text:
\section{Introduction}
\label{sec:Intro}
As one of the most intriguing fields in nuclear physics with available heavy ion beams, the
exploration of resonant structure especially from an alpha-clustering perspective in light nuclei has
attracted much attention in recent years \cite{Freer2007, Oertzen2006, He2014, He2014N, Tao2013, Ye2014}.
Measurements in inverse kinematics, for instance, were made to determine the exotic structure or decay mode of the compound
nuclei $^{18}$F, $^{18}$Ne, $^{11,14}$C, $^{11}$N \cite{Bailey2014, Campo2000, Freer2012, Freer2014, Markenroth2000}.
For the $^{11}$B nucleus, which was identified as an alpha-clustering compound system at several low-lying excited states,
there are many measurements using different reactions endeavored for extracting
the information of resonant structure and energy levels, while the ambiguousness for its high excitation state properties of
isospins T = 3/2 and T = 1/2 still remains. Just as shown in Figure \ref{b11levelspre}, little spectroscopic
knowledge of unbound excited states in $^{11}$B has been determined due to the difficulty of extracting the parameters of states
above the separation energies of proton (Q$_p$ = 11.2285 MeV), neutron (Q$_n$ = 11.4541 MeV) and
triton (Q$_t$ = 11.2235 MeV), where several particle channels are simultaneously open. As the analog of the 1.78 MeV state in $^{11}$Be, primary evidence for the 14.33 MeV state with $J^\pi$ = ($5/2^+$, $3/2^-$), T = 3/2 assignment comes from the $^{10}$Be(p, $\gamma$)$^{11}$B reaction \cite{Goosman1973},
which also yielded the population of 15.30 MeV, the analog state of 2.70 MeV in $^{11}$Be, but without spin-parity and T assignment. For the 15.30 MeV state, there is no definite $J^\pi$ information reported in Refs. \cite{Zwieglinski1979, Zwieglinski1982, Soic2004, Watson1971, Fletcher2003}, and its width and spin-parity determinations from the work \cite{Goosman1973, Aryaeinejad1985, Sadowski1990} are also questionable just as demonstrated by Refs. \cite{Fortune2006, Barker2007}.

In the present work, the measurement was undertaken in the hope
that the properties, i.e., the excitation energy E$_{x}$, the spin-parity J$^\pi$, and the proton-decay partial width of those unbound excited states lied over the energy region of present interest shown in Fig. \ref{b11levelspre} could be extracted and determined by employing the elastic resonance reaction $^{10}$Be + p with the thick-target technique in inverse kinematics
(TTIK) \cite{Artemov1990, Uribarri2000, Hernandez1998}. Comparing with those traditional approaches,
the TTIK method takes advantage of the fact that a continuous excitation function spectrum
can be achieved when the incident energy of beam particles on the target remains unchanged. In the following, the experiment details are described in Sec. \ref{sec:expintro}, the results and discussion are presented in Sec. \ref{sec:resultanddiss}, and finally the summary is given in Sec. \ref{sec:sum}.

\begin{figure}[htbp]
\resizebox{8.6cm}{!}{\includegraphics{Fig1_b11previous.eps}}
\vspace{-0.8cm}
\caption{(Color online) The partial energy level diagram of  $^{11}$B from Refs.~\cite{Ajzenberg1975, Ajzenberg1985, Kelly2012}, with energy levels in MeV and the possible spin-parities and isospin component (T) shown in parenthesis.}
\label{b11levelspre}
\vspace{+0.0cm}
\end{figure}

\section{details of the experiment}
\label{sec:expintro}

The measurement of $^{10}$Be + p elastic resonance scattering was carried out at the Heavy Ion Research
Facility in Lanzhou (HIRFL) \cite{Zhan2008, Zhan2010}, China. A secondary ion beam of $^{10}$Be was
produced by the Radioactive Ion Beam Line in Lanzhou (RIBLL) through the projectile fragmentation
of a 59.62$A$ MeV $^{18}$O primary beam with an average intensity of 100 enA bombarded on a solid  853.96 mg/cm$^{2}$-thick target of $^{9}$Be. The schematic view of RIBLL and the experiment setup for the present measurement are shown in Fig. \ref{expset}. At the first momentum-dispersive focal plane (C1), a 3060 $\mu$m-thick aluminium achromatic degrader was installed to reject unwanted ion species from the $^{10}$Be beam. Two nominal horizontal slits at focal planes C1 (S1: $\pm$ 15 mm) and C2 (S2: $\pm$ 25 mm) were used to restrict the momentum spread of $^{10}$Be beam. The time of flight (TOF) from the first achromatic focal point T1 to the second one T2 was measured by two 50 $\mu$m-thick plastic scintillators.

\begin{figure}[htbp]
\resizebox{8.6cm}{!}{\includegraphics{Fig2_expsetup_RIBLL.eps}}
\vspace{-0.8cm}
\caption{(Color online) The schematic view of RIBLL and the experiment setup for the measurement of $^{10}$Be + p reaction. See text for details. }
\label{expset}
\vspace{+0.0cm}
\end{figure}

At the scattering chamber (see Fig. \ref{expset}), the incident angles of beam particles on the target were determined by extrapolating the hit-positions recorded by two parallel-plate avalanche counters (PPAC), i.e., PPACa and PPACb, which were 500 mm and 100 mm away from the target, respectively. Consequently, the $^{10}$Be beam of about 92\% purity with an intensity of about 5$\times$10$^{3}$ pps and an energy of 7.2$A$ MeV impinged on a 41.85 mg/cm$^{2}$-thick polyethylene (PE) target. In addition, a 280 $\mu$m-thick silicon detector (Si) was inserted in front of PPACa during the beam tuning to perform the particle identification with TOF. As shown in Fig. \ref{de_tof}, $^{10}$Be nuclei can be distinguished unambiguously with the help of TOF.

\begin{figure}[htbp]
\resizebox{10.6cm}{!}{\includegraphics{Fig3_dE_TOF.eps}}
\centering
\vspace{-0.8cm}
\caption{(Color online) The particle identification plot of E(Si) versus TOF detectors (top panel) and its projection on the TOF-axis (bottom panel). E is the energy deposited in the Si detector with 280 $\mu$m thickness, and TOF is the time of flight (ns) of particles from T1 to T2.}
\label{de_tof}
\vspace{+0.0cm}
\end{figure}

The scattered particles from the reaction target were detected by three downstream $\Delta$E-E silicon
telescopes at the end of the scattering chamber, which were placed at laboratory angles of $\theta_{lab}$
$\approx$ +16$^{o}$, -16$^{o}$, and 0$^{o}$ with respect to the beam axis (see Fig. \ref{expset}).
The distances from the telescopes to the reaction target, with solid angles of 21 msr, 20 msr, and
21 msr, were set to be about 344 mm, 349 mm, and 345 mm, respectively. For the sake of convenience in the
following description, hereafter the three $\Delta$E-E silicon telescopes are referred to Set1, Set2, and Set3.
The averaged scattering angles for recoiled particles were determined to be
$\theta_{c.m.}$ $\approx$ 148$^{o}$, 147$^{o}$, and 180$^{o}$ via the relation of
$\theta_{c.m.}$ = 180$^o$ - 2$\theta_{lab}$. The $\Delta$E detectors were
position-sensitive double-sided-strip detectors (16 $\times$ 16, 3 mm width of each strip),
measuring  energy and two-dimensional position information of the recoiled particles with
thickness of 301 $\mu$m, 300 $\mu$m, and 149 $\mu$m, respectively. The E silicon detectors, with thickness of 1533 $\mu$m and 1524 $\mu$m in Set1 and Set2, and 1528 $\mu$m + 1538 $\mu$m in Set3,
measured the partial or whole energy, depending on whether the scattered particles punching through
the $\Delta$E-E telescopes. As demonstrated in Fig. \ref{partidwithg4}, the combination
of $\Delta$E-E detectors clearly revealed the ability of separating protons from deuterons and tritons while heavier ions were stopped in the target. The time of flight and Si detectors were calibrated using the primary beam $^{18}$O, while the position calibration for PPACs and the energy calibration for three silicon telescopes were performed utilizing a standard double $\alpha$ source of energy 5.156 MeV from $^{239}$Pu and 5.499 MeV from $^{238}$Pu. Since the two points of $\alpha$ source are very close in energy, the back bending points for protons and tritons in the $\Delta$E (channel)-E (channel) plot were utilized to carry out the calibration for three silicon telescopes. As shown in Fig. \ref{partidwithg4}, the calibration result was in excellent agreement with that from GEANT4 \cite{Agostinelli2003} simulations.

\begin{figure}[htbp]
\resizebox{8.6cm}{!}{\includegraphics{Fig4_stsi2_pidwithG4.eps}}
\vspace{-0.8cm}
\caption{(Color online) The scattering plot of recoiled light particles (colored points) from Set1 $\Delta$E-E telescope, with the locus of protons, deuterons, and tritons labeled by arrows. The simulation results (red points) of the present reaction by GEANT4~\cite{Agostinelli2003} are also shown for comparison.}
\label{partidwithg4}
\vspace{+0.0cm}
\end{figure}

Additionally, in the last stage of the measurement, an evaluation of background contribution from the reactions of $^{10}$Be with carbon nuclei in the (CH$_{2}$)$_n$ target was performed through a separate run with a 50.69 mg/cm$^{2}$-thick carbon target and the same $^{10}$Be beam as described above. In the following spectra, the data was directly extracted from the locus of protons as shown in Fig. \ref{partidwithg4} except for the high-energy ones that were ceased by depositing the remaining energy into detector telescopes. The proton yield ratio of two runs was normalized by the number of beam particles and target thickness per unit energy loss of the incident beam in the target.

