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Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 8
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 75970)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 154, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 91, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Missing a closing quotation mark in string. in row 8
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1529, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1154, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1027, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1122, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
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\section{Introduction}
Although the charm of nature resides in the presence of symmetries,
lots of interesting and relevant phenomena are due to their
breaking. This can happen spontaneously when, despite the
corresponding conservation laws are respected by the theory, the
state of the system is not symmetric, or explicitly, when the
Hamiltonian that dictates the dynamics of the system
contains terms that do not respect the symmetry. Much work has been
done about many aspects of symmetry breaking in different
branches of physics. However, little attention has been paid to study
the non-equilibrium dynamics of a broken symmetry in a quantum
many-body system; for example, how is the time evolution of a global
symmetry if the system is initiated in a state that
breaks it and then it is let evolve with a Hamiltonian that does
preserve it. Regarding this, an important point is if the symmetry
can be dynamically restored and how fast it does.
Only some few works have analysed this question in spin chains both
for a global $U(1)$ symmetry~\cite{FCEC14, pvc-16} and for a discrete
$\mathbb{Z}_2$ group~\cite{cef-12, cef-12-2} using spin correlators.
The absence of studies on this problem is perhaps due to the lack
of a proper quantity that measures how much a symmetry is broken.
In extended quantum systems, this issue is intrinsically bound to
consider a specific subsystem. In the recent Ref.~\cite{amc22}, a
subsystem measure of symmetry breaking, dubbed \textit{entanglement
asymmetry}, has been introduced by employing tools from the theory of
entanglement in quantum many-body systems. In such work, the new
entanglement asymmetry is applied to study the time evolution of a
$U(1)$ symmetry in a spin-$1/2$ chain initiated in a tilted
ferromagnetic configuration, which breaks that symmetry, after a sudden global quench to the XX spin chain Hamiltonian, which respects it.
The analysis of the entanglement asymmetry reveals
not only that the symmetry is restored but also that the more the
symmetry is initially broken, the smaller is the time necessary to recover
it. This surprising and counterintuitive phenomenon is a sort of
quantum version of the still controversial Mpemba effect \cite{mpemba, kb-20}--- more
the system is initially out of equilibrium, the faster it relaxes.
The present work is a complement of the analysis done in Ref.~\cite{amc22}. Here we also use the entanglement
asymmetry to study the dynamics of the same $U(1)$ symmetry after a quench to the XX Hamiltonian,
but preparing the spin chain in a tilted Néel configuration instead of the tilted
ferromagnetic one. This change in the initial state of the quench protocol drastically modifies
the time evolution of the $U(1)$ symmetry, which now is not restored at large times.
\paragraph{Entanglement asymmetry:}
The setup we are interested in is an extended quantum system in a pure state $\ket{\Psi}$, defined in a bipartite Hilbert space $\mathcal{H}=\mathcal{H}_A\otimes \mathcal{H}_B$, where $\mathcal{H}_A$ and $ \mathcal{H}_B$ are respectively associated to two spatial regions $A$ and $B$. The state
of $A$ is described by the reduced density matrix $\rho_A$
obtained by taking the partial trace to the complementary subsystem
as $\rho_A=\mathrm{Tr}_B(\ket{\Psi}\bra{\Psi})$.
We further consider a charge operator $Q$ with integer eigenvalues
that generates a $U(1)$ symmetry group. For the given bipartition,
$Q$ is the sum of the charge in each region, $Q=Q_A+Q_B$.
If $\ket{\Psi}$ is an eigenstate of $Q$,
then $[\rho_A, Q_A]=0$ and $\rho_A$ displays a block-diagonal structure in the charge sectors of $Q_A$. On the other hand, if the
state $\ket{\Psi}$ breaks the symmetry, then $[\rho_A, Q_A]\neq 0$ and $\rho_A$ is not block-diagonal in the eigenbasis
of $Q_A$. Based on these observations, the entanglement asymmetry in the subsystem $A$ is defined as
\begin{equation}\label{eq:def}
\Delta S_{A}=S(\rho_{A,Q})-S(\rho_A),
\end{equation}
where
\begin{equation}\label{eq:vn}
S(\rho)=-{\rm Tr}(\rho\log \rho),
\end{equation}
is the von Neumann entropy associated to the density matrix $\rho$.
The matrix $\rho_{A,Q}$ is obtained from $\rho_A$ by projecting over all the symmetry sectors of $Q_A$ such that $\rho_{A,Q}=\sum_{q\in\mathbb{Z}}\Pi_q\rho_A\Pi_q$, where $\Pi_q$ is
the projector onto the eigenspace of $Q_A$ with charge
$q\in\mathbb{Z}$. Thus $\rho_{A, Q}$ is block-diagonal in the
eigenbasis of $Q_A$. In Fig.~\ref{fig:cartoon}, we schematically
represent the form of $\rho_{A}$ and $\rho_{A, Q}$.
As a measure of symmetry breaking, the entanglement asymmetry~\eqref{eq:def} satisfies two fundamental properties.
First, it is non-negative, $\Delta S_{A}\geq 0$, since it can be rewritten as the relative entropy between $\rho_A$ and $\rho_{A,Q}$, $\Delta S_{A}=\mathrm{Tr}[\rho_A(\log\rho_A-\log\rho_{A,Q})]$, which by definition can never be negative~\cite{ms-21}. Second, it vanishes, $\Delta S_{A}=0$, iif the state of subsystem $A$ respects the symmetry associated to $Q$, i.e. $[\rho_A, Q_A]=0$. In fact, in this case, $\rho_A$ is block diagonal
in the eigenbasis of $Q_A$, the projector $\Pi_q$ leaves it
invariant and, therefore, $\rho_{A, Q}=\rho_A$. When this occurs, the entanglement entropy $S(\rho_A)$, a key quantity in the study of
quantum many-body systems that measures the degree of entanglement between subsystems $A$ and $B$, can be resolved into the contribution of each charge sector~\cite{lr-14, goldstein, xavier}.
This fact has recently motivated an intense research activity on the interplay between entanglement and symmetries both theoretically\cite{brc-19, mgc-20, pbc-21, pbc-21-2, pvcc-22, bcckr-22} and experimentally~\cite{lukin-19, azses-20, neven-21, vitale-22, rath-22}.
\begin{figure}[t]
\centering
\includegraphics[width=0.65\textwidth]{rho2.pdf}
\caption{Schematic comparison of the form of the density matrices $\rho_A$ and $\rho_{A, Q}$ in the eigenbasis of the subsystem charge $Q_A$. If the full system has not a definite charge, the reduced density matrix $\rho_A$ contains both non-zero diagonal and off-diagonal blocks. By projecting over all the symmetry sectors, we obtain $\rho_{A,Q}$, where the off-diagonal blocks are annihilated. The difference $\Delta S_{A}^{(n)}$ between the entanglement entropies of these matrices gives the entanglement asymmetry~\eqref{eq:def}.}
\label{fig:cartoon}
\end{figure}
\paragraph{Replica trick:} The definition of Eq.~\eqref{eq:def} makes clear the connection between the entanglement asymmetry and the entanglement entropy. For this reason, in order to investigate the asymmetry, we can apply many of the techniques developed
for the analysis of the (symmetry-resolved) entanglement entropy in the bipartite setup described above; in particular, one of the most fruitful is the \textit{replica trick}~\cite{hlw-94, cc-04}. If we introduce the R\'enyi entropies,
\begin{eqnarray}\label{eq:renyiE}
S^{(n)}(\rho)=\frac{1}{1-n}\mathrm{log}\mathrm{Tr}(\rho^n),
\end{eqnarray}
then the von Neumann entropy~\eqref{eq:vn} can be obtained as the limit $n \to 1$ of Eq. \eqref{eq:renyiE}. With this in mind, the definition of Eq.~\eqref{eq:def} can be extended by replacing the von Neumann entropy $S(\rho)$ with the R\'enyi entropy $S^{(n)}(\rho)$,
\begin{equation}\label{eq:replicatrick}
\Delta S_{A}^{(n)}=S^{(n)}(\rho_{A, Q})-S^{(n)}(\rho_A).
\end{equation}
The main advantage of the R\'enyi entanglement asymmetry is that it
is easier to calculate for integer $n$, and Eq.~\eqref{eq:def} can be recovered by
taking the limit $\lim_{n\to 1}\Delta S_{A}^{(n)}=\Delta S_{A}$.
Moreover, for integer $n\geq 2$, it can be experimentally accessed
via randomised measurements in ion-trap setups~\cite{randomised-1, randomised-2, randomised-3, randomised-4}. As in the case
$n\to 1$, $\Delta S_A^{(n)}$ is always non-negative \cite{hms-22} and vanishes if
and only if $\rho_A$ respects the symmetry, i.e. $[\rho_A, Q_A]=0$.
\paragraph{Quench protocol:} As we already announced, the goal of this paper is to use the entanglement asymmetry
to analyse in a subsystem of an infinite spin-$1/2$ chain the
non-equilibrium dynamics of a broken $U(1)$ symmetry after a
global quantum quench. In particular, we take the transverse
magnetisation,
\begin{equation}\label{eq:trans_mag}
Q=\frac{1}{2}\sum_{j}\sigma_j^z,
\end{equation}
as the charge that generates the $U(1)$ symmetry and as subsystem
$A$ a set of $\ell$ contiguous spins. The specific quench
protocol that we consider is the following. We initially prepare the
spin chain in the cat state
\begin{equation}\label{eq:cat}
\ket{\Psi(0)}=\frac{\ket{{\rm N}, \theta}-\ket{{\rm N}, -\theta}}{\sqrt{2}},
\end{equation}
where $\ket{{\rm N},\theta}$ denotes the tilted Néel state,
\begin{equation}\label{eq:tiltedNéel}
\ket{{\rm N}, \theta}=e^{i\frac{\theta}{2} \sum_{j}\sigma_j^y}|\uparrow\downarrow\cdots\rangle,
\end{equation}
which breaks the $U(1)$ symmetry associated to the transverse
magnetisation~\eqref{eq:trans_mag}. We take as initial configuration
the linear combination of Eq.~\eqref{eq:cat}, instead of the state~\eqref{eq:tiltedNéel}, because the corresponding reduced density
matrix $\rho_A$ is Gaussian, a property that makes much easier both
the numerical and analytical study of the entanglement asymmetry.
Notice that the entanglement asymmetry in the tilted N\'eel state (both normal and cat version) is exactly the same as in the tilted ferromagnetic state (described in Ref. \cite{amc22}) because of the product-state structure.
Then we evolve it in time
\begin{equation}\label{eq:quench}
\ket{\Psi(t)}= e^{-i tH}\ket{\Psi(0)}
\end{equation}
with the Hamiltonian of the XX spin chain
\begin{equation}\label{eq:xx}
H=-\frac{1}{4}\sum_{j=-\infty}^{\infty}\left[\sigma_{j}^x\sigma_{j+1}^x
+\sigma_{j}^y\sigma_{j+1}^y\right],
\end{equation}
which does preserve the charge~\eqref{eq:trans_mag}, i.e. $[H, Q]=0$.
The tilting angle $\theta\in[0, \pi]$ tunes how much the symmetry is
initially broken. In fact, the tilted Néel state breaks the $U(1)$
symmetry associated to the charge~\eqref{eq:trans_mag} when $\theta\neq 0, \pi$. On the other hand, for $\theta=0, \pi$, the transverse
magnetisation has a defined value since all the spins point in the $z$-direction. Therefore, the initial entanglement
asymmetry is $\Delta S_A^{(n)}=0$ when $\theta =0,\pi$, and $\Delta S_A^{(n)}>0$ otherwise.
In particular, the symmetry is maximally broken for $\theta=\pi/2$, when all the spins
are aligned in the $x$-direction, and $\Delta S_A^{(n)}$ reaches its maximum value.
Between the extremes, $\Delta S_A$ is a monotonic function of $\theta$, a property
that is not true for all $n$.
The rest of the paper will be devoted to study the time evolution of the R\'enyi entanglement
asymmetry after the quench of Eq.~\eqref{eq:quench}. By applying the quasi-particle picture of entanglement,
we derive an exact analytic expression for $\Delta S_A^{(n)}(t)$ in the scaling limit $t,\ell\to \infty$
with $\zeta=t/\ell$ finite. We obtain that, at large times after the quench, the entanglement asymmetry
tends to a non-zero constant value, except at $\theta=\pi/2$, for which it does go to zero.
This implies that, after the quench~\eqref{eq:quench}, the $U(1)$ symmetry is not restored unless it is maximally broken
at time zero.
Using the generalised Gibbs ensemble description of the post-quech
relaxation, we will show that the reason behind the lack of symmetry
restoration is the existence of a set of charges of the evolution
Hamiltonian~\eqref{eq:xx} that do not commute with $Q$ and whose
initial state expectation value is not zero due to the
breaking of translational invariance of the tilted Néel state.
\paragraph{Outline:} The paper is organised as follows.
In Sec.~\ref{sec:basic_tools}, we describe the general
approach to compute the R\'enyi entanglement asymmetry~\eqref{eq:replicatrick} in terms of a generalised version
of the charged moments of the reduced density matrix, and we discuss
how to efficiently calculate the latter when the reduced density matrix is a fermionic Gaussian state. In this section, we also
review the results of Ref.~\cite{amc22} for the entanglement asymmetry when the initial configuration is the tilted ferromagnetic state. In Sec.~\ref{sec:charged_moments}, we obtain
the exact time evolution of the charged moments when the spin chain is quenched from the cat tilted Néel state. With this result, in Sec.~\ref{sec:ent_asymmetry}, we analyse the behaviour of the R\'enyi
entanglement asymmetry after the quench, finding that the initially broken $U(1)$ is generally not restored at large times. We check the analytical expressions derived in these sections with exact numerical calculations. In Sec.~\ref{sec:gge}, we explain the lack of symmetry
restoration in terms of the generalised Gibbs ensemble that describes
the post-quench stationary behaviour. We conclude in Sec.~\ref{sec:conclusion} with some remarks and future prospects.
We include an appendix where we discuss some properties of products
of block Toeplitz matrices that are relevant for our analytical computations.
\section{Entanglement asymmetry and charged moments}\label{sec:basic_tools}
In this section, we introduce the basic tools that we employ to calculate the R\'enyi entanglement asymmetry defined in Eq.~\eqref{eq:replicatrick}. We also review the known results for the
quench from the tilted ferromagnetic state, following Ref.~\cite{amc22}.
\subsection{Charged moments}
In order to evaluate Eq.~\eqref{eq:replicatrick}, we consider the Fourier representation of the projector $\Pi_q$. Then the projected density matrix $\rho_{A, Q}$ can be re-expressed in the form
\begin{equation}
\rho_{A, Q}=\int_{-\pi}^\pi \frac{d\alpha}{2\pi}e^{-i\alpha Q_A}\rho_A e^{i\alpha Q_A},
\end{equation}
and its moments as
\begin{equation}\label{eq:FT}
\mathrm{Tr}(\rho_{A, Q}^n)=\int_{-\pi}^\pi \frac{d\alpha_1\dots d\alpha_n}{(2\pi)^n} Z_n(\boldsymbol{\alpha}),
\end{equation}
where $\boldsymbol{\alpha}=\{\alpha_1,\dots,\alpha_n\}$ and
\begin{equation}\label{eq:Znalpha}
Z_n(\boldsymbol{\alpha})=
\mathrm{Tr}\left[\prod_{j=1}^n\rho_A e^{i\alpha_{j,j+1}Q_A}\right],
\end{equation}
with $\alpha_{ij}\equiv\alpha_i-\alpha_j$ and $\alpha_{n+1}=\alpha_1$. From the previous expressions, it is
straightforward to see that, if $[\rho_A, Q_A]=0$, then $Z_n(\boldsymbol{\alpha})=Z_n(\boldsymbol{0})$, which implies $\mathrm{Tr}(\rho_{A, Q}^n)=\mathrm{Tr}(\rho_A^n)$
and $\Delta S_{A}^{(n)}=0$, consistently with the properties of the asymmetry that we discussed in the introduction. We call
the objects $Z_n(\boldsymbol{\alpha})$
charged moments since they can be seen as a non-trivial generalisation of
the ones considered in the study of symmetry-resolved entanglement~\cite{goldstein}.
If $\rho_A$ and $e^{i\alpha Q_A}$ are Gaussian, i.e. they are the exponential of a quadratic fermionic operator, then the charged
moments $Z_n(\boldsymbol{\alpha})$
can be obtained from the fermionic two-point correlations. After a
Jordan-Wigner transformation, the XX spin chain Hamiltonian of Eq.~\eqref{eq:xx} is quadratic in terms of the fermionic operators $\boldsymbol{c}_j=(c_j, c_j^\dagger)$, which
satisfy the canonical anticommutation relations
$\{c_j, c_{j'}^\dagger\}=\delta_{j, j'}$. Then
it can be diagonalised by performing a Fourier transformation to momentum space~\cite{lms-61}. The one-particle dispersion relation is $\epsilon_k=-\cos(k)$. Therefore, if
the initial state $\ket{\Psi(0)}$ satisfies the
Wick theorem, then the reduced density matrix $\rho_A(t)=\mathrm{Tr}_B |\Psi(t)\rangle\langle \Psi(t)|$ after the quench~\eqref{eq:quench} is
a Gaussian operator for all values of $t$ and it can be obtained from the time-dependent two-point correlation matrix restricted to subsystem $A$~\cite{p-03}
\begin{equation}\label{eq:two_point_corr}
\Gamma_{jj'}(t)=2\bra{\Psi(t)}
\boldsymbol{c}_j^\dagger
\boldsymbol{c}_{j'}
\ket{\Psi(t)}-\delta_{j,j'},\quad j, j'\in A.
\end{equation}
If the subsystem $A$ is a single interval of $\ell$
contiguous sites, then $\Gamma(t)$ has dimension
$2\ell \times 2\ell$.
In terms of the fermionic operators $\boldsymbol{c}_j$, the transverse magnetisation~\eqref{eq:trans_mag} is
related to the fermionic number operator, $Q=\sum_j(c_j^\dagger c_j -1/2)$, which is
also quadratic. Therefore, Eq.~\eqref{eq:Znalpha} is the trace of the product of Gaussian fermionic operators, $\rho_A(t)$ and $e^{i\alpha_{j,j+1} Q_A}$. Applying the composition rules derived in Ref.~\cite{bb-68, FC10} for the trace of a product of Gaussian operators, we can calculate the charged moments of Eq.~\eqref{eq:Znalpha} from the correlation matrix $\Gamma(t)$ with the formula~\cite{amc22}
\begin{equation}\label{eq:numerics}
Z_n(\boldsymbol{\alpha}, t)=\sqrt{\det\left[\left(\frac{I-\Gamma(t)}{2}\right)^n
\left(I+\prod_{j=1}^n W_j(t)\right)\right]},
\end{equation}
where $W_j(t)=(I+\Gamma(t))(I-\Gamma(t))^{-1}e^{i\alpha_{j,j+1} n_A}$ and $n_A$ is a diagonal matrix with $(n_A)_{2j,2j}=1$, $(n_A)_{2j-1,2j-1}=-1$, $j=1, \cdots, \ell$. We will use the result in Eq.~\eqref{eq:numerics} to exactly compute the time evolution of $\Delta S^{(n)}_{A}(t)$ and verify the analytical predictions that we find throughout the manuscript.
\subsection{Quench from the tilted Ferromagnetic state}\label{sec:ferro}
It is illustrative to review what happens
when in a global quantum quench to the XX Hamiltonian the chain is initiated in the cat state of Eq.~\eqref{eq:cat} but building it with the tilted ferromagnetic state,
\begin{equation}\label{eq:tiltedferro}
\ket{{\rm F},\theta}=e^{i\frac{\theta}{2} \sum_{j}\sigma_j^y}|\uparrow\uparrow\cdots\rangle,
\end{equation}
instead of the tilted Néel configuration.
This
case was studied in Ref.~\cite{amc22} employing the formalism described above.
In order to compare it with our findings for the tilted Néel case, it will be enough to present the exact time evolution after the quench of
the charged moments~\eqref{eq:Znalpha} in the scaling limit $t, \ell \to \infty$ with $\zeta=t/\ell$ fixed,
\begin{equation}\label{eq:z_n_t_ferro_evolution}
Z_n(\boldsymbol{\alpha}, t)=Z_n(\boldsymbol{0}, t)e^{\ell(A_n(\boldsymbol{\alpha})+B_n(\boldsymbol{\alpha},\zeta))},
\end{equation}
where the functions $A_n(\boldsymbol{\alpha})$ and $B_n(\boldsymbol{\alpha},\zeta)$ read, respectively,
\begin{equation}\label{eq:Balpha}
\begin{split}
A_n(\boldsymbol{\alpha})=&\displaystyle \int_0^{2\pi}\frac{d k}{2\pi}\log\prod_{j=1}^nf_k^{{\rm F}}(\theta,\alpha_{j,j+1}),\\
B_n(\boldsymbol{\alpha}, \zeta)=&-\int_0^{2\pi}\frac{d k}{2\pi}\mathrm{min}(2\zeta |\epsilon'_k|,1)\log\prod_{j=1}^n f_k^{{\rm F}}(\theta,\alpha_{j,j+1}),
\end{split}
\end{equation}
and $f_k^{{\rm F}}(\theta,\alpha)$ is defined as
\begin{equation}
f_k^{{\rm F}}(\theta,\alpha)=i e^{i\Delta_k(\theta)} \sin\left(\frac{\alpha}{2}\right)+\cos\left( \frac{\alpha}{2}\right),
\end{equation}
with
\begin{equation}
e^{i\Delta_k(\theta)}=
\frac{2\cos\theta - (1 + \cos^2\theta) \cos k+ i \sin^2(\theta)\sin(k)}{1 - 2\cos(\theta)\cos(k)+\cos^2\theta}.
