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Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 7
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 54199)
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 7
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}
\label{sec:introduction}
Blazars are a class of rare radio-loud Active Galactic Nuclei (AGN) \citep{Sandage1965}, characterized by their flat radio spectra, rapid variability in multiwavelength emission, significant polarization, and bi-modal synchrotron/Compton spectral energy distributions (SEDs). These characteristics are likely consequences of shocks driven by a powerful relativistic jet pointing to a direction close to our line of sight (l.o.s) and roughly perpendicular to the accretion disc \citep{Urry1995}. Blazars may show strong broad emission lines in their optical spectra, similar to typical quasars, or be featureless in their optical spectra; these blazars are called Flat-Spectrum Radio Quasars (FSRQs) and BL Lacertae objects (BL Lacs), respectively.
Observations show that the majority of blazars, regardless of their type, are hosted by giant elliptical galaxies (for BL Lacs, e.g.,\citealt{Stickel1991,Kotilainen1998,Kotilainen1998a,Falomo1999,Urry2000,Falomo2000,Kotilainen2005,Hyvoenen2007,LeonTavares2011,Falomo2014}; for FSRQs, e.g.,\citealt{OlguinIglesias2016}). \citet{OlguinIglesias2016} presented deep NIR images of a sample of 19 ($0.3 < z < 1.0$) FSRQs, finding that the host galaxies of their sample are luminous and apparently to follow the $\mu_{e}-R_{eff}$ relation for ellipticals and bulges, consistent with the conclusion based on BL Lacs \citep{Stickel1991}.
As yet, whether a blazar can be hosted by a disk galaxy remains an open question. Only a handful of cases have been reported that BL Lacs are found to be hosted by disk-dominated galaxies (e.g.,\citealt{Halpern1986,Abraham1991,Wurtz1996}). \citet{Nilsson2003} analyzed 100 BL Lacs from the ROSAT-Green Bank (RGB) sample obtained by using the Nordic Optical Telescope (NOT), finding that all of their spatially resolved objects are better fitted by an elliptical galaxy model ($\beta$ = 0.25) than by a disk galaxy model ($\beta$ = 1.0), though with two exceptions that may be hosted by disk galaxies (\citealt{Abraham1991,Wurtz1996} for 1419+543 and \citealt{Urry1999} for 1540+147), whose bulk properties, however, are indistinguishable from normal elliptical galaxies.
The host of the two BL Lacs, 1415+255 and 1413+135, were originally identified to be disk-dominated galaxies (\citealt{Stocke1992,Halpern1986,Lamer1999}, respectively), but recent works, based on imaging with higher resolution, classify 1415+255 as an isolated giant elliptical galaxie \citep{Gladders1997}, highlighting the importance of high quality data. Furthermore, \citet{Scarpa2000} investigated 69 spatially-resolved BL Lacs hosts and found one case of disk (that of 0446+449) and two cases with disk models preferred (1418+546, 0607+711). Despite the efforts that have been made, a definitive evidence showing that a blazar can be hosted by a disk galaxy (especially for FSRQs) remains to be found. This is mostly due to the lack of high-resolution imaging or specially resolved spectra, and the brightness of blazars is so high in optical and IR bands that the hosts are outshined.
A combination of high-resolution imaging and spatially-resolved spectroscopy may shed light on the morphology and dynamics of the host galaxy. Hence, we conduct a case study on a confirmed FSRQ at $z$ = 0.229, known as B2 0003+38A (a.k.a. S4 0003+38 or J0005+3820) using the long-slit optical spectroscopy taken by the Echellette Spectrograph and Imager (ESI) at the Keck II observatory. This paper is structured as follows: after an overview of B2 0003+38A in previous studies, we describe observations and data reduction in Section 2. In Section 3, we present the analysis of the spectral data. In Section 4, we discuss the gas kinematics in the host galaxy and the extended emission line region. We conclude with a summary in Section 5. Throughout this paper, we adopt a cosmology with $H_0$ = 0.7 km s$^{-1}$ Mpc$^{-1}$, $\Omega_{m} = 0.3$, $\Omega_{\Lambda} = 0.7$. The wavelengths of all the spectral lines are given in vacuum.
\section{Observation and Data Reduction}
\label{sec:section_name}
\subsection{The FSRQ B2 0003+38A} \label{sec:dr}
B2 0003+38A is classified as an FSRQ due to its broad emission lines in the optical spectrum and the flat radio spectrum with $\alpha = -0.3$ \citep{Stickel1994,Healey2007,Massaro2009}. Resolved by a VLBI mapping at 2.29 GHz \citep{Morabito1982}, it shows a prominent core that emits more than 95 per cent of the 2.3 GHz flux, along with a weak jet pointing to the sourth-east \citep{Fey2000}, as shown in Figure ~\ref{fig:fig0}. Kinematic analyses show quasi-stationary knots at the jet base and relativistic motions downstream \citep{Hervet2016}. \citet{Aditya2018} detected HI 21-cm absorption towards this source, finding the velocity-integrated HI 21-cm optical depths to be 1.943 $\pm$ 0.057 km s$^{-1}$, and the HI column density to be 3.54 $\pm$ 0.11 $\times 10 ^{20}$ cm$^{-2}$.
\subsection{Optical Long-slit Spectrometry} \label{sec:dr}
The quasar B2 0003+38A was observed by Keck-II Echellette Spectrograph and Imager (ESI) long-slit spectrography in its echelle mode on 2015 Sep 9.
The spectrum was taken in the optical band (wavelength coverage $\lambda$ $\sim$ 3995 $\mathrm{\AA}$ - 10198 $\mathrm{\AA}$ in the observer's frame), covering 3250 $\mathrm{\AA}$ to 8297 $\mathrm{\AA}$ in the rest frame for our target at $z \sim 0.2$.
A 1.25$\arcsec$-wide slit was employed, resulting in an instrumental dispersion $\sigma$ of 22 km s$^{-1}$.
The position angle of the slit is 41 deg (north to east, Figure ~\ref{fig:fig0}). The slit width allows for both the blazar and its jet to be covered.
Two exposures were taken with an integration time of 10 minutes per each.
Arc lamps were used during the observing campaign for wavelength calibration, and the standard star BD+28 4211 was observed about one hour before the target with identical settings for the purpose of flux calibration.
\begin{figure}[htb]
\centering
\includegraphics[width=0.45\textwidth]{sy.pdf}
\caption{The VLBI 2.292GHz cutout centered on the B2 0003+38A \citep{Lister2013}. The orientation of the ESI/Keck slit is illustrated by the blue line. The 1.25$\arcsec$ slit width is larger than the size of the cutout. The slit covers the quasar and jet. }
\label{fig:fig0}
\end{figure}
\subsection{Data reduction} \label{sec:dar}
We use a customized routine based on the IDL package \textbf{XIDL}\footnote{http://www2.keck.hawaii.edu/inst/esi/ESIRedux/index.html} to obtain a two-dimensional (hereafter 2-D) spectrum.
Our data reduction consists of four steps:
(1) after bias corrections, flat-fielding and removing cosmic rays, we stack the two exposures;
(2) the nucleus of the target quasar is traced along 10 echelle orders; and for each order, we resample the 2-D spectrum so that a resultant pixel corresponds to 10 km s$^{-1}$ in the wavelength direction and 0.15$\arcsec$ in the spatial direction;
(3) sky subtraction and flux calibration are applied on these resampled 2-D spectra;
(4) we combine these resampled 2-D spectra from the 10 orders.
\section{Data analysis and results}
\label{sec:section_name}
We conduct kinematic analyses using both the 1-D and 2-D spectrum to investigate the gas motion in the host galaxy. In Section \ref{sec:nsa}, we perform a model fitting to the 1-D spectra to contextualize the quasar and stellar components, and the information that the fitting delivers is discussed in Section \ref{sec:2ds}. With the quasar and stellar contributions removed, we scrutinize the spatially resolved emission lines in the 2-D spectra, and the gas kinematics is described in Section \ref{sec:2dsnl}. We observe three components exist in the gas: a rotating component, an extended emission-line component, and an outflow component. The former two components are further analyzed in Sections \ref{sec:disk} and \ref{sec:eelr}.
\subsection{Spectrum of the nucleus} \label{sec:nsa}
We extracted the one-dimensional (hereafter 1-D) spectrum from the 2-D spectrum as a result of data reduction by applying an aperture with a size of 3$\arcsec$ $\times$ 1.25$\arcsec$ centered on the quasar nucleus, as shown in Figure ~\ref{fig:fig1} (black lines). AGNs features, including broad emission lines (BELs), narrow emission lines (NELs) and Fe II bumps, are seen therein. However, the existence of high-order Balmer absorption lines indicate that the continuum contribution from starlight.
To disentangle different contributions to the spectrum, we construct a model consisting of four components: a stellar, a power-law, a BEL and an Fe II component.\\
1. The stellar component: we assume a single simple stellar population (SSP), utilizing the library of \citet{Bruzual2003}. When fit to the data, the template is shifted to a redshift $z_\star$, broadened by convolving to a Gaussian parameterized by a velocity dispersion parameter $\sigma_\star$, and multiplied by the extinction curve of the Small Magellan Cloud (SMC) parameterized by $E(B-V)_\star$ \citep{Pei1992}.\\
2. The power-law component: we assume a formulation of $f_\lambda(\lambda)=C \lambda^\alpha$, where $\alpha$ is the exponent and $C$ is the normalization.\\
3. The BEL component: for this component, we mainly account for the Balmer series, He I $\lambda$5876 and He II $\lambda$4686. We assume that the BEL profiles can be represented by a linear combination of two Gaussians, whose mean and standard dispersion are fixed but the flux is allowed to vary. Furthermore, we fix the flux ratios of higher-order Balmer BELs to H$\gamma$ BEL to those under the ``Case B" situation \citep{Storey1995}, assuming a typical circumstance for broad line regions with $T_e=15,000$ K and $n_e=10^9$ cm$^{-3}$.\\
4. The Fe II component: we employ the Fe II template constructed by \citet{VeronCetty2004}.
After experiments with various combinations of different Fe II lines, we conclude that only narrow Fe II lines are necessary for fitting the observed Fe II complex. We further assume that these different narrow Fe II lines can be represented by single Gaussians with a fixed redshift and a fixed width. The flux ratios between different Fe II lines are taken from the template given in \citet{VeronCetty2004}. In addition, we assume the SMC extinction curve for the dust attenuation and reddening of the Fe II component parameterized by $E(B-V)_{\rm FeII}$.\\
We then fit the spectrum of the nuclear region, where the contamination from NELs is minimal, resulting in a minimized reduced Chi-square of 1.99. This fit can be further improved if the regions affected by telluric absorptions are dismissed (reduced $\chi^2$ = 1.66, Figure ~\ref{fig:fig1}). The individual components of the best-fit spectra are depicted by colored lines in the same figure, and the best-fit parameters are tabulated in Table ~\ref{tab:tab1}.
\begin{table}[htb]\footnotesize
\caption{Parameters for modelling the nuclear spectrum.}
\begin{tabular}{ccc}
\hline
\hline
Parameter & Value & Unit\\
\hline
\multicolumn{3}{c}{stellar component}\\
\hline
age & $450$ & Myr \\
$M_V$ & -22.77 & mag \\
$z_\star$ & $0.22911\pm0.00001$ \\
$\sigma_\star$ & $150\pm20$ & km s$^{-1}$ \\
$E(B-V)_\star$ & $0.62\pm0.02$ & mag\\
\hline
\multicolumn{3}{c}{power-law component} \\
\hline
$f_{\lambda5100}$ & $9.84\pm0.06$ & $10^{-17}$ erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$ \\
$\alpha$ & $1.32\pm0.02$ & \\
\hline
\multicolumn{3}{c}{BEL component} \\
\hline
$\Delta v_1$ & $263\pm3$ & km s$^{-1}$ \\
$\sigma_1$ & $974\pm4$ & km s$^{-1}$\\
$\Delta v_2$ & $250\pm10$ & km s$^{-1}$\\
$\sigma_2$ & $2960\pm20$ & km s$^{-1}$\\
frac$_1$ & $0.598\pm0.003$ & \\
$f_{\rm H\alpha}$ & $4740\pm20$ & $10^{-17}$ erg s$^{-1}$ cm$^{-2}$\\
$f_{\rm H\beta}$ & $587\pm3$ & $10^{-17}$ erg s$^{-1}$ cm$^{-2}$\\
$f_{\rm H\gamma}$ & $103\pm4$ & $10^{-17}$ erg s$^{-1}$ cm$^{-2}$\\
$f_{\rm HeI5876}$ & $307\pm3$ & $10^{-17}$ erg s$^{-1}$ cm$^{-2}$\\
$f_{\rm HeII4686}$ & $55\pm3$ & $10^{-17}$ erg s$^{-1}$ cm$^{-2}$\\
\hline
\multicolumn{3}{c}{Fe II component}\\
\hline
$\Delta v$ & $430\pm20$ & km s$^{-1}$\\
$\sigma$ & $720\pm20$ & km s$^{-1}$\\
$f_{\lambda4590}$ & $1.7\pm0.2$ & $10^{-17}$ erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$\\
$E(B-V)_{\rm FeII}$& $0.19\pm0.04$ & mag \\
\hline
\end{tabular}
\label{tab:tab1}
\end{table}
\begin{figure*}[htb]
\centering
\includegraphics[width=0.95\textwidth]{fig_spec_fit_small-eps-converted-to.pdf}
\caption{The observed nuclear spectrum (black) and the best-fitted spectrum(red) are shown in regions where the NELs affect little. The four components of the best-fitted model are shown in different colored lines: the stellar component in blue, the power-law component in green, the BEL component in cyan and the Fe II component in yellow. The spectrum in regions affected by the telluric absorption is plotted in grey line.}
\label{fig:fig1}
\end{figure*}
\subsubsection{A Narrow Absorption Line System} \label{sec:nabsya}
The nuclear spectrum of B2 0003+38A shows deep and narrow Na I $\lambda\lambda$5890,5896 absorption lines ( Figure ~\ref{fig:fig2}), corresponding to a redshift of $\sim$0.22883, which is about 70 km s$^{-1}$ blueshifted relative to the stellar redshift (0.22911).
Unlikely to originate from the stellar populations, these are likely absorption lines from absorbers that happen to lie in our line of sight towards the quasar nucleus.
Narrow Ca II $\lambda$3934 absorption is detected at the same redshift as that of NaI absorption, implying for the same absorption system, though we cannot affirm the existence of the Ca II $\lambda$3969 absorption line due to the influence of [NeIII] $\lambda$3869 and H$\epsilon$ emission lines. Assuming that the starlight is not affected by the absorber mentioned above, we consider the case of partial coverage:
\begin{equation}
f(\lambda) = f_{\rm qso}(\lambda) \times (1-C_f+C_f \times e^{-\tau(\lambda)}) + f_{\rm stellar},
\end{equation}
where $f_{\rm qso}$ is the flux density of the emission from the quasar nucleus (including the power-law continuum, the BEL and the FeII components), $f_{\rm stellar}$ is the flux density of stellar emission, $C_f$ is the covering factor, and $\tau(\lambda)$ is the optical depth of absorption.
The $\tau(\lambda)$ profile that we adopt for each absorption line is Gaussian with a fixed velocity dispersion. We perform this nuclear spectrum fitting in the vicinity of the narrow absorption lines, and the resultant best-fit model and the corresponding parameters are given in Figure ~\ref{fig:fig2} and Table ~\ref{tab:tab2}, respectively.
We find the redshift of these absorption lines to be $z = 0.228827\pm0.000002$, highly consistent with that of the H I 21-cm absorption ($z\sim0.2288$; \citealt{Aditya2018}).
In addition, the H I 21-cm absorption line is spectrally unresolved, in line with the narrowness of Na I and Ca II absorption lines ($\sigma\sim23$ km s$^{-1}$).
These consistencies strongly imply for a relation between the absorption of Na I/Ca II and that of H I.
\citet{Aditya2018} measured the H I column density to be $(3.54\pm0.11)\times (T_s/100\ {\rm K})\times 10^{20}$ cm$^{-2}$, where $T_s$ is the spin temperature.
The H I column density $N({\rm H})$ and the Na I column density $N({\rm Na I})$ are related by the following equation (e.g., \citealt{Rupke2005}):
\begin{equation}
N({\rm H}) = N({\rm Na I}) (1-y)^{-1} 10^{-(a+b)},
\end{equation}
where $y$ is the ionization fraction, $a$ is the Na abundance, and $b$ describes the depletion onto dust.
If we adopt $y=0.9$ following \citet{Rupke2005}, a solar abundance (so that $a=-5.69$), and the canonical Galactic depletion value of $b=-0.95$, then we see that the measured $N({\rm Na I})$ corresponds to $N({\rm H}) \sim$ 1.1$\times10^{21}$ cm$^{-2}$ , a value close to the result of radio spectral analysis.
\begin{table}[htb]\footnotesize
\caption{Parameters for modelling the narrow absorption lines.}
\begin{tabular}{ccc}
\hline
\hline
Parameter & Value & Unit\\
\hline
z & $0.228827\pm0.000002$ & \\
$\sigma$ & $23.4\pm0.6$ & km s$^{-1}$ \\
$C_f$ & 0.78 & \\
$N_{\rm NaI}$ & $26\pm2$ & $10^{12}$ cm$^{-2}$ \\
$N_{\rm CaII}$ & $3.4\pm0.7$ & $10^{12}$ cm$^{-2}$ \\
\hline
\end{tabular}
\label{tab:tab2}
\end{table}
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\textwidth]{ca2_na1_abs_small-eps-converted-to.pdf}
\caption{ The observed spectrum near the Ca II (left) and Na I (right) narrow absorption lines are shown in black lines. The best-fitted spectra are shown in colored lines based on their components. The best-fitted spectra of the stellar component and stellar plus quasar components, and deep narrow absorption lines are shown in blue, red, and green, respectively.
\label{fig:fig2}}
\end{figure}
\subsection{The nature of the quasar} \label{sec:2ds}
BELs in the optical spectrum are unambiguous emitted from the quasar nucleus.
We measure the Balmer decrement of the BELs to be $8.07\pm0.05$, significantly higher than the theoretical ``Case" B value of 2.7 \citep{Gaskell2017}, indicating heavy dust reddening towards the quasar nucleus.
Assuming an SMC extinction curve results in an estimation that $E_{\rm B-V} \sim 1.2$.
The power-law component contributes 60\% to 70\% of the total continuum flux in the wavelength range of 4000-7000 \AA.
The quasar continuum is remarkably red with a power-law index $\alpha$ of $1.32\pm0.02$, probably a result of the synchrotron emission from the radio jet, and/or the reddened thermal emission from quasar nucleus.
In view of the heavy dust extinction in optical bands, we use the infrared continuum to estimate the bolometric luminosity of the quasar. As the first step, we obtain the 5$\mu$m monochromatic luminosity using the infrared spectral energy distribution constructed from the ALLWISE photometry of the quasar, finding that $\nu L_\nu$(5$\mu$m) $= 8\times10^{44}$ erg s$^{-1}$. We then follow \citet{Richards2006} to apply a bolometric correction factor of 8, reaching a bolometric luminosity of $6\times10^{45}$ erg s$^{-1}$, though this value may have been overestimated due to the possible contribution of synchrotron emission from the jet at 5 $\mu$m.
\subsection{2-D Spectra of Narrow Emission Lines and Spatial Decomposition} \label{sec:2dsnl}
\begin{figure*}[htb]
\centering
\includegraphics[width=0.9\textwidth]{oisiichong44.pdf}
\includegraphics[width=0.9\textwidth]{oiiihabchong44.pdf}
\caption{Top two rows: the observed 2-D spectra of the \hbox{[O\,{\sc i}]}, \hbox{[S\,{\sc ii}]}, \hbox{H$\beta$}, \hbox{[O\,{\sc iii}]}\ and \hbox{H$\alpha$}\ after the continuum and \hbox{[Fe\,{\sc ii}]}\ subtractions. Only spaxels where the flux is detected with S/N $>$ 4 are plotted. The color of spaxels are scaled by flux density (squared scale, in units of 10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ $\rm \AA^{-1}$). The spectral direction is horizontal, and the spatial direction is vertical (southwest is in positive). The 2-D spectra of \hbox{[O\,{\sc i}]}- and \hbox{[S\,{\sc ii}]}-emitting NELs show a velocity gradient across the spatial extent, indicating a rotation-dominated disk. The 2-D spectra of \hbox{H$\alpha$}\ and \hbox{H$\beta$}\ NELs imply the existence of an extended region located $\sim$4--20 kpc southwest to the quasar nucleus, whose position and velocity is reminiscent of extended emission line regions (EELRs) around quasars. The 2-D spectra of \hbox{[O\,{\sc iii}]}\ NEL is more complicated: besides the two components mentioned above, an additional blue-shifted component centered on the quasar nucleus is evident. Such a profile is conventionally considered to be suggestive of the existence of outflowing gas. Our multiple-Gaussian decomposition of these emission lines allows us to isolate the contribution from these components. Third row: the reconstructed 2-D spectra of blue peaks colored by the flux density. This shows the rotating disk plus the outflow component. Bottom row: the reconstructed 2-D spectra of red peaks colored by the flux density. This reveals the EELR extended to Southwest $\sim$ 20 kpc.
