Entropy, fidelity, and double orthogonality for resonance states in two-electron quantum dots

Resonance states of a two-electron quantum dots are studied using a variational expansion with both real basis-set functions and complex scaling methods. The two-electron entanglement (linear entropy) is calculated as a function of the electron repulsion at both sides of the critical value, where the ground (bound) state becomes a resonance (unbound) state. The linear entropy and fidelity and double orthogonality functions are compared as methods for the determination of the real part of the energy of a resonance. The complex linear entropy of a resonance state is introduced using complex scaling formalism.

Figure 1

(a) Behavior of the variational eigenvalues $E_j^{(\alpha )}(\lambda )$ (black lines) for $N=14$ and nonlinear parameter $\alpha =2$. The red-dashed line corresponds to the threshold energy $\varepsilon \simeq -1.091$. Note that the avoided crossings between the variational eigenvalues are fairly visible. (b) The same variational eigenvalues as in (a) (black lines) and the energy calculated using the complex scaling method (green line) for a parameter $\theta =\pi /10$.

Figure 2

(a) First variational state energy versus $\lambda$ for different values of the variational parameter $\alpha$. From bottom to top $\alpha$ increases from $\alpha =2$ (blue-dashed line) to $\alpha =6$ (orange-dashed line). The real part of the resonance eigenvalue obtained using complex scaling ($\theta =\pi /10$) is also shown (green line). (b) Same as (a), but for the second-state energy.

Figure 3

Density of states $\rho (E)$ for $\lambda =2.25$ and basis-set size $N=14$. The results were obtained using Eq. (13) and correspond to, from top to bottom, the second (black line), third (red-dashed line), fourth (green line), and fifth (blue-dashed line) levels.

Figure 4

The upper panel shows the behavior of ${\mathcal{G}}_n$, for $n=1,\dots ,7$. Each function ${\mathcal{G}}_n$ has two peaks, except for $n=1$. Since one of the peaks of ${\mathcal{G}}_n$ coincides with one of the peaks of ${\mathcal{G}}_{n+1}$, only one peak for each level $n$ is apparent. From left to right the visible line at each peak corresponds to $n=2,\dots ,7$ (cyan, red, yellow, blue, grey, and brown lines, respectively). The $n=1$ line has no visible peak (black). The lower panel shows the variational eigenlevels $n=1,\dots ,7$ (with the same color convention used as in the upper panel), and $E_r(\lambda )$ (green-dashed line). The black vertical dashed lines connecting both panels show the value of $\lambda$ where each ${\mathcal{G}}_n$ has its minimum, ${\lambda }_n^f$. The red dots in the lower panel correspond to $E_n({\lambda }_n^f)$.

Figure 5

The lower panel shows the variational energy levels $E_n(\lambda )$, from bottom to top, for $n=1,\dots ,7$ (the black, green, red, yellow, blue, grey, and brown continuous lines, respectively); $E_r(\lambda )$ (the dark-green-dashed line); $E_n({\lambda }_{\mathit{DO}}^n)$ (blue squares); and $E_n({\lambda }_n^f)$ (red dots). The upper panel shows the behavior of ${\mathcal{D}}_n$ versus $\lambda$ for $n=2,\dots ,7$. The color convention for the ${\mathcal{D}}_n$ is the same as used in the lower panel. The black-dot-dashed vertical lines show the location of the points ${\lambda }_{\mathcal{D}}^n$.

Figure 6

Behavior of $S_{\mathrm{lin}}[{\Psi }_j^{(\alpha )}]$, where the ${\Psi }_j^{(\alpha )}$ are the variational eigenstates corresponding to the first seven energy levels shown in Fig. 1 for $N=14$ and $\alpha =2.0$. All the curves $S_{\mathrm{lin}}[{\Psi }_j^{(\alpha )}]$, except for the corresponding to $S_{\mathrm{lin}}[{\Psi }_1^{(\alpha )}]$, have a single minimum located at ${\lambda }_j^S$; that is, $S_{\mathrm{lin}}[{\Psi }_j^{(\alpha )}({\lambda }_j^S)]={min}_{\lambda }{S}_{\mathrm{lin}}[{\Psi }_j^{(\alpha )}(\lambda )]$. If $i<j$, then ${\lambda }_i^S<{\lambda }_i^S$.

Figure 7

Expectation values of the Coulomb repulsion for the variational states ${\Psi }_n^{(\alpha )}$, $n=1,\dots ,8$ with $N=14$ and $\alpha =2.0$. Also shown is the curve $\partial E_r/\partial \lambda$ obtained from the complex energy of the complex scaling method.

Figure 8

Expectation value $\langle 1/r_{12}{\rangle }_2^{(\alpha )}$ versus $\lambda$ for a basis size $N=14$ and $\alpha =2,\dots ,3.5$ in steps of 0.1 and for $\alpha =4,\dots ,5.5$ in steps of 0.5 (black solid lines). The real (cyan dotted) and imaginary (orange line) parts of $\langle 1/r_{12}\rangle {}_{\theta }$ ($\theta =\pi /10$) and the derivative of the real part of the complex-scaled energy are also shown.

Figure 9

Panel (a) shows the linear entropy for the same values as in Fig. 6 (magenta-dashed lines). Also shown is the real part of the complex linear entropy for several values of the complex rotation angle $\theta =\frac{\pi }{5},\frac{\pi }{10},\frac{\hspace{0.5em}\mathit{pi}}{20},\frac{\pi }{30},\frac{\pi }{40}$ (black-empty diamonds, red dots, green squares, blue triangle, yellow-empty dots). Panel (b) shows the imaginary part of the complex linear entropy for the same values as in (a).