\end{equation}
This result has been obtained by combining two ingredients: the first one is that in this quench protocol, the $U(1)$ symmetry is restored in the large time limit \cite{FCEC14,pvc-16}, and $B_n(\boldsymbol{\alpha}, \zeta)\to -A_n(\boldsymbol{\alpha})$ as $\zeta\to\infty$ such that $\Delta S_{A}^{(n)}(t) \to 0$. The second one consists in adapting the quasi-particle picture of entanglement in order to reconstruct the behaviour of $B_n(\boldsymbol{\alpha}, \zeta)$ for finite $\zeta$. Taking the Fourier transform~\eqref{eq:FT} of Eq. \eqref{eq:z_n_t_ferro_evolution}, we obtain the result for $\Delta S_{A}^{(n)}(t)$. In particular, it is found that this quantity vanishes for large times as $t^{-3}$ for any value of $\theta$.
Another feature, which follows from having a space-time scaling, is that larger subsystems require more time to recover the symmetry. Finally, we have observed the very odd and unexpected feature that the more the symmetry is initially broken, i.e. the larger $\theta$, the smaller the time to restore it, the aforementioned quantum Mpemba effect.
\section{Charged moments after the quench from the tilted Néel state}\label{sec:charged_moments}
In this section, we study the time evolution of the
charged moments $Z_n(\boldsymbol{\alpha}, t)$ defined in Eq.~\eqref{eq:Znalpha} after the quench
\eqref{eq:quench} from the tilted Néel state. To this
end, we employ the determinant formula of Eq.~\eqref{eq:numerics} in terms of the fermionic two-point correlation matrix~\eqref{eq:two_point_corr}. Therefore, in the first part of this section, we calculate the latter
in our specific quench. We then introduce some useful properties of determinants involving products of block Toeplitz matrices and their inverse. Finally, with these results and the quasi-particle picture of entanglement, we derive
an exact analytic expression for the evolution $Z_n(\boldsymbol{\alpha}, t)$ after the quench.
\subsection{Correlation functions}
Since the tilted Néel state \eqref{eq:tiltedNéel} is invariant by two-site translations, it is useful to rearrange the entries of the two-point correlation matrix $\Gamma(t)$ in $4\times 4$ blocks of the form
\begin{equation}\label{eq:4}
\Gamma_{ll'}(t)=2\left\langle \Psi(t) \left| \left(\begin{array}{c} c_{2l-1}^\dagger \\ c_{2l}^\dagger\\ c_{2l-1} \\ c_{2l} \end{array}\right)
\left(c_{2l'-1}, c_{2l'}, c_{2l'-1}^\dagger \\ c_{2l'}^\dagger\right)\right| \Psi(t) \right\rangle-\delta_{l, l'},
\quad l, l'=1, \dots, \ell/2.
\end{equation}
In this way, for an infinite spin chain, the correlation matrix $\Gamma(t)$ at time zero can be cast as a block Toeplitz matrix,
\begin{equation}
\Gamma_{ll'}(0)=\int_{0}^{2\pi}\frac{dk}{2\pi}\mathcal{G}_0(k, \theta)e^{-i k(l-l')}, \qquad l,l'=1,\dots \ell/2,
\end{equation}
where the symbol $\mathcal{G}_0(k, \theta)$ is the $4\times 4$ matrix
\begin{equation}\label{eq:symbol}
\mathcal{G}_0(k, \theta)=\left(\begin{array}{cccc}
g_{11}(k, \theta) & e^{ik/2}g_{12}(k, \theta) & -if_{11}(k,\theta) & -i e^{ik/2}f_{12}(k, \theta) \\
e^{-i k/2} g_{12}(k,\theta) & -g_{11}(k, \theta) & -ie^{-ik/2}f_{12}(k, \theta) &if_{11}(k, \theta)\\
i f_{11}(k, \theta) & i e^{ik/2}f_{12}(k, \theta) & -g_{11}(k, \theta) & -e^{ik/2}g_{12}(k, \theta) \\
i e^{-ik/2} f_{12}(k,\theta) & -i f_{11}(k, \theta) & -e^{-ik/2} g_{12}(k, \theta) & g_{11}(k, \theta)\\
\end{array}\right),
\end{equation}
whose entries are given by
\begin{eqnarray}
g_{11}(k, \theta)&=&-\cos(\theta)-
\frac{\cos\theta\sin^2\theta(\cos k+\cos^2\theta)}
{(1+2\cos k \cos^2\theta+\cos^4\theta)},\\
g_{12}(k, \theta)&=&-\frac{\cos(k/2)(1-\cos^4\theta)}
{1+2\cos k \cos^2\theta+\cos^4\theta},\\
f_{11}(k,\theta)&=&-\frac{\cos\theta\sin^2\theta\sin k}
{1+2\cos k \cos^2\theta+\cos^4\theta},\\
f_{12}(k, \theta)&=&-\frac{\sin(k/2)\sin^4\theta}
{1+2\cos k\cos^2\theta+\cos^4\theta}.
\end{eqnarray}
As we mentioned before, the $U(1)$ symmetry generated by the transverse magnetisation corresponds to particle number conservation in fermionic language. This implies that the correlations $\bra{\Psi(t)}c_j^{\dagger} c_{j'}^\dagger\ket{\Psi(t)}$ and $\bra{\Psi(t)}c_j c_{j'}\ket{\Psi(t)}$ vanish when the symmetry is not broken, as actually happens for $\theta=0,\pi$ at time zero.
After the quench to the XX spin chain, the correlation matrix $\Gamma(t)$ is also block Toeplitz, given that the post-quench Hamiltonian~\eqref{eq:xx} is
translationally invariant. In this case, it is useful
to study separately the correlation functions $\bra{\Psi(t)} c^{\dagger}_j c_{j'} \ket{\Psi(t)}$
and $\bra{\Psi(t)} c^{\dagger}_j c_{j'}^\dagger\ket{\Psi(t)}$. For the former, we find that
\begin{equation}\label{eq:sym1}
\bra{\Psi(t)}\left(
\begin{array}{cc}
c_{2l-1}^\dagger \\
c_{2l}^\dagger
\end{array}
\right)\left(c_{2l'-1}, c_{2l'}\right)\ket{\Psi(t)}=
\frac{\delta_{ll'}}{2}+ \int_{0}^{2\pi}\frac{dk}{4\pi}\mathcal{C}_t(k, \theta)
e^{-ik(l-l')},
\end{equation}
where the symbol is provided by
\begin{equation}
\mathcal{C}_t(k, \theta)=
\left(\begin{array}{cc}
\cos(2t\epsilon_{k/2}) g_{11}(k, \theta) &
e^{ik/2}(g_{12}(k,\theta)+i\sin(2t\epsilon_{k/2})g_{11}(k,\theta)) \\
e^{-ik/2}(g_{12}(k,\theta)-i\sin(2t\epsilon_{k/2})g_{11}(k,\theta)) & -\cos(2t\epsilon_{k/2})g_{11}(k,\theta)
\end{array}\right).
\end{equation}
On the other hand, the terms involving the correlation functions that vanish when the symmetry is respected, $\bra{\Psi(t)} c_j^\dagger c_{j'}^\dagger \ket{\Psi(t)}$, are described by
\begin{equation}\label{eq:sym2}
\bra{\Psi(t)}\left(
\begin{array}{c}
c_{2l-1}\\
c_{2l}
\end{array}\right)
\left(c_{2l'-1}, c_{2l'}\right)\ket{\Psi(t)}=
\int_{-\pi}^\pi\frac{dk}{4\pi}
\mathcal{F}_t(k,\theta) e^{-ik(l-l')},
\end{equation}
with
\begin{equation}
\mathcal{F}_t(k,\theta)=
\left(\begin{array}{cc}
i f_{11}(k,\theta)-f_{12}(k,\theta)\sin(2t\epsilon_{k/2}) &
ie^{ik/2}\cos(2t\epsilon_{k/2})f_{12}(k,\theta)\\
i e^{-ik/2}\cos(2t\epsilon_{k/2})f_{12}(k,\theta) &
-if_{11}(k,\theta)-f_{12}(k,\theta)\sin(2t\epsilon_{k/2})
\end{array}\right).
\end{equation}
Now, combining Eqs. \eqref{eq:sym1} and \eqref{eq:sym2}, we obtain the expression for the full correlation matrix as a function of time $t$,
\begin{equation}
\Gamma_{ll'}(t)=\int_{-\pi}^\pi \frac{dk}{2\pi}e^{-ik(l-l')}
\mathcal{G}_t(k,\theta),
\end{equation}
where
\begin{equation}\label{eq:symbol_t_Néel}
\mathcal{G}_t(k,\theta)=
\left(\begin{array}{cc}
\mathcal{C}_t(k,\theta) & \mathcal{F}_t(k,\theta)^\dagger\\
\mathcal{F}_t(k,\theta) & -\mathcal{C}_t(-k,\theta)^*
\end{array}\right).
\end{equation}
To derive the stationary value of the charged moments and of the entanglement asymmetry at large times, we can average the time
dependent terms in the symbol $\mathcal{G}_t(k, \theta)$ of $\Gamma(t)$. At $t\to \infty$, the functions $\sin(2t\epsilon_{k/2})$ and $\cos(2t\epsilon_{k/2})$ in Eq.~\eqref{eq:symbol_t_Néel}
average to zero and the symbol simplifies,
\begin{equation}\label{eq:averaged_symbol_quench_t_Néel}
\mathcal{G}_{t\to\infty}(k,\theta)=\begin{pmatrix}
0 & e^{ik/2} g_{12}(k, \theta) & -if_{11}(k,\theta)& 0 \\
e^{-ik/2} g_{12}(k,\theta) & 0 & 0 & i f_{11}(k, \theta) \\
i f_{11}(k,\theta) & 0 & 0 & - e^{i k/2} g_{12}(k, \theta) \\
0 & -i f_{11}(k,\theta) & -e^{-ik/2} g_{12}(k) & 0 \\
\end{pmatrix}.
\end{equation}
Note that, when we take the time average, some of the correlation functions $\bra{\Psi(t)} c_j c_{j'} \ket{\Psi(t)}$ and $\bra{\Psi(t)} c^{\dagger}_j c^{\dagger}_{j'} \ket{\Psi(t)}$ do not vanish for $\theta\neq 0,\pi/2$ and $\pi$. This is the first indicator that,
in such case, the broken symmetry is not restored at large times after the quench, as we will see in the following sections. This is the main difference with respect to the quench from the tilted ferromagnetic state reviewed in Sec. \ref{sec:ferro}.
\subsection{Useful properties of block Toeplitz matrices}
Before proceeding, we report two important properties of block Toeplitz matrices that will be useful to calculate
the charged moments from Eq.~\eqref{eq:numerics}. The determinant of that expression involves the product of the block Toeplitz matrices
$(I+\Gamma(t))e^{i\alpha_{j, j+1}n_A}$ as well as the
inverse matrix $(I-\Gamma(t))^{-1}$, which do not commute. In general, the latter is not block Toeplitz,
and the same occurs with the product of block Toeplitz matrices. Therefore, we cannot in principle apply the well-known results on the asymptotic behaviour
of block Toeplitz matrices, e.g. the Widom-Szeg\H{o} theorem~\cite{w-74} or the Fisher-Hartwig conjecture~\cite{aefq-18, fh-68, b-79}, usually
employed to study the entanglement entropy and other quantities in free fermionic systems. However, we
formulate the following conjectures on the asymptotics of the determinant of a product of block
Toeplitz matrices that may also contain the inverse of block Toeplitz matrices.
Let us denote by $T_\ell[g]$ a block Toeplitz matrix of dimension $d\cdot \ell$ with symbol
the $d\times d$ matrix $g(k)$ defined on $k\in[0, 2\pi)$. That is, the entries of $T_\ell[g]$
are the Fourier coefficients of $g(k)$,
\begin{equation}\label{eq:block_toep}
(T_\ell[g])_{l l'}=\int_{0}^{2\pi} \frac{dk}{2\pi} e^{-ik(l-l')}g(k),\quad l, l'=1,\dots,\ell.
\end{equation}
If we consider the product of $n$ different block Toeplitz matrices $T_{\ell}[g_j]$, then we conjecture
that for large $\ell$
\begin{equation}\label{eq:conj_prod}
\log\det\left[I+\prod_{j=1}^n T_\ell[g_j]\right]\sim A\ell,
\end{equation}
where the coefficient $A$ is
\begin{equation}
A=\int_{0}^{2\pi}\frac{dk}{2\pi} \log\det\left[I+\prod_{j=1}^n g_j(k)\right],
\end{equation}
provided $\det\left[I+\prod_{j=1}^n g_j(k)\right]\neq 0$. We refer the reader to Appendix~\ref{app:block_toep} for a discussion on the intuition behind this result.
The second relevant property for our computations concerns the inverse matrix $T_\ell[g]^{-1}$. In general,
the inverse of a block Toeplitz matrix is not a block Toeplitz matrix. However, we have checked numerically the following result.
If we further include in the product of matrices $T_\ell[g_j]$ the inverse $T_\ell[g_j']^{-1}$ of other
block Toeplitz matrices with invertible symbol $g_j'(k)$, i.e.
$\det[g_j'(k)]\neq 0$ for all $j$, then we conjecture that
\begin{equation}\label{eq:conj_prod_inv}
\log \det\left[I+\prod_{j=1}^n T_\ell[g_j]T_\ell[g'_j]^{-1}\right]
\sim A'\ell,
\end{equation}
where $A'$ can be calculated from
\begin{equation}
A'=\int_0^{2\pi}\frac{dk}{2\pi}\log \det\left[I+\prod_{j=1}^n
g_j(k) g_j'(k)^{-1}\right].
\end{equation}
We stress that this result holds only in the limit $\ell \to \infty$ and we have tested its validity numerically for arbitrary choices of the symbols $g_j(k), g_j'(k)$.
To derive the time evolution after the quench of the charged moments $Z_n(\boldsymbol{\alpha}, t)$
and, therefore, of the entanglement asymmetry $\Delta S_A^{(n)}(t)$, we apply the following strategy.
From the determinant of Eq.~\eqref{eq:numerics}, we can analytically deduce the asymptotic behaviour for large $\ell$ of
$Z_n(\boldsymbol{\alpha}, t)$ at $t=0$ and $t\to\infty$ using the properties~\eqref{eq:conj_prod} and \eqref{eq:conj_prod_inv} described above. With these results and applying the quasi-particle picture, we
then obtain the exact analytic expression of $Z_n(\boldsymbol{\alpha}, t)$ in the scaling
limit $t,\ell\to \infty$ with $\zeta=t/\ell$ fixed. In what follows, we
discuss in detail the case $n=2$, and then we generalise the results
to any integer $n\geq 2$.
\subsection{Calculation of the time evolution}
For $n=2$, Eq.~\eqref{eq:numerics} simplifies to
\begin{equation}\label{eq:ent_asymm_det_corr_n_2}
Z_2(\alpha, t)=\sqrt{\det\left(\frac{I+\Gamma_\alpha(t)\Gamma_{-\alpha}(t)}{2}\right)},
\end{equation}
where $\Gamma_\alpha(t)=\Gamma(t)e^{i\alpha n_A}$. If the $U(1)$ symmetry is broken,
then $\Gamma_\alpha(t)$ and $\Gamma_{-\alpha}(t)$ do not commute and, therefore,
$Z_2(\alpha, t)\neq Z_2(0,t)$. While the matrix $\Gamma_\alpha(t)$ is block Toeplitz with symbol
$\mathcal{G}_{\alpha, t}(k, \theta)=\mathcal{G}_t(k,\theta)e^{i\alpha (\sigma_z\otimes I)}$, the same is not
true for the product $\Gamma_\alpha(t)\Gamma_{-\alpha}(t)$. Therefore, we have to apply the conjecture
of Eq.~\eqref{eq:conj_prod} to determine $Z_2(\alpha, t)$ before the quench and its stationary value when $t\to\infty$.
At $t=0$, if we employ the conjecture~\eqref{eq:conj_prod} in Eq.~\eqref{eq:ent_asymm_det_corr_n_2}, we have
\begin{equation}
\log Z_2(\alpha, t=0) \sim \frac{\ell}{4} \int_0^{2\pi} \frac{dk}{2\pi}
\log\det\left[\frac{I+\mathcal{G}_{\alpha, 0}(k,\theta)\mathcal{G}_{-\alpha, 0}(k,\theta)}{2}\right].
\end{equation}
Inserting the explicit expression of the symbol $\mathcal{G}_{\alpha, 0}(k, \theta)$, which
is given in Eq.~\eqref{eq:symbol_t_Néel}, we directly find
\begin{equation}\label{eq:z_2_t_Néel_initial}
\log Z_2(\alpha, t=0) \sim \frac{\ell}{2}\int_0^{2\pi} \frac{dk}{2\pi}
\log\left[1-\sin^2\alpha\left(f_{11}(k,\theta)^2+f_{12}(k, \theta)^2\right)\right].
\end{equation}
On the other hand, to obtain the stationary value of $Z_2(\alpha, t)$ at large times, we
can average the time dependent terms in the symbol $\mathcal{G}_t(k, \theta)$ of $\Gamma(t)$, as we did in Eq.~\eqref{eq:averaged_symbol_quench_t_Néel}.
Thus employing again the conjecture of Eq.~\eqref{eq:conj_prod}, we have
\begin{equation}
\log Z_2(\alpha, t\to \infty) \sim \frac{\ell}{4} \int_0^{2\pi} \frac{dk}{2\pi}
\log\det\left[\frac{I+\mathcal{G}_{\alpha, t\to\infty }(k,\theta)\mathcal{G}_{-\alpha, t\to\infty}(k,\theta)}{2}\right].
\end{equation}
Using the time-averaged symbol of Eq.~\eqref{eq:averaged_symbol_quench_t_Néel}, we finally
get the stationary behaviour of the charged moments $Z_2(\alpha, t)$ at large times after
the quench,
\begin{equation}\label{eq:z_2_t_Néel_stationary}
\log Z_2(\alpha, t\to\infty)\sim \frac{\ell}{2}
\int_0^{2\pi}\frac{dk}{2\pi}\log\left[h_2(n_+(k,\theta)) h_2(n_-(k, \theta))-f_{11}(k,\theta)^2\sin^2\alpha\right],
\end{equation}
where $n_\pm(k,\theta)=(g_{12}(k,\theta)\pm f_{11}(k,\theta)+1)/2$, note that $n_-(k,\theta)=n_+(-k,\theta)$, and
\begin{equation}
h_n(x)=x^n+\left(1-x\right)^n.
\end{equation}
For integer $n>2$, we cannot remove the inverse matrix $(I-\Gamma(t))^{-1}$
in Eq.~\eqref{eq:numerics}, as happened for $n=2$, c.f. Eq.~\eqref{eq:ent_asymm_det_corr_n_2}. Therefore, one may
resort to the conjecture of Eq.~\eqref{eq:conj_prod_inv} to derive the asympotic behaviour
of $Z_n(\boldsymbol{\alpha}, t)$ at the initial time and its stationary value
after the quench. Unfortunately, the symbol $I-\mathcal{G}_t(k,\theta)$ of the matrix $I-\Gamma(t)$ at $t=0$
is not invertible and we cannot apply Eq.~\eqref{eq:conj_prod_inv} in that case. Nevertheless,
since the R\'enyi entanglement asymmetries of the tilted Néel and
ferromagnetic states are equal, we expect that the charged moments
of the tilted Néel state take a very similar form as the ones of Eq.~\eqref{eq:z_n_t_ferro_evolution} for the tilted ferromagnetic configuration; that is,
\begin{equation}\label{eq:z_n_t_Néel_initial}
\log Z_n(\boldsymbol{\alpha}, t=0)\sim \frac{\ell}{2} \int_0^{2\pi} \frac{dk}{2\pi}
\log \prod_{j=1}^n f_k^{{\rm N}}(\theta, \alpha_{j, j+1}).
\end{equation}
The function $f_k^{{\rm N}}(\theta, \alpha)$ can be straightforwardly
deduced from the case $n=2$ analysed before, see Eq.~\eqref{eq:z_2_t_Néel_initial},
\begin{equation}
f_k^{\rm N}(\theta, \alpha)=1+m_{{\rm N}}(k, \theta)\sin(\alpha),
\end{equation}
where
\begin{equation}
m_{{\rm N}}(k,\theta)=\sqrt{f_{11}(k,\theta)^2+f_{12}(k,\theta)^2}=
\frac{\sin(k/2) \sin^2(\theta) \sqrt{4\cos^2(k/2) \cos^2\theta + \sin^4\theta}}{1 + 2 \cos(k) \cos^2\theta +
\cos^4\theta}.
\end{equation}
On the other hand, to calculate the stationary value of $Z_n(\boldsymbol{\alpha}, t)$
at $t\to\infty$, we can take the time average of the correlation $\Gamma(t)$, whose symbol
$\mathcal{G}_{t\to\infty}(k)$ was obtained in Eq.~\eqref{eq:averaged_symbol_quench_t_Néel}. It turns out that the time-averaged
symbol $I-\mathcal{G}_{t\to\infty}(k,\theta)$ of the matrix $I-\Gamma(t)$ --- whose inverse
enters in the determinant formula~\eqref{eq:numerics} for $Z_n(\boldsymbol{\alpha}, t)$--- is invertible.