\section{results and discussions}
\label{sec:resultanddiss}

For the elastic resonance scattering, the relation between the energy of recoiled protons detected at a laboratory scattering angle $\theta_{sca}$ with the center-of-mass (C.M.) energy, is expressed as:
\begin{equation}
E_{c.m.}=\frac{A_P+A_T}{4A_P\cos^{2}\theta_{sca}}E_p,
\label{ecm}
\end{equation}
where A$_{P}$ and A$_{T}$ are the mass numbers of projectile and target nuclei, respectively, and E$_{p}$ is total proton
energy derived using an energy loss program \cite{Ziegler1980} within an uncertainty of about 5\%.
The energy loss of particles in the target was deduced from a Monte Carlo simulation. The $\theta_{sca}$ represents the scattering angle between the beam direction and the outgoing direction of protons. The proton energy spectra for Set1 and Set2 are shown in Fig.~\ref{Ep}. The deduced E$_{c.m.}$ resolution, consisting of the energy width of the secondary beam, the energy resolution of the silicon telescope, the angular resolution of the scattering angle, and the energy straggling in the target, is estimated to be 40 $\sim$ 100 keV, depending on the relative energies of the reaction system.

\begin{figure}[htbp]
\resizebox{8.6cm}{!}{\includegraphics{Fig5_Ep.eps}}
\vspace{-0.8cm}
\caption{(Color online) The energy spectra of protons for Set1 (a) and Set2 (b), in which the energy straggling in the target have been corrected. The contributions of protons from the reactions of $^{10}$Be with carbon nuclei in the (CH$_2$)$_n$ target are indicated by red curves.}
\label{Ep}
\vspace{+0.0cm}
\end{figure}

It is possible that the proton-decays from compound nuclei $^{11}$B to the first excited state
(3.368 MeV, 2$^{+}$) or higher excited states in $^{10}$Be contribute to proton spectra.
However, according to the available data of elastic and inelastic proton scatterings from a $^{10}$Be
target \cite{Auton1970} and using a distorted-wave Born approximation (DWBA)  to the energy of interest, the inelastic scattering
contributions in our case from the channels of p + $^{10}$Be$^{\ast}$ can be neglected, thus the energy
derived from Eq. (\ref{ecm}) can be reasonably recognized as the elastic scattering events.
In addition, the background contribution of protons from the reactions of $^{10}$Be with carbon nucleus
in the (CH$_2$)$_n$ target is found to be negligible in comparison with those of the p + $^{10}$Be reactions as shown in Fig. \ref{Ep}.

The achieved differential cross section through a transformation from the laboratory to the C.M. frame energy is presented by
\begin{equation}
\left(\frac{d\sigma}{d\Omega}\right)_{c.m.}(\theta_{c.m.}, E_{c.m.})=\frac{1}{4\cos\theta_{lab}}\left(\frac{d\sigma}{d\Omega}\right)_{lab}(\theta_{lab}, E_{p}),
\label{lab2cmdiffcro}
\end{equation}
where E$_{c.m.}$ is the C.M. energy of p + $^{10}$Be system calculated by Eq. (\ref{ecm}). The quantity on the right-hand side of Eq. (\ref{lab2cmdiffcro}) is expressed as
\begin{equation}
\left(\frac{d\sigma}{d\Omega}\right)_{lab}(\theta_{lab}, E_{p})=\frac{N_{p}}{N_{0}N_{t}\Delta\Omega},
\label{labdiffcro}
\end{equation}
representing the nominal laboratory differential cross section at recoiled proton energy E$_{p}$ and laboratory angle $\theta_{lab}$ for silicon telescopes placed with respect to the reaction target, with N$_{0}$ being the total number of $^{10}$Be bombarded on the (CH$_{2}$)$_{n}$ target, N$_{t}$ being the number of hydrogen atoms per unit area (cm$^{2}$) per energy bin (dE$_{p}$) in the target, and N$_{p}$ being the number of protons detected per energy bin by the $\Delta$E-E telescope with a solid angle $\Delta\Omega$.

\begin{figure}[htbp]
\centering
\resizebox{10.9cm}{!}{\includegraphics{Fig6_excitfunc.eps}}
\vspace{-0.8cm}
\caption{(Color online) The extracted excitation function of cross sections for Set1 (a) and Set2 (b), with the vertical error bars
 showing statistical errors only, the horizontal error bars representing systematic uncertainty of energy levels, and arrows indicating the populated resonant states with energy level in MeV. The solid curves are the R-matrix fits to the data. See text for details.}
\label{excitfun}
\vspace{+0.2cm}
\end{figure}

The excitation energies in $^{11}$B are deduced using the relation of E$_{x}$ = E$_{r}$ + Q$_p$ where the proton separation energy Q$_p$ is 11.2285 MeV and the resonant energy E$_r$ is obtained by the following R-matrix analysis of the data, the resulting proton excitation functions are depicted in Fig. \ref{excitfun} in which the background contributions of protons from the contamination of other reactions were subtracted. Small discrepancy between the two data sets respectively shown in Fig. \ref{excitfun} (a) and (b) can be seen, and it mainly originates from the counting statistics, the finite size of the detectors, and the contamination of deuterons and tritons as shown in Fig. \ref{partidwithg4}.

\begin{table*}[htbp]
\caption{The extracted resonant level parameters from the R-matrix best-fit to the present data. Those tentative assignments of J$^\pi$ or probably populated states are put in parentheses. For comparison, the extracted total or partial widths of states ($\Gamma_{p}$) and corresponding widths of Wigner limit ($\Gamma_{W}$) are shown in unit of keV. Errors of overall uncertainty for energy levels and fitting uncertainty for resonant partial widths are given.}
\label{TBestPar}
\flushleft
  \begin{tabular}{p{100pt}   p{100pt}    p{60pt}   p{65pt}  p{90pt}   p{70pt} }
  \hline
  \hline
          E$_x$ (MeV)        &  E$_{r}$ (MeV)   & l$_p$  &  J$^\pi$            & $\Gamma_{p}$ (keV)  & $\Gamma_{W}$ (keV)\\
          \hline
          13.46 $\pm$ 0.13       &  2.23 $\pm$ 0.13    &  0     &  $1/2^+$            & 608 $\pm$ 242       &  1537 \\
          14.55 $\pm$ 0.07       &  3.32 $\pm$ 0.07    &  2     &  $3/2^+$            & 475 $\pm$ 80        &  547\\

          14.74 $\pm$ 0.09       &  3.51 $\pm$ 0.09    &  1     &  $3/2^-$            & 830 $\pm$ 145       &  1507\\
          (16.18) $\pm$ 0.21     &  4.95 $\pm$ 0.21    &  1     &  ($1/2^-$, $3/2^-$) & $<60$               &  206\\
  \hline
  \hline
  \end{tabular}
\end{table*}

Analysis of the data was performed by a multichannel R-matrix program $Multi$ \cite{Lane1958, Brune2002, Descouvemont2010} to extract resonant parameters such as the resonant energy E$_{r}$, the spin-parity J$^{\pi}$, and the proton-decay partial width $\Gamma_{p}$. In the present work, a channel radius of 4.606 fm given by R$_{c}$ = 1.46(A$_T$$^{1/3}$ + A$_P$$^{1/3}$) has been utilized. The fit results were insensitive to the choice of the radius in the present energy region of interest. The inelastic channel and $\gamma$ capture channel widths are neglected. Therefore, the single particle width is derived by $\Gamma_{sp}$ = 2P$_{l}\gamma^2$, where P$_l$ is the barrier penetrability factor and $\gamma^2$  is the reduced particle partial width. In practice, the calculated particle widths are frequently compared with the Wigner limit width as a measure of the partial width of a resonance in terms of $\Gamma_W$ = 2P$_{l}\gamma_W ^2$ with $\gamma_W^2$ = 3$\hbar^2$/2$\mu$R$_c^2$ in which $\mu$ is the reduced mass \cite{Teichmannand1952}.

Since the spin-parity of protons and the ground state of $^{10}$Be are 1/2$^+$ and 0$^+$, respectively, the incident channel spin is determined to be $s = 1/2$. Regarding conservation of the total angular momentum $J$ obtained with relative orbit angular momentum $l$ coupling to the channel spin $s$, many R-matrix fits with all possible spin-parity combinations for observed resonances were attempted to reproduce the data. Consequently, the best-fit parameters are summarized in Table \ref{TBestPar} and the resulting fits are displayed in Fig. \ref{excitfun}. In the following subsections, the detailed discussions for the observed states are presented.

\subsection{Bump around 13.46 MeV and Peak at 14.55 MeV}

Because of the close proximity and the proton-decay channel, the peak at E$_x$ = 14.55 MeV is assumed to represent a resonance as the well-known state at 14.34 MeV, and the large proton-decay width of this resonance precludes the possibility of the previously known state at 14.56 MeV \cite{Kelly2012}. Attempts were made to fit the peak around E$_x$ = 14.50 MeV with J$^\pi$ = 1/2$^+$, 3/2$^\pm$, and 5/2$^+$, which are assignments made by previous measurements \cite{Goosman1973, Zwieglinski1982, Zwieglinski1979, Soic2004}. Calculations were also guided by the most recently tabulated levels for $^{11}$B \cite{Kelly2012} and the isobaric analog states in $^{11}$Be. This state was firstly observed at E$_x$ = 14.33 MeV in the $^{10}$Be(p, $\gamma$)$^{11}$B reaction \cite{Goosman1973} and identified as the analog state of 1.78 MeV in $^{11}$Be with the most probable J$^\pi$ = 5/2$^+$ assignment~\cite{Pullen1962}, but the possibility of J$^\pi$ = 3/2$^+$ was firmly excluded on the basis of quadrupole-to-dipole amplitude analysis. Later, other studies on this state were totally based on Goosman's measurement. Therefore, the preferred assignment for this state is adopted to be J$^\pi$ = 5/2$^+$ in the recent compilation of A = 11 nuclei \cite{Kelly2012}. Our result demonstrates that the curve with J$^\pi$ = 3/2$^+$ assignment is in accordance with the data shown in Fig. \ref{levelsfit}(b) as the solid curve, while the combination of J$^\pi$ = 5/2$^+$ with other states yields good description of the overall shape except for the height of the prominent peak shown in Fig. \ref{levelsfit}(b) as the black dotted curve.

\begin{figure}[htbp]
\resizebox{9.2cm}{!}{\includegraphics{Fig7_levels_enlarge.eps}}
\vspace{-0.8cm}
\caption{(Color online) R-matrix fits to the present data with resonant energy levels near 13.46 MeV (left panel), 14.55 MeV, and 14.74 MeV (right panel).}
\label{levelsfit}
\vspace{+0.0cm}

\end{figure}

Most of previous studies have measured the state J$^\pi$ = 5/2$^+$, and excitation energies of E$_x$ = 14.34 \cite{Goosman1973}, 14.47 \cite{Soic2004}, 14.40 \cite{Watson1971,Zwieglinski1982}, and 14.30 MeV \cite{Fletcher2003} have been reported. In the current measurement, the experimental resolution and counting statistics prevent us from making a firm assignment, thus the candidate J$^\pi$ = 5/2$^+$ can not be excluded. Instead, it may be a strong candidate as suggested by the DWBA and shell-model calculations in Refs. \cite{Zwieglinski1982, Teeters1977, Kohen1970}. A high-resolution study is required to make a definite assignment for this state.

