text:
\section{S-I. Model}
We use a modified version of Model II of Ref.~\cite{Bauer2013}: In the central region, described by $N=2N'+1$ sites, with $i=-N', \dots, N'$, the on-site potential is zero, and the hopping elements vary from site to site according to
\begin{equation}
	\tau_j = \tau-\tfrac 12 \tilde V_{\rm c} \exp \left(-\frac{x_j^2}{1-x_j^2} \right); \; x_j = \frac{2j+1}{N-1},
\label{eq:hopping}
\end{equation}
where $j$ runs from $-N'$ to $N'-1$.
The on-site interaction in the central region is given by
\begin{equation}
\label{eq:interaction}
	U_i = U_0 \exp \left(-\frac{l_i^6}{1-l_i^2} \right); \; l_i = \frac{i}{N'+\tfrac 12},
\end{equation}
The hopping and interaction Eqs.~\eqref{eq:hopping},\eqref{eq:interaction} lead to a Hamiltonian
\begin{align}
\label{eq:define_H_general}
	\mathcal H &= -\sum_{\sigma, i} \tau_{i} \left( c^\dagger_{i+1,\sigma} c_{i,\sigma} + {\rm h.c.} \right) + \sum_{i} \left( U_i c^\dagger_{i\uparrow} c_{i\uparrow} c^\dagger_{i\downarrow} c_{i\downarrow} \right), \nonumber \\
	           &=: \sum_{\sigma, i,j} \left( \tilde H^\sigma_{ji} c^\dagger_{j,\sigma} c_{i,\sigma} + {\rm h.c.} \right) + \sum_{i} \left( U_i c^\dagger_{i\uparrow} c_{i\uparrow} c^\dagger_{i\downarrow} c_{i\downarrow} \right),
\end{align}
where we use the tilde to indicate that the indices of the Hamiltonian matrix $\tilde H^\sigma$ run over $\mathbbm Z$.
$\tilde H^\sigma_{ij}$ is invariant under transposition and parity ${\cal P}$, which we implement as $ {\cal P}: i \mapsto -i$. We will explicitly assume the presence of these symmetries in the following.
Note that for our description of the central region, the effect of the tight-binding leads with hopping $\tau$ coupling to sites $-N'$ and $N'$ is fully included in the self-energy contribution
\begin{align}
\label{eq:selfenergy_lead}
	{\Sigma^{R}_{\rm lead}}_{ij} (\omega) =& (\delta_{i,-N'} \delta_{j,-N'} + \delta_{i,N'} \delta_{j,N'}) \nonumber \\
	& \times
	\begin{cases}
		\frac{\omega}{2} \left( 1 - \sqrt{1-\left( \frac{2 \tau}{\omega} \right)^2 } \right), \, \vert \omega \vert > 2 \tau \\
		\frac{\omega}{2} - i \tau \sqrt{1 - \left( \frac{\omega}{2 \tau} \right)^2},          \, \vert \omega \vert < 2 \tau,
	\end{cases} \\
	{\Sigma^{K}_{\rm lead}}_{ij} (\omega) =& (1-2n_F(\omega))({\Sigma^R_{\rm lead}}_{ij}-{\Sigma^A_{\rm lead}}_{ij}).
\end{align}
Here, the superscript $R$($K$,$A$) denotes the retarded (Keldysh, advanced) component of the self energy and $n_F$ is the Fermi distribution function.

As stated in the main text, we use $U_0 = 0.7 \tau$ and $\tilde V_{\rm c} \in [0.44, 0.58] \tau$.

\section{S-II. Keldysh fRG}
\label{sec:fRG_flow_equations}
The model is solved by employing the functional renormalization group (fRG) \cite{Jakobs2010,Jakobs2010-2,Karrasch2006a,Metzner2012} on the Keldysh-contour to obtain real-frequency information. The flow is truncated perturbatively, i.e. we set the three-particle vertex (and all higher vertices) to zero during the flow and approximate the two-particle vertex by the three usual channels ($P$, $X$, and $D$) \cite{Bauer2013,Jakobs2010}, assuming a local and static inter-channel mixing (coupled-ladder-approximation). The computation is then exact to second order in the interaction. It may be viewed as extension of the flow used in Ref.~\cite{Jakobs2010-2} to multiple sites (neglecting the $D^{\sigma \bar \sigma}$-channel, which in our case is of order $U_0^3$) or an extension of the flow used in Ref.~\cite{Bauer2013} to real frequencies. As flow parameter we use an artificial, on-site broadening of the spectrum (c.f.~Eq.~\eqref{eq:green_function_with_flow_param}, and Ref.~\cite{Jakobs2010}).
This flow parameter respects fluctuation-dissipation theorems, so that in equilibrium it is unnecessary to compute the Keldysh components of the self energy ($\Sigma^K$) and the channels ($b^P$, $b^X$, $b^D$).
The conventions on the Keldysh-contour used are those of Ref.~\cite{Jakobs2010-2}, with the difference that after the Keldysh rotation we use the labels c(lassical) and q(uantum), instead of $2$ and $1$. In particular, this means that the Keldysh rotation used for the fermions is the same as the one usually used for bosons.
We use $\sigma = \uparrow, \downarrow$ to denote spin, and $\bar \sigma$ to denote the spin opposite to $\sigma$. Letters from the middle of the roman alphabet ($i$,$j$) refer to spatial sites, while letters from the beginning of the Greek alphabet ($\alpha$, $\beta$) refer to the Keldysh indices.

\subsection{A. The Single-Scale Propagator}
The flow parameter is determined by the bare retarded Green's function
\begin{equation}
\label{eq:green_function_with_flow_param}
	\tilde G^R_{0,\Lambda, \sigma} (\omega) = \frac{1}{\omega {\mathbbm 1}- \tilde H^\sigma + i \left( \tfrac 12 \Lambda \right) {\mathbbm 1}},
\end{equation}
where $\tilde H^\sigma$ is the non-interacting Hamiltonian matrix extracted from Eq.~\eqref{eq:define_H_general}. $\Lambda$ is the flow parameter, ranging from $\infty$ (start of flow) to $0$ (end of flow). $\mathbbm 1$ is the unit matrix in the space of the sites, which we will omit from now on. Once the leads have been projected out, we drop the tilde on the restricted Hamiltonian matrix $H^\sigma$ and the spatial indices then only run from $-N'$ to $N'$. We use the artificial on-site broadening for all sites (including the leads) to avoid artifacts at the transition from the lead to the central region.

The retarded single-scale propagator $\tilde S^R$ is
\begin{equation}
	\tilde S^R(\omega) = \left(\tilde G \tilde G_0^{-1} \partial_\Lambda \tilde G_0 \tilde G_0^{-1} \tilde G \right)^R = -\frac{i}{2} \tilde G^R_\Lambda \cdot \tilde G^R_\Lambda,
\end{equation}
where we omit the site and spin labels.

After the integration over the leads' degrees of freedom has been performed, the Green's function projected onto the central part acquires an additional self-energy term
\begin{equation}
	G^{R(\sigma)}_0(\omega) = \frac{1}{\omega^{(\sigma)} - \mathcal H^{(\sigma)} - {\Sigma^{(\sigma)}_{\rm lead}}(\omega, \Lambda) + i\Lambda/2},
\end{equation}
where $\omega^{(\sigma)} = \omega + \tfrac \sigma 2 B$ and
\begin{align}
\label{eq:Sigma_lead_flow}
	{\Sigma^{(\sigma)}_{\rm lead}}_{ij} (\omega, \Lambda) &= \frac{1}{2}\left(\omega^{(\sigma)} +i \tfrac{\Lambda}{2}-i\sqrt{4\tau^2-\left(\omega^{(\sigma)}+
i \tfrac{\Lambda}{2}\right)^2} \right) \nonumber \\
	                                &\times (\delta_{i,-N'} \delta_{j,-N'} + \delta_{i,N'} \delta_{j,N'}).
\end{align}
This self-energy is also reflected in the projected single-scale
propagator, which now takes the form
\begin{align}
	S^R(\omega) &= \left( G G_0^{-1} \partial_\Lambda G_0 G_0^{-1} G \right)^R \nonumber \\
	            &= G^R_\Lambda \cdot \left(-\frac{i}{2} + \partial_\Lambda \Sigma_{\rm lead} (\omega, \Lambda) \right) \cdot G^R_\Lambda.
\end{align}