This means that, in the large time limit, we can apply the conjecture \eqref{eq:conj_prod_inv} to Eq. \eqref{eq:numerics},
\begin{equation}
\log Z_n(\boldsymbol{\alpha}, t\to\infty)\sim
\frac{\ell}{4}\int_0^{2\pi} \frac{dk}{2\pi}\log \det\left[\left(\frac{I-\mathcal{G}_{t\to\infty}(k)}{2}\right)^n
\left(I+\prod_{j=1}^n\mathcal{W}_j(k)\right)\right],\end{equation}
where $\mathcal{W}_j(k)$ is the $4\times 4$ symbol
$\mathcal{W}_j(k)=(I+\mathcal{G}_{t\to\infty}(k))(I-\mathcal{G}_{t\to\infty}(k))^{-1}e^{i\alpha_{j, j+1}(\sigma_z\otimes I)}$.
Plugging the explicit expression \eqref{eq:averaged_symbol_quench_t_Néel} of $\mathcal{G}_{t\to\infty}(k)$ and calculating
directly the determinant, we find
\begin{equation}\label{eq:z_n_t_Néel_stationary}
\log Z_n(\boldsymbol{\alpha}, t\to\infty)\sim
\frac{\ell}{2} \int_0^{2\pi}\frac{dk}{2\pi} \log\left[h_n(n_+(k, \theta)) h_n(n_-(k, \theta))-f_{11}(k, \theta)^2 \tilde{h}_n(\boldsymbol{\alpha}, k, \theta)\right],
\end{equation}
where
\begin{multline}
\tilde{h}_n(\boldsymbol{\alpha}, k, \theta)=\sum_{j=1}^{\frac{n-{\rm mod}(n, 2)}{2}}
\frac{f_{11}(k, \theta)^{2j-2}(n_+(k,\theta)+n_-(k,\theta)-2n_+(k,\theta)n_-(k,\theta))^{n-2j}}{2^{n-2}}\\
\times\sum_{1\leq p_1< p_2<\cdots<p_{2j}\leq n} \sin^2\left(\alpha_{p_1}-\alpha_{p_2}+\cdots-\alpha_{p_{2j}}\right).
\end{multline}
This result agrees with the one obtained for the
case $n=2$ in Eq.~\eqref{eq:z_2_t_Néel_stationary}.
Moreover, when $\boldsymbol{\alpha}=\boldsymbol{0}$, we must recover the
stationary value of the R\'enyi entanglement entropies.
For free systems, it is expected that, at large times after
the quench, the R\'enyi entanglement entropy behaves as~\cite{FC-08}
\begin{equation}\label{eq:renyi_n_stationary}
S^{(n)}(\rho_A(t\to\infty))=\frac{1}{1-n}\log Z_n(\boldsymbol{0}, t\to\infty) \sim
\frac{\ell}{1-n} \int_0^{2\pi} \frac{d k}{2\pi} \log[h_n(n(k))],
\end{equation}
where $n(k)$ is the density of occupied modes in the post-quench stationary state. In fact, it is easy to check that
Eq.~\eqref{eq:z_n_t_Néel_stationary} leads to
Eq.~\eqref{eq:renyi_n_stationary} with $n(k)=n_+(k, \theta)$,
see also Sec.~\ref{sec:gge}.
Once we have the expression of the charged moments $Z_n(\boldsymbol{\alpha}, t)$ at
the initial time and its stationary behaviour at large times, Eqs.~\eqref{eq:z_n_t_Néel_initial} and \eqref{eq:z_n_t_Néel_stationary},
we can
exploit the quasi-particle picture of entanglement to reconstruct its full
time evolution~\cite{cc-05, ac-17, ac-18}. According to it, the quench creates quasi-particle excitations,
in particular pairs of entangled quasi-particles emitted from the same point
that propagate ballistically in opposite directions with momentum $\pm k$.
Therefore, the entanglement generated after the quench is proportional to the pairs
of entangled quasi-particles produced in the quench that are shared by the
subsystem $A$ and its complementary $B$. Then the integrand of Eq.~\eqref{eq:renyi_n_stationary} is the contribution to the R\'enyi entropy of
a pair of entangled excitations with momentum $k$. Since the quasi-particles propagate
at a finite velocity $v_k=|\epsilon'_{k/2}|$, the number of entangled pairs of excitations
with momentum $k$ shared by $A$ and $B$ at time $t$ is given by $\min(2tv_k, \ell)$.
Then one finds~\cite{FC-08}
\begin{equation}\label{eq:renyi_n_evolution}
S^{(n)}(\rho_A(t)) \sim
\frac{\ell}{1-n} \int_0^{2\pi}\frac{d k}{2\pi} \min(2\zeta v_k, 1) \log[h_n(n_+(k,\theta))].
\end{equation}
The same idea can be applied to deduce the time evolution of the charged moments $Z_n(\boldsymbol{\alpha}, t)$. In this case,
we have that $\log Z_n(\boldsymbol{\alpha}, t)$ does not vanish at $t=0$. Therefore, if we consider the
difference between $\log Z_n(\boldsymbol{\alpha}, t)$ at $t=0$ and $t\to \infty$, obtained in Eqs.~\eqref{eq:z_n_t_Néel_initial} and \eqref{eq:z_n_t_Néel_stationary} respectively,
\begin{equation}
\log\left(\frac{Z_n(\boldsymbol{\alpha},t\to\infty)}{Z_n(\boldsymbol{\alpha}, t=0)}\right)\sim
\frac{\ell}{2}\int_0^{2\pi}\frac{dk}{2\pi}\log\left[\frac{h_n(n_+(k,\theta)) h_n(n_-(k, \theta))
-f_{11}(k,\theta)^2 \tilde{h}_n(\boldsymbol{\alpha}, k,\theta)}{\prod_{j=1}^n(1+m_{{\rm N}}(k,\theta)\sin\alpha_{j,j+1})}\right],
\end{equation}
then, in the light of the quasi-particle picture, the integrand of this expression
can be interpreted as the contribution of an entangled pair of excitations with
momentum $\pm k$ created in the quench. Counting the number of such pairs shared
between $A$ and $B$ at time $t$, as we have done for the R\'enyi entanglement entropy, one
can conclude that
\begin{multline}\label{eq:chargedn}
\log\left(\frac{Z_n(\boldsymbol{\alpha},t)}{Z_n(\boldsymbol{\alpha}, t=0)}\right)\sim\\
\frac{\ell}{2}\int_0^{2\pi}\frac{dk}{2\pi}\min(2\zeta v_k, 1)\log\left[\frac{h_n(n_+(k,\theta)) h_n(n_-(k, \theta))
-f_{11}(k,\theta)^2 \tilde{h}_n(\boldsymbol{\alpha}, k,\theta)}{\prod_{j=1}^n(1+m_{{\rm N}}(k,\theta)\sin\alpha_{j,j+1})}\right].
\end{multline}
Observe that, when we take $\boldsymbol{\alpha}=\boldsymbol{0}$, this result agrees
with the time evolution of the R\'enyi entanglement entropy reported in Eq.~\eqref{eq:renyi_n_evolution}.
In conclusion, we obtain that, in the scaling limit $t,\ell\to\infty$ with $\zeta=t/\ell$ finite, the
exact time evolution of the charged moments $Z_n(\boldsymbol{\alpha}, t)$ after the quench from
the tilted Néel state is
\begin{equation}\label{eq:z_n_t_Néel_evolution}
Z_n(\boldsymbol{\alpha}, t)=Z_n(\boldsymbol{0}, t) e^{\ell(A_n(\boldsymbol{\alpha})+B_n(\boldsymbol{\alpha})+B_n'(\boldsymbol{\alpha},\zeta))},
\end{equation}
where
\begin{equation}
A_n(\boldsymbol{\alpha})=\int_0^{2\pi} \frac{dk}{4\pi} \log\prod_{j=1}^n \left[1+m_{{\rm N}}(k,\theta)\sin(\alpha_{j, j+1})\right],
\end{equation}
\begin{equation}
B_n(\boldsymbol{\alpha},\zeta)=-\int_0^{2\pi} \frac{dk}{4\pi}\min(2\zeta v_k, 1)
\log\prod_{j=1}^n \left[1+m_{{\rm N}}(k,\theta)\sin(\alpha_{j, j+1})\right],
\end{equation}
and
\begin{equation}
B_n'(\boldsymbol{\alpha},\zeta)=\int_0^{2\pi}\frac{dk}{4\pi}
\min(2\zeta v_k, 1)\log\left[1-\frac{f_{11}(k, \theta)^2\tilde{h}_n(\boldsymbol{\alpha}, k,\theta)}{h_n(n_+(k,\theta))h_n(n_-(k,\theta))}\right].
\end{equation}
It is interesting to compare the result of Eq.~\eqref{eq:z_n_t_Néel_evolution}
with the one of Eq.~\eqref{eq:z_n_t_ferro_evolution} for a chain initially prepared in the tilted ferromagnetic state. The terms $A_n(\boldsymbol{\alpha})$ and
$B_n(\boldsymbol{\alpha}, \zeta)$ display the same behaviour as
in the tilted ferromagnet and $B_n(\boldsymbol{\alpha},\zeta)\to-A_n(\boldsymbol{\alpha})$ when $\zeta\to\infty$. However, the most remarkable
difference is the appearance of the extra
term $B_n'(\boldsymbol{\alpha},\zeta)$, which in general does
not vanish in the limit $\zeta\to \infty$. Therefore, for the tilted Néel state, $Z_n(\boldsymbol{\alpha}, t)$ does not tend to the neutral moments $Z_n(\boldsymbol{0}, t)$ at large times; except if $\theta=\pi/2$, for which $f_{11}(k, \pi/2)=0$ and the term $B_n'(\boldsymbol{\alpha}, \zeta)$ cancels for any $\zeta$.
As we discuss in the following sections, this fact implies that the entanglement asymmetry
$\Delta S_A^{(n)}(t)$ does not cancel when $t\to \infty$, which indicates that the $U(1)$ symmetry is not restored if $\theta\neq \pi/2$.
In Fig.~\ref{fig:charged}, we check numerically the time evolution of the charged moments
predicted by Eq.~\eqref{eq:z_n_t_Néel_evolution} in the cases $n=2$ and $n=3$. The points are
the exact numerical values of $Z_n(\boldsymbol{\alpha}, t)$ computed using
the determinant formula of Eq.~\eqref{eq:numerics}, while the solid curves correspond to Eq.~\eqref{eq:z_n_t_Néel_evolution}.
We obtain an excellent agreement.
\begin{figure}[t]
\centering
{\includegraphics[width=0.49\textwidth]{fcharged_n_2.pdf}}
{\includegraphics[width=0.49\textwidth]{fcharged_n_3_2.pdf}}
\caption{Time evolution of the charged moments $Z_n(\boldsymbol{\alpha}, t)$ as a function of $t/\ell$ for $n=2$ (left panel) and $n=3$ (right panel) after the quench from the tilted Néel state. The curves represent the quasi-particle prediction of Eq.~\eqref{eq:z_n_t_Néel_evolution} for several tilting angles $\theta$
and $\alpha_{j, j+1}$. The points are the exact numerical values of the charged moments obtained using Eq.~\eqref{eq:numerics} and taking different subsystem lengths $\ell$.}
\label{fig:charged}
\end{figure}
\section{Entanglement asymmetry after the quench from the tilted Néel state}\label{sec:ent_asymmetry}
In this section, we employ the results for the charged moments previously obtained to analyse the time evolution of the R\'enyi entanglement asymmetry~\eqref{eq:replicatrick} after the quench~\eqref{eq:quench} from the tilted Néel state.
Combining Eqs.~\eqref{eq:replicatrick}~and~\eqref{eq:FT}, the entanglement asymmetry $\Delta S_A^{(n)}(t)$ can be derived from the Fourier transform of the charged moments $Z_n(\boldsymbol{\alpha}, t)$. In Fig.~\ref{fig:ent_asymm}, the solid curves represent the resulting entanglement asymmetry for different tilting angles $\theta$ when we insert the analytic
expression~\eqref{eq:z_n_t_Néel_evolution}
for $Z_n(\boldsymbol{\alpha}, t)$ in Eq.~\eqref{eq:FT}, which is exact for any $\zeta=t/\ell$ in the limit $t,\ell\to\infty$. The symbols
in the plots correspond to the exact numerical values of the asymmetry, showing a very good agreement with the quasi-particle
prediction.
As clear from Fig.~\ref{fig:ent_asymm}, the most remarkable feature of the dynamics of the
entanglement asymmetry after the quench is
that it does not vanish when $t\to\infty$, but it saturates to a value that depends on the tilting angle $\theta$. The exception
is the case $\theta=\pi/2$, in which $\Delta S_A^{(n)}(t)$ does tend to zero at large times. In other words, the $U(1)$
symmetry associated to the transverse magnetisation $Q$ is not restored after
the quench, unless the symmetry is initially maximally broken, i.e. when
$\theta=\pi/2$.
\begin{figure}[t]
\centering
{\includegraphics[width=0.49\textwidth]{ent_asymm_quench_tilted_neel_diff_l.pdf}}
{\includegraphics[width=0.49\textwidth]{ent_asymm_quench_tilted_neel.pdf}}
\caption{Time evolution of the R\'enyi entanglement asymmetry after the quench~\eqref{eq:quench} from the tilted Néel state.
In the left panel, we fix the initial tilting angle to
$\theta=\pi/3$ and we consider several subsystem sizes.
In the right panel, we take different tilting angles $\theta$ and R\'enyi index $n$ for the same subsystem size $\ell=100$.
The curves have been obtained by plugging in Eq.~\eqref{eq:replicatrick} the Fourier transform \eqref{eq:FT} of our quasi-particle prediction \eqref{eq:z_n_t_Néel_evolution} for $Z_n(\boldsymbol{\alpha}, t)$. The symbols are the exact numerical values.}
\label{fig:ent_asymm}
\end{figure}
The initial and the stationary value of
$\Delta S_A^{(n)}(t)$ for large subsystem sizes $\ell$ can be determined from the analytic expression \eqref{eq:z_n_t_Néel_evolution} of the charged moments as follows.
In these two cases, once we plug Eq.~\eqref{eq:z_n_t_Néel_evolution} into Eq.~\eqref{eq:FT}, we can solve in the large
$\ell$ limit the multi-dimensional integral using the saddle point approximation.
In principle, this would require to find the solutions $\boldsymbol{\alpha}^*$
of
\begin{equation}
\nabla_{\boldsymbol{\alpha}}[A_n(\boldsymbol{\alpha})+B_n(\boldsymbol{\alpha}, \zeta)
+B_n'(\boldsymbol{\alpha},\zeta)]=0,
\end{equation}
at $\zeta=0$ and $\zeta\to\infty$ respectively, and verify that they are maxima of $Z_n(\boldsymbol{\alpha}, t)$ in each case. Since the solution to this equation is not an easy goal to pursue,
in order to fix the ideas, we focus on the case $n=2$.
At $t=0$, we can employ the expression for the charged moments obtained in Eq.~\eqref{eq:z_n_t_Néel_initial}.
For $n=2$, if $\alpha\in [-\pi, \pi]$, the function $Z_2(\alpha, 0)$ has saddle points
at $\alpha^*=0, \pi$. Performing a series expansion around them up to quadratic order,
we obtain
\begin{equation}\label{eq:saddle_n_2}
\log Z_2(\alpha, t=0)= \log Z_2(\alpha^*, t=0) - \frac{(\alpha-\alpha^*)^2}{2}\ell g_0(\theta)+O((\alpha-\alpha^*)^4),
\end{equation}
where
\begin{equation}
g_0(\theta)=\int_0^{2\pi} \frac{dk}{2\pi} (f_{11}(k, \theta)^2+f_{12}(k,\theta)^2).
\end{equation}
By doing the Fourier transform~\eqref{eq:FT} of the quadratic approximations of Eq.~\eqref{eq:saddle_n_2} around each saddle
point, we get
\begin{equation}
\mathrm{Tr}(\rho_{A, Q}^2(t=0))=\frac{\mathrm{Tr}(\rho_A^2(t=0))}{\sqrt{\pi\ell g_0(\theta)/2}}+O(\ell^{-3/2})
\end{equation}
and, plugging it into Eq.~\eqref{eq:replicatrick}, we obtain that the $n=2$ entanglement asymmetry behaves at
time $t=0$ as
\begin{equation}
\Delta S_A^{(2)}(t=0)=\frac{1}{2}\log \ell +\frac{1}{2}\log \frac{\pi g_0(\theta)}{2}+O(\ell^{-1}).
\end{equation}
For larger integer values of $n$, one can follow a similar reasoning. We then find
\begin{equation}\label{eq:ent_asymm_t_0}
\Delta S_A^{(n)}(t=0)=\frac{1}{2}\log \ell + \frac{1}{2} \log \frac{\pi n^{1/(n-1)} g_0(\theta)}{4}.
\end{equation}
It is interesting to note that the term $g_0(\theta)$ is given by the square of the eigenvalues
of the symbol $\mathcal{F}_t(k,\theta)$ at $t=0$ that generates the correlations
$\bra{\Psi(0)} c_j^\dagger c_{j'}^\dagger\ket{\Psi(0)}$, see Eq.~\eqref{eq:sym2}.
In the large $t$ limit, we can repeat the same steps. Now we can use the stationary value
of the charged moments provided by Eq.~\eqref{eq:z_n_t_Néel_stationary}. If $n=2$, this function presents two saddle points,
at $\alpha^*=0, \pi$, whose leading contributions in $\alpha$ read
\begin{equation}\label{eq:fgamma}
\log Z_2(\alpha, t\to\infty)=\log Z_2(\alpha^*, t\to\infty)- \frac{(\alpha-\alpha^*)^2}{2}\ell g_\infty^{(2)}(\theta)+O((\alpha-\alpha^*)^4),
\end{equation}
where
\begin{equation}\label{eq:g_infty_2}
g_\infty^{(2)}(\theta)=\int_0^{2\pi}\frac{dk}{2\pi} \frac{f_{11}(k,\theta)^2}{h_2(n_+(k,\theta))h_2(n_-(k,\theta))}.
\end{equation}
Notice that this expansion is analogous to the one of Eq.~\eqref{eq:saddle_n_2} for
$t=0$. Then we can directly conclude that the stationary $n=2$ entanglement asymmetry after the quench has the form
\begin{equation}\label{eq:Deltasp}
\Delta S_A^{(2)}(t\to\infty)=\frac{1}{2}\log \ell+\frac{1}{2} \log \frac{\pi g_\infty^{(2)}(\theta)}{2}+O(\ell^{-1}).
\end{equation}
By repeating similar steps, we can also obtain an analytical prediction for the stationary value of the R\'enyi entanglement
asymmetry for larger integer $n$, which is given by
\begin{equation}\label{eq:n3}
\Delta S_A^{(n)}(t\to\infty)=\frac{1}{2}\log\ell+
\frac{1}{2}\log\frac{\pi n^{1/(n-1)}g_\infty^{(n)}(\theta)}{4}+O(\ell^{-1}).
\end{equation}
As at initial time, the stationary entanglement asymmetry grows
logarithmically with the subsystem length $\ell$. The $\ell$-independent term is instead different and, moreover, it
shows a non-trivial dependence on the R\'enyi index $n$; for example, for $n=3$,
\begin{equation}\label{eq:g_infty_3}
g_\infty^{(3)}(\theta)=\int_0^{2\pi}\frac{dk}{2\pi}
\frac{f_{11}(k,\theta)^2(n_+(k,\theta)+n_-(k,\theta)-2n_+(k,\theta)n_-(k,\theta))}
{h_3(n_+(k, \theta))h_3(n_-(k,\theta))}.
\end{equation}
Although for any integer $n$ we can reconstruct an expression for $g_\infty^{(n)}$, it gets more and more cumbersome as $n$ increases and a closed analytic form cannot be obtained.
\begin{figure}[t]
\centering
{\includegraphics[width=0.49\textwidth]{ent_asymm_t_neel_time_0.pdf}}
\subfigure
{\includegraphics[width=0.49\textwidth]{S_n_vstheta.pdf}}
\caption{R\'enyi entanglement asymmetry
as a function of the tilting angle $\theta\in[0,\pi]$
at initial time (left) and at a fixed large time after the quench (right). We consider different R\'enyi index $n$ and
subsystem lengths $\ell$.
The black curves in the left panel correspond to the asymptotic expression
of the entanglement asymmetry at $t=0$ obtained in Eq.~\eqref{eq:ent_asymm_t_0} while in the right panel they are the
prediction~\eqref{eq:Deltasp}
for its stationary value with $g_\infty^{(n)}$ given by Eq.~\eqref{eq:g_infty_2} for $n=2$ and by Eq.~\eqref{eq:g_infty_3} for $n=3$. The symbols are the exact numerical values.}
\label{fig:ent_asymm_vsthetal}
\end{figure}
In Fig.~\ref{fig:ent_asymm_vsthetal}, we
report the profile of the entanglement
asymmetry for $n=2,3$ in terms of the tilting
angle $\theta$ both before (left panel) and at large times after the quench (right panel).
The symbols indicate the exact numerical value
and the solid curves represent the expressions
for large $\ell$ found in
Eq.~\eqref{eq:ent_asymm_t_0} for $t=0$ (left
panel) and in Eq.~\eqref{eq:n3} for the
stationary regime (right panel).
