\section{Introduction}

Phase transitions induced by a non-equilibrium drive are a topic of fundamental importance and current experimental interest \cite{Aver:opt-rev,Basov:review,Mitra06,Aver:optCu,Cav:K3C60,laser:gold}. Transient and steady-state non-equilibrium drives may allow access to many phases, some of which are absent in equilibrium \cite{Kogar:CDW,FE:SrTiO,CMA:trsc,CMA:neg,Sun:neq,Mazur,VO2:science}. One important class of non-equilibrium transitions is the insulator-metal transition in a correlated electron insulator subject to a dc electric field or current \cite{Assamitsu97,Maecit:7,Maecit:8,Maecit:9,Maecit:10,Maecit:11}. The theoretical consensus is that the insulating phase may be destabilized when the driving field is such that the voltage drop over a unit cell is a significant fraction (greater than a few percent) of the insulating gap \cite{Amaricci12}. At these drives enough valence band carriers are excited over the gap to destroy the insulator, by either Landau-Zener tunneling or a renormalization of the electronic temperature \cite{CM:cdw,Han:cdw,Mazza:PRL,Matthies:PRB}. Another mechanism, experimentally confirmed in several cases \cite{VO2:PRB,Stefanovich_2000,VO2:heat}, is Joule heating to temperatures above the transition temperature.

However, the behavior of the current-driven transition observed in Ca$_2$RuO$_4$ appears to be inconsistent with these expectations \cite{Maeno:mainp,Fuku:CRO,Cao2:CRO,Dia:CRO,Xray:CRO,Cao:CRO,Friedt:CROPT,Pavarini:CRO,Cao2:CRO}. In this material the insulating phase is destroyed at threshold fields ($E_{\rm{th}}\sim40\um{V/cm}$) that are several orders of magnitude smaller than the fields required for a significant excitation of carriers, while it has been reported that the global temperature of the system remains below the equilibrium critical temperature \cite{Fuku:CRO}. Furthermore, a recent nano-imaging experiment \cite{Mengkun:CRO} observed coexistence of metallic and insulating phases when applying an increasing current to Ca$_2$RuO$_4$, and found that the metallic phase always nucleates out of the negative electrode, meaning that the phase switching depends on the direction of the current flow; another paper \cite{Cao2:CRO} reported a dependence of the hysteresis cycle on current direction. These findings indicate that Joule heating, which is proportional to the square of the current and hence does not change when the current polarity is reversed, is not the dominant effect.

In this work we reconsider heating effects in connection with metal-insulator transitions in correlated materials, with emphasis on a very simple point: in the presence of a current, a spatial variation in the Seebeck coefficient $S$ acts as a  heat source or a heat sink, depending on the direction of the current flow with respect to the Seebeck gradient. In the most common realization, a modulation in $S$ is produced by a temperature gradient ($\mathbf{\nabla} S\sim(dS/dT)\mathbf{\nabla} T$), while in a thermocouple the modulation occurs at a device boundary. Here we observe that in a system consisting of an inhomogeneous mixture of metal and insulating phases, similar effects may occur at interfaces between metallic and insulating phases, leading in appropriate cases to a marked dependence of the position of the phase boundaries on the direction of the current. We further show that because the difference in Seebeck coefficient between metal and insulating phases of correlated materials is typically large, the effect may be comparable to or larger than Joule heating. We  present a study of an idealized geometry that shows how these ideas may account for the essential features of the Ca$_2$RuO$_4$ data \cite{Mengkun:CRO}.

The rest of the paper is organized as follows. In Section \ref{sec:macroscopic} we use transport arguments to derive a heat balance equation that accounts for Joule heating, the Peltier (interface) effect, heat diffusion and heat dissipation into a reservoir. Section \ref{sec:Ca2RuO4} presents an analysis of  a specific geometry that is an idealized version of the Ca$_2$RuO$_4$ experiments. Section \ref{sec:Conclusions}  summarizes the results and outlines  directions for further research. Appendices provides a microscopic derivation of the heat balance based on the electronic quantum kinetic equation, and details of our estimations of experimental parameters

\section{Heat balance equation}\label{sec:macroscopic}

We consider a system of electrons with electric charge $q=-e$, chemical potential $\mu$ and temperature $T$. We assume that a charge (electric) current density $\mb j_c$ exists in the system, along with an electric field $\mb E$. The relevant quantities are functions of the position $\mb x$, which we typically do not explicitly notate here. We write a steady state heat balance equation that relates $T$ to $\mb j_c$, beginning with consideration of the electronic energy density.

In the absence of magnetic fields, the total energy density $u_{tot}$ of the electronic system is the sum of the internal electron energy $u_{el}$ and of the electric field energy, and satisfies the equation of state $\mathrm{d}u_{tot}=\mathrm{d}u_{el}+\mb E\cdot\mathrm{d}\mb D/4\pi$, with $\mb D$ the displacement vector \cite{LL8}, implying that the continuity equation is
\begin{equation}\label{Ebal}
\partial_tu_{\rm{el}}+\frac{\mb E}{4\pi}\partial_t\mb D+\mathbf{\nabla}\cdot\mb j_e=-\dot Q_{\rm{d}},
\end{equation}
where $\mb j_e$ is the energy current and $\dot Q_{\rm{d}}$ describes the rate of energy dissipation into non-electronic degrees of freedom, such as the lattice modes in our case. The dissipation rate depends on the  temperature $T_l$ of the non-electronic degrees of freedom and the electronic temperature $T$: $\dot Q_{\rm{d}}=\dot Q_{\rm{d}}(T,T_l)$.

We use the fourth Maxwell equation to relate the displacement vector to the electrical current $\partial_t\mb D=-4\pi\mb j_c$, and introduce the electrochemical potential $\Phi$ via $-\mathbf{\nabla}\Phi\equiv\mb E-\mathbf{\nabla}\mu/q$. In a steady state $\partial_tu_{el}=0$ and $\mathbf{\nabla}\cdot\mb j_c=0$, we add and subtract $\mathbf{\nabla}\cdot(\mb j_c\mu/q)=\mb j_c\cdot\mathbf{\nabla}\mu/q$ in Eq. \eqref{Ebal} and introduce the heat current $\mb j_h=\mb j_e-\mu\mb j_c/q$ so that Eq. \eqref{Ebal} becomes
\begin{equation}\label{Hbal}
0=-\mb j_c\cdot\mathbf{\nabla}\Phi-\mathbf{\nabla}\cdot\mb j_h-\dot Q_{\rm d}.
\end{equation}
Equation \eqref{Hbal} represents the steady-state heat balance, relating the divergence of the heat current to Joule heating $-\mb j_c\cdot\mathbf{\nabla}\Phi$ and to heat dissipation into the reservoir.

We now use the linear theory of thermoelectric transport \cite{Kreuzer, Ashcroft} to write the heat and charge currents as functions of $-\nabla \Phi$ and of the electronic temperature gradient $\nabla T$. A standard form of this relation is
\begin{gather}\label{jcjs}
\begin{pmatrix}
\mb j_c \\
\mb j_h/T
\end{pmatrix}=\begin{pmatrix}
\sigma & \sigma S \\
\sigma S & \kappa_e/T+\sigma S^2
\end{pmatrix}\begin{pmatrix}
-\mathbf{\nabla}\Phi \\
-\mathbf{\nabla} T
\end{pmatrix},
\end{gather}
where $\sigma$ is the electric conductivity, $S$ the Seebeck coefficient (or thermopower) and $\kappa_e$ the electronic thermal conductivity. In Eq. \eqref{jcjs} the transport coefficients matrix is symmetric thanks to the Onsager relations. An equivalent form, based on charge and energy currents rather than charge and heat currents would involve the generalized forces $-\mathbf{\nabla}(\Phi/T)$ and $\mathbf{\nabla}(1/T)$ rather than $-\mathbf{\nabla}\Phi$ and $-\mathbf{\nabla} T$. The two formulations of course lead to equivalent results

We now rearrange Eq. \eqref{jcjs} to obtain an expression for $\mathbf{\nabla}\Phi$ and $j_h$ in terms of $\mb j_c$ and $\mathbf{\nabla} T$:
\begin{equation}\label{Ejs}
\begin{pmatrix}
-\mathbf{\nabla}\Phi \\
\mb j_h
\end{pmatrix}=\begin{pmatrix}
\rho & -S \\
\Pi & \kappa_e
\end{pmatrix}\begin{pmatrix}
\mb j_c \\
-\mathbf{\nabla} T
\end{pmatrix},
\end{equation}
with $\Pi=TS$ the Peltier coefficient, $\rho=\sigma^{-1}$ the electrical resistivity and $\kappa_e$ is the thermal diffusion coefficient. Combining Eq. \eqref{Hbal} and \eqref{Ejs} and noting that the steady-state condition $\nabla\cdot\mb j_c=0$ implies $\mathbf{\nabla}\cdot(\Pi\mb j_c)=S\mb j_c\cdot\mathbf{\nabla} T+T\mb j_c\cdot\mathbf{\nabla} S$, yields
\begin{equation}\label{Heat}
0=-\dot Q_{\rm d}+\rho\mb j_c^2-T\mb j_c\cdot\mathbf{\nabla} S+\mathbf{\nabla}\cdot(\kappa_e\mathbf{\nabla} T).
\end{equation}
The first two terms in Eq. \eqref{Heat} are the dissipation into the reservoir and the Joule heating respectively, and the last term represents thermal diffusion. The remaining term, $T\mb j_c\cdot\mathbf{\nabla} S$ shows how spatial structure in the Seebeck coefficient in the presence of a current gives a thermal effect that may be either heating or cooling depending on the direction of current flow relative to the gradient of $S$. Heat is generated when the current flows from the phase with higher Seebeck coefficient to the phase with the lower one. In particular, a sharp interface separating two materials with  different thermoelectric coefficients is a localized heat source or sink.

