\section{\label{sec:introduction}Introduction}
Spin qubits~\cite{Loss1998,Barenco1995,Brum1997,Kane1998}, whose dephasing time can be on the order of microseconds, thus several orders longer than those of charge qubits, are promising proposals for scalable solid-state quantum bits. Spin qubits based quantum gate mechanism was first proposed by Loss and Divincenzo~\cite{Loss1998}. Later, Burkard and his collaborators~\cite{Burkard1999} showed that the ground state spin configuration of two electrons in coupled quantum dots can switch between a spin-singlet ($S=0$) and a spin-triplet state ($S=1$). The manipulation of the switching between the two spin states can be realized by changing the applied gate voltage in the presence of a finite applied magnetic field~\cite{Kyriakidis2002,Lee2004}. Such spin qubits can also be realized with two electron concentric quantum rings. As compared to quantum dots, quantum rings have a very different topology due to the presence of the central hole. As a result, the single particle energy levels of quantum rings have a unique magnetic field dispersion, giving rise to the Aharonov-Bohm (AB) effect where the quantized ground state total angular momentum increases with magnetic field~\cite{Fuhrer2001,Lorke2000}. Since in a multi-ring configuration the electrons may switch between different rings, resulting in different patterns of AB oscillations in the same structure~\cite{Szafran2005}. A few electron system in such structures may exhibit a fractional Aharonov-Bohm effect~\cite{Chakraborty1994,Niemela1996} due to the correlations~\cite{Niemela1996} between the electrons.

Quantum rings can be realized by using local oxidation techniques (lithography)~\cite{Fuhrer2001,Wiel2003a}, as well as self-assembly with~\cite{Garcia1997} or without~\cite{Gong2005,Mano2005,Kuroda2005} a capping layer, such as droplet-epitaxial growth techniques~\cite{Mano2005,Somaschini2009}. By changing the substrate temperature during the quantum ring growth, it is possible to fabricate not only concentric double quantum rings (CDQRs), but also multiple concentric quantum rings (MCQRs)~\cite{Somaschini2009,Somaschini2010}. The radii of the rings, together with the confinement in each ring, may strongly affect the spectrum~\cite{Szafran2005,Climente2006,Planelles2005}. The magnetic dependence of the two and few electron spectra in a single quantum ring\cite{Chakraborty1994,Niemela1996}, CDQRS~\cite{Szafran2005,Climente2006,Escartin2009} and MCQRs\cite{Escartin2010} has already been studied theoretically. Previous studies showed that transitions between singlet and triplet spin states in quantum dots can be controlled by the applied gate voltage which controls the confinement\cite{Wagner1992,Kyriakidis2002}. Earlier in 2005, Szafran and Peeters~\cite{Szafran2005} studied the few electron eigenstates in CDQRs in case of different inter-ring coupling (in their paper, it depends on the distance between the rings), while they did not focus on the details of the confinement (or coupling) dependent spin-singlet spin-triplet transition. Later in 2010, Escart\'{\i}n reported interesting few electrons ground state and persistent current results in triple concentric quantum rings with strong confinement\cite{Escartin2010}. However, studies with full detail of the spin singlet spin triplet transition and their dependence on an external magnetic field or confinement is still lacking for MCQRs, while it is essential for their eventual application in a practical implementation for quantum information. Such calculations will be presented in this paper. We will show an unusual ordering of angular momentum transitions in MCQRs that strongly depend on the confinement, and we studied the period and amplitude of the spin singlet - spin triplet gap and its dependence on both the magnetic field and the confinement potential in the rings.

We restricted ourselves to the two electron system of two-dimensional triple concentric quantum rings where a perpendicular magnetic field is applied, and for two extremes of confinement potential in the rings. We will first present our physical model, and then in the second part of this paper we will study the single electron states in triple concentric quantum rings by using the finite difference method. After that we calculate, using the configuration interaction method, the two electron spin singlet and spin triplet ground and excited state energies, and the spin singlet spin triplet energy gap as a function of the applied magnetic field for both strong and weak confinement. In the last part of our paper, we will discuss the influence of the spin Zeeman energy, by taking different values of the Land\'{e} $g$-factor, on the spin singlet - triplet transition.
\section{\label{sec:2}Model}
In two-dimensional circularly symmetric concentric rings, the single particle Hamiltonian in the presence of a perpendicular magnetic field $B$, using the symmetric gauge $A=-\frac{B}{2}(-y,x,0)$, is given by (in polar coordinates)
\begin{eqnarray}
H_{e}= &-&\frac{\hbar^2}{2m_{e}}\left(\frac{d^2}{d\rho^2}+\frac{1}{\rho}\frac{d}{d\rho}\right)+\frac{\hbar^2l^2}{2m_{e}\rho^2}\nonumber\\
&+&\frac{m_{e}\omega_c^2\rho^2}{8}+\frac{1}{2}\hbar\omega_c l++\mu_B g s_z+V_{e}(\rho)
\label{eq:SHamil},
\end{eqnarray}
where $m_{e}$ is the effective electron band mass, $l$ is the angular momentum of the considered state, and $\omega_c=q_{e}B/m_{e}$ is the cyclotron frequency. Because of circular symmetry the single particle wave function is assumed to be $\Phi_{n,l}=1/\sqrt{2\pi}e^{-il\varphi}\psi_n(\rho) $ where the angular momentum is a conserved quantity. Eq.~(\ref{eq:SHamil}) is the Hamiltonian for the radial part and the problem is thus reduced to an effective one dimensional ($1$D) one. The last term $V_{e}(\rho)$ is the confinement potential of the electron in the concentric rings which is modeled by a parabolic confinement potential and taken of the form:
\begin{equation}
V(\rho)=\frac{m_{e}\omega^2}{2}min[(\rho-R_1)^2,(\rho-R_2)^2,(\rho-R_3)^2,...]
\end{equation}
where $R_i$ stands for the ring radii, $\omega$ is the harmonic oscillator frequency of the lateral confinement for the electrons in each ring\cite{Szafran2005}, and $\mu_B$ is the Bohr magneton. The confinement is taken the same for all rings in order to limit the number of parameters (The outer ring in a real system may have a larger confinement potential, we will discuss its effect at the end of this paper). The second, third, fourth, and fifth terms of the Hamiltonian (\ref{eq:SHamil}) are the centrifugal, diamagnetic, orbital Zeeman terms, and the spin Zeeman term, respectively. The spin-orbit coupling is not considered here as we are dealing with quantum rings of size hundred nanometers. Moreover, the spin Zeeman interaction is decoupled from the orbital degree of freedom, it does not influence the tunnel coupling and can be trivially accounted for as an energy shift linear in $B$. Meanwhile, the Land\'{e} $g$-factor is not a constant in nano-structure (even the sign can changed), but can be controlled through the material concentration\cite{Porras-Montenegro2010,Salis2001}, structure size\cite{Dios-Leyva2006,Babayev2009,Hannak1995}, temperature~\cite{Oestreich1995} and the magnetic field~\cite{Porras-Montenegro2010} applied to the structure. Therefore, in the first part of our paper, we will not take the spin Zeeman energy into account, but the influence of the spin is not neglected and enters through symmetry of the wave function. However, the spin Zeeman energy will have a considerable effect on the spectrum especially for large value of the magnetic field, it lowers~\cite{Wagner1992} (depending on the value of the Land\'{e} $g$-factor) the spin triplet state energy level. Therefore, we will come back to this term and study the influence of it on our results in the last part of the paper.

The single particle Schr\"{o}dinger equation can not be solved analytically. Therefore, we used the finite difference method. The single electron energy is $E_{n,l}$ with wavefunction $\Phi_{n,l}\left(\vec{r}\right)$ where the quantum number $n$ is for the radial part and $l$ is the angular momentum. After obtaining the single electron eigen-states, we can construct the two electron wave functions as a linear combination of products of single electron states $\Psi\left(\vec{r}_1,\vec{r}_2\right)=\Sigma_iC_i\Phi_{n_{1i},l_{1i}}\left(\vec{r}_1\right)\Phi_{n_{2i},l_{2i}}\left(\vec{r}_2\right)$. Then by using the configuration interaction method we can find the eigen-states for two electrons whose Hamiltonian is given by:
\begin{equation}
H_t=H_1+H_2+V_c\left(\vec{r}_1,\vec{r}_2\right)
\end{equation}
where $V_c\left(\vec{r}_1,\vec{r}_2\right)=q_1q_2/4\pi\epsilon\epsilon_0|\vec{r}_1-\vec{r}_2|$ is the Coulomb
potential. We adopted the same method we used previously~\cite{Li2011} to transform the 4-D Coulomb integral into a 2-D one. Note that for the two electron singlet state we should have $\Psi\left(\vec{r}_1,\vec{r}_2\right)=-\Psi\left(\vec{r}_2,\vec{r}_1\right)$, while for the two electron triplet case we have $\Psi\left(\vec{r}_1,\vec{r}_2\right)=\Psi\left(\vec{r}_2,\vec{r}_1\right)$.

