A rotatory power reversion is not usual in cold plasma theory. However, it is
observed in rotating plasmas [100] and in magnetized chiral cold plasmas ruled
by the MCFJ electrodynamics [86, 87]. Such a reversion is also reported in
graphene systems [92] and bi-isotropic dielectrics supporting chiral magnetic
current [101].

Figure 13: The rotatory power (37) defined for the refractive indices $n_{E}$
and $n_{R}$. It is represented by the solid blue line for the condition (25)
and $V_{0}=(3/2)\omega_{p}$, and by the brown line for the condition (26) and
$V_{0}=(5/2)\omega_{p}$. Here, we have used $\omega_{p}=1$ $\mathrm{rad}$
$s^{-1}$.

The RP can also be evaluated for the refractive indices $n_{L}$ and $n_{M}$,
yielding

$\delta_{LM}=-\frac{\omega}{2}\mathrm{Re}\left[V_{0}/\omega+2\sqrt{R\left(\omega\right)}\right].$
(38)

that is ploted in Fig. 14, for the conditions (25) and (26). As noticed, the
RP is always negative, with no sign reversion.

Figure 14: The rotatory power (38) defined for the refractive indices $n_{L}$
and $n_{M}$. It is represented by the solid blue line for the condition (25)
and $V_{0}=(3/2)\omega_{p}$, and by the brown line for the condition (26) and
$V_{0}=(5/2)\omega_{p}$. Here, we have used $\omega_{p}=1$ $\mathrm{rad}$
$s^{-1}$.

###  III.2 Dichroism coefficients

The dichroism circular effect occurs when circularly polarized modes undergo
absorption at different degrees in the zones where the refractive indices are
complex. Taking into account the imaginary pieces of the refractive indices,
the dichroism for LCP and RCP waves is expressed in terms of the coefficient

$\delta_{d}=-\frac{\omega}{2}\left(\mathrm{Im}[n_{LCP}]-\mathrm{Im}[n_{RCP}]\right).$
(39)

As already seen, under the condition (25) the refractive indices $n_{L}$ and
$n_{R}$ are complex for $\omega<\omega_{\mathit{rad}}$ and there occurs
absorption. The dichroism coefficient, however,

$\delta_{dLR}=-\frac{\omega}{2}\left(\mathrm{Im}[n_{L}]-\mathrm{Im}[n_{R}]\right)=-\frac{1}{2}\mathrm{Im}\left[V_{0}\right]=0,$
(40)

is null in this window, $\delta_{dLR}=0$, since the chiral parameter $V_{0}$
is considered real. Physically, it means that both RCP and LCP waves are
absorbed at the same degree in the frequency $\omega<\omega_{\mathit{rad}}$,
where the indices are complex. This behavior differs from the chiral
magnetized plasma scenario, where the coefficient $\delta_{dLR}$ is non-null
for different frequency ranges; see Ref. [86].

For the indices $n_{E}$ and $n_{R}$, under the condition (25), the dichroism
coefficient is

$\delta_{dER}=\begin{cases}\text{$\omega\sqrt{R\left(\omega\right)}$},&\quad\text{for
$0<\omega<\omega_{rad}$},\\\ \text{$0$},&\quad\text{for
$\omega>\omega_{rad}$}.\\\ \end{cases}$ (41)

Such coefficient is non-null only for $\omega<\omega_{rad}$, being illustrated
in Fig. 15. This coefficient $\delta_{dER}$ is also different from the one of
a chiral magnetized plasma, which can be non-null for one or more intervals,
as discussed in Ref. [86].

The dichroism coefficient for the refractive indices $n_{L}$ and $n_{M}$,
under the condition (25), is

$\delta_{dLM}=\begin{cases}\text{$-\omega\sqrt{R\left(\omega\right)}$},&\quad\text{for
$0<\omega<\omega_{rad}$},\\\ \text{$0$},&\quad\text{for
$\omega>\omega_{rad}$},\\\ \end{cases}$ (42)

presenting non-null and negative dichroism only for $\omega<\omega_{rad}$, as
shown by the blue line in Fig. 15, which corresponds to the mirrored behavior
(opposite sign) of the coefficient (41).

Figure 15: Red line: dichroism coefficient (41), defined by the refractive
indices $n_{E}$ and $n_{R}$ under the condition (25). Blue line: dichroism
coefficient (42), defined by the refractive indices $n_{L}$ and $n_{M}$ under
the condition (25). Here, we have used $V_{0}=(3/2)\omega_{p}$, and
$\omega_{p}=1$ $\mathrm{rad}$ $s^{-1}$.

