text:
\section{Introduction}

Iron-based pnictides and chalcogenides attract much attention due to the presence of unconventional superconducting state and peculiar normal state properties \cite{SadovskiPhUsp2008, IzyumovReview2008, IvanovskiiReview2008, PaglioneReview, MazinReview, WenReview, StewartReview, Hirschfeld2011}. Electronic structure of these Fe-based high-$T_c$ superconductors (FeBS) demonstrate several features putting them apart from other conventional and unconventional superconductors. First of all, most of FeBS have similar topology of the Fermi surface that consists of two or three hole pockets centered at $(0,0)$ point and two electron pockets centered at $(\pi,\pi)$ point of the Brillouin zone corresponding to one Fe per unit cell. Exceptions are the cases of the extreme electron or hole doping, where only one group of sheets remains. Second important feature is that all five iron $3d$ orbitals contribute to the formation of the Fermi surface and even a single Fermi pocket is formed by several $d$-orbitals. Thus the system can only be described within a multiorbital and, respectively, multiband model. Minimal model would be a two-band model, such as suggested in Refs.~\cite{Raghu, Efremov2011, KorshunovPhUsp2016}. The third interesting feature is connected with the unconventional superconductivity. While, in general, there are some controversy in  discussion  of the gap symmetry in unconventional superconductors, growing   evidences  in  iron-based  materials  support  sign-changing gap. The structure of the Fermi surface provides practically ideal nesting at the antiferromagnetic wave vector $\mathbf{Q}=(\pi,\pi)$. Long-range spin-density wave antiferromagnetic state is destroyed by doping, however, the enhanced spin fluctuations remain. These fluctuations are claimed to be the main source of the Cooper pairing thus contributing to the formation of the superconducting ground state \cite{Mazin2010, Hirschfeld2011, KorshunovPhUsp2014}. Since spin fluctuations with the large wave vector $\sim\mathbf{Q}$ lead to the repulsive interband Cooper interaction, the gap function have to possess the momentum dependence and change sign between the hole and the electron Fermi surface pockets to compensate the sign of the interaction in the gap equation. The simplest solution is called the $s_\pm$ state and corresponds to the $A_{1g}$ representation with the gap having one sign at hole pockets and opposite sign at electron pockets. At the same time, bands near the Fermi level have a mixed orbital character. Therefore, orbital fluctuations enhanced by, for example, electron-phonon interaction may also lead to a superconductivity \cite{Kontani2010, Onari2012, Yamakawa2017}. These two mechanisms of superconducting pairing differs by the dominating superconducting gap structure: sign-changing $s_{\pm}$ state for the spin fluctuations and sign-preserving $s_{++}$ state for the orbital fluctuations. Alas, superconducting gap structure has not been convincingly determined in experiments yet. Nevertheless, series of experimental observation such as the spin resonance peak in inelastic neutron scattering \cite{Maier2008, Korshunov2008, Christianson2008, Inosov2009}, a quasiparticle interference in tunneling experiments \cite{Wang2009, Gonnelli2009, Szab2009, Zhang2012}, the NMR spin-lattice relaxation rate \cite{Nakai2009, Fukazawa2009}, and the temperature dependence of the penetration-depth \cite{Ghigo2017, GhigoPRB2017, Teknowijoyo2018} are conveniently explained assuming the $s_{\pm}$ state.

Scattering on nonmagnetic impurities has different effect on superconductors with different gap symmetries and structures. The pure attractive interaction, both in the intraband and interband channels, results in the $s_{++}$ state. Its reaction to the disorder is well known since the earlier studies of the conventional $s$-wave superconductivity \cite{Anderson1959, Morosov1979} and the modern treatment of the $s_{++}$ state within the multiband models \cite{Onari2009}. According to the so-called Anderson's theorem \cite{Anderson1959}, in a single-band $s$-wave superconductor, a nonmagnetic disorder does not affect the superconducting critical temperature $T_c$. In the case of unconventional superconductors, Anderson's theorem is violated and the critical temperature is suppressed by a nonmagnetic disorder \cite{Golubov1997} similar to the $T_c$ suppression according to the Abrikosov-Gor'kov theory for magnetic impurities \cite{AbricosovGorkovJETP}. However, series of experiments on FeBS reveals that this suppression is less intensive then predicted by the Abrikosov-Gor'kov theory \cite{Karkin2009, Cheng2010, Li2010, Nakajima2010, Prozorov2014}. One possible reason is that the $s_{\pm}$ state transforms to the $s_{++}$ state at some critical value of the impurity scattering rate $\Gamma^\mathrm{crit}$ \cite{Efremov2011, Yao2012, Chen2013, KorshunovPhUsp2016}. The latter is proportional to the impurity potential and the concentration of impurities $n_\mathrm{imp}$. The transition occurs when one of the gaps, the smaller one, decreases and then changes its sign going through zero. By and large, this happens due to the presence of the subdominant attractive intraband interactions besides the dominant interband repulsion resulting in the $s_\pm$ state in the clean case \cite{KorshunovPhUsp2016}.