In our numerical calculation, we use the GaAs value for the effective mass $m_e=0.067m_0$, the dielectric constant $\epsilon=12.4$, and assume the lateral confinement to be $\hbar\omega=3$ meV, for strong coupling between the different rings and $\hbar\omega=30$ meV for weak coupling for fixed radial positions $R_i$ of the rings.
\section{\label{sec:3}Single particle states}
\begin{figure}[h]
\includegraphics[width=8cm]{Eee3r
\vspace{-0.1cm}\caption{\label{fig:Eee3r}(Color online) Single electron spectrum for triple concentric rings with $\hbar\omega=3$ meV (a) and $\hbar\omega=30$ meV (b), as a function of magnetic field. Here solid curves are for the ground state energy of each angular momentum, while dashed and dot-dashed curves are for the first and second excited states, respectively.}
\end{figure}
The results of the single electron eigen-energies for triple concentric rings are shown in Figs.~\ref{fig:Eee3r}(a) (the weak confinement case $\hbar\omega=3$ meV) and \ref{fig:Eee3r}(b) ($\hbar\omega=30$ meV). For the ring radius we took typical values that are realizable experimentally: $R_1=120$ nm, $R_2=180$ nm and $R_3=240$ nm. In Figs.~\ref{fig:Eee3r}(a) and \ref{fig:Eee3r}(b), the solid lines correspond to the ground states of each angular momentum $l$, while the dashed and dot-dashed lines are the corresponding first and second excited states, respectively. It is remarkable that the spectrum of the electron shows a quite different behavior for different confinement potential. The spectrum for $\hbar\omega=30$ meV is similar to a combination of the spectrum of three 1D rings (i.e. rings with zero width). While for $\hbar\omega=3$ meV, it exhibits clear finite width effects (ground state energy far lower than $1/2\hbar\omega$). Comparing with the results of Ref.~\onlinecite{Szafran2005} for CDQRS, we notice that the spectrum shown in Fig.~\ref{fig:Eee3r}(a) is similar to Fig. 3(c) of Ref.~\onlinecite{Szafran2005}, which implies a strong coupling between different rings, while the latter one looks rather like Fig. 3(a) of Ref.~\onlinecite{Szafran2005}, except with a smaller period of the level crossings. This difference results from the different confinement potential which makes the width of the considered rings for the previous case $d_1=2\sqrt{2\hbar/m_e\omega}=54.9$ nm  much larger (for the latter case we have $d_2=17.4$ nm). As a consequence the coupling between the rings is much stronger for the weak confinement case. This is the reason why the electron ground state energy is much lower than $1/2\hbar\omega$.

In both cases, the energy levels of the same angular momentum switch their order through avoided crossings at different magnetic field. And with increasing magnetic field, the energy levels for different angular moment $l$ (except some states with small value of $l$ in the case $\hbar\omega=3$ meV) possess three minima (e.g., see the short horizontal line in Fig.~\ref{fig:Eee3r}(b) for the $l=2$ state), before and after the two avoided level crossings. At those magnetic fields (where levels cross) the corresponding wave functions change. The spatial location of the wave functions change from the outer ring (region) to the inner ring with increasing magnetic field, but this behavior for the two cases is different. This difference is clearly seen from the effective radius of the electron $<\rho_e>$ as shown in Fig.~\ref{fig:effrho3r}. We notice that for the weak confinement case, as the coupling between the rings is strong, the ground states with small value of $l$ (like $l=0,1,2$, which have small centrifugal potentials) exhibit only one clear minimum. As a result, the wave functions locate predominately in the region between the middle and the inner ring at small magnetic field. For states with larger angular momentum $l$, larger centrifugal potentials result in an electron localization that is mainly in the middle ($l=3$, two minima for the energy), or between the outer and the middle ring ($l>3$ and $<6$, two clear minima), or in the outer ring ($l>6$, three minima). For states with larger angular momentum, a larger magnetic field is needed to make the electron wave function leave the outer ring and to transfer entirely to the inner ring, which is clearly shown in Fig.~\ref{fig:effrho3r}(a).
\begin{figure}
\includegraphics[width=7.8cm]{effrhor3}
\vspace{-0.2cm}\caption{\label{fig:effrho3r}(Color online) Effective radius $<\rho_e>$ of the electron for $\hbar\omega=3$ meV (a) and $\hbar\omega=30$ meV (b) in the lowest energy states of the different angular momentum. The thin dashed lines in (a) shows the position of the rings. The bold dashed dark curves in both figures (a) and (b) show the effective radius of the ground state, while the numbers specify the angular momentum.}
\end{figure}
\begin{figure}
\includegraphics[width=8cm]{singwave}
\vspace{-0.2cm}\caption{\label{fig:singwave}(Color online) The wave function of the ground (blue solid), first excited (green dashed) and second excited (red dot-dashed) electron states for both weak (upper two figures) and strong confinement (lower two figures) cases for magnetic field $B=0.1$ T, and different angular momentum. The vertical dashed lines indicate the position of the rings (i.e. the minima of the confinement potential).}
\end{figure}

For case $\hbar\omega=30$ meV, the magnetic field dependence of the spectrum is quite different. Small effective width of the rings makes the confinement extremely large and the coupling between the rings very small. As a result the lowest three states for each angular momentum are entirely localized in the three different rings. Fig.~\ref{fig:singwave} shows the lowest three states of the electron wave functions for both the strong and weak confinement cases. Notice that the wave function in the strong confinement case is much more localized, in contrast to the weak confinement case where the wave functions are more extended. We also find from Fig.~\ref{fig:effrho3r}(b) that the $l=0$ state is always localized in the inner ring. While for the non-zero angular momentum states, the electron is entirely in the outer ring (smallest centrifugal potential) for small magnetic field. After reaching the magnetic field where the first avoided level crossing takes place (e.g., $B=0.06$ T for $l=2$ as shown in Fig.~\ref{fig:Eee3r}(b)), the electron switches to the middle ring. And finally, the electron will localize in the inner ring. This can be clearly seen in Fig.~\ref{fig:effrho3r}(b) where $<\rho_e>$ exhibits a steep change after the two avoided level crossings.

There are several additional differences between Figs.~\ref{fig:Eee3r}(a) and \ref{fig:Eee3r}(b): 1) As a result of the extremely small coupling and large ring radii, the distance between the anti-crossing energy levels in the latter case is much smaller (almost zero). 2) For the strong confinement case, there are more frequent angular momentum transitions slightly above the ground state transition, which correspond to the situation where the electron is entirely in the outer or middle ring. Moreover, parts of these transitions can become the ground state transition at some magnetic field (e.g. black dotted region in Fig.~\ref{fig:Eee3r}(b)). 3) Different from the weak confinement case, the angular momentum transition does not exhibit uniform behaviour, it is not always from the electron localized in the same ring and not always with increasing angular momentum. For example, Fig.~\ref{fig:effrho3r}(b) clearly shows that the angular momentum changes in the sequence of ($0$, inner ring) (electron has a zero angular momentum and localized mainly in the inner ring), ($1$, middle ring), ($1$, inner ring), ($3$, middle ring), ($6$, outer ring), ($2$, inner ring) and so on. The state ($0$, inner ring) is the ground state at very small magnetic field. With increasing magnetic field, the Zeeman term decreases more the energy for states with larger angular momentum, while the centrifugal term and the diamagnetic term increase the energy. The centrifugal term decreases the energy of the state that is localized in the ring with larger radius, while the diamagnetic term influences it in the reverse way. Thus, there is a competition between the different wave function configurations in the three rings, the sum of the centrifugal, the diamagnetic and the Zeeman terms in Hamiltonian (\ref{eq:SHamil}) decides which one has the smallest energy (Zeeman term results in an angular momentum transition, and the diamagnetic and centrifugal term determines in which ring the electron is localized).