Appendix ~\ref{Appendix:Derivation} gives a derivation of Eq. \eqref{Heat} from a microscopic approach, starting from the equations for the Keldysh Green functions and writing a kinetic equation for the electron distribution function.

Crucial to the solution of Eq. \eqref{Heat} is the dissipation of the generated heat into the thermal reservoir. In the situation of most interest here, the thermal reservoir is provided by the lattice degrees of freedom of the material. For simplicity, we assume that the heat transfer is proportional to an electron-lattice coupling $\gamma_{\rm{e-l}}$ (which for simplicity we take to be structureless) and to the difference in electron and lattice temperatures: $\dot Q_{\rm d}=\gamma_{\rm{e-l}}(T(\mb x)-T_l(\mb x))$. The heat balance equation for the lattice is then
\begin{equation}\label{HL1}
0=\gamma_{\rm{e-l}}(T-T_l)+\mathbf{\nabla}\cdot(\kappa_l\mathbf{\nabla} T_l),
\end{equation}
with $\kappa_l$ the lattice thermal conductivity. Note that there is neither Joule heating nor thermoelectric effects for the lattice. The heat that flows into the lattice will be dissipated into the environment, typically at the boundaries of the sample, leading to boundary conditions on $T_l$. A specific example will be discussed below.

Equation \eqref{HL1} implies that the length scale for variations in $T_l-T$ is $\sim \sqrt{\kappa_l/\gamma_{\rm{e-l}}}$. Typically this scale is $10-100\um{nm}$ (see Appendix \ref{Appendix:Ca2RuO4Parameters} for details) and is much shorter than the relevant geometrical length scales, so that to sufficient approximation $|T-T_l|\ll T_l\Rightarrow T_l\approx T$  and we can combine Eqs. \eqref{Heat} and \eqref{HL1} into an equation for just one temperature:
\begin{equation}\label{HT}
0=\rho\mb j_c^2-T\mb j_c\cdot\mathbf{\nabla} S+\mathbf{\nabla}\cdot(\kappa\mathbf{\nabla} T),
\end{equation}
with $\kappa\equiv\kappa_l+\kappa_e$ the total thermal conductivity, and with boundary conditions taken from those for Eq. \eqref{HL1}.

Equation \eqref{HT} determines the temperature, given the spatial dependence of $\rho$, $S$, $\kappa$ and the current. We suppose that the state of the system, and thus the values of the transport coefficients, only depends on the local temperature $T(\mb x)$. Given a certain dependence of the transport coefficients on $T(\mb x)$, then the continuity equation $\mathbf{\nabla}\cdot\mb j_c=0$, the third Maxwell equation $\mathbf{\nabla}\times\mb E=0=\mathbf{\nabla}\times(\rho\mb j_c)$ and Eq. \eqref{HT} completely determine the current and temperature. \footnote{Note that the boundary conditions on $\mb j_c$ and $\mb \mathbf{\nabla} T$ are that they vanish at the system surfaces, since no electrons can flow out of the system.}

\section{Results and Application to Ca$_2$RuO$_4$ }\label{sec:Ca2RuO4}

\subsection{Overview and parameters}

In this section we present analytical and numerical solutions of Eq. \eqref{HT}. When specific parameters are required we use values reasonable for Ca$_2$RuO$_4$.

Correlation-driven metal-insulator transitions are typically first order with narrow hysteresis regimes, and we assume that the transport coefficients take metallic or insulating values according to whether the local temperature is greater or less than the critical temperature $T_{\rm{MIT}}$. Thus the conductivity (resistivity) and thermopower take the values $\sigma_{M,I}$ ($\rho_{M,I}$) and $S_{M,I}$; we assume $\kappa_e\ll\kappa_l$ for simplicity so that $\kappa\approx\kappa_l$. Thus $\nabla S$ vanishes except at the metal-insulator phase boundaries, where it has delta-function singularities proportional to $\Delta S\equiv S_I-S_M$; this means that the Peltier heating is an interface effect while Joule heating is a bulk effect.

For  Ca$_2$RuO$_4$ at room temperature, the thermal conductivity is $\kappa\sim10^{-3}\um{W/cmK}$ (see Appendix \ref{Appendix:Ca2RuO4Parameters}), the insulating state resistivity is $\rho_I\sim10\um{\Omega cm}$ and the metal-insulator transition temperature is $T_{MIT}\approx 360K$; we define $\Delta T\equiv T_{\rm{MIT}}-T_0\sim60\um K$. The metal phase of Ca$_2$RuO$_4$ has a very low Seebeck coefficient while the insulating phase has a high and positive Seebeck coefficient \cite{Seeb:CRO}: $S_M\approx0$ and $S_I\approx400-1000\,\mu \rm{V/K}$; the sign of $S_I$ is consistent with the large particle-hole asymmetry found in dynamical mean field calculations \cite{QH:AJM,Ricco:CRO}. Thus in Ca$_2$RuO$_4$, $\mathbf{\nabla} S$ points from the metal to the insulating phase so the interface is heated when the current flows from the insulator to the metal, i.e. when the metal phase nucleates out of the negative electrode. This agrees with the experimental reports from Ref. \cite{Mengkun:CRO} regarding the dependence of the nucleation process on the direction of the electric current.

We study the geometry shown in Fig. \ref{IV}a, an idealized representation of a typical experimental geometry: a film of length $L$, width $W$ and thickness $h$ grown on a substrate, which we assume to be held at temperature $T_0$ and to act as a heat sink. We choose the $x$ axis to be along $L$, the $y$ axis to be along $W$ and the $z$ axis to be along $h$. Since for typical experiments $h\sim0.2\um{mm}\gg\sqrt{\kappa_l/\gamma_{\rm{e-l}}}$, we can use Eq. \eqref{HT} with boundary conditions $\partial_zT(z=0)=0$ and $T(z=h)=T_0$

\subsection{Analytic Solutions}

Equation \eqref{HT} can be solved numerically, but analytical insight can be gained in particular limits. Although it is not directly relevant to most experimental situations, we assume $W\gg L\gg h\gg \sqrt{\frac{\kappa}{\gamma_{\rm{e-l}}}}$. We first consider a current $\mb j_c=j_0\hat x$ uniform in $x,y$, introduced at $x=0$ and removed at $x=L$. If the sample is entirely insulating the current does not depend $z$; the only source of heat is Joule heating and the temperature profile is given by (see Eqs. \eqref{TC1}-\eqref{TC3}
\begin{equation}\label{TIunif}
T(z)=T_0+\rho_Ij_0^2(h^2-z^2)/2\kappa.
\end{equation}

When $j_0$ reaches the critical current
\begin{equation}\label{jcrit}
j_{cr}\equiv\sqrt{2\Delta T\kappa/\rho_Ih^2}\sim5\um{A/cm^2},
\end{equation}
the temperature of the top surface reaches $T_{\rm{MIT}}$. For $j_0>j_{cr}$ a metallic region appears at the top surface. The estimate of $j_{cr}$ for Ca$_2$RuO$_4$ is in agreement with experimental reports \cite{Maeno:mainp,Fuku:CRO,Mengkun:CRO}. The appearance of a metallic phase near the top surface means that  the current becomes dependent on $z$. In the limit $h\ll L$, the region of $x$ in which $\mb j_c$ is not parallel to $\hat x$ is very small (of order $\sim h$), so that over most of the sample the current flows parallel to the interface and we can ignore any Peltier effect and focus on Joule heating only. In this geometry metal and insulator are essentially two resistors in parallel, so that the respective current densities $j_M$ and $j_I$ are related by $\rho_Ij_I=\rho_Mj_M$. Following Eqs. \eqref{jMjI}-\eqref{TdUnif} in Appendix \ref{Appendix:parallel current} we find the temperature at the interface $z=d$
\begin{equation}\label{Td}
T(d)=T_0+\Delta T\left(\frac{j_0}{j_{cr}}\right)^2\left[1-\frac{d^2}{(d+\frac{\rho_M}{\rho_I}(h-d))^2}\right].
\end{equation}
The condition $T(d)=T_{\rm{MIT}}$ determines the thickness of the metallic layer:
\begin{equation}\label{d}
d=h\frac{\rho_M/\rho_I\sqrt{1-j_{cr}^2/j_0^2}}{1-(1-\rho_M/\rho_I)\sqrt{1-j_{cr}^2/j_0^2}}.
\end{equation}
As expected $d(j_0=j_{cr})=0$, while $d$ approaches $h$ for $j_0\sim\sqrt{\rho_I/\rho_M}j_{cr}\gg j_{cr}$. From the value of $d$ we calculate the total current and find the V-I curve, shown in Fig. \ref{IV}b. The calculated curve exhibits the usual ``N'' shaped behavior expected for heating-driven insulator-metal transitions.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{Fig1.png}
\caption{\scriptsize{(a) Sketch of the geometry showing system (in blue) of length $L$, width $W$ and thickness $h$ and  substrate (red) which is held at temperature $T_0$. (b) Plot of the $I$-$V$ curve resulting from Eq. \eqref{d}; the cusp point is a consequence of the approximation used. The inset shows the geometry of the metallic and insulating phase. (c) Sketch of the side view and top view of the metallic and insulating phase in the situation considered for Eqs. \eqref{TPel} and \eqref{rc}.}}\label{IV}
\end{figure}

We now turn to an idealization of the situation studied in Ref. \cite{Mengkun:CRO}, in which a total current $I$ is injected through a point electrode at $x=L$, $y=W/2$, $z=0$ and removed at $x=0$, $y=W/2$, $z=0$. We focus on what happens near the $x=0$ electrode; in the limit $L\ll W$ and for points close to the electrode, the current decreases as $\mb j_c(r)\approx I/(\pi hr)\hat r$, with $r=\sqrt{x^2+(y-W/2)^2}$, with $I$ the total current. Crucially given the point like electrodes, all of the injected current must cross the insulator-metal phase boundary, and Peltier heating will play a role. The depth $d$ of the metallic region depends on $r$ and beyond the critical distance $r_c$ at which $d=0$ the sample is insulating at all $z$, as qualitatively sketched in Fig. \ref{IV}c.