\label{fig:oisii}}
\end{figure*}
\begin{figure*}[htb]
\centering
\includegraphics[width=0.9\textwidth]{testnnnn.pdf}
\caption{Examples of \hbox{H$\beta$}, \hbox{[O\,{\sc iii}]}\ and \hbox{H$\alpha$}\ velocity profiles at junction of ``inclined" structure and ``tail" structure. The red and blue dashed lines are the best-fitted spectra of red and blue peaks, respectively, with the sum of them in orange solid lines. The red and blue peaks reveal the ``tail" and ``inclined" structure, respectively.
\label{fig:habo}}
\end{figure*}
To investigate the gas kinematics of the host galaxy through the 2-D profiles of narrow emission lines, we remove the emission from the quasar nucleus and the stars, each of which is represented by double Gaussians in the spatial regions where the NELs are negligible. The two top panels of Figure ~\ref{fig:oisii} show the restframe 2-D spectra in the neighborhood of \hbox{[O\,{\sc i}]}, \hbox{[S\,{\sc ii}]}\ $\lambda\lambda$6717,6731 doublet, and \hbox{H$\alpha$}, \hbox{N\,{\sc ii}}\ doublet, \hbox{H$\beta$}, \hbox{[O\,{\sc iii}]}\ $\lambda\lambda$5007,4969 doublet (note that we employ the stellar redshift, $z = 0.22911$). The NEL structure extends to a size of $\sim$2.2$\arcsec$, greater than the FWHM of the point spread function (PSF) 0.71$\arcsec$ (see Appendix A), and thus is spatially resolved.
These 2-D NEL spectra reveal three components.
In particular, the 2-D spectra of \hbox{[O\,{\sc i}]}- and \hbox{[S\,{\sc ii}]}-emitting NELs show a velocity gradient across the spatial extent, indicating a rotation-dominated disk. The velocity gradient can be seen in the 2-D NEL spectra produced using both the two independent exposures (see Appendix B), and thus is reliable.
The 2-D spectra of \hbox{H$\alpha$}\ and \hbox{H$\beta$}\ NELs (Figure ~\ref{fig:oisii}) imply the existence of an extended region located $\sim$ 4--20 kpc southwest to the quasar nucleus, whose position and velocity is reminiscent of extended emission line regions (EELRs) around quasars (e.g., \citealt{Stockton1987,Fu2009}).
The 2-D spectra of \hbox{[O\,{\sc iii}]}\ NEL is more complicated: besides the two components mentioned above, an additional blue-shifted component centered on the quasar nucleus is evident.
Such a profile is frequently seen in 2-D spectra of quasars' \hbox{[O\,{\sc iii}]}\ emission and is conventionally considered to be suggestive of the existence of outflowing gas.
Therefore, the three components of the 2-D NEL spectra include a rotating disk, an EELR, and an outflow. In addition, we see double-peak profiles in \hbox{[O\,{\sc iii}]}\, \hbox{H$\alpha$}\ and \hbox{H$\beta$}\ lines in the Southwest of the nuclear, indicating multiple components.
\begin{figure*}[htb]
\centering
\includegraphics[width=0.48\textwidth]{rooisiikenn.pdf}
\includegraphics[width=0.48\textwidth]{rohabken.pdf}
\caption{The observed and best-fitted spectra of the \hbox{[O\,{\sc i}]}, \hbox{[S\,{\sc ii}]}, \hbox{H$\beta$}\ and \hbox{H$\alpha$}\ at different spaxels, from southwest to northeast from top to bottom. The project distance from the nuclear is shown on the upper right corner in each panel. The velocity of the rotating component is shown at the upper right corner in blue. For those spectra best fitted by a single gaussian, the median velocity is marked by a blue dotted line, and the zero point of the velocity which is measured by stellar component is in grey dashed dotted line. For those spectra best-fitted by two gaussians, the median velocity of the main gaussian (i.e., larger peak value) is marked by a blue dotted line. We note the \hbox{H$\beta$}\ spectrum at the nuclear (r=0$\arcsec$) and \hbox{H$\alpha$}\ spectra within 0.3$\arcsec$ of the nuclear (r= 0.3, 0 and -0.3) are best fitted by two gaussians, due to the outflow component. The green dashed lines, in the second column, are the two gaussians from the double \hbox{[S\,{\sc ii}]}\ lines. The orange dashed lines, in the third and fourth columns, are the two gaussians of \hbox{H$\beta$}\ and \hbox{H$\alpha$}, respectively. The green dashed dotted lines, in the fourth column, are the best-fitted double \hbox{N\,{\sc ii}}\ lines.
\label{fig:rospec}}
\end{figure*}
To delineate the gas kinematics, we perform a two-step spectral fit to decomposition these three components using the Python package MPFIT. In the first step, for each spatial element (spaxel), we fit a double Gaussian to the profile of \hbox{H$\beta$}, \hbox{[O\,{\sc iii}]}, \hbox{H$\alpha$}, and \hbox{N\,{\sc ii}}. The resultant best-fit Gaussian models for a single spaxel are demonstrated in Figure ~\ref{fig:habo}. Our reconstructed 2-D spectra of the red Gaussian show the EELR only, while the blue Gaussian is a superposition of a rotating disk and an outflow (Figure ~\ref{fig:oisii}, the third and fourth rows). In the second step, for each individual spaxel, we perform a single-Gaussian fit to the \hbox{[O\,{\sc i}]}\ emission line and the \hbox{[S\,{\sc ii}]}\ $\lambda\lambda$6717,6731 doublet. The best-fit spectra of \hbox{[O\,{\sc i}]}\ and the \hbox{[S\,{\sc ii}]}\ doublet in seven spaxels are shown in Figure ~\ref{fig:rospec} (left panels), from northeast to southwest. Due to the higher complication of their profiles, we fit the \hbox{H$\alpha$}\ and \hbox{H$\beta$}\ profiles to a single or double Gaussian, for which the decision is made in a way similar to that in \citet{Liu2014}. It turns out that the 5 spectra closest to the nucleus demand for a double Gaussian. These fits allow us to successfully decompose the contribution from the rotating disk and a nearly spherical outflow structure, and the best-fit \hbox{H$\alpha$}\ and \hbox{H$\beta$}\ spectra of seven spaxels are shown in Figure ~\ref{fig:rospec} (right panels), from northeast to southwest.
\subsection{The Rotating Disk} \label{sec:disk}
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\textwidth]{oshabnvvs66.pdf}
\caption{Long-slit line-of-sight velocity and velocity dispersion profiles obtained with ESI/Keck. The velocity (top) and velocity dispersion (bottom) of J0005+3820 (PA = -139$^{\circ}$) measured by \hbox{[O\,{\sc i}]}, \hbox{[S\,{\sc ii}]}, \hbox{H$\beta$}, \hbox{H$\alpha$}\ and \hbox{N\,{\sc ii}}\ respectively are marked using grey pentagons, purple rhombus, green squares, red stars and blue triangles. And the black dots mark the median of the velocity and velocity dispersion measured by these emission lines. And the errors on median are calculated as $\sigma=\sigma_{\rm NMAD}/\sqrt{n - 1}$, where $\sigma_{NMAD}$ is the normalized median of the absolute deviations and n is to be 5 on behalf of the number of the sample.The cyan solid lines show the best fit of the data. The abscissa zero corresponds to the brightest-continuum bin along the slit (to the galactic nucleus).
\label{fig:rovw}}
\end{figure}
We scrutinize the rotating disk by measuring the velocity relative to the redshift of host galaxy and the velocity dispersions of best-fit spectra of the rotation components in the NELs of \hbox{[O\,{\sc i}]}, \hbox{[S\,{\sc ii}]}, \hbox{H$\beta$}, \hbox{H$\alpha$}\ and \hbox{N\,{\sc ii}}. In Figure ~\ref{fig:rovw}, we plot them as a function of the distance from the quasar nucleus (positive is to the southwest and negative to the northeast). In addition, we only show those spaxels with high signal to noise ratio (S/N of \hbox{[O\,{\sc i}]}\ $\gtrsim$ 3), corresponding to $\sim$ 5 kpc from the nucleus.
We see that the overall velocity distribution and dispersion of these NELs are in line with typical galactic disks \citep{Rix1992,Proshina2020}. We find that the velocity profiles measured from different NELs are consistent, and thus use the median of the measurement results from these different NELs, following the method employed by \citet{Courteau1997}, \citet{Weiner2006} and \citet{Drew2018}. In detail, assuming an axis ratio of $b/a = 1$, the distribution of velocity is formulated as:
\begin{equation}
V_{\rm obs}(r) = (\frac{2}{\pi}V_{\rm a}\arctan(\frac{r_{\rm obs}}{r_{\rm t}\cos(i)}) + c\frac{r_{\rm obs}}{\cos(i)}) \times \sin(i) + V_{\rm rel},
\end{equation}
where $V_{\rm a}$ is the asymptotic velocity, $r_{t}$ is the knee radius, $r_{\rm obs}$ is the projection distance from the center of the galaxy, $c$ is the outer-galaxy slop, and $V_{\rm rel}$ is the velocity of the ionized gas relative to the stars along the line of sight, and $i$ is the inclination angle of the galaxy. Here $V_{\rm a}$ and $r_{\rm t}$ are dimensional scaling parameters, whereas $c$ characterizes the shape of the rotation curve.
Meanwhile, if a Gaussian fit is used, the profile of the velocity dispersion is given by:
\begin{equation}
\Sigma(r) = A {\rm exp}(-\frac{(r-m)^{2}}{2\omega^{2}} + \sigma_{0}),
\end{equation}
where $\Sigma(r)$ is the value of the fitted Gaussian at each radius, $m$ is the central position (if the ionized gas is isotropically distributed around the center, then $m$ is 0), $\omega$ is the width of the Gaussian, and $\sigma_{0}$ is the isotroptic component of the velocity dispersion. Here we correct the observed velocity dispersion for the intrinsic instrmental dispersion using $\sigma = \sqrt{\sigma_{\rm obs}^{2} - \sigma_{\rm inst}^{2}}$, where $\sigma$ reported in Figure ~\ref{fig:rovw}, and $\sigma_{\rm inst}$ is the combined instrumental and spectral seeing dispersion that we measure to be $\sim$ 23 km s$^{-1}$. The resultant best-fit velocity dispersion $\sigma$ is 126 $\pm$ 10 km s$^{-1}$.
We fit the velocity and velocity dispersion profiles to the median values of the measurements results from different NELs (\hbox{[O\,{\sc i}]}, \hbox{[S\,{\sc ii}]}, \hbox{H$\beta$}, \hbox{H$\alpha$}\ and \hbox{N\,{\sc ii}}), utilizing a Monte-Carlo simulation that we run 50 times to calculate the errors of parameters. Errors on the median are calculated as $\sigma=\sigma_{\rm NMAD}/\sqrt{n - 1}$, where $\sigma_{\rm NMAD}$ is the normalized median of the absolute deviations \citep{Hoaglin1983} and $n$ is the number of the sample, here $n = 5$. The best-fit profiles are shown in Figure ~\ref{fig:rovw}, along with the corresponding parameters listed in Table ~\ref{tab:tab3}. The velocity profile reveals a roughly symmetric gas motion pattern. Gas in the quasar's nuclear region has a velocity of $\sim$ -75 km s$^{-1}$, and is blue-/red-shifted on the SW/NE side, respectively. The velocity profile flattens beyond a radius of 4 kpc in both of these two directions, with a velocity of -150 to -180 km s$^{-1}$ (SW) and $\sim$ 40 km s$^{-1}$ (NE) within the distance range of 4 to 6 kpc. The rotating component is blueshifted relative to the systemic (stellar) velocity with $V_{\rm rel} =$ -71 km s$^{-1}$. Since the narrow absorption line system is blue-shifted with a similar relative velocity (-85 km s$^{-1}$; see Sections \ref{sec:nabsya}), this kinetic consistency is suggestive of the same origin of the two.
In the central region, we note that a velocity difference of $\sim$ 50 km s$^{-1}$ exists at a $> 3\sigma$ significant level between \hbox{[S\,{\sc ii}]}\ and other NELs. This is likely due to the outflow, though the possibility of influence from the sky emission line in the blue outskirt of \hbox{[S\,{\sc ii}]}\ cannot be fully ruled out; but if \hbox{[S\,{\sc ii}]}\ is excluded from these analyses, our results change minimally.
\begin{table}[htb]\footnotesize
\caption{Parameters for modelling the velocity shift and velocity dispersion.}
\begin{tabular}{ccc}
\hline
\hline
Parameter & Value & Unit\\
\hline
\multicolumn{3}{c}{velocity shift}\\
\hline
$V_{\rm a}$ & $-196\pm48$ & km s$^{-1}$ \\
$r_{\rm t}$ & $4.1\pm1.1$ & kpc \\
$c $ & $-0.8\pm4.8$ \\
$V_{\rm rel} $ & $-71\pm4$ & km s$^{-1}$ \\
$i$ & $0.84\pm0.07$ & rad \\
\hline
\multicolumn{3}{c}{velocity dispersion} \\
\hline
$A$ & $24\pm2$ \\
$m$ & $0.26\pm0.01$ & kpc \\
$\omega$ & $0.63\pm0.06$ & kpc \\
$\sigma_{0}$ & $126\pm3$ & km s$^{-1}$ \\
\hline
\end{tabular}
\label{tab:tab3}
\end{table}
\subsection{The Extended Emission Line Region} \label{sec:eelr}
The EELR extends to the southwest of the nucleus (see Figure~\ref{fig:oisii}), which is most evident in the \hbox{[O\,{\sc iii}]}\ emission, where two compact knots and a diffuse region are seen. For the line spectra at each position, we measure the integrated flux, median velocity, and line width ($W_{80}$, as detailed in \citealt{Liu2013b}) of \hbox{H$\alpha$}, \hbox{H$\beta$}\ and \hbox{[O\,{\sc iii}]}\ emission lines by fitting their profiles to Gaussian models, and conclude that the three NELs depict consistent velocity profiles (Figure ~\ref{fig:ep}).
The integrated flux profile of all three NELs unambiguously show a series of three peaks at galactocentric distances of about 7, 13 and 18 kpc. Hence, we divide the EELR into three annuli, with a radius range of 2 - 8 kpc, 8 - 16 kpc and 16 - 25 kpc, respectively, and for each of which, we extract a 1-D emission line spectra (Figure ~\ref{fig:esp}). The best-fit integrated fluxes, median velocities and line widths are listed in Table~\ref{tab:tab4}, where it can be seen that the emission lines from the 2 - 16 kpc are relatively narrow and are redshifted relative to the stellar redshift, while at 16 - 25 kpc they are broader and blue-shifted.
The intensity ratios of the emission lines facilitate our analysis on the physical conditions of the EELR. In specific, we use the \hbox{[O\,{\sc iii}]}/\hbox{H$\beta$}\ and \hbox{H$\alpha$}/\hbox{H$\beta$}\ ratios to quantify the degree of ionization and the dust attenuation, respectively. To obtain higher S/N ratios, we bin the spatial pixels within 0.45 arcsec before measuring the median values of these line ratios, for which the uncertainties are, again, calculated using the MC method (Figure ~\ref{fig:er}). We find that \hbox{[O\,{\sc iii}]}/\hbox{H$\beta$}\ is larger than 10 at all galactocentric radii under our consideration (top panel therein), implying for a high-ionization state in general. Meanwhile, considering the theoretical ratio \hbox{H$\alpha$}/\hbox{H$\beta$}\ $\sim$ 2.86 for ``case B" ($n_{e}$ = 100 cm$^{-3}$, $T_{e}$ = 10000 K;\citealt{Storey1995}), we plot the result in Figure ~\ref{fig:er} (blue dash line, bottom panel). The \hbox{H$\alpha$}/\hbox{H$\beta$}\ ratio at galactocentric radii of 2 - 16 kpc are roughly constant, corresponding to $A_V$ $\sim$1. However beyond 16 kpc, \hbox{H$\beta$}\ is too weak for accurate measurement of dust attenuation.
The electron density is achievable from \hbox{[O\,{\sc ii}]}\ $I(3729)/I(3726)$ and \hbox{[S\,{\sc ii}]}\ $I(6717)/I(6731)$ ratios. For this purpose, we stack the spectra taken from locations 2-16 kpc away from the center, and fit the [O ii] and [S ii] doublet emission lines by fixing the kinematics of the two line (Figure ~\ref{fig:eoii}). Assuming an electron temperature of 10000K and at a 68\% confidence level, we estimate the electron density to be $n_{e}$ = 137 $^{+36}_{-30}$ cm$^{-3}$. This result is generally lower than that of typical narrow line regions ($n_{e}$ $\sim$ 1000 cm$^{-3}$;\citealt{Greene2011}), but close to that of EELR \citep{Fu2009}.
\begin{figure*}[htb]
\centering
\includegraphics[width=0.9\textwidth]{fvwhabo.pdf}
\caption{Long-slit line-of-sight velocity, velocity dispersion and normalized flux density profiles obtained with ESI/Keck. The velocity, velocity dispersion and normalized flux measured by \hbox{[O\,{\sc iii}]}, \hbox{H$\beta$}\ and \hbox{H$\alpha$}\ respectively are marked using red, yellow and blue.
\label{fig:ep}}
\end{figure*}
\begin{figure}[htb]
\centering
\includegraphics[width=0.45\textwidth]{extendhabo.pdf}
\caption{Spectra of the \hbox{H$\alpha$}, \hbox{H$\beta$}\ and \hbox{[O\,{\sc iii}]} at three different regions. Red line shows the best fitting, and the blue dotted line marks the median velocity, the velocity zero is in grey dashed dotted line.\label{fig:esp}}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=0.45\textwidth]{xbt.pdf}
\caption{Line ratio of the \hbox{[O\,{\sc iii}]}/\hbox{H$\beta$}\ and \hbox{H$\alpha$}/\hbox{H$\beta$}\ of the extended ionized gas profiles along the long-slit line-of-sight. Blue dash lines mark the typical \hbox{[O\,{\sc iii}]}/\hbox{H$\beta$}\ ratio and ``Case B" value, respectively.
\label{fig:er}}
\end{figure}
\begin{figure*}[htb]
\centering
\includegraphics[width=0.9\textwidth]{ap1oisii.pdf}
\caption{Spectra of the \hbox{[O\,{\sc ii}]}\ and \hbox{[S\,{\sc ii}]}\ emission line in the rest frame. The fitted line is in red. The blue dashed line shows the two gaussians from the double \hbox{[O\,{\sc ii}]}\ lines. The grey shaded region are influenced by sky line.