For $\Lambda \rightarrow \infty$ the model is exactly solvable and the irreducible part of the full vertex is simply the bare vertex \cite{Jakobs2010-2}. Since we only consider equilibrium situations in this paper and the flow parameter respects fluctuation-dissipation theorems, the Keldysh Green's function $G^K$ [and single scale $S^K$] is determined simply via the fluctuation-dissipation theorem
\begin{equation}
	G^K = (1-2 n_F)(G^R-G^A),~S^K = (1-2 n_F)(S^R-S^A).
\end{equation}

\subsection{B. The Vertex}
The vertex is assumed to consist \textit{only} of a two-particle contribution. This contribution is approximated by a structure compatible with a decomposition into three channels (with only static and local interchannel feedback). This approximation yields a consistent set of flow equations. We
use the following parametrization:

We decompose the $2$-particle vertex into three channels, according to
\begin{equation}
	\gamma(\omega_1', \omega_2'; \omega_1, \omega_2) \approx \bar v + \varphi^P(\omega_1 + \omega_2) + \varphi^X(\omega_2 - \omega_1') + \varphi^D(\omega_2 - \omega_2'),
\end{equation}
where we have suppressed all indices other than frequency, and primed quantities denote outgoing legs. $\bar v$ denotes the bare vertex.
The Keldsh structure is arranged according to the convention
\begin{equation}
	\gamma^{\alpha \beta | \gamma \delta} =
	\begin{pmatrix}
		(qq|qq) & (qq|cq) & (qq|qc) & (qq|cc) \\
		(cq|qq) & (cq|cq) & (cq|qc) & (cq|cc) \\
		(qc|qq) & (qc|cq) & (qc|qc) & (qc|cc) \\
		(cc|qq) & (cc|cq) & (cc|qc) & (cc|cc)
	\end{pmatrix}.
\end{equation}
The channels are labelled as (the Keldysh structure corresponds to Eqs.~(A8,A11,A17) of Ref.~\cite{Jakobs2010}, while the spatial structure is that of Eq.~(S48) of Ref.~\cite{Bauer2013})
	\begin{equation}
		(\varphi^P)_{(\sigma \bar \sigma | \sigma \bar \sigma), (ii|jj)}(\Pi) =
		\begin{pmatrix}
		0           & a^{P*}_{ji} & a^{P*}_{ji} & 0          \\
		a^{P}_{ij}  & b^P_{ij}    & b^P_{ij}    & a^P_{ij}   \\
		a^{P}_{ij}  & b^P_{ij}    & b^P_{ij}    & a^P_{ij}   \\
		0           & a^{P*}_{ji} & a^{P*}_{ji} & 0
		\end{pmatrix}^{(\sigma \bar \sigma)} (\Pi),
	\end{equation}
	\begin{equation}
		(\varphi^X)_{(\sigma \bar \sigma | \sigma \bar \sigma), (ji|ij)}(X) =
		\begin{pmatrix}
		0           & a^{X*}_{ji} & a^{X}_{ij}  & b^X_{ij}   \\
		a^{X}_{ij}  & b^X_{ij}    & 0           & a^{X*}_{ji}\\
		a^{X*}_{ji} & 0           & b^X_{ij}    & a^X_{ij}   \\
		b^X_{ij}    & a^{X}_{ij}  & a^{X*}_{ji} & 0
		\end{pmatrix}^{(\sigma \bar \sigma)} (X),
	\end{equation}
	\begin{equation}
		(\varphi^D)_{(\sigma \sigma | \sigma \sigma), (ij|ij)}(\Delta) =
		\begin{pmatrix}
		0           & a^{D}_{ij}  & a^{D*}_{ji} & b^D_{ij}   \\
		a^{D}_{ij}  & 0           & b^D_{ij}    & a^{D*}_{ii}\\
		a^{D*}_{ji} & b^D_{ij}    & 0           & a^D_{ij}   \\
		b^D_{ij}    & a^{D*}_{ji} & a^{D}_{ij}  & 0
		\end{pmatrix}^{(\sigma \sigma)} (\Delta).
	\end{equation}
Each channel is labelled by only two spatial indices and one frequency. Conceptually, it can be thought of as the propagator of a Hubbard-Stratonovitch particle of the corresponding channel with retarded ($a^P$, $a^D$, and $a^{X*}$) and Keldysh ($b^P$, $b^{D}$, and $b^{X}$) components. From this point of view it is not surprising that in equilibrium the channels satisfy the fluctuation-dissipation theorems (c.f. Eqs.~(A10,A13,A19) of Ref.~\cite{Jakobs2010}:
\begin{subequations}
\begin{align}
	{b^P}^{(\sigma \bar \sigma)}_{(ij)} (\Pi)    & =   2 i \coth\left[\beta \left( \frac \Pi 2 - \mu \right)\right] {\rm Im} \, {a^{P}}^{(\sigma \bar \sigma)}_{(ij)} (\Pi) \\
	{b^X}^{(\sigma \bar \sigma)}_{(ij)} (X)      & = - 2 i \coth\left[\frac {\beta X} 2 \right]                     {\rm Im} \, {a^{X}}^{(\sigma \bar \sigma)}_{(ij)} (X) \\
	{b^D}^{(\sigma      \sigma)}_{(ij)} (\Delta) & =   2 i \coth\left[\frac {\beta \Delta} 2 \right]                {\rm Im} \, {a^{D}}^{(\sigma      \sigma)}_{(ij)} (\Delta)
\end{align}
\end{subequations}

