Avoided level crossings, diabolic points, and branch points in the complex plane in an open double quantum dot

We study the spectrum of an open double quantum dot as a function of different system parameters in order to receive information on the geometric phases of branch points in the complex plane (BPCP). We relate them to the geometrical phases of the diabolic points (DPs) of the corresponding closed system. The double dot consists of two single dots and a wire connecting them. The two dots and the wire are represented by only a single state each. The spectroscopic values follow from the eigenvalues and eigenfunctions of the Hamiltonian describing the double dot system. They are real when the system is closed, and complex when the system is opened by attaching leads to it. The discrete states as well as the narrow resonance states avoid crossing. The DPs are points within the avoided level crossing scenario of discrete states. At the BPCP, width bifurcation occurs. Here, different Riemann sheets evolve and the levels do not cross anymore. The BPCP are physically meaningful. The DPs are unfolded into two BPCP with different chirality when the system is opened. The geometric phase that arises by encircling the DP in the real plane, is different from the phase that appears by encircling the BPCP. This is found to be true even for a weakly opened system and the two BPCP into which the DP is unfolded.

Figure 1

Two single state QDs are connected to the wire w of length $L$ with the coupling constants $u$ and to the continuum of scattering wave functions (“reservoir”) with the coupling constants $v$. The energy of the mode in the wire is $ϵ\left(L\right)$ and those of the two single dots are ${\epsilon }_1^R$ and ${\epsilon }_1^L$, respectively. Mostly the two single dots are assumed to be equal: ${\epsilon }_1^R={\epsilon }_1^L\equiv {\epsilon }_1$.

Figure 2

The surfaces of the BPCP in the four-manifold parameter space $\Delta \epsilon \left(L\right)\equiv ({\epsilon }_1-ϵ\left(L\right))∕2,u,v,E$, defined by Eqs. (5, 6).

Figure 3

The line of the BPCP in the three-manifold parameter space $L,u,v$ (solid line). The three arrows show paths as a function of $v$ (by keeping $u$ and $L$ constant) which start at $v=0$, $u=u_c$ and different $L$. The path starting at $L=L_c$ (shown by an open circle) crosses the line of the BPCP at $v=v_c$ (shown by a full circle). The paths which start at $L=L_c\pm \Delta L$ (shown by a star and a cross, respectively) do not cross the line of the BPCP.

Figure 4

The evolution of the eigenvalues $z_1$ (solid lines) and $z_3$ (dashed lines) (a, b) and of the components $a=\mid a\mid e^{i\alpha }$ (dashed lines) and $b=\mid b\mid e^{i\beta }$ (solid lines) of the eigenfunctions ∣1) and ∣3) (c, d) of the effective Hamiltonian $H_{\mathrm{eff}}$ as a function of $v$ for $E=E_c=\sqrt{2}$. The parameters $u=u_c=1∕4$, $L=L_c-0.01$, $L_c=1.4645$ correspond to the point shown in Fig. 3 at $v=0$ by a star. The evolution corresponds to the path shown in Fig. 3 by the dashed-dotted arrow. ${\epsilon }_1=1$, $ϵ\left(L\right)=2-L∕5$. At the critical value of $v$, $\mid a\mid ⪢1$, $\mid b\mid ⪢1$ and the phases jump by $-\pi ∕4$.

Figure 5

The same as Fig. 4 but $L=L_c+0.01$. At $v=0$, the evolution starts at the point shown in Fig. 3 by a cross. The evolution corresponds to the path shown in Fig. 3 by the dashed arrow. At the critical value of $v$, it is $\mid a\mid ⪢1$, $\mid b\mid ⪢1$ and the phases jump by $+\pi ∕4$.

Figure 6

The evolution of the eigenvalues $z_1$ (solid lines) and $z_3$ (dashed lines) (a, b) and of the components $a=\mid a\mid e^{i\alpha }$ (dashed lines) and $b=\mid b\mid e^{i\beta }$ (solid lines) of the eigenfunctions ∣1) and ∣3) (c, d) of the effective Hamiltonian $H_{\mathrm{eff}}$ as a function of $E$ for $v\approx v_c=1$ $(v<v_c)$. The other parameters of the system satisfy the Eqs. (5, 6): $L=L_{c_1}=1.4645$, $u=u_c=1∕4$, ${\epsilon }_1=1$, $ϵ\left(L\right)=2-L∕5$. At the critical value of $E$, it is $\mid a\mid ⪢1$, $\mid b\mid ⪢1$ and the phases jump by $-\pi ∕4$.

Figure 7

The encircling of the diabolic point $u=0$, $\Delta \epsilon =0$, given by Eq. (10), (left) and its mapping onto the plane $X,Y$ (right) according to Eq. (13).

Figure 8

The encircling of the branch point $X=0$, $Y=0$, given by Eq. (12), (left) and its mapping onto the plane $\Delta \epsilon$, $u$ (right) according to Eq. (15) for $E=0$, $v=0.5$. Dashed line: $R=v^4∕4$, solid line: $R=v^4∕6$. The point marked by an open circle corresponds to the branch point $X=0$, $Y=0$. The point marked by a full circle corresponds to the DP $\Delta \epsilon =0$, $u=0$.