\label{fig:eoii}}
\end{figure*}
\begin{table}[htb]\footnotesize
\caption{Nonparametric measurements of three parts.}
\begin{tabular}{ccc}
\hline
\hline
Parameter & Value & Unit\\
\hline
\multicolumn{3}{c}{part one}\\
\hline
$f_{\hbox{[O\,{\sc iii}]}}$ & $28.8\pm0.10$ & 10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ \\
$v_{\hbox{[O\,{\sc iii}]}}$ & $123\pm0.12$ & km s$^{-1}$ \\
$W_{\hbox{[O\,{\sc iii}]}}$ & $77\pm0.31$ & km s$^{-1}$ \\
$f_{\hbox{H$\alpha$}}$ & $8.6\pm0.32$ & 10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ \\
$v_{\hbox{H$\alpha$}}$ & $119\pm0.61$ & km s$^{-1}$ \\
$W_{\hbox{H$\alpha$}}$ & $79\pm1.58$ & km s$^{-1}$ \\
$f_{\hbox{H$\beta$}}$ & $2.1\pm0.12$ & 10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ \\
$v_{\hbox{H$\beta$}}$ & $130\pm2.11$ & km s$^{-1}$ \\
$W_{\hbox{H$\beta$}}$ & $82\pm5.82$ & km s$^{-1}$ \\
$A_{V}$ & $1.186\pm0.233$ & mag \\
\hline
\multicolumn{3}{c}{part two} \\
\hline
$f_{\hbox{[O\,{\sc iii}]}}$ & $52.6\pm0.72$ & 10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ \\
$v_{\hbox{[O\,{\sc iii}]}}$ & $108\pm2.33$ & km s$^{-1}$ \\
$W_{\hbox{[O\,{\sc iii}]}}$ & $170\pm3.03$ & km s$^{-1}$ \\
$f_{\hbox{H$\alpha$}}$ & $12.3\pm1.60$ & 10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ \\
$v_{\hbox{H$\alpha$}}$ & $114\pm6.75$ & km s$^{-1}$ \\
$W_{\hbox{H$\alpha$}}$ & $149\pm8.44$ & km s$^{-1}$ \\
$f_{\hbox{H$\beta$}}$ & $3.4\pm0.17$ & 10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ \\
$v_{\hbox{H$\beta$}}$ & $117\pm10.08$ & km s$^{-1}$ \\
$W_{\hbox{H$\beta$}}$ & $150\pm25.15$ & km s$^{-1}$ \\
$A_{V}$ & $0.860\pm0.475$ & mag \\
\hline
\multicolumn{3}{c}{part three} \\
\hline
$f_{\hbox{[O\,{\sc iii}]}}$ & $21.3\pm0.23$ & 10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ \\
$v_{\hbox{[O\,{\sc iii}]}}$ & $-122\pm7.75$ & km s$^{-1}$ \\
$W_{\hbox{[O\,{\sc iii}]}}$ & $348\pm8.38$ & km s$^{-1}$ \\
$f_{\hbox{H$\alpha$}}$ & $3.3\pm1.70$ & 10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ \\
$v_{\hbox{H$\alpha$}}$ & $-171\pm4.59$ & km s$^{-1}$ \\
$W_{\hbox{H$\alpha$}}$ & $175\pm11.87$ & km s$^{-1}$ \\
$f_{\hbox{H$\beta$}}$ & $2.4\pm0.16$ & 10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ \\
$v_{\hbox{H$\beta$}}$ & $-116\pm6.56$ & km s$^{-1}$ \\
$W_{\hbox{H$\beta$}}$ & $238\pm17.35$ & km s$^{-1}$ \\
\hline
\end{tabular}
\tablecomments{Flux of the \hbox{H$\alpha$}, \hbox{H$\beta$}\ and \hbox{[O\,{\sc iii}]}\ are directly measured, not do dust attenuation correction. The velocity is the median velocity measured by \hbox{H$\alpha$}\, \hbox{H$\beta$}\ and \hbox{[O\,{\sc iii}]}, and $W$ is the $W_{80}$\citep{Liu2013b}. We use the Milky Way attenuation curve starbust galaxies and the average extinction \citep{Calzetti2000} to reddening relation at the $V$ band of $A_{V} = 4.05E(B-V)$}
\label{tab:tab4}
\end{table}
\section{Discussion}
\label{sec:discussion}
\subsection{host galaxy} \label{sec:dr}
As we have introduced in the introduction section, there were abundant works studying the host galaxies of blazars, particularly to distinguish whether the host galaxies are disk or elliptical galaxies. Most of them focused on imaging BL Lac, while, none of them successfully found a disk galaxy hosted FSRQ. Because blazars are rare and can only be found in the distant universe, even using HST with the highest spatial resolution in optical and NIR bands, cannot spatially decompose the nuclei and the host galaxies well. However, we could use spatially resolved spectrum to constrain the host galaxy properties. In disk galaxies, young stars and interstellar gas and dust rotate in disks around bulging nuclei, while in elliptical galaxies, old stars randomly swarm and gas and dust are lack. Thus, the properties and kinematics of stars and gases can be used to distinguish between disk and elliptical galaxies.
In Section \ref{sec:disk}, we find that the kinematics of gas in the host galaxy of B2 0003+38A is dominated by rotating, and the curvatures of velocity and velocity dispersion are similar with those of disk galaxies \citep{Ho2020}.
In addition, there are other hints that supports B2 0003+38A being hosted by a disk galaxy. The velocity of the ionized gas (rotating component) relative to the systemic (stellar) velocity with $V_{\rm rel} =$ -71 km s$^{-1}$ is similar with the velocity of the narrow absorption line system (-85 km s$^{-1}$), suggesting the same origin of these two. Furthermore, emission lines from host galaxy can be detected significantly, indicating a gas-rich host galaxy.
The spectrum lacks the features of old stars, such as TiO molecular bands, indicating that old stars are not the dominant. Meanwhile, the spectrum is red, suggesting that the starlight is dust reddened. Young stars and rich interstellar dust are also characteristic of disk galaxies.
This is consistent with the analysis of nuclear spectrum prefers young stellar population ($\sim$450 Myr) with large dust reddening ($E(B-V)_\star=0.62$), under the assumption of a single SSP and quasar spectrum. Therefore, we conclude that the B2 0003+38A is mostly likely hosted by a disk galaxy.
\subsection{The Extended Emission Line Region} \label{sec:dr}
We find the EELR in southwest of the nucleus that extends to a projected distance up to 25 kpc.
EELRs with such sizes were found around a substantial fraction of radio-loud quasars.
Before further discussing the origin of the EELR, we estimate some parameters of the EELR.
The gases in part one and part two have redshifted velocities of $v_{0} \sim 120$ km s$^{-1}$, and their distance to the galaxy nucleus is $R_{0} \sim 15$ kpc (see Figure ~\ref{fig:esp}).
We estimate a dynamical time scale to be $t \sim 1.2 \times 10^{8}$ yr, i.e. the time taken by the gas from the nucleus to reach such a distance with an average velocity of $v_{0}$.
The mass of gas in the two parts can be estimated using \hbox{H$\beta$}\ luminosity $L_{\hbox{H$\beta$}}$ and electron density $n_{e}$ (e.g \citealt{Liu2013,Harrison2014}).
After corrected for dust attenuation, the summed luminosity in the two parts is $\sim$ 2.4$\times10^{40}$ erg s$^{-1}$.
The total mass of these ionized extended gas can be estimated as:
\begin{equation}
\frac{M_{\rm gas}}{2.82 \times 10^{9} \rm ~ M_\odot} = (\frac{L_{\hbox{H$\beta$}}} {10^{43} \rm ~ erg ~ s^{-1}}) (\frac{n_e} {100 \rm ~ cm^{-3}}).
\end{equation}
We measure an electron density of $n_{e}$ $\sim$ 137 cm$^{-3}$ using \hbox{[O\,{\sc ii}]}\ and \hbox{[S\,{\sc ii}]}\ doublet, if the electron temperature is 10000K.
We find $M_{\rm gas} \sim 9.3 \times 10^{6}$ $M_\odot$.
Combining with the average velocity of 120 km s$^{-1}$, the total kinetic energy of the gas can be estimated as:
\begin{equation}
E_{\rm kin} = \frac{1}{2}M_{\rm gas} v_{\rm gas}^{2} = 1.33 \times 10^{54} \rm ~ erg.
\end{equation}
If assuming the lifetime of the structure is the dynamical time scale, we can also estimate a mass rate $\dot{M}$ is 7.8 $\times 10^{-2}$ M$_\odot$ yr$^{-1}$, and a kinetic energy rate $\dot{E}_{\rm ink} \sim$ 3.5 $\times 10^{38}$ erg s$^{-1}$.
However, this electron density value may not be the bulk density of the EELR gas.
The EELR might be illuminated by the quasar.
If so, the ionization parameter $U$ of the EELR can be estimated as:
\begin{equation}
U = \frac{ Q_H }{ 4\pi r^2 c n_H },
\end{equation}
where $Q(H)$ is the rate of the hydrogen ionization photon from the quasar, $r$ is the distance from the quasar nucleus to the EELR, $n_H$ is the hydrogen number density.
Assuming that the intrinsic SED of the quasar is that given in \citet{Mathews1987}, we estimate a $Q_H$ of $1.2\times10^{56}$ s$^{-1}$ using the bolometric luminosity estimated in Section \ref{sec:2ds}.
Assuming a distance of 10 kpc, and assuming $n_e=1.2 n_H$ for highly ionized plasma, the inferred $U$ is about $10^{-2.5}$.
However, the observed \hbox{[O\,{\sc iii}]}/\hbox{H$\beta$}\ values of 12--16 indicates a higher $U$ value of $\sim10^{-1}$.
This may be because the \hbox{[O\,{\sc iii}]}\ emitting gas has a lower density than previously estimated value.
Using the mixed-medium model \citep{Robinson2000}, \cite{Stockton2002} divided the EELR of 4C 37.43 into two components, and found that one component with a density of several hundred cm$^{-3}$, another with a density near 1 cm$^{-3}$.
Most of the \hbox{[O\,{\sc iii}]}, \hbox{H$\beta$}\ and \hbox{H$\alpha$}\ come from regions with low density, \hbox{[O\,{\sc ii}]}\ and \hbox{[S\,{\sc ii}]}\ come from the regions with density several hundred times higher. If this is the case for the EELR in B2 0003+38A, actual value of $M_{\rm gas}$ is two orders of magnitude higher.
There are various possibilities for the origin of an EELR.
In the case of an isolated galaxy, the gas can be inflow, i.e. cold accretion flow from the intergalactic medium, outflow triggered by AGN or starburst, or recycling gas (of the outflow).
In the case of galaxy with interactions, the gas can also belong to tidal features.
Firstly, inflows are generally isotropic, which excludes the possible of EELR being inflow driven.
Secondly, EELRs are common in radio quasars and these EELRs are generally related with outflows driven by radio jets (e.g., \citealt{Fu2009}), so the EELR in B2 0003+38A may also be the same. Thirdly, verifying the possibility of recycling gas requires a panoramic image of the circum-galactic gas.
The long-slit spectrum is not sufficient, and future Integral Field Unit (IFU) observation data is needed.
We did not see any companion galaxy around B2 0003+38A, and did neither see any asymmetry from the brightness profile of the starlight. As for the possible of galaxy interactions, we do not find any direct evidence including any signature of asymmetry from the brightness profile of the starlight and any companion galaxy around it.
Though, we note that the current data quality is not enough to rule out the possibility.
The EELR may also originate in tidal features.
If so, starlight accompanied with EELR should be seen.
This cannot be tested using the existing data. Future observations, such as IFU, with higher quality are required to fully investigate the origin of EELR.
\section{Conclusions}
\label{sec:conclusions}
In this paper, we present a long-slit observations taken from the ESI/Keck to study the gas in the host galaxy of FSRQ B2 0003+38A at redshift $z =$ 0.22911. Based on multiple Gaussian fitting processes, we separate the 2-D NEL spectra into three components, indicating a rotation disk, an EELR and an outflow, respectively. To model the rotating disk, we measure and analyse the curves of velocity and velocity dispersion. We also analyse the EELR, which extends to a projected distance up to 25 kpc from the nuclear. We summarize our results below.
(i) For the first time, we discover a rotating gasous disk from optical spectroscopy in a FSRQ host galaxy. The curvatures of velocities and the velocity dispersions derived from different emission lines agree with an identical rotating disk model. The rotating gas disk has a mean velocity of $v$ = $-$75 km s$^{-1}$ relative to the stellar redshift. The velocity has little difference with that of the absorber seen in Na I, Ca II, and H I lines ($v$ = $-$85 km s$^{-1}$), suggesting that the rotating gas disk and the absorber are related.
(ii) According to the kinematics and morphology of the EELR, we divide them into three parts, including two knots and a diffused region. We calculate that the two knots have an averaged electron density $n_{e}$ of 137 $^{+36}_{-30}$ cm$^{-3}$ and dust attenuation $A_{V}$ of 1.19 $\pm$ 0.23/0.86 $\pm$ 0.48, respectively. After correcting for the dust attenuation, we estimate a corresponding mass to be 9.3 $\times 10^{6}$ M$_\odot$. The velocity of ionized gas in these two knots is redshifted 120 km s$^{-1}$. The dynamical timescale of the knots can be estimated at $\sim 1.2 \times 10^{8}$ yr as the travel time of clouds to reach the observed distances from the centre. There are various possibilities for the origin of an EELR, including inflows, outflows, recycling gas and galaxy interactions. If the EELR is from an outflow/inflow, the mass rate is 7.8 $\times 10^{-2}$ M$_\odot$ yr$^{-1}$, and the kinetic energy carried by ionized gas is estimated as 1.46 $\times 10^{54}$ erg.
\acknowledgements
QZ, LS and GL acknowledge the grant from the National Key R\&D Program of China (2016YFA0400702), the National Natural Science Foundation of China (No. 11673020 and No. 11421303), and the Fundamental Research Funds for the Central Universities. We acknowledge the support from Chinese Space Station Telescope (CSST) Project.
\begin{appendices}
\section{Measuring the Point Spread Function}
We measure the PSF due to seeing effect and instrumental effect using the spatial brightness profiles of BELs.
We can do in this way because the BELR, with a typical size less than 1 pc, is a point source at $z\sim0.2$.
We extract the spatial brightness profiles of the H$\alpha$ and H$\beta$ BELs (Figure ~\ref{fig:psf}.
Both the two profiles can be well fit using a Gaussian function with a FWHM of 0.71$\arcsec$.
\begin{figure}[htb]
\centering
\includegraphics[width=0.45\textwidth]{psf-eps-converted-to.pdf}
\caption{The spatial brightness profiles of H$\alpha$ and H$\beta$ BELs, and the Gaussian functions that fit them.
\label{fig:psf}}
\end{figure}
\section{Verifying the velocity gradient of NELs across the spatial extent}
One may doubt that velocity gradient of NELs across the spatial extent seen from the 2-D spectra might be artificial: it might be caused by a tiny inclined movement of the target along the slit during the observation.
As there were two exposures of this target, we test this presumption by independently analyzing the data from the two exposures.
We reduce the data and extract the 2-D NEL spectra again following the methods described in Sections \ref{sec:dar} and \ref{sec:2dsnl}, while this time we do not stack the two exposures.
The results are shown in Figure ~\ref{fig:ctwo}.
The velocity gradient of NELs are seen in both the two 2-D NEL spectra, indicating that the rotation-dominated disk structure in B2 0003+38A is reliable.
\begin{figure*}[htb]
\centering
\includegraphics[width=0.9\textwidth]{ctwo.pdf}
\caption{Same as Figure ~\ref{fig:oisii}. First row: the observed 2-D spectra of the \hbox{[O\,{\sc i}]}, \hbox{H$\alpha$}, \hbox{N\,{\sc ii}}\ and \hbox{[S\,{\sc ii}]}\ from the data of the first exposure, second row: those from the second exposure. Note that the cosmic rays are not removed.
\label{fig:ctwo}}
\end{figure*}
\end{appendices}
|
{
"timestamp": "2021-06-09T02:03:23",
"yymm": "2106",
"arxiv_id": "2106.03940",
"language": "en",
"url": "https://arxiv.org/abs/2106.03940"
}
|
\section{Introduction}
The AdS/CFT correspondence conjectures a remarkable equivalence between large-$N$ gauge theories and string/M-theory on asymptotically AdS backgrounds. In this context, Chern-Simons-matter theories are of particular interest in regards to the dynamics of M2-branes \cite{Bagger:2006sk,Bagger:2007jr,Bagger:2007vi,Gustavsson:2007vu,Gustavsson:2008dy,VanRaamsdonk:2008ft,Aharony:2008ug}. In particular, the worldvolume theory of $N$ coincident M2-branes probing the singularity of a $\mathbb{C}^4/\mathbb{Z}_k$ orbifold was constructed in \cite{Aharony:2008ug} and is known as the Aharony-Bergman-Jafferis-Maldacena (ABJM) theory. ABJM theory is an $\mathcal N=6$, $U(N)_k\times U(N)_{-k}$ Chern-Simons-matter theory, and in the large-$N$ limit is dual to either M-theory on AdS$_4\times S^7/\mathbb{Z}_k$ or IIA string theory on AdS$_4\times\mathbb{CP}^3$, depending on the limit taken.
ABJM theory and its holographic dual provide an excellent opportunity to probe the dynamics of string/M-theory as well as quantum gravity and AdS$_4$ black holes. However, as AdS/CFT is a strong/weak coupling duality, it is highly non-trivial to make direct comparisons on both sides of the duality. Nevertheless, certain path integrals in superconformal Chern-Simons-matter theories reduce to matrix models via supersymmetric localization \cite{Pestun:2007rz,Kapustin:2009kz}. Such localization techniques have long been studied in the context of supersymmetric and topological QFTs, and the application of \cite{Pestun:2007rz,Kapustin:2009kz} to superconformal field theories have proven a powerful technique to analyze observables via matrix models. In particular, ABJM theory can be localized to a two-matrix model \cite{Kapustin:2009kz}, which can then be studied via standard methods of random matrix theory or by novel methods such as the ideal Fermi gas approach \cite{Marino:2011eh}.
Many important results have been obtained for the supersymmetric partition function and Wilson loop observables in ABJM theory \cite{Marino:2009jd,Marino:2011eh,Hatsuda:2012hm,Putrov:2012zi} and the ABJ generalization \cite{Awata:2012jb,Honda:2014npa,Hatsuda:2016rmv,Cavaglia:2016ide}. In particular, the $S^3$ partition function at fixed Chern-Simons levels $k$ and $-k$ was shown to have the form of an Airy function
\begin{equation}
Z_{\mathrm{ABJM}}^{S^3}=\left(\fft2{\pi^2k}\right)^{-1/3}e^{A(k)}\operatorname{Ai}\left[\left(\fft2{\pi^2k}\right)^{-1/3}\left(N-\fft1{3k}-\fft{k}{24}\right)\right]+Z_{\mathrm{np}},
\end{equation}
where $A(k)$ encodes certain quantum corrections and $Z_{\mathrm{np}}$ is a non-perturbative contribution. Taking $F=\log Z$ then leads to a fixed $k$ expansion of the free energy as
\begin{equation}
F_{\mathrm{ABJM}}=\fft{\pi\sqrt2}3k^{1/2}N^{3/2}-\fft\pi{\sqrt{2k}}\left(\fft{k^2}{24}+\fft13\right)N^{1/2}+\fft14\log N+\mathcal O(1).
\end{equation}
In the M-theory dual, the $N^{3/2}$ term can be matched to the on-shell classical supergravity action, while the $N^{1/2}$ term is related to eight-derivative couplings in M-theory \cite{Bergman:2009zh,Aharony:2009fc} which reduce to four-derivative couplings in AdS$_4$ supergravity \cite{Bobev:2020egg,Bobev:2021oku}. The Airy function form of the partition function holds for a wide range of Chern-Simons-matter theories beyond ABJM theory. Then, by expanding the Airy function at large $N$, one can see that the $\fft14\log N$ term is universal to this full set of theories. As an important test of quantum gravity, this log term has been reproduced successfully by a one-loop calculation in eleven-dimensional supergravity on AdS$_4\times X^7$ \cite{Bhattacharyya:2012ye}.
Given the remarkable successes of precision tests of ABJM holography, we wish to extend such investigations to the Gaiotto-Tomasiello (GT) case \cite{Gaiotto:2009mv}. The GT theory is an $\mathcal{N}=3$ Chern-Simons-matter theory, and can be thought of as a generalization of the ABJM theory to arbitrary Chern-Simons levels, $k_1$ and $k_2$, with $F_0=k_1+k_2\ne0$. This model is dual to massive IIA supergravity with $F_0$ playing the role of the Romans mass \cite{Romans:1985tz}. The leading order behavior of GT free energy is \cite{Suyama:2010hr,Suyama:2011yz,Jafferis:2011zi}
\begin{equation}
F_{\mathrm{GT}}=\fft{3^{5/3}\pi }{5\cdot2^{4/3}}e^{-i\pi/6}(k_1+k_2)^{1/3}N^{5/3}+\cdots.
\label{eq:FGT0}
\end{equation}
The $N^{5/3}k^{1/3}$ scaling is in contrast to the $N^{3/2}k^{1/2}$ scaling of the ABJM free energy, and has confirmed on the supergravity side \cite{Aharony:2010af}. While this leading-order behavior is well established and generalizes to a large class of $\mathcal N=3$ necklace quiver models with $F_0\ne0$, less is known about its subleading corrections, which is the focus of this paper.
Although the partition function for GT theory can also be mapped to a corresponding ideal Fermi gas system, unlike for the ABJM model, the resulting expression does not take the form of an Airy function \cite{Marino:2011eh,Hong:2021bsb}. Furthermore, the mapping to the quantum Fermi gas system promoted in \cite{Marino:2011eh} involves taking
\begin{equation}
\fft{4\pi}\hbar=\fft1{k_1}-\fft1{k_2}.