##  IV Final remarks

In this paper, the propagation and absorption of the electromagnetic waves in
unmagnetized plasma were analyzed in the case of the chiral axion
electrodynamics stemming from the MCFJ timelike sector, which determines
optical activity, negative refraction and RP sign reversion, a richer scenario
than the one of a usual magnetized plasma.

The Ampère’s law of this theory includes the CME current term,
$\mathbf{J}_{CME}=\left(K_{AF}\right)_{0}\mathbf{B}$, with the scalar
background $\left(K_{AF}\right)_{0}$ representing the magnetic conductivity.
The dispersion relation and corresponding refractive indices were derived by
using the usual unmagnetized plasma permittivity and the modified Maxwell
equations.

In Sec. II, we obtained four modified refractive indices, $n_{R,M}$ and
$n_{L,E}$, associated with RCP and LCP modes, respectively. The properties of
such indices were analyzed and highlighted in the Secs. II.1 \- II.4 and Figs.
1 \- 8, where one can see that $n_{E}$ is the oppositive (negative) of
$n_{R}$, and $n_{M}$ is the oppositive (negative) of $n_{L}$. Such indices
manifest significant modifications in relation to the ones of a typical non-
magnetized plasma. For example, the refractive index $n_{R}$, illustracted in
Fig. 1, possesses a negative refraction behavior in the range
$0<\omega<\omega_{p}$, occurring a free negative propagation in the window
$\omega_{\mathit{rad}}<\omega<\omega_{p}$. This feature is different from what
takes place in the usual unmagnetized plasmas, where pure absorption happens
in the interval $0<\omega<\omega_{p}$. Furthermore, in a magnetized chiral
plasma scenario, negative refraction also emerges in the index associated with
the RCP wave, as seen in Ref. [86]. The very low-frequency regime was analyzed
in Sec. II.5, where RCP and LCP helicons were obtained, given by Eqs. (31) and
(32), respectively. In the present context, both RCP and LCP helicons can
propagate simultaneously, a common point with the magnetized case [86].

Optical effects, including rotatory power and dichroism, were discussed in
Sec. III.1 under the conditions $V_{0}^{2}/4<\omega_{p}^{2}$ and
$V_{0}^{2}/4>\omega_{p}^{2}$. Despite yielding isotropic refractive indices,
see Eqs. (19) and (20), the chiral factor induces birefringence (absent in
usual unmagnetized plasmas), mimicking the optical role played by the magnetic
field in cold plasmas. This birefringence is expressed in terms of the RP. For
the indices $n_{R}$ and $n_{L}$, the associated RP is constant,
$\delta_{LR}=-V_{0}/2$, under any condition. Such a value represents the high-
frequency limit of the corresponding RP in a magnetized chiral plasma context,
as shown in Eq. (77) of Ref. [86]. On the other hand, a dispersive RP is
defined for the indices $n_{R}$ and $n_{E}$, which is negative at low
frequencies and positive at high frequencies, undergoing a sign reversion at
the plasma frequency $\omega_{p}$, under both conditions, as depicted in Fig.
13. The RP was also evaluated for the refractive indices $n_{L}$ and $n_{M}$,
implying a negative coefficient, as illustrated in Fig. 14.

RP reversion was also reported in the more involved scenario of an axion-like
MCFJ magnetized chiral plasma as shown in Figs. 15 and 16 of Ref. [86]. The
axion term is thus identified as RP reversion inductor factor (in the absence
or presence of the magnetic field). The effect of RP reversion has been also
reported in rotating plasmas [100], where the RP undergoes sign reversion and
behaves with $1/\omega^{2}$ for high frequencies.

The dichroism effect was examined in Sec. III.2, with the coefficient
$\delta_{dLR}$, associated with the indices $n_{L}$ and $n_{R}$, being null,
which is in full compatibility with the constant RP (35). For the indices
$n_{E}$ and $n_{R}$, the coefficient $\delta_{dER}$ is non-null in the range
$0<\omega<\omega_{rad}$ and null for $\omega>\omega_{rad}$, under the
condition (25). This behavior is very different from the magnetized case,
where both coefficients, $\delta_{dLR}$ and $\delta_{dER}$, are non-null for
varied intervals. See Sec. VI B in Ref. [86].

