Figure 1

Parameter-space diagram identifying the different types of solution for Eq. (9). In region I there are no solutions for $z$ when $\theta =0$ and one solution for $z$ when $\theta =N\pi /4$. In region II there are two solutions for $z$ when $\theta =0$ and one solution for $z$ when $\theta =N\pi /4$. In region III there exists one solution for $z$ when $\theta =0$, one solution for $z$ when $\theta =N\pi /4$, and two solutions for $\theta$ when $z=-1$. In region IV there is one solution for $z$ when $\theta =0$ and no solution for $z$ when $\theta =N\pi /4$. In this region the global minima occur at $z=-1$ and all values of $\theta$. The boundary separating regions I and II from region III is given by $\lambda =\alpha +1$, whereas the equation $\lambda =\alpha -1$ separates region III from region IV. The boundary between regions I and II has been obtained numerically.Reuse & Permissions

Figure 2

Dimensionless energy gap between the first excited state and the ground state as a function of $\alpha ={\mu }_a/\Omega \sqrt{2N}$ for different values of $N$ and $\lambda =0$. Here $\Omega =1$, ${\mu }_b=0,$ and $U_a=U_b/4=0.25$.Reuse & Permissions

Figure 3

Dimensionless energy gap between the first excited state and the ground state as a function of $\alpha ={\mu }_a/\Omega \sqrt{2N}$ for different values of $\lambda$ and $N=2000$. Values of $\Omega ,{\mu }_b$ and $U_a$,$U_b$ are the same as in the previous figure. These results indicate that the minimal values lie approximately on the line $\lambda =\alpha -1$.Reuse & Permissions

Figure 4

Ground-state expectation values for the atomic number fraction $N_a/N$ and the molecular number fraction $N_b/N$ as a function of $\alpha ={\mu }_a/\Omega \sqrt{2N}$ for $\lambda =0$ and $N=2000$. We have set $\Omega =1$, ${\mu }_b=0$ and $U_a=U_b/4=0.25$. There is a sharp transition at $\alpha =1$.Reuse & Permissions

Figure 5

Entropy of entanglement of the ground state as a function of $\alpha ={\mu }_a/\Omega \sqrt{2N}$ for different values of $N$ and $\lambda =0$. Here $\Omega =1$, ${\mu }_b=0,$ and $U_a=U_b/4=0.25$.Reuse & Permissions

Figure 6

Entropy of entanglement of the ground state as a function of $\alpha ={\mu }_a/\Omega \sqrt{2N}$ for different values of $\lambda$ and $N=2000$. Here $\Omega =1$, ${\mu }_b=0,$ and $U_a=U_b/4=0.25$.Reuse & Permissions

Figure 7

First (inset) and second derivative of the entropy of entanglement of the ground state as a function of $\alpha ={\mu }_a/\Omega \sqrt{2N}$ for $N=1000$ and $\lambda =0$. Here $\Omega =1$, ${\mu }_b=0,$ and $U_a=U_b/4=0.25$.Reuse & Permissions

Figure 8

Fidelity of the ground state as a function of $\alpha ={\mu }_a/\Omega \sqrt{2N}$ for $\Omega =1$, $\lambda =0$, $N=1000,$ and $U_a=U_b/4=0.25$. We keep ${\mu }_b=0$ for the first state and vary ${\mu }_b=\gamma$ for the second state.Reuse & Permissions

Figure 9

Fidelity of the ground state as a function of $\alpha ={\mu }_a/\Omega \sqrt{2N}$ for $\lambda =0$,${\mu }_b=0$ (first state), and ${\mu }_b=1$ (second state) with varying $N$. Here $\Omega =1$ and $U_a=U_b/4=0.25$. The inset shows the minima close to the critical point $\alpha =1$.Reuse & Permissions

Figure 10

Fidelity of the ground state as a function of $\alpha ={\mu }_a/\Omega \sqrt{2N}$ for $N=2000$, ${\mu }_b=0$ (first state), and ${\mu }_b=1$ (second state) and varying $\lambda$. Here $\Omega =1$ and $U_a=U_b/4=0.25$.Reuse & Permissions