Pair condensation in a finite Fermi system

The lowest seniority-zero eigenstates of an exactly solvable multilevel pairing Hamiltonian for a finite Fermi system are examined at different pairing regimes. After briefly reviewing the form of the eigenstates in the Richardson formalism, we discuss a different representation of these states in terms of the collective pairs resulting from the diagonalization of the Hamiltonian in a space of two degenerate time-reversed fermions. We perform a two-fold analysis by working both in the fermionic space of these collective pairs and in a space of corresponding elementary bosons. On the fermionic side, we monitor the variations which occur, with increasing the pairing strength, in the structure of both these collective pairs and the lowest eigenstates. On the bosonic side, after reviewing a fermion-boson mapping procedure, we construct exact images of the fermion eigenstates and study their wave function. The analysis allows a close examination of the phenomenon of pair condensation in a finite Fermi system and gives new insights into the evolution of the lowest (seniority-zero) excited states of a pairing Hamiltonian from the unperturbed regime up to a strongly interacting one.

Figure 1

Pair energies relative to the ground state (solid lines). The energies are real up to the points where $E_{2\lambda -1}=E_{2\lambda }$ ($\lambda =1,2,3$). From there on, they turn into complex-conjugate pairs of the type of Eq. (6) and only their real part ${\xi }_{\lambda }$ is shown (dash-dotted lines). All values are in units of $d$.

Figure 2

Pair energies relative to the first excited state (solid lines). Similarly to Fig. 1, dash-dotted lines refer to their real part ${\xi }_{\lambda }$ [see Eq. (7)]. All values are in units of $d$.

Figure 3

Amplitudes $p(k,\rho )$ characterizing the collective pairs ${\Pi }_{\rho }^{†}$ ($1\hspace{1em}\le \hspace{1em}\rho \hspace{1em}\le \hspace{1em}12$) for two values of the pairing strength $g$. The pairs ${\Pi }_{\rho }^{†}$ are ordered for increasing values of their energy ${\tilde{\epsilon }}_{\rho }$.

Figure 4

Energies ${\tilde{\epsilon }}_{\rho }$ of the pairs ${\Pi }_{\rho }^{†}$ as defined in Eq. (12) (solid lines). The dashed lines show the energies of the same pairs at $g=0$. All values are in units of $d$.

Figure 5

The quantity $R(\rho ,1)$ discussed in the text for the states $|{\rho }_w^{(1)}\rangle ={\Pi }_1^{†}{\Pi }_2^{†}{\Pi }_3^{†}{\Pi }_4^{†}{\Pi }_5^{†}{\Pi }_6^{†}|0\rangle$ and $|{\rho }_s^{(1)}\rangle =({\Pi }_1^{†}){}^6|0\rangle$. The dash-dotted line shows the maximum values reached by the remaining $R(\rho ,1)$'s in region II.

Figure 6

Relative difference between the correlation energy of the state $|{\rho }_s^{(1)}\rangle =({\Pi }_1^{†}){}^6|0\rangle$ (properly normalized) and the corresponding energy of the ground state (in percent).

Figure 7

Amplitudes of the pairs ${\Pi }_1^{†}$ (circles) and ${\Pi }_{\mathrm{BCS}}^{†}$ (triangles) for two values of the pairing strength $g$.

Figure 8

The quantity $R(\rho ,2)$ discussed in the text for the states $|{\rho }_w^{(2)}\rangle ={\Pi }_1^{†}{\Pi }_2^{†}{\Pi }_3^{†}{\Pi }_4^{†}{\Pi }_5^{†}{\Pi }_7^{†}|0\rangle$ and $|{\rho }_s^{(2)}\rangle =({\Pi }_1^{†}){}^5{\Pi }_7^{†}|0\rangle$. The dash-dotted line shows the maximum values reached by the remaining $R(\rho ,2)$'s in region II. The dashed line refers to $R(\rho ,2)$ for $|\rho \rangle =({\Pi }_1^{†}){}^5{\Pi }_2^{†}|0\rangle$.