An interesting point is to establish a comparison among the present scenario,
the usual magnetized plasma and chiral MCFJ plasmas. Indeed, Table 1 compares
four scenarios of cold plasmas: i) usual unmagnetized plasma, ii) usual
magnetized plasma, iii) unmagnetized plasma ruled by the chiral MCFJ theory
with $\mathbf{J}_{B}=K_{AF}^{0}\mathbf{B}$, iv) magnetized cold plasma ruled
by the MCFJ theory with $\mathbf{J}_{B}=K_{AF}^{0}\mathbf{B}$, with the
magnetized cases defined for the Faraday configuration,
$\mathbf{B}\parallel\mathbf{k}$. It points out relevant optical aspects, such
as propagating modes, birefringence, RP reversion, absorption, and helicon
modes. As well known, usual magnetized cold plasma presents birefringence and
absorption, expressed in terms of RP and dichroism coefficient, but no RP
reversion. It differs from magnetized and unmagnetized cold plasma in MCFJ
electrodynamics, where RP reversion occurs in both scenarios. Another
difference is observed in propagating helicons, which occurs only for RCP
waves in usual magnetized plasma, while both RCP and LCP can appear for
magnetized and unmagnetized cold plasma governed by the MCFJ theory with
$\mathbf{J}_{B}=K_{AF}^{0}\mathbf{B}$. For the Voigt configuration, both
magnetized cold plasma (usual and chiral MCFJ) exhibits elliptical modes,
manifesting birefringence (phase shift) and absorption, as discussed in Ref.
[86].

Table 1: Propagation properties of four distinct scenarios of cold plasmas. The symbol “$...$” means that the entry does not apply to the mentioned feature and plasma model. The magnetized scenarios are for the Faraday configuration, $\mathbf{B}\parallel\mathbf{k}$. |  |  |  |
---|---|---|---|---
|  Unmagnetized plasma in usual electrodynamics |  Magnetized plasma in usual electrodynamics |  Unmagnetized chiral MCFJ plasma with $\mathbf{J}_{B}=k_{AF}^{0}\mathbf{B}$ |  Magnetized chiral MCFJ plasma with $\mathbf{J}_{B}=k_{AF}^{0}\mathbf{B}$
|  |  |  |
Propagating modes |  linear mode |  RCP for $n_{-}$, LCP for $n_{+}$ |  RCP for $n_{R,M}$, LCP for $n_{L,E}$ |  RCP for $n_{R,M}$, LCP for $n_{L,E}$, see Ref. [86]
|  |  |  |
|  |  |  |
Birefringence |  … |  RP |  RP (35) and (37) |  RP $\delta_{LR}$ and $\delta_{ER}$, see Ref. [86]
|  |  |  |
|  |  |  |
RP reversion |  … |  no |  yes |  yes
|  |  |  |
|  |  |  |
Absorption |  yes (conventional) |  yes (dichroism) |  yes, $\delta_{dER}$ (41) |  yes (dichroism)
|  |  |  |
Helicons |  … |  RCP |  RCP and LCP, enabled by the chiral factor |  RCP and LCP, see Ref. [86]

###### Acknowledgements.

The authors express their gratitude to FAPEMA, CNPq, and CAPES (Brazilian
research agencies) for their invaluable financial support. M.M.F. is supported
by FAPEMA APP-12151/22, CNPq/Produtividade 317048/2023-6 and
CNPq/Universal/422527/2021-1. P.D.S.S. is grateful to grant CNPq/PDJ 150584/23
and FAPEMA APP-12151/22. Furthermore, we are indebted to CAPES/Finance Code
001 and FAPEMA/POS-GRAD-02575/21.