As shown in Figs. \ref{excitfun} and \ref{levelsfit}, there is a bump in the low-energy region of
the excitation function. This structure was the result of low-energy cutoff
imposed by the $\Delta$E-E telescope except for the first two points, since no protons with energy lower
than E$_x$ = 13.0 MeV could be detected. As shown in Fig. \ref{levelsfit}(b), exclusion of this bump around 13.46 MeV results in a much less satisfactory fit by the blue dashed line for the overall shape of the spectrum. To improve the fitting, the bump is introduced as a resonance. Attempts were made to fit the bump with $J^\pi$ = $1/2^{\pm}$, $3/2^{\pm}$, and 5/2$^{\pm}$. As demonstrated by the blue dashed curve in Fig. \ref{levelsfit}(a), the bump can be well described by a Breit-Wigner form \cite{Freer2012M} with transferred angular momentum l$_p$ = 0. The excitation functions were well reproduced by using an assignment of J$^\pi$ = 1/2$^{+}$ for this state in combination with other states observed, but the extracted proton-decay width was roughly estimated to be $\Gamma_{p}$ = 608 keV with large uncertainty due to the paucity of the low-energy data.

For excitation energies at E$_x$ = 13.30 MeV \cite{Groce1963} and 13.63 MeV \cite{Cusson1966}, no information of the spin-parity and the particle width has been reported so far. In this work, the resonance at E$_x$ = 13.46 MeV was tentatively assumed to be the 13.30 MeV or 13.63 MeV state but not the member of the doublet, i.e., the 13.137 MeV (9/2$^+$) state and 13.16 MeV (5/2$^+$, 7/2$^{+}$) state \cite{Zwieglinski1982}, due to the restriction of the largest transferred angular momentum (l$_{max}$$\leq3$). Alternatively, this resonant state may be an unnatural-parity state that was not observed previously. Further studies above the $^{10}$Be + p threshold are needed to obtain a firm assignment for this state.

\begin{table*}[htbp]
\caption{The energy levels of $^{11}$B with proton-decay channels observed by the previous and present reactions in the energy range covered by the present measurement.}
\label{B11table}
\flushleft
          \begin{tabular}{p{130pt}   p{90pt}    p{115pt}      p{156pt} }
          \hline
          \hline
       E$_x$ (MeV)  &  J$^\pi$            &  $\Gamma_{p}$ (keV)   &Reactions (year)\\
          \hline
          13.30     &                     &                 &  (1963) \cite{Groce1963}\\
          13.63     &                     &                 & $^{7}$Li($\alpha$, $\alpha$$^\prime$)$^{7}$Li, $^{7}$Li($\alpha$, $\alpha$)$^{7}$Li (1966) \cite{Cusson1966}\\
          $\textbf{13.46 $\pm$ 0.13}$     &1/2$^+$          & 608 $\pm$ 242 & \textbf{this work}\\

          \hline
          14.34$^*$ &   5/2$^+$           &  253            &   \cite{Kelly2012}\\
          14.33     &  (5/2$^+$, 3/2$^-$) &  255            &   $^{10}$Be(p, $\gamma$)$^{11}$B (1973) \cite{Goosman1973}\\
          14.33     &  (1/2$^+$, 3/2$^+$, 5/2$^+$)$^a$&     &   $^9$Be(t, p)$^{11}$Be (1962) \cite{Pullen1962}\\
          1.78$^b$  &  (3/2$^+$, 5/2$^+$) &                 &   $^{10}$Be(d, p)$^{11}$Be (1979) \cite{Zwieglinski1979}\\
          14.40     &  (5/2$^+$)$^c$      &  261            &   $^{9}$Be($^3$He, p), $^{9}$Be($\alpha$, d) (1982) \cite{Zwieglinski1982}\\
          14.47     &                     &                 &   $^{16}$O($^9$Be, $\alpha^7$Be), $^7$Li($^9$Be, $\alpha$ $^7$Li) (2004) \cite{Soic2004}\\
          14.40     &                     &  220 $\pm$ 50.   &   $^{11}$B(t, t$^{\prime}$)$^{11}$B (1974) \cite{Watson1971}\\
          14.30     &                     &                 &   $^{7}$Li($^7$Li, t)$^{11}$B (2003) \cite{Fletcher2003}\\
          $\textbf{14.55 $\pm$ 0.07}$   &   (3/2$^+$, 5/2$^+$)  & 475 $\pm$ 80 & \textbf{this work}\\

          \hline
          15.29$^*$ &  (3/2$^-$)          &  282            &   \cite{Kelly2012}\\
          15.3      &                     & 635 $\pm$ 180  &   $^{10}$Be(p, $\gamma$)$^{11}$B (1973) \cite{Goosman1973}\\
          15.12     &                     &  750            &   $^{10}$B(n, n$^{\prime}$)$^{10}$B, $^{7}$Li($\alpha$, n)$^{10}$B (1968) \cite{Ajzenberg1968}\\
          15.2      &  (3/2$^+$, 5/2$^+$, 7/2$^+$)$^d$      &   700  &$^{10}$B(n, n$^{\prime}$)$^{10}$B (1974) \cite{Hausladen1973}\\
          15.29     &  (3/2$^-$, 5/2$^-$) &   282           &   $^{14}$C(p, $\alpha$)$^{11}$B (1985) \cite{Aryaeinejad1985}\\
          $\textbf{14.74 $\pm$ 0.09}$     &   3/2$^-$       &   830 $\pm$ 145    & \textbf{this work} \\

          \hline
          16.43$^*$ &  (5/2$^-$)          &   $<$ 30        &   \cite{Kelly2012}  \\
          16.43     &                     &                 &   $^{9}$Be(d, p)$^{10}$Be (1974) \cite{Annegarn1974}\\
          16.5, 16.2&                     &                 &   $^{11}$B($\gamma$, p)$^{10}$Be (1970) \cite{Sorokin1970}\\
          16.5      &                     &                 &   $^{11}$B($\gamma$, n)$^{10}$B (1965) \cite{Hayward1965}\\
          16.44     &                     &   $\leq$ 30     &   $^{9}$Be($^3$He, p), $^{9}$Be($\alpha$, d) (1982) \cite{Zwieglinski1982}\\

          16.5$^e$  &  (5/2$^-$)          &   201           &   $^{14}$C(p, $\alpha$)$^{11}$B (1985) \cite{Aryaeinejad1985}\\

          \textbf{(16.18 $\pm$ 0.21)}     &   (1/2$^-$, 3/2$^-$) &   $<$ 60   & \textbf{this work}\\

          \hline
          \hline
       \end{tabular}
 $^{*}$ Energy levels from the compilation for A = 11 light nuclei \cite{Kelly2012}.

 $^{a}$ Taken from the $^9$Be(t, p)$^{11}$Be experiment at 10 MeV and 14 MeV triton energy, where the observed E$_x$ = 1.78 MeV state in $^{11}$Be corresponds to the analog state in $^{11}$B at E$_x$ = 14.33 MeV with J$^\pi$ = (3/2$^-$, 5/2$^+$) \cite{Pullen1962}.

 $^{b}$ The tentative assignment of J$^\pi$ = (5/2$^+)$ for the state at E$_x$ = 1.78 MeV in $^{11}$Be (corresponding to the analog state E$_x$ = 14.33 MeV in $^{11}$B) from the comparison of the $^{10}$Be(d, p)$^{11}$Be experimental data with the DWBA calculation \cite{Zwieglinski1979} can not rule out the possibility of 3/2$^+$ state since the purity of  T = 3/2 in $^{11}$B (14.33 MeV state) is in debate.

 $^{c}$ This value is adopted from the $^{10}$Be(p, $\gamma$)$^{11}$B reaction \cite{Goosman1973}.

 $^{d}$ The J$^\pi$ = (1/2$^+$, 11/2$^+$) were excluded since they give too low and too high peak heights over the energy region \cite{Hausladen1973}.

 $^e$ It may not be the 16.44 MeV state \cite{Zwieglinski1982}. See also Ref. \cite{Kurath1975}.
 \end{table*}

\subsection{Bump around 14.74 MeV}

As shown in Fig. \ref{excitfun}(a), there is a bump around 14.74 MeV. The fitting to the data with exclusion
of the bump as a resonance results in an obvious deviation as shown in Fig. \ref{levelsfit}(b) by the blue dashed curve. In previous experiments, the characteristics of this resonance were observed at 15.30 MeV \cite{Goosman1973}, 15.29 MeV \cite{Aryaeinejad1985}, and 15.2 MeV \cite{Hausladen1973}, and were considered as the analog state of 2.78 MeV in $^{11}$Be. Earlier work associated with this state at 15.12 MeV was the studies of $^{10}$B(n, n$^{\prime}$)$^{10}$B, $^7$Li($\alpha$, n)$^{10}$B, and $^{10}$B(n, $\alpha$)$^7$Li reactions compiled in Ref. \cite{Ajzenberg1968}. However, no more information on spin-parities among these works was reported except for Refs. \cite{Hausladen1973} and \cite{Aryaeinejad1985}.

Based on the analysis of R-matrix fits to the data of neutron scattering from $^{10}B$ \cite{Hausladen1973},
Hausladen {\it et al.} suggested an assignment of J$^\pi$ = (3/2$^+$, 5/2$^+$, 7/2$^+$), where fits with
a p-wave state did not reproduce any distributions of the Legendre polynomial expansion coefficients.
Since then, Aryaeinejad {\it et al.} suggested a spin-parity assignment J$^\pi$ = $3/2^-$ through the study of $^{14}$C(p, $\alpha$)$^{11}$B
reaction \cite{Aryaeinejad1985} in which only negative-parity states with T = 3/2 were strongly excited
in $^{11}$B. In the current measurement, we speculated that this bump as a resonance was populated,
  and attempts with various combinations of excited energies and spin-parities were made to derive the resonant parameters. Consequently, the data was found to be reproduced in a satisfactory way with the J$^\pi$ = 3/2$^-$ assignment. Obviously, our result supports Aryaeinejad's assignment but with a larger particle width because of the present systematic resolution.

\subsection{Alternative Structure around 16.18 MeV}

In the previous measurements as listed in Table \ref{B11table}, the state with E$_{x}=16.432$ MeV and J$^\pi$ = (5/2$^-$) of $^{11}$B compiled in Ref. \cite{Kelly2012} was reported and has the resonant energy of 16.44 MeV in Ref.~\cite{Zwieglinski1982} and 16.43 MeV in Refs.~\cite{Annegarn1974, Sorokin1970}, respectively. R-matrix fits to the data were attempted for this state formed by l$_p$ = 0, 1, 2, and 3 protons. Calculations with p-wave states using a particle width smaller than 60 keV could naively produce a character of the tail of the excitation function, but the $\chi^2$ of the fit to the data was almost unchanged with the exclusion of this state. It is assumed that we have populated the well-known resonance, i.e., the analog of the 3.889 MeV state in $^{11}$Be, and a tentative assignment of J$^\pi$ = (1/2$^-$, 3/2$^-$) is set to this state. The characteristic behavior of angular momentum transfer l$_p$ = 1 for this state disagrees with the previously observed one at 16.50 MeV \cite{Aryaeinejad1985} with a width of 201 keV, which was not represented by any analog states in $^{11}$Be.