In our previous studies \cite{Efremov2011, KorshunovMagn2014, KorshunovPhUsp2016, Shestakov2018}, we have discussed mostly the low temperature behavior of the superconducting gaps. Here we study the effect of nonmagnetic disorder on the two-band $s_\pm$ superconductor for a wide temperature range up to $T_c$. We show that there is a range of $\Gamma$'s near $\Gamma^\mathrm{crit}$, where  although at low temperatures the smaller gap changed its sign ($s_{++}$ state), at higher temperatures the sign is changed back and the system again becomes of $s_\pm$ type staying in this state up to $T_c$.

\section{Model and Method}

Here we use the same two-band model as in Refs.~\cite{Efremov2011, KorshunovPhUsp2016, Shestakov2018} with the following Hamiltonian,

\begin{equation}
H = \sum_{{\mathbf{k}}, \alpha, \sigma}\xi_{{\mathbf{k}},\alpha}c_{{\mathbf{k}}\alpha\sigma}^{\dagger}c_{{\mathbf{k}}\alpha\sigma} + \sum_{\mathbf{R}_i, \sigma, \alpha, \beta}{\mathcal{U}_{\mathbf{R}_i}^{\alpha \beta}c_{\mathbf{R}_i\alpha\sigma}^{\dagger}c_{\mathbf{R}_i\beta\sigma} } + H_{SC}, \label{twoBandHimp}
\end{equation}
where operator $c_{{\mathbf{k}}\alpha\sigma}^{\dagger}$ ($c_{{\mathbf{k}}\alpha\sigma}$) creates (annihilates) electron with momentum ${\mathbf{k}}$, spin $\sigma$, and band index $\alpha = a,b$, $\xi_{{\mathbf{k}},\alpha} = \mathbf{v}_{F \alpha}({\mathbf{k}} - {\mathbf{k}}_{F \alpha})$ is the quasiparticles' dispersion linearized near the Fermi level with ${\mathbf{v}}_{F\alpha}$ and ${{\mathbf{k}}}_{F\alpha}$ being the Fermi velocity and the Fermi momentum of the band $\alpha$, respectively. Hamiltonian contains the term with the impurity potential $\mathcal{U}$ at site ${\mathbf{R}}_i$ and the term responsible for superconductivity, $H_{SC}$. The exact form of the latter is not important for the present discussion but we assume that it provides superconducting pairing due to the exchange of spin fluctuations (repulsive interaction) and electron-phonon coupling (attractive interaction), see details in Ref.~\cite{KorshunovPhUsp2016}.

To describe the effect of disorder, we use the Eliashberg approach for multiband superconductors \cite{AllenMitrovic}. The connection between the full Green's function $\hat{\mathbf{G}}({\mathbf{k}}, \omega_n)$, the self-energy matrix $\hat{\Sigma}({\mathbf{k}},\omega_n)$, and the `bare' Green's function,

\begin{equation}
\hat{\mathrm{G}}^{(0)\alpha\beta}({\mathbf{k}}, \omega_n) = \left( i\omega_n\hat{\tau}_0 \otimes \hat{\sigma}_0 - \xi_{{\mathbf{k}} \alpha}\hat{\tau}_3 \otimes \hat{\sigma}_0 \right)^{-1} \delta_{\alpha \beta}, \label{G02band}
\end{equation}
is established via the Dyson equation, $\hat{\mathbf{G}}({\mathbf{k}}, \omega_n) = \left( \hat{\mathbf{G}}^{(0)-1}({\mathbf{k}}, \omega_n) - \hat{\mathbf{\Sigma}}({\mathbf{k}}, \omega_n) \right)^{-1}$, where $\omega_n = (2n+1)\pi T$ is Matsubara frequency. Green's function is the matrix in the band space (denoted by bold face) and combined Nambu and spin spaces (denoted by hat). The Pauli matrices $\hat{\tau}_i$ and $\hat{\sigma}_i$ refers to Nambu and spin spaces, respectively.

Assuming the isotropic superconducting gaps at the Fermi surface sheets, we set the impurity self-energy to be independent on the momentum ${\mathbf{k}}$ but preserving the dependence on the frequency and the band indices,

\begin{equation}
\hat{\mathbf{\Sigma}}(\omega_n) = \sum_{i = 0}^{3}{ \Sigma_{(i)\alpha \beta}(\omega_n)\hat{\tau}_i }.
\end{equation}
Therefore the task is simplified by averaging over ${\mathbf{k}}$. As a result, all equations are written in terms of the quasiclassical $\xi$-integrated Green's functions,

\begin{equation}
\hat{\mathbf{g}}(\omega_n) = \int{ d\xi({\mathbf{k}}) \hat{\mathrm{G}}({\mathbf{k}}, \omega_n) } = \left( \begin{matrix} \hat{\mathrm{g}}_{an} & 0 \\ 0 & \hat{\mathrm{g}}_{bn} \end{matrix} \right), \label{xiIntegrated}
\end{equation}
where