\subsection{C. The Flow Equations}
When all vertices higher than the $2$-particle vertex are set to zero, the resulting truncated flow equations are (c.f. Eqs.~(27,28) of Ref.~\cite{Jakobs2010})
\begin{align}
	\frac{d}{d\Lambda} \Sigma^\Lambda_{1'1}       =
& -\! \SumInt{2' 2}{}{4mm}  \frac{i}{2\pi} \gamma^\Lambda_{1' 2' 1 2} S^\Lambda_{2 2'} \nonumber \\
	\frac{d}{d\Lambda} \gamma^\Lambda_{1' 2' 1 2} =
&+\! \SumInt{3' 4' 3 4}{}{6.5mm}  \!\!\!
\frac{i}{2\pi} \gamma^\Lambda_{1' 2' 3 4} S^\Lambda_{3 3'} G^\Lambda_{4 4'} \gamma^\Lambda_{3' 4' 1 2} \nonumber
\\
&+\! \SumInt{3' 4' 3 4}{}{6.5mm} \!\!\!
\frac{i}{2\pi} \gamma^\Lambda_{1' 4' 3 2} \left[S^\Lambda_{3 3'} G^\Lambda_{4 4'} + S^\Lambda_{4 4'} G^\Lambda_{3 3'} \right] \gamma^\Lambda_{3' 2' 1 4} \nonumber\\
&- \! \SumInt{3' 4' 3 4}{}{6.5mm} \!\!\!
\frac{i}{2\pi} \gamma^\Lambda_{1' 3' 1 4}
\left[S^\Lambda_{3 3'} G^\Lambda_{4 4'} + S^\Lambda_{4 4'} G^\Lambda_{3 3'} \right]
\gamma^\Lambda_{4' 2' 3 2}.
\end{align}
Here, $1$, $1'$ etc. are multi-indices encompassing spin, site and
frequency. In the flow of the vertex, each summand corresponds to a
single channel. The vertex of each summand will be approximated by the
contribution of the corresponding channel for all frequencies and the
feedback of the other channels at a specific frequency ($2\mu$ for the
P-channel, $0$ for the X- and D-channels).  Inserting the channel
decomposition with the above notations into the flow equations, the
flow of the self-energy is given by [compare Eqs.~(B3,B4) of
Ref.~\cite{Jakobs2010}]:
\begin{widetext}
\begin{flalign}
	\partial_\Lambda \Sigma_{(kl)}^{q|c(\sigma)} (\omega) = -\frac{i}{2\pi}
	                                                               \int d\omega' &\left[   S_{(lk)}  ^{c|c(\bar \sigma)} (\omega') {a^P}^{(\sigma \bar \sigma)}_{(kl)} (\omega + \omega')   \right.
	                                                                               + S_{(kl)}  ^{c|c(\bar \sigma)} (\omega') {a^X}^{(\sigma \bar \sigma)}_{(lk)} (\omega' - \omega)
	                                                                               - S_{(kl)}  ^{c|c(     \sigma)} (\omega') {a^D}^{(\sigma \sigma)}_{(kl)}  (\omega - \omega')       \nonumber\\
	                                                                             &       + S_{(lk)}  ^{q|c(\bar \sigma)} (\omega') {b^P}^{(\sigma \bar \sigma)}_{(kl)} (\omega + \omega')
	                                                                               + S_{(kl)}  ^{c|q(\bar \sigma)} (\omega') {b^X}^{(\sigma \bar \sigma)}_{(lk)} (\omega' - \omega)
	                                                                               - S_{(kl)}  ^{c|q(     \sigma)} (\omega') {b^D}^{(\sigma \sigma)}_{(lk)} (\omega - \omega')     \nonumber\\
	                                                                             &       + S_{(lk)}    ^{c|c(\bar \sigma)} (\omega') U_k/2 \delta_{kl}
	                                                                               + \left. \sum_m S_{(mm)}^{c|c(\sigma)} (\omega') {a^D}^{(\sigma \sigma)}_{(km)} (0) \delta_{kl} \right]          \nonumber\\
\end{flalign}
and
\begin{flalign}
	\partial_\Lambda \Sigma_{(kl)}^{q|q(\sigma)} (\omega) = -\frac{i}{2\pi}
	                                                                        \int d\omega &'\left[                  S_{(kl)}  ^{c|q(\bar \sigma)}(\omega') {a^X}^{(\sigma \bar \sigma)}_{(lk)} (\omega' - \omega)     \right.
	                                                                               -                 S_{(kl)}  ^{c|q(     \sigma)}(\omega') {a^D}^{(\sigma \sigma)}_{(kl)}      (\omega - \omega')
	                                                                               +                 S_{(lk)}  ^{q|c(\bar \sigma)}(\omega') {a^P}^{(\sigma \bar \sigma)}_{(kl)} (\omega + \omega')            \nonumber\\
	                                                                               &       +                 S_{(lk)}  ^{c|q(\bar \sigma)}(\omega') {a^{P*}}^{(\sigma \bar \sigma)}_{(lk)} (\omega' + \omega)
	                                                                               +                 S_{(kl)}  ^{q|c(\bar \sigma)}(\omega') {a^{X*}}^{(\sigma \bar \sigma)}_{(kl)} (\omega' - \omega)
	                                                                               -                 S_{(kl)}  ^{q|c(     \sigma)}(\omega') {a^{D*}}^{(\sigma \sigma)}_{(lk)} (\omega - \omega')              \nonumber\\
	                                                                               &       +                 S_{(lk)}  ^{c|c(\bar \sigma)}(\omega') {b^P}^{(\sigma \bar \sigma)}_{(kl)} (\omega + \omega')
	                                                                               +                 S_{(kl)}  ^{c|c(\bar \sigma)}(\omega') {b^X}^{(\sigma \bar \sigma)}_{(lk)} (\omega' - \omega)
	                                                                               -                 S_{(kl)}  ^{c|c(     \sigma)}(\omega') {b^D}^{(\sigma \sigma)}_{(kl)}      (\omega - \omega')            \nonumber\\
	                                                                               &       + \left.\left( S_{(lk)}  ^{c|q(\bar \sigma)}(\omega') + S_{(lk)}  ^{q|c(\bar \sigma)} (\omega') \right) U_k/2 \delta_{kl}\right].\nonumber\\
\end{flalign}
The flow of the vertex contains two bubbles
\begin{align}
	I^{pp}_{ab|a'b'} &(\omega)^{(\sigma_1 \sigma_2)}_{(ij|kl)} = \frac{i}{2\pi} \int d\omega' \left[ G_{(i|k)}^{a|a'(\sigma_1)}(\omega/2 + \omega')S_{(j|l)}^{b|b'(\sigma_2)}(\omega/2 - \omega') +  S_{(i|k)}^{a|a'(\sigma_1)}(\omega/2 + \omega')G_{(j|l)}^{b|b'(\sigma_2)}(\omega/2 - \omega') \right],
\\
	I^{ph}_{ab|a'b'}& (\omega)^{(\sigma_1 \sigma_2)}_{(ij|kl)} = \frac{i}{2\pi} \int d\omega' \left[ G_{(i|k)}^{a|a'(\sigma_1)}(-\omega/2 + \omega')S_{(j|l)}^{b|b'(\sigma_2)}(\omega/2 + \omega') + S_{(i|k)}^{a|a'(\sigma_1)}(-\omega/2 + \omega')G_{(j|l)}^{b|b'(\sigma_2)}(\omega/2 + \omega') \right],
\end{align}
and is given by (compare Eqs.~(C3,C6,C9) of Ref.~\cite{Jakobs2010})
\begin{align}
\partial_\Lambda& (\varphi^P)^{qq|cq}_{(\sigma \bar \sigma \vert \sigma \bar \sigma)(ii \vert jj)} (\Pi) = \partial_\Lambda {a^{P*}}^{(\bar \sigma \sigma)}_{(ij)}(\Pi) \nonumber\\
		=& \sum_{km} \left(\tfrac 12 U_k \delta_{ki} + a^{P*}(\Pi)^{(\bar \sigma \sigma)}_{(ki)} + \tfrac{1}{2}{U^X}^{(\bar \sigma \sigma)}_{(ki)} \right) \left( I^{pp}_{cq|cc}(\Pi)^{(\sigma \bar \sigma|\sigma \bar \sigma)}_{(kk|mm)} + I^{pp}_{qc|cc}(\Pi)^{(\sigma \bar \sigma|\sigma \bar \sigma)}_{(kk|mm)}\right) \left( \tfrac 12 U_j \delta_{jm} + a^{P*}(\Pi)^{(\bar \sigma \sigma)}_{(jm)} + \tfrac{1}{2}{U^X}^{(\bar \sigma \sigma)}_{(jm)} \right)