The agreement between the curves and the numerical points worsens as $\Delta S_A^{(n)}$ tends to zero, i.e. when the symmetry is restored. As we have already
discussed, this happens for $\theta=0, \pi$ at
$t=0$ and for $\theta=0,\pi/2, \pi$ when $t\to\infty$. This can be understood also from our analytical prediction in Eqs. \eqref{eq:ent_asymm_t_0} and \eqref{eq:n3}. The functions $g_0(\theta)$ and $g_\infty^{(n)}(\theta)$ vanish at $\theta=0,\pi$
and $\theta=0,\pi/2,\pi$ respectively and Eqs. \eqref{eq:ent_asymm_t_0}, \eqref{eq:n3}
are not well defined when the symmetry is
respected. In fact, the limits $\ell\to \infty$ and $\theta \to 0, \pi/2, \pi$ do not commute: to properly get $\Delta S_A^{(n)}=0$ when the symmetry is recovered, one should fix first the value of $\theta$ in Eq. \eqref{eq:z_n_t_Néel_evolution} and then consider the large $\ell$ regime.
In order to verify the logarithmic behaviour in
$\ell$ of Eq.~\eqref{eq:n3}, we compare it with the exact numerical results in Fig.~\ref{fig:ent_asymm_vs_l} by varying the interval length. As the subsystem size increases, we observe that the agreement between the numerics and our theoretical predictions improves, despite we are using finite values both for $t$ and $\ell$ and our predictions are valid in the scaling limit $t,\ell \to \infty$ with $t/\ell$ finite.
\begin{figure}[t]
{\includegraphics[width=0.49\textwidth]{S_n_vsl_t200.pdf}}
\centering
{\includegraphics[width=0.49\textwidth]{S_n_vsl_t200log.pdf}}
\caption{R\'enyi entanglement asymmetry at a large time after the quench~\eqref{eq:quench} from the tilted Néel state as a function of the subsystem size $\ell$ (left panel) and $\log \ell$ (right panel) for different R\'enyi index $n$ and initial tilting angle $\theta$.
The black curves correspond to the prediction~\eqref{eq:Deltasp} for its stationary value with $g_\infty^{(n)}$ given by Eq.~\eqref{eq:g_infty_2} for $n=2$ and by Eq.~\eqref{eq:g_infty_3} for $n=3$ while the symbols are the exact numerical values.}
\label{fig:ent_asymm_vs_l}
\end{figure}
\section{Description in terms of the post-quench stationary state}\label{sec:gge}
The quasi-particle picture employed in the previous sections makes use of the density $n(k)$ of occupied modes in the post-quench stationary state. In free or integrable models, the latter can be obtained by a description of the stationary state in terms of a Generalised Gibbs Ensemble (GGE), taking account all of the conserved local or quasilocal charges.
As we will see now, the situation here is particularly subtle, due to the presence of a non-abelian set of conserved charges for the Hamiltonian \eqref{eq:xx}, a feature also known as {\it superintegrability} \cite{superintegrability}. Non-abelian charges also found application in other contexts such as quantum thermodynamics~\cite{hfow-16, hbk-20, kranzl-22, hm-22, mlhh-23} and time crystals~\cite{mbj-20, mpz-20}.
The charges commuting with the Hamiltonian \eqref{eq:xx} can be split in four families, which we write in the thermodynamic limit as \cite{fagotti,bertinifagotti,vernierZn}
\begin{eqnarray}
H_m &=& i \sum_j \left(e^{i \frac{m \pi}{2}} c_j^\dagger c_{j+m} - e^{-i \frac{m \pi}{2}} c_{j+m}^\dagger c_{j} \right)
= \int_{0}^{2\pi} \frac{dk}{2\pi} \sin\left(m \left(\frac{\pi}{2}- k \right)\right) c_k^\dagger c_{k},
\label{eq:Hm}
\\
Z_m &=& \sum_j \left( e^{i \frac{m \pi}{2}} c_j^\dagger c_{j+m} + e^{-i \frac{m \pi}{2}} c_{j+m}^\dagger c_{j} \right)
= \int_{0}^{2\pi} \frac{dk}{2\pi} \cos\left(m \left(\frac{\pi}{2}- k \right)\right) c_k^\dagger c_{k},
\label{eq:Zm}
\\
Y_m^\dagger &=& \sum_j (-1)^{j+1} c_j^\dagger c_{j+m}^\dagger
= \int_{0}^{2\pi} \frac{dk}{2\pi} e^{i k m} c_k^{\dagger} c_{\pi-k}^\dagger,
\\
Y_m &=& \sum_j (-1)^j c_j c_{j+m}
= \int_{0}^{2\pi} \frac{dk}{2\pi} e^{-i k m} c_{\pi-k} c_{k},
\end{eqnarray}
(for a system of finite size $L$ with periodic boundary conditions fermion bilinears with $j\leq L < j+m$ come with a different prefactor depending on the global charge $Q$, but we will not need to worry about this here). Note in particular that $H_1$ is (proportional to) the Hamiltonian $H$ and $Z_0$ is the number operator generating the $U(1)$ symmetry.
The charges $H_m$ commute with all others, as is made clear by the Fourier transform to momentum space: since they have a dispersion relation which is odd under $k\to \pi - k$, fermions of momenta $k$ and $\pi-k$ come with opposite $H_m$-eigenvalue, and can therefore be created or destroyed in pairs without affecting the $H_m$ charges \cite{vernierZn}. In contrast, the charges $Z_m$, $Y_m$ and $Y_m^\dagger$ form three commuting families, but do not commute with one another (we note in passing that the same structure of non-commuting conserved charges can be found for interacting models, namely for the XXZ models at the ``root-of-unity'' points $\Delta=\cos\frac{\pi l}{m}$, $m,l \in \mathbb{Z}^*$ \cite{korff1,korff2,lenart,ilievskireview,miao};
historically, these charges appeared first in the field theory literature \cite{bl-91,l-93}).
Another important point is that the charges $H_m$ and $Z_m$ commute with the $U(1)$ charge $Q=Z_0$, while $Y_m$ and $Y_m^\dagger$ do not. Let us now see how these charges affect the GGE description of the post-quench relaxation.
\subsection{Abelian case}
Let us start by briefly reviewing the case of models with a commuting family of conserved charges $\{H_m\}$, as happens for instance in the XXZ chain with generic interaction parameter $\Delta$.
It is now well-established in such cases that the late-time steady state following a quantum quench should be described by a GGE entirely specified by the expectation values of all local or quasilocal conserved charges \cite{rigol2007relaxation,vidmar2016generalized,essler2016quench}.
Consequently, local observables relax at late time to
\begin{equation}
\lim_{t\to \infty} \lim_{L\to \infty} \langle \Psi(0) |e^{-i H t} \mathcal{O}_j e^{i H t} | \Psi(0) \rangle
=
\mathrm{Tr} \left( \rho_{\rm GGE} \mathcal{O}_j \right)
=
\langle \Psi_{\rm GGE}| \mathcal{O}_j | \Psi_{\rm GGE} \rangle\,,
\end{equation}
where $\mathcal{O}_j$ is an operator localized around position $j$, $\rho_{\rm GGE}$ is a density matrix of the form $\rho_{\rm GGE} \propto \exp(- \sum_m \beta_m H_m)$ where all the Lagrange multipliers $\beta_m$ are characterised by the expectation values of (quasi)local conserved charges, and the last equality states the equivalence of the latter with a {\it representative state}, namely an eigenstate of all the conserved charges characterised by their expectation values on the initial state \cite{quenchaction,ilievski,pvc-16}.
In Bethe ansatz-integrable models, the representative state is described by a set of densities associated with the various types of quasi-particles, which reduce in the XX chain/free fermionic case to the mode occupation function
\begin{equation}
n(k) = \langle \Psi(0) |c_k^\dagger c_k | \Psi(0) \rangle \,.
\label{eq:modeoccup}
\end{equation}
\subsection{Non-abelian case} We now investigate the effect of the additional non-abelian set of conserved charges in the XX model, namely $\{ Z_m\}, \{ Y_m\}, \{ Y_m^\dagger\}$, in addition to the usual $\{H_m\}$. We point out that related questions have been addressed in \cite{fagotti,bertinifagotti}, where the focus was however on a weak perturbation breaking the non-abelian symmetry; here instead no perturbation is introduced, and the symmetry is exact.
The non-abelian symmetry splits the spectrum of the charges $H_m$ into degenerate subspaces, where the charges $Z_m, Y_m$ and $Y_m^\dagger$ act non-commutatively \cite{vernierZn}.
Therefore, specifying the expectation values of the charges $H_m$ does not uniquely fix a representative state, as can be seen more directly by observing that Bogoliubov rotations of the form
\begin{equation}
\left(
\begin{array}{c}
c_k \\ c_{\pi-k}^\dagger
\end{array}
\right)
\longrightarrow
\left(
\begin{array}{cc}
\cos r_k & - e^{-i \varphi_k}\sin r_k \\ e^{i \varphi_k}\sin r_k & \cos r_k
\end{array}
\right)
\left(
\begin{array}{c}
c_k \\ c_{\pi-k}^\dagger
\end{array}
\right) \,, \qquad
r_k\,, \varphi_k \in [0,2\pi]
\label{eq:bogo}
\end{equation}
leave the charges $H_m$ invariant, while changing the mode occupation function \eqref{eq:modeoccup}. Alternatively, we can view those transformations as a rotation under the unitary operator
\begin{equation}
\mathcal{U}_k = \exp\left(i r_k (e^{i \varphi_k} c_k^\dagger c_{\pi-k}^\dagger + e^{-i \varphi_k} c_{\pi-k} c_k ) \right) \,.
\label{eq:rotation}
\end{equation}
A complete GGE should therefore include, in addition to the charges $H_m$, a maximal abelian subset of the remaining charges. Different choices of complete GGEs are conjugated to one another by arbitrary products of rotations of the type \eqref{eq:rotation}, and, as we will see now, which choice is to be made crucially depends on the quench protocol under consideration.
\paragraph{Quench from the tilted ferromagnetic state}
It is instructive to start by revisiting the case of a quench from the tilted ferromagnetic state, recently considered in \cite{amc22}, see also Section \ref{sec:ferro}. There it was shown that the $U(1)$ symmetry generated by $Q$ is restored at large time. This can be viewed as a consequence of the fact that the charges $Y_m$ and $Y_m^\dagger$ have zero expectation value, as they are odd under translation while the initial state is translationally invariant.
It is then natural to look for a GGE built out of the maximal set of charges commuting with $Q$, namely, the $\{H_m\}$ and $\{Z_m\}$ charges (this is the same type of GGE as what has been considered in \cite{deluca}, in an interacting setup). As is clear from their expression \eqref{eq:Hm}, \eqref{eq:Zm} in terms of the fermionic mode operators, specifying the expectation of those charges amounts to specifying the mode occupation numbers \eqref{eq:modeoccup}, which are found to be \cite{amc22}
\begin{equation}
2 n(k) +1
= \frac{2 \cos \theta - (1+(\cos \theta)^2) \cos k }{1-2 \cos \theta \cos k + (\cos \theta)^2} \equiv \cos\Delta_k \,.
\end{equation}
Indeed, plugging the above expression in the quasi-particle picture for the entanglement asymmetry yields the correct result.
\paragraph{Quench from the tilted Néel state}
We now turn back to the case of a quench from the tilted Néel state, which is the main focus of this paper.
As for the tilted ferromagnet, it is easy to compute the mode occupation numbers by evaluating the various bilinear combinations of fermionic operators in the initial state (see also \eqref{eq:symbol}), yielding:
\begin{equation}
\langle \Psi(0) | c_k^\dagger c_k | \Psi(0) \rangle
= \frac{1}{2} \left( 1+ \frac{(1- (\cos \theta)^4)\cos k}{1+2 (\cos \theta)^2 \cos 2 k + (\cos \theta)^4} \right) \,.
\end{equation}
However, plugging the resulting mode occupation numbers $n(k)$ into equation \eqref{eq:renyi_n_stationary} for the stationary value of the Rényi entanglement entropy does not recover the result obtained from numerics or from the calculations presented in the previous sections.
The reason for this is that in the present case the $U(1)$ symmetry generated by $Q$ is not restored, in other terms the charges $Y_m^{(\dagger)}$ have a non-zero expectation value, as can be seen by computing the off-diagonal conserved charges
\begin{eqnarray}
\label{eq:offdiagmodes}
\langle \Psi(0) | c_{\pi-k} c_k | \Psi(0) \rangle
=
-\langle \Psi(0) | c_k^\dagger c_{\pi-k}^\dagger | \Psi(0) \rangle
&=& \frac{i}{2} \frac{ (\sin \theta)^2 \cos \theta \sin 2k}{1+2 (\cos \theta)^2 \cos 2 k + (\cos \theta)^4}
\,.
\end{eqnarray}
Let us however make the following observation: applying to all values of $k$ a rotation of the form \eqref{eq:bogo}, \eqref{eq:rotation} with $r_k = \frac{\pi}{4}$, $\varphi_k = \frac{\pi}{2}$, namely defining the rotated fermionic operators
\begin{equation}
\widetilde{c}_k = \frac{c_k + i c_{\pi-k}^\dagger}{\sqrt{2}} \,, \qquad
\widetilde{c}_{\pi-k}^\dagger = \frac{c_{\pi-k}^\dagger + i c_{k}}{\sqrt{2}}
\,,
\end{equation}
we have
\begin{eqnarray}
\label{eq:cctilde}
\langle \Psi(0) | \widetilde{c}_{\pi-k} \widetilde{c}_k| \Psi(0) \rangle
= \frac{1}{2} \langle \Psi(0) | \left(
{c}_{\pi-k} {c}_k
+ {c}^\dagger_{k} {c}^\dagger_{\pi-k}
+ i ( {c}_{\pi-k} {c}_{\pi-k}^\dagger
- {c}_{k}^\dagger {c}_{k} )
\right)| \Psi(0) \rangle
= 0 \,, \\
\label{eq:cdcdtilde}
\langle \Psi(0) | \widetilde{c}_{\pi-k}^\dagger \widetilde{c}_k^\dagger| \Psi(0) \rangle
= \frac{1}{2} \langle \Psi(0) | \left(
{c}^\dagger_{\pi-k} {c}^\dagger_k
+ {c}_{k} {c}_{\pi-k}
+ i ( {c}^\dagger_{\pi-k} {c}_{\pi-k}
- {c}_{k} {c}_{k}^\dagger )
\right)| \Psi(0) \rangle
= 0 \,,
\end{eqnarray}
as resulting from \eqref{eq:offdiagmodes} and from the fact that
\begin{equation}
\langle \Psi(0) | c^\dagger_k c_k |\Psi(0) \rangle
= \langle \Psi(0) | c_{\pi-k} c_{\pi-k}^\dagger| \Psi(0) \rangle \,.
\label{symmetryNeel}
\end{equation}
What Eqs. \eqref{eq:cctilde} and \eqref{eq:cdcdtilde} mean, is that the rotated $U(1)$ symmetry generated by
\begin{equation}
\widetilde{Q} = \int_{0}^{2\pi}\frac{dk}{2\pi} \widetilde{c}_k^\dagger \widetilde{c}_k
\end{equation}
should be restored at late times after the quench.
Following the logic discussed above for the tilted ferromagnetic case, we therefore look for a GGE built out a maximal set of charges commuting with $\widetilde{Q}$. Equivalently this can be expressed in terms of the rotated mode occupations $\widetilde{n}(k)$, obtained from:
\begin{multline}
\langle \Psi(0) | \widetilde{c}_k^\dagger \widetilde{c}_k | \Psi(0) \rangle =
\langle \Psi(0) | c_k^\dagger c_k | \Psi(0) \rangle- i \langle \Psi(0) | c_{\pi- k} c_k | \Psi(0) \rangle
\\
=
\frac{1}{2} \left( 1+ \frac{\sin ( \theta )^2 \cos
(k)}{1+ \cos(\theta)^2-2
\cos ( \theta ) \sin
k} \right) \,.
\end{multline}
The corresponding distribution function $\widetilde{n}(k)$ is precisely the function $n_+(k,\theta)$ which was found in the previous sections to enter the quasi-particle picture.
\section{Conclusions}
\label{sec:conclusion}
In this paper, we considered the quantum quench in the XX spin chain starting from the cat-version of the tilted N\'eel state given by Eq.
\eqref{eq:cat}. This state (both in normal and cat version) explicitly breaks the $U(1)$ symmetry of the XX Hamiltonian.
We found that, surprisingly, the $U(1)$ symmetry is not generically restored at large time and this can be traced back to the activation of a non-abelian set of charges which all break it.
We characterised quantitatively the breaking of the symmetry by the recently introduced entanglement asymmetry \cite{amc22}.
By a combination of exact calculations and quasi-particle picture arguments, we have been able to exactly describe the behaviour of the asymmetry at any time after the quench.
We obtained that, at large times after the quench, the entanglement asymmetry
tends to a non-zero constant value, except at $\theta=\pi/2$, for which it does go to zero.
Hence, the $U(1)$ symmetry is not restored unless it is maximally broken
at initial time.
Finally, we showed that the stationary behaviour is completely captured by a non-abelian generalised Gibbs ensemble.
We conclude this paper with some outlooks. The first natural question is whether in interacting integrable models there are integrable initial states (in the sense of Ref. \cite{ppv-17}) for which symmetries of the post-quench Hamiltonian are not restored.
This entirely depends on the structure of the GGE and on the activation of possibly existing non-abelian charges. For example, it is known that the XXZ spin chain at root of unity (i.e. only in the regime $|\Delta|<1$) there are non-abelian charges \cite{ilievskireview,miao} and they are activated by the tilted N\'eel.
Then, in this case, we expect the non-restoration of the $U(1)$ symmetry.
Conversely, in the regime $|\Delta|>1$ the $U(1)$ symmetry is expected to be always restored.
The calculation of the entanglement asymmetry for the XXZ spin-chain should be possible by generalising the approach already used for non-equilibrium symmetry resolved entanglement in these models \cite{bcckr-22,pvcc-22,bka-22} and work in this direction is already in progress.
The situation is instead less clear
for other integrable models,
such as for example Hubbard or Gaudin-Yang, for which integrable initial states have been recently proposed \cite{rbc-22,rcb-22b}.
Another more difficult question concerns how to describe a (weak) integrability breaking which eventually always leads to symmetry restoration, but with a pre-thermal regime in which the symmetry is broken. It should be possible to study this problem with pre-thermalisation techniques \cite{MarcuzziPRL13,EsslerPRB14,BEGR:PRL,BEGR:PRB,bc-20}.
\section*{Acknowledgments}
We thank Colin Rylands and Olexei I. Motrunich for useful discussions.
PC and FA acknowledge support from ERC under Consolidator grant number 771536 (NEMO). SM thanks support from Caltech Institute for Quantum Information and Matter and the Walter Burke Institute for Theoretical Physics at Caltech.
\begin{appendices}
\section*{Appendix}
\section{Determinant involving a product of block Toeplitz
matrices}\label{app:block_toep}
One of the main results on the theory of block Toeplitz determinants
is the Widom-Szeg\H{o} theorem~\cite{w-74}. According to it, the determinant of
a block Toeplitz matrix $T_\ell[g]$ with symbol $g$, see Eq.~\eqref{eq:block_toep}, behaves
for large $\ell$ as
\begin{equation}\label{eq:w-s}
\log \det T_\ell[g]~\sim \ell \int_0^{2\pi} \frac{d k}{2\pi} \log \det g(k)
\end{equation}
if $\det g[k]\neq 0$ with zero winding number.
If $T_\ell[g]$ and $T_\ell[g']$ are two arbitrary block Toeplitz matrices with
a symbol given by, respectively, $g(k)$ and $g'(k)$, then the entries of
the product $T_\ell[g] T_\ell[g']$ behave in the limit $\ell \to \infty$ as
\begin{eqnarray}\label{eq:prod_toep}
(T_\ell[g] T_\ell[g'])_{ll'} &=&
\sum_{j=1}^{\ell} (T_\ell[g])_{lj} (T_\ell[g'])_{jl'}\nonumber \\
&\sim&
\sum_{j=1}^{\infty} (T_\ell[g])_{lj} (T_\ell[g'])_{jl'}=\int_0^{2\pi}\frac{dk}{2\pi}g(k)g'(k)e^{ik(l-l')},
\end{eqnarray}
where we have extended the sum over $j$ until $\infty$ and we have used
$\sum_{j=1}^\infty e^{i(k-k')j}=2\pi\delta(k-k')$. Thus, from Eq.~\eqref{eq:prod_toep}, we may conclude
that the entries of the product $T_\ell[g]T_\ell[g']$ behave as the ones of the block Toeplitz matrix
$T_\ell[gg']$ in the limit $\ell\to\infty$. This observation,
combined with Widom-Szeg\H{o} theorem, allows to derive the asymptotic behaviour
with $\ell$ of the determinant of matrices of the form $I+T_\ell[g]T_\ell[g']$, which are relevant
in the analysis performed in this paper. In fact, according to it, $\det(I+T_\ell[g]T_\ell[g'])\sim\det(I+T_\ell[gg'])$
and applying Widom-Szeg\H{o} theorem of Eq.~\eqref{eq:w-s} we conclude
that
\begin{equation}
\log\det(I+T_\ell[g]T_\ell[g'])\sim \ell\int_0^{2\pi} \frac{dk}{2\pi} \log\det(1+g(k)g'(k)).
\end{equation}
This result also extends to the product of more than two block Toeplitz matrices, as we do
in Eq.~\eqref{eq:conj_prod}.