The above two cases are just two opposite extremes for triple concentric quantum rings. The spectra and effective radii show how the electron behaves at these extreme conditions. We may find an interesting spectrum for the case with medium confinement potential, as shown in Fig~\ref{fig:Ee83r}. The spectrum is just a combination of the two spectra depicted in Figs.~\ref{fig:Eee3r}(a) and (b). For small magnetic field, we have a spectrum like in the weak confinement case, while much more complicated angular momentum transitions occur when the magnetic field is large, just like in the strong confinement case. This also implies that the coupling between the rings becomes smaller with increasing magnetic field.

\begin{figure}
\includegraphics[width=8cm]{Ee83r
\vspace{-0.2cm}\caption{\label{fig:Ee83r}(Color online) Single electron spectrum for triple concentric rings with $\hbar\omega=8$ meV, as a function of magnetic field.}
\end{figure}
\section{two electrons confined in triple concentric rings}
We first calculate the two electron energy for different total angular momentum in triple concentric rings with $\hbar\omega=30$ meV. The two electron wave function with fixed total angular momentum $L$ are constructed from linear combinations of the one electron wave functions as follows:
\begin{eqnarray}
\Psi&=&\sum _{n_1=1}^{n_m} \sum_{n_2=1}^{n_m} \sum_{\quad l=-l_m}^{l_m}\!\!\!\!\!\!^\prime C_{n_1n_2}^l\left[\Phi_{n_1,(L+l)/2}(\vec{r_1})\Phi_{n_2,(L-l)/2}(\vec{r_2}) \right.\nonumber\\
&&\left.\pm\Phi_{n_2,(L-l)/2}(\vec{r_1})\Phi_{n_1,(L+l)/2}(\vec{r_2})\right],
\label{eq:phi}
\end{eqnarray}
where the subscript $n_1$ and $n_2$ correspond to the energy levels of the one electron problem with fixed angular momentum. The sum $\sum^\prime$ denotes that only even values of $l$ are taken when $L$ is even and odd values otherwise. Because the spatial and the spin parts of the wave function decouple, the value of the total spin will determine the symmetry of the spatial wave function under particle permutation. The plus sign in Eq.(~\ref{eq:phi}) corresponds to the singlet state with total spin $S=0$, and the minus sign corresponds to the triplet state with $S=1$. Notice that for the spin triplet state, the quantum numbers $n$ and $l$ for the two electrons can not be equal simultaneously. The value $l_m$ is taken sufficiently large such that the energy levels are determined within a given tolerance. As a result of the strong confinement potentials in the rings, the third and fourth excited states have a much larger energy than the lowest three levels. We found that taking $n_m=3$ gives already good accuracy.

\begin{figure}
\includegraphics[width=8cm]{E2e3r30mev
\vspace{-0.2cm}\caption{\label{fig:E2eL}(Color online) The spin singlet (solid line) and spin triplet (dashed line) states ground state and first exited state energies for different value of total angular momentum $L$, as a function of magnetic field. Different colors correspond to different value of $L$. Here $\hbar\omega=30$ meV.}
\includegraphics[width=8cm]{EL07
\vspace{-0.2cm}\caption{\label{fig:EofL07}(Color online) The lowest four states for two electrons in triple concentric rings for $\hbar\omega=30$ meV.  The blue curves are for total angular momentum $L=0$ and green curves for $L=7$. Here S (T) stands for spin singlet (triplet) state, while G (F) stands for the ground and first excited state for the corresponding spin singlet and spin triplet states.}
\end{figure}
The results of the two electron ground and three lowest excited states for different total angular momentum $L$ are shown in Fig.~\ref{fig:E2eL} (always with two singlet states and two triplet states for fixed $L$). The solid (spin singlet) and dashed (spin triplet) line in the same color correspond to the same total angular momentum. Notice that the first and second excited states are degenerate here, however this degenerate behavior only exists in some region of magnetic field (compare Figs.~\ref{fig:E2eL} and \ref{fig:EofL07} for the $L=7$ states). We should mention, as a result of the exchange energy, that for even total angular momentum the spin singlet state is the two electron ground state, while for odd total angular momentum the spin triplet is the ground state. By increasing the magnetic field, the angular momentum of the ground state shifts from $0$ to a larger value and exhibits a fractional Aharonov-Bohm effect. Moreover, the ground state switches between spin singlet and spin triplet states and exhibits a clear spin singlet - spin triplet transition. This transition exists not only in the ground state, but also in the excited states. Notice that from a further inspection of the spectrum we see that the ground state angular momentum transition has the same period (indicated by the vertical dashed line) as the third excited state, but smaller than the overlapped first and second excited states (dotted line). The period for the ground and third excited states is about $B=0.0113$ T, while the period for the first and second excited states is $B=0.0147$ T. Comparing these periods with the single electron spectrum, we find that the period of the ground state $B=0.0113$ T is almost the same as one half the period for the single electron in the most outer ring ($B=0.0224$ T, see the third lowest angular momentum transition in Fig.~\ref{fig:Eee3r}). This indicates that the electrons are in the most outer ring for the two electron ground and third excited states, and is also a fingerprint of the so called fractional Aharonov-Bohm effect. On the other hand, the period of the first and second excited states are much smaller (larger) than one half of the period for the single electron in the middle (outer) ring (which is $0.0407$ T for the middle ring), which means that the electrons are in the area between the middle and outer rings (since the confinement is extremely strong, an electron may be partly in the middle ring and partly in the outer ring).

Figure~\ref{fig:EofL07} shows the lowest four states for two electrons: the spin-singlet ground state (solid curves), the spin-singlet first excited state (dashed line), the spin-triplet ground state (dot-dashed line) and the spin-triplet first excited state (dotted curves). The blue curves (also the ones specified in the inset) are for total angular momentum $L=0$ (even value) and green curves for $L=7$ (odd value). We found that the ground state for $L=0$ is a spin singlet at small magnetic field, and almost overlap with the spin triplet when $B$ reaches $0.074$ T. The inset shows that there are three anti-crossing points between the ground (spin singlet) and the first excited states (spin triplet). The case for $L=7$ is a little different. The spin triplet takes the place of the spin singlet in case of $L=0$, and instead of an anti-crossing, the spin triplet state overlaps with the spin singlet state in some regions of field $B$.

The result for $\hbar\omega=3$ meV is shown in figure~\ref{fig:E2eL3}. As the coupling between the different rings is stronger we take $n_m=6$ instead. Just like the case of strong confinement, the ground state shows a fractional Aharonov-Bohm effect and a similar spin singlet - spin triplet transition. The differences from the previous case are as follows: 1) The second and third angular momentum transition are never degenerate. The degeneracy of the first and second excited states which we have in the strong confinement case, within the given magnetic field region, is lifted; 2) The two electron energy has a distinctly increscent tendency with increasing magnetic field for both ground and excited states. The reason is that for the weak confinement case many more single electron levels have to be taken into account, especially at large magnetic field, and the Coulomb interaction between them strongly increases the total energy. 3) From the inset of Fig.~\ref{fig:E2eL3} we observe that the period for these different transitions are different. The lowest angular momentum transition (ground state) has the largest period, which means that the two electrons occupy the region with the smallest radius as compared to the excited states. Moreover, the period of the ground state transition for $\hbar\omega=30$ meV almost does not change with increasing magnetic field (up to $0.5$ T). While the period for the case of $\hbar\omega=3$ meV has a slight but observable change. For example, the period is $B=0.0211$ T for the transition from $L=1$ to $L=2$, but $B=0.0185$T which is smaller for the transition from $L=13$ to $L=14$ state. This indicates that, in contrast to the case of strong confinement, the two electrons could move slowly to the outer region of the rings with increasing magnetic field.
\begin{figure}
\includegraphics[width=8cm]{E2e3r3mev
\vspace{-0.2cm}\caption{\label{fig:E2eL3}(Color online) The spin singlet (solid line) and spin triplet (dashed line) ground state and first exited state energies for the case of $\hbar\omega=3$ meV. In the inset we indicate the period for the lowest three angular momenta transition}
\end{figure}
This can be more clearly seen from the electron density in Fig.~\ref{fig:denp}. The electron probability density in the radial direction is defined by
\begin{equation}
n(\rho)=\sum_{i=1}^2<\delta(\vec{\rho}-\vec{\rho}_i)>.
\end{equation}
We show here the electron probability density of the ground state in the radial direction, for three different values of the magnetic field $B=0$ T (red solid curve), $B=0.1$ T (green dashed curve) and $B=0.2$ T (blue dash-dotted curve). We also give in Fig.~\ref{fig:denp} the value of the angular momentum and total spin of the two electrons ground state at these values of the magnetic field. We find that the electron probability density for strong confinement rings (Fig.~\ref{fig:denp}(b)) overlap for different values of the magnetic field, and it is always concentrated in the most outer ring. While for weak confinement rings, as shown in Fig.~\ref{fig:denp}(a), the probability density peaks of the electron moves slightly to the region with larger $\rho$ with increasing magnetic field. Moreover, the electron probability extend into all the rings as a result of the strong coupling between the different rings. It is predicted that the coupling between the different rings could decrease as a result of the large energy difference in the rings when the magnetic field is very large. Thus we may have a similar spectrum as in the strong confinement case.
\begin{figure}
\includegraphics[width=7.5cm]{eledens3r
\vspace{-0.2cm}\caption{\label{fig:denp}(Color online) The electron density in the radial direction of the ground state at three different values of the magnetic field for both (a) $\hbar\omega=3$ meV and (b) $\hbar\omega=30$ meV. The total angular momentum and spin of the corresponding states are also given in the figures. }
\end{figure}