\section{summary}
\label{sec:sum}

To summarize, the elastic resonance scattering of protons from $^{10}$Be using the thick-target technique in inverse kinematics was performed for the first time to investigate the resonant structure of $^{11}$B above the proton-decay threshold. The excitation function for the resonant protons decayed from the compound nucleus $^{11}$B was analysed using the multichannel R-matrix procedure. Four resonant states at E$_x$ = 13.46 MeV, 14.55 MeV, 14.74 MeV, and 16.18 MeV were populated, where the latter three levels are considered to be the previously known ones with isospin value T = 3/2, and the data was reproduced well with the combination of those resonant parameters as listed in Table.~\ref{TBestPar}. The extracted parameters seem reliable within uncertainty except for the 14.55 MeV resonance as implied by the large ratio of $\Gamma_p/\Gamma_W$ arising from the counting statistics. Alternatively, the 16.18 MeV state which is located at the end of the higher excitation energy is tentatively assigned to be the one observed previously. The extracted parameters for the resonant energy, the spin parity, and the proton-decay partial width are summarized in Table \ref{B11table}. The present results suffer from the large uncertainty in the level width, the spin-parity assignments, and even the isospin components, and further studies based on high-resolution experiments are imperative to draw firm conclusions for the structure of $^{11}$B.

$Acknowledgements:$ The authors would like to thank the HIRFL staffs for their operation of the accelerators and Jun Xu for helping with polishing the manuscript. This work was partially supported by the National Natural Science Foundation of China under Grant Nos. 11421505, 11075195, 11475245, and 11305239 and the Major State Basic Research Development Program in China under Contract No. 2013CB834405.

redpajama_set_name: RedPajamaArXiv


text:
\section{INTRODUCTION}
The increasing interest in quantum information processing is
boosting the search for mechanisms to produce and control
entanglement in devices of different nature. \cite{qe} Photonic
devices are routinely used for preparing and detecting entangled photon
pairs.\cite{inter} The main limitations of such devices is the non-existence
of deterministic sources as well as the difficulty in controlling the
interaction between the photons. Entanglement mechanisms have also been
proposed in solid state devices, like quantum dots, \cite{entqdot}
and superconductors.\cite{entsup}  The fundamental ingredient
behind all these mechanisms is a many-body interaction. More recently, it was determined
that entanglement is also possible in systems of non-interacting
electrons.\cite{been1, been2} In particular, it was shown that electron-hole entangled
pairs can be produced by biasing a tunneling barrier.

The edge states of systems in the quantum Hall regime can be employed
in solid state devices to produce electron beams with properties similar
to those of a photon beam in optical setups.
Exploiting this analogy, several theoretical and experimental proposals
of electronic interferometers have been reported. \cite{elin}
Interestingly, the electronic counterpart of the Hanbury-Brown-Twiss
device has been analyzed in a configuration of  edge Hall states that do not have
interfering orbits. \cite{hbt,but1}
For this reason, Aharonov-Bohm  (AB) effect takes
place only at the level of two-particle correlation functions,
while the single-particle AB effect is not present.
This entanglement seems to be related with an asymmetry in the
device which favors the internal production of particle-hole
pairs and manifest itself in the behavior of the current-current correlation functions.
The relevance of the current-current correlation functions as entanglement
witness has been also discussed in other fermionic systems. \cite{noise-ent}

The aim of this work is to establish the existence of orbital
entanglement in AB systems where single-particle
interference is present. Using a microscopic model,
we show that when coupling the AB-ring to four leads (two on the right
and two on the left), the post select two-particle states of electrons at opposite
leads are typically
entangled. We name it {\em ''chiral-mediated entanglement''} (CME) because its creation  is possible by the existence of intermediate
states in the AB-ring which are coherent superpositions
of two different chiralities for the electronic motion. Remarkably, this kind of entanglement can be also defined in
a transport setup, which is identical to the one that was proposed
to define the electron-hole entanglement, \cite{been1,been2,but1,but2,frust1,frust2}.
In fact, we also show here that the noise current-current
correlation function can be a good witness for the entanglement in our
setup.

The paper is organized as follows. In Sec. II, we present
the model in detail and we sketch the formulation of the theory,
where we calculate the reduced density-matrix of a two-particle system.
The results of this work are presented in Sec. III. The conclusions are presented
in Sec. IV.
\
\section{THEORETICAL TREATMENT}

\subsection{Model}

We consider the AB single-channel ring with
a magnetic flux. The ring is connected to four terminals
\cite{been1,been2,but1,but2,frust1} as shown in Fig. \ref{fig1}.
Two of the terminals, those labeled by $\alpha=1,2$, at the
left side of the ring, are at a higher voltage $V$ with
respect to the ones at the right, labeled
by $\alpha=3,4$. All terminals are ordinary single-channel
metallic leads where electrons can move either to the right or to
the left. For simplicity, we consider spinless
electrons and we describe the setup by the Hamiltonian
\begin{equation} \label{ham}
H=\sum_{\alpha=1}^4 \left( H_{\alpha} + H_{c,\alpha} \right) + H_{ring},
\end{equation}
where
$H_{\alpha} = \sum_{k_{\alpha}} \varepsilon_{k_{\alpha}}
c^{\dagger}_{k_{\alpha}} c_{k_{\alpha}}$ are Hamiltonians of non-interacting electrons
representing the
leads. For the AB-ring we use a non-interacting  model, where
electrons move with velocity $v$ either clockwise ($+$ chirality)
or anticlockwise ($-$ chirality). The Hamiltonian is
\begin{equation}\label{ring}
 H_{ring} = \sum_{\lambda=\pm}  \int_0^L dx    v \lambda
\Psi^{\dagger}_{\lambda}(x) {\cal D}_x \Psi_{\lambda}(x) ,
\end{equation}
being $\lambda$ the chirality, $ {\cal D}_x = - i \partial_x -  \phi, \;\; \phi= \Phi/(L \Phi_0)$,
with $\Phi=2 \pi \Phi_B$, where $\Phi_B$ is the magnetic flux, $\Phi_0=h c/e$ is the flux quantum
and $L$ is the length of the ring.
The contacts between the leads and the ring are modeled by
tunneling terms of the form
$H_{c \alpha} = \sum_{k_{\alpha}, \lambda=\pm } w_{k_{\alpha}} [c^{\dagger}_{k_{\alpha}}
\Psi_{\lambda}(x_{\alpha})+ H.c.]$,
where $x_{\alpha}$ define the positions of the ring to which the
leads are attached. We consider the leads to be at zero temperature.
The chemical potentials enforce a bias voltage between
left and right leads, i.e. $\mu_1=\mu_2=\mu_L$;
$\mu_3=\mu_4= \mu_R$ and $\mu_L-\mu_R= eV$, which we assume to be very small
$eV \sim 0$.
\
\subsection{Reduced Density Matrix and Concurrence}

In a setup as the one in Fig. \ref{fig1}, electrons tunnel from the
left leads to the right ones. We aim to define the effective
density matrix describing the quantum state post-selected from the total
two--electron state by projecting out the components where both electrons are
either in the right or in the left leads. \cite{been2,but2,frust1}
We introduce the operators $A^\dagger_{00} \equiv c_{k_{1}}^{\dagger}c_{k_{3}}^{\dagger},
 A^\dagger _{01} \equiv c_{k_{1}}^{\dagger}c_{k_{4}}^{\dagger},
 A^\dagger _{10} \equiv c_{k_{2}}^{\dagger}c_{k_{3}}^{\dagger},
 A^\dagger _{11} \equiv c_{k_{2}}^{\dagger}c_{k_{4}}^{\dagger} $, which
 create one particle in one of the left leads and a second particle at one of the
 right leads.  The ensuing
the $4\times 4$ density matrix describes a system of two qubits with
elements
\begin{eqnarray}\label{rho}
[\rho^{(2)}(\varepsilon)]_{ab,a'b'}&=&
\frac{1}{{\cal N}_0}\prod_{\alpha}\sum_{k_{\alpha}} \delta(\varepsilon-\epsilon_{k_{\alpha}})
\langle A_{ab}^\dagger A_{a'b'}  \rangle,\nonumber
\end{eqnarray}
where ${\cal N}_0$ is a normalization factor, while the mean value is taken in
the nonequilibrium state
with a net current flowing from the left to the right.
We remark that in our approach matrix elements of $\rho^{(2)}$ are
obtained in terms of operators appearing in the Hamiltonian $H$. Expectation values of four time dependent creation and/or annihilation operators
are computed using the nonequilibrium Green function formalism and Wick theorem, \cite{jauho}
\begin{widetext}
\begin{eqnarray}
\langle c_{k_{\alpha}}^{\dagger}(t)c_{k_{\beta}}^{\dagger}(t)c_{k_{\lambda}}(t)c_{k_{\delta}}(t) \rangle
=\int_{-\infty}^{\infty} d\varepsilon \int_{-\infty}^{\infty} d\varepsilon^{\prime} \left[G_{k_{\lambda},k_{\alpha}}^{<}(\varepsilon-\varepsilon^{\prime})G_{k_{\delta},k_{\beta}}^{<}(\varepsilon^{\prime}) - G_{k_{\delta},k_{\alpha}}^{<}(\varepsilon-\varepsilon^{\prime})G_{k_{\lambda},k_{\beta}}^{<}(\varepsilon^{\prime})\right], \label{matrixelement}
\end{eqnarray}
\end{widetext}
where $G_{k_\alpha,k_\beta}^{<} (\varepsilon)$ is the Fourier transform with respect to $t-t^{\prime}$ of the lesser Green function
\begin{equation}
G_{k_\alpha,k_\beta}^{<} (t-t^{\prime}) = i \langle c_{k_{\beta}}^{\dagger}(t^{\prime}) c_{k_{\alpha}}(t) \rangle .
\end{equation}
We  assume that $eV \sim 0$ and we are interested in analyzing $\varepsilon \sim \mu=(\mu_L + \mu_R)/2$.
We present the corresponding expression of $\rho^{(2)}$ in Appendix A.