\end{appendices}
|
{
"timestamp": "2023-02-16T02:17:31",
"yymm": "2302",
"arxiv_id": "2302.03330",
"language": "en",
"url": "https://arxiv.org/abs/2302.03330"
}
|
\section{Introduction}
The rotation period (hereafter $P_\mathrm{rot}$) is an important characteristic of a star, as is its effective temperature ($T_\mathrm{eff}$\,), mass, radius and corresponding gravity ($\log g$), metallicity ($\left[M/H\right]$), and age. In the context of exoplanet search by velocimetry (measure of the radial velocity, RV), the knowledge of the stellar rotation period avoids attributing a periodogram peak to an exoplanet orbital period by taking the systematic noise that is introduced by stellar activity into account \citep[e.g.,][]{queloz01}. Several publications identified a periodic signal as due to activity and/or proposed methods to distinguish RV jitter from Keplerian variation (see, e.g., \citealt{bonfils07, figueira10, boisse11}) or identified a signal that was previously announced as due to a planet as actually being caused by activity (e.g., \citealt{huelamo08, robertson14a, robertson14b, kane16, faria20}).
Stellar rotation periods for magnetic stars have been measured using the small-scale surface magnetic field, which is detected through the broadening and intensification of some stellar lines due to the Zeeman effect \citep[e.g.,][]{babcock47} or the large-scale surface magnetic field derived from the polarimetric signal of spectral lines, as pioneered by \citet{preston71, landstreet80} for early-type stars. A periodic variation in the longitudinal magnetic field is interpreted as due to the periodic appearance of magnetic regions at the surface of the star, which allows measuring its rotation period; see \citet{landstreet92} for a review. An application to M dwarfs can be found in \citet{donati08, morin08, morin10, hebrard16}.
Rotation periods can also be measured from long photometric sequences obtained with small telescopes such as ASAS \footnote{All Sky Automated Survey}\citep{pojmanski97}, APT \footnote{Automatic Photoelectric Telescope}\citep{henry99}, HATNet \footnote{Hungarian-made Automated Telescope Network}\citep{bakos04}, NSVS \footnote{Northern Sky Variability Survey}\citep{wozniak04}, SuperWASP \footnote{Wide Angle Search for Planets}\citep{pollacco06}, and MEarth \citep{NC08, irwin09}: examples can be found in \citet{ks07, irwin11, suarez16, diez19}, among others, or more recently, from space telescopes such as Kepler, K2, or TESS \footnote{Transiting Exoplanet Survey Satellite}\citep{diez19}.
An alternative technique uses sequences of stellar activity indicators such as H$\alpha$, Ca II H\&K, the Ca II infrared triplet, or values computed at the same time as the radial velocities using the cross-correlation function (CCF) such as the full width at half maximum (FWHM), contrast, or the bisector \citep[e.g.,][]{noyes84, queloz01, bonfils07, west08, boisse11, bonfils13, suarez15, nelson16, suarez18, toledo19, lafarga21}.
In the framework of the SPIRou legacy survey (hereafter SLS), a sample of about 50 nearby low-mass stars is regularly monitored to detect exoplanets using the SPIRou spectropolarimeter \citep{donati20}. Some of these targets lack a measured rotation period, and some have several measurements that sometimes disagree. To clarify the situation and increase the sample of known rotation periods, we focus in this paper on determining the rotation period by exploiting the large-scale magnetic field information extracted from the Stokes $V$ profiles of spectral lines.
The paper is organized as follows: Section \ref{sec:observations} briefly describes SPIRou\footnote{ \url{http://spirou.irap.omp.eu} and \url{https://www.cfht.hawaii.edu/Instruments/SPIRou/}} and the SPIRou Legacy Survey (SLS\footnote{\url{http://spirou.irap.omp.eu/Observations/The-SPIRou-Legacy-Survey}}; id P40 and P42, PI: Jean-Fran\c{c}ois Donati). Section \ref{sec:reductionandanalysis} describes the generic software called {\it A PipelinE to Reduce Observations} APERO (\footnote{\url{https://github.com/njcuk9999/apero-drs}} v0.7.232, \citealt{cook22}), which is used to reduce the SPIRou observations, in particular, the spectropolarimetric analysis and the derivation of the longitudinal magnetic field. Section \ref{sec:results} explains the results for the stars in our sample, separating them into three categories according to their spectral type. Two examples are treated in detail, and a comparison is made with results from the literature. Finally, Section \ref{sec:conclusions} examines our general results and further perspectives.
\section{Observations}
\label{sec:observations}
Observations are part of the CFHT large program SPIRou Legacy Survey. SPIRou is a stabilized high-resolution near-infrared (NIR) spectropolarimeter \citep{donati20} mounted on the 3.6~m CFHT at the top of Maunakea, Hawaii. It is designed for high-precision velocimetry and spectropolarimetry to detect and characterize exoplanets and stellar magnetic fields. It provides full coverage of the NIR spectrum from 950~nm to 2500~nm at a spectral resolving power of $\lambda / \Delta \lambda \sim 70\,000 $.
The SLS was allocated 310 nights over seven semesters (February 2019 to June 2022). It covers three different science topics, called work packages (see \citealt{donati20} for details). The analysis of this paper is restricted to work package 1 (WP1), dedicated to a blind planet search. More than 50 M dwarfs were regularly monitored during the SPIRou runs (typically ten contiguous nights per month). We excluded monitored active stars with short and well-known rotation periods from this study ($P_\mathrm{rot}$ < 5~d): Gl\,388 (AD\,Leo), Gl\,406 (CN\,Leo), Gl\,873 (EV\,Lac), GJ\,1111 (DX\,Cnc), GJ\,1154, GJ\,1245B, GJ\,3622, and PM\,J18482+0741. We therefore selected 43 stars in the original WP1 sample for which about 150 (50 at least, 250 at most) polarimetric sequences (of four individual subexposures each) have been secured. This represents about 6800 visits, corresponding to more than 27\,000 individual spectra secured for the considered sample. The stars belonging to our sample and their stellar characteristics are given in Table \ref{tab:samplechars}: {\it Gaia} absolute magnitudes and colors use {\it Gaia} DR3 \citet{gaia21}; effective temperatures and metallicities come from \citet{cristofari22}, who used the same set of observations and the same sample (except for Gl\,581, which does not satisfy our criterion of the minimum number of visits); and masses were computed from absolute 2MASS $K_{\rm s}$ magnitude and metallicity using the \citealt{mann19} relations. The relative precision of these masses is 2-3\% according to \citet{mann19}, and the internal errors amount to 30\,K and 0.1\,dex for $T_\mathrm{eff}$\, and [M/H], according to \citet{cristofari22}. We divided our sample into three subsamples according to a proxy of stellar mass. These three mass bins correspond to the three types of magnetic behavior identified among active M dwarfs by \citet{morin10} (see Figure~15 therein). We prefer to define regions according to the absolute $G$ magnitude rather than using the spectral types. Although they match on average, a low metallicity makes the star fainter at a given spectral type: Gl\,412A (M1.0V, $\left[M/H\right]$$=-0.42$) is 1\,mag fainter in $G$ than Gl\,410 (M1V, $\left[M/H\right]$$=+0.05$), therefore both should not belong to the same group. The horizontal lines in Table \ref{tab:samplechars} separate the three subsamples. The number of visits corresponds to the initial number, before some polarimetric sequences were rejected for the reasons explained in Sec. \ref{sec:spectropolarimetry}. Finally, we list the FWHM of the median Stokes $I$ profile of each star because it was used to define the velocity range on which we measured the longitudinal magnetic field from the Stokes $V$ profile, as explained in Sec. \ref{sec:spectropolarimetry}.
\begin{table*}
\caption[]{Stellar characteristics of the sample of 43 M dwarfs. For uncertainties on mass, $T_\mathrm{eff}$\, , and [M/H], see text. The three sections correspond to early-, mid-, and late-type M dwarfs from top to bottom}
\label{tab:samplechars}
\begin{center}
\begin{tabular}{lcccccccc}
\hline
\noalign{\smallskip}
Star & spectral type & $M_{\rm G}$ & $G_{\rm BP} - G_{\rm RP}$ & mass & $T_{\rm eff}$ & [M/H] & visits & FWHM Stokes $I$ \\
Units & & mag & mag & M$\mathrm{_\odot}$ & K & dex & & km\,s$^{-1}$\, \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Gl\,338B & M0V & 8.046 & 1.846 & 0.58 & 3952 & $-0.08$ & 58 & $6.95 \pm 0.25$ \\
Gl\,846 & M0.5V & 8.282 & 1.967 & 0.57 & 3833 & 0.07 & 194 & $6.65 \pm 0.26$ \\
Gl\,205 & M1.5V & 8.327 & 2.122 & 0.58 & 3771 & 0.43 & 160 & $6.54 \pm 0.16$ \\
Gl\,410 & M1V & 8.426 & 2.006 & 0.55 & 3842 & 0.05 & 131 & $7.73 \pm 0.21$ \\
Gl\,880 & M1.5V & 8.611 & 2.151 & 0.55 & 3702 & 0.26 & 166 & $6.37 \pm 0.17$ \\
Gl\,514 & M1.0V & 8.793 & 2.088 & 0.50 & 3699 & $-0.07$ & 177 & $5.84 \pm 0.29$ \\
Gl\,382 & M2.0V & 8.898 & 2.234 & 0.51 & 3644 & 0.15 & 124 & $6.34 \pm 0.31$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Gl\,752A & M3.0V & 9.240 & 2.379 & 0.47 & 3558 & 0.11 & 130 & $6.06 \pm 0.23$ \\
Gl\,48 & M2.5V & 9.364 & 2.459 & 0.46 & 3529 & 0.08 & 194 & $5.99 \pm 0.26$ \\
Gl\,617B & M3V & 9.459 & 2.483 & 0.45 & 3525 & 0.20 & 149 & $6.09 \pm 0.21$ \\
Gl\,412A & M1.0V & 9.460 & 2.104 & 0.39 & 3620 & $-0.42$ & 224 & $5.50 \pm 0.24$ \\
Gl\,15A & M1.5V & 9.460 & 2.164 & 0.39 & 3611 & $-0.33$ & 256 & $5.76 \pm 0.26$ \\
Gl\,849 & M3V & 9.511 & 2.542 & 0.46 & 3502 & 0.35 & 203 & $6.17 \pm 0.22$ \\
Gl\,411 & M2.0V & 9.522 & 2.216 & 0.39 & 3589 & $-0.38$ & 182 & $4.93 \pm 0.18$ \\
Gl\,480 & M3.5V & 9.565 & 2.591 & 0.45 & 3509 & 0.26 & 104 & $6.14 \pm 0.22$ \\
Gl\,436 & M3.0V & 9.631 & 2.449 & 0.42 & 3508 & 0.03 & 90 & $5.89 \pm 0.33$ \\
Gl\,687 & M3.0V & 9.739 & 2.518 & 0.39 & 3475 & 0.01 & 227 & $5.69 \pm 0.20$ \\
Gl\,408 & M2.5V & 9.831 & 2.430 & 0.38 & 3487 & $-0.09$ & 179 & $6.17 \pm 0.33$ \\
Gl\,317 & M3.5V & 9.859 & 2.664 & 0.42 & 3421 & 0.23 & 79 & $5.39 \pm 0.21$ \\
GJ\,4063 & M3.5V & 9.982 & 2.770 & & 3419 & 0.42 & 220 & $5.96 \pm 0.22$ \\
Gl\,725A & M3.0V & 10.119 & 2.461 & 0.33 & 3470 & $-0.26$ & 219 & $5.43 \pm 0.22$ \\
Gl\,251 & M3.0V & 10.129 & 2.561 & 0.35 & 3420 & $-0.01$ & 187 & $5.73 \pm 0.26$ \\
GJ\,4333 & M3.5V & 10.233 & 2.818 & 0.37 & 3362 & 0.25 & 193 & $5.70 \pm 0.16$ \\
GJ\,1012 & M4V & 10.268 & 2.710 & 0.35 & 3363 & 0.07 & 142 & $5.26 \pm 0.17$ \\
GJ\,1148 & M4V & 10.370 & 2.778 & 0.34 & 3354 & 0.11 & 105 & $5.45 \pm 0.16$ \\
Gl\,876 & M3.5V & 10.527 & 2.809 & 0.33 & 3366 & 0.15 & 88 & $5.65 \pm 0.28$ \\
PM J09553-2715 & M3V & 10.629 & 2.667 & 0.29 & 3366 & $-0.03$ & 75 & $5.77 \pm 0.25$ \\
PM J08402+3127 & M3.5V & 10.739 & 2.703 & 0.28 & 3347 & $-0.08$ & 142 & $5.33 \pm 0.25$ \\
Gl\,725B & M3.5V & 10.790 & 2.625 & 0.25 & 3379 & $-0.28$ & 212 & $5.34 \pm 0.22$ \\
GJ\,1105 & M4V & 10.931 & 2.792 & 0.27 & 3324 & $-0.04$ & 171 & $5.44 \pm 0.17$ \\
Gl\,445 & M3.5V & 10.948 & 2.702 & 0.24 & 3356 & $-0.24$ & 94 & $5.21 \pm 0.19$ \\
GJ\,3378 & M3.5V & 10.975 & 2.791 & 0.26 & 3326 & $-0.05$ & 177 & $5.65 \pm 0.22$ \\
Gl\,169.1A & M4V & 10.994 & 2.896 & 0.28 & 3307 & 0.13 & 185 & $5.66 \pm 0.28$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
GJ\,1289 & M3.5V & 11.556 & 3.011 & 0.21 & 3238 & 0.05 & 209 & $7.60 \pm 0.37$ \\
PM J21463+3813 & M4V & 11.591 & 2.814 & 0.18 & 3305 & $-0.38$ & 187 & $5.21 \pm 0.26$ \\
GJ\,1103 & M4.5V & 11.817 & 3.116 & 0.19 & 3170 & $-0.03$ & 70 & $5.44 \pm 0.28$ \\
Gl\,699 & M4.0V & 11.884 & 2.834 & 0.16 & 3311 & $-0.37$ & 249 & $6.04 \pm 0.32$ \\
Gl\,15B & M3.5V & 11.928 & 2.836 & 0.16 & 3272 & $-0.42$ & 189 & $6.33 \pm 0.37$ \\
Gl\,447 & M4.0V & 11.960 & 3.033 & 0.18 & 3198 & $-0.13$ & 60 & $6.24 \pm 0.24$ \\
GJ\,1151 & M4.5V & 12.158 & 3.142 & 0.17 & 3178 & $-0.16$ & 157 & $6.42 \pm 0.37$ \\
Gl\,905 & M5.5V & 12.881 & 3.529 & 0.15 & 3069 & 0.05 & 220 & $6.13 \pm 0.29$ \\
GJ\,1286 & M5.5V & 13.344 & 3.706 & 0.12 & 2961 & $-0.23$ & 113 & $7.46 \pm 0.46$ \\
GJ\,1002 & M5.5V & 13.347 & 3.675 & 0.12 & 2980 & $-0.33$ & 146 & $6.92 \pm 0.34$ \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{center}
\end{table*}
\section{SPIRou data reduction and analysis}
\label{sec:reductionandanalysis}
\subsection{ APERO reduction}
\label{sec:aperoreduction}
Our SPIRou data were reduced with the software APERO. APERO first performs some initial preprocessing of the 4096$\times$4096~pixel images of the HAWAII\,4RG$^{\rm {\small TM}}$ (H4RG), applying a series of procedures to correct detector effects, remove background thermal noise, and identify bad pixels and cosmic-ray impacts.
It then uses exposures of a quartz halogen lamp (flat) to calculate the position of 49 of the 50 \'echelle spectral orders recorded on the detector. It optimally extracts \citep{horne86} spectra of the two science channels (fibers A and B) and the simultaneous reference channel (fiber C). This APERO extraction takes the non-Gaussian shape of the instrument profile generated by the pupil slicer into account. APERO corrects the spectra for the blaze signature of the \'echelle orders obtained from the flat-field exposures, as described in \citet{cook22}. Both a 2D order-by-order and 1 D order-merged spectrum are produced for each channel of each scientific exposure.
The pixel-to-wavelength calibration is obtained from exposures of both a uranium-neon hollow cathode lamp and a Fabry-P\'erot etalon, generally following the procedure given in \citet{hobson21}, but with differences described in \citet{cook22}. We refer to Sections 5.4 and 6.6 of the latter paper for details about the use of a reference night for the whole survey and updates for each survey night, with subtle differences in the wavelength calibration of fibers AB with respect to fibers A and B separately. This procedure provides wavelengths in the rest frame of the observatory, but APERO also calculates the barycentric Earth radial velocity (BERV) and the barycentric Julian date (BJD) of each exposure using the code \texttt{barycorrpy}\footnote{\url{https://github.com/shbhuk/barycorrpy}}\citep{wright14, kanodia18}. These can then be used to reference the wavelength and time to the barycentric frame of the Solar System.
APERO calculates the spectrum of the telluric transmission using a novel technique based on a model obtained from the collection of standard star observations carried out since the beginning of SPIRou operations in 2019 and a fit made for each individual observation using a revised technique based on the principles explained in \cite{artigau14}, as briefly described in \citet{cook22} and in more details in \citet{artigau22}. APERO also calculates the Stokes parameters describing the polarization state, as defined, for instance, in \citet{landi92}. Each orthogonal polarization state is recorded on one science fiber. Four sub-exposures are taken with different positions of the quarter-wave rhombs: the method is described in \cite {donati97}. The two polarization states of each subexposure are combined to define the Stokes parameters, as explained in Sec. 10.1 of \citet{cook22}) using equations defined in \citet{bagnulo09}: the Stokes $I$ parameter measures the intensity, the Stokes $V$ parameter measures the circular polarization, and the Stokes $Q$ and $U$ measure the linear polarization (not used in thiswork). Another combination defines a null measurement to quantify the quality of the Stokes $V$ detection. The continuum for Stokes $I$ and $V$ can be modeled by an iterative sigma-clip algorithm to fit a polynomial to the data. The spectra are normalized to this continuum before an LSD analysis is performed. More details are given in \citet{martioli20, cook22}.
\subsection{Spectropolarimetry analysis}
\label{sec:spectropolarimetry}
We further analyzed the SPIRou polarized spectra using the code \texttt{spirou-polarimetry}\footnote{\url{https://github.com/edermartioli/spirou-polarimetry}} , which we applied to the telluric-corrected 2D spectra. We assumed that the APERO telluric correction is good enough so that we did not need to exclude zones of telluric water bands in this code. The Stokes $I$, Stokes $V$, and null-polarization spectra were compressed to one-line profiles using the least-squares deconvolution (LSD) method of \cite{donati97}. The line masks used in our LSD analysis were computed using the VALD3 catalog \citep{piskunov95, ryabchikova15} and a MARCS model atmosphere \citep{gustafsson08} with a grid of effective temperatures between 3000~K and 4000~K by steps of 500~K, a constant surface gravity of $\log g=5.0$~dex, and a micro-turbulent velocity $\upsilon_{\mathrm mic}$\, of 1 km\,s$^{-1}$\,. We selected all lines deeper than 3\% and with a Land\'{e} factor of $g_{\rm eff}>0$ for a total of 2460, 1335, and 956 atomic lines for 4000\,K, 3500\,K, and 3000~K, respectively. We normalized the Stokes $V$ profile using a mean wavelength, a mean depth, and a mean Land\'e factor of the lines in the mask, but we only used the mean depth to normalize the Stokes $I$ profile. For the 4000\,K mask (3500~K and 3000~K), the mean values are 1651.2\,nm (1617.9\,nm and 1604.8\,nm) for the wavelengths, 0.135 (0.133 and 0.155) for the depths, and 1.249 (1.242 and 1.235) for the Land\'e factors. To ensure that Stokes $I$ LSD profiles have a continuum at 1, we used a linear regression of a significant number of bins located well outside of the line for each profile to improve the continuum normalization. We measured the standard deviation of the values in these bins and rejected the profiles for which this standard deviation significantly deviated from the average over all the profiles. This linear regression gives us a better defined Voigt profile, allowing a more precise measure of the FWHM of the median Stokes $I$ LSD profile of each star.
Not all polarimetric sequences led to a useful measurement of the longitudinal magnetic field. There are several steps of data rejection: the telluric correction of some spectra may fail, some LSD profiles are noisy enough that they cannot be fit by a Voigt profile, and others can be fit, but present a noisier continuum than the rest (see above) and are thus rejected. We only rejected data that might contaminate our results, especially, to build a clean median Stokes $I$ profile on which we measured the FWHM and defined the velocity limits to integrate the Stokes $V$ profile (using 6 FWHM as the measuring window, meaning $\pm 3$ FWHM, generally about $\pm 20$ km\,s$^{-1}$\,). The final number of polarimetric sequences used in our study is about 6500, compared to the 6800 initially available sequences for our sample of stars. This is an acceptable loss of about 4\% that generally corresponds to spectra with a low signal-to-noise ratio (S/N).
As an example, we took the LSD profiles of the early-M dwarf Gl\,410 (DS\,Leo), which displays mild signs of magnetic activity. Fig.~\ref{fig:GL410_lsd_timeseries} shows the resulting LSD profiles at each observing epoch, stacked vertically. Dynamical maps from left to right correspond to the intensity (Stokes $I$), the circular polarization (Stokes $V$), and the null LSD profiles. A nonzero Stokes $V$ profile indicates the presence of magnetic fields, while the null profile allows us to assess the reliability of the Stokes $V$ detection. Finally, Fig. \ref{fig:GL410_lsd_median_profiles} shows the median profile of our time series, where the Zeeman signature is clearly visible in the detection of the Stokes $V$ profile. We also display a median null profile to assess that the Stokes $V$ signature is reliably detected. For comparison, we display the Stokes $I$, $V$ and null dynamical maps (Fig.~\ref{fig:GL905_lsd_timeseries}) and median profiles (Fig.~\ref{fig:GL905_lsd_median_profiles}) of the late-M dwarf Gl\,905, whose longitudinal magnetic field has a similar amplitude as Gl\,410. Here again, the Stokes $V$ signal is clearly detected, with a shape corresponding at first order to the first derivative of the Stokes $I$ profile.