We assume that over most of the relevant $r$ range in both metal and insulating regimes the current flow is perpendicular to $z$ and that the curvature in the $x-y$ plane of the interface may be neglected. Then the parallel resistor arguments imply that the total current in the metal (integrated along the $z$ direction) at distance $r$ is
\begin{equation}\label{IMt}
I_M(r)=\frac{d(r)}{d(r)+(h-d(r))\rho_M/\rho_I}\frac{I}{\pi r}.
\end{equation}
The current $I_M$ changes with $r$ thanks to the $r$ dependence of $d(r)$ so that some current must flow across the interface at $r$, giving rise to Peltier heating; this is a consequence of the point-like geometry of the electrodes which ensures an inhomogeneity in the $x-y$ plane.

We account for the Peltier contribution in the heat balance and require $T(d(r))=T_{\rm{MIT}}$, obtaining an equation for $\partial_rd$ which may be solved to find $d(r)$. If Joule heating is neglected we find (see Eqs. \eqref{IM}-\eqref{TdPH2} in Appendix \ref{Appendix:PeltierDetails})
\begin{equation}\label{TPel}
\Delta T=T_{\rm{MIT}}\Delta S\frac{I}{\pi r}\frac{\rho_M/\rho_I h^2}{\kappa(d+\rho_M/\rho_Ih)^2}\frac{\partial_rd}{\sqrt{1+(\partial_rd)^2}},
\end{equation}
where $\Delta S>0$. We observe that for $I<0$, i.e. current flowing into the metal, $\partial_rd$ is negative as it should be since $d(r=0)>0$ and $d(r_c>0)=0$; on the other hand, if $I>0$ then $\partial_rd\geq0$ and no stable interface can exist. In other words, for one direction of $I>0$ Peltier cooling contracts the interface to the very near vicinity of the electrode, while for the opposite direction it pushes the interface away from the electrode, consistently with observations in Ca$_2$RuO$_4$ that the metallic phase always nucleates out of the negative electrode, in such a way that the current flows from insulator into metal.

We solve Eq. \eqref{TPel} with $d(r=r_c)=0$ and integrating backwards to find $d(r)$. As explained in Appendix \ref{Appendix:PeltierDetails} (Eqs. \eqref{rd}-\eqref{rc2}), we can determine the critical radius by examining the solution for $r\rightarrow0$ under the assumption $d(r\rightarrow0)>\rho_M/\rho_Ih$ (corresponding to a larger current flow in the metal)
\begin{equation}\label{rc}
r_c\approx\sqrt{2hT_{\rm{MIT}}|I|\Delta S/\pi\kappa\Delta T}.
\end{equation}
Note that the metal phase disappears into the insulator at a nonzero angle, since $\partial_rd$ is nonzero for $r\rightarrow r_c$. This is in agreement with the wedge-shaped metallic phase considered in the elastic theory of stripe formation in Ref. \cite{Mengkun:CRO}. For $I\sim10\um{mA}$ and parameters compatible with Ca$_2$RuO$_4$ we find $r_c\sim0.2\um{mm}$ which is a sizable fraction of the length $L\sim1\um{mm}$; we thus show that even in a very simple limit our model reproduces the qualitative experimental features reported in Ref. \cite{Mengkun:CRO}.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{Fig2.png
\caption{\scriptsize{(a) Color map of the temperature $T(x,z)$ evaluated at $y=W/2$ and normalized to the external temperature $T_0$ for $h/L=0.2$, $W=L$, $\rho_M/\rho_I=0.1$, $\rho_II^2/(\kappa L^2T_0)=0.2$, $\rho_M/\rho_I=0.1$, $I\Delta S/(\kappa L)=-0.2$, with $I\sim10\um{mA}$ the total current; these parameters correspond to a realistic simulation of Ca$_2$RuO$_4$ experiments. The negative/positive electrode is located at $(0<x<0.1,y=W/2,z=0)$/$(0.9<x<1,y=W/2,z=0)$; the dashed black line shows the trial phase boundary used, the $M/I$ labels indicate the metal/insulator phase. The inset shows the plot of $T(x,y=W/2,z=0)/T_0$ as function of $x$ compared to the transition temperature $T_{\rm{MIT}}$. (b) Color map of $T(x,y,z=0)/T_0$ for the parameters of (a). (c) Color map of $T(x,y=W/2,z)/T_0$ for the same parameters as (a), except for a small Seebeck coefficient $I\Delta S/(\kappa L)=-0.02$ and a larger current $\rho_II^2/(\kappa L^2T_0)=1.2$; from the approximate symmetry of the metal phase we can see that Joule heating is the dominant effect. (d) Color map of $T(x,y,z=0)/T_0$ for the same parameters of (a) and (b), but $W=L/2$; in a rectangular geometry the metal phase can extend from $y=0$ to $y=W/2$. (e) Shape of the phase boundary for the parameters of (d) and different values of the current $I=5,10,25\um{mA}$.}}\label{T2D}
\end{figure*}

\subsection{Numerical Solution}

We now consider both Joule heating and Peltier effect, and iteratively solved the coupled system given by $\mathbf{\nabla}\cdot\mb{j}_c=0$, $\mathbf{\nabla}\times(\rho\mb{j}_c)=0$ and Eq. \eqref{HT}, using the parameters listed above which are appropriate to Ca$_2$RuO$_4$. We assume that the transport coefficients are determined by the local temperature $T(\mb x)$, i.e. $\sigma(T(\mb x)>T_{\rm{MIT}})=\sigma_M$ and $\sigma(T(\mb x)<T_{\rm{MIT}})=\sigma_I$. The system has dimensions $L=1\um{mm}$, $W\sim L$ and $h/L=0.2$ and the negative (positive) electrode is located at $0<x/L<0.1$ ($0.9<x/L<1$), $y=W/2$, $z=0$

We present in Fig. \ref{T2D} two different cases, for the point electrodes geometry. Panels (a) and (b) shows computations for $S_I\sim1000\um{\mu V/K}$,  which is a value representative of Ca$_2$RuO$_4$, and a square geometry $W=L$; the other parameters are given in the figure caption. The simulation clearly shows that a current $I\sim10\um{mA}$ can induce a metal phase extending over part of the sample (up to $x/L\sim0.3$; see inset in Fig. 2a). Note also that the metal phase exists only on one side of the sample, showing that the interface Peltier effect leads to a pronounced spatial asymmetry of the metallic domains, consistent with observations of Ref. \cite{Mengkun:CRO}. Panel (b) shows the top view of the surface temperature $T(x,y)$; we see that for distances from the electrode less than $\sim 0.3L$ the temperature is above the metal-insulator transition temperature, while for the rest of the sample the temperature is below. Panel (c) shows results of analogous computations, in which the Seebeck coefficient is reduced to a much smaller value, showing that in this case two metallic phases nucleate around both electrodes in a nearly symmetric way; this occurs when the Joule heating plays is more relevant and the effect is thus almost symmetric in the direction of the current. These results show that Peltier heating at interfaces can explain the appearance of a stable metal phase in typical Ca$_2$RuO$_4$ experiments even for modest values of the applied current.

We notice that the thickness $d$ of the metallic layer is a sizeable fraction of $h$, and depends on the value of $S$ and on of the resistivity ratio $\rho_I/\rho_M$; in particular $d$ decreases for larger $\rho_I/\rho_M$. Notice that since $h$ is not much smaller than $L$ as assumed in the derivation of Eq. \eqref{TPel}, the boundary geometry is only qualitatively a wedge.

In Fig. 2d and 2e we consider a rectangular geometry $W=L/2$, observing that the maximum temperature does not change appreciably, while the interface geometry is modified; in Fig. 2e we study the evolution of the phase boundary for increasing currents and find that it extends to larger values of $x/L$, in agreement with the results of Ref. \cite{Mengkun:CRO}, and can stretch along the entire width of the sample. We also considered the case of a wider geometry $W=2L$ and found that the phase boundary has a geometry similar to that of the square $W=L$ case and is more stretched along the $x$ axis rather than along the $y$ axis.

We can also calculate the average temperature of the system; for the parameters of Fig. 2a we find an average temperature $\bar T=1.05T_0$ and an average top surface temperature $\bar T(z=0)=1.07T_0$. This shows that even for modest increases of the global temperature of the system, $T$ can be locally larger than the critical temperature.

\section{Conclusions}\label{sec:Conclusions}

We considered a correlated system with an electric current flowing through an  interface between its metallic and insulating phases.  Using macroscopic arguments based on entropy production and linear transport theory, we wrote a heat balance equation that can be solved for the temperature given the current. The equation accounts for Joule heating, heat diffusion, heat dissipation, and crucially, includes the Peltier effect, which linearly couples the current to the discontinuity $\Delta S$ in thermopower across an interface between a metal and an insulator. This term leads to heating or cooling at the interface depending on the direction of the current with respect to the change in thermopower. The magnitude of the interface Peltier effect depends on the current, the thermopower change across the interface, and the efficiency of heat dissipation (determined here by the thermal conductivity and the distance to the nearest heat sink). Because the Peltier heating is linear in the current while Joule heating is quadratic, the Peltier heating dominates at small currents, and leads to interesting physics if the Peltier effect remains dominant at currents large enough to heat the system above its insulator-metal transition temperature. This condition is equivalent to a thermoelectric-like figure of merit $ZT\equiv T\Delta S^2/\rho\kappa$ being larger than one. Our analysis showed that for Ca$_2$RuO$_4$ in geometries similar to those considered in recent experiments \cite{Mengkun:CRO}, heating effects can stabilize a non-equilibrium metal phase in a region of the sample that depends on the direction of the current. Other materials that exhibit a large discontinuity in thermopower at the metal insulator transition and may also exhibit a similar polarity dependence include VO$_2$ microbeams \cite{Seeb:VO2} and Cu$_2$Se \cite{Seeb:Cu2Se}.