\end{equation}
This demonstrates that a small $\hbar$ expansion is in tension with taking $k_1\approx k_2$, which is the natural realm for exploring the free energy in (\ref{eq:FGT0}). We thus find it more natural to work directly with the GT theory partition function written as a two-matrix model. While a saddle point analysis was performed in \cite{Hong:2021bsb}, here we use a standard resolvent approach and compute the genus-zero partition function as an expansion in inverse powers of the 't~Hooft parameter $t=g_sN$ with $g_s=2\pi i/k$ where $k$ is an effective overall Chern-Simons level. For equal levels, $k=k_1=k_2$, we find (at genus zero)
\begin{equation}
F_{\mathrm{GT}}^{k_1=k_2}=\fft1{g_s^2}\left[\fft35\left(\fft{3\pi^2}2\right)^{2/3}\left(t+\fft{\zeta(3)}{2\pi^2}\right)^{5/3}-\fft{\pi^2}{12}t+\mbox{const.}\right],
\label{eq:FGT=}
\end{equation}
up to exponentially small corrections in the large $|t|$ limit.
To highlight the first subleading corrections to the planar free energy, we substitute $t=2\pi iN/k$ into (\ref{eq:FGT=}) and expand to obtain
\begin{equation}
F_{\mathrm{GT}}^{k_1=k_2}=\fft{3^{5/3}\pi}{10}e^{-i\pi/6}k^{1/3}N^{5/3}+\fft{i\pi}{24}kN+\fft{3^{2/3}}{8\pi^2}e^{-2\pi i/3}\zeta(3)k^{4/3}N^{2/3}+\mathcal O(1).
\end{equation}
The leading order $N^{5/3}$ term matches (\ref{eq:FGT0}), while the linear-$N$ term was previously obtained in \cite{Hong:2021bsb}, and is pure imaginary for real Chern-Simons levels. At the next order, we find a $N^{2/3}$ term with a coefficient proportional to $\zeta(3)$. This term is of $\mathcal O(1/t)$ compared to the leading order, and has a natural interpretation in the massive IIA supergravity dual as originating from a tree-level $\alpha'^3R^4$ coupling.
This paper is organized as follows. In Section~\ref{sec:GTreview}, we predominantly follow \cite{Suyama:2010hr} in summarizing important results about the planar limit and the resolvent in GT theory. We then proceed in Section~\ref{sec:GTfree} to obtain the planar free energy from the resolvent in the limit of large 't~Hooft coupling, and further check our results against numerical data. Finally, we conclude in Section~\ref{sec:disc} with some open questions. Some of the more technical calculations are relegated to two appendices.
\section{GT theory and the planar resolvent}
\label{sec:GTreview}
GT theory is an $\mathcal{N}=3$ superconformal Chern-Simons-matter theory with $U(N_1)_{k_1}\times U(N_2)_{k_2}$ gauge group and quiver diagram given in Figure~\ref{fig:quiver}. It was originally constructed as a deformation of ABJM theory in \cite{Gaiotto:2009mv} by allowing the two $U(N)$ quivers to take on arbitrary ranks and levels, which in turn knocks the supersymmetry down from $\mathcal{N}=6$ to $\mathcal{N}=3$. On the dual gravity side, which was constructed to first order in perturbation theory in \cite{Gaiotto:2009yz}, this corresponds to turning on a nonzero Romans mass $F_0=k_1+k_2$, which is a 0-form R-R flux sourced by D8-branes. The supergravity description then corresponds to the massive IIA theory where the 2-form NS-NS $B$-field acquires a mass precisely equal to $F_0$ by ``eating'' the 1-form gauge field in a Higgs-like mechanism \cite{Romans:1985tz}. It is generally believed that there is no M-theory limit \cite{Aharony:2010af} when this mass is non-vanishing.
\begin{figure}[t]
\centering
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\draw (235.16,23.26) node [anchor=north west][inner sep=0.75pt] [color={rgb, 255:red, 208; green, 2; blue, 27 } ,opacity=1 ] {$A_{1}$};
\draw (309.11,21.96) node [anchor=north west][inner sep=0.75pt] [color={rgb, 255:red, 208; green, 2; blue, 27 } ,opacity=1 ] {$A_{2}$};
\draw (247.09,232.04) node [anchor=north west][inner sep=0.75pt] [color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] {$B_{1}$};
\draw (324.62,230.74) node [anchor=north west][inner sep=0.75pt] [color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] {$B_{2}$};
\draw (76.8,117.47) node [anchor=north west][inner sep=0.75pt] {$U( N_{1})_{k_{1}}$};
\draw (428.67,118.78) node [anchor=north west][inner sep=0.75pt] {$U( N_{2})_{k_{2}}$};
\end{tikzpicture}
\caption{The $\mathcal{N}=3$ GT quiver diagram. $A_1$ and $A_2$ are bifundamental hypermultiplets and $B_1$ and $B_2$ are anti-bifundamental hypermultiplets coupling the nodes of the quiver.}
\label{fig:quiver}
\end{figure}
Since GT theory still retains $\mathcal{N}=3$ supersymmetry, its partition function can be localized following \cite{Kapustin:2009kz}, just as in the AJBM case. The resulting matrix model takes the form
\begin{equation}
Z=\fft1{N_1!N_2!}\int\prod_{i=1}^{N_1}\frac{\dd{u_i}}{2\pi}\prod_{j=1}^{N_2}\frac{\dd{v_j}}{2\pi}e^{-S(u_i,v_j)},
\end{equation}
where the action is given by
\begin{equation}
e^{-S}=\exp\qty[\frac{ik_1}{4\pi}\sum_{i=1}^{N_1}u_i^2+\frac{ik_2}{4\pi}\sum_{i=1}^{N_2}v_i^2]\frac{\prod_{i<j}^{N_1}\sinh^2\qty(\frac{u_i-u_j}{2})\prod_{i<j}^{N_2}\sinh^2\qty(\frac{v_i-v_j}{2})}{\prod_{i=1}^{N_1}\prod_{j=1}^{N_2}\cosh^2\qty(\frac{u_i-v_j}{2})}.
\end{equation}
Since there are two independent Chern-Simons levels, $k_1$ and $k_2$, we can define two 't~Hooft couplings, $\lambda_1=N_1/k_1$ and $\lambda_2=N_2/k_2$. However, to highlight the planar limit, we find it more convenient to follow \cite{Suyama:2010hr} by introducing an auxiliary parameter $k$ and defining
\begin{equation}
t_1=\fft{2\pi iN_1}k,\qquad t_2=\fft{2\pi iN_2}k,\qquad\kappa_1=\fft{k_1}k,\qquad\kappa_2=\fft{k_2}k.
\end{equation}
The planar limit is then taken by sending $k\to\infty$ while holding $t_i$ and $\kappa_i$ fixed.
Written in terms of the above quantities, the action now takes the form%
\footnote{Note that this choice of parameters differs from that of \cite{Suyama:2010hr} in the choice of sign of $t_2$ and $\kappa_2$. In particular, $(t_2)_{\mathrm{there}}=(-t_2)_{\mathrm{here}}$ and $(\kappa_2)_{\mathrm{there}}=(-\kappa_2)_{\mathrm{here}}$.}
\begin{align}
S&=\fft1{g_s^2}\Bigl[\fft{\kappa_1t_1}{2N_1}\sum_{i=1}^{N_1}u_i^2+\fft{\kappa_2t_2}{2N_2}\sum_{i=1}^{N_2}v_i^2-\fft{t_1^2}{N_1^2}\sum_{i<j}^{N_1}\log\sinh^2\fft{u_i-u_j}2-\fft{t_2^2}{N_2^2}\sum_{i<j}^{N_2}\log\sinh^2\fft{v_i-v_j}2\nonumber\\
&\kern14em+\fft{t_1t_2}{N_1N_2}\sum_{i=1}^{N_1}\sum_{j=1}^{N_2}\log\cosh^2\fft{u_i-v_j}2\Bigr],
\label{eq:action}
\end{align}
where we have introduced $g_s=2\pi i/k$. While the physical Chern-Simons levels $k_1$ and $k_2$ are real, below we will analytically continue to imaginary levels such that the couplings $t_i$ and $\kappa_i$ are real. This will allow us to work with a real action and corresponding real saddle point equations. In particular, varying the action, (\ref{eq:action}), with respect to $u_i$ and $v_j$ gives the saddle-point equations
\begin{subequations}
\begin{align}
\kappa_{1} u_{i} &=\frac{t_{1}}{N_{1}} \sum_{j \neq i}^{N_{1}} \operatorname{coth} \frac{u_{i}-u_{j}}{2}-\frac{t_{2}}{N_{2}} \sum_{j=1}^{N_{2}} \tanh \frac{u_{i}-v_{j}}{2}, \\
\kappa_{2} v_{i} &=\frac{t_{2}}{N_{2}} \sum_{j \neq i}^{N_{2}} \operatorname{coth} \frac{v_{i}-v_{j}}{2}-\frac{t_{1}}{N_{1}} \sum_{j=1}^{N_{1}} \tanh \frac{v_{i}-u_{j}}{2}.
\end{align}
\label{eq:unexponentatedSPE}
\end{subequations}
At this stage, it is convenient to switch to exponentiated coordinates
\begin{equation}
z_i:=e^{u_i},\qquad w_i:=-e^{v_i}.
\end{equation}
Making note of the sign in the definition of the $\{w_i\}$, the saddle-point equations then take the form
\begin{subequations}
\begin{align}
\kappa_{1} \log z_{i} &=\frac{t_{1}}{N_{1}} \sum_{j \neq i}^{N_{1}} \frac{z_i+z_{j}}{z_{i}-z_{j}}-\frac{t_{2}}{N_{2}} \sum_{j=1}^{N_{2}} \frac{z_i+w_{j}}{z_{i}-w_{j}}, \\
\kappa_{2} \log(- w_{i}) &=\frac{t_{2}}{N_{2}} \sum_{j \neq i}^{N_{2}} \frac{w_i+w_{j}}{w_{i}-w_{j}}-\frac{t_{1}}{N_{1}} \sum_{j=1}^{N_{1}} \frac{w_i+z_{j}}{w_{i}-z_{j}}.
\end{align}
\label{eq:mmspe}
\end{subequations}
We now define the planar resolvent in terms of the exponentiated variables
\begin{equation}
v(z):=v_1(z)-v_2(z)=\fft{t_1}{N_1}\sum_{i=1}^{N_1}\fft{z+z_i}{z-z_i}-\fft{t_2}{N_2}\sum_{i=1}^{N_2}\fft{z+w_i}{z-w_i},
\label{eq:resolventi}
\end{equation}
where the eigenvalues $\left\{z_i\right\}_{i=1}^{N_1}$ and $\left\{w_i\right\}_{i=1}^{N_2}$ solve the saddle-point equations \eqref{eq:mmspe}. In the planar limit, $k\to\infty$, we expect the eigenvalue distributions $\{z_i\}_{i=1}^{N_1}$ to localize to a cut $[c,d]\subset\mathbb{R}^+$ and
$\{w_i\}_{i=1}^{N_2}$ to localize to a cut $[a,b]\subset\mathbb{R}^-$. We thus introduce eigenvalue densities $\rho(x)$ and $\tilde\rho(x)$ and write the planar resolvent as
\begin{equation}
v(z):=t_1\int_c^d\dd{x}\rho(x)\frac{z+x}{z-x}-t_2\int_a^b\dd{x}\tilde{\rho}(x)\frac{z+x}{z-x}.
\label{eq:expResolvent}
\end{equation}
Note that $v(z)$ has branch-cut discontinuities along $[a,b]$ and $[c,d]$ where the eigenvalues condense. In terms of this resolvent, we can rewrite the saddle-point equations quite simply as
\begin{subequations}
\begin{align}
\kappa_{1} \log z& =\ft12[v(z+i 0)+v(z-i 0)],\qquad y\in[c,d] \\
-\kappa_{2} \log (-z) & =\ft12[v(z+i 0)+v(z-i 0)],\qquad y\in[a,b]
\end{align}\label{eq:expSPE}
\end{subequations}
These equations can be solved by standard methods that have been developed in random matrix theory (see \textit{e.g.}~\cite{Marino:2011nm}).
In fact, the planar resolvent for GT theory was already worked out in \cite{Suyama:2010hr} by solving the Riemann-Hilbert problem. The idea is to convert the saddle-point equations, (\ref{eq:expSPE}), which correspond to the principal value of the resolvent along the two cuts, into corresponding discontinuity equations by introducing
\begin{equation}
f(z)=\fft{v(z)}{\sqrt{(z-a)(z-b)(z-c)(z-d)}}.
\end{equation}
We then use Cauchy's theorem to write
\begin{equation}
f(z)=\oint\fft{d\zeta}{2\pi i}\fft{f(\zeta)}{\zeta-z},
\end{equation}
where the contour is a small circle surrounding $z$. By deforming the contour to go around the two cuts and using the saddle-point equations, we can obtain an integral expression for $f(z)$. Converted back to the resolvent, $v(z)$, we finally obtain \cite{Suyama:2010hr}
\begin{align}
v(z)&=\frac{\kappa_1}{\pi}\int_c^d\dd{x}\frac{\log\qty(x)}{z-x}\frac{\sqrt{(z-a)(z-b)(z-c)(z-d)}}{\sqrt{|(x-a)(x-b)(x-c)(x-d)|}}\nonumber \\
&\quad+\frac{\kappa_2}{\pi}\int_a^b\dd{x}\frac{\log\qty(-x)}{z-x}\frac{\sqrt{(z-a)(z-b)(z-c)(z-d)}}{\sqrt{|(x-a)(x-b)(x-c)(x-d)|}}.
\label{eq:vzint}
\end{align}
This is the starting point for the subsequent analysis.
\subsection{Fixing the endpoints}
While the GT theory is parametrized by the couplings $t_1$ and $t_2$, the expression (\ref{eq:vzint}) for the resolvent is instead parametrized by the endpoints $a,b,c,d$ of the two cuts. We thus want to relate these two sets of parameters. The problem can be simplified by noticing that the saddle-point equations, \eqref{eq:mmspe}, are invariant under $z\to z^{-1}$ and $w\to w^{-1}$. This suggests that the eigenvalue distributions should also be invariant under this map, which leads to an ansatz
\begin{equation}
ab=1,\ \ cd=1.\label{eq:endpointAnsatz}
\end{equation}
It was shown in \cite{Suyama:2010hr} that this ansatz is consistent with the constraints imposed by the asymptotic behavior of the resolvent $v(z)$ in the limits $z\to\infty$ and $z\to0$. We still need to relate the two undetermined parameters (say $a$ and $d$) to the couplings $t_1$ and $t_2$. This can be done using the relations
\begin{subequations}
\begin{align}
t_1&=\fft1{4\pi i}\oint_{\mathcal{C}_1} {\dd{z}}\frac{v(z)}{z},\\
t_2&=\fft1{4\pi i}\oint_{\mathcal{C}_2} {\dd{z}}\frac{v(z)}{z},
\end{align}
\label{eq:endpointContour}%
\end{subequations}
which can be derived directly from the expression \eqref{eq:expResolvent} for the resolvent. Here $\mathcal{C}_1$ and $\mathcal{C}_2$ are contours enclosing the branch cuts $[c,d]$ and $[a,b]$, respectively.
While the resolvent, (\ref{eq:vzint}), does not appear to admit a simple analytic form, we can work with it as an integral expression. This is facilitated in the strong coupling limit $t_1,t_2\gg1$, where it was shown in \cite{Suyama:2011yz} that the endpoints of the two cuts scale uniformly when $t_1\approx t_2\to\infty$. In particular, making note of (\ref{eq:endpointAnsatz}), we let
\begin{equation}
a=-e^\alpha,\qquad b=-e^{-\alpha},\qquad c=e^{-\beta},\qquad d=e^\beta.
\label{eq:abcd}
\end{equation}
Since the strong coupling limit is taken with $\alpha\approx\beta$, we find it convenient to further parametrize the endpoints by
\begin{equation}
\alpha=\gamma+\delta,\qquad\beta=\gamma-\delta.
\label{eq:abgd}
\end{equation}
The symmetric case, $t_1=t_2$ and $\kappa_1=\kappa_2$, corresponds to $\delta=0$ and
\begin{equation}
t_1=t_2\sim\frac{\kappa_1+\kappa_2}{3\pi^2}\gamma^3,
\label{eq:endptIntermed}
\end{equation}
at least to leading order \cite{Suyama:2011yz}. More generally, the scaling $t_i\sim\gamma^3$ continues to hold, while $\delta$ is of subleading order compared with $\gamma$. The relation between $\{\gamma,\delta\}$ and $\{t_1,t_2\}$ will be worked out in more detail below.
\subsection{Computing the free energy}\label{subsec:freeEnergy}
While the leading order free energy, (\ref{eq:FGT0}), can be obtained directly from a large-$N$ saddle point solution \cite{Jafferis:2011zi}, since we are interested in subleading corrections, we will instead work with the resolvent, following \cite{Suyama:2010hr,Suyama:2011yz}. In particular, making the identification $g_s={2\pi i}/{k}$, the free energy can be written in the form of a genus expansion
\begin{equation}
F=\sum_{g=0}^\infty g_s^{2g-2}F_g(t).
\end{equation}
It has long been known that the genus-zero free energy, $F_0(t)$, for such matrix models can be written as an integral of the planar resolvent over a particular contour \cite{Dijkgraaf:2002fc,Halmagyi:2003fy,Halmagyi:2003ze}%
\footnote{Note that the convention for the free energy in this paper differs from that in \cite{Halmagyi:2003fy,Halmagyi:2003ze} in that we take the free energy to be $F=-\log Z$.}.
The basic idea is to look at the change in the leading order free energy from adding one eigenvalue to the branch cut, and use this to deduce the derivative of the genus-zero free energy with respect to the 't~Hooft parameter. The resulting expression can then be shown to an integral of the resolvent around the $B$-cycle, a contour that starts at infinity on one Riemann sheet, passes through the branch cut, and goes off to infinity on the other Riemann sheet. This results in a beautiful geometric picture, where the $A$-cycle determines the endpoints and the $B$-cycle determines the free energy; this is depicted in Figure \ref{fig:Bcycle} for the Chern-Simons matrix model. This is the strategy we will employ for GT theory.
\begin{figure}[t]
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\draw (392,111.4) node [anchor=north west][inner sep=0.75pt] {$\textcolor[rgb]{0.29,0.56,0.89}{B}$};
\draw (231,196.4) node [anchor=north west][inner sep=0.75pt] {$\textcolor[rgb]{0.82,0.01,0.11}{A}$};
\draw (271,269.4) node [anchor=north west][inner sep=0.75pt] [font=\scriptsize] {$-i\pi $};
\draw (274,72.4) node [anchor=north west][inner sep=0.75pt] [font=\scriptsize] {$i\pi $};
\end{tikzpicture}
\caption{The $A$ and $B$-cycle contours for Chern-Simons theory. Note that the Riemann sheets are curled up due to the $2\pi i$ periodicity of the resolvent.}
\label{fig:Bcycle}
\end{figure}
The two-node GT theory has two gauge groups whose eigenvalues condense along separate cuts in the complex plane. As a result, there are two B-cycle integrals to consider. We start by taking the genus-zero free energy $F_0=\left.g_s^2S\right|_{N\to\infty}$ from the effective action, (\ref{eq:action}). For the first gauge group, we play the trick of adding one more $\hat u$ eigenvalue to the first branch cut (\textit{i.e.} we take $N_1\to N_1+1$). The 't~Hooft parameter correspondingly changes by $\Delta t_1={2\pi}/{k}$. This gives
\begin{equation}
\fft{\Delta F_0}{\Delta t_1}=\fft{\kappa_1}2\hat u^2-t_1\fft1{N_1}\sum_i^{N_1}\log\sinh^2\fft{\hat u-u_i}2+t_2\fft1{N_2}\sum_i^{N_2}\log\cosh^2\fft{\hat u-v_i}2.
\end{equation}
Integrating the resolvent, (\ref{eq:resolventi})
\begin{equation}
v_1(z)=\fft{t_1}{N_1}\sum_{i=1}^{N_1}\fft{z+z_i}{z-z_i},
\end{equation}
we then obtain
\begin{equation}
\fft{t_1}{N_1}\sum_{i=1}^{N_1}\log\sinh^2\fft{\hat u-u_i}2=-\int_{e^{\hat u}}^{e^\Lambda}v_1(z)\fft{dz}z+t_1(\Lambda-\log4),
\end{equation}
where $\Lambda$ is a large cutoff and we have dropped exponentially small terms of the form $e^{-\Lambda}$. Using this expression and a similar one for the integral of $v_2(z)$ gives, in the large-$N$ limit
\begin{equation}
\fft{\partial F_0}{\partial t_1}=\fft{\kappa_1}2\hat u^2+\int_{e^{\hat u}}^{e^\Lambda}v(z)\fft{dz}z-(t_1-t_2)(\Lambda-\log4).