To have a better comparison of the two cases and to have a better understanding of the spin singlet - spin triplet transition, we show in Fig.~\ref{fig:E3rsplitE}, the splitting energy $J$ (the energy gap of the spin singlet ground state and spin triplet ground state) as a function of the magnetic field. Notice that in both cases the splitting energy is very small (around $0.01$ meV), and oscillates with increasing magnetic field. The splitting energy for $\hbar\omega=3$ meV has a larger amplitude at small value of the magnetic field, but a decreasing amplitude when $B$ increases. The amplitude of the splitting energy for rings with strong confinement is almost constant with increasing magnetic field. As the confinement energy in concentric rings can be changed through an external gate voltage, we could modify both the period of the spin singlet - spin triplet transition and amplitude of the splitting energy by changing the gate voltage. This implies that the singlet - triplet transition can be tuned by both the confinement potential and the magnetic field, and thus it tunes the ground state entanglement of the two electron system. A similar study was done in Ref.~\onlinecite{Kyriakidis2002} for lateral quantum dots with parabolic and non-parabolic confinement, where the authors found a singlet-triplet energy gap that depends on the gate voltage. However the influence of the magnetic field on the singlet-triplet gap amplitude was not discussed. We believe that the decreasing amplitude for the weak confinement rings mainly results from the concentric geometry of the ring where the electrons can extend more to the outer rings by increasing magnetic field. Therefore, similar results may not exist in quantum dots. Our results for TCQRs with strong confinement potential confirm the behavior of the persistent current studied by local-spin density-functional theory (LSDET) in Ref.~\onlinecite{Escartin2010} where TCQRs with only strong confinement were chosen.
\begin{figure}
\includegraphics[width=8cm]{SplitE3r
\vspace{-0.2cm}\caption{\label{fig:E3rsplitE}(Color online) The splitting energy of the spin singlet and spin triplet ground states as a function of the magnetic field. The blue solid line is for $\hbar\omega=3$ meV, while red dashed line for the case of $\hbar\omega=30$ meV. Insets are the ground state transition for these two cases.}
\end{figure}
\subsection{Influence of Spin Zeeman energy}
The triple concentric GaAS/AlGaAs rings we studied can be both fabricated through lithography and self-assembly approaches. In the first case, the magnetic length is typically smaller than the ring radius, and the Land\'{e} $g$-factor is related to those of quantum wells with a value from around $-0.5$ to $0.5$ and depends strongly on the width~\cite{Porras-Montenegro2010,Jeune1997,Hannak1995} of the well and magnetic field~\cite{Porras-Montenegro2010}. For the second case, the Land\'{e} $g$-factor is also not fixed, and was previously taken with completely different values, such as in Refs.~\onlinecite{Kyriakidis2002} and \onlinecite{Escartin2010} where the value $g=-0.44$ of bulk GaAs was taken. But the value of the Land\'{e} $g$-factor is actually strongly material concentration~\cite{Salis2001} and magnetic field~\cite{Porras-Montenegro2010} dependent. Here we assumed an isotropic Land\'{e} $g$-factor with values of $-0.44$ as in bulk GaAs, $-0.05$ and $-0.1$ to calculated the shifted two electron ground state energy, and compare them to the one without the spin Zeeman term. The result is shown in Fig.~\ref{fig:E3r3meVg} for rings with weak confinement potential and in Fig.~\ref{fig:E3r30meVg} for the strong confinement case.
\begin{figure}
\includegraphics[width=7.5cm]{E3r3meVg
\vspace{-0.2cm}\caption{\label{fig:E3r3meVg}(Color online) The two electron spin singlet (solid curves) and spin triplet (dashed curves) ground states as a function of the magnetic field, in the presence of the spin Zeeman energy. Here the confinement is $\hbar\omega=3$ meV, and the Land\'{e} $g$-factor is chosen to be $g=-0.05$ (upper figure) and $g=-0.44$ (lower figure).}
\end{figure}
\begin{figure}
\includegraphics[width=8cm]{E3r30meVg
\vspace{-0.2cm}\caption{\label{fig:E3r30meVg}(Color online) Same as in Figure \ref{fig:E3r3meVg} but now for $\hbar\omega=30$ meV.}
\end{figure}

In the presence of the spin Zeeman energy, the two electron spin triplet state energy has an additional term $g\mu_BBS_z$. Therefore, the triplet state splits into three distinct levels, here only the lowest triplet state is considered and shown, as it lowers the triplet state energy thus it may change the previous ground state transitions. We found that when we just take the bulk value of $g$ the spin Zeeman energy may largely lower the triplet state energy, especially when the magnetic field is strong. The spin singlet state in this case is strongly suppressed, and even not present in the ground state at large magnetic field (e.g., after $B=0.16$ T for the strong confinement case). When the Land\'{e} $g$-factor is considerable small, for example $g=-0.05$ as shown here, the spin Zeeman energy effect is negligible for a large range of magnetic fields. For a more clear investigation, the splitting energies of the spin singlet and spin triplet ground states in the presence of spin Zeeman energy for different values of Land\'{e} $g$-factor are shown in Fig.~\ref{fig:Esplit3rg}. Here positive $J$ implies that the spin triplet state is the two electron ground state. It shows, beside the case $g=-0.44$, that the spin Zeeman energy does not have a large influence on the splitting energy, especially for small magnetic field. But for the case $g=-0.44$ the effect is considerably large and the triplet state always has the lowest energy at large magnetic field. Thus the total angular momentum does not change continuously, e.g., for the case $\hbar\omega=3$ meV, the total angular momentum of the ground state increase in the sequence $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $9$, $11$ (this is specified in the upper figure together with the corresponding total spin). Notice that as the spin Zeeman energy only introduces a magnetic dependent linear term to the triplet state and this shift is the same for different angular momentum, the spin Zeeman energy will not change the period of the spin singlet (or triplet) angular momentum transition. As a result, the period of the splitting energy is independent of the Land\'{e} $g$-factor, this is clearly shown in Fig.~\ref{fig:Esplit3rg}. But the amplitude decreases linear as a function of magnetic field. Moreover, we found that the spin Zeeman energy has a slightly stronger effect on the weak confinement case.
\begin{figure}
\includegraphics[width=8cm]{Esplit3rvsg
\vspace{-0.2cm}\caption{\label{fig:Esplit3rg}(Color online) The splitting energy of the spin singlet and spin triplet ground states as a function of the magnetic field, for four different values of the Land\'{e} $g$-factor $g=0$ (black dotted curce), $-0.05$ (red solid curve), $-0.1$ (green dashed curve), and $-0.44$ (blue dot-dashed curve). While figure (a) is for the weak confinement case with $\hbar\omega=3$ meV and figure (b) for the strong confinement $\hbar\omega=30$ meV.}
\end{figure}

\section{\label{sec:5}conclusions}
In this paper we studied the coupling between triple concentric rings and its consequences on the two electron spin singlet - spin triplet transition. The effects of the magnetic field and spin-Zeeman energy, by choosing different value of the Land\'{e} $g$-factor, on the splitting energy are also studied. We found that when the confinement potential is weak ($3$ meV) the coupling between the different rings is strong, while the coupling is considerable weaker for the strong confinement case and the spectrum exhibits a richer structure. In contrast to the single ring case, the angular momentum transitions are not always from small to large angular momentum, especially when the confinement is strong. The ground state angular momentum could change from the outer ring with large angular momentum to the inner ring but with small angular momentum, as shown in Fig.~\ref{fig:Eee3r}(b) and Fig.~\ref{fig:Ee83r}. For the strong confinement case, there are angular momentum transitions with different period which correspond to different rings in TCQRs. The sum of Zeeman energy and diamagnetic shift (centrifugal energy is also important, especially for rings with small radii) determines which angular momentum transition becomes the ground state transition. In medium confined rings, we can have an angular momentum transition which is just the combination of the two extremes.