\begin{equation}
\hat{\mathrm{g}}_{\alpha n} = g_{0\alpha n}\hat{\tau}_0 \otimes \hat{\sigma}_0 + g_{2\alpha n}\hat{\tau}_2 \otimes \hat{\sigma}_2, \label{gAlpha}
\end{equation}
Here $g_{0\alpha n}$ and $g_{2\alpha n}$ are the normal and anomalous (Gor'kov) $\xi$-integrated Green's functions in the Nambu representation,

\begin{align}
&g_{0 \alpha n} = -\frac{ {\mathrm{i}} \pi N_{\alpha} \tilde{\omega}_{\alpha n} }{ \sqrt{\tilde{\omega}_{\alpha n}^2 + \tilde{\phi}_{\alpha n}^2} }, &g_{2 \alpha n} = -\frac{ \pi N_{\alpha} \tilde{\phi}_{\alpha n} }{ \sqrt{\tilde{\omega}_{\alpha n}^2 + \tilde{\phi}_{\alpha n}^2} }. \label{g0g2}
\end{align}
They depend on the density of states per spin at the Fermi level of the corresponding band ($N_{a,b}$), and on the order parameter~$\tilde{\phi}_{\alpha n}$ and frequency $\tilde{\omega}_{\alpha n}$. Both the order parameter and the frequency are renormalized by the self-energy,

\begin{align}
&{\mathrm{i}}\tilde{\omega}_{\alpha n} = {\mathrm{i}}\omega_n - \Sigma_{0 \alpha}(\omega_n) - \Sigma_{0 \alpha}^\mathrm{imp}(\omega_n), \label{tildeOmega} \\
&\tilde{\phi}_{\alpha n} = \Sigma_{2 \alpha}(\omega_n) + \Sigma_{2 \alpha}^\mathrm{imp}(\omega_n), \label{tildePhi}
\end{align}
where $\Sigma_{0(2)\alpha}$ and $\Sigma_{0(2)\alpha}^\mathrm{imp}$ are the parts of the self-energy coming from the pairing interaction (spin fluctuations, electron-phonon coupling, etc.) and impurity scattering, respectively. Impurity scattering will be considered in the $s$-wave channel only. Treating a more complicated impurity potential is a separate cumbersome task, although this goal was pursued by some authors~\cite{Hoyer2015,Scheurer2015,Pogorelov2018}. The order parameter $\tilde{\phi}_{\alpha n}$ is connected with the gap function $\Delta_{\alpha n}$ via the renormalization factor $Z_{\alpha n} = \tilde{\omega}_{\alpha n} / \omega_{n}$, i.e.

\begin{equation}
 \Delta_{\alpha n} = \tilde{\phi}_{\alpha n} / Z_{\alpha n}.
 \label{eq:Delta}
\end{equation}

The part of the self-energy due to the pairing interaction can be written as

\begin{align}
&\Sigma_{0\alpha}(\omega_n) = T\sum_{n',\beta}{ \lambda_{\alpha \beta}^{Z}(n-n') \frac{g_{0\beta n'}}{N_{\beta}} }, \label{Sigma0} \\
&\Sigma_{2\alpha}(\omega_n) = -T\sum_{n',\beta}{ \lambda_{\alpha \beta}^{\phi}(n-n') \frac{g_{2\beta n'}}{N_{\beta}} }, \label{Sigma2}
\end{align}
where $\lambda_{\alpha \beta}^{\phi,Z}(n-n')$ is a coupling function,

\begin{equation}
\lambda_{\alpha \beta}^{\phi,Z}(n-n') = 2\lambda_{\alpha \beta}^{\phi,Z}\int_{0}^{\infty}{ d\Omega \frac{ \Omega B(\Omega) }{ \left( \omega_n - \omega_{n'} \right)^2 + \Omega^2 } },
\end{equation}
that is defined by coupling constants $\lambda_{\alpha \beta}^{\phi,Z}$ and includes density of states $N_{\beta}$ and the normalized bosonic spectral function $B(\Omega)$. The matrix elements $\lambda_{\alpha \beta}^{\phi}$ may be positive (attractive) or negative (repulsive) due to Coulomb repulsion, spin fluctuations and electron-phonon interaction, while matrix elements $\lambda_{\alpha \beta}^{Z}$ are always positive. For simplicity we set $\lambda_{\alpha \beta}^{Z} = \left| \lambda_{\alpha \beta}^{\phi} \right| \equiv \left| \lambda_{\alpha \beta} \right|$.

To calculate the impurity part of the self-energy $\hat{\mathbf{\Sigma}}^\mathrm{imp}$, we use noncrossing diagrammatic approximation that is equivalent to the $\mathcal{T}$-matrix approximation,

\begin{equation}
\hat{\mathbf{\Sigma}}^\mathrm{imp}(\omega_n) = n_\mathrm{imp}\hat{\mathbf{U}} + \hat{\mathbf{U}}\hat{\mathbf{g}}(\omega_n)\hat{\mathbf{\Sigma}}^\mathrm{imp}(\omega_n), \label{Tmatrix}
\end{equation}
where $\hat{\mathbf{U}} = \mathbf{U} \otimes \hat{\tau}_3$ is the matrix of impurity potential, $\left( \mathbf{U} \right)_{\alpha \beta} = \mathcal{U}_{\mathbf{R}_i}^{\alpha \beta}$. Without loss of generality we set $\mathbf{R}_i = 0$. Intra- and interband parts of the impurity potential are given as $v$ and $u$ respectively, $\left( \mathbf{U} \right)_{\alpha \beta} = \left( v - u \right)\delta_{\alpha \beta} + u$. The relation between intra- and interband impurity scattering is set by a parameter $\eta = v / u$.