## References

  * [1] A. Zangwill, Modern Electrodynamics (Cambridge University Press, New York, 2012).
  * [2] J.D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, New York, 1999).
  * [3] D.A. Gurnett and A. Bhattacharjee, Introduction to Plasma Physics (Cambridge University Press, Cambridge, 2005).
  * [4] P.A. Sturrock, Plasma Physics: An Introduction to the Theory of Astrophysical, Geophysical and Laboratory Plasmas, (Cambridge University Press, Cambridge, England, 1994).
  * [5] J. A. Bittencourt, Fundamentals of Plasma Physics, 3rd ed. (Springer, New York, 2004).
  * [6] P. Chabert and N. Braithwaite, Physics of Radio-Frequency Plasmas (Cambridge University Press, Cambridge, 2011).
  * [7] T. H. Stix, Waves in Plasmas (Springer, New York, 1992).
  * [8] T.J.M. Boyd and J.J. Sanderson, The Physics of plasmas (Cambridge University Press, New York, 2003).
  * [9] A. Piel, Plasmas Physics - An Introduction to Laboratory, Space, and Fusion Plasmas (Springer, Heidelberg, 2010).
  * [10] E. V. Appleton and G. Builder, The ionosphere as a doubly refracting medium. Proc. Phys. Soc. 45 208, (1932); E. V. Appleton, Wireless studies of the ionosphere, J. Inst. Electr. Eng. 7, 257 (1932).
  * [11] D. R. Hartree, The propagation of electromagnetic waves in a stratified medium, Mathematical Proc. Cambridge Philos. Soc. 25, 97 (1929).
  * [12] J. A. Ratcliff, The formation of the ionosphere. Ideas of the early years (1925-1955). J. Atmos. Terr. Phys. 36, 2167 (1974).
  * [13] H. Li, J. Liu, W. Ding, L. Bai, and Z. Wu, Propagation of arbitrarily polarized terahertz Bessel vortex beam in inhomogeneous unmagnetized plasma slab, Phys. Plasmas 25, 123505 (2018).
  * [14] H. Li, W. Ding, J. Liu, L. Bai, and Z. Wu, OAM crosstalk of multiple coaxial THz vortex beams propagating through an inhomogeneous unmagnetized plasma slab, Phys. Plasmas 26, 123507 (2019).
  * [15] J. Wang, Z. Lu, J. Tu; Z. Huang, and D. Deng, Propagation properties of the Pearcey Gaussian vortex electron plasma wave, Phys. Plasmas 30, 102107 (2023).
  * [16] L. Wu, Y. Liu, J. Wu, S. Hong, L. Wu, Y. Chen, J. Xie, and D. Deng, Airy vortex electron plasma wave, Phys. Plasmas 26, 092111 (2019).
  * [17] S. V. Vladimirov, Amplification of electromagnetic waves in dusty nonstationary plasmas, Phys. Rev. E 49, R997(R) (1994).
  * [18] S. V. Vladimirov, Scattering of electromagnetic waves in dusty plasmas with variable charges on dust particles, Phys. Rev. E 50, 1422 (1994).
  * [19] R. K. Varma, P. K. Shukla, and V. Krishan, Electrostatic oscillations in the presence of grain-charge perturbations in dusty plasmas, Phys. Rev. E 47, 3612 (1993).
  * [20] P. K. Shukla and I. Sandberg, Radiation-condensation instability in a self-gravitating dusty astrophysical plasma, Phys. Rev. E 67, 036401 (2003).
  * [21] M. R. Amin, T. Ferdous, and M. Salimullah, Decay instability of an electron plasma wave in a dusty plasma, Phys. Rev. E 53, 2740 (1996).
  * [22] I. Y. Dodin and N. J. Fisch, Nonlinear Dispersion of Stationary Waves in Collisionless Plasmas, Phys. Rev. Lett. 107, 035005 (2011).
  * [23] C. D. Decker and W. B. Mori, Group velocity of large-amplitude electromagnetic waves in a plasma, Phys. Rev. E 51, 1364 (1995).
  * [24] J. Vranješ, G. Marić, and P. K. Shukla, Nonlinear magnetic electron tripolar vortices in streaming plasmas, Phys. Rev. E 61, 7009 (2000).
  * [25] S. Mahmood and H. Ur-Rehman, Electrostatic solitons in unmagnetized hot electron–positron–ion plasmas, Phys. Lett. A 373, 2255 (2009).
  * [26] N. Chakrabarti and , M.S. Janaki, Nonlinear evolution of ion-acoustic waves in unmagnetized plasma, Phys. Lett. A 305, 393 (2002).
  * [27] M. Ghosh, S. Pramanik, and S. Ghosh, Nonlinear coherent structures of electron acoustic waves in unmagnetized plasmas, Phys. Lett. A 396, 127242 (2021).