In the appendix, we display the Stokes $V$ temporal evolution for each star of our sample. The Stokes $I$ temporal evolution and the median Stokes profiles carry less important information for the present work, and the null profiles are very similar to each other. We decided not to publish them for brevity.
To confirm the consistency of our measurements, we also obtained an independent polarimetric reduction and LSD analysis of our SPIRou data using the pipeline \texttt{Libre-Esprit} \citep{donati97,donati20}. The results of this alternative analysis will be given in a forthcoming paper (Donati et al., in prep).
\begin{figure*}
\centering
\includegraphics[width=\hsize]{Gl410_DynamicalPlot.png}
\caption{Dynamical maps of Gl\,410 constructed from individual observations (horizontal bands), stacked vertically. The first plot shows Stokes $I$ LSD profiles, where the continuum is given in yellow and the different absorption depths are shown in green and blue shades. The following plots show the Stokes $V$ and null LSD profiles, where positive values are given as red shades, and negative values are shown as blue shades. Note that Stokes $V$ and null LSD amplitudes are expressed in percent. All plots are given in the rest frame of Gl\,410 and display a velocity window of 10 FWHM (see Table~\ref{tab:samplechars}).
}
\label{fig:GL410_lsd_timeseries}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=\hsize]{GL410_lsd_median_profiles.png}
\caption{Median of all LSD profiles in the Gl\,410 SPIRou time series. The top panel shows Stokes $I$ LSD (red points) with a Voigt profile model fit (green line), the middle panel shows Stokes $V$ (blue points), and the bottom panel shows the null polarization profile (orange points).
}
\label{fig:GL410_lsd_median_profiles}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=\hsize]{Gl905_DynamicalPlot.png}
\caption{Similar to Fig.~\ref{fig:GL410_lsd_timeseries}, but for the dynamical maps of Stokes $I$, $V$ and null LSD profiles collected for Gl\,905 using SPIRou.
}
\label{fig:GL905_lsd_timeseries}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=\hsize]{GL905_lsd_median_profiles.png}
\caption{Median of all LSD profiles in the Gl\,905 SPIRou time series. The top panel shows Stokes $I$ LSD (red points) with a Voigt profile model fit (green line), the middle panel shows Stokes $V$ (blue points), and the bottom panel shows the null polarization profile (orange points).
}
\label{fig:GL905_lsd_median_profiles}
\end{figure}
\subsection{Longitudinal magnetic fields}
\label{sec:stellaractivity}
To diagnose the magnetic field in our sample of stars, we calculated the longitudinal magnetic field $B_\ell$ in the LSD profiles of SPIRou following the same prescription as in \cite{rees79, donati97, moutou20, martioli20}. $B_\ell$ is defined as the brightness-weighted line-of-sight-projected component of the vector magnetic field integrated over the visible hemisphere of the star, and is given in G by
\begin{equation}
\label{eq:Bl}
B_\ell=-2.142 \times 10^{11} \frac{\int v \, V (v) \, dv}{\lambda_0 \cdot g_{\rm eff} \cdot c \cdot \int \left[ 1 - I(v) \right] dv},
\end{equation}
where $c$ is the speed of light in the same unit as $v$ (km\,s$^{-1}$\,), $I(v)$ is the Stokes $I$ LSD profile normalized by the continuum, and $V(v)$ is the Stokes $V$ LSD profile, both as functions of the velocity $v$ in the stellar frame, $\lambda_{0}$ is the mean wavelength in nm, and $g_{\rm eff}$ is the mean Land\'e factor of the lines included in the LSD analysis.
As the spatial distribution of the magnetic regions may be nonaxisymmetric, $B_\ell$ can be modulated by the stellar rotation. This allows deriving the stellar rotation period if any periodicity is detected in its time series \citep[e.g.,][]{preston71, landstreet80}. Examples focusing more on K and M dwarfs can be found in \citep[e.g.,][]{morin08, moutou17, petit21}. Fig. \ref{fig:GL410_blong_periodogram} and Fig. \ref{fig:GL905_blong_periodogram} show the generalized Lomb-Scargle (GLS) periodogram \citep{zechmeister09} for the $B_\ell$ data calculated using the tool \texttt{astropy.timeseries}\footnote{\url{https://docs.astropy.org/en/stable/timeseries/lombscargle.html}} for our two example stars Gl\,410 and Gl\,905. We find a maximum power at a period of 13.9~d for Gl\,410 and at 108.6~d for Gl\,905, both with a false-alarm probability (FAP) below 0.001\%. In the case of Gl\,905, three other peaks below an FAP of 0.001\% can be noted in the periodogram, and they are due to harmonics of the main period.
\begin{figure}
\centering
\includegraphics[width=1.0\hsize]{GL410_Blong_period0_v07232c.png}
\caption{Generalized Lomb-Scargle periodogram analysis of the longitudinal magnetic field ($B_\ell$) time series of Gl\,410. The dashed blue line shows the highest power at a period of 13.9~d, and the dot-dashed lines indicate possible harmonics at half and twice the period.
}
\label{fig:GL410_blong_periodogram}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1.0\hsize]{Gl905_Blong_period0_v07232c.png}
\caption{GLS periodogram analysis of the longitudinal magnetic field ($B_\ell$) time series of Gl\,905. The dashed blue line shows the highest power at a period of 108.6~d, and the dot-dashed lines indicate possible harmonics at half and twice the period.
}
\label{fig:GL905_blong_periodogram}
\end{figure}
As the magnetic field of M dwarfs is likely to evolve with time, this study requires a flexible model to account for the variability. We employed a Gaussian process (GP) regression analysis \citep[e.g.,][]{haywood14,aigrain15} using the code \texttt{george}\footnote{\url{http://dfm.io/george/current/}} \citep{ambikasaran15}, where we assumed that the rotationally modulated stellar activity signal in $B_\ell$ is quasi-periodic (QP). Thus, we adopted a parameterized covariance function (or kernel) as in \cite{angus18}, which is given by
\begin{equation}
k(\tau_{ij}) = \alpha^{2} \exp{\left[ -\frac{\tau_{ij}^2}{2l^2} - \frac{1}{\beta^2} \sin^{2}{\left( \frac{\pi \tau_{ij}}{P_{\rm rot}} \right)} \right]} + \sigma^{2} \delta_{ij},
\end{equation}
\noindent where $\tau_{ij} = t_{i} - t_{j}$ is the time difference between data points $i$ and $j$, $\alpha^{2}$ is the amplitude of the covariance, $l$ is the decay time, $\beta$ is the smoothing factor, $P_{\rm rot}$ is the star rotation period, and $\sigma$ is an additional uncorrelated white noise, which adds a jitter term to the diagonal of the covariance matrix. This kernel combines a squared exponential component describing the overall covariance decay and a component that describes the periodic covariance structure, the amplitude of which is controlled by the smoothing factor\footnote{The definition of the decay timescale and smoothing factor vary among researchers. The former is sometimes defined as $\sqrt{2} \, l$ \citep[e.g.,][]{petit21}, and $1/\beta^2$ may appear as $\Gamma$ and be called harmonic complexity \citep[e.g.,][]{nicholson22}.}. Typical values of $\beta$ vary between 0.25 and 1.25. The lower values correspond to multiple harmonics, and the higher values emphasize single sinusoidal variation. Similarly, the decay parameter varies between a few tens to a few hundred days. As pointed out by \cite{angus18}, the flexibility of this model can easily lead to an overfitting of the data. To avoid this, we adopted a uniform prior distribution of the parameters (see, e.g., Table \ref{tab:earlygpfitparams}) that restricts the search range to realistic values. In addition, when the posterior distribution of the smoothing factor and decay time did not display a clear peak, we restricted the fit from six to five parameters by fixing the value of the smoothing factor at 0.7, which corresponds to the middle of the range of explored values and to the median value of stars with a constrained smoothing factor. We also explored a four-parameter fit in which both the smoothing factor and the decay time were fixed at 0.7 and 200 days, respectively. We found no significant differences with respect to the five-parameter fit. The value of the fixed decay time is a guess at best and may depend on the spectral type (early-M stars seem to have a shorter decay time, about 70 days): \citet{giles17} reported based on Kepler light curves that cooler stars have spots that last much longer, in particular for stars with longer rotational periods.
We used this GP framework to model the $B_\ell$ data, where we first fit the GP model parameters by maximizing the likelihood function defined in (Eq. 5.8)\footnote{http://gaussianprocess.org/gpml/chapters/RW5.pdf} of \citet{rasmussen06} and as implemented in \texttt{george}, and then we sampled the posterior distribution of the free parameters using a Bayesian Markov chain Monte Carlo (MCMC) framework with the package \texttt{emcee} \citep{foreman13}. We set the MCMC with 50 walkers, 1000 burn-in samples, and 5000 samples. The results of our analysis are illustrated in Fig. \ref{fig:GL410_blong_GP} and Fig. \ref{fig:GL905_blong_GP}, where we present the observed $B_\ell$ data and the best-fit GP model for Gl\,410 and Gl\,905, respectively. Fig. \ref{fig:GL410_blong_CP} and Fig. \ref{fig:GL905_blong_CP} show the MCMC samples and posterior distributions of the GP parameters for a six-parameter fit of the $B_\ell$ time series for Gl\,410 and Gl\,905, respectively.
\begin{figure*}
\centering
\includegraphics[width=1.0\hsize]{GL410_Blong_GP_v07232c.png}
\caption{GP analysis of the SPIRou $B_\ell$ data of Gl\,410. In the top panel, the black points show the observed $B_\ell$ data, and the orange line shows the best-fit quasi-periodic GP model. The bottom panel shows the residuals with an RMS dispersion of 7.6~G.}
\label{fig:GL410_blong_GP}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=1.0\hsize]{GL410_Blong_CP_v07232c.png}
\caption{MCMC samples and the posterior distributions of parameters in the quasi-periodic GP analysis of the stellar activity in the SPIRou $B_\ell$ data of Gl\,410. The blue crosses mark the mode of the distribution, and the vertical dashed lines in the histograms indicate the median and the 16 and 84 percentiles of the posterior PDF. The shaded regions correspond to uncertainties of 1, 2, and 3 $\sigma$ in order of increasing radius.}
\label{fig:GL410_blong_CP}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=1.0\hsize]{GL905_blong_GP_v07232c.png}
\caption{GP analysis of the SPIRou $B_\ell$ data of Gl\,905. In the top panel, the black points show the observed $B_\ell$ data, and the orange line shows the six-parameter best-fit quasi-periodic GP model. The bottom panel shows the residuals with an RMS dispersion of 13~G.}
\label{fig:GL905_blong_GP}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=1.0\hsize]{GL905_Blong_CP_v07232c.pdf}
\caption{MCMC samples and the posterior distributions of the six-parameter fit of a quasi-periodic GP on the SPIRou $B_\ell$ data of Gl\,905. The same format is used as in Fig. \ref{fig:GL410_blong_CP}.}
\label{fig:GL905_blong_CP}
\end{figure*}
\section{Results} \label{sec:results}
\subsection{Early-type M dwarfs}
\label{sec:early}
Our sample includes seven early-M dwarfs (M0V to M2V) with absolute magnitudes in the {\it Gaia} EDR3 $G$ band between 8 and 9. With masses between 0.5-0.6 M$\mathrm{_\odot}$\ , they are partially convective. To compute the LSD profiles of these stars, we used a mask with atomic lines of the known Land\'e factor at 4000\,K and $\log g$=5.0. For six of them, the longitudinal magnetic field clearly varies, and accordingly, the rotation period is measured quite accurately. We used the six-parameter fit of the quasi-periodic GP model for them, but the decay time could not be well constrained for two of them. They are reported in Table \ref{tab:earlygpfitparams}, which lists the best-fit parameters measured as the median of the distribution, with uncertainties given by the 0.16 and 0.84 quantiles. Here and in Tables \ref{tab:middlegpfitparams} and \ref{tab:lategpfitparams}, the number of visits corresponds to the final number after poor polarimetric sequences were rejected, and it therefore differs from the numbers listed in Table \ref{tab:samplechars}. Results for Gl\,205 were already published in \citet{cortes23} based on a similar data set, but with the \texttt{Libre-Esprit} data reduction and analysis package: the agreement of the measured parameters is encouraging (see Table \ref{tab:gl205}). The rotation periods lie between 11 and 38~d for this sample. As the radius of early-type M dwarfs is in the range 0.5-0.6 R$\mathrm{_\odot}$, these rotation periods translate into equatorial velocities below 3 km\,s$^{-1}$\,. This $v\,\mathrm{sin}\,i$ \ is below the detectability threshold of our spectrograph (2-3 km\,s$^{-1}$\,). A broadening in the FWHM of the Stokes $I$ profile may be due to several factors when the rotation is negligible: for active stars, it may come from a Zeeman broadening effect, but for the quiet stars studied in this sample, it is probably due to other factors. In Sec. \ref{sec:conclusions} we return to this topic after the full sample is analyzed. For the earliest spectral-type star Gl\,338B, the Stokes V signal is clearly detected (see Fig. \ref{fig:gl338b}). However, we failed to detect any periodicity in the longitudinal magnetic field variation, probably due to the low number of visits (58 observations).
\begin{table*}
\caption[]{Best six-parameter fit of a quasi-periodic GP model obtained in our analysis of the stellar activity from the SPIRou $B_\ell$ data of Gl\,205, compared to the results from \citet{cortes23} using a similar data set, but the \texttt{Libre-Esprit} data reduction and analysis package. }
\label{tab:gl205}
\begin{tabular}{lccccccccc}
\hline
\noalign{\smallskip}
Package & rotation period & mean $B_\ell$ & white noise & amplitude & decay time & smoothing factor & rms & $\chi^2_{\rm red}$ & visits \\
& $P_{\rm rot}$ [d] & $\mu$ [G] & $\sigma$ [G] & $\alpha$ [G] & $l$ [d] & $\beta$ & [G] & & \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
APERO & $34.3^{+0.4}_{-0.4}$ & $2.5^{+2.0}_{-2.0}$ & $1.0^{+0.5}_{-0.6}$ & $7.1^{+1.3}_{-1.0}$ & $67^{+15}_{-11}$ & $0.54^{+0.09}_{-0.07}$ & 2.5 & 0.79 & 152 \\
\noalign{\smallskip}
Libre-Esprit & $34.4^{+0.5}_{-0.4}$ & $1.3^{+0.9}_{-0.9}$ & $0.4^{+0.2}_{-0.2}$ & $3.1^{+0.6}_{-0.4}$ & $63^{+13}_{-8}$ & $0.57^{+0.10}_{-0.08}$ & 0.94 & 0.84 & 153 \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{table*}
The GP parameters appear to show that the decay time is pretty well constrained, with values clustering around 75~d. The smoothing parameter varies more widely, between 0.5 (several harmonics) and 1.0 (smooth variation). For two stars with poorly constrained smoothing parameter and decay time, we used a five-parameter fit by fixing the smoothing parameter to 0.7. The white noise is always compatible with 0 within $2 \sigma$, meaning that the GP does not need an additional variable to explain the dispersion, and the reduced $\chi^2$ are close to 1. We reach an rms of the fit of a few G for these stars. The results are summarized in Table \ref{tab:earlygpfitparams}.
\begin{table*}
\caption[]{Best six-parameter fit of a quasi-periodic GP model obtained in our analysis of the stellar activity from the SPIRou $B_\ell$ data of the early-type M dwarfs in our sample. }
\label{tab:earlygpfitparams}
\begin{tabular}{lccccccccc}
\hline
\noalign{\smallskip}
Star & rotation period & mean $B_\ell$ & white noise & amplitude & decay time & smoothing factor & rms & $\chi^2_{\rm red}$ & visits \\
& $P_{\rm rot}$ [d] & $\mu$ [G] & $\sigma$ [G] & $\alpha$ [G] & $l$ [d] & $\beta$ & [G] & & \\
\noalign{\smallskip}
Priors & $\mathcal{U}(2,300)$ & $\mathcal{U}(-\infty,+\infty)$ & $\mathcal{U}(0,+\infty)$ & $\mathcal{U}(0,+\infty)$ & $\mathcal{U}(50,1000)$ & $\mathcal{U}(0.25,1.25)$ & & & \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Gl\,846 & $11.01^{+0.17}_{-0.22}$ & $-0.6^{+2.0}_{-2.1}$ & $2.6^{+1.0}_{-1.3}$ & $6.2^{+1.4}_{-1.2}$ & $57^{+18}_{-5}$ & $0.96^{+0.19}_{-0.22}$ & 6.1 & 1.05 & 188 \\
\noalign{\smallskip}
Gl\,205 & $34.3^{+0.4}_{-0.4}$ & $2.5^{+2.0}_{-2.0}$ & $1.0^{+0.5}_{-0.6}$ & $7.1^{+1.3}_{-1.0}$ & $67^{+15}_{-11}$ & $0.54^{+0.09}_{-0.07}$ & 2.5 & 0.79 & 152 \\
\noalign{\smallskip}
Gl\,410 & $13.87^{+0.08}_{-0.07}$ & $-1.1^{+10.8}_{-10.5}$ & $3.7^{+2.0}_{-2.2}$ & $28.9^{+6.7}_{-4.7}$ & $78^{+16}_{-14}$ & $0.73^{+0.13}_{-0.10}$ & 7.6 & 0.86 & 126 \\
\noalign{\smallskip}
Gl\,880 & $37.7^{+0.8}_{-0.6}$ & $4.0^{+3.4}_{-3.1}$ & $0.8^{+0.8}_{-0.6}$ & $11.4^{+2.4}_{-1.7}$ & $94^{+24}_{-20}$ & $0.56^{+0.10}_{-0.08}$ & 3.8 & 0.65 & 162 \\
\noalign{\smallskip}
Gl\,514 & $30.45^{+0.13}_{-0.14}$ & $-4.2^{+5.9}_{-5.4}$ & $2.1^{+1.3}_{-1.4}$ & $8.8^{+3.6}_{-2.4}$ & & $0.99^{+0.18}_{-0.25}$ & 8.2 & 1.10 & 165 \\
\noalign{\smallskip}
Gl\,382 & $21.32^{+0.04}_{-0.03}$ & $-0.7^{+4.2}_{-4.2}$ & $2.8^{+1.2}_{-1.5}$ & $7.9^{+3.2}_{-1.9}$ & & $0.62^{+0.27}_{-0.18}$ & 6.7 & 1.11 & 114 \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{table*}
Rotation periods were already known for all these stars, based on a variety of indicators: ZDI analysis, photometry (noted "phm" in Tables \ref{tab:earlylit}, \ref{tab:midlit}, and \ref{tab:latelit}), and activity indicators (noted "act" in the same tables). In Table \ref{tab:earlylit} we compare our new values to those from the literature. The agreement is good in general, but some literature values are discrepant for unclear reasons (see, e.g., Gl\,846) or reveal a harmonic (twice the frequency of rotation: Gl\,382 for \citet{sabotta21}). Photometry and activity measurements may be affected by sporadic variability (flares), and poor sampling may lead to differences or detection of harmonics of the true rotation period. A case-by-case discussion is difficult as the literature values are based on different techniques with varying precision, time sampling, and total time coverage. For our method, the time sampling and total coverage are generally adequate, and the precision is quite uniform after rejecting low S/N measurements.
\begin{table*}
\caption[]{Comparison of the rotation periods given in the literature with our measured values for the early-M stars.}
\label{tab:earlylit}
\begin{center}
\scalebox{0.9}{
\begin{tabular}{lccccccc}
\hline
\noalign{\smallskip}
Reference & Category & Gl\,205 & Gl\,382 & Gl\,410 & Gl\,514 & Gl\,846 & Gl\,880 \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
This work & $B_\ell$ & $34.3 \pm 0.4$ & $21.32 \pm 0.04$ & $13.87 \pm 0.08$ & $30.45 \pm 0.14$ & $11.01 \pm 0.20$ & $37.7 \pm 0.7$ \\
\citet{ks07} & phm & 33.61 & 21.56 & & & & \\
\citet{donati08} & ZDI & & & 14.0 & & & \\
\citet{bonfils13} & act & 32.8, 39.3 & & & & 10.7 & \\
\citet{suarez15} & act & $35.0 \pm 0.1$ & $21.7 \pm 0.1$ & & $28.0 \pm 2.9$ & $31.0 \pm 0.1$ & $37.5 \pm 0.1$ \\
\citet{suarez16} & phm & $33.4 \pm 0.1$ & $21.2 \pm 0.1$ & & & & \\
\citet{hebrard16} & ZDI & $33.63 \pm 0.37$ & & $13.83 \pm 0.10$ & & $10.73 \pm 0.10$ & \\
\citet{suarez17} & act & $34.8 \pm 1.3$ & $21.8 \pm 0.1$ & & $30.0 \pm 0.9$ & $26.3 \pm 5.6$ & $37.2 \pm 6.7$ \\
\citet{diez19} & phm & $33.8 \pm 0.6$ & $21.6 \pm 0.2$ & $14.6 \pm 0.2$ & & $29.5 \pm 0.1$ & $39.5 \pm 0.2$ \\
\citet{sabotta21} & act & 37.08 & 10.65, 21.4 & & & & \\
\hline
\end{tabular}}
\end{center}
\end{table*}
\subsection{Mid-type M dwarfs}
\label{sec:middle}
The most numerous subcategory in our sample comprises 26 stars with M$_{\mathrm{G}}$~ between 9 and 11, spectral types ranging from M1V to M4V, and masses between 0.2 and 0.5 M$\mathrm{_\odot}$. This group crosses the so-called Jao gap \citep{jao18} at about M$_{\mathrm{G}}$=10.2, G$_{\mathrm{BP}}$-G$_{\mathrm{RP}}$=2.3, and includes the transition between partially convective and fully convective M dwarfs. As the transition mass between these two regimes is poorly defined (0.2 to 0.35\,M$\mathrm{_\odot}$) and depends on metallicity (see \citealt{feiden21} for a more detailed discussion), we preferred not to try to define a finer grid of magnitudes or spectral types with the risk of a possibly inhomogeneous subcategory. To compute the LSD profiles of these stars, we used a mask with atomic lines of the known Land\'e factor at 3500\,K and $\log g$=5.0.