It is convenient to introduce the generalized cross-section parameter

\begin{equation}
\sigma = \frac{\pi^2 N_a N_b u^2}{1 + \pi^2 N_a N_b u^2} \to \begin{cases} 0, & \mbox{Born limit}, \\ 1, & \mbox{unitary limit} \end{cases}
\end{equation}
and the impurity scattering rate

\begin{equation}
\Gamma_{a(b)} = 2 n_\mathrm{imp} \pi N_{b(a)} u^2 \left(1-\sigma\right) = \frac{2 n_\mathrm{imp} \sigma}{\pi N_{a(b)}} \to  \begin{cases} 2n_\mathrm{imp}\pi N_{b(a)} u^2, \mbox{Born limit}, \\ \dfrac{2n_\mathrm{imp}}{\pi N_{a(b)}}, \mbox{unitary limit}. \end{cases} \label{GammaAB}
\end{equation}

Both are shown to have special values in the limiting cases: (i) the Born limit corresponding to the weak impurity potential ($\pi u N_{a(b)} \ll 1$), and (ii) the unitary limit corresponding to the strong impurity potential ($\pi u N_{a(b)} \gg 1$).

Equations presented above are general and can be applied to a wide range of multiband systems. Below we concentrate on iron-based superconductors within the minimal two-band model. Bands are labelled as $a$ and $b$. Spin fluctuation spectrum enters through the bosonic spectral function $B(\Omega)$ \cite{Parker2008, Popovich2010, Charnukha2011}; the superconducting interaction is also determined by the choice of the coupling constants $\lambda_{\alpha \beta}$.

\section{Results and Discussions}

For the calculations, we choose the ratio between densities of states of different bands as $N_{b}/N_{a} = 2$ and set the values of the coupling constants to be ($\lambda_{aa};\lambda_{ab};\lambda_{ba};\lambda_{bb}$) $=$ ($3;-0.2;-0.1;0.5$), so the averaged coupling constant is positive, $\langle \lambda \rangle = \left( \lambda_{aa} + \lambda_{ab} \right)N_{a}/N + \left( \lambda_{ba} + \lambda_{bb} \right)N_{b}/N > 0$, where $N = N_a+N_b$. This set of parameters leads to the $s_{\pm}$ superconducting state with unequal gaps -- larger positive (band $a$) and smaller negative (band $b$) gap -- and the critical temperature $T_{c0} = 40$ K in the clean limit.

We present the smaller gap $\Delta_{b,n}$ for the lowest Matsubara frequency, $n=0$, as a function of the scattering rate $\Gamma_a$ and the temperature $T$ in two different cases: (i) weak scattering with $\sigma = 0$ (Born limit), see Figure~\ref{fig:DeltaGammaTsigma0}, and (ii) intermediate scattering limit with $\sigma = 0.5$, see Figure~\ref{fig:DeltaGammaTsigma05}. The larger gap, $\Delta_{a,n}$, is always positive and, therefore, there is a line of $s_\pm \to s_{++}$ transition corresponding to the change of $\Delta_{b,n}$ sign from negative to positive. The transition goes through the gapless state with the finite larger gap and the vanishing smaller gap \cite{Efremov2011}. Note that the line of transition is not vertical in Figures~\ref{fig:DeltaGammaTsigma0} and~\ref{fig:DeltaGammaTsigma05}. Therefore, there is no single critical scattering rate but a temperature-dependent $\Gamma_a^\mathrm{crit}(T)$. Moreover, if we stay at a fixed $\Gamma_a$ in the range $1.1T_{c0} < \Gamma_a < 1.6T_{c0}$ for $\sigma=0$ or $1.5T_{c0} < \Gamma_a < 3.2T_{c0}$ for $\sigma = 0.5$ and increase the temperature, we observe an interesting behavior of the gap. Namely, while at low temperatures the transition to the $s_{++}$ state already took place ($\Delta_{b,n} > 0$), at higher temperatures the system goes back to the $s_\pm$ state ($\Delta_{b,n} < 0$). Thus, there is a temperature-dependent $s_{++} \to s_\pm$ transition. With the increasing $\Gamma_a$, the temperature of this transition is shifted to $T_c$ and the system becomes $s_{++}$ for the whole temperature range.