\end{equation}
We take the last eigenvalue $\hat u$ at the right endpoint of the cut, (\ref{eq:abcd}), and write
\begin{equation}
\fft{\partial F_0}{\partial t_1}=\fft{\kappa_1}2\beta^2+\int_\beta^\Lambda v(e^u)du-(t_1-t_2)(\Lambda-\log4).
\label{eq:dF0dt1}
\end{equation}
Geometrically, this is the $B_1$-cycle integral, which we have graphically depicted in Figure~\ref{fig:contours}. By swapping the two gauge groups, we can obtain a similar $B_2$-cycle integral for $\partial F_0/\partial t_2$. This integral will be worked out perturbatively in the next section.
\begin{figure}[t]
\centering
\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
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\draw (170.97,61.23) node [anchor=north west][inner sep=0.75pt] [color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] {$B_{2}$};
\draw (454.92,73.89) node [anchor=north west][inner sep=0.75pt] [color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] {$B_{1}$};
\draw (416.36,118.08) node [anchor=north west][inner sep=0.75pt] [color={rgb, 255:red, 208; green, 2; blue, 27 } ,opacity=1 ] {$\mathcal{C}_{1}$};
\draw (224.91,117.38) node [anchor=north west][inner sep=0.75pt] [color={rgb, 255:red, 208; green, 2; blue, 27 } ,opacity=1 ] {$\mathcal{C}_{2}$};
\end{tikzpicture}
\caption{The integration contours (in exponentiated coordinates) used in the derivation of the genus-zero free energy. Note that we no longer have the $2\pi i$ periodicity of the Riemann sheets because we are in exponentiated coordinates.}
\label{fig:contours}
\end{figure}
\section{Subleading Corrections to the Free Energy}
\label{sec:GTfree}
We now turn to an evaluation of the free energy beyond leading order. As we have seen above in (\ref{eq:vzint}), the planar resolvent for the GT model can be written down in integral form. While the integral is challenging to perform analytically, the general expression will be sufficient when working out the free energy.
Our goal is to compute the derivative of the free energy, (\ref{eq:dF0dt1}), up to exponentially small terms in the large 't~Hooft parameter limit. To do so, we insert the integral expression for the resolvent, (\ref{eq:vzint}), into (\ref{eq:dF0dt1}) and work out the double integral in the large $t_1$ and $t_2$ limit. However, since this gives an expression for $\partial F_0/\partial t_1$ as a function of the endpoints of the cuts, (\ref{eq:abcd}), we additionally need to relate the endpoints to the 't~Hooft couplings using the $A$-cycle integrals (\ref{eq:endpointContour}). We will work this out first and then return to the free energy integral.
\subsection{Correction to the endpoints}
At leading order, the endpoints of the cuts scale with the 't~Hooft couplings according to (\ref{eq:endptIntermed}). However this will pick up corrections, both for $t_1\ne t_2$ and subleading in the couplings. We explicitly work out the $A$-cycle integral for $t_1$; then the $t_2$ expression follows from symmetry under $t_1\leftrightarrow t_2$ and $\kappa_1\leftrightarrow\kappa_2$ interchange.
Substituting the integral expression for the resolvent, (\ref{eq:vzint}), into (\ref{eq:endptIntermed}), then explicitly writing out the $A$-cycle integral as an integral over the discontinuity across the cut and finally interchanging the order of integration gives
\begin{equation}
t_1=\fft{\kappa_1}{2\pi^2}J_1+\fft{\kappa_2}{2\pi^2}J_2,
\end{equation}
where
\begin{subequations}
\begin{align}
J_1=\int_c^ddx\fft{\log x}{\sqrt{(x-a)(x-b)(x-c)(d-x)}}I(x),\\ J_2=\int_a^bdx\fft{\log(-x)}{\sqrt{(x-a)(b-x)(c-x)(d-x)}}I(x),
\end{align}
\label{eq:J1J2ints}%
\end{subequations}
with
\begin{equation}
I(z)=\int_c^d\fft{dy}y\fft{\sqrt{(y-a)(y-b)(y-c)(d-y)}}{z-y}.
\end{equation}
Here the principal value of $I(x)$ has to be taken in the $J_1$ integral. We proceed by rewriting these expressions in terms of exponentiated variables:
\begin{subequations}
\begin{align}
J_1&=\int_{-\beta}^\beta dv\fft{vI(e^v)}{4\sqrt{\cosh\fft{\alpha+v}2\cosh\fft{\alpha-v}2\sinh\fft{\beta+v}2\sinh\fft{\beta-v}2}},\\
J_2&=\int_{-\alpha}^\alpha dv\fft{vI(-e^v)}{4\sqrt{\sinh\fft{\alpha+v}2\sinh\fft{\alpha-v}2\cosh\fft{\beta+v}2\cosh\fft{\beta-v}2}},
\end{align}
\end{subequations}
and
\begin{equation}
I(z)=\int_{-\beta}^\beta du\fft{4\sqrt{\cosh\fft{\alpha+u}2\cosh\fft{\alpha-u}2\sinh\fft{\beta+u}2\sinh\fft{\beta-u}2}}{ze^{-u}-1}.
\end{equation}
Note that the $\cosh$ terms are never vanishing, while the $\sinh$ terms vanish at the endpoints. Moreover, the square-root factors are all even under $v\to-v$ or $u\to-u$. This suggests that we split up the regions of integration into half intervals and write
\begin{subequations}
\begin{align}
J_1&=\int_0^\beta dv\fft{vI_1(v)}{4\sqrt{\cosh\fft{\alpha+v}2\cosh\fft{\alpha-v}2\sinh\fft{\beta+v}2\sinh\fft{\beta-v}2}},\\
J_2&=\int_0^\alpha dv\fft{vI_2(v)}{4\sqrt{\sinh\fft{\alpha+v}2\sinh\fft{\alpha-v}2\cosh\fft{\beta+v}2\cosh\fft{\beta-v}2}},
\end{align}
\end{subequations}
where
\begin{subequations}
\begin{align}
I_1(v)&=\int_0^\beta du\,4\sqrt{\textstyle\cosh\fft{\alpha+u}2\cosh\fft{\alpha-u}2\sinh\fft{\beta+u}2\sinh\fft{\beta-u}2}\left(\coth\fft{v-u}2+\coth\fft{v+u}2\right),\\
I_2(v)&=\int_0^\beta du\,4\sqrt{\textstyle\cosh\fft{\alpha+u}2\cosh\fft{\alpha-u}2\sinh\fft{\beta+u}2\sinh\fft{\beta-u}2}\left(\tanh\fft{v-u}2+\tanh\fft{v+u}2\right).
\end{align}
\end{subequations}
Here we see explicitly that the integrand of $I_1$ has a pole when $v-u$ vanishes, so the principal value ought to be taken when evaluating the integral.
So far, these expressions are still exact, as far as the planar resolvent is concerned. However, the integrals are not easy to evaluate. To proceed, we now focus on the large 't~Hooft coupling limit, where $\alpha,\beta\gg1$. Since the integrals are over half intervals, we can approximate $\alpha+v\gg1$, $\beta+v\gg1$ and similarly for $v$ replaced by $u$. As a result, up to exponentially suppressed terms, we have
\begin{align}
J_1=\int_0^\beta dv\fft{ve^{-\fft12(\gamma+v)}I_1(v)}{2\sqrt{\cosh\fft{\alpha-v}2\sinh\fft{\beta-v}2}},\qquad
J_2=\int_0^\alpha dv\fft{ve^{-\fft12(\gamma+v)}I_2(v)}{2\sqrt{\sinh\fft{\alpha-v}2\cosh\fft{\beta-v}2}},
\label{eq:J1intdef}
\end{align}
with
\begin{subequations}
\begin{align}
I_1(v)&=\int_0^\beta du\,2e^{\fft12(\gamma+u)}\sqrt{\textstyle\cosh\fft{\alpha-u}2\sinh\fft{\beta-u}2}\left(\coth\fft{v-u}2+\coth\fft{v+u}2\right),\\
I_2(v)&=\int_0^\beta du\,2e^{\fft12(\gamma+u)}\sqrt{\textstyle\cosh\fft{\alpha-u}2\sinh\fft{\beta-u}2}\left(\tanh\fft{v-u}2+\tanh\fft{v+u}2\right).
\end{align}
\label{eq:I12intdef}%
\end{subequations}
Recall that we have defined $\gamma=(\alpha+\beta)/2$ and $\delta=(\alpha-\beta)/2$, following (\ref{eq:abcd}).
The $I_1$ and $I_2$ integrals can be performed explicitly and then substituted into the integrands for $J_1$ and $J_2$. The remaining integrals are more challenging, and we have been unable to obtain a closed form expression for $J_1$ and $J_2$. Nevertheless, they can be reduced to polynomial expressions in $\gamma$ up to exponentially suppressed terms. The integration is worked out in Appendix~\ref{appendix:endpoints}, and the result is a relation between the 't~Hooft couplings $t_1$ and $t_2$ and the endpoints of the cuts as parametrized by $\gamma$ and $\delta$. After defining convenient combinations of $t_1$ and $t_2$,
\begin{equation}
\bar t=\ft12(t_1+t_2),\qquad\Delta=\ft12(t_1-t_2),
\end{equation}
we find
\begin{subequations}
\begin{align}
\bar t&=\fft{\kappa_1+\kappa_2}{4\pi^2}\biggl[\fft43(\gamma-\log\ft12\cosh\delta)^3+4\gamma\tan^{-1}\sinh\delta(\tan^{-1}\sinh\delta-\xi)\nonumber\\
&\kern5em+\fft43\log^3(\ft12\cosh\delta)+j_{1,e}(\delta)+j_{2,e}(\delta)+\fft{2\xi}\pi j_{1,o}(\delta)\biggr],\\
\Delta&=\fft{\kappa_1+\kappa_2}{4\pi^2}\Bigl[-2\pi\gamma(\tan^{-1}\sinh\delta-\xi)+j_{1,o}(\delta)+2\pi \xi\log\ft12\cosh\delta\Bigr],
\end{align}
\label{eq:tbD}%
\end{subequations}
where
\begin{equation}
\xi:=\fft\pi2\fft{\kappa_1-\kappa_2}{\kappa_1+\kappa_2}=\fft\pi2\fft{k_1-k_2}{k_1+k_2},
\label{eq:xdef}
\end{equation}
is the relative difference in Chern-Simons levels. Here $j_1(\delta)$ and $j_2(\delta)$ are particular functions explicitly defined in Appendix \ref{appendix:endpoints}, and the subscripts $e$ and $o$ denote their even and odd parts, respectively.
For the most part, we are interested in the case of equal ranks, $N_1=N_2$, in which case the difference $\Delta$ vanishes. Setting $\Delta=0$ in (\ref{eq:tbD}) then gives a straightforward expression for $\gamma$ in terms of $\delta$
\begin{equation}
2\pi\gamma(\tan^{-1}\sinh\delta-\xi)=j_{1,o}(\delta)+2\pi \xi\log\ft12\cosh\delta.
\label{eq:2pgeqn}
\end{equation}
However, we are actually more interested in obtaining $\delta$ in terms of $\gamma$ since we are focused on the large coupling expansion generalizing (\ref{eq:endptIntermed}) where $t_i\sim\gamma^3$ with $\delta$ being subdominant. Working to leading order in $\gamma$, we can disregard the last two terms in the expression for $\Delta$ in (\ref{eq:tbD}), so that
\begin{equation}
\delta\approx\sinh^{-1}\tan\left(\xi\right).
\end{equation}
However, we can do better than this. Since we assume $\gamma\gg1$, we can expand perturbatively
\begin{equation}
\delta\approx\sinh^{-1}\tan\left(\xi\right)+\frac{\delta_1}{\gamma}+\frac{\delta_2}{\gamma^2}+\mathcal{O}\qty(\frac{1}{\gamma^3}).
\label{eq:delta}
\end{equation}
Solving the $\mathcal{O}(\gamma^0)$ expression in $\Delta$ gives
\begin{equation}
\delta_1=\sec \left(\xi\right)\operatorname{Cl}_2\left(\pi+2\xi\right),
\end{equation}
where $\operatorname{Cl}_2(x)$ denotes the Clausen function
\begin{equation}
\operatorname{Cl}_2(x)=\Im\operatorname{Li}_2(e^{ix}).
\end{equation}
The expression for $\delta_2$ is rather more involved
\makeatletter
\newcommand{\bBigg@{4}}{\bBigg@{4}}
\makeatother
\begin{align}
\delta_2&=\frac{1}{2}\sec\qty(\xi)\operatorname{Cl}_2\qty(\pi+2\xi)
\bBigg@{4}[\tan \left(\xi\right) \left(2\xi-2 \text{gd}\left(\sinh ^{-1}\left(\tan \left(\xi\right)\right)\right)+\operatorname{Cl}_2\qty(\pi+2\xi)\right)\nonumber\\
&-4 \sinh ^{-1}\left(\tan \left(\xi\right)\right)-2 \log \left(\frac{8\sec\qty(\xi)}{\left(\qty(\tan \left(\xi\right)+\sec \left(\xi\right))^2+1\right){}^2}\right)\bBigg@{4}],
\end{align}
where $\text{gd}$ denotes the Gudermannian function
\begin{equation}
\mathrm{gd}(z)=2\arctan\tanh\qty(\tfrac{1}{2}z).
\end{equation}
This is a rather messy expression, but for $\xi\ll 1$, it takes the nice perturbative form
\begin{equation}
\delta_2\approx 2\xi \log ^2(2)+\xi^3\left(3\log ^2(2)-\fft43 \log (2)\right)+\mathcal{O}\qty(\xi^5).
\end{equation}
Having obtained $\delta$, at least perturbatively, we now proceed to related $\gamma$ and $\bar t$. Keeping $\Delta=0$, we first eliminate the second term in the $\bar t$ expression in (\ref{eq:tbD}) to obtain
\begin{align}
\bar t&=\fft{\kappa_1+\kappa_2}{4\pi^2}\biggl[\fft43(\gamma-\log\ft12\cosh\delta)^3+\fft2\pi(\tan^{-1}\sinh\delta+\xi)j_{1,o}(\delta)\nonumber\\
&\kern5em+4\log(\ft12\cosh\delta)\left(\fft13\log^2(\ft12\cosh\delta)+\xi\tan^{-1}\sinh\delta\right)
+j_{1,e}(\delta)+j_{2,e}(\delta)\biggr].
\end{align}
This expression is useful since now the only $\gamma$ dependence shows up in the first term. We can now substitute the perturbative expression (\ref{eq:delta}). To first non-trivial order, we find
\begin{align}
\bar t&=\fft{\kappa_1+\kappa_2}{4\pi^2}\biggl[\fft43(\gamma-\log\ft12\cosh\delta)^3-4\log(2\cos \xi)\left(\fft13\log^2(2\cos \xi)+\xi^2\right)\nonumber\\
&\kern5em+\fft{4\xi}\pi j_{1,o}(\delta)+j_{1,e}(\delta)+j_{2,e}(\delta)+\mathcal O(\gamma^{-1})\biggr].
\end{align}
The transcendental functions on the second line are a bit troublesome to work with. However, by studying the series expansion of $j_1(\delta)$ and $j_2(\delta)$, we can determine empirically that
\begin{equation}
\bar t=\fft{\kappa_1+\kappa_2}{4\pi^2}\left[\fft43(\gamma-\log\ft12\cosh\delta)^3-4(\operatorname{Cl}_3(\pi-2\xi)+\zeta(3))+\mathcal O(\gamma^{-1})\right],
\label{eq:tbgeqn}
\end{equation}
where
\begin{equation}
\operatorname{Cl}_3(x)=\Re\operatorname{Li}_3(e^{ix}).
\end{equation}
We will make use of this expression below when computing the planar free energy.
\subsection{The free energy}
We now turn to the evaluation of the free energy which can be obtained from the integral expression, (\ref{eq:dF0dt1}). The $B$-cycle integral can be evaluated in a similar manner as the $A$-cycle integral performed above for computing the endpoint relation. In particular using the integral expression for the resolvent, (\ref{eq:vzint}), we can write
\begin{equation}
\fft{\partial F_0}{\partial t_1}=\fft{\kappa_1}2\beta^2-(t_1-t_2)(\Lambda-\log4)-\fft{\kappa_1}\pi K_1-\fft{\kappa_2}\pi K_2,
\end{equation}
where
\begin{subequations}
\begin{align}
K_1=\int_c^ddx\fft{\log x}{\sqrt{(x-a)(x-b)(x-c)(d-x)}}I_B(x),\\ K_2=\int_a^bdx\fft{\log(-x)}{\sqrt{(x-a)(b-x)(c-x)(d-x)}}I_B(x).
\end{align}
\end{subequations}
These integrals are similar to the $J_1$ and $J_2$ integrals in (\ref{eq:J1J2ints}), except that now $I_B(x)$ is a $B$-cycle integral
\begin{equation}
I_B(z)=\int_d^{e^\Lambda}\fft{dy}y\fft{\sqrt{(y-a)(y-b)(y-c)(y-d)}}{z-y}.
\end{equation}
These integrals can be evaluated up to exponentially small terms in a similar manner as was done for the endpoint integrals. Combining $\partial F_0/\partial t_1$ and the corresponding expression for $\partial F_0/\partial t_2$, we find the relatively compact expression
\begin{equation}
\fft{\partial F_0}{\partial\bar t}=\fft{\kappa_1+\kappa_2}2\Bigl[(\gamma-\log\ft12\cosh\delta)^2+(\tan^{-1}\sinh\delta-\xi)^2-\ft1{12}\pi^2-\xi^2\Bigr].
\label{eq:dF0dtb}
\end{equation}
Details of the calculation are given in Appendix \ref{appendix:freeEnergy}.
We now have everything we need to obtain the planar free energy from the resolvent. Since the derivative $\partial F_0/\partial\bar t$ is given in terms of the endpoint parameters $\gamma$ and $\delta$, the general procedure is to first obtain these parameters from the 't~Hooft couplings $t_1$ and $t_2$ by inverting the endpoint relations (\ref{eq:tbD}). After doing so, it becomes straightforward to integrate (\ref{eq:dF0dtb}) to obtain the planar free energy $F_0$ up to a $\bar t$ independent constant which remains to be fixed.
Focusing on the case $\Delta=0$, the relation (\ref{eq:2pgeqn}) demonstrates that the combination $(\tan^{-1}\sinh\delta-\xi)$ is of $\mathcal O(\gamma^{-1})$. As a result, (\ref{eq:dF0dtb}) can be written as
\begin{equation}
\fft{\partial F_0}{\partial\bar t}=\fft{\kappa_1+\kappa_2}2\Bigl[(\gamma-\log\ft12\cosh\delta)^2-\ft1{12}\pi^2-\xi^2+\mathcal O(\gamma^{-2})\Bigr].
\end{equation}
Making use of the $\bar t$ versus $\gamma$ relation, (\ref{eq:tbgeqn}), and integrating then gives the genus zero free energy
\begin{equation}
F_0=\bar\kappa^2\left[\fft35\left(\fft{3\pi^2}2\right)^{2/3}\left(\fft{\bar t}{\bar\kappa}+\fft2{\pi^2}\left(\operatorname{Cl}_3(\pi-2\xi)+\zeta(3)\right)\right)^{5/3}-\left(\fft{\pi^2}{12}+\xi^2\right)\fft{\bar t}{\bar\kappa}+\mathcal O(\bar t^{1/3})+\mbox{const.}\right],
\label{eq:f0fin}
\end{equation}
where we have defined
\begin{equation}
\bar\kappa=\fft{\kappa_1+\kappa_2}2=\fft{k_1+k_2}{2k}.
\end{equation}
Several points are now in order. Firstly, the ``constant'' term is independent of $\bar t$ but can depend on the fractional difference of Chern-Simons levels, $\xi$. However, it cannot be obtained directly from integrating the derivative of the free energy%
\footnote{In contrast, the $\mathcal{O}(\ol{t}^{1/3})$ part can, in principle, be obtained term-by-term from higher-order perturbation theory. We denote the $\mathcal{O}(\ol{t}^{1/3})$ and constant terms separately to emphasize this distinction.}.
In addition, the leading term in the large-$\bar t$ expansion of this expression matches what we expect from \eqref{eq:FGT=}. Finally, note that the $\mathcal O(\bar t^{1/3})$ term vanishes in the $\xi=0$ limit (ie, for $k_1=k_2$). In this case, expression (\ref{eq:f0fin}) is exact up to exponentially small terms in $\bar t$.