The ground state of the two electrons exhibits an spin singlet - spin triplet transition, and this transition is strongly depended on the confinement potential of the rings (or the related coupling between different rings). By decreasing the confinement energy, we can increase the period of the transition and the amplitude of the anti-crossing energy gap. We also found that this anti-crossing energy gap (splitting energy) for the strong confinement case has an almost constant amplitude (at least within the magnetic field range we have considered). But it has a clear decreasing tendency with increasing magnetic field for rings with weak confinement. As a result, by choosing multiple concentric rings with different confinement potential, we can have distinct ground state entanglements of the two electron system. And this ground state entanglement can be tuned by a magnetic field. This magnetic field and confinement tunable singlet-triplet transition is of paramount importance in previous proposals for quantum-gate operation, because it allows for a controllable spin swap. This behavior can be observed by detecting the persistent current or from a far-infrared study. Similar results for the strong confinement case were previously presented by Escart\'{\i}n et al.~\cite{Escartin2010}, who reported few electron ground states in TCQRs by using local-spin density-functional theory. They restricted themselves to rings with strong confinement and showed a similar magnetic field dependent behavior in the persistent current. Notice that this spin singlet - triplet transition mainly depends on the confinement or the coupling between the rings. The size of rings is also a critical parameter but it rather determines the period of the fractional AB effect. We may find in Ref.~\onlinecite{Szafran2005} for CDQRs with weak confinement that the two electron spectrum has a similar pattern as in TCQRs, although they used different ring radii. The differences of TCQRs from the CDQRs are: 1) The outer ring has a considerable larger radius in TCQRs, thus a smaller magnetic field is required to realize the spin singlet - spin triplet transition. 2) As a result of the increased number of rings, the two electron ground and excited states have more optional configurations with quite different related singlet - triplet periods, and the difference between the ground and first excited state energy should be more sensitive to the magnetic field.

The spin Zeeman energy is also taken into account by choosing a different value of the Land\'{e} $g$-factor. We found that the spin Zeeman energy has a considerable large influence on the magnetic dependent spin singlet - spin triplet transition, especially when the Land\'{e} $g$-factor is large. However, the influence of the spin Zeeman energy is small enough to be neglect when the magnetic field is very small, even for a large Land\'{e} $g$-factor. By increasing the applied magnetic field, the spin triplet state lowers more its energy that the spin singlet state which is now suppressed as the ground state, and finally the triplet state persists to be the ground state. We should mention here that the period of the splitting energy $J$ was independent of the choice of the Land\'{e} $g$-factor, while the amplitude decreased linearly as a function of the magnetic field. Moreover, the TCQR with weak confinement was affected more by the spin Zeeman energy.

The outer ring can exhibit a larger confinement than the inner ring as one expects to be the case in case of self-assembled concentric rings. Such a large confinement may strongly repel the electrons from the most outer ring. In such a case, two main features appear. If the confinement is not very strong such that the electrons can still be localized in the most outer ring, the two electron spectrum possess a similar structure as for a normal TCQRs, but the period of the ground state transitions will be larger. And as the wave function is repelled towards the region between the middle and outer rings, the coupling between these two rings will be stronger, thus each splitting (i.e., ant-crossings) as shown in Fig.~\ref{fig:E3rsplitE} will be just like in the weaker confinement case. But if the confinement in the outer ring is so strong that the electrons can not localize in this ring, the TCQRs will behave like a normal concentric double quantum ring with a less rich structure of the energy spectrum.

\begin{acknowledgments}
This work was supported by the EU-NoE: SANDiE, and the Flemish Science Foundation (FWO-Vl).
\end{acknowledgments}


\section{Introduction}

{\em Introduction.---}The transition from the hadronic phase to the
quark-gluon plasma phase has been a recurrent topic in hadronic
physics~\cite{Fukushima:2011jc}. In pure gluodynamics or, equivalently, for
infinitely heavy quarks, this is a true phase transition. The order parameter
is identified as the thermal Wilson line or Polyakov loop
\cite{Polyakov:1978vu,Susskind:1979up,Svetitsky:1985ye},
\begin{equation}
\Omega = {\sf P} e^{i \int_0^{1/T}\! A_0\, dx_0}
,\quad
L(T) =  \langle  {\rm tr} \,\Omega \rangle
,
\end{equation}
where $A_0 $ is the gluon field, $T$ is the temperature and ${\sf P}$ denotes
path ordering. $L(T)$ changes abruptly from zero to near $N_c$ (the number of
colors) due to the breaking of the center symmetry $\mathbb{Z}(N_c)$ for $T
> T_c$.  In QCD, i.e., for dynamical quarks, the center symmetry is
explicitly broken by the quarks and one has instead a smooth crossover
\cite{Aoki:2006we} and the critical temperature $T_c$ is usually defined by
the condition $L^{\prime\prime}(T_c)=0$. Lattice simulations show that chiral
symmetry is restored when quarks and gluons are deconfined. These theoretical
insights strongly suggested the experimental quest for the quark-gluon plasma
in current facilities. The Polyakov loop also serves as a gluonic effective
degree of freedom in the successful Polyakov--Nambu--Jona-Lasinio (PNJL) and
Polyakov--quark-meson models (PQM) to describe hot and/or dense QCD
\cite{Meisinger:1995ih,Fukushima:2003fw,Megias:2004hj,Ratti:2005jh,Schaefer:2007pw}.

Since hadrons (and possibly glueballs) are the physical states in the confined
phase it should be expected, by quark-hadron duality, that physical quantities
admit a representation in terms of hadronic states. The QCD pressure presents
a prime example of this, through the hadronic resonance gas model (HRGM)
\cite{Hagedorn:1984hz, Agasian:2001bj,Tawfik:2004sw, Megias:2009mp,
  Huovinen:2009yb,Borsanyi:2010cj, Bazavov:2011nk},
\begin{equation}
\frac{1}{V}\log Z =
-\int \frac{d^3 p}{(2\pi)^3} \sum_\alpha \zeta_\alpha g_\alpha
\log \left( 1 - \zeta_\alpha e^{-\sqrt{p^2+M_\alpha^2}/T} \right) \,,
\label{eq:rgm}
\end{equation}
with $g_\alpha$ the degeneracy factor, $\zeta_\alpha=\pm 1$ for bosons and
fermions respectively, and $M_\alpha$ the hadron mass. These resonances are
 the low-lying states listed in the review by the Particle Data Group
(PDG)~\cite{Nakamura:2010zzi}. Actually, in the large $N_c$ limit this
expectation becomes a true theorem in QCD since the flavor resonances become
narrow $\Gamma/M=O(1/N_c)$ (see
e.g. \cite{Cohen:2004cd,Masjuan:2012gc}). After some controversies, lattice
calculations seem to suggest that this is also a good approximation in the
physical case $N_c=3$ \cite{Borsanyi:2010bp}. This problem has been addressed
within a strong coupling expansion for heavy quarks in
\cite{Langelage:2010yn}.

Despite its prominent theoretical role, $\Omega({\bm{x}})$ does not appear to be
directly accessible in the laboratory, being most naturally defined in the
imaginary-time formalism of field theory at finite temperature
\cite{Matsubara:1955ws,Landsman:1986uw} (see \cite{MoralGamez:2011en} for its
definition within the real-time formalism). The realization of the Polyakov
loop through a static (or heavy) quark and its relation with the heavy-quark
self-energy (or even with the binding energy between a static and a dynamical
quark \cite{Petreczky:2012rq}), is not new
\cite{Svetitsky:1985ye,Kaczmarek:2002mc}, but the phenomenological
consequences of this fact have not yet been extracted at a quantitative level.
Here we argue that the Polyakov loop in the confined phase can also be
represented in terms of hadronic properties in a direct and quantitative way,
similarly to the HRGM for the pressure.

{\em Polyakov loop and hadronic spectrum.---} In the Hamiltonian formulation
of QCD~\cite{Kogut:1974ag,Luscher:1976ms}, the gauge is partially fixed by the
condition $A_0=0$ and the dynamical degrees of freedom are contained in the
spatial gluons ${\bm{A}}$ and the quarks.  The time-independent gauge
transformations $g({\bm{x}})$ are still a residual symmetry of the Hamiltonian
acting in the Hilbert space $\mathcal H$ of functionals
$\Psi({\bm{A}},q,\bar{q})$. The gauge group ${\rm SU}(N_c)$ decomposes $\mathcal H$ into
invariant subspaces labeled by an irrep $r$ at each point ${\bm{x}}$: $ \mathcal{H}
= \bigoplus_{\{r({\bm{x}})\}} \mathcal{H}_{\{r({\bm{x}})\}} $.  In the Euclidean lattice
formulation, the role of the integration over $A_0$, or equivalently
integration over $\Omega({\bm{x}})$ with the Haar measure, is to project onto the
physical subspace, which requires a color singlet at every point ${\bm{x}}$.