Figure 9

The quantity $R(\rho ,+)$ discussed in the text for the states $|{\rho }_w^{(+)}\rangle ={\Pi }_1^{†}{\Pi }_2^{†}{\Pi }_3^{†}{\Pi }_4^{†}{\Pi }_5^{†}{\Pi }_8^{†}|0\rangle$ and $|{\rho }_s^{(-)}\rangle =({\Pi }_1^{†}){}^5{\Pi }_8^{†}|0\rangle$. The dash-dotted line shows the maximum values reached by the remaining $R(\rho ,+)$'s in region II.

Figure 10

The quantity $R(\rho ,-)$ discussed in the text for the states $|{\rho }_w^{(-)}\rangle ={\Pi }_1^{†}{\Pi }_2^{†}{\Pi }_3^{†}{\Pi }_4^{†}{\Pi }_6^{†}{\Pi }_7^{†}|0\rangle$ and $|{\rho }_s^{(-)}\rangle =({\Pi }_1^{†}){}^5{\Pi }_6^{†}|0\rangle$. The dash-dotted line shows the maximum values reached by the remaining $R(\rho ,-)$'s in region II.

Figure 11

Absolute values of the amplitudes $c_w^{(1)}$ and $c_s^{(1)}$ associated with the components $b_1^{†}{b}_2^{†}{b}_3^{†}{b}_4^{†}{b}_5^{†}{b}_6^{†}|0)$ and $\frac{1}{\sqrt{6!}}(b_1^{†}){}^6|0)$, respectively, of the boson ground state $|{\Psi }_1)$. The dashed-dotted line shows the maximum value reached by the remaining amplitudes $\left|c_i^{(1)}\right|$ in region II.

Figure 12

Absolute values of the amplitudes $c_w^{(2)}$ and $c_s^{(2)}$ associated with the components $b_1^{†}{b}_2^{†}{b}_3^{†}{b}_4^{†}{b}_5^{†}{b}_7^{†}|0)$ and $\frac{1}{\sqrt{5!}}(b_1^{†}){}^5{b}_7^{†}|0)$, respectively, of the boson first excited state $|{\Psi }_2)$. The dashed-dotted line shows the maximum values reached by the remaining amplitudes $\left|c_i^{(2)}\right|$ in region II.

Figure 13

Absolute values of the amplitudes $c_w^{(+)}$ and $c_s^{(+)}$ associated with the components $b_1^{†}{b}_2^{†}{b}_3^{†}{b}_4^{†}{b}_5^{†}{b}_8^{†}|0)$ and $\frac{1}{\sqrt{5!}}(b_1^{†}){}^5{b}_8^{†}|0)$, respectively, of the boson state $|{\Psi }^{(+)})$ (see text). The dashed-dotted line shows the maximum values reached by the remaining amplitudes $\left|c_i^{(+)}\right|$ in region II.

Figure 14

Absolute values of the amplitudes $c_w^{(-)}$ and $c_s^{(-)}$ associated with the components $b_1^{†}{b}_2^{†}{b}_3^{†}{b}_4^{†}{b}_6^{†}{b}_7^{†}|0)$ and $\frac{1}{\sqrt{5!}}(b_1^{†}){}^5{b}_6^{†}|0)$, respectively, of the boson state $|{\Psi }^{(-)})$ (see text). The dashed-dotted line shows the maximum values reached by the remaining amplitudes $\left|c_i^{(-)}\right|$ in region II.

Figure 15

Overlaps between the ground state and the condensate $({\Pi }_1^{†}){}^N|0\rangle$ (solid line) and between the ground state and the state ${\Pi }_1^{†}{\Pi }_2^{†}\dots {\Pi }_N^{†}|0\rangle$ (dashed-dotted line) for systems with different number of particles (shown next to each line).