Interesting directions for future theoretical research include extension of our analysis to other experimental geometries, in particular to the very thin sample regime and to the filamentary conduction pattern observed in other correlated electron systems.  Experimental observations of  our predicted distribution of the local temperature of the sample surface would be of great interest. A very recent work \cite{Mattoni:CRO} by Mattoni et al. studied the local temperature in Ca$_2$RuO$_4$ under current and reported data compatible with our predictions. Equally of great interest would be the investigation of different electrode geometries (e.g. introducing the current uniformly across a sample face rather than in a small region) and sample thicknesses. Finally, our analysis is primarily based on a macroscopic theory, and a more detailed microscopic analysis of the physics of the metal-insulator interface, in the presence of current flow, would be of considerable interest.

\textit{Acknowledgements} We thank A. Georges, A. McLeod, and, particularly, Mengkun Liu for very helpful discussions. This work was supported by the US Department of Energy under grant DE-SC0012375.

\bibliographystyle{apsrev4-1}


\section{Introduction}

Highly blueshifted absorption lines of ionised material have recently been detected in the X-ray spectra of a number of active galactic nuclei (AGN), reaching velocities from $\sim$0.1c to as high as 0.5c \citep[e.g.][]{Pounds+03, Reeves+03, Tombesi+10b, Tombesi+10a, Tombesi+15, Nardini+15, Parker+16, Kosec+18c, Walton+19}. These absorption lines could originate from accretion disc winds, launched at high velocities from the accretion flow of AGN by radiation pressure \citep{Proga+00} or magnetic forces \citep{Fukumura+10}. The large velocity and the consequent significant kinetic power makes these outflows strong candidates for being the drivers of AGN feedback in galaxies \citep{Fabian+12, King+15}. Alternatively, these blueshifted absorption lines could originate from low-density material co-rotating with the inner accretion flow without the need for an outflow from the system \citep{Gallo+13, Fabian+20a}.

In most cases the signatures of these so-called `ultra-fast outflows' (UFOs) were identified using the highly ionised features of \ion{Fe}{XXV} and \ion{Fe}{XXVI} \citep{Tombesi+10b} in the iron K band (7-10 keV). The iron K band, however, is often on the upper edge of the energy range of the current instruments like XMM-Newton (EPIC pn and MOS) and Chandra, where the collecting area of the instruments steeply drops off, resulting in poor counts and signal-to-noise ratios. Additionally, this band commonly contains strong features of relativistically blurred reflection of the AGN coronal emission from the accretion disc \citep{Tanaka+95, Fabian+09}. Therefore the observed absorption features can be hard to interpret as outflow signatures and it is difficult to infer the wind properties such as the velocity and the ionisation level \citep{Zoghbi+15}. Furthermore, spurious low significance features or lines of instrumental origin can be interpreted as real UFO signatures.

Less frequently, fast ionised wind absorption lines are also observed in softer X-rays ($<5$ keV), for example using the \ion{Si}{XIV}, \ion{S}{XVI} and \ion{O}{VIII} transitions \citep{Jiang+18, Pinto+18, Reeves+18a}. The soft X-ray band (0.3-2.0 keV) can however be dominated by the commonly observed warm absorption from low ionisation material at larger distances from the AGN \citep[e.g.][]{Kaastra+00,Kaspi+02,Blustin+05,Detmers+11}, which complicates line identification.

It is therefore important to study UFOs in different objects and across a broad X-ray band, comparing their properties to the source X-ray continuum properties in order to understand the true nature of UFOs and their relationship to AGN.

Here we present the detection of an ultrafast outflow with a velocity of 0.09c in the spectrum of a Narrow Line Seyfert 1 galaxy PG 1448+273 observed with the \textit{XMM-Newton}\ observatory. The signatures of the outflow are observed in both hard X-rays (iron K band) and soft X-rays (0.3-2.0 keV), and we constrain its physical properties systematically using an automated multi-parameter search of the X-ray spectrum with photo-ionised absorption grids.

PG 1448+273 is a Narrow Line Seyfert 1 AGN \citep{Boller+96, Grupe+04} located at a redshift of z=0.0645 \citep{Alam+15}. Its mass is $(9 \pm 2) \times 10^{6} M_{\odot}$ \citep[][from single-epoch optical spectroscopy]{Vestergaard+06}. PG 1448+273 has an estimated bolometric luminosity of $\sim 10^{45.5}$ erg/s \citep{Grupe+04}, resulting in an Eddington ratio above unity ($L/L_{\rm Edd}\sim3$). PG 1448+273 could therefore be a super-Eddington accretor \citep{Kawaguchi+03}. At such a high Eddington fraction, radiation pressure is an important component of the accretion flow and is expected to drive powerful outflows from the accretion disc \citep{Shakura+73}. PG 1448+273 is therefore a natural candidate for a spectroscopic search of UFO signatures.

Previous X-ray studies have remarked on the complex shape of the soft X-ray excess \citep{Innoue+07} of PG 1448+273, requiring more than the usual $\sim$0.1 keV blackbody component for an accurate description. The AGN is however not heavily absorbed by neutral absorption or a strong warm absorber. Its X-ray spectrum is also highly variable \citep{Ponti+12b}.

This paper consists of 5 sections. The observation and data reduction is described in Section \ref{data}. Details of our analysis and the results are shown in Section \ref{results}, followed by a discussion of the findings in Section \ref{discussion}. Section \ref{conclusions} contains the conclusions of this study.

\section{Observations and Data Reduction}
\label{data}

The \textit{XMM-Newton}\ \citep{Jansen+01} data were obtained from the XSA\footnote{nxsa.esac.esa.int/nxsa-web/} archive. The main dataset consists of a single observation taken on 24/01/2017 (ID: 0781430101, proposal PI: A. Zoghbi) with a total raw exposure of 126 ks. There is also a second, much shorter (22 ks) observation, which was taken on 08/02/2003 (PI: Kawaguchi).

We reduced the main dataset using standard pipelines with SAS v17. There were no high background periods of time during the observation and thus no time intervals needed to be filtered out.

The observation with the European Photon Imaging Camera (EPIC) pn and MOS instruments was split into two exposures, a scheduled (`S') and an unscheduled (`U') part. Initially we extracted the spectra of the two exposures separately, but found no significant differences between them. We therefore combined the scheduled and unscheduled parts of the observation into a single exposure.

\begin{figure}
	\includegraphics[width=\columnwidth]{ds9_regions.pdf}
    \caption{Image of the EPIC pn Small Window Mode exposure. The source region used in the analysis is the green circle (20 arcsec radius), the 2 polygons are the regions used to extract the background spectrum. The non-uniform background count distribution (in the vertical direction) is evident and is likely caused by the wings of the source point spread function. }
    \label{PN_image}
\end{figure}

The EPIC pn \citep{Struder+01} instrument was operated in Small Window mode during the observation to limit pile-up. An image of the pointing was made using standard routines. We noticed a strong non-uniformity of background counts across the image (Fig. \ref{PN_image}), likely due to source counts from the wing of the source point spread function (PSF). To avoid including source counts in the background spectrum, we chose background regions to be two small polygons as far from the source as possible. Furthermore, we decreased the importance of background subtraction by choosing a small (20 arcsec) source region. This boosted the signal-to-noise ratio in the critical $6-10$ keV energy band where the Doppler-shifted absorption features of highly ionised iron (\ion{Fe}{XXV/XXVI}) are expected. Only events of PATTERN <= 4 (single/double) were accepted. The average EPIC pn count rate was 2.2 ct/s, but the source is strongly variable, varying between 1 ct/s and 4 ct/s over the course of the observation.

EPIC/MOS 1 and 2 \citep{Turner+01} were operated in Large Window mode. The source region was chosen to be a circle with a radius of 20 arcsec, similar to pn data. The background covered the rest of the central chip area, at least 110 arcsec away from the source, avoiding other bright point sources. We did not notice any background non-uniformity as observed in EPIC pn data, confirming that the non-uniformity observed in the EPIC pn image is caused by the wings of the PSF. Events of PATTERN <= 12 (single/double) were accepted. The average EPIC MOS 1 count rate was 0.44 ct/s and the average MOS 2 count rate was 0.43 ct/s. We stacked data from MOS 1 and MOS 2 into a single spectrum with the \textsc{epicspeccombine} routine.

Data from both EPIC/pn and EPIC/MOS were grouped using the \textsc{specgroup} procedure to at least 25 counts per bin, and to oversample the detector resolution by a factor of at most 3. The clean exposure time of the pn dataset is 76 ks, and the exposure of the combined MOS 1 and MOS 2 data is 108 ks (pn exposure is smaller due to a lower Live time in Small Window mode).

Reflection Grating Spectrometer \citep[RGS,][]{denHerder+01} data were reduced using standard routines with default values and with standard observational background. Data were binned by a factor of 3 directly within the SPEX fitting package to oversample the real grating resolution by roughly a factor of 3. The clean exposure of RGS 1 and RGS 2 data is 115 ks.

After reduction, the data were converted into \textsc{spex fits} format using the \textsc{trafo} tool. \textsc{spex} fitting package \citep{Kaastra+96} was used for spectral fitting as it contains the physically-motivated ionised absorption grids \textsc{xabs} and \textsc{pion}. All spectral models were fitted using Cash statistics \citep{Cash+76} and the errors are stated at 1$\sigma$ level. We use 3 cross-calibration constants to account for calibration differences between the four spectral datasets (pn, MOS, RGS1 and RGS2). The differences are smaller than 10 per cent in all cases.

EPIC pn and MOS data were initially ignored below 0.3 keV and above 10 keV following standard guidelines. RGS data were used in the range between 7.5 \AA\ (1.7 keV) and 28 \AA\ (0.44 keV), limited by strong background on both ends of this interval. We found a 10-15 per cent discrepancy between EPIC and RGS data in the 0.3-1.0 keV range which could be caused by poor spectral resolution of the EPIC instruments in this range, or by calibration differences. At these energies EPIC lacks the resolution to resolve the fine structure of absorption lines but would drive the spectral fits due to much higher count rates than RGS (resulting in much smaller errorbars compared to RGS data). To avoid confusion between different unresolved spectral components, we ignored EPIC pn and MOS data below 1.7 keV altogether in the main part of our analysis.