An infinitely heavy quark (of a new flavor) sitting at ${\bm{x}}_0$ is a spectator
with spin and color degrees of freedom only. For the gluons and dynamical (as
opposed to heavy) quarks, this is equivalent to living in the subspace
$\mathcal{H}_{r({\bm{x}}_0)={\bf 3}}$: a color singlet at every point except
${\bm{x}}_0$, which is in the fundamental representation (${\bf 3}$ for three
colors). The projector onto this subspace is obtained by adding the factor
$N_c{\rm tr}(\Omega({\bm{x}}_0))$ to the Haar measure \cite{Tung:1985bk}.  For the
expectation value of the Polyakov loop this immediately implies the relation
\cite{Luscher:2002qv,Jahn:2004qr}
\begin{equation}
L_{\rm bare}(T)
=
\frac{1}{2}\frac{{\rm Tr}_{h,{\bm{x}}_0} (e^{-H/T})}{{\rm Tr}_{\rm phys} ( e^{-H/T})}
\,.
\label{eq:31}
\end{equation}
The factor $N_c$ in the projector represented the trivial degeneracy of the
system formed by gluons plus dynamical quarks in the fundamental
representation at ${\bm{x}}_0$, and is canceled when this is combined with the
spectator quark to form a color singlet. The factor $1/2$ removes the double
counting from the two spin states of the spectator quark.  The l.h.s. is
independent of the heavy quark spin (as the Polyakov loop carries no spin) and
this is fully consistent with the well-known heavy-quark spin symmetry present
in QCD \cite{Isgur:1989vq,Neubert:1993mb}.  The (infinite) mass of the
spectator quark is not included in $H$.

\Eq{31} is exact for the bare Polyakov loop and the partition
functions in $\mathcal{H}_{{\rm phys}}$ and $\mathcal{H}_{r({\bm{x}}_0)=3}$ on the
lattice. In the renormalized continuum limit the relation still holds, after
removing the additional specific UV divergence introduced by the heavy quark
self-energy in $L(T)$ and ${\rm Tr}_{h,{\bm{x}}_0} (e^{-H/T})$. Such removal leaves a
nonperturbative ambiguity by an additive constant in the Polyakov loop free
energy $F(T)=-T\log L(T)$
\cite{Kaczmarek:2002mc,Dumitru:2003hp,Gupta:2007ax,Gavai:2010qd,%
Borsanyi:2010bp,Mykkanen:2012ri}.

The (renormalized) partition functions in \Eq{31} are saturated by states of
the spectrum. Since the spectator quark can be reached smoothly by taking the
infinite mass limit of a heavy quark at rest, this implies
\begin{equation}
L(T)
=
\lim_{m_h\to\infty}
\frac{1}{2}\frac{\sum^\prime_\alpha g_{h\alpha}  e^{-(E_{h\alpha}-m_h)/T}}
{\sum_\alpha g_\alpha  e^{-E_\alpha /T}} \,,
\label{eq:32}
\end{equation}
where $m_h$ denotes the heavy quark mass. Here, the sum in the denominator is
just the QCD partition function and so it includes all possible states made of
gluons and dynamical quarks (labeled by $\alpha$). On the other hand, the sum
in the numerator includes all possible QCD states with exactly one heavy
quark $h$ at rest,\footnote{Such statement would be meaningless for dynamical
  quarks, but not for infinitely heavy quarks.} plus gluons and dynamical
quarks (jointly labeled by $h\alpha$). The difference $E_{h\alpha}-m_h$
explicitly removes the heavy quark mass from the total energy of the state.

For temperatures well below the crossover, we expect the previous states to be
of hadronic type (and possibly glueballs). In particular, the heavy quark $h$
will form a hadron with the dynamical quarks, typically, a meson of hybrid
type, i.e., formed by the heavy quark and a dynamical antiquark, $h\bar{q}$,
or a hybrid baryon with the heavy quark and two dynamical quarks, $hqq$.

The HRGM for the QCD partition function follows naturally from assuming that
the QCD interaction primarily confines quarks into hadrons and that purely
hadronic interactions can be neglected. Under this assumption, the numerator
of \Eq{32} contains one hybrid heavy-light hadron at rest plus exactly the
same multi-hadron states occurring in the denominator. This yields a
cancellation between numerator and denominator. Therefore, within the same
approximations leading to the HRGM, we expect the following relation to hold
between the Polyakov loop expectation value in the confined phase and the
hadronic spectrum
\begin{equation}
L(T) \approx
\frac{1}{2}\sum_\alpha \!{}^{{}^{\prime\prime}} g_{h\alpha}\,  e^{-\Delta_{h\alpha}/T} \,,
\quad
\Delta_{h,\alpha} = M_{h\alpha}-m_h \,.
\label{eq:34}
\end{equation}
Here the sum is over all states made just of a single hybrid hadron at rest
(with exactly one heavy flavor quark, mass $M_{h\alpha}$ and degeneracy
$g_{h\alpha}$), and no additional hadrons.

Of course, neither the sum rule in \Eq{34} nor the HRGM can be accurate when
unconfined states of the spectrum start to play a role, that is, for
temperatures in the crossover region or above.  For instance, $L(T)$ is a
decreasing function at high enough temperatures, in the perturbative regime
\cite{Gava:1981qd}, while the Boltzmann distribution in the r.h.s of \Eq{34}
is always increasing as a function of $T$. On the other hand, the mass of the
heavy-light hadron is an observable (a renormalization invariant) but depends
on the heavy flavor, while $\Delta_{h\alpha}$ is universal in the heavy quark
limit but has some running from $m_h$, which is itself not an observable. The
Polyakov loop is renormalization invariant and well defined, modulo the
abovementioned shift ambiguity in the free energy. This can be compensated by
a corresponding shift in the heavy quark mass.

\begin{figure}[t]
\begin{center}
\epsfig{figure=fig1.eps,height=6cm,width=9.0cm}
\end{center}
\caption{(color online). $T \log(L(T))$ as a function of $T$ (units in MeV)
  from simulations on the lattice with $2+1$ physical dynamical quark
  masses. Lower set of lines: data after a common shift $C=-150\,{\rm MeV}$ (just
  for displaying purposes) for continuum extrapolated stout (black solid line,
  ``circle'') \cite{Borsanyi:2010bp}, HISQ/tree action $N_t=12$ scale set
  $r_1$ (blue solid line, ``square'') and $f_k$ (blue dashed line, ``up
  triangle''), and asqtad action $N_t=12$ scale set $r_1$ (red solid line,
  ``down triangle'') and $f_k$ (red dashed line, ``rhombus'')
  \cite{Bazavov:2011nk}. Upper set of lines: same data (except asqtad
  scale set $r_1$) with shifts $C=0$, $0$, $-10\,{\rm MeV}$ and $-30\,{\rm MeV}$, for stout,
  HISQ/tree scale $r_1$, HISQ/tree scale $f_k$, and asqtad scale $f_k$,
  respectively.  }
\label{fig:1}
\end{figure}

\begin{figure}[t]
\begin{center}
\epsfig{figure=fig2.eps,height=6cm,width=9.0cm}
\end{center}
\caption{(color online). Extraction of the slope $\frac{d}{dT}(T\log(L(T)))$
  (dimensionless) as a function of $T$ (in MeV) from the same four lattice
  data sets of Fig.~\ref{fig:1}. The error bars in the slope were obtained
  from assuming a linear interpolation between measured points. The black
  thick line indicates the average in the range of temperatures common to the
  four sets. The yellow strip indicates the uncertainty.}
\label{fig:2}
\end{figure}

{\em Estimates from the physical spectrum.---}In order to compare different
Polyakov loop determinations with the hadronic sum rule, they have to be
brought to a common renormalization condition. Two such determinations are
related by $L^\prime(T)= e^{\,C/T}L(T)$, for some constant energy shift
$C$. In Fig.~\ref{fig:1} we compile five Polyakov loop data sets, obtained
with physical quark masses and three flavors on the lattice
\cite{Borsanyi:2010bp,Bazavov:2011nk}. The plot shows that four of them agree
after applying suitable finite renormalizations (no attempt has been made to
optimize the agreement), in a wide range of temperatures. Unfortunately, the
agreement deteriorates at lower temperatures, below $T\approx 150\,{\rm MeV}$. This
region is relevant for comparison with the hadronic sum rule. Since there is
no ``true'' value of $L(T)$, a finite renormalization, or choice of heavy
quark mass, should be admitted too in the hadronic sum rule. Nevertheless,
large renormalizations (compared with the hadronic scale) would be unnatural,
and would probably signal an inadequacy of the sum rule, or of the
renormalization prescription used for the Polyakov loop. A neat way to remove
any ambiguities is to work with the derivative of $T \log(L(T))$ with respect
to $T$. This slope is sensitive to the effective number of states at a given
temperature. Although with some noise, Fig.~\ref{fig:2} shows that a signal
can be extracted in this way.