  * [28] K. Walia, Self-focusing of high power beam in unmagnetized plasma and its effect on Stimulated Raman scattering process, Optik 225, 165592 (2021).
  * [29] K. Walia, Self-focusing of q-Gaussian beam in unmagnetized plasma and its impact on second harmonic generation, Optik 277, 170681 (2023).
  * [30] C. P. Dettmann, N. E. Frankel, and V. Kowalenko, Plasma electrodynamics in the expanding Universe, Phys. Rev. D 48, 5655 (1993).
  * [31] S. Koide, Charge separation instability in an unmagnetized disk plasma around a Kerr black hole, Phys. Rev. D 83, 023003 (2011).
  * [32] V. Perlick and O. Yu. Tsupko, Light propagation in a plasma on Kerr spacetime: Separation of the Hamilton-Jacobi equation and calculation of the shadow, Phys. Rev. D 95, 104003 (2017).
  * [33] J. Badía and E. F. Eiroa, Shadows of rotating Einstein-Maxwell-dilaton black holes surrounded by a plasma, Phys. Rev. D 107, 124028 (2023).
  * [34] G. Briozzo, E. Gallo, and T. Mädler, Shadows of rotating black holes in plasma environments with aberration effects, Phys. Rev. D 107, 124004 (2023).
  * [35] S. Kumar, A. Uniyal, and S. Chakrabarti, Shadow and weak gravitational lensing of rotating traversable wormhole in nonhomogeneous plasma spacetime, Phys. Rev. D 109, 104012 (2024).
  * [36] W. S. Porter and E. M. Bock Jr., Faraday effect in a plasma, Am. J. Phys. 33, 1070 (1965).
  * [37] Y. Tang and A. E. Cohen, Optical Chirality and Its Interaction with Matter, Phys. Rev. Lett. 104, 163901 (2010).
  * [38] A. H. Sihvola and I. V. Lindell, Bi-isotropic constitutive relations, Microw. Opt. Technol. Lett. 4 295 (1991); Properties of bi-isotropic Fresnel reflection coefficients, Opt. Commun. 89, 1 (1992); S. Ougier, I. Chenerie, A. Sihvola, and A. Priou, Propagation in bi-isotropic media: Effect of different formalisms on the propagation analysis, Prog. Electromagn. Res. 09, 19 (1994).
  * [39] P. Hillion, Manifestly covariant formalism for electromagnetism in chiral media, Phys. Rev. E 47, 1365 (1993); Y. Itin, Dispersion relation for electromagnetic waves in anisotropic media, Phys. Lett. A 374, 1113 (2010); N.J. Damaskos, A.L. Maffett, and P.L.E. Uslenghi, Dispersion relation for general anisotropic media, IEEE Trans. Antennas Propagat. 30, 991 (1982).
  * [40] J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986).
  * [41] Y. T. Aladadi and M. A. S. Alkanhal, Classification and characterization of electromagnetic materials, Sci. Rep. 10, 11406 (2020).
  * [42] W. Mahmood and Q. Zhao, The double Jones birefringence in magneto-electric medium, Sci. Rep. 5, 13963 (2015).
  * [43] V. A. De Lorenci and G. P. Goulart, Magnetoelectric birefringence revisited, Phys. Rev. D 78, 045015 (2008).
  * [44] P. D. S. Silva, R. Casana, and M. M. Ferreira Jr., Symmetric and antisymmetric constitutive tensors for bi-isotropic and bi-anisotropic media, Phys. Rev. A 106, 042205 (2022).
  * [45] G. R. Fowles, Introduction to Modern Optics, 2nd ed. (Dover Publications, INC., New York, 1975); A. K. Bain, Crystal Optics: Properties and Applications (Wiley-VCH Verlag GmbH & Co. KGaA, Germany, 2019).
  * [46] S.M. Carroll, G.B. Field, and R. Jackiw, Limits on a Lorentz- and parity-violating modification of electrodynamics, Phys. Rev. D 41, 1231 (1990).
  * [47] D. Colladay and V.A. Kostelecký, CPT violation and the standard model, Phys. Rev. D 55, 6760 (1997); Lorentz-violating extension of the standard model, Phys. Rev. D 58, 116002 (1998); S.R. Coleman and S.L. Glashow, High-energy tests of Lorentz invariance, Phys. Rev. D 59, 116008 (1999).