In contrast with the group of early-M dwarfs, here only half of the stars have a detected rotation period. We first tried a six-parameter fit of the quasi-periodic GP model for them, but only three stars have a well-constrained decay time at about 100\,days. For the other stars for which we were able to determine a rotation period, we use a five or even a four parameters fit, fixing the smoothing parameter at 0.7 (five-parameter fit), and when necessary, the decay time to 200\,days (four-parameter fit). We detect a very long periodic variation of about 450\,days for Gl\,411 in this group, which clearly disagrees with the shorter period reported from photometry by \citet{diaz19} of $56.15 \pm 0.27$d, and is unexpectedly long compared to all the M dwarfs with known stellar rotation periods (see Fig.~\ref{fig:protmassdiagram}). It would imply an unknown mechanism of angular momentum loss. We show the GP fit in Fig.~\ref{fig:GL411_blong_GP} that corresponds to this particular star, and its associated corner plot is presented in Fig.~\ref{fig:GL411_blong_CP}. The measured period may correspond to a cyclic variation in the magnetic field that is more related to the variation of the activity than to the stellar rotation.
We were able to measure the rotation period for half of the mid-M dwarfs compared to 100\% for the early-M dwarfs. Furthermore, a six-parameter fit could only be measured for three stars compared to four out of six for the early-M dwarfs. This may come from a longer decay time that is not constrained enough by our three-year survey. The shape of the variation in the longitudinal magnetic field varies strongly, from almost sinusoidal variations for Gl\,169.1A (smoothing factor of 1) to a variation featuring only a few harmonics for Gl\,48, GJ\,3378 or GJ\,4333 (smoothing factor of 0.6). The amplitude of the variations ranges from very weak (5~G for Gl\,411) to about 20 G for GJ\,4333 and Gl\,876. For this group, the white-noise component of the GP is always compatible with 0 at $2 \sigma$, as was the case for the early-M dwarfs.
\begin{table*}
\caption[]{Best six to four parameters fit of a quasi-periodic GP model obtained in our analysis of the stellar activity from the SPIRou $B_\ell$ data of the mid-range M dwarfs in our sample. The asterisk after a star name means that the measured period is more uncertain and was obtained by a five or four parameters fit only.}
\begin{center}
\label{tab:middlegpfitparams}
\scalebox{0.9}{
\begin{tabular}{lccccccccc}
\hline
\noalign{\smallskip}
Star & rotation period & mean $B_\ell$ & white noise & amplitude & decay time & smoothing factor & rms & $\chi^2_{\rm red}$ & visits \\
& $P_{\rm rot}$ [d] & $\mu$ [G] & $\sigma$ [G] & $\alpha$ [G] & $l$ [d] & $\beta$ & [G] & & \\
\noalign{\smallskip}
Priors & $\mathcal{U}(2,300)$ & $\mathcal{U}(-\infty,+\infty)$ & $\mathcal{U}(0,+\infty)$ & $\mathcal{U}(0,+\infty)$ & $\mathcal{U}(50,1000)$ & $\mathcal{U}(0.25,1.25)$ & & & \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Gl\,752A & $53.2^{+5.5}_{-3.0}$ & $0.2^{+3.4}_{-3.2}$ & $1.6^{+1.3}_{-1.1}$ & $8.6^{+2.9}_{-1.9}$ & $93^{+55}_{-29}$ & $0.89^{+0.24}_{-0.29}$ & 6.2 & 0.87 & 128 \\
\noalign{\smallskip}
Gl\,48 & $51.2^{+1.4}_{-1.4}$ & $-7.0^{+3.6}_{-3.7}$ & $1.8^{+1.6}_{-1.2}$ & $10.2^{+3.2}_{-2.1}$ & $112^{+55}_{-36}$ & $0.62^{+0.22}_{-0.16}$ & 9.4 & 0.86 & 188 \\
\noalign{\smallskip}
Gl\,15A & $44.3^{+2.0}_{-2.0}$ & $-1.0^{+1.8}_{-1.7}$ & $1.0^{+0.9}_{-0.7}$ & $5.4^{+1.4}_{-1.0}$ & $85^{+36}_{-22}$ & $0.71^{+0.28}_{-0.23}$ & 5.6 & 0.82 & 246 \\
\noalign{\smallskip}
Gl\,849 & $41.4^{+0.4}_{-0.4}$ & $4.7^{+6.3}_{-5.9}$ & $2.4^{+1.6}_{-1.6}$ & $12.1^{+5.1}_{-3.2}$ & & $0.87^{+0.24}_{-0.23}$ & 10.0 & 1.01 & 185 \\
\noalign{\smallskip}
Gl\,411 & $471^{+41}_{-40}$ & $5.9^{+2.4}_{-2.1}$ & $0.8^{+0.8}_{-0.6}$ & $4.2^{+1.8}_{-1.1}$ & & $0.68^{+0.32}_{-0.27}$ & 6.3 & 0.94 & 212 \\
\noalign{\smallskip}
Gl\,687 & $56.5^{+1.3}_{-0.5}$ & $4.9^{+8.9}_{-6.4}$ & $1.7^{+1.3}_{-1.1}$ & $13.3^{+7.3}_{-4.3}$ & & $0.97^{+0.19}_{-0.26}$ & 8.2 & 1.10 & 214 \\
\noalign{\smallskip}
Gl\,725A & $103.5^{+4.6}_{-5.1}$ & $-14.9^{+3.4}_{-3.1}$ & $1.2^{+1.2}_{-0.8}$ & $8.7^{+2.4}_{-1.6}$ & & & 8.1 & 0.81 & 214 \\
\noalign{\smallskip}
Gl\,251$^{\ast}$ & $98.7^{+11.5}_{-4.8}$ & $17.4^{+3.6}_{-3.6}$ & $1.7^{+1.6}_{-1.2}$ & $8.3^{+2.4}_{-1.9}$ & & & 9.9 & 0.85 & 177 \\
\noalign{\smallskip}
GJ\,4333$^{\ast}$ & $72.0^{+0.9}_{-1.2}$ & $6.8^{+8.9}_{-8.0}$ & $2.6^{+2.1}_{-1.7}$ & $18.5^{+8.9}_{-4.1}$ & & $0.55^{+0.25}_{-0.17}$ & 12.7 & 0.83 & 190 \\
\noalign{\smallskip}
Gl\,876 & $82.8^{+2.0}_{-0.7}$ & $3.5^{+13.3}_{-13.5}$ & $2.0^{+1.9}_{-1.4}$ & $23.0^{+12.3}_{-7.1}$ & & $0.72^{+0.24}_{-0.19}$ & 8.5 & 0.92 & 88 \\
\noalign{\smallskip}
PM J09553-2715$^{\ast}$ & $70.5^{+5.7}_{-1.9}$ & $15.4^{+9.8}_{-11.7}$ & $2.7^{+2.4}_{-1.9}$ & $15.1^{+8.7}_{-5.1}$ & & $0.82^{+0.30}_{-0.33}$ & 11.8 & 0.91 & 74 \\
\noalign{\smallskip}
GJ\,3378$^{\ast}$ & $92.1^{+4.1}_{-5.3}$ & $13.6^{+5.6}_{-5.4}$ & $4.6^{+2.3}_{-2.7}$ & $12.0^{+5.1}_{-2.8}$ & & $0.61^{+0.32}_{-0.23}$ & 13.9 & 1.00 & 174 \\
\noalign{\smallskip}
Gl\,169.1A & $91.9^{+4.1}_{-2.6}$ & $2.3^{+8.4}_{-7.2}$ & $1.9^{+1.9}_{-1.3}$ & $11.4^{+5.3}_{-3.3}$ & & $1.04^{+0.15}_{-0.23}$ & 13.4 & 0.91 & 172 \\
\noalign{\smallskip}
\hline
\end{tabular}}
\end{center}
\end{table*}
In Table \ref{tab:midlit} we compare our new values of the rotation period to those from the literature. The agreement is good in general, but some literature values are discrepant. Excluding possible sampling problems or a low number of measurements in one case (Gl\,876), there may be more fundamental reasons for some techniques to succeed or fail on a given star: photometric variations may be affected by flares, while the longitudinal magnetic field of the star may stay constant due to an axisymmetric field topology that prevents us from detecting a periodic variation even when the Stokes $V$ profiles show a clear detection. At least in one case, Gl\,411, the very long measured period may in fact reflect a cyclic variation of activity rather than the stellar rotation period. We also show at the end of Sec. \ref{sec:late} that some stars have a clear detection of the Stokes $V$ profile, but no rotation period detected when the magnetic field is axisymmetric.
\begin{table*}
\caption[]{Comparison of the rotation periods given in the literature with our measured values for the mid-M stars.}
\label{tab:midlit}
\begin{center}
\scalebox{0.7}{
\begin{tabular}{lcccccccccccc}
\hline
\noalign{\smallskip}
Reference & Category & Gl\,752A & Gl\,48 & Gl\,15A & Gl\,849 & Gl\,411 & Gl\,687 & Gl\,251 & GJ\,4333 & Gl\,876 & GJ\,3378 \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
This work & $B_\ell$ & $53 \pm 4$ & $51.2 \pm 1.4$ & $44.3 \pm 2.0$ & $41.4 \pm 0.4$ & $470 \pm 40$ & $56.5 \pm 0.9$ & $99 \pm 8$ & $72.0 \pm 1.0$ & $82.8 \pm 1.4$ & $92 \pm 5$ \\
\citet{rivera05} & phm & & & & & & & & & $96.7 \pm 1.0$ \\
\citet{bonfils13} & act & & & & 2000 ? & & & & & 61.0, 30.1 \\
\citet{burt14} & phm & & & & & & $61.8 \pm 1.0$ \\
\citet{howard14} & phm & & & $43.82 \pm 0.56$ & & & & & & & \\
\citet{suarez15} & act & $46.5 \pm 0.3$ & & & $39.2 \pm 6.3$ & & & & & $87.3 \pm 5.7$ \\
\citet{nelson16} & act & & & & & & & & & $95 \pm 1$ \\
\citet{suarez16} & phm & $46.0 \pm 0.2$ & & & & & & \\
\citet{moutou17} & ZDI & & & & & & & $90 \pm 10$ & & & \\
\citet{suarez17} & act & & & & & & & & & $90.9 \pm 16.5$ \\
\citet{suarez18} & act & & & $45.0 \pm 4.4$ & & & & \\
\citet{diaz19} & phm & & & & & $56.15 \pm 0.27$ & & \\
\citet{diez19} & phm & $46.0 \pm 0.2$ & $51.5 \pm 2.6$ & & & & & $18.1 \pm 0.3$ & $74.7 \pm 0.7$ & $81.0 \pm 0.8$ & \\
\citet{reinhold20} & phm & & & & & & & & & $31.31 \pm 8.15$ & \\
\citet{stock20} & phm & & & & & & & $122.1 \pm 2.2$ & & & \\
\citet{sabotta21} & act & 174.48 & 43.39 & & & & & 67.59, 119.48 & & & 83.39 \\
\hline
\end{tabular}}
\end{center}
\end{table*}
We also tested the results of the \texttt{Libre-Esprit} pipeline as given in Carmona et al. (in prep) for Gl\,388 (AD\,Leo), even though this star was discarded from our sample because it is too active. Table \ref{tab:gl388} compares the results, which agree excellently well.
\begin{table*}
\caption[]{Best six-parameter fit of a quasi-periodic GP model obtained in our analysis of the stellar activity from the SPIRou $B_\ell$ data of Gl\,388 (AD\,Leo), compared to the results from Carmona et al. (in prep) using a similar data set, but the \texttt{Libre-Esprit} data reduction and analysis package. }
\label{tab:gl388}
\begin{tabular}{lccccccccc}
\hline
\noalign{\smallskip}
Package & rotation period & mean $B_\ell$ & white noise & amplitude & decay time & smoothing factor & rms & $\chi^2_{\rm red}$ & visits \\
& $P_{\rm rot}$ [d] & $\mu$ [G] & $\sigma$ [G] & $\alpha$ [G] & $l$ [d] & $\beta$ & [G] & & \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
APERO & $2.2301^{+0.0019}_{-0.0017}$ & $-156^{+43}_{-42}$ & $4.6^{+3.3}_{-3.0}$ & $68^{+21}_{-14}$ & $220^{+41}_{-40}$ & $1.35^{+0.11}_{-0.19}$ & 12.7 & 0.91 & 70 \\
\noalign{\smallskip}
Libre-Esprit & $2.2304^{+0.0015}_{-0.0013}$ & $-156^{+47}_{-44}$ & $5.4^{+3.7}_{-3.6}$ & $67^{+23}_{-15}$ & $255^{+96}_{-77}$ & $1.36^{+0.10}_{-0.17}$ & 14.7 & 0.96 & 69 \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{table*}
\subsection{Late-M dwarfs}
\label{sec:late}
The last group contains ten stars with $M_G$ above 11 (up to 13.5), spectral types from M3.5V to M5.5V, and masses between 0.1 and 0.2 M$\mathrm{_\odot}$. These stars are fully convective. Stars with later spectral types (M6.0V and M6.5V) in the WP1 sample are all active stars with short rotation periods and are therefore excluded from this study. To compute the LSD profiles of these late-type stars, we used a mask with atomic lines of the known Land\'e factor at 3000~K and $\log g$=5.0. We generally had to use a four-parameter fit of the quasi-periodic GP model for them, fixing the smoothing parameter to 0.7 and the decay time to 200 days, except for two stars in this group, for which a six-parameter fit gives well-constrained values of the smoothing factor and the decay time. The others are marked with an asterisk in Table \ref{tab:lategpfitparams}. For only one star, Gl\,447 (Ross\,128, FI\,Vir), are we unable to confirm the long rotation period measured from ASAS photometry by \cite{suarez16} ($165.1 \pm 0.8$~d) and \citet{diez19} ($163 \pm 3$~d). This is probably due to the low number and distribution of the visits (57, clustered into two groups) compared to the other late-M dwarfs, which prevents us from determining the rotation period of Gl\,447. This highlights once again that our set limit to 50 visits with adequate sampling is essential. Below this value, the determination of the rotation period becomes difficult.
\begin{table*}
\caption[]{Best six, five, or four parameters fit of a quasi-periodic GP model obtained in our analysis of the stellar activity from the SPIRou $B_\ell$ data of the late-M dwarfs in our sample. The asterisk after a star name means that the measured period is more uncertain and was obtained by a four-parameter fit only.}
\label{tab:lategpfitparams}
\begin{center}
\begin{tabular}{lccccccccc}
\hline
\noalign{\smallskip}
Star & rotation period & mean $B_\ell$ & white noise & amplitude & decay time & smoothing factor & rms & $\chi^2_{\rm red}$ & visits \\
& $P_{\rm rot}$ [d] & $\mu$ [G] & $\sigma$ [G] & $\alpha$ [G] & $l$ [d] & $\beta$ & [G] & & \\
\noalign{\smallskip}
Priors & $\mathcal{U}(2,300)$ & $\mathcal{U}(-\infty,+\infty)$ & $\mathcal{U}(0,+\infty)$ & $\mathcal{U}(0,+\infty)$ & $\mathcal{U}(50,1000)$ & $\mathcal{U}(0.25,1.25)$ & & & \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
GJ\,1289 & $74.0^{+1.5}_{-1.3}$ & $47^{+25}_{-25}$ & $2.4^{+2.5}_{-1.7}$ & $67^{+21}_{-12}$ & $142^{+33}_{-26}$ & $0.64^{+0.17}_{-0.11}$ & 14.7 & 0.67 & 180 \\
\noalign{\smallskip}
GJ\,1103$^{\ast}$ & $139^{+22}_{-23}$ & $8.2^{+10.1}_{-10.1}$ & $4.1^{+4.0}_{-2.9}$ & $18.2^{+8.9}_{-5.9}$ & & & 16.4 & 0.83 & 62 \\
\noalign{\smallskip}
Gl\,699$^{\ast}$ & $137.1^{+6.8}_{-4.0}$ & $3.0^{+5.7}_{-5.8}$ & $1.1^{+1.1}_{-0.8}$ & $14.7^{+3.0}_{-2.4}$ & & & 8.7 & 0.77 & 243 \\
\noalign{\smallskip}
Gl\,15B$^{\ast}$ & $116.5^{+6.4}_{-4.7}$ & $-0.3^{+6.3}_{-6.2}$ & $2.1^{+2.1}_{-1.4}$ & $13.8^{+3.8}_{-2.8}$ & & & 16.5 & 0.78 & 184 \\
\noalign{\smallskip}
GJ\,1151$^{\ast}$ & $158^{+14}_{-9}$ & $-9.6^{+13.4}_{-13.2}$ & $2.7^{+2.6}_{-1.9}$ & $29.6^{+8.2}_{-5.5}$ & & & 15.7 & 0.83 & 153 \\
\noalign{\smallskip}
Gl\,905 & $109.5^{+4.9}_{-5.4}$ & $-14^{+11}_{-11}$ & $1.7^{+1.8}_{-1.2}$ & $29.4^{+8.1}_{-5.3}$ & $149^{+26}_{-25}$ & $0.71^{+0.21}_{-0.17}$ & 12.6 & 0.73 & 216 \\
\noalign{\smallskip}
GJ\,1286$^{\ast}$ & $203^{+14}_{-21}$ & $36^{+19}_{-20}$ & $5.7^{+4.5}_{-3.8}$ & $44^{+14}_{-10}$ & & & 25.0 & 0.89 & 108 \\
\noalign{\smallskip}
GJ\,1002 & $93.0^{+1.4}_{-1.7}$ & $-6.4^{+12.5}_{-14.2}$ & $4.1^{+3.4}_{-2.8}$ & $20.4^{+9.7}_{-5.9}$ & & $0.90^{+0.24}_{-0.31}$ & 19.9 & 1.05 & 154 \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{center}
\end{table*}
In Table \ref{tab:latelit} we compare our new rotation periods to those from the literature. The agreement is not as good as for earlier M dwarfs, probably because their rotation periods are longer and the uncertainties are larger. It becomes naturally harder to determine very long periods as we are limited by the time range of the SPIRou observations.
\begin{table*}
\caption[]{Comparison of the rotation periods given in the literature with our measured values for the late Ms.}
\label{tab:latelit}
\begin{center}
\begin{tabular}{lccccccc}
\hline
\noalign{\smallskip}
Reference & Category & GJ\,1289 & Gl\,699 & GJ\,1151 & Gl\,905 & GJ\,1286 & GJ\,1002 \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
This work & $B_\ell$ & $74.0 \pm 1.4$ & $137 \pm 5$ & $158 \pm 12$ & $110 \pm 5$ & $203 \pm 18$ & $93.0 \pm 1.6$ \\
\citet{benedict98} & phm & & 130.4 & & & & \\
\citet{irwin11} & phm & & & 132 & & & \\
\citet{suarez15} & act & & $148.6 \pm 0.1$ & & & & \\
\citet{moutou17} & ZDI & $54 \pm 4$ & & & & & \\
\citet{newton18} & phm & & & & & 88.92 & \\
\citet{toledo19} & phm & & $145 \pm 15$ & & & & \\
\citet{diez19} & phm & $83.6 \pm 7.0$ & & $125 \pm 23$ & $106 \pm 6$ & & \\
\citet{sabotta21} & act & & 311.25 & & 178.74 & & \\
\citet{suarez22} & act & & & & & & $126 \pm 15$ \\
\hline
\end{tabular}
\end{center}
\end{table*}
Finally, in Table \ref{tab:undetectedgpfitparams}, we present the data of the 16 stars for which we were unable to measure a rotation period. For these stars, we arbitrarily fixed the decay parameter to 200~d and the smoothing factor to 0.7 to obtain some indications about the mean $B_\ell$, residual white noise, and amplitude of the field. The GP also gives a value of the rotation period, but we do not list it as it is not well constrained. We assumed that the values of the other parameters of the GP still bear some information, although the fact that the rotation period is not constrained and that two parameters of the GP are arbitrarily fixed limits the value of this information. For completeness, we list a period and its reference, when available in the literature, but this information was not used in our GP fit.