\end{align}
\begin{align}
	\partial_\Lambda &(\varphi^X)^{qq|cq}_{(\sigma \bar \sigma \vert \sigma \bar \sigma)(ji|ij)} (X) = \partial_\Lambda a^{X*}(X)^{(\bar \sigma \sigma)}_{(ji)} \nonumber \\
	         =& \sum_{kl} \left(\tfrac{1}{2}U_j \delta_{jk} + \tfrac{1}{2}{U^P}^{(\bar \sigma \sigma)}_{(jk)} + a^{X*}(X)^{(\bar \sigma \sigma)}_{(jk)} \right) \left( I^{ph}_{qc|cc}(X)^{(\sigma \bar \sigma| \sigma \bar \sigma)}_{kl|lk} + I^{ph}_{cc|cq}(X)^{(\sigma \bar \sigma| \sigma \bar \sigma)}_{kl|lk} \right) \left( \tfrac{1}{2}U_i \delta_{il} + \tfrac{1}{2}{U^P}^{(\bar \sigma \sigma)}_{(li)} + a^{X*}(X)^{(\bar \sigma \sigma)}_{(li)} \right)
\end{align}
\begin{align}
	\partial_\Lambda &(\varphi^D)^{cq|qq}_{(\sigma \sigma)(ij|ij)} (\Delta) = \partial_\Lambda a^D(\Delta)^{(\sigma \sigma)}_{(ij)} \nonumber \\
	              =& -\sum_{kl} \left[ \left(-\tfrac 12 {W^D}^{(\sigma \sigma)}_{(ik)} + a^D(\Delta)^{(\sigma \sigma)}_{(ik)} \right) \left( I^{ph}_{qc|cc}(\Delta)^{(\sigma \sigma | \sigma \sigma)}_{(lk|kl)} + I^{ph}_{cc|cq}(\Delta)^{(\sigma \sigma | \sigma \sigma)}_{(lk|kl)} \right) \left(-\tfrac 12 {W^D}^{(\sigma \sigma)}_{(lj)} + a^D(\Delta)^{(\sigma \sigma)}_{(lj)} \right) \right.\nonumber \\
	               &+ \left. \left( \tfrac 12 {U_i} + \tfrac 12 {U^P}^{(\sigma \bar \sigma)}_{(ik)} + \tfrac 12 {U^X}^{(\sigma \bar \sigma)}_{(ik)} \right) \delta_{ik} \left( I^{ph}_{qc|cc}(\Delta)^{(\bar \sigma \bar \sigma | \bar \sigma \bar \sigma)}_{(lk|kl)} + I^{ph}_{cc|cq}(\Delta)^{(\bar \sigma \bar \sigma | \bar \sigma \bar \sigma)}_{(lk|kl)} \right) \delta_{jl} \left( \tfrac 12{U_j} + \tfrac 12 {U^P}^{(\bar \sigma \sigma)}_{(jl)} + \tfrac 12 {U^X}^{(\bar \sigma \sigma)}_{(jl)} \right) \right]
\end{align}
\begin{align}
	\partial_\Lambda &(\varphi^P)^{cq|cq}_{(\sigma \bar \sigma)(ii|jj)} (\Pi) = \partial_\Lambda b^P(\Pi)^{(\sigma \bar \sigma)}_{(ij)} \nonumber \\
		=& \sum_{km} \left[ \left(\tfrac{1}{2}U_i \delta_{ik} + a^P(\Pi)^{(\sigma \bar \sigma)}_{(ik)} + \tfrac{1}{2}{U^X}^{(\sigma \bar \sigma)}_{(ik)} \right) \left( I^{pp}_{cc|cc}(\Pi)^{(\sigma \bar \sigma|\sigma \bar \sigma)}_{(kk|mm)} + I^{pp}_{qq|cc}(\Pi)^{(\sigma \bar \sigma|\sigma \bar \sigma)}_{(kk|mm)} + I^{pp}_{cc|qq}(\Pi)^{(\sigma \bar \sigma|\sigma \bar \sigma)}_{(kk|mm)} \right) \right. \nonumber \\
		& \times \left(\tfrac{1}{2}U_j \delta_{jm} + a^{P*}(\Pi)^{(\sigma \bar \sigma)}_{(jm)} + \tfrac{1}{2}{U^X}^{(\sigma \bar \sigma)}_{(jm)} \right)\nonumber \\
		 &+ b^P(\Pi)^{(\sigma \bar \sigma)}_{(ik)} \left( I^{pp}_{qc|cc}(\Pi)^{(\sigma \bar \sigma|\sigma \bar \sigma)}_{(kk|mm)} + I^{pp}_{cq|cc}(\Pi)^{(\sigma \bar \sigma|\sigma \bar \sigma)}_{(kk|mm)} \right) \left(\tfrac{1}{2}U_j \delta_{jm} + a^{P*}(\Pi)^{(\sigma \bar \sigma)}_{(jm)} + \tfrac{1}{2}{U^X}^{(\sigma \bar \sigma)}_{(jm)} \right) \nonumber \\
		 &+ \left .\left(\tfrac{1}{2}U_i \delta_{ik} + a^P(\Pi)^{(\sigma \bar \sigma)}_{(ik)} + \tfrac{1}{2}{U^X}^{(\sigma \bar \sigma)}_{(ik)} \right) \left( I^{pp}_{cc|qc}(\Pi)^{(\sigma \bar \sigma|\sigma \bar \sigma)}_{(kk|mm)} + I^{pp}_{cc|cq}(\Pi)^{(\sigma \bar \sigma|\sigma \bar \sigma)}_{(kk|mm)} \right) b^P(\Pi)^{(\sigma \bar \sigma)}_{(mj)} \right]
\end{align}
\begin{align}
	 \partial_\Lambda &(\varphi^X)_{ji|ij}^{qq|cc} (X) = \partial_\Lambda b^X(X)^{(\sigma \bar \sigma)}_{(ij)} \nonumber \\
	         =& \sum_{kl} \left[ \left(\tfrac{1}{2}U_k \delta_{kj} + \tfrac{1}{2}{U^P}^{(\sigma \bar \sigma)}_{(kj)} + a^X(X)^{(\sigma \bar \sigma)}_{(kj)} \right) \left( I^{ph}_{cc|cc}(X)^{(\sigma \bar \sigma | \sigma \bar \sigma)}_{kl|lk} + I^{ph}_{qc|cq}(X)^{(\sigma \bar \sigma | \sigma \bar \sigma)}_{kl|lk} + I^{ph}_{cq|qc}(X)^{(\sigma \bar \sigma | \sigma \bar \sigma)}_{kl|lk} \right) \right. \nonumber \\
				 &\, \, \times  \left(\tfrac{1}{2}U_l \delta_{il} + \tfrac{1}{2}{U^P}^{(\sigma \bar \sigma)}_{(il)} + a^{X*}(X)^{\sigma \bar \sigma}_{(li)}\right) \nonumber \\
	          &+ b^X(X)^{(\sigma \bar \sigma)}_{(kj)} \left( I^{ph}_{qc|cc}(X)^{(\sigma \bar \sigma | \sigma \bar \sigma)}_{kl|lk} + I^{ph}_{cc|cq}(X)^{(\sigma \bar \sigma | \sigma \bar \sigma)}_{kl|lk} \right) \left(\tfrac{1}{2}U_l \delta _{il} + \tfrac{1}{2}{U^P}^{(\sigma \bar \sigma)}_{(li)} + a^{X*}(X)^{\sigma \bar \sigma}_{li}\right) \nonumber \\
	          &+ \left. \left(\tfrac{1}{2}U_j \delta_{jk} + \tfrac{1}{2}{U^P}^{(\sigma \bar \sigma)}_{(jk)} + a^X(X)^{(\sigma \bar \sigma)}_{(kj)}\right) \left( I^{ph}_{cq|cc}(X)^{(\sigma \bar \sigma | \sigma \bar \sigma)}_{kl|lk} + I^{ph}_{cc|qc}(X)^{(\sigma \bar \sigma | \sigma \bar \sigma)}_{kl|lk} \right) b^X(X)^{(\sigma \bar \sigma)}_{(il)} \right]
\end{align}
\begin{align}
	\partial_\Lambda &(\varphi^D)^{cc|qq}_{(\sigma \sigma)(ij|ij)} (\Delta) = \partial_\Lambda b^D(\Delta)^{(\sigma \sigma)}_{(ij)}  \nonumber \\
	              =& -\sum_{kl} \left[ \left(-\tfrac 12 {W^D}^{\sigma \sigma}_{ik} + a^D(\Delta)^{(\sigma \sigma)}_{(ik)}\right) \cdot \left( I^{ph}_{cc|cc}(\Delta)^{(\sigma \sigma | \sigma \sigma)}_{(lk|kl)} + I^{ph}_{qc|cq}(\Delta)^{(\sigma \sigma | \sigma \sigma)}_{(lk|kl)} + I^{ph}_{cq|qc}(\Delta)^{(\sigma \sigma | \sigma \sigma)}_{(lk|kl)} \right) \left(-\tfrac 12 {W^D}^{(\sigma \sigma)}_{(lj)} + a^{D*}(\Delta)^{(\sigma \sigma)}_{(jl)}\right) \right. \nonumber \\
	               &+ \left(-\tfrac 12 {W^D}^{(\sigma \sigma)}_{(ik)} + a^D(\Delta)^{(\sigma \sigma)}_{(ik)}\right) \left( I^{ph}_{qc|cc}(\Delta)^{(\sigma \sigma | \sigma \sigma)}_{(lk|kl)} + I^{ph}_{cc|cq}(\Delta)^{(\sigma \sigma | \sigma \sigma)}_{(lk|kl)} \right) b^D(\Delta)^{(\sigma \sigma)}_{(lj)} \nonumber \\
	               &+ b^D(\Delta)^{(\sigma \sigma)}_{(ik)} \left( I^{ph}_{cq|cc}(\Delta)^{(\sigma \sigma | \sigma \sigma)}_{(lk|kl)} + I^{ph}_{cc|qc}(\Delta)^{(\sigma \sigma | \sigma \sigma)}_{(lk|kl)} \right) \left(-\tfrac 12 {W^D}^{(\sigma \sigma)}_{(lj)} + a^{D*}(\Delta)^{(\sigma \sigma)}_{(jl)}\right) \nonumber \\
	               &+ \left. \left(\tfrac 12 {U_i} \delta_{ik} + \tfrac 12 {U^P}^{\sigma \bar \sigma}_{(ik)} + \tfrac 12 {U^X}^{\sigma \bar \sigma}_{(ik)} \right) \left( I^{ph}_{cc|cc}(\Delta)^{(\bar \sigma \bar \sigma | \bar \sigma \bar \sigma)}_{(lk|kl)} + I^{ph}_{qc|cq}(\Delta)^{(\bar \sigma \bar \sigma | \bar \sigma \bar \sigma)}_{(lk|kl)} + I^{ph}_{cq|qc}(\Delta)^{(\bar \sigma \bar \sigma | \bar \sigma \bar \sigma)}_{(lk|kl)} \right)
\right.\nonumber \\ & \phantom{+}
\left.  \times
\left(\tfrac 12 {U_l} \delta_{lj} + \tfrac 12 {U^P}^{(\bar \sigma \sigma)}_{(lj)} + \tfrac 12 {U^X}^{(\bar \sigma \sigma)}_{(lj)} \right) \right]
\end{align}
\end{widetext}
The relative signs between the $X$- and the $D$-channel stem from the fact that they are related through exchange of two fermionic legs.