A natural step is to check to what extend the hadronic sum rule is fulfilled
by experimental states compiled in the PDG. Several sources of error should be
kept in mind when doing this, among other, that not all needed states may have
been compiled, that the heavy quarks in nature have a finite mass, that their
current mass is scale dependent, and that the quark masses on the lattice may
not be identical to the physical ones. Hadrons with a bottom quark would be
optimal, due to the large quark mass compared to $\Lambda_{\rm QCD}$, but the
available data are scarce, so we turn to charmed hadrons. Specifically, we
consider the lowest lying single-charmed mesons and baryons with $u$, $d$, and
$s$ as the dynamical flavors, with quarks in relative $s$-wave inside the
hadron. For mesons, these are usually identified with the states (spin-isospin
multiplets) $\bar{D}$, $\bar{D}_s$, $\bar{D}^*(2010)$ and $\bar{D}_s^*$, and
for baryons, with $\Lambda_c$, $\Sigma_c(2455)$, $\Xi_c$, $\Xi_c^\prime$,
$\Omega_c$, $\Sigma_c(2520)$, $\Xi_c(2645)$, and $\Omega(2770)$. A total of 12
meson states and 42 baryon states.  \ignore{ $IJ^\pi=\frac{1}{2}0^-$
  $\bar{D}$, the $00^-$ $\bar{D}_s$, the $\frac{1}{2}1^-$ $\bar{D}^*(2007)$
  and the $01^-$ $\bar{D}_s^{*}$. For baryons, with the states
  $0\frac{1}{2}^+$ $\Lambda_c$, the $1\frac{1}{2}^+$ $\Sigma_c(2455)$, the
  $\frac{1}{2}\frac{1}{2}^+$ $\Xi_c$, the $\frac{1}{2}\frac{1}{2}^+$
  $\Xi_c^\prime$, the $0\frac{1}{2}^+$ $\Omega_c$, the $1\frac{3}{2}^+$
  $\Sigma_c(2520)$, the $\frac{1}{2}\frac{3}{2}^+$ $\Xi_c(2645)$, and the
  $0\frac{3}{2}^+$ $\Omega(2770)$. For the charm quark mass, to be subtracted
  from the hadron masses, we take the $\overline{\rm MS}$ scheme value
  $m_c=1290^{+50}_{-110}\,{\rm MeV}$, also directly from~\cite{Nakamura:2010zzi}.}

\begin{figure}[t]
\begin{center}
\epsfig{figure=fig3.eps,height=6cm,width=9.0cm}
\end{center}
\caption{(color online). Comparison of $\frac{d}{dT}(T\log(L(T)))$ (yellow
  strip) with $\frac{d}{dT}(T\log(\frac{1}{2}\sum^{\prime\prime}_\alpha
  g_{h\alpha} e^{-\Delta_{h\alpha}/T}))$ from hadronic states (mesons plus
  baryons): lowest-lying hadrons from PDG (solid brown line, label $a$), RQM
  states from \cite{Godfrey:1985xj,Capstick:1986bm} with quark $c$ (solid red
  line, label $b$), and with quark $b$ (dashed red line, label $c$), and bag
  model estimate including states up to $\Delta=5500\,{\rm MeV}$ (solid blue line,
  label $d$).}
\label{fig:3}
\end{figure}

The plot in Fig.~\ref{fig:3} shows that the lowest-lying states fall short to
saturate the sum rule, regardless of the choice of mass of the charmed quark,
$m_c$. This is not surprising as any model predicts many excited states on top
of the lowest-lying ones, as is also the case for light-quark hadrons. Adding
more states from the PDG does not seem practical due to the fragmentary
information available. Instead we turn to hadronic models. The aim is not so
much to have a detailed description of the various states but to give a
sufficiently good overall description of the whole spectrum. To this end we
consider the relativized quark model (RQM)
\cite{Godfrey:1985xj,Capstick:1986bm}, and the bag model
\cite{Chodos:1974je,Bernotas:2008bu}.  We have verified that the RQM provides
a good account of the trace anomaly in \cite{Borsanyi:2010bp}. The total
number of hadron states computed in \cite{Godfrey:1985xj,Capstick:1986bm} with
one $c$ quark is $117$ for mesons and $1470$ for baryons, corresponding to a
maximum value of $\Delta=M-m_c$ about $1500\,{\rm MeV}$. For hadrons with one $b$
quark, $87$ mesonic and $1740$ baryonic states, with a similar upper bound for
$\Delta$. In both cases we have supplemented missing states with strange
quarks by means of the equal spacing rule \cite{Savage:1995dw} and a $s$ quark
mass of $109\,{\rm MeV}$ (extracted from the lowest-lying hadrons masses). The
prediction based on these hadronic states is displayed in
Fig.~\ref{fig:3}. The two sum rules are closer to the Polyakov loop result but
still tend to stay below it in the range $T<175\,{\rm MeV}$, a consequence of the
truncation of states to $\Delta<1500\,{\rm MeV}$. It is noteworthy that the bottom
sum rule gives a better value, as it would be expected due to the larger mass
of the $b$ quark.

The other model we consider is more schematic but allows us to easily include
a larger number of states. This is a simplified MIT bag
model \cite{Chodos:1974je,Bernotas:2008bu}, in order to correctly count the
number of states without fine details such as multiplet splittings. As bag
energy we take
\begin{equation}
\Delta=\frac{\sum_i {n_i\omega_i}-Z}{R}+\frac{4\pi}{3}R^3 B+\sum_i m_i,
\end{equation}
where $n_i$, $\omega_i$ and $m_i$ are the occupation number, bag frequency and
current quark mass ($m_u=m_d=0$, $m_s=109\,{\rm MeV}$). The model gives directly
$\Delta$ after adjustment of $R$, without center of mass corrections, since
the heavy quark has actually infinite mass (not included), sits at the center
and plays no active role. The sum over $i$ runs over just one antiquark for
$h\bar{q}$ mesons and two quarks for $hqq$ baryons. We set $B=(166\,{\rm MeV})^4$
\cite{Bernotas:2008bu}, and find $m_c=1390\,{\rm MeV}$ and $Z=0.5$ from a fit to the
single-charmed lowest-lying mesons and baryons masses. The result for the bag
model with $\Delta<5500\,{\rm MeV}$ is displayed in Fig.~\ref{fig:3}. The more
complete description of the hadronic spectrum gives a better account of the
Polyakov loop data in the range $145\,{\rm MeV} < T < 175\,{\rm MeV}$. We have checked that:
i) For this range of temperatures heavier states become irrelevant for the sum
rule. We see this by projecting the cumulative number of states assuming a
power law dependence for mesons and for baryon \cite{Kapusta:1981ue}. ii)
Truncation to $\Delta<1500\,{\rm MeV}$ gives a result quite consistent with that of
RQM. And iii) A similar power-law projection of the spectrum for the RQM, to
estimate the effect of adding the states above $\Delta=1500\,{\rm MeV}$, also
reproduces quantitatively the bag model result. Fig.~\ref{fig:3} also shows
that the hadronic sum rule eventually overshoots the Polyakov loop result, as
the crossover to the deconfined phase sets in. This mimics the same behavior
in the HRGM \cite{Bazavov:2009zn,Borsanyi:2010bp}.

\begin{figure}[t]
\begin{center}
\epsfig{figure=fig4.eps,height=6cm,width=9.0cm}
\end{center}
\caption{(color online). $T\log(L(T))$ (in MeV) for the four lattice data sets
  of Fig.~\ref{fig:1} compared to the hadronic sum rule from the bag
  model. Solid blue line: estimate from conventional mesons and baryons
  ($C=0$). Dashed green line: estimate when exotic hadrons are included,
  applying a shift $C=-40\,{\rm MeV}$.}
\label{fig:4}
\end{figure}

In Fig.~\ref{fig:4} we display lattice data for $T \log L(T)$ vs the hadronic
sum rule. The hadronic estimate (with no additional finite renormalization in
the case of the bag model) describes well the various lattice data sets in the
range $150\,{\rm MeV}<T<190\,{\rm MeV}$, and even lower temperature data from
\cite{Bazavov:2011nk}. However, the steeper slope displayed by the stout
action data \cite{Borsanyi:2010bp} for $T<150\,{\rm MeV}$ cannot possibly be
saturated with conventional mesons and hadrons, since all these states have
already been accounted for. Inclusion of exotic hadrons, $hq\bar{q}\bar{q}$
(tetraquarks) and $hqqq\bar{q}$ (pentaquarks), do actually produce the
required slope (see Fig.~\ref{fig:4}), and it would imply a much shorter
temperature range of applicability of the hadronic sum rule. At present, the
various sets of lattice data disagree at the lowest temperatures, and also it
is unsettled whether exotic hadrons are present in the QCD spectrum or not
\cite{Jaffe:2004ph}. Our analysis implies that resolution of one of these
issues would shed light on the other.