Figure 9

Mapping of the encircling of the BPCP (open circle) in the plane $E=\sqrt{2}+r\sin \theta$, $L=1.4645+r\cos \theta$ onto the plane $X=R\cos \varphi$, $Y=R\sin \varphi$ given by Eq. (17) for $r=0.25$. $E_c=\sqrt{2}$, $L_c=1.4645$, $v=v_c=1$, $u=u_c=1∕4$, ${\epsilon }_1=1$, $ϵ\left(L\right)=2-L∕5$.

Figure 10

The evolution of $\mathrm{Re}\left(z_k\right)$ (a) and $\mathrm{Im}\left(z_k\right)$ (b) of the two eigenvalues $z_1$ (solid lines) and $z_3$ (dashed lines) of the effective Hamiltonian $H_{\mathrm{eff}}$ as a function of the angle $\theta$. The corresponding evolution of the components $a=\mid a\mid e^{i\alpha }$ (solid lines) and $b=\mid b\mid e^{i\beta }$ (dashed lines) of the two eigenvectors ∣1) and ∣3) (c, d). The encircling is around the BPCP in the plane $E$, $L$ given by (16). The parameters are the same as in Fig. 9. The eigenvalues $z_k$ and the moduli $\mid a\mid$, $\mid b\mid$ are restored after two cycles while the phases $\alpha$, $\beta$ are restored after four cycles.

Figure 11

Mapping of the circular encircling of the BPCP (open circle) in the plane $L=L_c+0.5\cos \theta$, $v=v_c+0.5\sin \theta$ (dashed line) and of the elliptic encircling $L=L_c+2\cos \theta$, $v=v_c+0.5\sin \theta$ (solid line) onto the plane $X=R\cos \varphi$, $Y=R\sin \varphi$ for $r=0.5$. It is $v=v_c=1$, $E=E_c=\sqrt{2}$, $L=L_c=1.4645$, $u=u_c=1∕4$, ${\epsilon }_1=1$, $ϵ\left(L\right)=2-L∕5$.

Figure 12

The evolution of $\mathrm{Re}\left(z_k\right)$ (a) and $\mathrm{Im}\left(z_k\right)$ (b) of the two eigenvalues $z_1$ (solid lines) and $z_3$ (dashed lines) of the effective Hamiltonian $H_{\mathrm{eff}}$ as a function of the angle $\theta$. The corresponding evolution of the components $a=\mid a\mid e^{i\alpha }$ (solid lines) and $b=\mid b\mid e^{i\beta }$ (dashed lines) of the two eigenvectors ∣1) and ∣3) (c, d). The elliptical encircling is around the BPCP in the plane $L$, $v$ shown in Fig. 11 by the solid lines. The parameters are the same as in Fig. 11. The eigenvalues $z_k$ and the moduli $\mid a\mid$, $\mid b\mid$ are restored after two cycles while the phases $\alpha$, $\beta$ are restored after four cycles.

Figure 13

The evolution of the two eigenvalues $E_{1,3}^B$, Eq. (4) [10], (a) and eigenvectors ${\psi }_m\left(k\right)$, $m$, $k=1,3$, Eq. (6) [10], (b, c) of the Hamiltonian $H_B$ of the closed system for the elliptic encircling $L=5+0.25\cos \theta$, $u=0.1\sin \theta$. $E_1$ is shown in (a) by the solid line and $E_3$ by the dashed line. The components ${\psi }_1\left(1\right)$, ${\psi }_3\left(1\right)$ are shown by the dashed lines in (b, c), and the components ${\psi }_1\left(3\right)$, ${\psi }_3\left(3\right)$ by the solid lines. The eigenvalues are restored after every cycle while the eigenvectors are restored after two cycles.

Figure 14

The same as Fig. 13 but for a double QD consisting of two different single QDs with ${\epsilon }_1^L=1$, ${\epsilon }_1^R=0$, and a wire of length $L$. $E_2$ is shown in (a) by the solid line, $E_3$ by the dashed line. In (b, c), the components ${\psi }_m\left(k\right)$, $m=2,3$, $k=1,2,3$ are shown. Dotted-dashed line: $k=1$, solid line: $k=2$ and dashed line: $k=3$. The eigenvalues are restored after every cycle while the eigenvectors are restored after two cycles.

Figure 15

The evolution of the eigenvalues $z_{1,3}$ (a, b), and of the components $a$ and $b$ of the eigenvectors (c, d) of the effective Hamiltonian for the circular encircling of the DP given by Eq. (19). $v=0.2$, ${\epsilon }_1=1$, $ϵ\left(L\right)=2-L∕5$, $E=\sqrt{2}$, $r=1$. The eigenvalues and eigenvectors are restored after every cycle.

Figure 16

The same as Fig. 15, but the BP is encircled by the elliptic path $L=5+0.25\cos \theta$, $u=0.1\sin \theta$ as in Figs. 13, 14. The eigenvalues and eigenvectors are restored after every cycle.