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\title{Supplemental Material to ``Decay law of magnetic turbulence with helicity balanced by chiral fermions''}

\author{by Axel Brandenburg, Kohei Kamada, \& Jennifer Schober}
\date{\today}
\begin{document}
\maketitle

\section{Decay law with non-vanishing total chirality}

The Hosking integral, as defined in Eq.~(5) of the main paper, applies
to the case where the total chirality vanishes.
When this is not the case, the decay laws predicted by the conservation
of the Hosking integral do not apply.
The details of this will be discussed in a forthcoming paper.
Here we just highlight the main point, namely that the Hosking integral
no longer governs the decay when $\aver{h_{\rm tot}}\neq0$.

\begin{figure}[h]\begin{center}
\includegraphics[width=\columnwidth]{penerg_comp}
\end{center}\caption[]{
Time dependence of $\EEM$ (black), $\xiM^{-1}$ (orange), $\HHM$ (red),
and $-2\mu_5/\lambda$ (blue), for (a) Run~A with smaller and (b) Run~B
with larger magnetic helicity than in the balanced case.
Dashed lines indicate negative values; at late times
$-2\mu_5/\lambda$ changes sign in (a), and
$\HHM$ changes sign in (b).
In (a) the dotted line denotes the $t^{-1/2}\log(t/t_{\log})$ scaling
of Schober+20 [17] with $t_{\log}=3$.
}\label{penerg_comp}\end{figure}

To illustrate how the decay laws change when magnetic helicity and
fermion chirality no longer balance, we plot in \Fig{penerg_comp} the
time dependencies of $\EEM$, $\xiM$, $\HHM$, and $-2\mu_5/\lambda$,
for (a) Run~A with 20\% smaller and (b) Run~B with 20\% larger magnetic
helicity than in the balanced case.
In both cases, we see a tendency of the decays of $\EEM$ and $\xiM^{-1}$
to slow down while those of $\HHM$ and $-2\mu_5/\lambda$ follow separate
evolutions.
Especially in the case of Run~B, where the magnetic helicity
dominates of the fermion chirality, we see a tendency toward
a $\EEM\propto\xi^{-1}\propto t^{-2/3}$ evolution, as expected
from magnetic helicity conservation.

\section{Simulations with spin flipping}

When spin flipping is included, the evolution equation for $\mu_5$
attains an extra term,
\begin{equation}
\frac{\partial\mu_5}{\partial t}=...-\Gamma\mu_5.
\end{equation}
Again, the details of this will be discussed in a forthcoming paper.
However, to get an idea of the modifications, we show
in \Fig{p_comp5} the evolution of $\mu_{\rm M}\blue{\equiv\aver{\AAA\cdot\BB}\lambda/2}$,
$\mu_5$, and their sum for models where $\Gamma=10^{-2}$
(i) for $t\geq10^2$ or (ii) during $10^2\leq t\leq10^3$.
The other parameters are $\eta=2\times10^{-4}$ and 
$\lambda_5=2\times10^4$, as in the main paper.

\begin{figure}[h]\begin{center}
\includegraphics[width=\columnwidth]{p_comp5}
\end{center}\caption[]{
Evolution of $\mu_{\rm M}$ (red), $\mu_5$ (blue), and their sum (black)
with $\Gamma=10^{-2}$ for (i) $t\geq10^2$ (solid lines) or (ii) during
$10^2\leq t\leq10^3$ (dotted lines).
}\label{p_comp5}\end{figure}

We see that spin flipping causes $\mu_5$ to drop to zero.
Because the total chirality is \blue{no longer} conserved,
the drop of $\mu_5$ to zero means that the average,
$\mu_5+(\lambda/2)\,\aver{\AAA\cdot\BB}$, now shifts away from zero.
In our first experiment (i), we have $\Gamma\neq0$ for $t\geq10^2$.
This halts the decay of $\HHM$ for a little while, but eventually
it also decays \blue{via magnetic diffusion while $\mu_5$ stays
vanishingly small owing to the spin-flipping interaction}.
However, in our second experiment (ii), where $\Gamma\neq0$ only during
the time interval $10^2\leq t\leq10^3$, the decay of $\HHM$ is prevented
for all later times \blue{because the total chirality is now conserved again and 
the magnetic diffusion is less effective due to the inverse cascade}.

\section{Chiral plasma instability for different total chiralities}

When magnetic helicity is balanced by fermion chirality, the chiral
plasma instability is strongly suppressed near the wave numbers were
the instability operates; see \Fig{rspec_select_1024a_mu10_k002o}.
This corresponds to the run discussed in the main paper, which is
here referred to as Run~O.

\begin{figure}[h]\begin{center}
\includegraphics[width=\columnwidth]{rspec_select_1024a_mu10_k002o}
\end{center}\caption[]{
Magnetic energy (solid lines) and normalized helicity spectra
$k\HM(k)/2$ (dotted lines with red and blue symbols for positive
and negative helicity spectra, respectively) for Run~A at
times $\eta=5$, 32, $320$, 1500, 4600, 15,000, and 46,000.
}\label{rspec_select_1024a_mu10_k002o}\end{figure}