The Stokes $V$ profiles in Appendix \ref{sec:stokesV} may tentatively explain these nondetections by their magnetic topology: if the magnetic field is axisymmetric, we cannot detect the rotation modulation. To test this interpretation, we measured the rate of detection of Stokes V profiles for a given star by comparison to the noise measured in a velocity region well outside of the line. We find that this metric quantifies the visual impression of the time series of Stokes V profiles quite well, as shown in Appendix \ref{sec:stokesV}. Then, we subtracted a median Stokes V profile for each star and measured the detection rate again. For some stars, it changes dramatically, and we tentatively interpret this change by the subtraction of a constant component due to an axisymmetric magnetic field. This is the case of Gl\,408, Gl\,338B, Gl\,436, GJ\,4063, and Gl\,617B. For other stars, the detection rate does not significantly change when the median-subtracted profiles are compared to the original Stokes V profiles. This is the case of PM\,J21463+3813, Gl\,412A, PM\,J08402+3127, Gl\,480, and GJ\,1148 among the stars without a detected rotation period. We plan a more detailed study of the magnetic topology of the stars in our sample to further investigate the reasons for these nondetections.
\begin{table*}
\caption[]{Best four-parameter fit of a quasi-periodic GP model obtained in our analysis of the stellar activity from the SPIRou $B_\ell$ data of the M dwarfs in our sample without a clear periodic variation detection.}
\label{tab:undetectedgpfitparams}
\begin{center}
\scalebox{0.8}{
\begin{tabular}{lcccccccc}
\hline
\noalign{\smallskip}
Star & mean $B_\ell$ & white noise & amplitude & rms & $\chi^2_{\rm red}$ & number of visits & rotation period from literature & reference \\
& $\mu$ [G] & $\sigma$ [G] & $\alpha$ [G] & [G] & & & $P_{\rm rot}$ [d] & \\
\noalign{\smallskip}
Priors & $\mathcal{U}(-\infty,+\infty)$ & $\mathcal{U}(0,+\infty)$ & $\mathcal{U}(0,+\infty)$ & & & & & \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Gl\,338B & $-8.1^{+1.6}_{-1.6}$ & $1.3^{+1.0}_{-0.9}$ & $3.1^{+1.6}_{-1.4}$ & 4.4 & 1.03 & 58 & 16.66 & \citet{sabotta21} \\
\noalign{\smallskip}
Gl\,617B & $19.3^{+4.0}_{-4.0}$ & $1.4^{+1.5}_{-1.0}$ & $8.9^{+2.9}_{-2.1}$ & 8.2 & 0.71 & 142 & \\
\noalign{\smallskip}
Gl\,412A & $13.7^{+4.1}_{-4.3}$ & $4.0^{+1.2}_{-1.4}$ & $10.7^{+3.5}_{-2.5}$ & 9.7 & 1.23 & 174 & $100.9 \pm 0.3$ & \citet{suarez18} \\
\noalign{\smallskip}
Gl\,480 & $6.6^{+3.5}_{-3.3}$ & $5.5^{+1.8}_{-2.1}$ & $6.9^{+2.8}_{-2.3}$ & 11.0 & 1.28 & 104 & & \\
\noalign{\smallskip}
Gl\,436 & $-10.4^{+2.2}_{-2.0}$ & $1.6^{+1.6}_{-1.1}$ & $4.4^{+2.1}_{-1.7}$ & 7.5 & 0.81 & 85 & $44.09 \pm 0.08$ & \citet{bourrier18} \\
\noalign{\smallskip}
Gl\,408 & $-43.8^{+3.6}_{-3.6}$ & $1.4^{+1.4}_{-1.0}$ & $8.7^{+2.6}_{-2.0}$ & 9.1 & 0.79 & 168 & & \\
\noalign{\smallskip}
Gl\,317 & $-10.8^{+8.4}_{-8.8}$ & $2.7^{+2.5}_{-1.9}$ & $14.5^{+8.2}_{-4.1}$ & 11.9 & 0.97 & 76 & & \\
\noalign{\smallskip}
GJ\,4063 & $17.8^{+3.3}_{-3.2}$ & $2.9^{+1.9}_{-1.9}$ & $7.1^{+2.3}_{-1.8}$ & 11.4 & 0.96 & 204 & $40.2 \pm 0.8$ & \citet{diez19} \\
\noalign{\smallskip}
GJ\,1012 & $6.4^{+2.6}_{-2.6}$ & $3.2^{+2.7}_{-2.2}$ & $4.0^{+3.8}_{-2.7}$ & 15.9 & 0.99 & 135 & & \\
\noalign{\smallskip}
GJ\,1148 & $-9.1^{+3.9}_{-3.5}$ & $2.3^{+2.4}_{-1.6}$ & $7.1^{+4.1}_{-2.9}$ & 13.9 & 0.81 & 101 & $71.5 \pm 5.1$ & \citet{diez19} \\
\noalign{\smallskip}
PM J08402+3127 & $16^{+20}_{-21}$ & $4.6^{+3.8}_{-3.1}$ & $39^{+16}_{-9}$ & 18.8 & 0.98 & 139 & $118 \pm 14$ & \citet{diez19} \\
\noalign{\smallskip}
Gl\,725B & $-4.6^{+5.2}_{-5.1}$ & $1.2^{+1.3}_{-0.8}$ & $12.1^{+3.7}_{-2.8}$ & 10.7 & 0.75 & 209 \\
\noalign{\smallskip}
GJ\,1105 & $1.2^{+3.6}_{-3.1}$ & $2.4^{+2.2}_{-1.7}$ & $7.3^{+3.2}_{-2.7}$ & 13.5 & 0.93 & 161 \\
\noalign{\smallskip}
Gl\,445 & $-1.8^{+2.7}_{-2.6}$ & $3.4^{+3.2}_{-2.4}$ & $3.2^{+3.7}_{-2.3}$ & 17.4 & 0.97 & 90 & & \\
\noalign{\smallskip}
PM J21463+3813 & $-0.4^{+7.5}_{-7.7}$ & $3.1^{+3.1}_{-2.2}$ & $16.3^{+6.6}_{-4.6}$ & 20.7 & 0.84 & 176 \\
\noalign{\smallskip}
Gl\,447 & $20.5^{+9.1}_{-9.5}$ & $4.8^{+3.0}_{-3.0}$ & $16.6^{+8.5}_{-5.3}$ & 14.9 & 1.13 & 57 & $165.1 \pm 0.8$ & \citet{suarez16} \\
\noalign{\smallskip}
\hline
\end{tabular}}
\end{center}
\end{table*}
\section{Discussion and conclusions}
\label{sec:conclusions}
We have shown that spectropolarimetry is a useful technique for measuring the rotation period of a star, even for quiet M dwarfs. In our sample of 43 such stars, we were able to reliably measure the rotation period for 27 stars, 8 of which were previously unknown. The rotation periods cover a wide range in this sample of quiet stars, from 10 to 450 days. The agreement with other techniques, such as photometry or stellar activity indicators is good, except for a few stars (e.g., Gl\,251, Gl\,411, and Gl\,846) for which some techniques find a harmonic of the rotation period or converge toward very different values. The amplitude of variation in the longitudinal magnetic field for these quiet M dwarfs ranges from 3 G (Gl\,338B, Gl\,445) to 20 G for the majority of the stars in our sample, but it can reach up to 70 G in some cases (GJ\,1289), which is comparable to more active stars such as AD\,Leo (Gl\,388: $45 \pm 6$\;G, Carmona et al., in prep) or EV\,Lac (Gl\,873: $149 \pm 17$\;G) over the same period of observations. The FWHM of the median Stokes $I$ profile is another important measurement, as it may reveal a broadening due to the Zeeman effect for active stars. As the FWHM is sensitive to the total magnetic field (small and large scale),while the longitudinal magnetic field is more sensitive to the large-scale field, a large amplitude of variation in the longitudinal field does not necessarily correlate with a high FWHM, and both measures are therefore important. The average value over our sample is 6.07 km\,s$^{-1}$\, , with a standard deviation of 0.66 km\,s$^{-1}$\,. This is clearly larger than the mean uncertainty of each FWHM in Table \ref{tab:samplechars}. This shows a real dispersion among the mean FWHM values, which will be further investigated by using different line lists (low and high Land\'e factors) and by studying the time variation in the measured FWHM for a given star, as in Bellotti et al. (in prep) for AD\,Leo.
We divided our sample into three sub-categories according to the absolute magnitude in {\it Gaia} $G$ band, roughly corresponding to early-, mid- and late-M dwarfs. There is a clear tendency to lower values of the FWHM of the median Stokes $I$ profile toward mid-type stars, with averages and standard deviations for the three groups of 6.65 and 0.59 km\,s$^{-1}$\, (early type), 5.73 and 0.34 km\,s$^{-1}$\, (mid type), and 6.49 and 0.75 km\,s$^{-1}$\, (late type), respectively. The larger dispersion in the early-type and late-type groups may be due to a few active outliers: removing Gl\,410 from the early-type group gives new average and standard deviation of 6.45 and 0.40 km\,s$^{-1}$\,, respectively, while for the late-type group, removing GJ\,1286 and GJ\,1289 gives 6.14 and 0.52 km\,s$^{-1}$\,, respectively, now more compatible with the mid-type group, given the small size of the early- and late-type groups. The most extreme values correspond to Gl\,411 ($4.93\pm0.18$ km\,s$^{-1}$\,) and Gl\,410 ($7.73 \pm 0.21$ km\,s$^{-1}$\,). A test on GJ\,1289 shows that the measured FWHM clearly depends on the minimum adopted depth (0.03 in this work) and the LSD mask (3000 and 3500\,K give different results for this 3238\,K star), and does not clearly show an increase when using masks limited to low Land\'e factors ($g_{\rm eff}$\, $\leqslant 1.2$) versus high Land\'e factors ($g_{\rm eff}$\, $> 1.2$). The possible effect of the magnetic field on the Stokes I FWHM of the stars in this sample will be studied in detail in Donati et al. (in prep) and Cristofari et al. (in prep).
We observe that it is easier to constrain the rotation period and the other parameters of the GP for early-type stars, even if the amplitude of variation of the longitudinal magnetic field is similar in the other two subcategories. This may be due to the fact that they are brighter than later M dwarfs on average. Additionally, the decay time seems well constrained between 50 and 100 days for the early-type stars. For the mid-M dwarfs, it lies around 100 days, and for the late-M dwarfs, it seems to be longer at about 150 days, although these results are only based on three and two measurements, respectively. There is a tendency for longer decay times for a longer stellar rotation period. Clearly, the quasi-periodic GP fit converges more easily for the early-type stars than for the others: it converges for only 60\% of the mid-type stars, and although the convergence rate is similar for the late- and early-type stars, we have to use a four-parameter fit for the late-type stars (fixing the smoothing factor and the decay time), while we can use a six-parameter fit for the early-type stars with good constraints on the smoothing factor and on the decay time.
\begin{figure}
\centering
\includegraphics[width=\hsize]{NewtonComparison.png}
\caption{Rotation period vs stellar mass diagram. Stars with rotation periods determined in this work are shown as larger circles, where the symbol sizes and colors are proportional to the amplitude of variation of $B_\ell$ (see $\alpha$ values given in Tables~\ref{tab:earlygpfitparams}, \ref{tab:middlegpfitparams}, and \ref{tab:lategpfitparams}). Rotation periods and mass determinations for inactive (white circles) and active (gray circles) stars from \citet{newton17} are shown for comparison.}
\label{fig:protmassdiagram}
\end{figure}
Our sample is too small to infer general properties of M stars. Therefore, we compare in Fig.~\ref{fig:protmassdiagram} the sample of stars analyzed in this paper to more complete nonmagnetic studies in which active and inactive stars are well represented. We include a total of 212 M stars from \citet{newton17}, whose mass and rotation periods are similar to the range explored in this paper. Our sample of quiet stars nicely fits the sequence of inactive stars, as expected. The long rotation period of Gl\,411 appears as an exception, and may in fact not reflect the stellar rotation period, as discussed in Sec. \ref{sec:middle}.
Low-mass stars arrive rapidly rotating on the main sequence and spin down with time as angular momentum is lost through magnetized winds \citep{barnes07}. Empirically calibrated relations between age and period can be used to estimate the ages of individual field stars. \citet{gaidos23} combined the 4\,Gyr M-dwarf gyrochrone of the open cluster M 67 \citep{dungee22} with those of younger, previously published gyrochrones \citep{curtis20} to assign ages to M dwarf host stars with $T_{\rm eff}=$3200-4200\,K. We used the same gyrochronology to estimate the ages of 20 stars in our sample with well-measured rotation periods and $T_{\rm eff}$ in that range. Mean values and standard deviations were calculated from distributions constructed by 10000 Monte Carlo calculations, incorporating uncertainties from $P_{\rm rot}$ (from Tables \ref{tab:middlegpfitparams} and \ref{tab:lategpfitparams}), the gyrochrones, $T_{\rm eff}$ (assumed $\pm$75K), [Fe/H] (assumed to be $\pm$0.1 dex), and variation in the initial rotation periods of the stars on the zero-age main sequence. The results are listed in Table \ref{tab:ages}: most stars are 4--10 Gyr old, and a few younger stars have ages from 0.5 to 2\,Gyr. The rotation-age relations are only valid for stars of approximately solar metallicity, but we list in the table (col. 6) a correction computed using a theory-based model and the stellar metallicity, to be added to the age given in col. 4. We report the age for members of binaries, but we emphasize that these stars may have distinct rotational histories due to tides and rapid dissipation of primordial disks \citep{fleming19, messina19}.
\begin{table*}
\caption[]{Gyrochronological ages of the stars in our sample. Errors on $T_{\rm eff}$ and [M/H] are assumed to be 75\,K and 0.1 dex. The correction in the last column corresponds to the value to be added to the age due to the nonsolar metallicity of the star.}
\label{tab:ages}
\begin{center}
\begin{tabular}{lccccc}
\hline
\noalign{\smallskip}
Star & $T_{\rm eff}$ & $P_{\rm rot}$ & Age & [M/H] & Metallicity correction \\
& K & d & Gyr & dex & Gyr \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Gl\,846 & 3833 & $11.01 \pm 0.20$ & $<0.53$ & 0.07 & 0.1 \\
Gl\,410 & 3842 & $13.87 \pm 0.08$ & $0.89 \pm 0.12$ & 0.05 & 0.1 \\
Gl\,382 & 3644 & $21.32 \pm 0.04$ & $1.9 \pm 0.6$ & 0.15 & 0.2 \\
Gl\,514 & 3699 & $30.45 \pm 0.14$ & $3.8 \pm 0.6$ & $-0.07$ & $-0.1$ \\
GJ\,1289 & 3238 & $74.0 \pm 1.4$ & $3.8 \pm 1.2$ & 0.05 & 0.0 \\
Gl\,849 & 3502 & $41.4 \pm 0.4$ & $4.2 \pm 1.0$ & 0.35 & 0.4 \\
Gl\,205 & 3771 & $34.3 \pm 0.4$ & $5.2 \pm 0.7$ & 0.43 & 0.6 \\
PM J09553-2715 & 3366 & $70.5 \pm 3.8$ & $5.2 \pm 2.0$ & $-0.03$ & $-0.0$ \\
Gl\,880 & 3702 & $37.7 \pm 0.7$ & $5.5 \pm 0.8$ & 0.26 & 0.4 \\
Gl\,169.1A & 3307 & $91.9 \pm 3.4$ & $5.5 \pm 2.4$ & 0.13 & 0.1 \\
Gl\,687 & 3475 & $56.5 \pm 0.9$ & $5.9 \pm 1.8$ & 0.01 & 0.0 \\
Gl\,15A & 3611 & $44.3 \pm 0.2$ & $6.0 \pm 1.2$ & $-0.33$ & $-0.5$ \\
GJ\,3378 & 3326 & $92.1 \pm 4.7$ & $6.0 \pm 2.7$ & $-0.05$ & $-0.0$ \\
Gl\,48 & 3529 & $51.2 \pm 1.4$ & $6.1 \pm 1.6$ & 0.08 & 0.1 \\
Gl\,15B & 3272 & $116.5 \pm 5.6$ & $6.1 \pm 3.0$ & $-0.42$ & $-0.2$ \\
Gl\,876 & 3366 & $82.8 \pm 1.4$ & $6.2 \pm 2.7$ & 0.15 & 0.1 \\
Gl\,752A & 3558 & $53.2 \pm 4.2$ & $7.1 \pm 1.9$ & 0.11 & 0.2 \\
Gl\,699 & 3311 & $137.1 \pm 5.4$ & $9.4 \pm 5.0$ & $-0.37$ & $-0.2$ \\
Gl\,251 & 3420 & $98.7 \pm 8.2$ & $10.8 \pm 4.7$ & $-0.01$ & $-0.0$ \\
Gl\,725A & 3470 & $103.5 \pm 4.8$ & $14.8 \pm 5.4$ & $-0.26$ & $-0.3$ \\
\hline
\end{tabular}
\end{center}
\end{table*}
Our study only exploited the longitudinal magnetic field measured from the spectropolarimetric data of the SPIRou SLS. The same data can be used to achieve much more, for example, to investigate the magnetic topology of these stars and its evolution using ZDI maps.
\begin{acknowledgements}
Based on observations obtained at the Canada-France-Hawaii Telescope (CFHT) which is operated from the summit of Maunakea by the National Research Council of Canada, the \emph{Institut National des Sciences de l'Univers} of the \emph{Centre National de la Recherche Scientifique} of France, and the University of Hawaii. The observations at the Canada-France-Hawaii Telescope were performed with care and respect from the summit of Maunakea which is a significant cultural and historic site. \\
This work has made use of data from the European Space Agency (ESA) mission {\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia} Data Processing and Analysis Consortium (DPAC, \url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC has been provided by national institutions, in particular, the institutions participating in the {\it Gaia} Multilateral Agreement.\\
This research has also made intensive use of the SIMBAD database and of the VizieR catalog access tool, operated at CDS, Strasbourg, France and of NASA's Astrophysics Data System (ADS). \\
We made an extensive use of the DACE (Data Analysis Center for Exoplanets) software from the University of Geneva \citep{diaz14, delisle16}. \\
This work has made use of the VALD database, operated at Uppsala University, the Institute of Astronomy RAS in Moscow, and the University of Vienna. \\
X.D., A.C., P.C.Z., E.M. and S.B. and more generally most of the French authors of this paper acknowledge funding from the French National Research Agency (ANR) under contract number ANR\-18\-CE31\-0019 (SPlaSH). This work is supported by the ANR in the framework of the \emph{Investissements d'Avenir} program (ANR-15-IDEX-02), through the funding of the ``Origin of Life'' project of the Grenoble-Alpes University. \\
E.M. acknowledges funding from the \emph{Funda\c{c}\~{a}o de Amparo \`{a} Pesquisa do Estado de Minas Gerais} (FAPEMIG) under the project number APQ-02493-22. \\
B.Z. acknowledges funding from the \emph{Programa de Internacionaliza\c{c}\~{a}o da Coordena\c{c}\~{a}o de Aperfei\c{c}oamento de Pessoal de Nível Superior} (CAPES-PrInt \#88887.683070/2022-00) and FAPEMIG (APQ-01033-22). \\
J.-F.D. acknowledges funding from the European Research Council (ERC) under the H2020 research \& innovation programme (grant agreement \#740651 NewWorlds). \\
N.J.C., E.A. and R.D. wish to thank the Natural Sciences and Engineering Research Council of Canada and the \emph{Fonds Qu\'eb\'ecois de Recherche - Nature et Technologies}, the \emph{Observatoire du Mont-M\'egantic} and the Institute for Research on Exoplanets and acknowledge funding from \emph{D\'eveloppement Economique Canada, Quebec's Minist\`ere de l'Education et de l'Innovation}, the Trottier Family Foundation and the Canadian Space Agency. \\
This research made use of the hosting service \texttt{github} and, among others, of the following software tools: \verb|matplotlib| \citep{hunter07}; \verb|NumPy| \citep{harris20}; \verb|SciPy| \citep{scipy20}; \verb|Astropy| \citep{astropy13,astropy18}; \verb|emcee| \citep{foreman13}; \verb|corner| \citep{foreman16}; \verb|george| \citep{ambikasaran15}; \verb|barrycorpy| \citep{wright14}.
\end{acknowledgements}
\bibliographystyle{aa}
|
{
"timestamp": "2023-02-09T02:12:22",
"yymm": "2302",
"arxiv_id": "2302.03377",
"language": "en",
"url": "https://arxiv.org/abs/2302.03377"
}
|
"\\section{Introduction}\n\n\nFormal argumentation has been proved to be a successful approach to no(...TRUNCATED)
| {"timestamp":"2023-02-08T02:09:50","yymm":"2302","arxiv_id":"2302.03305","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\n\nAs a basic and important mathematical model, the diffusion equation has (...TRUNCATED)
| {"timestamp":"2023-02-08T02:11:14","yymm":"2302","arxiv_id":"2302.03333","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\\label{s:1}\nAt the moment, the concept of fractional Brownian motion is\na(...TRUNCATED)
| {"timestamp":"2023-02-08T02:12:39","yymm":"2302","arxiv_id":"2302.03363","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\\label{section:introduction}\n\\emph{Federated learning} (FL) is an algorit(...TRUNCATED)
| {"timestamp":"2023-02-08T02:10:24","yymm":"2302","arxiv_id":"2302.03314","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\n\n\n\n\n\n\n\nDeep learning approximation methods -- usually consisting (...TRUNCATED)
| {"timestamp":"2023-02-08T02:09:02","yymm":"2302","arxiv_id":"2302.03286","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction} \n\\label{sec:intro}\nFU~Orionis (FUor) and EX~Lupi (EXor) type objects are(...TRUNCATED)
| {"timestamp":"2023-02-08T02:12:58","yymm":"2302","arxiv_id":"2302.03371","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\n\\label{sect:intro}\nWith the new results from ATLAS \\cite{ATLAS:2022oge}(...TRUNCATED)
| {"timestamp":"2023-02-08T02:11:00","yymm":"2302","arxiv_id":"2302.03324","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\n\nOrdinary differential equations (ODEs) and partial differential equation(...TRUNCATED)
| {"timestamp":"2023-02-08T02:12:33","yymm":"2302","arxiv_id":"2302.03358","language":"en","url":"http(...TRUNCATED)
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