\begin{figure}[H]
\centering
\includegraphics[width=0.7\linewidth]{Gamma_T_dep_sigma00}
\caption{Dependence of the lowest-frequency Matsubara gap function $\Delta_{b,n=0}$, indicated by the color code, for the band $b$ on the scattering rate $\Gamma_a$ and the temperature $T$ in the Born limit, $\sigma = 0.0$. All quantities are normalized by $T_{c0}$. Green color marks the state with the vanishingly small gap, $\Delta_{b,n}<10^{-3} T_{c0}$.}
\label{fig:DeltaGammaTsigma0}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=0.7\linewidth]{Gamma_T_dep_sigma05}
\caption{Dependence of the lowest-frequency Matsubara gap function $\Delta_{b,n=0}$, indicated by the color code, for the band $b$ on $\Gamma_a$ and $T$ in the intermediate scattering limit, $\sigma = 0.5$. All quantities are normalized by $T_{c0}$. Green color marks the state with the vanishingly small gap, $\Delta_{b,n}<10^{-3} T_{c0}$.}
\label{fig:DeltaGammaTsigma05}
\end{figure}

To illustrate the mentioned points, in Figures~\ref{fig:DeltaTsigma0} and~\ref{fig:phiZsigma0} we present results for the temperature dependence of the gap function $\Delta_{\alpha,n=0}$, order parameter $\tilde{\phi}_{\alpha,n=0}$, and the renormalization factor $Z_{\alpha,n=0}$ for several fixed values of $\Gamma_a$ for $\sigma=0$. Temperature dependencies for $\sigma=0.5$ are shown in Figures~\ref{fig:DeltaTsigma05} and~\ref{fig:phiZsigma05}. The gap in band $a$ has the same positive sign for all values of $\Gamma_a$ and vanishes at $T_c$, see Figures~\ref{fig:DeltaTsigma0}(a) and~\ref{fig:DeltaTsigma05}(a). There is, however, a small range of $\Gamma_a$ values around $\Gamma_a^\mathrm{crit}$, for which the gap is an increasing function of $T$ at low temperatures. At the same time, the order parameter behaves quite conventionally and decreases towards $T_c$, see Figures~\ref{fig:phiZsigma0}(a) and~\ref{fig:phiZsigma05}(a). The unusual temperature dependence of the gap at the lowest Matsubara frequency is due to the renormalization factor $Z_{\alpha,n=0}$ that is shown in Figures~\ref{fig:phiZsigma0}(b) and~\ref{fig:phiZsigma05}(b). That is, according to Equation~(\ref{eq:Delta}), large values of $Z_{\alpha,n}$ taking place at low temperatures lead to the decrease of $\Delta_{\alpha,n}$. For the band $b$ the same effect is not seen in the Born limit, but becomes pronounced in the intermediate scattering limit; compare the gap function in Figure~\ref{fig:DeltaTsigma05}(b) and the order parameter in Figure~\ref{fig:phiZsigma05}(c), and notice also the low-temperature behavior of $Z_{\alpha,n}$ in Figure~\ref{fig:phiZsigma05}(d). Similar behavior of gaps, in principle, may occur from the functional form of gaps at real frequencies, see Eq.~(7) in Ref. \cite{Mikhailovsky1991}.

\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{Delta_a_b_T_dep_sigma00}
\caption{Temperature dependence of the lowest-frequency Matsubara gap $\Delta_{\alpha,n=0}$ normalized by $T_{c0}$ for fixed values of $\Gamma_a$ in the Born limit with the band index $\alpha=a$ (\textbf{a}) and $\alpha=b$ (\textbf{b}).}
\label{fig:DeltaTsigma0}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{phi_Z_a_b_T_dep_rev_sigma00}
\caption{Temperature dependence of the lowest-frequency Matsubara order parameter $\tilde\phi_{\alpha,n=0}$ (\textbf{a}),(\textbf{c}) and the renormalization factor $Z_{\alpha,n=0}$ (\textbf{b}),(\textbf{d}), both normalized by $T_{c0}$, for fixed values of $\Gamma_a$ in the Born limit with the band index $\alpha=a$ (\textbf{a}),(\textbf{b}) and $\alpha=b$ (\textbf{c}),(\textbf{d}).}
\label{fig:phiZsigma0}
\end{figure}

In the clean limit and for the small $\Gamma_a$, the sign of the smaller gap $\Delta_{b,n}$ is negative at all temperatures, see Figures~\ref{fig:DeltaTsigma0}(b) and~\ref{fig:DeltaTsigma05}(b). With the increase of the impurity scattering rate, the gap at low temperatures changes sign while at higher temperatures the sign is either reversed again (small $\Gamma_a$) or the gap vanishes ($\Gamma_a \gtrsim 2 T_{c0}$). Therefore, the transition from the $s_\pm$ state to the $s_{++}$ state is characterized by two parameters, namely, the critical scattering rate $\Gamma_a^\mathrm{crit}$ and the critical temperature $T^\mathrm{crit}$. The latter changes from zero to $T_c$. Thus the $s_{++}$ state becomes dominant in the initially clean $s_\pm$ system for $\Gamma_a > \Gamma_a^\mathrm{crit}$ and $T < T^\mathrm{crit}$. This is true in both Born limit and the intermediate scattering limit. Also, we have checked that the similar behavior holds for the higher Matsubara frequencies, see Figure~\ref{fig:DeltaTsigma05n1n10} for the illustration of the gaps behavior for $n=1$ and $n=10$.