In equilibrium, we set
\begin{align}
{U^P}^{(\sigma \bar \sigma)}_{ij}   &= 2 {\rm Re} \, a^P(2 \mu)^{(\sigma \bar \sigma)}_{(ij)} \delta_{ij}, \nonumber \\
{U^X}^{(\sigma \bar \sigma)}_{(ij)} &= 2 {\rm Re} \, a^X (0)^{(\sigma \bar \sigma)}_{(ij)} \delta_{ij}, \nonumber \\
{W^D}^{(\sigma \sigma)}_{(ij)}      &= 2 {\rm Re} \, a^D (0)^{(\sigma \sigma)}_{(ij)} \delta_{ij}.
\end{align}
Note that in equilibrium, $a^P(2 \mu)$, $a^X(0)$, and $a^D(0)$ are all real matrices.

In order to fully specify the flow, it remains to fix the initial conditions at large but finite Lambda:
\begin{align}
	\Sigma_{ij} & = \delta_{ij} U_i/2, \\
	\phi^X      & = \phi^P = \phi^D = 0.
\end{align}

\section{S-III. The frequency structure of the spin-susceptibility}
\label{sec:SS_freq}
\begin{figure}
\includegraphics[width=\columnwidth]{electron-hole-pairs_newML-crop.pdf}
\caption{(a,b) LDOS of a non-interacting homogenous system and (c,d) LDOS at the central site of an interacting QPC in the open regime. The filled (empty) circles show electrons (holes) of an electron-hole pair contributing to the spin susceptibility Eq.~\eqref{eq:nonint_local_spin_noise}, (a,c) for $\omega<\epsilon_F$ and (b,d) for $\omega>\epsilon_F$. Electron (or hole) energies lie between the chemical potential $\mu$ (solid black line) and $\mu+\omega$ (or $\mu-\omega$), indicated by the dotted green (or red) line. The frequency $\omega_{\rm max}$, at which the LDOS is maximal, is indicated by the black dashed-dotted line.}
\label{fig:eh-pairs}
\end{figure}
In this section, we substantiate the claim of the main text that the characteristic frequency $\omega_{\rm spin}$ of spin fluctuations is given by the distance between the chemical potential, $\mu$, and the lower effective band edge, $\omega_{\rm max}$.
To do so, we consider the local non-interacting spin-susceptibility, defined in Eq.~\eqref{eq:spin-noise_definition}, which at zero temperature can be written as
\begin{equation}
\label{eq:nonint_local_spin_noise}
	\chi^{U=0}_{ii} (\omega) = 2 \pi^2 \int_{\mu}^{\mu+\omega} d\Omega \mathcal A_i(\Omega) \mathcal A_i(\Omega-\omega).
\end{equation}
Let us begin by analyzing its properties for a homogenous tight-binding model with hopping $\tau$ and Fermi energy $\epsilon_F$ close to the lower band edge, i.e.~$\epsilon_F =\mu + 2 \tau \ll D, \omega \ll D$, where $D=4 \tau$ is the band width. This choice of $\epsilon_F$ most closely resembles the situation in the center of a QPC.
$\mathcal A_i (\Omega)$ is zero for frequencies below the band edge, shows a divergence at the band edge and subsequently decreases monotonically with increasing frequencies [Fig.~\ref{fig:eh-pairs} (a,b)].
$\chi^{U=0}_{ii}$ essentially counts the number of available electron-hole excitations, where the electrons have an energy of $\Omega \in [\mu, \mu+\omega]$ and the holes an energy $\Omega-\omega \in [\mu-\omega, \mu]$ [Fig.~\ref{fig:eh-pairs} (a,b)].

Consider $\omega < \epsilon_F$ [Fig.~\ref{fig:eh-pairs}(a)]. Then
\begin{align}
\label{eq:spin_noise_estimate1}
	\partial_\omega \chi^{U=0}_{ii} (\omega) = - 2 \pi^2 &\int_{\mu}^{\mu+\omega} d\Omega \mathcal A_i(\Omega) \mathcal A_i'(\Omega-\omega) \nonumber \\
	                                                    &+ 2 \pi^2 \mathcal A_i(\mu+\omega) \mathcal A_i(\mu) > 0.
\end{align}
Here, the prime denotes a derivative.
Thus $\chi^{U=0}_{ii} (\omega)$ is a monotonically increasing function for $\omega < \epsilon_F$.
This can be understood intuitively by considering the effects of an infinitesimal increase in $\omega$: The first term in Eq.~\eqref{eq:spin_noise_estimate1} describes how, if the \textit{electron} remains at energy $\Omega$, the weight of the hole at energy $\Omega-\omega$ increases [$\mathcal A_i'(\Omega-\omega)$]. The second term in Eq.~\eqref{eq:spin_noise_estimate1} describes the appearance of additional electron-hole pairs.

For $\epsilon_F < \omega$ [Fig.~\ref{fig:eh-pairs}(b)] Eq.~\eqref{eq:spin_noise_estimate1} is not useful, as the derivative of $\cal A$ is ill-defined at the band edge. We thus rewrite Eq.~\eqref{eq:nonint_local_spin_noise} as
\begin{equation}
\label{eq:nonint_local_spin_noise2}
	\chi^{U=0}_{ii} (\omega) = 2 \pi^2 \int_{\mu-\epsilon_F}^{\mu} d\Omega \mathcal A_i(\Omega+\omega) \mathcal A_i(\Omega),
\end{equation}
where we have used the fact that $\cal A$ vanishes for arguments below the band edge to restrict the range of integration.
Using Eq.~\eqref{eq:nonint_local_spin_noise2} we obtain
\begin{flalign}
\label{eq:spin_noise_estimate2}
	\partial_\omega \chi^{U=0}_{ii} (\omega) = 2 \pi^2 \int_{\mu-\epsilon_F}^\mu d\Omega \mathcal A'_i(\Omega+\omega) \mathcal A_i(\Omega) < 0.
\end{flalign}
For $\epsilon_F < \omega$, $\chi^{U=0}_{ii}(\omega)$ is thus monotonically decreasing.
This can again be understood intuitively by considering the effects of an infinitesimal increase in $\omega$: consider an electron-hole pair with fixed \textit{hole} energy $\Omega$. The weight of the electron states near $\Omega+\omega$ [described by $\mathcal A_i'(\Omega+\omega)$] diminishes, reducing the spin susceptibility.

The above analysis and Eqs.~\eqref{eq:spin_noise_estimate1} and \eqref{eq:spin_noise_estimate2}, together, lead to the following important conclusion:
For the homogenous system considered so far, $\chi^{U=0}_{ii} (\omega)$ exhibits a local maximum at an energy, $\omega_{\rm spin}$, that corresponds to the Fermi energy, i.e.~to the distance between the chemical potential $\mu$ and the lower band edge $\omega_{\rm max}$, $\omega_{\rm  spin} = \mu - \omega_{\rm max}$.

We now switch to a QPC geometry in the presence of interactions. The inhomogenuity of the QPC potential changes the divergence of the bare LDOS at the band bottom into a broadened peak, but leaves the other features of the LDOS qualitatively unchanged [compare Fig.~\ref{fig:eh-pairs}(a) and (c) or (b) and (d)]. Within a Fermi liquid picture, where all of the above arguments still apply, albeit with renormalized parameters, we thus expect in the interacting QPC that $\omega_{\rm spin} \simeq \mu-\omega_{\rm max}$, where both $\omega_{\rm spin}$ and $\omega_{\rm max}$ are renormalized quantities.