We followed the same steps to reduce and prepare the second observation (0152660101) for analysis. The statistics were naturally much worse compared to the main dataset, albeit the AGN was brighter in 2003. The average EPIC pn count rate was 4 ct/s, and the average MOS 1/2 count rate was 0.9 ct/s. Since EPIC pn was operated in Large Window mode (with a pile-up limit of 3 ct/s), it is possible that the pn data are affected by pile-up. MOS was operated in Large Window mode (pile-up limit of 1.5 ct/s) and its data should not be affected. Nevertheless, we do not observe any significant differences between pn and MOS spectra. We ignored EPIC data below 1.7 keV, only using RGS data in the soft X-ray band, following the same procedure as with the main dataset.

Fig. \ref{PN_015_078} contains the $1.7-10$ keV EPIC pn spectra from both observations and shows the longterm variation of the source and the dataset quality. Especially striking is the difference in the iron K region ($5-8$ keV). We first focus on the analysis of the higher quality observation (0781430101) which also shows more intriguing spectral features. The second observation is analysed separately in Sect. \ref{oldobs}. It would not be appropriate to stack the two datasets given the long period of time and strong variability between the two observations.

\begin{figure}
	\includegraphics[width=\columnwidth]{PG1448_078_015_comparison.pdf}
    \caption{$1.7-10$ keV EPIC pn spectra of PG 1448+273 during the two observations. Data from the main observation (0781430101) are in black colour, data from the shorter archival observation (0152660101) are shown in red.}
    \label{PN_015_078}
\end{figure}

\section{Results from the main \textit{XMM-Newton}\ observation}
\label{results}

\subsection{Broadband X-ray continuum}
\label{continuummodelling}

First we fit the broadband 0.3-10 keV spectrum with a phenomenological X-ray AGN spectral model. We fit the RGS, EPIC pn and MOS data simultaneously with the same spectral model except for the cross-calibration constant. The model consists of a powerlaw (\textsc{pow} in SPEX) model describing the Inverse Compton coronal emission, a blackbody (\textsc{bb}) model describing the soft excess below 2 keV, and a relativistically blurred iron K emission line. Blurred reflection is described with a \textsc{laor} model applied to a Gaussian line (\textsc{ga}) with the energy fixed to 6.4 keV to avoid too much model freedom with the available data quality. All of these models are subject to any interstellar absorption in the host galaxy of the AGN described by a \textsc{hot} model within SPEX. The model is then redshifted by z=0.0645 of the host galaxy (found on the NED database) using the \textsc{reds} model. Finally, Galactic interstellar absorption is applied with a second \textsc{hot} spectral model with a fixed value of $N_{\textrm{H}}=$2.5$\times10^{20}$ cm$^{-2}$\ \citep{Kalberla+05}. The final spectral model has the following form: \textsc{hot(reds(hot(pow+bb+laor$\times$ga)))}.

We find that PG 1448+273 shows properties very similar to other Narrow Line Seyfert 1 AGN (for a summary of the continuum parameters, see Table \ref{Contparams}). The powerlaw slope is measured to be $2.07 \pm 0.03$, and the soft excess can be described with a blackbody of temperature $0.097 \pm 0.003$ keV. The black hole appears to have a high spin value with a minimum disc radius of just $1.52_{-0.29}^{+0.19}$ R$_{\textrm{G}}$, but we draw no firm conclusions since the relativistic iron K reflection is described only with a simple phenomenological model. The emissivity slope is q$=4.6 \pm 0.5$ and the measured disc inclination $44_{-3}^{+2}$ degrees. We also find a significant non-zero interstellar absorption in the host galaxy of PG 1448+273 with a column density of $(9.8 \pm 2.1) \times 10^{20}$ cm$^{-2}$. The fit statistics are C-stat=1594.43 for 1278 degrees of freedom (D.o.F.).

\begin{figure}
	\includegraphics[width=\columnwidth]{PG1448_cont_EPIC.pdf}
    \caption{EPIC pn (black) and MOS (red, MOS 1 and MOS 2 stacked) data ($4.5-10$ keV) of PG 1448+273 fitted with the phenomenological AGN continuum model from Section \ref{continuummodelling}. Apparent is the strong absorption residual observed at $\sim$7 keV. Black (pn) and red (MOS) solid lines show the best-fitting continuum models, and the dashed lines show the subtracted background spectra.}
    \label{PN_spectrum}
\end{figure}

\begin{figure*}
	\includegraphics[width=\textwidth]{PG1448_cont_RGS_v2.pdf}
    \caption{Background-subtracted RGS data (7.5-28 \AA) of PG 1448+273, stacked and overbinned for plotting purposes only, fitted with the phenomenological AGN continuum model from Section \ref{continuummodelling} (blue colour). The model particularly fails to reproduce the strong residuals around 18-19 \AA, however further residuals are seen around 12, 15.5 and 24 \AA. The edge at 24.5 \AA\ and the strong absorption line at 25 \AA\ suggest the presence of neutral absorption intrinsic to the AGN. The subtracted background is shown in orange.}
    \label{RGS_spectrum}
\end{figure*}

Upon visual inspection of the spectrum, strong residuals in EPIC pn and MOS data within the iron K band are obviously evident (Fig. \ref{PN_spectrum}), as are residuals in the RGS band (Fig. \ref{RGS_spectrum}). The iron K band residuals suggest the presence of a highly ionised, high-velocity outflow, the flux drop above 7 keV is very similar to that of another NLS1 1H 0707$-$495 with a known high-velocity outflow \citep{Boller+02,Kosec+18c}. The RGS data point to a more complex situation. There is a prominent absorption residual at 18-19 \AA\ (observed wavelength), which could correspond to \ion{O}{VIII} absorption, blueshifted by $\sim$0.1c. Further structures are also seen around 12 \AA\ and 24 \AA\ in observed wavelength.

\subsection{Ionised wind search}
\label{wind_scan}

\subsubsection{Wind search set up and results}

To determine the blueshift and physical parameters of any ionised outflow component as well as its significance in the present data, we systematically scan the X-ray spectrum of PG 1448+273 with an automated ionised absorber search. In principle, we use a routine to scan a large parameter space of a photoionisation absorber grid, to locate the best-fitting ionised outflow properties. Similar methods were recently successfully used to detect and study ultrafast outflows in Ultraluminous X-ray Sources \citep{Kosec+18a, Kosec+18b, Pinto+20}.

The photoionisation model we use here to describe the spectral features (absorption lines) of an ionised outflow is called \textsc{xabs} in the SPEX fitting package. The model calculates the ionising balance and absorption line strengths based on the Spectral Energy Distribution (SED) shape of the AGN NGC 5548 \citep{Steenbrugge+03, Steenbrugge+05}. The SED of NGC 5548 should be a reasonable approximation for the PG 1448+273 SED, although the former is likely more massive at $(9 \pm 3) \times 10^7 M_{\odot}$, and has a smaller Eddington ratio \citep{Lu+16}. The coronal powerlaw slope of NGC 5548 is $\Gamma\approx1.8$ versus $\Gamma=2.07 \pm 0.03$ in PG 1448+273, so PG 1448+273 has a softer spectrum but not as soft as the more extreme NLS1 AGN like IRAS 13224$-$3809 where $\Gamma$ varies between 2.2 and 2.9 \citep{Pinto+18}. At each step, we also check our results for consistency with the \textsc{pion} model in the \textsc{spex} fitting package. \textsc{pion} is a similar model of photo-ionised plasma as \textsc{xabs} but takes into account the actual continuum SED of the currently used spectral model, albeit at a much higher computational cost, which is the limiting factor in our analysis.

The main parameters of the \textsc{xabs} model are the column density $N_{\textrm{H}}$, ionisation parameter $\log \xi$, systematic velocity $z$ and turbulent velocity (or simply velocity broadening) $v_{\textrm{turb}}$ of the ionised absorber. An outflow in the X-ray spectrum of the AGN can in principle have any blueshift, ionisation parameter, column density and turbulent velocity. This is a large parameter range which has to be searched in an automated way to avoid missing the parameter space with the best-fitting solution. As an added bonus, performing a systematic search in this way will allow us to constrain the statistical significance of any outflow detection in the AGN spectrum.

We therefore create a grid of photoionisation models which spans ionisation parameters between $\log (\xi/ \textrm{erg cm s}^{-1})=-1.0$ and $\log (\xi/ \textrm{erg cm s}^{-1})=4.8$ with a grid spacing of $\Delta\log\xi=0.2$ \citep[in line with the recommendation of][]{Reynolds+12}, with three fixed turbulent velocities of $v_{\textrm{turb}}=100$ km/s, $v_{\textrm{turb}}=1000$ km/s and $v_{\textrm{turb}}=10000$ km/s. The turbulent velocity space grid is chosen to be coarse to decrease the computational cost of the automated search.

Afterwards a range of realistic systematic absorber velocities (blueshifts) is chosen. We choose to search for an absorber with a systematic velocity between +20000 km/s (to account for possible infalling gas) and -100000 km/s (blueshift of roughly 0.4c, after relativistic correction). The step size is 150 km/s for turbulent broadening of 100 km/s and 300 km/s for turbulent broadening of 1000 km/s. This is chosen so that in each case the step size is not much larger than the broadening and at the same time is not much larger than the spectral resolution of our detectors, RGS being the best of our instruments with a spectral resolution of roughly $\sim$300 km/s in velocity units. In the case of turbulent velocity equal to 10000 km/s, the features are so broad that such detailed sampling is not necessary, instead we sample by 1000 km/s.