\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{Delta_a_b_T_dep_sigma05}
\caption{Temperature dependence of the lowest-frequency Matsubara gap $\Delta_{\alpha,n=0}$ normalized by $T_{c0}$ for fixed values of $\Gamma_a$ in the intermediate scattering limit ($\sigma=0.5$) with the band index $\alpha=a$ (\textbf{a}) and $\alpha=b$ (\textbf{b}).}
\label{fig:DeltaTsigma05}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{phi_Z_a_b_T_dep_rev_sigma05}
\caption{Temperature dependence of the lowest-frequency Matsubara order parameter $\tilde\phi_{\alpha,n=0}$ (\textbf{a}),(\textbf{c}) and the renormalization factor $Z_{\alpha,n=0}$ (\textbf{b}),(\textbf{d}), both normalized by $T_{c0}$, for fixed values of $\Gamma_a$ in the intermediate scattering limit ($\sigma=0.5$) with the band index $\alpha=a$ (\textbf{a}),(\textbf{b}) and $\alpha=b$ (\textbf{c}),(\textbf{d}).}
\label{fig:phiZsigma05}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{Delta_ab_n1n10_T_dep_sigma05}
\caption{Temperature dependence of the gap for higher Matsubara frequencies, $\Delta_{\alpha,n=1}$ (\textbf{a}) and $\Delta_{\alpha,n=10}$ (\textbf{b}), normalized by $T_{c0}$ for fixed values of $\Gamma_a$ in the intermediate scattering limit ($\sigma=0.5$). Gaps corresponding to the band index $a$ (band index $b$) are shown by dashed (solid) curves.}
\label{fig:DeltaTsigma05n1n10}
\end{figure}

Previously we have found a steep change in the smaller gap as a function of the scattering rate in the weak scattering limit \cite{Shestakov2018}. Here we observe that the same jump is present in the temperature dependence of the gap, see Figure~\ref{fig:DeltaTsigma0jump}, where $\Delta_{b,n}$ in the Born limit is shown. Discontinuous jump in the temperature dependence of the smaller gap appears at $T<0.1T_{c0}$.

\begin{figure}[H]
\centering
\includegraphics[width=0.6\linewidth]{Delta_b_step}
\caption{Temperature dependencies of lowest-frequency Matsubara gap $\Delta_{b,n=0}$ normalized by $T_{c0}$ in the Born limit for several values of $\Gamma_a$. Solid curves correspond to a smooth evolution of the gap in the $s_\pm$ state and across the $s_{++} \to s_\pm$ transition while the temperature dependencies with the discontinuous jump of the gap are shown by symbols.}
\label{fig:DeltaTsigma0jump}
\end{figure}

\section{Conclusions}

We considered the two-band model for FeBS that has the $s_\pm$ superconducting ground state in the clean limit. Here we studied dependence of the superconducting gaps $\Delta_{\alpha,n}$ on both the temperature and the nonmagnetic impurity scattering rate. We show that the disorder-induced transition from $s_\pm$ to $s_{++}$ state is temperature-dependent. That is, in a narrow region of scattering rates, while the ground state is $s_{++}$, it transforms back to the $s_\pm$ state at higher temperatures up to $T_c$. With the increasing impurity scattering rate, temperature of such a $s_{++} \to s_{\pm}$ transition shifts to the critical temperature $T_{c}$. The $s_\pm \to s_{++}$ transition is characterized by two parameters: (i) the critical scattering rate $\Gamma_a^\mathrm{crit}$ and (ii) the critical temperature $T^\mathrm{crit} \leq T_c$. The $s_{++}$ state becomes dominant in the initially clean $s_\pm$ system for $\Gamma_a > \Gamma_a^\mathrm{crit}$ and $T < T^\mathrm{crit}$.
The similar situation takes place for the case of the magnetic disorder where the $s_\pm$ state at low temperatures occurring due to the $s_{++} \to s_\pm$ transition \cite{KorshunovMagn2014}, at higher temperatures may transform back to the $s_{++}$ state.

Experimentally, one can observe the reentrant $s_{\pm}$ state by increasing the temperature for the fixed amount of disorder that results in the low-temperature $s_{++}$ state. For example, the spin resonance peak in the inelastic neutron scattering should be absent in the low-temperature $s_{++}$ state, but have to appear in the $s_{\pm}$ state at higher temperatures \cite{Maier2008, Korshunov2008, Hirschfeld2011, KorshunovPhUsp2014}. Temperature dependence of the penetration depth should also bear specific signatures of the gapless behavior accompanying the $s_{++} \to s_\pm$ transition \cite{KorshunovPhUsp2016}.

\vspace{6pt}

\authorcontributions{All author contributed equally.