\section{S-IV. Implementational Details}
The central region consists of $N=61$ sites. We use $\sim 1500$ frequencies to sample the real frequency axis.
One third of the frequencies is sampled exponentially in the region $\vert \omega \vert > 4 \tau$, the rest is sampled homogeneously in the region $\omega \in [-4\tau, 4 \tau]$. An additional $~100$ frequencies are included in windows of size $4T$ around $\mu$ and $2 \mu$.
In order to numerically perform the integrals, it is useful to map the real axis to a finite region. We thus represent $\omega \in {\mathbbm R}$ in terms of the variable $\tilde y = y/\tau \in (-7, 7)$ via
\begin{equation}
\omega =
\begin{cases}
	-2 \tau \frac{(\tilde y+6) (1+\Lambda)}{(\tilde y+6 )^2-1} -6 \tau,& \text{for } (\tilde y<-6) \\
	-2 \tau-\tau (\tilde y+2)^2/4,& \text{for } (-6 <\tilde y<-2) \\
	\tau \tilde y \sqrt{\frac{4}{\tilde y^2}-\frac{\tilde y^2-4^2}{4\tilde y^2}},& \text{for } (-2 < \tilde y<2) \\
	2 \tau+\tau (\tilde y-2)^2/4,& \text{for } (2 < \tilde y<6) \\
	-2 \tau \frac{(\tilde y-6)(1+\Lambda)}{(\tilde y-6)^2-1}+6 \tau,& \text{for } (6 < \tilde y).
\end{cases}
\end{equation}
The structure of this substitution is chosen such that the van Hove divergence at the band edges is trivially integrated (i.e.~the integral $\int d\omega (\omega \pm 2\tau)^{-1/2}$ is mapped to the integral ${\rm const.} \times \int d\tilde y 1$ for $\omega$ close to the band edges), while the large-frequency region is scaled with the flow parameter $\Lambda$ and substituted such that the integral $\int d\omega \omega^{-2}$ (the most diverging integral that occurs) is mapped to the integral ${\rm const.} \times \int d\tilde y 1$ for $\omega \gg \Lambda$. For convenience, $y=\pm 2 \tau, \pm 6 \tau$ is mapped to $\omega = \pm 2 \tau, \pm 6\tau$.
Continuous frequency information is obtained by linearly interpolating in $y$-space.

The flow equation is solved with a $6$th-order Runge-Kutta ODE solver with adaptive step size, while the integrals over internal frequencies are computed using Patterson sets.
The integrals over internal frequencies are split into multiple intervals, such that a strong dependence on the internal frequency occurs near the integration boundaries, as the sampling is more dense there.
The boundaries are determined by either the unsubstituted frequency of a Green's function or single scale propagator taking the value $\pm 2 \tau$, $\mu$, $\mu \pm 10 T$, $\pm (-2 \tau + V_{\rm c})$ or $\pm 2\tau \pm \Lambda$, or by the argument of the P-channel (X-channel, D-channel) taking the value $2 \mu$ ($0$).
The flow parameter used is not $\Lambda$, but rather $u := \log \left( \frac{\Lambda}{1+\Lambda} \right)$. This improves the dynamic choice of step size within the ODE-solver. The flow starts at $\Lambda \approx 10^5$ and goes down to $\Lambda \approx 10^{-9}$.
To minimize runtime, the Green's function and single scale propagator are computed at $\sim 30000$ frequencies, and a linear interpolation in $y$-space is used when either of them is required in an integrand.
In equilibrium, the matrices appearing are symmetric under an exchange of sites. Further, the model considered here has a left-right parity symmetry. Both symmetries are exploited by using symmetric matrices to store the self-energy and the vertex, and by using a parity basis in the computation of the Green's function and the single scale propagator.

\section{S-V. DMRG calculations}
\begin{figure}
\includegraphics[width=\columnwidth]{dmrg_compare_total_newML.pdf}
\caption{Comparison between Keldysh-fRG and DMRG results. (a) The local density as a function of position and (b) the LDOS as a function of frequency of a closed (green), subopen (red), and open (orange) QPC, computed without interactions (dashed lines), and with interactions (solid lines), using Keldysh-fRG (colored) and DMRG (black), respectively.}
\label{fig:compare_DMRG}
\end{figure}
The results in the main text are obtained using Keldysh-fRG, which is
based on a perturbative ansatz. To verify the validity of the fRG data, we also employ
density-matrix-renormalization-group (DMRG) calculations
\cite{White1992,*White1993,*Schollwoeck2005}.  DMRG represents one of
the most powerful quasi-exact numerical method for describing
one-dimensional quantum many-body systems regarding their static
ground-state, dynamic, as well as thermodynamic properties. In
particular, DMRG can treat fermionic systems with arbitrary
interaction strength due to its non-perturbative character.
Specifically, we have used DMRG to compute the local density $n$ [Fig.~\ref{fig:compare_DMRG}(a)] and the LDOS [Fig.~\ref{fig:compare_DMRG}(b)], obtaining good qualitative agreement between our DMRG (black) and Keldysh-fRG (colored) results.

Below, we first elaborate on some peculiarities of our
matrix-product-state (MPS) implementation \cite{Schollwoeck2011},
which could be of interest to practitioners, and then discuss the choice of model parameters used for this comparison.

\subsection{A. DMRG details}

The QPC model in Eq.~\eqref{eq:define_H} poses a particular challenge to DMRG since, in contrast to fRG, it is not possible to incorporate the non-interacting leads to the left and right of the interacting region by an additional term in the self-energy. Instead, a finite-size chain representation of both leads is necessary as a prerequisite to make the model accessible for DMRG. The simplest ansatz is to replace the semi-infinite leads by a finite-length tight-binding chain with open boundary conditions (OBC). However, this setup is not practicable as it requires to go to very large system sizes in order to avoid strong finite-size artefacts in physical properties in the interacting part of the QPC. Instead, we here employ the concept of smooth boundary conditions (SBC) \cite{Vekic_PRL_1993,Vekic_PRB_1996}, which enable us to minimize finite-size effects in the interacting region of the QPC. Implementing SBC, the parameters of the non-interacting tight-binding chains are smoothly decreased to zero towards both ends of the chain to avoid having a sharp and rigid boundary as in the OBC setup. Thus for the interacting region of the QPC, the system's size is no longer fully determinable. SBC enable us to mimic very large leads with only $\mathcal{O}(10)$ sites.

In practice, we scale the Hamiltonian parameters in the non-interacting regions (which we label symmetrically by $I=1, \dots, N_L$ for both the left and right lead; $I=1$ corresponds to the left- or rightmost boundary, $I=N_L$ to the lead sites closest to the central region) with a smoothing function $f_I$ such that $\tau_I=\tfrac \tau 2 (f_I + f_{I+1})$ and $\mu_I= \mu f_I$. Following Ref.~\cite{Vekic_PRB_1996}, we choose $f_I = y (1-I/[N_L+1]])$, and the smoothing function $y(x) = \tfrac 12 \big( 1- \tanh \frac{x-1/2}{x(1-x)}\big)$ for $0<x<1$, which interpolates between $1$ at the edge of the central region and $0$ and the boundary.

In this setup, we first determine the ground state of the QPC using standard DMRG formulated in terms of MPS. The LDOS ${\cal A}_i (\omega) = -\frac 1\pi {\rm Im} G^{R}_{ii} (\omega)$ is then determined using time-dependent DMRG \cite{Vidal04,*Daley04,*WhiteFeiguin04}. To this end, we carry out two independent tDMRG runs to determine the retarded correlator in the time domain, ${G^{R}_{ii} (t) = -i[\langle c_i^\dagger(t) c_i \rangle+ \langle c_i(t) c_i^\dagger \rangle^*]}$. The entanglement in the MPS increases linearly during the real-time evolution, thus the number of states $D$ kept in simulation needs to be continuously increased to kept the numerical error constant. This implies that the simulation is bound to some maximum time $T_{\rm max}$ at which the simulation is no longer numerically feasible. A finite-time cutoff typically introduces artificial oscillations in the Fourier transform, requiring some artificial broadening to obtain a smooth and positive definite LDOS. However, we can avoid incorporating a broadening function by extending $T_{\rm max}$ to much larger times by means of linear prediction \cite{White_PRB_2008_LinPre,Barthel_PRB_2009}. The extrapolation scheme is expected to work for the present model since the correlator $G^{R}_{ii} (t)$ decreases exponentially over time scales smaller than the inverse mean level spacing and larger than the lifetime of excitations in the central region.

We end this section with some technical notes.
All DMRG calculations in this work are performed with the QSpace tensor library of A. Weichselbaum \cite{weichselbaum12a}. We studied
a QPC with an interacting region consisting of $N=31$ sites and two non-interacting regions to the left and right containing $N_L=50$ sites each, yielding a total of $N^{\rm DMRG}_{\rm tot} = 131$ sites, whose parameters are tuned in terms of SBCs (see above). The DMRG ground-state calculation employs a two-site update keeping up to $D=1600$ states. Convergence was typically reached after 10 to 40 sweeps, 40 being required particularly for an almost closed QPC, where the low particle density slows down convergence and the algorithm can get stuck in local minima during early iterations. In the tDMRG simulations we use a second-order Trotter-Suzuki decomposition with a time step $\Delta t=0.05/\tau$ and adapt the number of states in the MPS dynamically by truncating all singular values smaller than $\epsilon_{\rm SVD} = 5 \cdot 10^{-5}$. We stop the simulation when the number of kept states in the MPS exceeds $D=4000$. In this setting, we typically reach time scales $T_{\rm max} \cdot \tau = 60 - 65$ before applying linear prediction.