We generate an absorption grid for every possible systematic velocity, ionisation parameter and broadening within the selected range, add this model to the baseline continuum from Section \ref{continuummodelling}, and fit for the column density of the absorber (with the remaining absorber parameters fixed), leaving the continuum parameters free to vary. If an absorber of such parameters is disfavoured by the data, the resulting column density will be 0, i.e. the best-fitting solution is the null (baseline continuum) model. Otherwise we recover the best-fitting absorber column density as well as the $\Delta$C-stat\ fit improvement obtained by adding the photoionisation grid to the (null) baseline continuum model. Any improvement in the $\Delta$C-stat\ fit statistics compared to the null model is recorded for every point in the multi-dimensional search grid, and determines how strongly the spectral model including the wind absorption is preferred to the baseline continuum model.

\begin{figure*}
	\includegraphics[width=\textwidth]{2D_Scan_cont_allwidths_2.pdf}
    \caption{Systematic scan for ionised outflow features in the spectrum of PG 1448+273. The spectrum is scanned with ionised absorption model grids with systematic velocities between +20000 km/s and -100000 km/s (on the X axis), ionisation parameters $\log \xi$ between -1.0 and +4.8 (on the Y axis), with turbulent velocities of 100 km/s (top subplot), 1000 km/s (middle subplot) and 10000 km/s (bottom subplot). The colour shows the $\Delta$C-stat\ fit improvement upon adding the absorption grid to the baseline continuum model (according to the colour bar on the right), the black contours show $\Delta$C-stat\ fit improvement of 30, 50 and 70, respectively.}
    \label{2D_Scan_cont}
\end{figure*}

We search the large parameter space for signatures of an ionised outflow as described above, using the full dataset (RGS + pn + MOS) on PG 1448+273. The results of the search are shown in Fig. \ref{2D_Scan_cont}. The wind search achieves a strong detection of ionised plasma with a systematic outflow velocity of $\sim30000$ km/s, with a fit improvement of $\Delta$C-stat$>$70. The plasma is highly ionised with $\log \xi$ of $4.0-4.2$ and the fit improvement is much higher in the scans with a 1000 or 10000 km/s turbulent velocity compared to the one with 100 km/s velocity width. All the other $\Delta$C-stat\ peaks in the systematic scans are much smaller, reaching at most $\Delta$C-stat$\sim$30-35, hence the `primary' (strongest) solution is highly preferred.

The fact that our detection is above $\Delta$C-stat=70 already strongly suggests the significance of this outflow component, comparing with previous similar analyses \citep[e.g.][]{Kosec+18b}. However, due to the look-elsewhere effect, since we are searching through a large parameter space, it is not trivial to assign directly a `wind detection significance' to a specific value of $\Delta$C-stat\ \citep{Protassov+02}. Monte Carlo simulations must be employed to rigorously `map' the C-stat fit improvement to a specific false alarm probability for each wind detection. The results of Monte Carlo simulations are shown in Appendix \ref{MCSims}. They indicate that the significance of the primary peak detection is well above $3-4\sigma$ and thus our UFO detection is robust.

\subsubsection{Secondary detections in the wind search}

\begin{figure*}
	\includegraphics[width=\textwidth]{PG1448_cont_xabs_warm_EPIC_v2.pdf}
    \caption{EPIC pn (left subplots) and MOS (right subplots) data ($1.7-10$ keV) of PG 1448+273 fitted with the phenomenological AGN continuum model from Sect. \ref{continuummodelling}, plus two wind components: one ultrafast, highly ionised component outflowing at $\sim$27000 km/s, and a warm absorber component in the rest-frame of the AGN. The top subplots show the spectra with the best-fitting baseline continuum (blue colour) and the continuum + wind models (red colour), the bottom subplots contain the residuals to the continuum model as well as the best-fitting wind solution (red). The orange dashed lines show the subtracted background spectra. Green labels name the strongest absorption lines of the UFO absorption.}
    \label{PN_xabs_warm}
\end{figure*}

\begin{figure*}
	\includegraphics[width=\textwidth]{PG1448_cont_xabs_warm_RGS_v2.pdf}
    \caption{RGS data between 7.5 \AA\ (1.7 keV) and 28 \AA\ (0.4 keV) of PG 1448+273, stacked and overbinned for plotting purposes only, fitted with the phenomenological AGN continuum model from Section \ref{continuummodelling}, plus two ionised components: one ultrafast, highly ionised component moving at $\sim$27000 km/s, and a warm absorber component in the rest-frame of the AGN. The top subplot contains the RGS spectrum fitted with the baseline continuum (blue colour) and continuum + wind (red colour) models, the bottom subplot shows the residuals to the continuum model and the best-fitting wind solution (red). The subtracted background in orange colour. Green labels name the most notable absorption lines of the UFO absorption and the magenta label shows the spectral features of the warm absorber.}
    \label{RGS_xabs_warm}
\end{figure*}

We also find a large number of secondary peaks with $\Delta$C-stat\ between 25-35 which by themselves could be considered significant detections, if they were detected alone. Their presence suggests that there could be further wind structure hiding in the spectrum of PG 1448+273. However, at this point it is not certain that the secondary peaks are independent of the primary peak. These secondary solutions could in fact be fitting some of the same spectral residuals to the baseline continuum as the primary peak, but with a different set of atomic lines, resulting in a shallower $\Delta$C-stat\ improvement.

Finally, we also notice a $\Delta$C-stat\ peak around the rest-frame of the AGN in the 100 km/s velocity width search. The peak is not particularly strong with $\Delta$C-stat$\sim$20 at the ionisation parameter of $\log \xi = -0.6$, but could mean the presence of a weak warm absorber. Absorbers with similar ionisation parameters are very commonly observed in other AGN \citep[e.g.][]{Blustin+05}, it is thus not unlikely than one could also appear in the spectrum of PG 1448+273. If this is the case, the same look-elsewhere effect as for the UFO detection does not apply. Warm absorbers in particular only appear over a very small systematic velocity parameter space (velocities from hundreds of km/s to a few thousands of km/s at maximum). The same $\Delta$C-stat\ significance rules for UFO detections hence do not apply as the available parameter space is much smaller, and a significantly weaker warm absorber detection might still represent real absorption features. For this reason we fit both the features of the UFO primary peak as well those of the warm absorber in the next section.

\subsection{Ultrafast outflow and warm absorption}
\label{Sec_xabs_warm}

To learn more about the properties of the ionised absorbers in PG 1448+273, we fit its spectrum directly with a model which includes the original baseline continuum (with all the parameters freed), plus two ionised absorbers described with the \textsc{xabs} model. The first \textsc{xabs} model is supposed to fit the primary peak in the systematic absorber search at around 30000 km/s, the second one should fit the possible features of a warm absorber in the rest-frame of the AGN.

The primary ultrafast component is found with an outflow velocity of $26900 \pm 600$ km/s (including the relativistic correction). Its column density is $0.28_{-0.07}^{+0.12} \times 10^{24}$ cm$^{-2}$, with an ionisation parameter of $\log (\xi / \textrm{erg cm s}^{-1})=4.03_{-0.08}^{+0.10}$ and a velocity width of $2100_{-500}^{+600}$ km/s. The final fit improvement of adding this component to the baseline model is very strong with $\Delta$C-stat=83.27, confirming the significance of the ultrafast outflow features in the spectrum of PG 1448+273. The strongest spectral features of the UFO absorption are the \ion{Fe}{XXV} and \ion{Fe}{XXVI} lines at $\sim$7 keV (in the EPIC band) and the \ion{O}{VIII} line at 18.5 \AA\ ($\sim$0.7 keV, in the RGS band).

The warm absorber has a column density of $(3.8 \pm 1.7) \times 10^{20}$ cm$^{-2}$\ and an ionisation parameter of $\log (\xi / \textrm{erg cm s}^{-1})=-0.4_{-0.4}^{+0.3}$. Its systematic velocity is $-100 \pm 200$ km/s in the rest-frame of the AGN so it is consistent with being at rest, with a velocity width of $50_{-30}^{+40}$ km/s. It is similar to warm absorbers found in other AGN \citep{Kaspi+02, Steenbrugge+05, Detmers+11}, apart from the column density which is much lower in PG 1448+273, resulting in rather weak spectral features, and a lower $\Delta$C-stat\ fit improvement of 15.19. The $\Delta$C-stat\ is smaller than the original value found in the automated wind search ($\sim$20), suggesting that the ultra-fast and warm absorber models are partly fitting the same residuals to the original baseline continuum. The spectral features of the warm absorber are located at around 24 \AA\ in the RGS band.

Since the strongest absorption lines of the warm absorption lie in the 24 \AA\ range, they overlap with the \ion{N}{VII} absorption of the 0.1c ultra-fast phase. It is therefore possible that these features instead could originate in \ion{N}{VII} absorption of the ultra-fast phase. We attempt to describe the features by freeing the abundance of N in our final spectral model (with the warm absorber column density fixed to 0), and reach fit statistic improvements at a $\Delta$C-stat\ $\approx10$ level (compared to a continuum + UFO with solar abundances spectral model), for a nitrogen over-abundance of $7^{+3}_{-2}$. We consider this to be a very high abundance value and prefer the warm absorption origin for the 24 \AA\ features as a far more likely explanation.

The baseline continuum parameters slightly shift (for a summary of the continuum parameters, see Table \ref{Contparams}), with the best-fitting powerlaw slope being $1.97 \pm 0.03$ and the soft excess blackbody temperature of $0.102 \pm 0.003$ keV, however they still agree with the original continuum fit values within $\sim3\sigma$ errors. The inner disc radius (from the iron K line) is $1.7 \pm 0.5$ R$_{\rm G}$, the emissivity slope is $3.5_{-0.5}^{+2.1}$ and the disc inclination is found to be $38_{-3}^{+16}$ degrees. Most notably, the best-fitting neutral absorber (in the rest-frame of PG 1448+273) is now weaker with a column density of $6.5_{-2.2}^{+2.0} \times 10^{20}$ cm$^{-2}$. This could be explained if the warm absorber (with a low ionisation parameter) is fitting similar spectral residuals as the neutral absorption. The final fit statistics are C-stat=1495.97 for 1270 D.o.F.