  * [48] A.A. Andrianov and R. Soldati, Lorentz symmetry breaking in Abelian vector-field models with Wess-Zumino interaction, Phys. Rev. D 51, 5961 (1995); Patterns of Lorentz symmetry breaking in QED by CPT-odd interaction, Phys. Lett. B 435, 449 (1998); A.A. Andrianov, R. Soldati, and L. Sorbo, Dynamical Lorentz symmetry breaking from a (3+1)-dimensional axion-Wess-Zumino model, Phys. Rev. D 59, 025002 (1999).
  * [49] L. C. T. Brito, J. C. C. Felipe, A. Yu. Petrov, and A. P. Baeta Scarpelli, No radiative corrections to the Carroll-Field-Jackiw term beyond one-loop order, Int. J. Mod. Phys. A36, 2150033 (2021); J. F. Assunção, T. Mariz, and A. Yu. Petrov, Nonanalyticity of the induced Carroll-Field-Jackiw term at finite temperature, Europhys. Lett. 116, 31003 (2016); J. C. C. Felipe, A. R. Vieira, A. L. Cherchiglia, A. P. Baêta Scarpelli, and M. Sampaio, Arbitrariness in the gravitational Chern-Simons-like term induced radiatively, Phys. Rev. D 89, 105034 (2014); T.R.S. Santos and R.F. Sobreiro, Lorentz-violating Yang–Mills theory: Discussing the Chern–Simons-like term generation, Eur. Phys. J. C 77, 903 (2017).
  * [50] R. Casana, M. M. Ferreira Jr., E. da Hora, and A. B. F. Neves, Maxwell-Chern-Simons vortices in a CPT-odd Lorentz-violating Higgs electrodynamics, Eur. Phys. J. C 74, 3064 (2014); R. Casana and L. Sourrouille, Self-dual Maxwell-Chern-Simons solitons from a Lorentz-violating model, Phys. Lett. B 726, 488 (2013).
  * [51] H. Belich, L. D. Bernald, Patricio Gaete, and J. A. Helayël-Neto, The photino sector and a confining potential in a supersymetric Lorentz-symmetry-violating model, Eur. Phys. J. C 73, 2632 (2013); L. Bonetti, L. R. dos Santos Filho, J A. Helayël-Neto, and A. D. A. M. Spallicci, Photon sector analysis of Super and Lorentz symmetry breaking: Effective photon mass, bi-refringence and dissipation, Eur. Phys. J. C 78, 811 (2018).
  * [52] L.H. C. Borges and A.F. Ferrari, External sources in a minimal and nonminimal CPT-odd Lorentz violating Maxwell electrodynamics, Mod. Phys. Lett. A 37, 2250021 (2022); Y. M. P. Gomes and P. C. Malta, Lab-based limits on the Carroll-Field-Jackiw Lorentz-violating electrodynamics, Phys. Rev. D 94, 025031 (2016); M.M. Ferreira Jr, J.A. Helayël-Neto, C.M. Reyes, M. Schreck, and P.D.S. Silva, Unitarity in Stückelberg electrodynamics modified by a Carroll-Field-Jackiw term, Phys. Lett. B 804, 135379 (2020); A. Martín-Ruiz and C. A. Escobar, Local effects of the quantum vacuum in Lorentz-violating electrodynamics, Phys. Rev. D 95, 036011 (2017).
  * [53] A. Sekine and K. Nomura, Axion electrodynamics in topological materials, J. Appl. Phys. 129, 141101 (2021).
  * [54] M. E. Tobar, B. T. McAllister, and M. Goryachev, Modified axion electrodynamics as impressed electromagnetic sources through oscillating background polarization and magnetization, Phys. Dark Universe 26, 100339 (2019).
  * [55] E. J. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics(Norht-Holland Publishing Company, Amsterdam, Dover Publications, 1997).
  * [56] Z. Qiu, G. Cao and X.-G. Huang, Electrodynamics of chiral matter, Phys. Rev. D 95, 036002 (2017).
  * [57] D.E. Kharzeev, The chiral magnetic effect and anomaly-induced transport, Prog. Part. Nucl. Phys. 75, 133 (2014); D.E. Kharzeev, J. Liao, S.A. Voloshin, and G. Wang, Chiral magnetic and vortical effects in high-energy nuclear collisions – A status report, Prog. Part. Nucl. Phys. 88, 1 (2016);
  * [58] D. Kharzeev, K. Landsteiner, A. Schmitt and H.U. Yee, Strongly Interacting Matter in Magnetic Fields, Lecture Notes in Physics. Vol. 871 (Springer-Verlag, Berlin $\cdot$ Heidelberg, 2013).
  * [59] K. Fukushima, D.E. Kharzeev, and H.J. Warringa, Chiral magnetic effect, Phys. Rev. D 78, 074033 (2008); D.E. Kharzeev and H. J. Warringa, Chiral magnetic conductivity,  Phys. Rev. D 80, 034028 (2009).
  * [60] Q. Li, D.E. Kharzeev, C. Zhang, Y. Huang, I. Pletikosić, A.V. Fedorov, R.D. Zhong, J.A. Schneeloch, G.D Gu, and T. Valla, Chiral magnetic effect in $\mathrm{ZrTe_{5}}$, Nat. Phys. 12, 550 (2016).