\subsection{B. Choice of model parameters}

Since DMRG solves a finite system, we need a way to estimate the 'optimal' system size:
We extract the LDOS as a Fourier-transform of the real time Green's function, computed by DMRG.
However, the resulting LDOS is only reliable if the Green's function is evolved up to time scales of the order of the traversal time $t_{\rm trav}$, as at shorter times the low-energy quasi-particles have yet to leave the central region.
This means that the system size must be chosen sufficiently large, such that the reflection time $t_{\rm refl} \sim N^{\rm DMRG}_{\rm tot}/(2 \tau)$ (the time until the first quasi-particles reflected at the boundary return to the center) is larger than the traversal time: $t_{\rm refl} \gtrsim t_{\rm trav}$.
For the setup of the main text this yields $N^{\rm DMRG}_{\rm tot} \gtrsim 500$.
Combined with the fact that we need to perform time-evolution up to the traversal time $t_{\rm trav} \approx 250 / \tau$, this would have required an unfeasible amount of resources in DMRG.

In order to reduce the traversal time, we shrink the system (i.e.~reduce $N$) and make the QPC potential steeper (i.e.~increase the curvature $\Omega_{\rm x}$):
If the curvature is larger, a larger interaction is necessary to observe the same physics, as the LDOS is smeared out more.
We have tried to compensate for this by choosing an appropriately larger interaction.
Comparing Figs.~\ref{fig:resonant_energy_structure}(a) and \ref{fig:compare_DMRG}(b), we see that the qualitative features of the fRG-LDOS are the same: There is a roughly constant energy-shift of the LDOS in the open region, in the sub-open region the LDOS peak is sharpened (the effective potential is flatter) and pinned to the chemical potential, while the LDOS in the closed region is almost unaffected by interactions.
Since the new parameters yield results that exhibit the same qualitative features as those shown in the main text, we consider them a reasonable proxy for a direct comparison between DMRG and Keldysh-fRG.

To be specific, the set of parameters used for this comparison is:
$N^{\rm DMRG}=31$, $U^{\rm DMRG}=0.94 \tau$, $V_c^{\rm DMRG} = \{-1.69, -0.56, 0.56\} \Omega_{\rm x}^{\rm DMRG}$, $\mu^{\rm DMRG} = - \tau$, and $\Omega_{\rm x}^{\rm DMRG} \approx 0.9 \tau$.
Since $\Omega_{\rm x}$ is $3$ times larger than in the main text, the traversal time should be reduced by a factor of roughly $3$.
We find $t_{\rm trav} \approx 70 / \tau$, and thus estimate $N_{\rm DMRG}^{\rm tot} \gtrsim 140$ (we use $N_{\rm DMRG}^{\rm tot} = 131$), which is still viable.

Finally, we remark that the choice of time $t_{\rm lp}$, after which linear prediction is applied, is a subtle issue:
The linear prediction method does not capture any physics that happens at time scales $t \gg t_{\rm lp}$ (this is an intended feature of the method, e.g.\ to mask finite-size effects).
However, this implies that for $t_{\rm trav} \gg t_{\rm lp}$ there may exist times at which linear prediction appears stable (i.e.~robust against variation of parameters used in linear prediction), while missing the finer details of the LDOS.
This happens in our system for times $t_{\rm lp} \sim 30 / \tau$, and is generically to be expected in a system with multiple time scales.
Once the largest time scale surviving the limit of infinite leads is reached (which in our case is $t_{\rm trav}$), and provided that time scale is still much shorter than the inverse level spacing, linear prediction appears to yield reasonable long-time results.

\end{document}

redpajama_set_name: RedPajamaArXiv


text:
Q: EF4 And POCO Class and Inheritance and Error Mapping and metadata information could not be found for EntityType I have a POCO Class with name TaskWeek in my business layer and in my presentation layer I defined a new Class:
public class TaskWeekUI : TaskWeek
{
    public DateTime EndDate { get; set; }

    public string PersianEndDate
    {
        get
        {
            return UIUtility.ConvertToPersianDate(EndDate);
        }
    }
}

Then I have a method  public void Save(List taskweeks) , When I want to call this method I convert a List and call this method:
            using (TaskWeekDA twa = new TaskWeekDA())
            {
                IEnumerable<TaskWeek> tw = MainObjects.twUi.OfType<TaskWeek>();
                twa.Save(tw.ToList());
            }

but instead to save I've got this error :
Mapping and metadata information could not be found for EntityType
Thanks

A: Instead of inheriting from TaskWeek is it possible for you to write your methods in a partial class?
For example, write a TaskWeek.part.cs like:
//namespace has to be the same as TaskWeek that your model generated
public partial class TaskWeek
{
    public DateTime EndDate { get; set; }

    public string PersianEndDate
    {
        get
        {
            return UIUtility.ConvertToPersianDate(EndDate);
        }
    }
}

This way the EF has info about the class type, and you have the same functionality as with inheriting (that is, if you're not using inheritance for some other stuff).

redpajama_set_name: RedPajamaStackExchange


text:
有跑马灯显示功能的TextView。

## 效果图
![](https://github.com/reallin/github.picture.io/blob/master/marqeen1.png)
![](https://github.com/reallin/github.picture.io/blob/master/marqeen2.png)
## 实现代码
```java

public class MarqueeTextView extends TextView {

    public MarqueeTextView(Context con) {
        super(con);
    }

    public MarqueeTextView(Context context, AttributeSet attrs) {
        super(context, attrs);
    }

    public MarqueeTextView(Context context, AttributeSet attrs, int defStyle) {
        super(context, attrs, defStyle);
    }

    @Override
    public boolean isFocused() {
        // TODO Auto-generated method stub
        if(getEditableText().equals(TruncateAt.MARQUEE)){
            return true;
        }
        return super.isFocused();
    }
}

```

## 使用方法
布局中直接添加:
```html

<com.lxj.marqueetextview.MarqueeTextView
                android:id="@+id/home_location_tv"
                android:layout_width="0dp"
                android:layout_height="wrap_content"
                android:layout_marginLeft="10dp"
                android:layout_marginRight="10dp"
                android:layout_weight="1"
                android:ellipsize="marquee"
                android:focusable="true"
                android:focusableInTouchMode="true"
                android:gravity="center"
                android:marqueeRepeatLimit="marquee_forever"
                android:scrollHorizontally="true"
                android:singleLine="true"
                android:text="正在定位..."
                android:textColor="#39ac69"
                android:textSize="18sp" />

```

redpajama_set_name: RedPajamaGithub


Worst article I've ever read. A year ago there was less content but it was decent wuality.
Terrible grammar. Terrible explanation of btc transactions. I am actually paying for this garbage? Time to unsubscribe..
Thank you for the comments. I will try my best to improve.


text:
This article is from the Encyclopedia of North Carolina edited by William S. Powell. Copyright © 2006 by the University of North Carolina Press. Used by permission of the publisher. For personal use and not for further distribution. Please submit permission requests for other use directly to the publisher.
Printer-friendly page
by Marian Dodd, 2006
See also: Women Suffrage; North Carolina Equal Suffrage Association.
The League of Women Voters of North Carolina was founded in 1920 by leaders of the movement for women suffrage to inform and encourage active participation by citizens in public policy issues. As a nonpartisan organization, the league does not support or oppose any candidate or political party. It encourages citizen participation and education through voter registration drives, public debate, and forums, and it fosters change through advocate lobbying efforts at all levels. It also studies and promotes a wide range of issues, such as universal voter registration and effective election laws; fair, honest, and open campaign practices; protection of natural resources; responsible local, state, and federal fiscal policies; safe, affordable housing; child care and women's issues; an effective criminal justice system; and fair and equitable taxes.
The North Carolina league is a three-tiered organization with a national office in Washington, D.C., a state office in Raleigh, and a state board of directors that serves as an administrative unit for 27 local leagues representing 23 counties in the state.
Celebrating a Quarter Century of Political Participation in North Carolina: A History of the League of Women Voters in North Carolina, 1951-1976 (1976).
Kathryn L. Nasstrom, "'More Was Expected of Us': The League of Women Voters and the Feminist Movement in the 1920s" (M.A. thesis, UNC-Chapel Hill, 1988).
"League of Women Voters, 1927." Courtesy of Duke University Archives. Available from https://www.flickr.com/photos/dukeyearlook/2108912472/ (accessed May 1, 2012).
Societies, clubs, and organizations
UNC Press
Dodd, Marian
Origin - location:
Encyclopedia of North Carolina, University of North Carolina Press.
1 January 2006 | Dodd, Marian
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RT @NCArchives: Interested in family history? Join UNC-TV on Thursday, Jan. 14, 7-9 pm for a FREE preview screening of the upcoming… https://t.co/5XvohL18LK — 1 day 9 hours ago