The best-fitting wind spectral model is shown in Fig. \ref{PN_xabs_warm} (EPIC pn and MOS data only) and in Fig. \ref{RGS_xabs_warm} (RGS data only).

\subsection{Residual structure in the X-ray spectrum - a multiphase outflow?}

\begin{figure*}
	\includegraphics[width=\textwidth]{2D_Scan_cont_xabs_warm_allwidths_2.pdf}
    \caption{Systematic scan for ionised outflow features in the spectrum of PG 1448+273, in addition to the primary ultrafast component outflowing at 27000 km/s and a warm absorber component in the AGN rest-frame. The spectrum is scanned with ionised absorption grids with systematic velocities between +20000 km/s and -100000 km/s (the X axis), ionisation parameters $\log \xi$ between -1.0 and +4.8 (the Y axis), with turbulent velocity 100 km/s (top subplot), 1000 km/s (middle subplot) and 10000 km/s (bottom subplot). The colour shows the fit improvement $\Delta$C-stat\ upon adding the absorption grid to the continuum model (according to the colour bar on the right), the black contours show $\Delta$C-stat\ fit improvement of 15 and 20, respectively.}
    \label{2D_Scan_xabs_warm}
\end{figure*}

The large C-stat value of the best-fitting spectral model given the number of degrees of freedom, and the secondary peaks in the systematic wind search suggest the presence of further ionised outflow components in the spectrum of PG 1448+273, assuming our baseline continuum model is accurate. However, the secondary peaks might not be independent of the primary wind detection, i.e. their spectral signatures could be shared. For this reason, it does not make sense to directly fit the secondary peaks based on the existing automated wind search (Fig. \ref{2D_Scan_cont}).

We therefore perform a new set of wind searches which includes the two outflow components described above (in Sec. \ref{Sec_xabs_warm}). We choose the best-fitting baseline continuum with 2 additional \textsc{xabs} components (ultra-fast outflow and warm absorber) to be the new baseline model. We then run the same wind search as described in Sec. \ref{wind_scan}, with all the baseline continuum parameters allowed to vary freely. The results are shown in Fig. \ref{2D_Scan_xabs_warm}.

There are no strong wind detections in the search with a turbulent velocity of 1000 km/s. On the other hand, there are roughly 5 peaks with fit improvements between $\Delta$C-stat\ of 15 and 22 in the search with 100 km/s turbulent velocity, suggesting tentative evidence of further ionised outflow components. Furthermore, there is a similarly strong peak ($\Delta$C-stat$\sim$20) in the 10000 km/s width search.

If only a single peak with a similar fit improvement ($\Delta$C-stat$\sim$20) was detected, it would be trivial to assign a false alarm probability to its detection based on our Monte Carlo simulation results from Appendix \ref{MCSims}. However, the fact that there are 6 peaks with similar $\Delta$C-stat\ values means that we likely cannot statistically differentiate which of these solutions are real and which are false detections as they will all result in similar statistical significances/false alarm probabilities. The situation is further complicated because these peaks are probably not independent, i.e. multiple potential peaks are fitting the same residuals to the baseline continuum, but with different outflow models.

We individually fit each of the peaks with $\Delta$C-stat\ $> 15$ but find no further fit improvement or any particularly strong outliers. The tentative solutions mostly fit residuals in the RGS band, in particular those at 12-13 \AA, at 15-16 \AA\ and in the 20-22 \AA\ range (Fig. \ref{RGS_xabs_warm}). At the current data quality these residuals can be fitted equally well with multiple wind models. Since multiple wind solutions fit the RGS residuals similarly well, we cannot identify a unique solution with the present data. We find that none of the tentative peaks fit the remaining residuals seen in the iron K band, which could instead be interpreted as our imperfect fitting of the relativistic reflection with just the phenomenological Laor model (omitting the reflection continuum).

The large abundance of peaks with $\Delta$C-stat\ of around 15-20 however still suggests a very likely presence of a multi-phase outflow in PG 1448+273 because such strong false detections should occur rarely by chance. Based on our MC simulations, even accounting for their limited parameter space, we can very roughly assign a 2$\sigma$ statistical significance (including the look-elsewhere effect) to a $\Delta$C-stat=16 outflow detection. Such strong fake detections should therefore only occur in 5 per cent of observations, suggesting that the secondary peaks are real. At this moment, unfortunately, identification of these potential components is just beyond the possibilities of the current data quality.

\section{Archival short \textit{XMM-Newton}\ observation}
\label{oldobs}

We apply the same continuum model as used in Sect. \ref{continuummodelling} to the second, archival dataset. Due to lower data quality, we fix the blurred iron K line parameters to the best-fitting parameters from the final fit in Sect. \ref{Sec_xabs_warm}, which included the broadband continuum as well as the ultrafast and warm absorption. We find that the coronal powerlaw slope is much higher with $\Gamma = 2.38 \pm 0.04$, and the average 2-10 keV luminosity is $(2.16 \pm 0.07) \times 10^{43}$ erg/s.

Since the statistics are much worse, the addition of neutral absorption intrinsic to the AGN is not significant. The best-fitting spectrum is shown in Fig. \ref{PN_old} (EPIC data) and Fig. \ref{RGS_old} (RGS data). The data are of lower quality than from the other observation, but we do not notice any strong absorption features at 7 keV or at 18 \AA\ as seen in the previous dataset, or any other absorption signature of a UFO. Interestingly, it appears that an emission feature is present at 18 \AA\ instead (Fig. \ref{RGS_old}). If the feature is real, it could be due to \ion{Fe}{XVII} emission (rest-frame wavelength 17.1 \AA) or blueshifted \ion{O}{VIII} emission (rest-frame wavelength 19.0 \AA).

\begin{figure*}
	\includegraphics[width=0.995\textwidth]{PG1448_015_EPIC_v2.pdf}
    \caption{EPIC pn (left subplots) and MOS (right subplots) data ($1.7-10$ keV) of the second, short observation of PG 1448+273 fitted with the baseline continuum model from Appendix \ref{oldobs} (blue colour). The model in red colour is the best-fitting baseline continuum plus one UFO component with the parameters of the outflow detected in the long \textit{XMM-Newton}\ observation (Sect. \ref{Sec_xabs_warm}). The top subplots show the fitted spectra, the bottom subplots contain the residuals. Green labels name the strongest absorption lines of the UFO absorption.}
    \label{PN_old}
\end{figure*}

\begin{figure*}
	\includegraphics[width=\textwidth]{PG1448_015_RGS_v2.pdf}
    \caption{RGS data between 7.5 \AA\ (1.7 keV) and 28 \AA\ (0.4 keV) of the second, short observation of PG 1448+273, stacked and overbinned for plotting purposes only, fitted with the baseline continuum model from Appendix \ref{oldobs} (blue colour). The model in red colour is the best-fitting baseline continuum plus one UFO component with the parameters of the outflow detected in the long \textit{XMM-Newton}\ observation (Sect. \ref{Sec_xabs_warm}). The top subplot shows the fitted spectra, the bottom subplot contains the residuals. Green labels name the strongest absorption lines of the UFO absorption.}
    \label{RGS_old}
\end{figure*}

As a further check, we add the same UFO absorption as observed in the long exposure dataset to the baseline continuum of the short observation. We use the best-fitting UFO parameters from Sect. \ref{Sec_xabs_warm}. The resulting spectral model is shown in Fig. \ref{PN_old} and \ref{RGS_old} (red colour), and is clearly rejected by the data. The 1$\sigma$ upper limit on the column density of such absorber is just $1.7 \times 10^{22}$ cm$^{-2}$, more than 10 times lower than the column density observed in the other \textit{XMM-Newton}\ observation. We cannot exclude a possibility that a UFO of the same column density is present, but with a higher ionisation parameter. However, the change in the ionisation parameter would have to be significant, with a lower limit on the ionisation parameter of about $\log (\xi / \textrm{erg cm s}^{-1})\approx 5$. Such a large change in the ionisation parameter ($10\times$ higher) would require a non-linear response between the absorber and the ionising luminosity of PG 1448+273.

Finally, we apply the same systematic wind search from Sect. \ref{wind_scan} to locate any potential absorption with parameters different to that of the UFO observed in the other observation. As the data quality is much lower compared to the first exposure and there are no strong spectral residuals, we do not find any significant detections ($\Delta$C-stat\ $> 15$).

\section{Discussion}
\label{discussion}

We have performed a systematic search for an ionised outflow in the X-ray spectrum of the Narrow Line Seyfert 1 AGN PG 1448+273. We significantly detect a highly ionised wind with a velocity of $(0.090 \pm 0.002)$c.

The absorption lines of the outflow are detected in both hard X-rays (iron K band) and soft X-rays (0.3-2.0 keV, particularly the \ion{O}{VIII} line). This suggests the wind is not as extremely ionised as in other AGN and X-ray binaries where often only \ion{Fe}{XXV/XXVI} features are observed \citep[e.g.][]{Tombesi+10b, Ponti+12}. This fact simplifies the inference of the wind parameters as it reduces the uncertainties in both the velocity and ionisation state of the outflow while simultaneously modelling the wind features as well as the iron K reflection.

The velocity of the outflow of $\sim$0.1c is also not as high as seen in many other AGN \citep[for a review of the recent UFO detections and their velocities see][]{Parker+18}, where velocities as high as 0.3c-0.5c were detected \citep[e.g.][are the most extreme examples]{Reeves+18b, Walton+19}, although velocities in the range of $0.05-0.1c$ have also been detected in some AGN \citep[e.g.][]{Pounds+16a}. Similarly, the velocity width of the observed features is not large at $2100_{-500}^{+600}$ km/s. This suggests that the AGN corona ionising the atoms within the outflow cannot be either too large or too close to where the wind absorption occurs as otherwise the toroidal motion within the wind (imprinted to the wind by the Keplerian rotation of the accretion disc) would broaden the absorption features \citep{Fukumura+19}.