{\em Final comments.---}From QCD considerations, we derive a Boltzmann
distribution formula in terms of hadrons for the expectation value of the
Polyakov loop in the confined phase, as required by quark-hadron duality,
where its real and positive character becomes manifest. Our derivation exposes
the obvious fact that the Polyakov loop gets its expectation value from the
dressing with dynamical quarks or antiquarks. Since ${\rm U}(N_f)$ is an exact
global symmetry, the numerator in \Eq{31}, and hence $L(T)$ itself, can be
decomposed into separate contributions from different flavors and different
baryon numbers. Such decompositions are in principle accessible to lattice
calculations (although with a difficulty similar to that of introducing a
chemical potential) and they would provide further information about the
interplay between the QCD thermal state and the heavy quark spectrum.

\begin{acknowledgments}We thank C. Garcia-Recio for comments on the manuscript.
This work has been supported by Plan Nacional de Altas Energ\'{\i}as
(FPA2008-01430 and FPA2011-25948), DGI (FIS2011-24149), Junta de
Andaluc{\'\i}a grant FQM-225, and the Spanish Consolider-Ingenio 2010
Programme CPAN (CSD2007-00042). E.M. would like to thank the Institute
for Nuclear Theory at the University of Washington, USA, and the Institut
f\"ur Theoretische Physik at the Technische Universit\"at Wien, Austria, for
their hospitality and partial support during the completion of this work. The
research of E.~Meg\'{\i}as is supported by the Juan de la Cierva Program of
the Spanish MICINN.
\end{acknowledgments}


\section{Introduction}
Byzantine fault tolerance (BFT) has become more and more practical over the last years.
Besides lot of theoretical BFT research, many research prototypes have made
important contributions. But still, there are only a few complete ready-to-use
prototypes, with \textit{UpRight} \cite{clement09upright} and \textit{BFT-SMART} \cite{sousa12bftsmart} being the most predominant examples. Practical use of
these prototypes in production systems is not common at all.

Clement et al. \cite{clement09making} have criticized the focus of lot of research
systems such as Q/U, HQ and Zyzzyva on optimizing performance in special best-case
situations.  The authors have shown that in common fault situations these systems
easily degrade seriously. They propose a different design of a BFT replication protocol
that guarantees good performance even if there are faulty servers or clients.

Amir et al. \cite{amir08bftattack} describe how timing attacks influence BFT systems. They have extended the two existing criteria \textit{liveness} and \textit{safety} for another criterion \textit{performance-oriented correctness}. They built a new replication protocol that meets this new criterion and handles timing attacks.

Veronese et al. \cite{veronese10ebawa} describe EBAWA, a BFT algorithm that
reduces the communication overhead by using the trusted platform module (TPM) for decreasing the amount of communication steps and replicas. This paper also shows a way to avoid timing attacks of malicious primary nodes by rotating them.

All above-mentioned works show good directions for future BFT developments
beyond the optimization for special best-case situations. However,
current ready-to-use implementations such as \textit{UpRight} and
\textit{BFT-SMART} do not make use of these results.

In addition, these implementations do not have an elaborate security architecture
for protecting confidentiality, integrity and availability.
For example, instead of secure mechanisms for dynamic key
management, participant keys typically are defined statically in configuration files.
More advanced mechanisms for access control and Denial of Service (DoS)
protection are desirable.
In previous work, Amir et al. \cite{amir2005secure} designed \textit{Secure Spread}, which
has an elaborate security concept that shows how to build an architecture that provides
these security mechanism in group communication systems. But it is not aimed for BFT systems.

In ongoing research work, we aim at making a contribution to this aspect of BFT
replication.  The basis we use for our work is \textit{BFT-SMART}, an open source Java library
that provides BFT state machine replication \cite{sousa12bftsmart}. The focus of \textit{BFT-SMART} is plainness and robustness. There are also mechanisms that provide authenticity and integrity protection by using preshared keys.
So we are designing and implementing SecureSMART, a security concept for \textit{BFT-SMART} that extends the library by advanced security mechanisms.

\section{Proposed Architecture}

Figure \ref{fig:architecture} shows the basic design of our security extension \textit{SecureSMART}.  This architecture meets the following goals:

\begin{itemize}
\item \textbf{Dynamic key distribution:} Whenever a new node wants to join the BFT system it can apply a certificate signing request (CSR) at a registration authority (RA)/certification authority (CA). After receiving the signed certificate the new node can propose its public key to all other nodes. After that a new group key needs to be created, encrypted and proposed to all nodes. In addition, new group keys also need
to be created if a node leaves the BFT system.
\item \textbf{Integrity and confidentiality of internal communication:} The internal communication system of \textit{BFT-SMART} should use the asymmetric key pair to sign and verify every proposed message. The shared group key should be used to encrypt every proposed message which is confidential.
\item \textbf{Integrity and confidentiality of external communication:} The communication system between the clients and the nodes should also be protected. Therefor we can use Transport Layer Security (TLS)
for client-server communication.
\item \textbf{Availability:}
BFT ensures availability by redundancy, tolerating the failure of a limited number of
$f$ nodes.  But if an adversary can attack the availability of the low-level communication
layer at many or all nodes, the whole system will fail. This means that the BFT system
needs elaborate protection mechanisms against DoS attacks.
\item \textbf{Access control:} There should be a mechanism to blacklist suspicious nodes and clients. One way to do this is a certificate revocation list (CRL). This implies that there is an own certification authority that distributes CRLs to the nodes.  This mechanism also influences the availability.
\end{itemize}

\begin{figure}[tbp]
	\centering
		\includegraphics[width=0.45\textwidth]{architecture.pdf}
	\caption{\textit{SecureSMART} architecture}
	\label{fig:architecture}
\end{figure}

\noindent
We have to address several challenges in order to achieve these
goals:
\balance

\begin{itemize}
\item The dynamic group management brings up some problems: Which nodes may join the BFT system? Is a valid certificate enough for this purpose? How can access control policies be distributed?\\
Another question is how to distribute the public keys? Only the leader node can propose messages including the public key and guarantee that the distributed public key
will be distributed correctly even in the presence of Byzantine faults.\\
What happens if a node is removed or replaced? The public key of that node is no longer valid for validating messages. This potentially has a big impact on internal validation
procedures within a BFT algorithm.
\item We are using symmetric and asymmetric encryption methods in this security concept which brings the following questions:\\ What are appropriate encryption algorithms? Should the symmetric encryption use block cipher algorithms or stream cipher algorithms? If using a block cipher algorithm, what kind of padding and which mode of operation should be used?
\item In a BFT scenario each node should receive the same information. Based on this fact, it is advisable to implement group keys for the message encryption. But from this point there raises the question of how the group keys should be generated. Does it make sense if all the nodes together generate the key using Diffie-Hellman (DH)? Is there an advantage by using Tree-based Group Diffie-Hellman (TGDH)? Or is it better if the primary node generates the key?
\item In the case that the group key has leaked there have to be some preventions: How important is perfect forward secrecy (PFS \cite{diffie92pfs}) in this context? When should a new key be established?
\item Measures to improve the protection against DoS attacks are Transmission Control Protocol (TCP) SYN-Cookies and blacklisting of packets source IP addresses. But there might be some problems with the prevention of TCP SYN-Flooding attacks by using SYN-Cookies:\\ At first SYN-Cookies are based deeply in the operating system kernel. Is it possible to enable this option or similar options from the view of the Java Virtual Machine (JVM)?\\ Another problem of using SYN-Cookies might be a performance issue.
\item The RA/CA is a single point of failure. What does this mean in the context of BFT? How can we use BFT replication for the RA/CA?
\end{itemize}

\section{Conclusion}
In terms of performance, BFT systems have already reached a level that
makes them well-suited for real-world applications.  There are still deficits
in other areas that stop BFT from happening in practice. This work addresses
parts of these problems in the area of security mechanisms.
In this abstract, we have described the basic architecture of \textit{SecureSMART},
a security extension for the \textit{BFT-SMART} library we are currently working on,
and we have explained some of the challenges we are handling in this
architecture.

\bibliographystyle{IEEEtran}