From the above density matrix we compute the concurrence, which is a
good measure of entanglement. \cite{woot} In this case it is given by
\begin{equation}
\mathcal{C}\left[\rho^{(2)} \right]=max\left\{0,\sqrt{\lambda_{1}}-
\sqrt{\lambda_{2}}-\sqrt{\lambda_{3}}-\sqrt{\lambda_{4}}\right\},
\end{equation}
where $\lambda_{i}$ are the eigenvalues of
$R:=\rho^{(2)}\sigma_{y}\bigotimes\sigma_{y}\rho^{(2)\ast}\sigma_{y}\bigotimes\sigma_{y}$ in decreasing order.
It is interesting to notice that the concurrence so  calculated is equivalent to the one obtained by the spin--dependent scattering formalism
for a single mode conductor ,\cite{been1,been2} which is
\begin{equation}
\mathcal{C}=2\frac{\sqrt{\tau_{1}(1-\tau_{1})\tau_{2}(1-
\tau_{2})}}{\tau_{1}+\tau_{2}-\tau_{1}\tau_{2}}.
\end{equation}
Here, $\tau_{1},\tau_{2}$ are transmission eigenvalues of the scattering matrices
$s_{\alpha,\beta}(\mu)=\delta_{\alpha,\beta}-
i\sqrt{\Gamma_{\beta} \Gamma_{\alpha}}G_{\alpha,\beta}^{R} (\mu)$,
where $G_{\alpha,\beta}^{R} (\mu)$ is the Fourier transform with respect to $t-t^{\prime}$ of the retarded Green function
\begin{equation}\label{green}
G_{\alpha,\beta}^{R} (t-t^{\prime}) =- i \Theta(t-t^{\prime}) \sum_{\lambda,\lambda^{\prime}} \langle \{\Psi_{\lambda}( x_{\alpha},t),
\Psi^{\dagger}_{\lambda^{\prime}}( x_{\beta},t^{\prime}) \} \rangle
\end{equation}
 evaluated at $\mu$, being $x_{\alpha}$ the position
of the ring at which the wire $\alpha$ is attached. while $\Gamma_{\alpha} =2 \pi \sum_{k_{\alpha}}
w_{k_{\alpha}}^2 \delta(\mu-\varepsilon_{k_{\alpha}}) $. \cite{fish-lee}
\
\subsection{Noise}

As pointed out in Refs. \onlinecite{been2, frust2} the concurrence
can be expressed in terms of correlators that quantify the degree of
violation of Bell inequalities. In transport setups the latter can
in turn be directly related to current-current correlation functions,
which are amenable to be experimentally detected. We, thus,
turn to analyze the connection between these correlation functions
and the above discussed entanglement. We first outline the procedure
to compute the current-current noise within our treatment.
The current passing through the contact to the terminal $\alpha$, can be expressed by
the following operator $ J_{\alpha} (t) = (e/ \hbar)  \sum_{k_{\alpha}}w_{k_{\alpha}}
[i c_{k_{\alpha}}^{\dagger} (t) \Psi(x_{\alpha},t) + h.c] $.
The zero frequency shot-noise is a measure of
the current-current correlations in different terminals. It reads
\begin{equation} \label{shotnoise}
{\cal S}_{\alpha,\beta}(0)=\frac{1}{2}\int d\tau \langle\{\delta J_{\alpha}(\tau),\delta J_{\beta}(0)\}\rangle,
\end{equation}
where $\delta J_{\alpha}=J_{\alpha}-\langle J_{\alpha}\rangle$. We calculate
(ec.(\ref{shotnoise})) by evaluating a bubble diagram in terms of nonequilibrium Green functions.
The corresponding expressions are presented in Appendix \ref{noise}.
\
\section{RESULTS}

\subsection{Qualitative Analysis}

To understand the origin of CME it is useful to begin
analyzing the electronic states of the isolated AB--ring.
The Hamiltonian can be diagonalized in momentum space: Defining
$\Psi_{\lambda}(x) = 1/\sqrt{{\cal N} } \sum_p e^{-i p x} c_{p,\lambda}$
with ${\cal N}$ a normalization factor, $p = 2 \pi n/L$ and
$n \in \mathbf{Z}$ we obtain
$H_{ring}=  \sum_{\lambda} \sum_p \varepsilon_{p,\lambda}(\Phi)
c^{\dagger}_{p,\lambda} c_{p,\lambda} $, with
$\varepsilon_{p,\lambda} = \lambda v (p - \phi) $
(a cutoff in the single-
particle energy spectrum is assumed).
The effect of the magnetic flux on the energies
$\varepsilon_{p,\lambda}(\Phi)$ is illustrated in Fig. \ref{fig1}.
Depending on the magnetic flux, there can be zero, one
or two single-particle states $|p,\lambda\rangle$ with
a given energy. States with different chirality $\lambda = \pm $
are degenerate only when the flux is an integer multiple
of $\pi \Phi_0$.

Let us consider first the case where two degenerate states with
opposite chiralities exist in the ring. These two states behaves as an intermediate qubit that couples to the qubits
defined by the leads. We will argue that in this case CME naturally emerges between electrons at the right and left leads.
The $N$-particle states with a Fermi energy $\epsilon_F$ can be
obtained from $|\overline{0} \rangle$, that represents the
Fermi sea with $N-2$ particles filling the states with
$\varepsilon_{p,\lambda}(\Phi) < \epsilon_F$,
where $v (p_F-\phi)=\epsilon_F$. Thus, we have
$|\Psi_{ring}\rangle = c^{\dagger}_{p_F,+} c^{\dagger}_{p_F,-}|\overline{0}\rangle$.
When the ring is in contact with the four leads, particles
can tunnel between the ring and the reservoirs.
For weak coupling
we can assume that each of the  chiral levels with $p_F$
hybridize with the levels of the leads having the same
energy $\epsilon_F$. This is described by the
following effective Hamiltonian
\begin{equation} \label{heff}
H_{eff}= w \sum_{\alpha=1}^4\sum_{\lambda=\pm} e^{i \lambda p_F x_{\alpha}}
[c^{\dagger}_{k_{\alpha}} c_{p_F,\lambda} + H. c.],
\end{equation}
where $w$ is the effective tunneling
parameter, $c^{\dagger}_{k_{\alpha}}$ creates an electron
in the single-particle state of the $\alpha$-lead
with energy $\varepsilon_{k_{\alpha}}=\epsilon_F$ (we take $\epsilon_F=0$ without
loss of generality). This Hamiltonian has four eigenstates of the
form
\begin{equation}
|\psi_n \rangle =[\sum_{\alpha=1}^4 \gamma_{n,\alpha} c^{\dagger}_{k_{\alpha}} + \sum_{\lambda=\pm} \gamma_{n,\lambda} c^{\dagger}_{p_F,\lambda} ]|0 \rangle,
\;\;\;\; n=1, \ldots, 4,
\end{equation}
 where the coefficients $\gamma_{n}$ are the weights of the eigenstates in the chosen base. It also has two additional degenerate
states of the form
$|\psi_n \rangle =\sum_{\alpha=1}^4 \gamma_{n,\alpha} c^{\dagger}_{k_{\alpha}} |0 \rangle, \;n=5,6$.
The latter correspond to states that do not hybridize with the ring.
When two particles are present, two different such states must be
occupied. It is simple to show that any state of this type
has a sizable projection on states of the form $\sum_{\alpha \neq \beta}
\Lambda_{\alpha,\beta} c^{\dagger}_{k_{\alpha}} c^{\dagger}_{k_{\beta}}|0\rangle $ for
some non-vanishing coefficients $\Lambda_{\alpha,\beta}$.
This two-particle state is typically entangled in the orbital
indices $\alpha,\beta$ of opposite leads.
Notice that it is also possible to use two AB--rings with degenerate levels as a two-qubit system coupled by two conducting leads (intermediate qubit).

On the other hand if we consider the case of a one chirality ring ($\epsilon_{p}=v(p-\phi)$ for example)
the situation drastically changes and no significant entanglement between left and right leads is attained.
This is can be seen because the effective Hamiltonian has a different level structure. In fact,
$H_{eff}$ can be naturally written  in terms of operators that
are linear combinations of the lead operators $c^{\dagger}_{k_{\alpha}} $ as
$H_{eff}= w (f^{\dagger} c_{p_F,\lambda} + H. c.)$, with
$f^{\dagger} = (1/2) \sum_{\alpha=1}^4 e^{i p_F x_{\alpha}} c^{\dagger}_{k_{\alpha}} $, while there are three additional orthogonal linear combinations
of these operators which do not hybridize with the ring.
The two eigenstates of $H_{eff}$ are linear combinations of a single-particle state of the leads and a single-particle state of the ring.
Thus,
 a two-particle state of this Hamiltonian has never the two particles in the leads and therefore no orbital
entanglement is possible.

The above argument suggests that by varying the magnetic flux, or the chemical potential of the leads (or equivalently, a gate voltage applied at the ring)
we can induce the system to switch from a situation with no orbital entanglement between the leads
onto another situation with inter-lead entanglement. This is done by varying $\Phi$ and/or $\mu$
in order to have degenerate chiral states of the ring at the Fermi energy.
We now present a rigorous calculation of the entanglement for
the states that are relevant for a transport experiment in the
coherent regime and explain how the CME depends on flux and chemical potential. We also discuss the way in which it may be detected in transport  experiments.

\begin{figure}
 \centering
 \includegraphics[width=5cm]{Fig1.eps}
 \caption{(a) Scheme of an Aharonov-Bohm ring with two chiralities, attached to four leads. The chemical potentials are such that there is a bias voltage between
the left and right leads. Electrons incoming from the left can tunnel to the ring and escape through leads.  (b) Linear dispersion relation of an isolated AB-ring, where
electrons move with velocity $v$ either clockwise ($+$ chirality) or anticlockwise ($-$ chirality). The magnetic flux determines whenever the Fermi level is two-degenerate or not.}
\label{fig1}
\end{figure}

\subsection{Numerical Results}

\subsubsection{Concurrence}

\begin{figure}
 \includegraphics[width=9cm]{Fig2.eps}
 \caption{ (Color online) Concurrence  as a function of $\Phi / \Phi_0$ and mean chemical
potential $\mu$ for a ring with length $L=20$ with wires connected at
$x_1=1,\;x_2=6, x_3= 11, x_4= 16$.   }
\label{fig2}
\end{figure}

\begin{figure}
 \includegraphics[width=9cm]{Fig3.eps}
 \caption{ (Color online) Concurrence  as a function of $\Phi / \Phi_0$ and mean chemical
potential $\mu$ for a ring with only one chirality. The ring parameters are the same as in Fig.\ref{fig2}.}
\label{fig2b}
\end{figure}

The behavior of $ \mathcal{C}$ as a function of the mean chemical potential $\mu$ and the magnetic flux $\Phi$ is shown in Fig. \ref{fig2}.
The concurrence is maximal at $\Phi/ \Phi_0=n \pi$ and for values $\mu$ close to the  energy of  two degenerate chiral states of the ring.
The same type of behavior is observed for other configurations of the wires, corresponding
to contacts at different positions $x_{\alpha}$.  For this Fig. we considered  wires with a
 bandwidth $W_{\alpha}$ and
$\Gamma_{\alpha}=  \Omega_{\alpha} \sqrt{ W_{\alpha}^2 - \mu^2}$, where $\Omega_{\alpha}$ is a constant, but the same behavior  is obtained for leads
with a constant density of states. For some chemical potentials ${\cal C}$ exhibits maxima at $\Phi/\Phi_0= 0, \mod(2 \pi)$, which corresponds
to the energy of two degenerate states of the ring, but achieves again the maximum value within a wide range of fluxes centered at  $\Phi/\Phi_0= \pi, \mod(2 \pi)$.
This feature is analyzed below in more detail.