In Run~A, where the magnetic helicity is weaker than in Run~O, the chiral
plasma becomes dominant and overcompensates the magnetic helicity.
The net chirality is then negative.
Eventually, all the sign of the magnetic helicity changes and the
remaining fermion chirality is converted to magnetic fields with negative
helicity; see \Fig{rspec_select_1024a_mu10_k002o_less}, where we show
the magnetic energy $\EEM(k,t)$ and the normalized magnetic helicity
spectra $k\HM(k)/2$ for Run~A at times $\eta=32$, $320$, 1000, 3200,
10,000, and 32,000.

\begin{figure}[h]\begin{center}
\includegraphics[width=\columnwidth]{rspec_select_1024a_mu10_k002o_less}
\end{center}\caption[]{
Magnetic energy (solid lines) and normalized helicity spectra
$k\HM(k)/2$ (dotted lines with red and blue symbols for positive
and negative helicity spectra, respectively) for Run~A at
times $\eta=32$, 100, $320$, 1000, 3200, 15,000, 22,000, and 32,000.
}\label{rspec_select_1024a_mu10_k002o_less}\end{figure}

On the other hand, when the fermion chirality is weak (Run~B),
the usual inverse magnetic cascade quickly gets established;
see \Fig{rspec_select_1024a_mu10_k002o_more}.
In either case, the fermion chirality always gets ultimately
converted into magnetic helicity.
It just takes a little longer than when the magnetic helicity is
initially weak.
At the end, however, the usual inverse cascade for a fully helical
magnetic field commences.
The sign of magnetic helicity can be positive or negative, depending
on the sign of the initial total chirality.

\begin{figure}[h]\begin{center}
\includegraphics[width=\columnwidth]{rspec_select_1024a_mu10_k002o_more}
\end{center}\caption[]{
Same as \Fig{rspec_select_1024a_mu10_k002o_more}, but for Run~B
at times $\eta=32$, 100, $320$, 1000, 3200, 10,000, and 32,000.
}\label{rspec_select_1024a_mu10_k002o_more}\end{figure}

\section{Other weight functions}

Following Zhou+22 [6], the function ${\cal I}_{\rm H}(t,R)$
can be recast as a weighted integral
\begin{equation}
{\cal I}_{\rm H}(t,R)=\frac{1}{V}\int\frac{\dd^3k}{(2\pi)^3}\ w_R(\kk)\ h^*(\kk)h(\kk).
\label{eqn:Iv_general}
\end{equation}
The weight function $w_R(\kk)$ depends on the shape of $V_R$.
For a cubic region with length $2R$, we have
\begin{equation}
w_R(\kk)=w_{\rm cube}^{\rm BC}(\kk)\equiv8R^3\prod_{i=1}^3
j_0^2(k_iR);
\end{equation}
see \Fig{psaff_Box_cube_1024a_mu10_k002b}.
Here $j_0(x)=\sin x/x$.
For a spherical region with radius $R$,
\begin{equation}
w_R(\kk)=w_{\rm sph}^{\rm BC}(k)\equiv\frac{4\pi R^3}{3}
\left[\frac{6j_1(kR)}{kR}\right]^2,
\label{eq:wsph}
\end{equation}
which is the version shown in the main paper.

\begin{figure}[t!]\begin{center}
\includegraphics[width=.9\columnwidth]{psaff_Box_cube_1024a_mu10_k002b}
\end{center}\caption[]{
Box counting method with cubic volumes showing
(a) ${\cal I}_H(R,t)$ versus $R$ for different times (the normalized
values of $t\eta k_0^2$ are indicated by different colors), and
(b) ${\cal I}_H(R,t)$ versus $t$ (normalized) for two choices of $R$,
indicated by solid ($k_0R\approx1.5$) and dashed ($k_0R\approx6$) lines
in both panels.
The $t^{-0.16}$ scaling is indicated as the dashed-dotted curve for
comparison.
}\label{psaff_Box_cube_1024a_mu10_k002b}\end{figure}

\begin{figure}[t!]\begin{center}
\includegraphics[width=.9\columnwidth]{psaff_Cor_cube_1024a_mu10_k002b}
\end{center}\caption[]{
Similar to \Fig{psaff_Box_cube_1024a_mu10_k002b} using the
correlation-integral method for cubic volumes.
}\label{psaff_Cor_cube_1024a_mu10_k002b}\end{figure}

\begin{figure}[t!]\begin{center}
\includegraphics[width=.9\columnwidth]{psaff_Cor_sph1024a_mu10_k002b}
\end{center}\caption[]{
Similar to \Fig{psaff_Box_cube_1024a_mu10_k002b} using the
correlation-integral method for spherical volumes.
}\label{psaff_Cor_sph1024a_mu10_k002b}\end{figure}

Alternatively, by the correlation-integral (CI) method, we have
\begin{equation}
w_R(\kk)=w_{\rm cube}^{\rm CI}(\kk)\equiv8R^3\prod_{i=1}^3
j_0(k_i R),
\label{eqn:w_spc}
\end{equation}
for a cubic region $V_R$; see \Fig{psaff_Cor_cube_1024a_mu10_k002b}, and
\begin{equation}
w_R(\kk)=w_{\rm sph}^{\rm CI}(k)\equiv\frac{4\pi R^3}{3}
\frac{6j_1(kR)}{kR}.
\end{equation}
for a spherical region; \Fig{psaff_Cor_sph1024a_mu10_k002b}.

\end{document}