redpajama_set_name: RedPajamaCommonCrawl


text:
Nabeel Rajab: a Lifelong Human Rights Defender
Posted on July 31, 2016 July 28, 2016 by ADHRB Admin
On 13 June 2016, Bahraini authorities arrested Nabeel Rajab on grounds that he spread "false news and rumors about the internal situation in a bid to discredit Bahrain." This arrest is the government's latest attempt to stifle Rajab's human rights activism. Despite its multiple attempts to jail Rajab and extinguish his activism, the Government of Bahrain has not been successful.
Over many years of activism, Rajab has demonstrated his passion for bringing human rights to Bahrain. His positive influence has inspired many other activists both in Bahrain and abroad to take up the cause and fight for rights the Bahraini government has been denying its citizens for decades. His presence and influence continues to impact other human rights defenders, and his continual arrests on trumped up charges of expression exemplifies the ongoing need for human rights in Bahrain.
Rajab has spent over fifteen years establishing and managing multiple institutions designed to address the human rights abuses affecting Bahraini citizens and residents. Rajab solidified his role as a human rights defender in 2001, when he co-founded the Bahrain Human Rights Society. In July 2002, Rajab co-founded the Bahrain Center for Human Rights (BCHR) along with other prominent human rights defenders, such as Abdulhadi Al-Khawaja,. Despite a governmental order to dissolve in 2004, the organization still actively promotes civil and human rights in Bahrain. BCHR has remained committed to advocating for Rajab's vision even during his multiple prison sentences. In 2003, Rajab was a founding member of the Migrant Workers Protection Society, which advocated for the human rights of the thousands of low-paid foreign workers in Bahrain.
Rajab also actively strives for justice in countries other than Bahrain. In April 2011, Rajab and Abdulhadi Al-Khawaja founded the Gulf Center for Human Rights (GCHR), another society striving to make the governments respect and protect human rights. Unlike the previous organizations he founded, GCHR is concerned with cultivating healthy human rights environments in the six Gulf Cooperation Council (GCC) member states as well as Yemen.
Because of his proven record of fighting for human rights, many organizations from around the world have sought Rajab for leadership positions. In October 2009, Coordination of Action Research on AIDS and Mobility (CARAM) Asia elected Rajab Chairperson due to his previous work promoting the rights of migrant workers. In late 2009, Human Rights Watch (HRW) chose him to be a member of the advisory board of the Middle East/North Africa division. He still holds this position. In July 2010, the International Federation for Human Rights (FIDH) appointed Rajab to Deputy Secretary-General. FIDH also appointed him to be their permanent representative in the GCC.
In addition to founding and holding leadership positions in organizations aimed at advancing human rights in Bahrain and around the world, Rajab peacefully advocated for positive government change by taking part in the 2011 Bahrain Pearl Roundabout protests. As a result of his participation in the movement, government authorities beat, arrested, and detained him for exercising his rights to free speech, free assembly, and free expression.
International organizations and charities have recognized Rajab's positive influence on Bahraini human rights, especially after his involvement in the 2011 protests. In July 2011, the Woodrow Wilson International Center for Scholars presented him with the Ion Ratiu Democracy Award, which aims to recognize individuals working "on behalf of democracy." In recognizing Rajab, CEO, director, and president of the Wilson Center Jane Harman said of him, "in such a tumultuous year, Nabeel Rajab's efforts to peacefully advance democratic freedoms for Bahraini citizens, even in the face of considerable personal peril, are truly exemplary." Later that year, in September 2011, the trustees of the Silbury Fund, a British charity, unanimously voted to award Rajab the Silbury Prize and 1,000 GBP to help "facilitate his on-going humanitarian and human rights work."
As his stature has grown around the world, Rajab continues to engage with the Bahraini government, urging officials to respect and protect the human, civil, and political rights of its citizens. The Government of Bahrain has arrested and charged Rajab many times in relation to his advocacy. In August 2012, courts sentenced him to three years in prison for protesting against the government's repression of human rights. After serving two years of the sentence, he was released, and spent two months campaigning in Europe against his treatment and the plight of human rights defenders in Bahrain. Following his return, Bahraini forces arrested him again over tweets critical of the government's security institutions. The Government of Bahrain has charged Rajab with multiple charges related to his exercise of the right to free speech since 2014. His most recent arrest, on 13 June 2016, is no exception.
Nabeel Rajab's commitment to ensuring the human rights of others, in- and outside Bahrain, is evident in his actions. Despite the Bahraini government's multiple attempts to silence his voice, Rajab has had too great an impact on Bahraini citizens and the international community to be effectively muted. His passion for creating safer human rights environments has positively impacted Bahraini citizens and residents. Through his activism, he has created numerous human rights institutions that strive to facilitate universal human rights and been recognized by human rights organizations across the globe. His work has impacted the lives of many individuals, both Bahraini citizens and members of the international community.
Brittany Hamzy is an Advocacy Fellow at ADHRB.
This entry was posted in Bahrain and tagged Bahrain, BCHR, Human Rights, Nabeel Rajab.


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Q: Escaping special characters with sed I have a script to generate char arrays from strings:
#!/bin/bash
while [ -n "$1" ]
do
    echo -n "{" && echo -n "$1" | sed -r "s/((\\\\x[0-9a-fA-F]+)|(\\\\[0-7]{1,3})|(\\\\?.))/'\1',/g" && echo "0}"
    shift
done

It works great as is:
$ wchar 'test\n' 'test\\n' 'test\123' 'test\1234' 'test\x12345'
{'t','e','s','t','\n',0}
{'t','e','s','t','\\','n',0}
{'t','e','s','t','\123',0}
{'t','e','s','t','\123','4',0}
{'t','e','s','t','\x12345',0}

But because sed considers each new line to be a brand new thing it doesn't handle actual newlines:
$ wchar 'test
> test'
{'t','e','s','t',
't','e','s','t',0}

How can I replace special characters (Tabs, newlines etc) with their escaped versions so that the output would be like so:
$ wchar 'test
> test'
{'t','e','s','t','\n','t','e','s','t',0}

Edit: Some ideas that almost work:
echo -n "{" && echo -n "$1" | sed -r ":a;N;;s/\\n/\\\\n/;$!ba;s/((\\\\x[0-9a-fA-F]+)|(\\\\[0-7]{1,3})|(\\\\?.))/'\1',/g" && echo "0}"

Produces:
$ wchar 'test\n\\n\1234\x1234abg
test
test'
{test\n\\n\1234\x1234abg\ntest\ntest0}

While removing the !:
echo -n "{" && echo -n "$1" | sed -r ":a;N;;s/\\n/\\\\n/;$ba;s/((\\\\x[0-9a-fA-F]+)|(\\\\[0-7]{1,3})|(\\\\?.))/'\1',/g" && echo "0}"

Produces:
$ wchar 'test\n\\n\1234\x1234abg
test
test'
{'t','e','s','t','\n','\\','n','\123','4','\x1234ab','g','\n','t','e','s','t',
test0}

This is close...
The first isn't performing the final replacement, and the second isn't correctly adding the last line

A: You can pre-filter before passing to sed. Perl will do:
$ set -- 'test1
> test2'
$ echo -n "$1" | perl -0777 -pe 's/\n/\\n/g'
test1\ntest2

A: This is a very convoluted solution, but might work for your needs. GNU awk 4.1
#!/usr/bin/awk -f
@include "join"
@include "ord"
BEGIN {
  RS = "\\\\(n|x..)"
  FS = ""
}
{
  for (z=1; z<=NF; z++)
    y[++x] = ord($z)<0x20 ? sprintf("\\x%02x",ord($z)) : $z
  y[++x] = RT
}
END {
  y[++x] = "\\0"
  for (w in y)
    y[w] = "'" y[w] "'"
  printf "{%s}", join(y, 1, x, ",")
}

Result
$ cat file
a
b\nc\x0a

$ ./foo.awk file
{'a','\x0a','b','\n','c','\x0a','\0'}