\funding{This work was supported in part by the Russian Foundation for Basic Research (grant 16-02-00098), Presidium of RAS Program for the Fundamental Studies \#12, and ``BASIS'' Foundation for Development of Theoretical Physics and Mathematics. MMK acknowledges support by the Gosbudget program \# 0356-2017-0030.}

\conflictsofinterest{The authors declare no conflict of interest.}

\abbreviations{The following abbreviations are used in this manuscript:\\

\noindent
\begin{tabular}{@{}ll}
FeBS & Fe-based superconductors \\
NMR & Nuclear Magnetic Resonance
\end{tabular}}

\reftitle{References}

\externalbibliography{yes}

redpajama_set_name: RedPajamaArXiv


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Investing in our future
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InterCultural, NextGen
The Commission for Mission's InterCultural unit is asking congregations and individuals to help 30 young people from the synod's diverse ethnic communities travel to China for a contextual learning experience.
InterCultural director Rev Swee-Ann Koh said he hoped the trip would build relationships and respect.
"It will encourage participants to learn about our partner's (the China Christian Council) ministry context in order to discern God's mission in, and between, the peoples," he said.
A primary focus of the InterCultural unit is to work with 'Next Generation' (NextGen) youth in the CALD (Culturally and Linguistically Diverse) communities within the synod.
"The NextGen are often caught between two cultures that do not completely accept them," Mr Koh said.
"At home, the NextGen learn the beliefs and values of their parents' first culture, but in school and in work they are enculturated into the new one. They have two worlds at the core of their being. This can lead to an identity crisis, which is compounded by the fact that their parents and sometimes their ministers do not understand them, and try to impose 'old ways' on them."
The InterCultural unit seeks to create a 'third space' where the NextGen can work through the issues that affect them in a group setting.
The trip is a joint project between UnitingWorld, the China Council and the CFM.
"This tripartite joint program will enable NextGen to connect and learn from the fastest growing Church in the world.
"The young people will have opportunity to share with the Church in China the multicultural nature of the Uniting Church and their faith journeys," Mr Koh said.
Upon their return, it is hoped the NextGen will share with the synod what they observed and learned.
"We see this as another critical step towards empowering them, finding their voice and enlarging their visions."
Each young person accepted into the program will travel to China, visiting four cities (Beijing, Shanghai, Mt Huangshan and Nanjing) and spend approximately two or three days in each city.
They will meet with young Chinese people from various Christian congregations, for discussions, participation in workshops, worship, and to enjoy their hospitality.
Adult leaders will accompany them from Australia and provide pastoral support and opportunities for reflections and debriefing.
Mr Koh would like congregations and individuals to encourage young people to apply, help financially where possible and pray for them as they work together to make the trip a reality.
For more information contact: KOH Swee-Ann, M: 0424260222 E: koh.sweeann@victas.uca.org.au
Church future questioned
Mixed emotions to future direction
Faith for the future