In fact, it is easily seen that $\delta$ vanishes in the $\xi=0$ limit. As a result, (\ref{eq:tbgeqn}) takes on the simple relation
\begin{equation}
t=\fft{\kappa}{2\pi^2}\left(\ft43(\gamma+\log2)^3-\zeta(3)\right),
\end{equation}
and the planar free energy, (\ref{eq:f0fin}) becomes
\begin{equation}
F_0=\kappa^2\left[\fft35\left(\fft{3\pi^2}2\right)^{2/3}\left(\frac{t}{\kappa}+\fft{\zeta(3)}{2\pi^2}\right)^{5/3}-\fft{\pi^2}{12}\frac{t}{\kappa}+\mbox{const.}+\mathcal O(e^{-t})\right].
\label{eq:F0equal}
\end{equation}
Here we have dropped the bars on $t$ and $\kappa$ as we are considering $t_1=t_2$ and $\kappa_1=\kappa_2$. If desired, this can be expanded in inverse powers of $t$
\begin{equation}
F_0(t)=-\frac{\pi^2}{12} t+\frac{3\cdot6^{2/3}}{40\pi^2}\qty(2\pi^2{t})^{5/3}\sum_{n=0}^\infty \frac{\qty(\tfrac{5}{3})_n}{n!}\qty(\frac{\zeta(3)}{2\pi^2t})^n,
\label{eq:F0expand}
\end{equation}
where $(\ )_n$ denotes the Pochhammer symbol. Since this expression holds for $k_1=k_2$, we have set $\kappa=1$ and $t=2\pi iN/k$. Note that $t$ is imaginary when we take $N$ and $k$ to be real. In this case, the first term, which is linear in $t$, does not contribute to the real part of the free energy.
\subsection{Numerical Analysis}
Our main result is the expression, (\ref{eq:f0fin}), for the genus zero free energy $F_0(N,k_1,k_2)$ at large 't~Hooft coupling $\bar t$. While the first term is complete, additional terms of $\mathcal O(\bar t^{1/3})$ and smaller will contribute when the Chern-Simons levels are different, as parametrized by $\xi$ defined in (\ref{eq:xdef}). In order to get an idea of the size of these terms, we carried out a numerical investigation of the large-$N$ partition function. In this limit, we solved the saddle-point equations in Mathematica for $N$ ranging from 100 to 340 at fixed (real positive) 't~Hooft coupling $\bar t$ and extrapolated $N\to \infty$, using a working precision of 50. This was done for various values of $\bar t$ and then fitted to extract the subleading coefficients $f_1(\xi)$ and $f_2(\xi)$ in the expansion
\begin{equation}
F_0(\bar t,\bar\kappa,\xi)=\fft35\left(\fft{3\pi^2}2\right)^{2/3}\bar t^{5/3}\bar\kappa^{1/3}+f_1(\xi)\bar t+f_2(\xi)\bar t^{2/3}+\cdots,
\label{eq:F0exp...}
\end{equation}
Throughout these fits, we hold $\bar\kappa=1$ fixed since changing the value of $\bar\kappa$ is equivalent to an overall rescaling of $k$. The coefficients $f_1(\xi)$ and $f_2(\xi)$ are then extracted from the numerical free energy for various values of $\xi=(\pi/4)(\kappa_1-\kappa_2)$. Due to the computational difficulty of this process, this was only done for five sample points corresponding to $\xi=\{0,\tfrac{\pi}{40},\tfrac{\pi}{20},\tfrac{\pi}{10},\tfrac{3\pi}{20}\}$.
We have verified that the leading order term in \eqref{eq:F0exp...} is reproduced numerically to very high precision and that no term of $\mathcal O(\bar t^{4/3})$ shows up within numerical uncertainties. As a result, we subtracted the analytic value of the leading term and fit only the subdominant coefficients. The coefficient $f_1(\xi)$ of the linear $\ol{t}$ term shows very good agreement, and is plotted in Figure~\ref{fig:linearTerm}.
\begin{figure}[t]
\includegraphics[width=0.75\textwidth]{linear_t_coeff.PNG}
\centering
\caption{Plot of the coefficient $f_1(\xi)$. The red line is the analytic prediction from \eqref{eq:f0fin}, and the blue dots are sample points for numerical simulations performed in Mathematica for $\xi=0$, $\tfrac{\pi}{40}$, $\tfrac{\pi}{20}$, $\tfrac{\pi}{10}$, and $\tfrac{3\pi}{20}$.}
\label{fig:linearTerm}
\end{figure}
We also plot the coefficient $f_2(\xi)$ of $\ol{t}^{2/3}$ in Figure~\ref{fig:t2/3term}. Here, the coefficient is slightly less numerically stable, and we cannot see the agreement quite as well. Nonetheless, we still see fairly good agreement with the data.
\begin{figure}[t]
\includegraphics[width=0.75\textwidth]{subleading_t_coeff.PNG}
\centering
\caption{Plot of the coefficient $f_2(\xi)$. The red line is the analytic prediction from \eqref{eq:f0fin}, and the blue dots are sample points for numerical simulations performed in Mathematica for $\xi=0$, $\tfrac{\pi}{40}$, $\tfrac{\pi}{20}$, $\tfrac{\pi}{10}$, and $\tfrac{3\pi}{20}$.}
\label{fig:t2/3term}
\end{figure}
\section{Discussion}
\label{sec:disc}
While the leading order $N^{5/3}k^{1/3}$ behavior of the free energy of GT theory was essentially known since the model was first introduced, the subleading corrections have been surprisingly difficulty to obtain analytically. The planar resolvent was constructed in \cite{Suyama:2010hr}. However its form did not readily lend itself to a simple expression for the free energy beyond the leading order. Even the remarkable Fermi-gas approach to Chern-Simons-matter theories \cite{Marino:2011eh} runs into limitations when exploring higher order corrections \cite{Hong:2021bsb}.
We were able to obtain the planar free energy up to exponentially small corrections in the limit of large 't~Hooft coupling by working with the resolvent (\ref{eq:vzint}) in integral form. The main technical observation is that the endpoints of the cuts can be obtained from $A$-cycle integrals of the resolvent integral while the derivative of the free energy can be obtained from $B$-cycle integrals. The order of the resulting double integrals can then be swapped, leading to expressions that can be more readily worked with. The key results are then the endpoint relations (\ref{eq:tbD}) and the free energy expression (\ref{eq:dF0dtb}).
The expressions (\ref{eq:tbD}) and (\ref{eq:dF0dtb}) in principle allow us to obtain the planar free energy $F_0(N_1,N_2,k_1,k_2)$ in the $\bar t\gg1$ limit directly in terms of the parameters of the model. However, inverting the endpoint equations is generally non-trivial. Nevertheless, for small differences in the Chern-Simons levels, $|k_1-k_2|\ll|k_1+k_2|$, these equations can be inverted perturbatively, assuming the self-consistent condition $|\delta|\ll1$ on the endpoints. Focusing on the equal rank case $N_1=N_2$, or equivalently $\Delta=0$,
we have found an explicit expansion of the free energy. If, in addition, the Chern-Simons levels are equal, we obtain the closed form expression (\ref{eq:F0equal}), which is exact up to exponentially suppressed terms.
While we have focused on the equal rank case, one can work with unequal ranks if desired. Here some care may be needed depending on how $N_1$ and $N_2$ scale in the large-$N$ limit, as there are now two independent 't~Hooft parameters. If the difference in ranks, $N_1-N_2$, is held fixed, then $\Delta$ is a constant, and the perturbative inversion of the endpoint equations (\ref{eq:tbD}) can be worked out as usual. However, if $\Delta$ is not fixed, then the inversion of $\{t_1,t_2\}\leftrightarrow\{\gamma,\delta\}$ becomes more involved and the free energy as a function of two independent 't~Hooft parameters becomes less obvious.
From a technical point of view, it is possible that the way we have chosen to break the integrals into intermediate functions is not necessarily the most efficient. Many of the expressions in Appendices \ref{appendix:endpoints} and \ref{appendix:freeEnergy} are quite complicated, and one may wonder if there is a simpler parameterization that makes the formulation more elegant. One possibility is to organize the expressions by the degree of transcendentality. However, it is not clear if this would actually make them simpler.
One of the motivations to examine the subleading behavior of the free energy is to compare with the holographic dual. From this point of view, it is interesting to observe that the expansion (\ref{eq:F0expand}) involves powers of $\zeta(3)/t$. From the supergravity point of view, this is suggestive of the $\alpha'$ expansion of the tree-level closed string effective action which starts with a term of the form $\zeta(3)\alpha'^3R^4$ \cite{Gross:1986iv}. More generally, at higher derivative order, one expects a series of corrections of the form $\alpha'^{3(n+1)}\zeta(3)^nD^{6n}R^4$, or equivalently $\alpha'^{3(n+1)}\zeta(3)^nR^{4+3n}$, which would provide an obvious source of corrections to the dual free energy.
Of course, for now this is only a heuristic picture, as many open questions remain to be addressed before the comparison can be made rigorous. For one thing, while the higher derivative couplings have been extensively studied for type II strings, the dual to GT theory is massive IIA supergravity, which may not receive exactly the same corrections as ordinary type II supergravity. Nevertheless, we expect the structure to be very similar, at least if we assume a common M-theory origin.
Perhaps more importantly, advances in computing open and closed tree-level string amplitudes have provided a clearer picture of the structure of higher derivative corrections beyond $\alpha'^3R^4$. In particular, it is known that the $\alpha'$ expansion yields terms of the form $\alpha'^{3+n}D^{2n}R^4$ (along with counterparts such as $\alpha'^{3+n}R^{4+n}$) multiplied by various combinations of $\zeta(n)$. Assuming the free energy can be expanded only in powers of $\zeta(3)$ then demands that these other terms not proportional to $\zeta(3)^n$ do not contribute to the free energy, and hence must vanish on-shell in the gravity dual.
Finally, the form of the planar free energy, (\ref{eq:F0equal}), where the large-$t$ expansion involves a linear function of $t$ raised to a fractional power, may hint at some underlying symmetry in the $\alpha'$ expansion. It would be interesting to study the dual massive IIA description of GT theory and to clarify some of these questions. One obstacle in doing so is the lack of an explicit construction of the dual supergravity background beyond the limit of
infinitesimally small Romans mass \cite{Gaiotto:2009yz}. However, we hope that such a solution may be found in the near future.
\section*{Acknowledgements}
We wish to thank J.~Hong for useful discussions. This work was supported in part by the U.S. Department of Energy under grant DE-SC0007859.
|
{
"timestamp": "2021-08-03T02:43:08",
"yymm": "2106",
"arxiv_id": "2106.03901",
"language": "en",
"url": "https://arxiv.org/abs/2106.03901"
}
|
\section{Introduction}
The magnetic field in the crust of strongly magnetised neutron stars evolves due to the Hall effect, which is the advection of the magnetic flux by the electron fluid. This process dominates over Ohmic decay, which is caused by the finite conductivity of the crust, for magnetic fields above $10^{12}-10^{13}$G
\citep{Goldreich:1992, Cumming:2004}. A key assumption in the Hall-Ohmic evolution scheme is that the forces acting on the crust are in balance and the system remains in equilibrium. This holds, provided that any deformation of the crust is below its elastic limit. At the upper end of neutron star magnetic fields, however, the magnetic field may become sufficiently intense to exceed the yield stress, at which point the crust will fail. Such crustal-failure events have been related to explosive activity in magnetars such as bursts and flares \citep{Thompson:1995, Turolla:2015,Kaspi:2017, Gourgouliatos:2018b, Esposito:2021}. In this context, the magnetic field of magnetars produces Maxwell stress leading to shear deformations close or beyond the elastic limit of the crust \citep{Horowitz:2009, Perna:2011, Pons:2011, Levin:2012, Lander:2016, Bransgrove:2018}. The outcome of these failures may lead to short-lived events of individual outbursts \citep{Beloborodov:2014, Li:2016, Beloborodov:2016}.
While it is reasonable to expect that these explosive events are connected with the crust's elastic limit being exceeded, it is not clear how the magnetic field evolves beyond that point. The Hall-MHD evolution scheme is not valid, as the assumption of an elastically deformed crust does not hold any more. A likely scenario is that once the elastic limit is reached the crust enters a state of slow deformation or flow, where the Maxwell stress of the magnetic field drives the evolution of the crust. At the same time, the electrons continue to advect the magnetic field and the two effects act simultaneously.
In previous work \citep{Lander:2019} we considered the combination of Hall-MHD evolution and a plastic flow in a Cartesian domain, with plane-parallel symmetry. The Cartesian domain represented a slab of the neutron star crust that becomes stressed by the intense magnetic field, eventually exceeding the elastic limit and failing. We modelled the plastic flow that was initiated once the slab of crust had failed, assuming that the field then evolved due to the combination of plastic flow and the Hall effect in the entire domain. Although this study could not capture the global impact of this evolution on the crustal magnetic field, it allowed us to explore some of the characteristics of magneto-plastic flow and avoid the additional complexity associated with the possibility of having plastic flow in some regions of the crust and not others.
In the work reported here we extend our study to a global axisymmetric model simulating the magnetic field evolution in the entire crust. We simulate the magnetic field evolution due to the Hall-Ohmic effect and account for the plastic flow if the crust fails. Our work complements and extends the recent study of \cite{Kojima:2020}, who studied global crustal field evolution under a viscous flow, using the formalism of \cite{Lander:2016, Lander:2019}. It was found that for a strong magnetic ($B>10^{14}$ G) field and low plastic viscosity $10^{36}-10^{37}$ g cm$^{-1}$s$^{-1}$ a significant fraction of the energy is transferred into the bulk flow energy. Furthermore, these effects can lead to the magnetic deformation of the crust \citep{Kojima:2021}.
The work of \cite{Kojima:2020, Kojima:2021} makes, however, two key simplifications that deserve further study. Firstly, they start their simulations assuming the crust is already in a plastic regime, avoiding the challenging issue of diagnosing and applying an elastic failure criterion in their simulations. Secondly, by simulating a viscous crust they effectively assume that any crustal failure is global, so that the entire crust yields together, whether or not the local value of the stress is always very high. If, instead, the plastic flow is confined to the region where the yield stress is exceeded, or perhaps also its environs, the evolutionary path taken by the magnetic field may be very different. In this context, the profiles of magnetar outbursts suggest that the emitting region is not the entire crust but a limited fraction of it \citep{Tiengo:2008, Alford:2016, CotiZelati:2018}. This suggests that most of the crust remains intact despite the magnetar being in active phase. We further note that whilst we adopted a global flow in our previous study \citep{Lander:2019}, the simulations were intrinsically local, with the domain being a square slab with sides of length $0.5$ km. Thus, a major aim of this paper is to explore the impact of different types of failure and consequently plastic flow, addressing the differences between global and local flows.
The plan of the paper is as follows. Section \ref{MATH} contains the setup of the problem, the relevant equations and the crust model. In section \ref{SIMULATIONS}, we discuss the numerical approach implemented for the integration of the differential equations of the system. In section \ref{RESULTS} we present the results of the simulations. We discuss their implications for strongly magnetised neutron stars in section \ref{DISCUSSION}. We conclude in section \ref{CONCLUSIONS}.
\section{Problem setup}
\label{MATH}
\subsection{Magnetic induction equation}
Let us consider an axisymmetric magnetic field in spherical coordinates $(r,\theta,\phi)$
\begin{equation}
{\bf B}= B_{r}(r,\theta){\bf \hat{r}}+B_{\theta}(r,\theta)\bm {\hat{\theta}}+B_{\phi}(r,\theta)\bm{\hat{\phi}}\,.
\end{equation}
We assume that the evolution is driven by the flow of the electron fluid and the plastic deformation of the lattice. The former effect is approximated by the Hall-Ohmic evolution \citep{Goldreich:1992}, whereas the latter is described by a plastic flow. Taking these into account, the magnetic field induction equation becomes:
\begin{equation}
\partial_{t} {\bm B} = -\nabla\times \left[\left(\frac{c}{4 \pi e n_e}\left(\nabla \times {\bm B}\right)-{\bm v}_{pl}\right)\times {\bm B} +\frac{c^2}{4 \pi \sigma}\nabla \times {\bm B}\right]\,,\
\label{HALL_EQ}
\end{equation}
where $n_e$ is the electron number density, $\sigma$ the electrical conductivity, $e$ is the elementary charge, ${\bm v}_{pl}$ the plastic flow velocity and $c$ the speed of light.
The first term in the bracket results from the electron-fluid motion, the second one is the plastic flow velocity ${\bm v}_{pl}$ and the last one is due to the Ohmic dissipation. We can estimate the ratio of the Hall to the Ohmic term through the dimensionless Hall parameter:
\begin{equation}
R_{H}=\frac{\sigma |B|}{e c n_{e}}\,.
\label{RH}
\end{equation}
In the absence of a plastic flow, the Hall-Ohmic equation can be solved by direct numerical integration, as the knowledge of the microphysics determining $n_e$ and $\sigma$ suffice, in principle, for a numerical solution. We note, however, that the non-linear nature of the equation makes the solution of this equation far from trivial.
Further complexity is added to the problem once the plastic flow is included. The details of how a neutron-star crust fails are not well understood - and so there is no longer an unambiguous route to understanding the crust's evolution in this case. In any case, an additional equation is needed to determine the plastic flow velocity that now appears in equation \eqref{HALL_EQ}. We follow the basic principles of the formulation of \cite{Lander:2019}, where we approximate the plastic flow as a Stokes flow. The Laplacian of the plastic flow velocity is equal to the divergence of the traceless part of the stress tensor of the current crust state, the stressed crust, minus the field arising from the unstressed crust, corresponding to the initial state, (we refer to this state as ``reference state" in \cite{Lander:2019}). This results in a Poisson partial differential equation
\begin{equation}
4 \pi \nu \nabla^2 \bm{v}_{pl}=(\bm{B_0}\cdot \nabla)\bm{B_0}-(\bm{B}\cdot \nabla)\bm{B}-\frac{1}{3}\nabla(B_0^2-B^2) \, ,
\label{Laplacian}
\end{equation}
where a subscript $0$ denotes the magnetic field of the initial state.
We assume that any motion of the crust is incompressible. Thus, the continuity equation for the plastic flow velocity becomes
\begin{equation}
\nabla \cdot \bm{v}_{pl} = 0\,.
\end{equation}
Given that the crust is stably stratified in the radial direction, any radial displacement due to the plastic flow is zero, leading to a zero plastic flow radial velocity $v_{pl,r}=0$. Combining this with the continuity equation and taking into account axisymmetry, it reduces to the following differential equation
\begin{equation}
\frac{1}{r\sin\theta}\partial_{\theta}\left(v_{pl,\theta}\sin\theta\right)=0\,,
\end{equation}
for which the only physically acceptable solution is $v_{pl,\theta}=0$. Therefore, the plastic flow is only along the azimuthal direction $\bm{v}_{pl}=v_{pl}\bm{\hat{\phi}}$.
The only component of equation \eqref{Laplacian} that needs to be evaluated is the azimuthal one $\left(\bm{B} \cdot \nabla \right) \bm{B}|_{\phi}$
and leads to determination of the flow velocity. We note that the magnetic pressure gradient term $\nabla B^2
does not have any component in the $\phi$ direction because of axisymmetry.
In axisymmetry, we can express the magnetic field in terms of two scalar functions proportional to the poloidal flux $2\pi \Psi(r, \theta)$ and electric current $cI(r, \theta)/2$ that pass through a spherical cap centered on the axis of the system, at distance $r$ from the origin and semi-opening angle $\theta$. The magnetic field takes the following form:
\begin{equation}
{\bf B} =\nabla \Psi(r,\theta) \times \nabla \phi + I(r, \theta) \nabla \phi\,.
\label{B_FIELD}
\end{equation}
Note that the toroidal magnetic field is expressed in terms of the poloidal current. The above expression satisfies by construction Gauss' law for the magnetic field $\nabla \cdot {\bf B}=0$. Substituting the expression of the magnetic field from equation \eqref{B_FIELD} into equation \eqref{HALL_EQ} we obtain two coupled partial differential equation for the scalars $\Psi$ and $I$. The first one for the poloidal field evolution is not directly affected by the plastic flow and is given by the following expression
\begin{equation}
\partial_t \Psi-r^2\sin^2\theta \chi \left(\nabla I \times \nabla \phi\right)\cdot \nabla \Psi =\frac{c^2}{4 \pi \sigma}\Delta^{\star} \Psi\,,
\label{dPsi}
\end{equation}
where we have set as in \cite{Reisenegger:2007}
\begin{equation}
\chi=\frac{c}{4\pi e n_e r^2\sin^2\theta}\,,
\end{equation}
and the Grad-Shafranov operator
\begin{equation}
\Delta^{\star}=\frac{\partial^2}{\partial r^2}+\frac{\sin \theta}{r^2}\frac{\partial}{\partial \theta}\left(\frac{1}{\sin\theta}\frac{\partial}{\partial \theta}\right)\, .