  * [61] A. Vilenkin, Equilibrium parity-violating current in a magnetic field, Phys. Rev. D 22, 3080 (1980); A. Vilenkin and D.A. Leahy, Parity nonconservation and the origin of cosmic magnetic fields, Astrophys. J. 254, 77 (1982).
  * [62] G. Inghirami, M. Mace, Y. Hirono, L. Del Zanna, D.E. Kharzeev, and M. Bleicher, Magnetic fields in heavy ion collisions: Flow and charge transport,  Eur. Phys. J. C 80, 293 (2020).
  * [63] J. Schober, A. Brandenburg and I. Rogachevskii, Chiral fermion asymmetry in high-energy plasma simulations, Geophys. Astrophys. Fluid Dyn. 114, 106 (2020).
  * [64] Y. Akamatsu and N. Yamamoto, Chiral Plasma Instabilities, Phys. Rev. Lett. 111, 052002 (2013); A. Boyarsky, O. Ruchayskiy, and M. Shaposhnikov, Long-Range Magnetic Fields in the Ground State of the Standard Model Plasma, Phys. Rev. Lett. 109, 111602 (2012).
  * [65] M. Dvornikov and V.B. Semikoz, Influence of the turbulent motion on the chiral magnetic effect in the early universe, Phys. Rev. D 95, 043538 (2017).
  * [66] G. Sigl and N. Leite, Chiral magnetic effect in protoneutron stars and magnetic field spectral evolution, J. Cosmol. Astropart. Phys. 01 (2016) 025.
  * [67] M. Dvornikov and V.B. Semikoz, Magnetic field instability in a neutron star driven by the electroweak electron-nucleon interaction versus the chiral magnetic effect, Phys. Rev. D 91, 061301(R) (2015).
  * [68] M. Dvornikov and V.B. Semikoz, Instability of magnetic fields in electroweak plasma driven by neutrino asymmetries, J. Cosmol. Astropart. Phys. 05 (2014) 002 ; M. Dvornikov, Electric current induced by an external magnetic field in the presence of electroweak matter, EPJ Web Conf. 191, 05008 (2018).
  * [69] A.A. Burkov, Chiral anomaly and transport in Weyl metals, J. Phys. Condens. Matter 27, 113201 (2015).
  * [70] E. Barnes, J. J. Heremans, and Djordje Minic, Electromagnetic Signatures of the Chiral Anomaly in Weyl Semimetals, Phys. Rev. Lett. 117, 217204 (2016).
  * [71] X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen, Z. Yang, H. Liang, M. Xue, H. Weng, Z. Fang, X. Dai, and G. Chen, Observation of the chiral-anomaly-induced negative magnetoresistance in 3D Weyl semimetal TaAs, Phys. Rev. X 5, 031023 (2015).
  * [72] Z.-X. Li, Yunshan Cao, and Peng Yan, Topological insulators and semimetals in classical magnetic systems, Phys. Rep. 915, 1 (2021).
  * [73] C.-X. Liu, P. Ye, and X.-L. Qi, Chiral gauge field and axial anomaly in a Weyl semimetal, Phys. Rev. B 87, 235306 (2013).
  * [74] F. D. M. Haldane, Berry Curvature on the Fermi Surface: Anomalous Hall Effect as a Topological Fermi-Liquid Property, Phys. Rev. Lett 93, 206602 (2004).
  * [75] D. Xiao, M. Chang, Q. Niu, Berry phase effects on electronic properties, Phys. Rev. Lett 82,1959 (2010).
  * [76] X. G. Huang, Simulating Chiral Magnetic and Separation Effects with Spin-Orbit Coupled Atomic Gases, Sci Rep 6, 20601 (2016).
  * [77] S. Nakatsuji, N. Kiyohara, T. Higo, Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature, Nature 527, 212-215 (2015).
  * [78] H. Chen, Q. Niu, A. H. MacDonald, Anomalous Hall Effect Arising from Noncollinear Antiferromagnetism, Phys. Rev. Lett 112, 017205 (2014).
  * [79] Y. Machida, S.Nakatsuji, S. Onoda, T. Tayama, T. Sakakibara, Time-reversal symmetry breaking and spontaneous Hall effect without magnetic dipole order, Nature 463, 210–213 (2010).
  * [80] R. Côté, R. N. Duchesne, G. D. Duchesne, and O. Trépanier, Chiral filtration and Faraday rotation in multi-Weyl semimetals, R. Physics 54, 107064 (2023).
  * [81] O. V. Bugaiko, E. V. Gorbar, and P. O. Sukhachov, Surface plasmon polaritons in strained Weyl semimetals, Phys. Rev. B 102, 085426 (2020).