redpajama_set_name: RedPajamaCommonCrawl


text:
Katherine M. Poldneff	admin	2018-05-14T16:36:50-04:00
Katherine M. Poldneff, Senior Counsel
Katherine M. Poldneff focuses her practice at Kaufman & Company on complex commercial litigation and employment disputes.
Her experience spans cases involving the False Claims Act, pharmaceutical pricing, Medicaid and Medicare fraud, fraud and breach of fiduciary duty in the healthcare receivables industry, securities fraud, defamation, and a variety of complex commercial contract disputes. She has also handled various employment discrimination matters both at the administrative and litigation stages.
Prior to joining Kaufman & Company, Katherine was a litigation associate at a nationally-ranked full-service law firm. Katherine obtained her Bachelor's degree from Vanderbilt University, graduating cum laude and with honors from the College Scholars Program, with a dual major of English and Communications. She earned her Juris Doctorate from the New York University School of Law in 2004. While at New York University, she was a part of the Civil Rights Clinic, Employment Discrimination Section.
Part of litigation team defending a large international law firm in a legal malpractice action brought by a Fortune 400 company related to the underlying trial of an employment retaliation case.
Part of litigation team representing a large investment holding company in a legal malpractice action against a large international law firm related to the trial of an underlying federal court action alleging claims of fraudulent transfer/conveyance and breaches of fiduciary duty.
Part of litigation team that successfully defended a private investment firm and favorably settled an action for breach of contract and fraudulent inducement related to a membership interest purchase agreement.
Part of litigation team that brought and favorably settled a case in the United States District Court for the Southern District of New York on behalf of a law firm against a former employee related to the employee's misappropriation of trade secrets and confidential information and intentional interference with client relationships after his termination from the firm.
Part of litigation team that successfully arbitrated a contractual dispute between two attorneys related to a co-counseling and fee-sharing agreement with respect to underlying class action cases.
Part of litigation team that defended a minor league baseball team against a municipality related to alleged breaches of a Stadium Lease. Drafted the key Motion to Dismiss that resulted in the dismissal of the case in the United States District Court for the Northern District of Ohio, which then led to a favorable settlement of the remaining dispute pursuant to a mandatory contractual dispute resolution process.
Part of litigation team that successfully defended pharmaceutical companies in multi-state, multi-defendant pharmaceutical pricing litigations involving both state and federal fraud claims based on Medicare and Medicaid law, as well as the federal False Claims Act.
Part of litigation team that successfully defended indenture trustee bank in multi-party fraud action in the healthcare receivables industry.
Part of litigation team that successfully settled complex commercial contract dispute involving the construction of a large, mixed-use real estate project.
Part of litigation team that successfully settled complex contract disputes involving breach of fiduciary duty claims between and among the principals of a partnership and an LLC.
Part of litigation team that successfully prosecuted and settled action based on fraud, conversion, and breach of contract on behalf of real estate development company specializing in FHA-insured, multi-family real estate projects.
Part of litigation team that arbitrated and then successfully settled a complex commercial contract dispute involving various breach of contract and tort claims related to a long-term licensing agreement between real estate development company and a large international corporation specializing in sports training and marketing.
Part of litigation team that successfully defended nationally prominent non-profit organization in case involving the Americans with Disabilities Act.
Part of litigation team that challenged hospital disciplinary action in sexual harassment matter and reached favorable settlement.
Part of team that successfully represented Department of Veterans' Affairs engineer in race discrimination case before the EEOC and ultimately led to successful recovery in federal court.
Part of litigation teams that successfully defended clients in various other employment discrimination cases, including age and religious discrimination claims before the Connecticut Human Rights Commission and the Second Circuit.
Part of litigation teams that both prosecuted and defended a number of complex commercial contract disputes.
Cleveland Metropolitan Bar Association, Women in Law Section Member
Volunteer for CMBA/Federal Bar Association Reach out for Non-Profits Program
Experience Highlights
Won a Week-Long Injunction Trial Allowing the Most Significant Public Development Project In Cleveland in 30 Years to Proceed
In the most significant Cleveland public development project in 30 years, Steve Kaufman as co-trial counsel won a TRO permitting the $275 million financing to close and then won a week-long trial, obtaining a preliminary injunction against the architectural firm who tried to block the $275 million Flats East Bank redevelopment project in a dispute over the ownership of the project drawings. A state court judge found that the developer had the right to maintain ownership of and use the drawings, and that not recognizing this right would irreparably harm the developer, the project, and the Greater Cleveland area.
New York University School of Law (J.D., 2004)
Vanderbilt University (B.A., 2001)
U.S. District Court, N.D. Ohio
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Kathy.Poldneff@kaufman-company.com


(Oslo/Helsinki, 2 October 2017) EVRY has entered into an agreement to deliver the entire value chain for card payment to the Finnish bank Aktia. As part of the agreement, EVRY will deliver virtual debit and credit cards for Aktia's digital wallet solution, Aktia Wallet.
The agreement, which runs for a period of five years, was won against strong competition from other technology providers. EVRY will deliver everything from physical debit and credit cards to digital card payments and related services, such as card administration, authorisation processing, transaction monitoring and security solutions that detect and prevent fraud.
Through the agreement EVRY will support Aktia's focus on virtual payment cards on mobile phones. With Aktia's digital wallet solution, Aktia Wallet, the bank's customers are able to make quick, simple and secure digital payments across a range of devices and channels – online, in-app and contactless in shops.
EVRY's card solutions are already in use at over 100 card issuers in Scandinavia and the UK. Aktia Bank will be one of the first customers to use EVRY's new software for digitising cards on mobile devices. This will mean cards will be issued in real-time, allowing customers to immediately use their mobile phones to pay at every payment terminal that accepts contactless payment in the world.
"Aktia continuously improves its financial services and products and their availability. We therefore need a collaboration partner that leads the field in the development of card and payment solutions and offers innovative and stable services. We are working hard to offer new solutions that will make it easier for our customers to manage their finances on a day-to-day basis, and this collaboration with EVRY will help accelerate developments in this area. The breadth of EVRY's deliveries mean that we only need to have a relationship with one collaboration partner in the card area", comments Anton Tulikoura, Director Loans and Cards in Aktia Bank.
"Aktia is a very important customer for EVRY and we are proud they have chosen to put their trust in us once again. We are now expanding our existing collaboration with Aktia with an increased focus on new and innovative services. Paying should be easy, quick and simple. This agreement will give the bank's customers exactly that, and will simplify Aktia's work. Digitalisation and new market entrants are creating a need for solutions to be even more personalised and customised. We will help with this through our specialist expertise and thus create digital advantage for Aktia", comments Christer Jonsson, SVP Card, EVRY Financial Services.
"EVRY has spent many years building up its unique position as a collaboration partner to the banking sector. This agreement strengthens our position in the Finnish market, which is a strategically important market for us. It is also further confirmation that our offer is attractive and we look forward to continuing our successful collaboration with Aktia in the years ahead", comments Wiljar Nesse, EVP Financial Services, EVRY.
EVRY has the market's most complete service portfolio in the card area, and delivers cards to more than 100 card issuers across 12 countries. EVRY processes up to 250 card transactions per second and issues 25 million cards every year. EVRY monitors over one billion card transactions per year from across the world to detect and prevent fraud.