Instead, if we evaluate ${\cal C}$ for the Hamiltonian (\ref{ring}) restricted to a single chirality,
we find negligibly orbital entanglement in the leads within the whole range of $\Phi$ and $\mu$. These results are presented in Fig.\ref{fig2b}. In this
case the ring behaves as a single-level system, and prevent the formation of orbital entangled states at the leads.

\begin{figure}
\centering
\includegraphics[width=8cm]{Fig4.eps}
\caption{(Color online)
Shot noise (squares) between left and right terminals, as a function of $\Phi/\Phi_0$  for two values of
the chemical potentials at which there are degenerate states.  (a) $\mu=0.3175$, (b) $\mu=0.7866$. The corresponding
plots for the concurrence are also shown for comparison (circles).}
\label{fig3}
\end{figure}

\subsubsection{Noise}

We now turn to present results on the behavior of the current-current correlation functions. Our aim is to analyze if the
signatures of entanglement found in the behavior of the concurrence can be also identified in the behavior of the noise.
Results for the total left-right noise
correlations $S= \sum_{\alpha=1,2,\beta=3,4}{\cal S}_{\alpha,\beta}(0)$
(equal to the self-correlation $ - \sum_{\alpha,\beta=1,2}{\cal S}_{\alpha,\beta}(0)$ ) are shown in Fig. \ref{fig3}.
The left (right) panel, corresponds to a chemical potential $\mu$ for which there are two degenerate chiral states in the ring for
$\Phi/\Phi_0=0, (\pi), \mbox{mod}(2\pi)$. The behavior of the concurrence is also plotted for comparison. In both cases,
 $S$, along with ${\cal C}$ exhibit maxima at the fluxes  for which the two degenerate
chiral states are resonant at the given $\mu$.
 This points to the idea that noise is indeed a reliable witness of orbital
entanglement, as discussed in the context of other electronic setups. \cite{been1,been2,but1,noise-ent} In the case of the left panel,
both quantities are maximum at $\Phi/\Phi_0=0, \mbox{mod}(2\pi)$. Within a range of fluxes which scans
the width of the resonant degenerate levels of the ring they first decrease and then increase, displaying a dip. As
the flux increases further, the behavior of these two quantities, however, depart one another. While $S$ tends to vanish around $\Phi/\Phi_0 = \pi$, the concurrence  displays a wide plateau
with height ${\cal C} \sim 1$. Qualitatively, the same type of
behavior is observed in the right panel.   In this case, both quantities exhibit a sharp maxima at resonance (see the peaks around $ \Phi/\Phi_0=\pi$). The  concurrence displays a plateau and another (lower) maximum around $\Phi/\Phi_0=0, \mbox{mod}(2 \pi)$ while
$S$ is vanishing small. On general grounds this is rather surprising, since one could easily imagine situations with a sizable noise without entanglement, but here we have
the converse situation.

\begin{figure}
\centering
\includegraphics[width=8cm]{Fig5.eps}
\caption{(Color online) Probability of finding two particles at the left and right leads  as a function of magnetic flux $\Phi/\Phi_0$.
Panels (a) and (b) correspond to the same parameters of Fig. \ref{fig3} (a) and (b), respectively .}
\label{fig4}
\end{figure}

In order to further understand the relation between the behavior of ${\cal C}$ and $S$ as well as the connection to the CME,
we analyze the probabilities
\begin{eqnarray}
P_{LL} &= &\prod_{\alpha}\sum_{k_{\alpha}} \delta(\varepsilon-\epsilon_{k_{\alpha}})\langle c^{\dagger}_{k_1}c_{k_1} c^{\dagger}_{k_2}c_{k_2} \rangle , \nonumber \\
P_{RR} &= &\prod_{\alpha}\sum_{k_{\alpha}} \delta(\varepsilon-\epsilon_{k_{\alpha}})\langle c^{\dagger}_{k_3}c_{k_3} c^{\dagger}_{k_4}c_{k_4} \rangle
\end{eqnarray}
of finding two particles at the left (right)  leads, respectively.
 In Fig. \ref{fig4} we show the  behavior of these two quantities for the same parameters of Fig. \ref{fig3}.
It is clear that $P_{LL}$ and $P_{RR}$ are both sizable when tunneling through the ring is allowed. For these chemical potentials, this corresponds
to the narrow window  of fluxes around $\Phi/\Phi_0=0 (\pi)$, for the case of the left (right) panels, respectively,
within which the degenerate levels of the ring remain resonant. Beyond these values, the transmission from left to right is blocked and
the two particles have a high probability of remaining within the left wires. A further analysis of Figs. \ref {fig2} and \ref{fig3} in the light of
the results shown in Fig. \ref{fig4}, then reveals that the high values of concurrence in the plateaus away from $\Phi/\Phi_0=0
(\pi), \mbox{mod}(2 \pi)$ in the case of the left (right) panels of Fig. \ref{fig3}, correspond to states
 which have a very low probability of taking place.  Instead, the resonant situation
with two degenerate chiral states of the ring at the Fermi energy, leads to a high orbital entanglement which is clearly
witnessed by a high noise amplitude $S$.
\
\section{CONCLUSION}

To summarize, we introduced a new mechanism for orbital entanglement. This type of entanglement is originated in the
spectral nature of the AB ring and  is
highly sensitive to the magnetic field.  Orbitally entangled electronic pairs can be produced by suitably tuning
the magnetic field, the chemical potential or a voltage gate at the ring, in order to have a degenerate pair of electronic states with different chiralities at the Fermi energy
(intermediate qubit). In fact, if the right and left leads are mediated by a single--level system (single-chirality ring) the orbital entanglement
 is negligible.
This type of entanglement  can be detected in transport experiments, the shot noise being a good witness.
The setup of Fig. \ref{fig1} could be experimentally realized in an architecture based on the quantum Hall regime of a 2D electron gas,
 by substituting each wire by a pair of incoming and outgoing edge states and the ring by a pair of edge states with different chiralities, separated by a narrow circular wall.
In such a setup it would  be possible to combine the main system of Fig. \ref{fig1} with beam splitters
connected at the wires,
in order
to test Bell inequalities and even perform the full quantum
tomography following the protocol of Ref. \onlinecite{but2}.
Two interesting possible generalizations are the combination between static and flying qubits
by combining with quantum dots, \cite{schom} as well as the introduction of dynamical single-particle emitters at the sources.\cite{spe}
\
\section*{ACKNOWLEDGMENTS}

We thank D. Frustaglia for useful conversations.
This work is supported by CONICET, MINCyT and UBACYT, Argentina.
LA thanks support from the J. S. Guggenheim Foundation.
\


text:
\section{Introduction}

When ferroelectric oxides are utilized in metal/oxide heterostructures and nanocapacitors, scaling of the active FE is required to improve performance \cite{Ishiwara:2004,1257065,doi:10.1080/00150199808009170}.
Nonetheless, deleterious nanoscale effects are amplified in these geometries \cite{Junquera/Ghosez:2003,Gerra:2006, Gerra:2006v2} and act to eliminate the functional electric polarization \cite{Ishikawa:1996,Tybell:1999,Jiang:2000}.
The loss of ferroelectricity in nanoscale capacitors frequently occurs when the polarization is perpendicular to the film surface \cite{Ghosezrabe:2000, Meyer:2001, Junquera/Ghosez:2003, Kornev:2004}, \emph{i.e.}, the desired polarization direction for field-tunable devices \cite{RameshSchlom:2002}.
Bound charges
are only partially screened at the interface, resulting in a strong depolarizing field that
suppresses the polarization.
Indeed,  numerous reports suggest a critical thickness, $t_\mathrm{FE}^*$ below which the electric polarization disappears.
Experimental studies on
PbTiO$_3$ find $t_\mathrm{FE}^*\sim20$\,\AA\,
at 300\,K \cite{Despont:2006}
whereas $t_\mathrm{FE}^*\sim4.0$\,nm,
in Pb(Zr$_{0.2}$Ti$_{0.8}$)O$_3$ films \cite{Tybell:1999}.
Furthermore, first-principles density functional theory (DFT) calculations predict $t_\mathrm{FE}^*\sim2.4$\,nm in single-domain BaTiO$_3$ films between SrRuO$_3$ electrodes \cite{Junquera/Ghosez:2003}, which reduces to $t_\mathrm{FE}^*\sim1.0$\,nm  after accounting for ionic relaxation in the electrodes \cite{Gerra:2006, Gerra:2006v2}.
With this limitation, ferroelectric based devices are unable to meet the continuous scaling changes demanded by higher density data storage technologies.

Although proposals to overcome the problem exist, a general solution remains elusive.
Most approaches focus on tuning the stability of the FE state.
Epitaxial strain engineering has been proposed; nonetheless, this strategy extends to a limited number of oxides, requires complex processing steps, and is limited by available commercial substrates. For example, although it is predicted that $t_\mathrm{FE}^*\rightarrow0$ in
thin films of the incipient ferroelectric BaZrO$_3$, a large epitaxial compressive strain of 4.25\% is required \cite{Zhang:2014},
which would produce deleterious misfit dislocations.
Integration with silicon is likely to also lead to uncontrolled interface states \cite{Hirth:2008}.
Furthermore, at this level of strain $t_\mathrm{FE}^*$ is still finite for BaTiO$_3$ (Ref.\ \onlinecite{Zhang:2014}).