  * [82] O. Trépanier, R. N. Duchesne, J. J. Boudreault, and R. Côté, Magneto-optical Kerr effect in Weyl semimetals with broken inversion and time-reversal symmetries, Phys. Rev. B 106, 125104 (2022).
  * [83] P. D. S. Silva, L. L. Santos, M. M. Ferreira, Jr., and M. Schreck, Effects of CPT-odd terms of dimensions three and five on electromagnetic propagation in continuous matter, Phy. Rev. D 104, 116023 (2021).
  * [84] B. Guo, Chirality-induced negative refraction in magnetized plasma, Phys. Plasmas 20, 093596 (2013).
  * [85] M. X. Gao, B. Guo, L. Peng, and X. Cai, Dispersion relations for electromagnetic wave propagation in chiral plasmas, Phys. Plasmas 21, 114501 (2014).
  * [86] Filipe S. Ribeiro, Pedro D.S. Silva, M.M.Ferreira Jr., Cold plasma modes in the chiral Maxwell-Carroll-Field-Jackiw electrodynamics, Phys. Rev. D 107, 096018 (2023).
  * [87] Filipe S. Ribeiro, Pedro D.S. Silva, M.M.Ferreira Jr., Anisotropic cold plasma modes in chiral vector Maxwell-Carroll-Field-Jackiw electrodynamics, Phys. Rev. D 109, 076003 (2024).
  * [88] D. G. Dimitriu and D. O. Dorohoi, New method to determine the optical rotatory dispersion of inorganic crystals applied to some samples of Carpathian Quartz, Spectrochim. Acta Part A 131, 674 (2014).
  * [89] L.A. Pajdzik and A.M. Glazer, Three-dimensional birefringence imaging with a microscope tilting-stage. I. Uniaxial crystals, J. Appl. Cryst. 39, 326 (2006).
  * [90] L. D. Barron, Molecular Light Scattering and Optical Activity, 2nd ed. (Cambridge University Press, New York, 2004).
  * [91] X. Liu, J. Yang, Z. Geng, and H. Jia, Simultaneous measurement of optical rotation dispersion and absorption spectra for chiral substances, Chirality 8, 1071 (2022).
  * [92] J.-M. Poumirol, P. Q. Liu, T. M. Slipchenko, A. Y. Nikitin, L. Martin-Morento, J. Faist, and A. B. Kuzmenko, Electrically controlled terahertz magneto-optical phenomena in continuous and patterned graphene, Nat. Commun. 8, 14626 (2017).
  * [93] I. Tutunnikov, U. Steinitz, E. Gershnabel, J-M. Hartmann, A. A. Milner, V. Milner, and I. Sh. Averbukh, Rotation of the polarization of light as a tool for investigating the collisional transfer of angular momentum from rotating molecules to macroscopic gas flows, Phys. Rev. Res. 4, 013212 (2022); U. Steinitz and I. Sh. Averbukh, Giant polarization drag in a gas of molecular super-rotors,  Phys. Rev. A 101, 021404(R) (2020).
  * [94] J. H. Woo, B. K. M. Gwon, J. H. Lee, D-W. Kim, W. Jo, D. H. Kim, and J. W. Wu, Time-resolved pump-probe measurement of optical rotatory dispersion in chiral metamaterial, Adv. Opt. Mater. 5, 1700141 (2017).
  * [95] Q. Zhang, E. Plum, J-Y. Ou, H. Pi, J. Li, K. F. MacDonald, and N. I. Zheludev, Electrogyration in metamaterials: Chirality and polarization rotatory power that depend on applied electric field, Adv. Opt. Mater. 9, 2001826 (2021).
  * [96] J. Mun et al., Electromagnetic chirality: From fundamentals to nontraditional chiroptical phenomena. Light Sci. Appl. 9, 139 (2020).
  * [97] J. Ma, and D. A. Pesin, Dynamic Chiral Magnetic Effect and Faraday Rotation in Macroscopically Disordered Helical Metals, Phys. Rev. Lett. 118, 107401 (2017).
  * [98] U. Dey, S. Nandy, and A. Taraphder, Dynamic chiral magnetic effect and anisotropic natural optical activity of tilted Weyl semimetals, Sci. Rep. 10, 2699 (2020).
  * [99] R. Gueroult, Y. Shi, J-M. Rax, and N. J. Fisch, Determining the rotation direction in pulsars, Nat. Commun. 10, 3232 (2019).
  * [100] R. Gueroult, J.-M. Rax, and N. J. Fisch, Enhanced tuneable rotatory power in a rotating plasma, Phys. Rev. E 102, 051202(R) (2020).
  * [101] P. D. S.Silva and M. M. Ferreira Jr., Rotatory power reversal induced by magnetic current in bi-isotropic media, Phys. Rev. B 106, 144430 (2022).