\label{GSop}
\end{equation}
The electron fluid angular velocity is
\begin{equation}
\Omega_e=\chi \Delta^{\star}\Psi,
\label{Omega}
\end{equation}
and we also define the plastic flow angular velocity:
\begin{equation}
\Omega_{pl}=\frac{v_{pl}}{r\sin\theta}.
\end{equation}
The second equation is related to the toroidal field evolution. In addition to the terms arising by the Hall effect, it is directly affected by the plastic flow velocity
\begin{eqnarray}
\frac{\partial I}{\partial t}+r^2\sin^2\theta\left\{\left[\nabla\left(\Omega_{e}+\Omega_{pl}\right)\times \nabla \phi\right]\cdot \nabla \Psi+I\left(\nabla \chi \times \nabla \phi\right)\cdot \nabla I\right\}\nonumber \\
=\frac{c^2}{4\pi \sigma}\left(\Delta^{\star} I-\frac{1}{\sigma}\nabla I \cdot \nabla \sigma\right)
\label{dI}
\end{eqnarray}
Thus, integration of equations (\ref{dPsi}), (\ref{dI}) and use of equation (\ref{Laplacian}) with an appropriate failure criterion for the evaluation of the plastic flow velocity will allow the determination of the magnetic field evolution.
\subsection{Plastic flow initiation}
\label{PFI}
Directly before a neutron star's crust freezes, the stellar structure (including its magnetic field) is that of a fluid body, with no shear stresses. It follows that the crust forms in an unstressed state. In this work we consider only stresses that build up over time due to the evolving crustal magnetic field deviating from its initial state, although rotation will generally also add to the crust's stress.
For a sufficiently weak magnetic field, stresses will never grow enough to induce failure of the star's crust, and so the plastic flow velocity will remain identically zero. In this case, any deformation stays within the crust's elastic limit and the Hall-Ohmic evolution suffices for the description of the magnetic field evolution in the crust. For typical magnetar-strength fields, however, it is quite likely that Maxwell stresses will become strong enough to lead to crustal failure. It is not fully resolved, however, how these failures start and progress.
In our approach, we use the modified von Mises criterion as described in \cite{Lander:2019}. In particular, the crust in our model fails if the following inequality is satisfied:
\begin{equation}
\tau_{el}\leq \frac{1}{4\pi} \sqrt{\frac{1}{3} B_0^4+\frac{1}{3}B^4+\frac{1}{3}B_0^2B^2-\left({\bm B}\cdot {\bm B_0}\right)^2}\,,
\label{FAILURE_EQ}
\end{equation}
where $\tau_{el}$ is the critical value of the stress the crust can support and is determined by the microphysics of the crust.
Even if the crust fails somewhere, we further need to answer how such failures propagate in the crust and whether they lead to a flow localised only in the region where the inequality (\ref{FAILURE_EQ}) holds, or extend over larger parts of the crust. Given these uncertainties, we simulate three types of failure: a local, an intermediate and a global one, which correspond to different treatments of equation \ref{Laplacian}.
In the case of a local failure, we assume that the plastic flow velocity is given by the solution of equation (\ref{Laplacian}) in the region where the failure criterion is satisfied. Apart from this region, the plastic flow velocity is set to zero everywhere else in the crust. This way, the only part that flows is the region where the condition described in inequality (\ref{FAILURE_EQ}) is satisfied.
In the intermediate case, we solve equation (\ref{Laplacian}) demanding that its right-hand-side is non-zero in the part of the crust where the failure criterion (\ref{FAILURE_EQ}) is satisfied, and zero elsewhere. Thus, the plastic flow velocity can be non-zero anywhere in the crust, however, a source term appears in equation (\ref{Laplacian}) only in the region where the failure criterion is satisfied. This provides a smoother transition of the plastic flow velocity between the regions of plastic flow and the rest of the star, whereas in the local flow, the transition between the failed region and the rest of the crust is rather sharp.
Finally, in the scenario of a global failure, a plastic flow begins everywhere in the crust if the criterion of equation \eqref{FAILURE_EQ} is fulfilled even in a single point in the crust. The plastic flow velocity corresponds to the solution of equation \eqref{Laplacian} everywhere in the crust and thus is non-zero everywhere, although will be largest in the failed region. This last case is rather extreme and physically unlikely, but it is worth studying to set the maximum possible plastic flow that can result from our formalism.
\begin{figure}
\includegraphics[width=0.45\textwidth]{Parameters.png}
\caption{Dominance of Hall, Ohmic and Plastic flow depending on the magnetic field and the density of the crust. The horizontal axis is either the radius of the star from the crust-core interface to the neutron drip point or the density (upper x-axis). The vertical axis is the intensity of the magnetic field.}
\label{FIG:1}
\end{figure}
\subsection{Neutron Star Properties}
For the numerical integration of partial differential equations (\ref{dPsi}) and (\ref{dI}) we require a model for the crust microphysical parameters. We approximate the crust with a spherical shell, starting from the crust-core boundary $R_{CC}$ up to the neutron drip point $R_{ND}$, where $R_{CC}/R_{ND}=0.95$. We use the density, electron number density, elastic limit and conductivity profiles from our previous work, \cite{Lander:2019}. In particular, we define a dimensionless coordinate
\begin{equation}
\mathcal{R}\equiv \frac{r-R_{CC}}{R_{ND}-R_{CC}},
\end{equation}
which is zero at the crust-core boundary and one at the neutron-drip point. We use the analytical approximation for $\rho$ with the scaled radius:
\begin{equation}
\tilde\rho\equiv\frac{\rho}{\rho_{CC}}=400\left(1-\frac{R_{CC}}{R_{ND}}\right)^2(1-\mathcal{R})^2+0.004,
\label{ne_eqn}
\end{equation}
which is close to the exact result in the region between the neutron drip and the crust-core interface, based on the equation of state of \cite{Douchin:2001}. In the above relation (\ref{ne_eqn}) the ratio of the density at the crust-core interface and the neutron drip point is 250. Based on the same equation of state, we use analytical fits to $Z,A,x_{fn}$ and $\rho$ throughout the crust, and calculate the electron number density $n_e$. This result is approximated to good accuracy by the following expression:
%
\begin{equation}
n_e=10^{36}(1.5\tilde\rho^{2/3}+1.9\tilde\rho^2)\textrm{ cm}^{-3}.
\end{equation}
We assume the electric conductivity scales as $\sigma \propto n_{e}^{2/3}$, and set the conductivity at the base of the crust to $\sigma(R_{CC})=10^{24}$s$^{-1}$, which has the same functional form as the expression used in \cite{GOURGOULIATOS:2014a}.
The Coulomb parameter is defined as follows
\begin{equation}
\Gamma=\frac{Z^2 e^2}{a_I k_B T}
\end{equation}
where the ion sphere radius is $a_I=(4\pi n_I/3)^{-1/3}$. For $\Gamma>175$ the envelope of the neutron star starts to crystallise.
For the implementation of our failure criterion, we use the fit of \citet{Chugunov:2010} giving the following expression for the critical stress:
\begin{equation}
\tau_{el}=\left(0.0195-\frac{1.27}{\Gamma-71}\right)\frac{Z^2 e^2 n_I}{a_I},
\end{equation}
which takes the following form once expressed in terms of the scaled density:
\begin{equation}
\tau_{el}=5.1\times 10^{29}(0.4\tilde\rho+0.5\tilde\rho^3)\textrm{ g\ cm}^{-1}\textrm{s}^{-2}.
\end{equation}
The plastic flow viscosity is a largely unknown quantity. We approximate it as a scaled form of the critical stress:
\begin{equation}
\nu=\nu_{0}(0.4\tilde\rho+0.5\tilde\rho^3).
\label{PLASTIC_NU}
\end{equation}
The scaling parameter is chosen to be
$\nu_0=2.5\times 10^{38}\textrm{ g\ cm}^{-1}\textrm{s}^{-1}$, using the estimate from \citet{Lander:2016} demanding that the corona of a magnetar is a persistent phenomenon leading to twists lasting for around $10$-yr \citep{Beloborodov:2007}. We have further explored two more values for the scaling parameter, where the plastic flow viscocity is higher: $\nu_0=2.5\times 10^{39}\textrm{ g\ cm}^{-1}\textrm{s}^{-1}$ and a lower one $2.5\times 10^{37}\textrm{ g\ cm}^{-1}\textrm{s}^{-1}.$
Using equations \eqref{RH} and \eqref{FAILURE_EQ} and the crust profile we have adopted, we can explore the parameter range where the field would be dominated by the Hall evolution, Ohmic decay and the plastic flow; these are plotted in Figure \ref{FIG:1}. In general the deeper part of the crust would be less likely to fail, as $\tau_{el}$ is higher there. Thus the magnetic field evolution in the inner crust is dominated by the Hall effect for fields in the range $2\times 10^{12}-2\times 10^{15}$ G, for fields lower than that the evolution is dominated by the Ohmic decay and for fields higher than that the effect of plastic flow becomes important. Close to the neutron drip point, magnetic field higher than $10^{14}$G will lead to crust failure and a plastic flow, whereas fields below $10^{12}$G are dominated by Ohmic decay in this region.
The crust density varies from $\sim 10^{14}$g cm$^{-3}$ at the crust-core boundary to $\sim 10^{7}$g cm$^{-3}$ at the surface. The inclusion of regions with such a range of densities would make the calculation extremely slow, as the timestep would be set by the fastest moving electrons, the ones located near the surface of the crust. Furthermore, a realistic treatment of this region requires the inclusion of the interaction between the crust and the magnetosphere \citep{Akgun:2018, Karageorgopoulos:2019}, which is beyond the scope of the current study. For this reason, in these simulations we consider the part of the crust extending from the crust-core interface to the neutron drip point. This part of the crust can also harbour far higher stresses and hence elastic energy than the weak outer crust, and so is likely to be more relevant for powering magnetar outbursts.
Regarding the core, we have adopted two basic approaches. In one of these we regard the magnetic field as having been expelled from the core and confined in the crust. In the other, some poloidal flux penetrates into the core, but the core field does not evolve within the timeframe of the simulation. Similarly to the crust-magnetospheric interplay, the physics of the crust-core interaction involve phenomena such as ambipolar diffusion and the evolution of the magnetic field within a superfluid and superconducting that deserve separate detailed studies \citep{Lander:2013a, Lander:2013b, Passamonti:2017a, Passamonti:2017b}. Therefore, in the current study we assume that the field of the core is a fixed boundary condition.
\section{Simulations}
\label{SIMULATIONS}
\begin{table*}
\centering
\caption{Numerical models implemented for a magnetic field that is confined in the crust. The first column is the name of the run, subsequent columns are the value of $\Psi_0$, the value of the field at the neutron star pole, the energy in the crust, whether the field threads the core or not, the value of $\nu_0$ and the type of failure of the run.}
\label{TAB:1}
\begin{tabular}{ccccccc}
\hline
Name & $\Psi_0$ &$B_{dip,0}$ ($10^{14}$ G)& $E_{mag}~(10^{46}{\rm erg})$ & Core &$\nu_0$(g cm$^{-1}{\rm s^{-1}}$) & Failure \\
\hline
LC-1 & $200$& $1$& $3.1$ & No & $2.5\times 10^{39}$ & Local \\
LC-2 &$200$& $1$& $3.1$ & No & $2.5\times 10^{38}$ & Local \\
LC-3 &$200$& $1$& $3.1$ & No & $2.5\times 10^{37}$ & Local \\
\hline
LT-1 & $5$& $0.5$& $1.95$ & Yes & $2.5\times 10^{39}$ & Local \\
LT-2 &$5$& $0.5$& $1.95$ & Yes & $2.5\times 10^{38}$ & Local \\
LT-3 & $10$& $1$& $7.8$ & Yes & $2.5\times 10^{39}$ & Local \\
LT-4 & $10$& $1 $& $7.8$ & Yes & $2.5\times 10^{38}$ & Local \\
\hline
IC-1 & $200$& $1$& $3.1$ & No & $2.5\times 10^{39}$ & Intermediate \\
IC-2 &$200$& $1$& $3.1$ & No & $2.5\times 10^{38}$ & Intermediate \\
IC-3 &$200$& $1$& $3.1$ & No & $2.5\times 10^{37}$ & Intermediate \\
\hline
IT-1 & $5$& $0.5$& $1.95$ & Yes & $2.5\times 10^{39}$ & Intermediate \\
IT-2 &$5$& $0.5$& $1.95$ & Yes & $2.5\times 10^{38}$ & Intermediate \\
IT-3 & $10$& $1$& $7.8$ & Yes & $2.5\times 10^{39}$ & Intermediate \\
IT-4 & $10$& $1$& $7.8$ & Yes & $2.5\times 10^{38}$ & Intermediate \\
\hline
GC-1 & $200$& $1$& $3.1$ & No & $2.5\times 10^{39}$ & Global \\
GC-2 &$200$& $1$& $3.1$ & No & $2.5\times 10^{38}$ & Global \\
GC-3 &$200$& $1$& $3.1$ & No & $2.5\times 10^{37}$ & Global \\
\hline
GT-1 & $5$& $0.5$& $1.95$ & Yes & $2.5\times 10^{39}$ & Global \\
GT-2 &$5$& $0.5$& $1.95$ & Yes & $2.5\times 10^{38}$ & Global \\
GT-3 & $10$& $1$& $7.8$ & Yes & $2.5\times 10^{39}$ & Global \\
GT-4 & $10$& $1$& $7.8$ & Yes & $2.5\times 10^{38}$ & Global \\
\hline
HC-1 & $200$& $1$& $3.1$ & No & N/A & No \\
\hline
HT-1 & $5$& $0.5$& $1.95$ & Yes & N/A & No \\
HT-2 & $10$& $1 $& $7.8$ & Yes & N/A & No \\
\hline
\end{tabular}
\end{table*}
\subsection{Numerical Setup}
We have discretised the numerical domain in radius in $r$ and $\mu=\cos\theta$. The typical resolution we use is $100^2$, but we have also experimented with higher resolution runs ($200^2$) to confirm the validity of our results and numerical convergence.
The simulation consists of a main loop for the time integration of the partial differential equations (\ref{dPsi}) and (\ref{dI}). Within the main loop, we test whether the failure criterion, Equation \eqref{FAILURE_EQ}, is satisfied or not. Should the failure criterion be satisfied, then depending on whether we consider a local, intermediate or global failure, we integrate (\ref{Laplacian}) in the appropriate domain and apply the relevant boundary conditions. We evaluate the plastic flow velocity, using the Gauss-Seidel iterative method, until it relaxes to a solution. This integration is the main bottleneck of the calculation, as it requires a large number of steps for the convergence of equation (\ref{Laplacian}). In practice, of the order $10^{4}$ iterations are needed for this calculation for a $100^2$ resolution, leading to a drastic increase of the integration time by four orders of magnitude if we update the plastic flow velocity at every single time-step. Fortunately we have found that this is not necessary, and updating every 10-100 steps is typically sufficient, since the plastic flow velocity -- being sourced by changes in the magnetic field -- changes only slowly with time. Simulations are thus typically a factor of 100-1000 slower than the same model under Hall-Ohmic evolution. Once the plastic flow velocity is known, we use the Adams-Bashforth 2nd-order method to integrate in time. Furthermore, we use an adaptable time-step evaluated by a Courant condition \citep{Courant:1952}, which is based on the maximum velocity of the system accounting both for the electron fluid and the plastic flow velocity.
Spatial derivatives are evaluated using a central difference scheme, a three-point stencil for the second derivative, and five-point stencil for the third derivative. The boundary conditions implemented are those of a multipolar current-free poloidal magnetic field on the surface of the star $(r=R_{ND})$ and with the poloidal current set to zero $I(R_{ND}, \theta)=0$, enforcing that no current flows from the star to the magnetosphere. On the axis of the star we set $\Psi(r, 0)=\Psi(r, \pi)=I(r,0)=I(r,\pi)=0$. The inner boundary condition for the poloidal current is $\frac{d I}{dr}=0$. The poloidal flux boundary condition is either $\Psi(R_{CC}, \theta)=0$ for the regime where the magnetic field has been expelled from the core or fixed to a given function at the initial conditions $\Psi(R_{CC}, \theta)=f(\theta)$, and not allowed to evolve in the core and at the crust-core boundary.
Regarding the plastic flow, the boundary conditions employed depend on the type of failure we have assumed. In the local failure simulations, the flow velocity is set to zero at the points where the failure criterion is not satisfied. In the intermediate and global flow case, we impose the following boundary conditions for the plastic flow velocity in the solution of equation (\ref{Laplacian}). We set $v_{pl}(r, 0)=v_{pl}(0, \pi)=0$ on the axis and at the crust-core interface $v_{pl}(R_{CC}, \theta)=0$; and at the surface of the integration domain we set free-slip boundary conditions $\partial_r v_{pl}(R_{ND}, \theta)=0$.
\subsection{Initial Conditions}
We have adopted the following initial conditions depending on whether the field threads the core or is confined in the crust. The fields that thread into the core have the following profile for $\Psi$:
\begin{eqnarray}
\Psi_{1} =\Psi_{0}\frac{r}{R_{ND}}\left(1.05-\frac{r}{R_{ND}}\right)\sin^2\theta\, ,
\end{eqnarray}
whereas the ones that are confined in the crust have:
\begin{eqnarray}
\Psi_{2} =\Psi_{0} \left(1.05-\frac{r}{R_{ND}}\right)\left(\frac{r}{R_{ND}}-\frac{R_{CC}}{R_{ND}}\right) \sin^2\theta\,.
\end{eqnarray}
We set the intensity of the magnetic field by the $\Psi_{0}$ parameter.
For all the families of plastic flow simulations we have performed, we have also simulated a case where only evolution through the Hall-Ohmic effect is allowed, for comparison.
\section{Results}
\label{RESULTS}
\begin{figure*}
a\includegraphics[width=0.3\textwidth]{Figures/LC3_VPL.png}
b\includegraphics[width=0.3\textwidth]{Figures/IC3_VPL.png}
c\includegraphics[width=0.315\textwidth]{Figures/GC3_VPL.png}
\caption{Plots of the poloidal magnetic field line structure (shown in black) and the plastic flow angular velocity (shown in colour), for models where crustal failure is local (a), intermediate (b), or global (c), all after 1 kyr of evolution. The specific runs used are LC-3 (a), IC-3 (b) and GC-3 (c).}
\label{FIG:2}
\end{figure*}
\begin{figure}
a\includegraphics[width=0.24\textwidth]{Figures/LC3_WE.png}
b\includegraphics[width=0.205\textwidth]{Figures/LC3_TAU.png}
\caption{Plots of the electron fluid angular velocity (a) and the ratio of the stress to the critical value (b), in colour, with the poloidal magnetic field lines shown in black, for model LC-3, at 1 kyr.}
\label{FIG:3}
\end{figure}
\begin{figure*}
a\includegraphics[width=0.23\textwidth]{Figures/LC3_TOR1.png}
b\includegraphics[width=0.23\textwidth]{Figures/IC3_TOR1.png}
c\includegraphics[width=0.245\textwidth]{Figures/GC3_TOR1.png}
d\includegraphics[width=0.23\textwidth]{Figures/HC1_TOR1.png}
e\includegraphics[width=0.23\textwidth]{Figures/LC3_TOR10.png}
f\includegraphics[width=0.23\textwidth]{Figures/IC3_TOR10.png}
g\includegraphics[width=0.245\textwidth]{Figures/GC3_TOR10.png}
h\includegraphics[width=0.23\textwidth]{Figures/HC1_TOR10.png}
\caption{Plots of the poloidal magnetic field lines shown in black and the toroidal field in colour, at 1 kyr (top row) and 10 kyrs (bottom row) for models LC-3 (a and e), IC-3 (b and f), GC-3 (c and g) and HC-1 (d and h). }
\label{FIG:4}
\end{figure*}
\begin{figure*}
a\includegraphics[width=0.3\textwidth]{Figures/LT4_VPL.png}
b\includegraphics[width=0.3\textwidth]{Figures/IT4_VPL.png}
c\includegraphics[width=0.3\textwidth]{Figures/GT4_VPL.png}
\caption{Plots of the poloidal magnetic field lines structure shown in black and the plastic flow angular velocity shown in colour, for models LT-4 (a), IT-4 (b) and GT-4 (c), at 1 kyr. }
\label{FIG:5}
\end{figure*}
\begin{figure}
\includegraphics[width=0.24\textwidth]{Figures/LT4_WE.png}
\includegraphics[width=0.23\textwidth]{Figures/LT4_TAU.png}
\caption{Plots of the electron fluid angular velocity (a) and the logarithm of the ratio of the stress to the critical value (b), in colour, with the poloidal magnetic field lines shown in black, for model LT-4, at 1 kyr.}
\label{FIG:6}
\end{figure}
\begin{figure}
\includegraphics[width=0.24\textwidth]{Figures/LT4_TOR1.png}
\includegraphics[width=0.24\textwidth]{Figures/HT2_TOR1.png}
\caption{Plots of the toroidal field in colour and the poloidal magnetic field lines in black, for models LT-4 (a) and HT-2 (b) at 1 kyr.}
\label{FIG:7}
\end{figure}
We have performed numerical simulations of the magnetic field evolution including plastic flow using the regimes outlined in section \ref{SIMULATIONS}. Given the vast parameter space related to the choice of the initial conditions and the plastic flow viscosity parameter $\nu$, we have focused on the impact of the following choices: whether the plastic flow is local, intermediate or global, the value of $\nu$, and the intensity and structure of the magnetic field -- in particular whether it threads the core or not. The simulation information is presented in Table \ref{TAB:1}. We have adopted the following naming convention. The first letter on the name of the run is either, L, I, G or H depending on the failure being local, intermediate, global or whether there is no failure and the evolution is only due to the Hall effect. The second letter signifies whether the field is confined in the crust (C) or threads the core (T). Finally, to differentiate from simulations that have the same type of failure prescription and structure, but different magnetic field intensity or plastic flow viscosity, we use a different number.