Alternative solutions change the type of ferroelectric and the active inversion symmetry lifting mechanism. Ab-initio calculations find that $t_\mathrm{FE}^*\!\rightarrow\!0$ using so-called hyperferroelectrics \cite{Garrity:2014}, which have a persistent polarization different from proper FE oxides,  or improper ferroelectrics \cite{Stengel:2012,PhysRevLett.102.107601}. However, hyperferroelectric bulk  materials (and thin films) remain to be synthesized  \cite{Bennett:2012}.
Another route relies on creating an enhanced interfacial FE state by controlling the covalency of the metal-oxygen bond at the heterointerface \cite{Stengel/Spaldin:2006,Stengel/Vanderbilt/Spaldin:2009a,Caimeng:2011,ADFM:ADFM201500371}.

\begin{figure}[t]
\includegraphics[width=0.95\columnwidth]{fig1
\caption{
Symmetric nanocapacitor consisting of polar-metal electrodes and a  ferroelectric oxide.} \textbf{a},  The  centrosymmetric nanocapacitor with insulating NaNbO$_3$ ($m$=2) between LiOsO$_3$ electrodes ($n$=6). \textbf{b}, The equilibrium structure of the ferroelectric capacitor  [LiO-(OsO$_2$-LiO)$_6$/NbO$_2$-(NaO-NbO$_2$)$_2$]. The direction of the polar displacements in the electrodes and the ferroelectric film ($\mathcal{A}_\mathrm{NNO}$) are indicated with arrows. Magnification of the LiO/NbO$_2$ interface of the  \textbf{c}, paraelectric aristotype and  \textbf{d}, the ground state structure.
The FE behavior at the interface is due to the Li, Nb, and O displacements. The direction of the polar displacements at the interfaces for the paraelectric and the ground state structure is indicated with red arrows.}
\label{fig1}
\end{figure}

Here we examine the critical thickness for ferroelectricity in nanocapacitors consisting of \emph{polar-metal} electrodes and conventional ferroelectric oxides under short-circuit boundary conditions using
 DFT calculations.
Recently, polar metals have garnered  considerable interest \cite{Shi:2013,Kim:2016} because they
exhibit simultaneously inversion-lifting displacements and metallicity.
In these compounds, the polar displacements are weakly coupled to the states at the Fermi level, which makes possible the coexistence of a polar structure and metallicity \cite{Puggioni:2013}.
Our main finding is that polar-metal  electrodes suppress the critical thickness limit through interfacial polar displacements, which stabilize the ferroelectric (polarized) state; this geometric effect does not rely on interfacial bond chemistry or `perfect' screening of the depolarizating field,  but rather results from the intrinsic broken parity present in the electrode.

\section{Methods}
We perform first-principles DFT calculations
within the local-density approximation (LDA) and hybrid functional (HSE06, Ref.~\onlinecite{Heyd2003,Heyd2006})
as implemented in the Vienna
{\it Ab initio} Simulation Package (VASP)
\cite{Kresse/Furthmuller:1996b}
with the projector augmented wave (PAW) approach \cite{Blochl/Jepsen/Andersen:1994}
to treat the core and valence electrons using
the following  electronic configurations:
$1s^2 2s^2$ (Li),
$5p^{6} 6s^2 5d^6$ (Os),
$2s^2 2p^4$ (O),
$5s^2 5p ^6 6s^2$ (Ba),
$3d^2 4s ^2$ (Ti),
$4s^2 4p ^6 5s^2$ (Sr),
$4d^7 5s ^1$ (Ru),
$2p^6 3s ^1$ (Na),
$4p^6 4d ^4 5s^1$ (Nb).
The Brillouin zone integrations are performed
with a  $13\times13\times1$
Monkhorst-Pack $k$-point mesh \cite{Monkhorst/Pack:1976}
and a 600~eV plane wave cutoff for the
\loonno\
and
\srobto\
capacitor structures.
We relax the atomic positions (force tolerance less than 0.1\,meV~\AA$^{-1}$) using Gaussian smearing (20\,meV width).

Below 150\,K NaNbO$_3$ and LiOsO$_3$ are isostructural with rhombohedral space group $R3c$ and pseudocubic lattice parameters of 3.907~{\AA} (Ref.~\onlinecite{Zeitschrift}) and 3.650~{\AA} (Ref.~\onlinecite{Shi:2013}), respectively.
Owing to the large lattice mismatch between the two compounds, we simulate a symmetric ferroelectric capacitor structure with an LiO/NbO$_2$ interfacial termination,
shown in \autoref{fig1}a, under an epitaxial  constraint that would be imposed by a (La$_{0.3}$Sr$_{0.7}$)(Al$_{0.65}$Ta$_{0.35}$)O$_3$ substrate\cite{Liu:2013} and we relax the out-of-plane lattice parameter.
This results in a compressive strain of $\sim$1\% for NaNbO$_3$ and a tensile strain of $\sim$6\% for LiOsO$_3$. Note that at the bulk level, we find that a tensile strain $\gtrsim$6\% suppresses the polar instability along the [001]-pseudocubic direction of LiOsO$_3$. Moreover, we  selected NaNbO$_3$ for the capacitor structures because to eliminate any charge transfer due to `polar catastrophe/charge mismatch' physics as the interface: [LiO]$^{1-}$, [NaO]$^{1-}$, [NbO2]$^{1+}$,  and [OsO2]$^{1+}$.

For the two ferroelectric capacitors, we adopt the layered-oxide notation used in Ref.~\onlinecite{Junquera/Ghosez:2003}, that is
\begin{itemize}\itemsep=0em
\item {[LiO-(OsO$_2$-LiO)$_n$/NbO$_2$-(NaO-NbO$_2$)$_m$]}  and  \item {[SrO-(RuO$_2$-SrO)$_n$/TiO$_2$-(BaO-TiO$_2$)$_m$]}
\end{itemize}
to clearly demarcate the interface composition in the \loonno\ and \srobto\ capacitors, respectively.
We use a LiO/NbO$_2$ electrode/ferroelectric interface for the \loonno\  capacitor and a SrO/TiO$_2$  interface termination for the \srobto\  capacitor.
For both
ferroelectric capacitors, we constrained the number of 5-atom perovskite units of the electrode at $n=6$
to ensure a thickness large enough to avoid interaction between the two interfaces, and $m$ ranged from 1 to 3.
The periodic boundary conditions naturally impose the required short-circuit condition between the electrodes.
Note that for \srobto\ capacitors, our geometry differs only slightly from that used by Junquera and Ghosez  \cite{Junquera/Ghosez:2003}, whereby the thickness of our electrode is greater.

The group theoretical analysis was aided by the \textsc{isodistort}  software \cite{Isodisplace:2006}.
It is used to evaluate the geometric-induced inversion symmetry-breaking displacements of the $P4mm$ structure with respect the $P4/mmm$ phase, reducing the polar structure into a set of symmetry-adapted modes associated with different irreducible representations of the $P4/mmm$ phase. The ``robust'' algorithm was used to match an atom in the undistorted structure to every atom in the distorted structure separated by a threshold distance less than 3\,\AA.

\section{Results and Discussion}
The first ferroelectric nanocapacitor we focus on consists of ferroelectric NaNbO$_3$ of varying thickness $m$ confined between electrodes of the experimentally known polar metal LiOsO$_3$ (see  \autoref{fig1}, $m=2$) \cite{Shi:2013,PhysRevB.90.195113}.
We adopt the layered-oxide notation used in Ref.~\onlinecite{Junquera/Ghosez:2003}, that is
[LiO-(OsO$_2$-LiO)$_n$/NbO$_2$-(NaO-NbO$_2$)$_m$]
to clearly demarcate the interface composition (see Methods).
We create two symmetric nanocapacitors with a polar and paraelectric configuration for both LiOsO$_3$ and NaNbO$_3$, respectively.
We then relax the out-of-plane lattice parameter and the atomics positions of the nanocapacitors for
$m$=1, 2, and 3.
The lowest energy heterostructures are polar with space group $P4mm$ and exhibit large Li ions displacements along the [001]-pseudocubic direction (\autoref{fig1}b). No zone-center dynamical instabilities are found in these heterostructures.
Note that structures with an initial paraelectric configuration relax into a centrosymmetric structure (space group $P4/mmm$) with Li atoms displaying large antipolar displacements in LiOsO$_3$ (\supf1) that decrease towards the interface (\autoref{fig1}c).

We use representation theory analysis to examine the inversion lifting distortions (see Methods), and find that the distortion vector corresponds to the irreps $\Gamma_1^+$ and $\Gamma_3^-$. The irrep $\Gamma_1^+$ reduces the antipolar displacements in
LiOsO$_3$ resulting in the centrosymmetric $P4/mmm$ structure depicted in \autoref{fig1}a. Differently, the irrep $\Gamma_3^-$ is a polar mode which involves mainly Li ion displacements---the maximum amplitudes being $\sim$1.3~\AA\ in LiOsO$_3$ with decreasing amplitude towards the LiO/NbO$_2$ interface (\autoref{fig1}b). It also consists of polar displacements of all ions in the dielectric NaNbO$_3$ layers with the Nb ions off-centering the most (\autoref{fig1}d)

For all thicknesses, the NaNbO$_3$ thin film maintains a ferroelectric ground state characterized by both Nb and Na displacements (\autoref{fig1}b).
A linear polarization-displacement model using  the Born effective charge from Ref.~\onlinecite{PhysRevLett.72.3618}
results in a 0.86\,C\,m$^{-2}$ polarization for NaNbO$_3$ in $m=2$.
An analysis of the differential ionic relaxations in the heterostructure reveals polar displacements at the LiO/NbO$_2$ interfaces---an interfacial ferroelectricity---which are a consequence of the polar metal used as an electrode.
This produces an enhanced polarization in the ferroelectric compared to that calculated using the aforementioned procedure in the experimental $R3c$ structure (0.59\,C\,m$^{-2}$) \cite{PhysRevB.76.024110}.
In particular, although the two interfaces of the paraelectric structures exhibit \emph{antiparallel} polar displacements (\autoref{fig1}c and  \supf1), the interfaces of the polar ground state structures have \emph{parallel} polar displacements as show in \autoref{fig1}d.

\begin{figure
\includegraphics[width=0.9\columnwidth]{fig2
\caption{Projected densities-of-states (DOS) for the NaNbO$_3$ layer in the \loonno\ ($m$=1) nanocapacitor within  \textbf{a}, LDA and \textbf{b}, HSE06. The inset shows the  \loonno\ ($m$=1) nanocapacitor.}
\label{fig_dos}
\end{figure}

\autoref{fig_dos} shows the electronic properties for the NaNbO$_3$ layer in the \loonno\ ($m$=1) nanocapacitor.  The LDA functional predicts that the NaNbO$_3$ film is metallic, rendering the ferroelectric capacitor unusable (\autoref{fig_dos}a). This behavior is artificial and due to the tendency of the LDA functional to underestimate the band gap in insulating compound \cite{Perdewgap:1985,stengelpuente:2011}. We solve this pathological problem for DFT by using a more sophisticated functional  which includes a fraction of exact exchange (HSE06). \autoref{fig_dos}b shows that the hybrid functional fully opens the gap between the O $2p$ and Nb $4d$ states. Moreover, we find that the HSE06-relaxed structures exhibit displacements similar to those obtained from the LDA functional; importantly, polar displacements at the LiO/NbO$_2$ interfaces. This result supports the conclusion that the interfacial ferroelectricity is induced by the polar crystal structure of the metallic electrode and not due to spurious shorting of the capacitor. In the remainder of this paper, we report results obtained using LDA owing to the similar crystal structure obtained with HSE06 functional.