We also note that the outflow velocity is lower than that of the UFO observed in IRAS 13224$-$3809 \citep[0.25c,][]{Parker+16}. At the same time, the inclination of PG 1448+273 ($\sim40^{\circ}$) is lower than the inclination of IRAS 13224$-$3809 \citep[about $70^{\circ}$,][]{Jiang+18}. A lower UFO velocity in PG 1448+273 would be expected in the framework of a model where the UFO absorption is produced in an atmosphere co-rotating with the accretion disc \citep{Fabian+20a} because the projected disc velocities are lower in PG 1448+273.

Curiously, the UFO is not detected in the short (20 ks) 2003 \textit{XMM-Newton}\ observation of PG 1448+273 (Appendix \ref{oldobs}). The dataset is good enough to significantly reject a UFO of parameters similar to that observed in the main dataset. It is possible that the UFO is still present but at much higher ionisation or with a much lower column density. A change in the outflow velocity alone cannot explain the disappearance of absorption lines from both EPIC and particularly RGS data. Interestingly, PG 1448+273 shows a higher flux during the shorter observation when the UFO is not seen. This is a similar behaviour to that of IRAS 13224$-$3809, which hosts a time-variable UFO, which anti-correlates in absorption strength with the AGN X-ray flux \citep{Parker+16, Pinto+18}. Further long exposures with \textit{XMM-Newton}\ will be required for a proper flux-resolved analysis to confirm the existence of this important trend in PG 1448+273.

We also detect a weak warm absorber in PG 1448+273. Warm absorbers were detected in multiple AGN with UFO detections, including PG 1211+143 \citep{Reeves+18a}, PG 1114+445 \citep{Serafinelli+19}, IRAS 17020+4544 \citep{Leighly+97, Longinotti+15} and IRAS 13349+2438 \citep{Sako+01, Parker+18b}. Comparing their energetics and mass outflow rates can be helpful in explaining their (possibly common) origin \citep{Tombesi+13} and their interaction with the surrounding environment. Unfortunately, the warm absorber in PG 1448+273 has a very small outflow velocity of $100 \pm 200$ km/s, consistent with being at rest. The limits placed on its energetics are therefore not useful.

We search the X-ray spectrum of PG 1448+273 for further wind structure as multiphase ultra-fast outflows were previously detected in the X-ray spectra of other similar systems such as PG 1211+143 \citep[two outflowing components at 0.06c and 0.13c,][]{Pounds+16a, Pounds+16b, Kriss+18}, 1H 0707$-$495 \citep[0.03c and 0.13c,][]{Kosec+18c} and PDS 456 \citep[$\sim$0.25c and $\sim$0.5c,][]{Boissay-Malaquin+19}. Evidence for even more complex wind structure was found in PG 1114+445 \citep{Serafinelli+19} and in IRAS 17020+4544 \citep{Longinotti+15}. The energetics of the different outflow phases can be used to infer the mode of interaction of the outflow with the surroundings of the AGN. \citet{Kosec+18c} found that three individual outflowing components (including one observed in the UV band) likely have comparable kinetic powers in the NLS1 galaxy 1H 0707$-$495, suggesting energy conservation at these scales.

Our second automated wind search locates a number of tentative outflow phases with a broad range of physical parameters, which would be difficult to identify with manual approach. However, the current data quality prevents us from claiming a significant detection of a multi-phase outflow in the present spectra. Future dedicated campaigns with \textit{XMM-Newton}\ or with future X-ray observatories, doubling or tripling the total count statistics on PG 1448+273, would be able to constrain these features or rule them out as false detections.

Assuming our measurement of the ionisation parameter of the outflowing material is accurate, we can calculate the kinetic power of the detected fast wind. Following the standard derivations \citep[e.g.][]{Gofford+15}, the mass outflow rate of a disc wind can be expressed as:

\begin{equation}
\dot{M}_{\textrm{out}} = \rho A v = ( C_{\textrm{V}} \mu m_{\textrm{p}} n ) ( \frac{\Omega}{4 \pi} 4 \pi R^{2} ) v
\end{equation}

\noindent where $\rho$ is the gas density and $A$ is the surface area into which the outflow is launched, $v$ is the outflow velocity, $C_{\textrm{V}}$ is the volume filling factor, $\mu$ defines the mean atomic mass ($\sim1.2$ assuming solar abundances), $m_{\textrm{p}}$ the proton mass, $n$ the ion density in the wind. $\frac{\Omega}{4 \pi}$ is the solid angle into which the wind is launched as a fraction of $4 \pi$ and $R$ is the distance of the outflow from the ionising source. The ionising parameter of the wind can be used to get rid of the $R$ and $n$ terms in the equation above since $\xi = \frac{L_{\textrm{ion}}}{nR^2}$, where $L_{\textrm{ion}}$ is the ionising luminosity of the AGN. The mass outflow rate can thus be expressed as:

\begin{equation}
\dot{M}_{\textrm{out}}=C_{\textrm{V}} \Omega \mu m_{\textrm{p}} v \frac{L_{\textrm{ion}}}{\xi}
\end{equation}

\noindent The kinetic power of the outflow is then:

\begin{equation}
\dot{E}_{\textrm{K}}=\frac{1}{2} \dot{M}_{\textrm{out}} v^{2} = \frac{1}{2} C_{\textrm{V}} \Omega \mu m_{\textrm{p}} v^{3} \frac{L_{\textrm{ion}}}{\xi}
\end{equation}

\noindent Using the results from the best-fitting two absorber model in Sect. \ref{Sec_xabs_warm} ($v=26900 \pm 600$ km/s,  $\log (\xi / \textrm{erg cm s}^{-1})=4.03_{-0.08}^{+0.10}$), we find:

\begin{equation}
\dot{E}_{\textrm{K}}= (1.8 \pm 0.5) ~ C_{\textrm{V}} \Omega L_{\textrm{ion}}
\end{equation}

\noindent Even if the clumping factor and the outflow solid angle are relatively small (unlikely as PG 1448+273 is not an edge-on AGN and yet we observe the wind), the kinetic power of the outflow can be comparable to its ionising (13.6 eV $-~13.6$ keV) luminosity. The wind is thus likely more than capable to drive AGN feedback in the host galaxy of PG 1448+273 \citep{diMatteo+05,King+10,Costa+14}. Its energy budget is as large as the UFOs detected in similar NLS1s such as 1H 0707$-$495 \citep{Kosec+18c} and IRAS 13224$-$3809 \citep{Parker+16}.

\section{Conclusions}
\label{conclusions}

Here we summarize our results:

\begin{itemize}
\item We systematically searched the X-ray spectrum of the Narrow Line Seyfert 1 galaxy PG 1448+273 for signatures of ionised outflows. We unambiguously detect an ultra-fast outflow with a velocity of $26900 \pm 600$ km/s $(0.090^{+0.002}_{-0.002}$c). Our Monte Carlo simulations of the systematic wind search on simulated data show that the UFO detection is highly statistically significant.

\item The wind material has an ionisation parameter of $\log (\xi / \textrm{erg cm s}^{-1})=4.03_{-0.08}^{+0.10}$ and a column density $2.8_{-0.7}^{+1.2} \times 10^{23}$ cm$^{-2}$. The ionisation of the material is similar to that of outflows found in other AGN. The projected wind velocity is on the lower end of the range  of the extreme outflows observed in other AGN.

\item The spectral features of the ionised outflow are observed in both hard X-rays (iron K band) with EPIC pn and MOS detectors, and in the soft X-ray band with RGS gratings onboard the \textit{XMM-Newton}\ observatory.

\item Our results underline the importance of the soft X-ray band in detecting high-velocity highly ionised outflows. Inferring the outflow properties from just the iron K band (7-8 keV) is often problematic due to the common presence of the blue wing of the relativistically blurred iron K reflection at similar energies \citep{Zoghbi+15}, as well as due to confusion between the \ion{Fe}{XXV} and \ion{Fe}{XXVI} absorption lines of the ionised outflow.

\item A UFO of similar parameters is not detected in a second, shorter (20ks) observation of PG 1448+273 during which the AGN was in a higher X-ray flux state. The outflow could still be present in the X-ray spectrum, but at a much higher ionisation or with a significantly lower column density. The UFO disappearance at higher fluxes suggests similarities between PG 1448+273 and IRAS 13224$-$3809, where the wind absorption anti-correlates with the X-ray flux \citep{Parker+16}.

\item We also detect a low ionisation warm absorber in soft X-rays. It is consistent with being at rest and has a column density of $(3.8 \pm 1.7) \times 10^{20}$ cm$^{-2}$\ and an ionisation parameter of $\log (\xi / \textrm{erg cm s}^{-1})=-0.4_{-0.4}^{+0.3}$.

\item We perform a systematic search for further ionisation and velocity phases of the outflow and tentatively detect one or more phases with potential outflow velocities in the range between 20000 and 90000 km/s. However, with the present data quality we are unable to significantly confirm or reject their presence in the spectrum of PG 1448+273.

\end{itemize}

\section*{Acknowledgements}

We are grateful to the anonymous referee for useful comments which improved the clarity and quality of the manuscript. P.K. acknowledges support from the Science and Technology Facilities Council. A.Z. acknowledges support by NASA under Grant No. NNX17AD60G issued through the XMM-Newton GO program. D.J.W. acknowledges support from an STFC Ernest Rutherford Fellowship. C.S.R. thanks the UK Science and Technology Facilities Council for support under the New Applicant grant ST/R000867/1, and the European Research Council for support under the European Union's Horizon 2020 research and innovation programme (grant 834203). This work is based on observations obtained with XMM-Newton, an ESA science mission funded by ESA Member States and USA (NASA). This research has made use of the NASA/IPAC Extragalactic Database (NED), which is funded by the National Aeronautics and Space Administration and operated by the California Institute of Technology.

\bibliographystyle{mnras}