\subsection{Evolution of crust-confined fields}
Let us first consider the simulations where the field is confined in the crust. In all simulations, the stress immediately exceeds the critical value at the equator of the crust and near the north and south poles close to the surface. Following the failure, the subsequent evolution of the plastic velocity profile depends on the type of failure, whether it is L, I and G; one example of each run, after 1 kyr, is shown in Figure \ref{FIG:2}. In the run where the failure is localised in the region where the stress exceeds the critical value (L-type), the overall plastic flow velocity is relatively slow, reaching $5\times 10^{-5}$ rad/yr for model LC-3; Figure \ref{FIG:2} left panel. It becomes somewhat higher for the intermediate case (IC-3) (Figure \ref{FIG:2}, middle panel) and a factor of 4 higher if the plastic flow is allowed to operate everywhere in the crust (model GC-3, Figure \ref{FIG:2} right panel). The plastic flow does not have any noticeable impact on the poloidal field, as can be seen from their similarity in all three panels of Figure \ref{FIG:2}. The angular velocity of the electron fluid depends on the poloidal field structure (see Equation \eqref{Omega}), and so it is also similar for these models.
Given its dependence on the electron number density, the electron angular velocity is highest near the surface of the star; Figure \ref{FIG:3} left panel. The ratio of the stress compared to its critical value is higher near the equator of the star, in the middle of the crust, and near the surface of the star at mid-latitudes; see Figure \ref{FIG:3} right panel. These details depend on the structure of the magnetic field and the changes that occur during its evolution and could vary for a different choice of initial conditions.
While the assumption of failure type has a mild and hardly noticeable impact on the poloidal field, its effect is rather drastic on the toroidal field, as is evident from Figure \ref{FIG:4}. Equation \eqref{dI} shows that the toroidal field is generated from the competition of the electron fluid angular velocity and the plastic flow. Thus, one can see that there is a difference in the intensity of the toroidal field: in the case of the global failure model (GC-3) roughly an order of magnitude weaker than all the others, with its maximum value only $3\times 10^{13}$G. In this model the plastic velocity efficiently opposes the electron velocity. On the contrary, in the LC-3 and IC-3 models the toroidal field is stronger. In the IC-3 model it is almost annulled in the region where the plastic flow is fastest, but its maximum value is even higher than that of the pure Hall evolution (HC-1). In the LC-3 model the toroidal field has a lower maximum value, but is spread over a larger region of the crust. After 10 kyr, none of the models with plastic flow has such a strong large-scale toroidal field as the pure Hall-Ohmic model. However, the models implementing the local- and intermediate-failure criteria display more complex field geometries than the Hall-Ohmic one, with fields roughly as intense locally as the highest intensity of toroidal field from the Hall-Ohmic model.
The above results refer to the lowest value of plastic flow viscosity we have simulated $\nu_0=10^{37}$g cm$^{-1}$ s$^{-1}$. Simulations with higher values e.g.~$\nu_0=10^{38}$g cm$^{-1}$ s$^{-1}$ have a plastic flow velocity which is lower by a factor of a few. In particular, in the LC-2 run the maximum value of the plastic flow angular velocity is $10^{-5}$rad/yr, and for $\nu_0=10^{39}$g cm$^{-1}$ s$^{-1}$ (LC-1) the maximum plastic flow angular velocity is $10^{-5}$rad/yr. A similar scaling behaviour is observed for intermediate and global failures. In these cases, the toroidal field is more mildly affected by the plastic flow compared to the cases with the lowest $\nu$, and the evolutions become increasingly similar to those without plastic flow; see \citet{Lander:2019} for examples of the effect of increasing $\nu$.
\subsection{Evolution of core-threading fields}
Next we consider a field that threads the core. These simulations fail near the surface of the star, and generate a flow in the $-\phi$ direction there, which is opposite to the electron fluid velocity. In all three types of failure the flow velocity has the same magnitude, $\sim 10^{-4}$rad/yr, considerably faster than the case of fields confined to the crust alone. This is due to the choice of the initial conditions leading to faster variation of the poloidal field near the surface in the core-threading fields compared to the crust confined ones. The plastic flow velocity patterns, however, are different deeper in the crust, depending on the type of failure. There is no plastic flow deeper in the crust when the failure is local (LT-4), Figure \ref{FIG:5} left panel. In the intermediate type (IT-4), the plastic flow peaks near the equator and at mid-latitudes and becomes zero near the poles, Figure \ref{FIG:5} middle panel. When the failure is global, the plastic flow velocity peaks near the equator and goes smoothly to zero near the poles. Moreover, in the intermediate and global failure, there is a plastic flow deeper in the crust. The electron fluid angular velocity near the surface of the crust peaks at $4\times 10^{-4}$rad/yr, Figure \ref{FIG:6} left panel. Thus the plastic flow velocity in general opposes the electron flow. Moreover, the ratio of the stress to the critical value exceeds unity in the outer half of the crust at the equator and mid-latitudes, Figure \ref{FIG:6} right panel. This demonstrates why in all models there is a plastic flow near the surface of the star as this region fails for all models. The difference in the plastic flow patterns deeper in the crust reflects the impact of the types of failure studied.
In simulation LT-4 the electron fluid flow in the outer region (left-hand side of Fig. \ref{FIG:6}) and the plastic flow there (left-hand side of Fig. \ref{FIG:5}) have similar magnitude in the outer part of the domain, but opposite direction. Therefore, as for the crustal-confined field in model GC-3 (Fig. \ref{FIG:4}), the Hall source terms from equation \eqref{dI} approximately cancel, and we would expect very limited generation of toroidal field. This is borne out in Fig. \ref{FIG:7}, where we see that the outer region from run LT-4 hosts a far weaker toroidal field than the corresponding model subject to Hall-Ohmic evolution alone (run HT-2).
\subsection{Generic features of evolutions}
Regardless of the details of the field geometry (confined to the crust or otherwise), some aspects of axisymmetric magneto-plastic evolution in a NS crust seem to be universal.
\begin{figure}
\includegraphics[width=0.45\textwidth]{Figures/Energy_tor_Conf.png}
\includegraphics[width=0.45\textwidth]{Figures/Energy_tor_thr.png}
\includegraphics[width=0.45\textwidth]{Figures/Energy_tor_thr_str.png}
\caption{Ratio of the toroidal energy to the total energy for the various runs.}
\label{FIG:8}
\end{figure}
Overall, we notice that the formation of the toroidal field is a direct consequence of the differential rotation of the electron fluid, in the context of the Hall effect. Once a plastic flow is present, it has the tendency to counteract this motion. We find this to be the case in the majority of the runs. The effect is more prominent when either the plastic viscosity is lower, or the magnetic field strength higher. When the plastic viscosity is lower, the plastic flow velocity becomes high enough to completely annul the impact of the differential flow of the electron fluid. A stronger magnetic field leads to the failure of a larger fraction of the crust, thus, even in the case of a local failure, the region where the plastic flow occurs is large enough to lead to drastic changes in the toroidal field.
While the plastic flow has a major impact on the toroidal field, it does not lead to a drastic decrease in the amount of stress in the crust.
This is mainly due to the fact that the plastic flow impacts the toroidal field directly through Eq. \eqref{dI}, but the poloidal field only indirectly. However, both the poloidal and the toroidal field contribute to the stress in the crust, thus, even if the toroidal field is annulled, any change in the poloidal field will still contribute to the stress.
At later times (t=10kyr), the simulation where the failure is local has a somewhat milder toroidal field near the surface compared to the simulation that evolves only due to the Hall effect. The intermediate and global failures retain their complex plastic flow velocity profile, with the intermediate case alternating from a plastic flow velocity in the $-\phi$ direction to $+\phi$ at mid-latitudes. This has a drastic effect on the toroidal field in the crust changing both its structure and its intensity, compared to systems that evolve only due to the Hall effect. Over very long times, $\gtrsim 10$ kyr, we see from figure \ref{FIG:8} that the fraction of magnetic energy in the toroidal component $E_{tor}/E_{mag}$ is always lower for models with plastic flow than for those evolved with the Hall-Ohmic prescription alone. Furthermore, in evolutions where the field is confined to the crust the maximum value of $E_{tor}/E_{mag}$ is about an order of magnitude smaller than for evolutions of core-threading fields.
\section{Discussion}
\label{DISCUSSION}
The inclusion of a plastic flow in the evolution of the crustal magnetic field adds extra elements of complexity compared to the pure Hall evolution, which is itself a complicated and non-linear problem. Studying an axisymmetric field, intuitively, one would expect that the plastic flow will annul the electron motion in the azimuthal direction. The majority of the simulations have a plastic flow velocity that in general opposes the electron fluid azimuthal velocity, but its profile is more complicated. For instance, in some simulations it even changes direction, leading to a more drastic twisting of the field compared to pure Hall evolution. This effect is caused because of the following reasons. First, the equation that is solved for the plastic flow velocity is inversely proportional to the plastic viscosity, while the electron velocity depends on the electron number density. While both the plastic viscosity and the electron number density are monotonic functions of the crust density, their functional forms differ drastically. Thus, even if an extended region of the crust fails and the overall Maxwell stress across this region is similar, the magnitude of the plastic flow velocity can be drastically different due to the large variation of $\nu$ in equation \eqref{Laplacian}. Second, the electron fluid angular velocity depends on $\Psi$ and not $B_{\phi}$. This can be clearly seen from equation \eqref{Omega}, where the electron fluid angular velocity is obtained by the action of the Grad-Shafranov operator (\ref{GSop}) on $\Psi$, without involving $B_{\phi}$ at all. On the contrary, the plastic flow motion depends on $B_{\phi}$ and its derivatives, as it is evident from equation (\ref{Laplacian}). This suggests that while the plastic flow velocity is driven by the magnetic tension term ${\bf B}\cdot \nabla {\bf B}$, the electron fluid angular velocity has a different dependence. A third issue is the dependence of the plastic flow on the type of failure. If the plastic flow is local or intermediate, it will only relieve part of the stress in the regions where the plastic flow is non-zero. Indeed, simulations IC and IT have a toroidal field whose polarity is opposite that of the Hall-only simulations. Similarly, when considering the total energy stored in the toroidal field, we notice that for some models with plastic flow the energy in the toroidal field is temporarily higher than that of the corresponding evolution driven exclusively by the Hall effect, as shown in Figure \ref{FIG:8}. In the long run however, the system that evolves exclusively due to the Hall effect is the one that develops the strongest toroidal field. Nevertheless, this implies that temporarily, the plastic flow is able to twist the field even more drastically and generate a stronger toroidal component.
The impact of the plastic flow strongly depends on the value of $\nu$, which is largely unknown. Our simulations have allowed us to assess the range of values of $\nu$ for which the plastic flow has a notable effect. In general for $\nu\lesssim 10^{38}$g cm$^{-1}$ s$^{-1}$, the plastic flow velocity is similar to that of the electron fluid. Even if this is the case though, the plastic flow velocity profile is drastically different depending on whether the flow is local or not. A local flow, where only a small fraction of the crust participates and the rest of the star remains intact, does not reach a high velocity. This is mainly due to the boundary conditions imposed, where the rest of the crust does not participate in the flow. Enforcing the flow velocity to be zero beyond the failed region leads to a smaller maximum velocity. On the contrary, if this condition is waived, then the plastic flow velocity can reach much higher values. This effect is very profound if the failed part of the crust is small, but the difference is moderate when the failure affects an extended part of the crust.
Although the toroidal field is most altered over the course of the evolutions reported here, the poloidal field is affected too, at a somewhat slower rate. These differences are more pronounced at later times ($t>10$kyr). They arise from the term $(\nabla I \times \nabla)
\cdot \nabla \Psi$ in equation \eqref{dPsi}. As the toroidal field is drastically different this will be reflected in $I$ and consequently in $\Psi$ through this equation. As this is a secondary effect, mediated by this term rather than appearing directly in the field-evolution equation, it is not as profound as the difference in $I$.
The flow velocity for the runs with $\nu_0 = 2.5\times 10^{38}$g cm$^{-1}$ s$^{-1}$ is in the range of $10^{-5}-10^{-4}$ rad yr$^{-1}$, which in linear velocity corresponds to $10-100$cm yr$^{-1}$. This velocity is consistent with the findings of our previous work \citep{Lander:2019} and those of \cite{Kojima:2020}. The velocity depends on the intensity of the field as well, with stronger fields leading to higher velocities for given $\nu_0$. We remark however that the presence of a plastic flow does not enhance the decay rate. This might initially seem counter-intuitive, since one would expect a viscosity term -- like the one in our evolution equations -- to be associated with a dissipative process. However, in our formulation of the problem -- following \cite{Lander:2016}, and based on a simple terrestrial theory of plasticity \citep{Prager:1961} -- $\nu$ simply plays the role of a source term regulating how fast crustal stresses are converted into plastic motion; see equation \eqref{Laplacian} and the discussion in section 2.6 of \cite{Lander:2019}. In fact, in our problem the plastic flow actually reduces the dissipation of magnetic energy, by opposing the formation of the kind of strong currents responsible for the Ohmic decay of the field.
We find that the magnetic-field geometry in a neutron-star crust is sensitive to details of how the crust fails. Depending on whether super-yield stresses cause a failure only in a small region of the crust or a more global collective effect, the magnetic field in the crust evolves in a different way, and in all cases the result is different from the standard Hall-Ohmic (electron MHD) evolution. These differences are not washed out over time, but persist into the crust's old age (for a magnetar) of order 10 kyr. Since high-energy magnetar bursts (and perhaps also fast radio bursts) are often thought to be associated with locally-intense toroidal field (e.g. \citet{Perna:2011}, there is a possibility of using magnetar observations to glean hints of the material physics of the crust. Suppose, for example, that a particular magnetar's activity seems to be associated with emission from its magnetic poles. Then, based on our evolutions from Fig. \ref{FIG:4}, such activity might favour a crust that fails in what we term an `intermediate' manner, since only in this case do we see a high concentration of toroidal field in the polar region alone.
Apart from their bursting activity, magnetars may be displaying the effects of plastic flow in other, less direct, ways. For example, an analysis by \cite{Beloborodov:2016} found mechanical heating due to plastic flow to be one of the most plausible mechanisms for explaining the high surface temperatures of magnetars. Our current model does not include such dissipative effects, but it would be a logical way to extend our work. Moreover, the impact of the plastic flow could be combined with studies of the magnetothermal evolution that has been explored in the context of the neutron star diversity \citep{VIGANO:2013, DeGrandis:2020, Igoshev:2021}.
It has long been known that magnetar rotation and spindown is far noisier and less regular than that of radio pulsars \citep{Melatos:1999,Dib:2008, Tsang:2013}. Their long-term spindown can be irregular, and on shorter timescales they undergo sudden spin-up glitches and potentially (but more controversially) anti-glitches. The key difference between magnetar and pulsar rotation could be that in the former case plastic flow plays a role in the crustal dynamics. Any patch of the inner crust (the region we simulate) will be the pinning site for a number of superfluid neutron vortices. If it begins to move plastically, one of a number of different things could happen to the vortices.
They could unpin immediately and cause a glitch, or they could remain pinned for some time as the crust shifts and produce a delayed glitch or a more gradual response. If the plastic flow is driven purely by magnetically-induced stresses, it is just as likely to move a patch of crust towards the rotational axis (increasing locally the rotational lag between the superfluid and the rest of the crust) as it is to move it away (decreasing the lag). This could result in a rich phenomenology of timing features; for example, a model for (crustquake-induced) inward vortex motion and its effect on spindown has been explored in relation to the high-magnetic field pulsar J1119-6127 \citep{Antonopoulou:2015,Akbal:2015}.
For now our understanding of magneto-plastic evolution is too rudimentary to make a reliable connection between evolutions and observations, such as the two examples we have outlined above, but we believe that more sophisticated modelling will start to make this a credible possibility.
\section{Conclusions}
\label{CONCLUSIONS}
We have studied the evolution of the magnetic field in the crusts of neutron stars in the presence of a plastic flow. Given that the plasticity of the crust is a largely unknown property, we have explored some regimes that are drastically different from each other, adopting an agnostic view. We find that the evolution of the magnetic field depends on the value of the plastic viscosity and the type of failure we have adopted.
In general a plastic flow with $\nu_0 =2.5 \times 10^{39}$ g cm$^{-1}$ s$^{-1}$ leads to rather low plastic velocity and has little impact on the magnetic-field evolution. Lower values, of the order $\nu_0 =2.5 \times 10^{38}$ g cm$^{-1}$ s$^{-1}$, lead to flows where the plastic velocity is comparable to the electron fluid velocity, and there is a drastic impact on the field evolution. We remark, though, that the flow does not simply annul the impact of the Hall effect but rather leads to more complex evolution, which depends on the type of failure.
The present study is a step forward from our previous Cartesian geometry work \citep{Lander:2019}. Nevertheless, being confined to an axisymmetric geometry and because of our imposed assumption of incompressibility, the type of flow can only be in the $\phi$ direction. While the lack of flow in the radial direction is dictated by the physics, an axisymmetric flow is a simplification allowing us to tackle a complicated problem. As has already been demonstrated, 3-D studies of the Hall evolution \citep{Gourgouliatos:2016a, Gourgouliatos:2018a} lead to results that are radically and qualitatively different from those found in the axisymmetric problem. We anticipate that a plastic flow in a three dimensional geometry will lead to a more realistic understanding of this effect.
\section*{Acknowledgments}
We thank Danai Antonopoulou for a helpful discussion about the implications of our results. KNG acknowledges grant FK81641 - Theoretical and Computational Astrophysics.
We thank an anonymous referee for their constructive comments.
\section*{Data availability statement}
The data underlying this article will be shared on reasonable request to the corresponding author.
\bibliographystyle{mnras}
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{
"timestamp": "2021-07-05T02:02:03",
"yymm": "2106",
"arxiv_id": "2106.03869",
"language": "en",
"url": "https://arxiv.org/abs/2106.03869"
}
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"\\section{Introduction}\nGiven an (undirected) graph $G=(V,E)$ with $n=|V|$ vertices and with maxim(...TRUNCATED)
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"\n\\section{Introduction}\n\nReinforcement learning (RL) is a promising field of current AI researc(...TRUNCATED)
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"\n\n\n\n\n\n\n\n\n\\section{Conclusion and outlook}\n\\label{sec:conclusion}\nIn summary, the resul(...TRUNCATED)
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"\\section{Introduction}\n\nThe Apollo 15 \\& 16 spacecrafts were launched on July 26, 1971 and Apri(...TRUNCATED)
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"\\section{Introduction}\n\nRelativistic jets appear to be common in a wide range of astrophysical s(...TRUNCATED)
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"\\section{Introduction}\n\nThe goal of spatial search algorithms is to find a marked item in a coll(...TRUNCATED)
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"\\section{Introduction}\n\nAs implanted medical devices become smaller, more numerous and more capa(...TRUNCATED)
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