\begin{figure}[t]
\centering
\includegraphics[width=0.9\columnwidth]{fig3
\caption
Energetic gain with increasing polar mode amplitude $\mathcal{A}_\mathrm{NNO}$ in the NaNbO$_3$ film for different thicknesses: $m$=1  (circles), $m$=2 (squares) and  $m$=3 (diamonds). A finite value of  $\mathcal{A}_\mathrm{NNO}$ leads to an energetic gain for all $m$, indicating the disappearance of the critical thickness. Note that the energy of the polar ground state structure with $\mathcal{A}_\mathrm{NNO}$=0~{\AA} is taken as reference for each value of $m$. As expected the energy gain increases as the thickness of the ferroelectric film increases owing to the reduced effects from the depolarizing field, which dominates thinner films.}
\label{fig2}
\end{figure}

\autoref{fig2} shows the evolution of the total energy of each capacitor
with mode amplitude $\mathcal{A}_\mathrm{NNO}$, which describes the atomic displacements involved in the soft mode of the NaNbO$_3$ film.
As expected the largest energy gain occurs when the thickness of the ferroelectric film increases. Note that the shape of the energy surface does not exhibit the characteristic double well behavior, because in our calculations we
fix the polar displacements in the metallic electrodes
and only change the amplitude of the polar displacements in NaNbO$_3$.
Independent of the NaNbO$_3$ film thickness, the energy is minimized for the ferroelectric ground state ($\mathcal{A}_\mathrm{NNO}\neq0$), indicating that an ideal ferroelectric capacitor can be reduced to an ultrathin (single unit cell) size, i.e., $t_\mathrm{FE}^*\rightarrow0$.

The disappearance of the critical-thickness limit to ferroelectricity is the result of the parallel polar displacements present at the electrode/dielectric interfaces (\autoref{fig1}b).
As proposed in Ref.\ \onlinecite{Stengel/Vanderbilt/Spaldin:2009a}, the ferroelectric state in ultrathin-film devices depends crucially on the nature of the chemical bonds at the metal/oxide interface.
Here, this interfacial bonding occurs and is an immediate consequence of the structure of the polar-metal electrodes. The enhanced and parallel interfacial polar displacements ``imprint'' and lead to an overall enhancement of the ferroelectric instability of the film, which we assess further below. We stress that differently from Ref.\ \onlinecite{Stengel/Vanderbilt/Spaldin:2009a}, the interfacial dipole distortions are due to a geometrical mechanism driven by the polar structure of the metallic electrode and not due to the stiffness of the electrode--oxide bonds.
\begin{figure
\includegraphics[width=\columnwidth]{fig4
\caption{{Typical behavior of a nanoscale ferroelectric capacitor below the critical thickness.} \textbf{a}, Nanocapacitor consisting of a BaTiO$_3$ film ($m$=2)  between  centrosymmetric SrRuO$_3$ ($n$=6) electrodes. The arrow indicates the direction of the polar displacements in BaTiO$_3$.
\textbf{b}, Energy as a function of the polar mode amplitude $\mathcal{A}_\mathrm{BTO}$ in BaTiO$_3$. The energy increases for all  thicknesses $m$ analyzed.
}
\label{fig3}
\end{figure}

We next examine a \srobto\ capacitor (\autoref{fig3}a), which as before may be written as
{[SrO-(RuO$_2$-SrO)$_n$/TiO$_2$-(BaO-TiO$_2$)$_m$]}
to reveal the layered monoxide planes in the structures.
We focus on the SrO/TiO$_2$ interface geometry to demonstrate the generality of this solution to the critical thickness problem.
We constrain the in-plane lattice parameters to that of SrTiO$_3$ (3.905\, \AA) and
we relax the out-of-plane lattice parameter and the atomic positions (see Methods), examining
capacitors with $m$=1, 2, and 3 that are well below the reported $m$=7 critical thickness \cite{Junquera/Ghosez:2003}.
With nonpolar SrRuO$_3$ electrodes the paraelectric configuration is  energetically more stable than the ferroelectric configuration for all BaTiO$_3$ film thicknesses  studied.
This is confirmed by the increase in total energy as a function of the polar mode amplitude $\mathcal{A}_\mathrm{BTO}$ (\autoref{fig3}b).
Indeed, the use of a centrosymmetric metal for the thinnest ferroelectric film results in an antisymmetric poling effect of the two interfaces, which forbids the possibility of a ferroelectric displacement \cite{Gerra:2006}.

\begin{figure
\centering
\includegraphics[width=0.9\columnwidth]{fig5
\caption{{Energetic landscape of a BaTiO$_3$ nanocapacitor with ``polar'' SrRuO$_3$ electrodes.} Energetic gain for different dielectric thicknesses: $m$=1  (circles), $m$=2 (squares) and  $m$=3 (diamonds). The Sr displacements given with respect to the centrosymmetric ground state are  fixed to \textbf{a}, 0.07~{\AA} (see inset) and \textbf{b},  0.14~{\AA}.
The energy of the polar  structure with $\mathcal{A}_\mathrm{BTO}$=0~{\AA} is taken as reference for each $m$.}
\label{fig4}
\end{figure}

Now we perform a computational experiment whereby we transmute centric SrRuO$_3$ into a hypothetical polar metal by following the design rules for noncentrosymmetric metals introduced in Ref.~\onlinecite{Puggioni:2013}. We do this by imposing a polar distortion in SrRuO$_3$,  which involves only the Sr atoms, as the orbital character at the Fermi level has a negligible  contribution from these atoms, with
parallel polar displacements
at the SrO/TiO$_2$ interfaces as suggested by our \loonno\ capacitor results. We point out that bulk SrRuO$_3$ does not exhibit polar distortions and here we make it artificially polar to isolate the interfacial geometric effect independent of chemistry with respect to the model with centrosymmetric electrodes.

In \autoref{fig4}a we show the energy evolution of this hypothetical capacitor
as a function of the mode amplitude $\mathcal{A}_\mathrm{BTO}$, with  parallel polar Sr displacements imposed uniformly at 0.07~{\AA} with respect to the centrosymmetric structure (see inset).
In contrast to \autoref{fig3}a, the ferroelectric state of BaTiO$_3$ is more stable than the paraelectric geometry for all thicknesses $m$=1, 2, and 3 as indicated by the energy gain for $\mathcal{A}_\mathrm{BTO}\ne0$. These results indicate that $t_\mathrm{FE}^*\rightarrow0$
 in the BaTiO$_3$ film between the polar-metal SrRuO$_3$.

The energy gain is strongly influenced by the polar displacements of the Sr atoms.
In particular, by doubling the amplitude of the polar displacements of the Sr atoms, from 0.07~{\AA} to 0.14~{\AA}  for the case $m$=1, we find that the energy gain increases from $\sim$3~meV to $\sim$10~meV
(\autoref{fig4}b).
Comparing \autoref{fig4}a and \autoref{fig4}b, we find a shift in the critical mode amplitude to larger values,
which suggests that the device containing a polar-metal electrode with larger Sr displacements displays a larger ferroelectric polarization.
Indeed, when we consider the other limit by fully removing the  polar displacements at the SrO/TiO$_2$ interface (setting them to 0\,\AA), the energy landscape presented in \autoref{fig3}b is restored.
Note that for the disappearance of the critical thickness, it is necessary that the polar direction (or a component of it) in the electrodes, and therefore that of the interfacial dipole, coincides with the direction of polarization of the ferroelectric film (\autoref{fig4}b, inset).

\begin{figure
\includegraphics[width=\columnwidth]{fig6
\caption
The planar and macroscopically averaged electrostatic potential of the BaTiO$_3$  ferrocapacitor ($m$=2) along [001] between
polar (solid line) and centrosymmetric  (broken line) SrRuO$_3$ electrodes. The difference, $\Delta$, is also shown.
The amplitude of the polar distortion in the BaTiO$_3$ film is fixed to $\sim0.6$~\AA, which corresponds to the minimum in \autoref{fig4}b. The configuration with paraelectric BaTiO$_3$ is used as reference.}
\label{fig6}
\end{figure}

One could argue that the findings here are a result of the polar metals better screening the ferroelectrics polarization; however, this is not the case. Indeed the electrostatic for the BaTiO$_3$ nanocapacitor with ``polar''  and nonpolar SrRuO$_3$ electrodes are almost the same (\autoref{fig6}).
Moreover, polar metals typically have longer screening lengths than conventional metals \cite{Giovannetti_etal:2016}.
This result further confirms the role of interfacial geometric effects induced by the polar structure of the metallic electrode in controlling
the critical thickness.

Lastly, we discuss how the proposed device can be switched.
Polar metals are not ferroelectrics. Indeed, the application of an electric field cannot switch the polar distortion in the metal because the free electrons will screen the electric field.
However, it has been show that the polar distortion in the metal can be switched
by applying an electric field to a superlattice composed of  an insulating ferroelectric material and a polar metal
by coupling to the ferroelectric polarization \cite{Xiang:2014}.
Similarly, when an electric field is applied to the aforementioned nanocapacitors, the polar distortion in the ferroelectric thin film should align along the direction of the electric field; then because of the interfacial coupling between the polar metallic electrode and the ferroelectric film \cite{Xiang:2014}, the polar displacements in the polar metal and the interfacial polar displacements should follow. Note that the geometric configuration required to sustain ferroelectricity, \emph{i.e.}, $t_\mathrm{FE}^*\rightarrow0$,
is preserved in the switching mechanism. Alternative approaches have also been applied to degenerately doped ferroelectrics.\cite{Nukala:2017}

\section{Conclusion}

In summary, we proposed a ferroelectric capacitor where the conventional metallic electrodes are replaced by \emph{noncentrosymmetric} metallic electrodes.
We showed that the polar displacements in the noncentrosymmetric metallic electrodes induce interfacial ferroelectricity, which supports a polar instability in the ferroelectric film  regardless of the dielectric thickness.
Although we utilized LiOsO$_3$ herein for simplicity, we point out that our result is completely general and the same conclusions may be achieved using other noncentrosymmetric metals as electrodes with our described geometric constrations  (see Ref.~\onlinecite{Benedek:2016} for a list of materials).
These polar-metal based nanoscale capacitors maintain the functionality of the ferroelectric film independent of the degree of miniaturization and could lead to device architectures with improved scalability

\begin{acknowledgements}
D.P.\ and J.M.R.\ acknowledge the Army Research Office under Grant No.\ W911NF-15-1-0017 for financial support and the DOD-HPCMP for computational resources.
\end{acknowledgements}