Real- and Complex-Energy Non-conserving Particle Number Pairing Solution

Abstract

Many-body open quantum systems are characterized by the correlations between bound and scattering states. In contrast to a closed system (i.e., a very well bound many-body system), the continuous part of the energy spectrum has to be considered explicitly due to the proximity of the Fermi level to the continuum’s threshold. In this work we show how to introduce these correlations through the continuum single-particle level density (CSPLD) in the pairing framework. By isolating the resonances of the system using an analytic continuation, we arrive at the Berggren (complex-energy) representation of the pairing solution.

Appendix

In this section we will deduce the BCS gap equation. As argued in Sect. 2, we use the following expressions for the singular contractions:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \widehat{a^+_{\nu m}(\varepsilon) a^+_{\nu' m'}(\varepsilon)} &\displaystyle =&\displaystyle \delta_{\nu \nu'} \, \delta_{m m'} \, v^2_\nu(\varepsilon) \, g_\nu(\varepsilon) \end{array} \end{aligned} $$

(30)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \widehat{a^+_{\nu m}(\varepsilon) a^+_{\nu' \bar{m}'}(\varepsilon)} &\displaystyle =&\displaystyle \delta_{\nu \nu'} \, \delta_{m,- m'} \, (-)^{j-m} \,u_\nu(\varepsilon) \, v_\nu(\varepsilon)\, g_\nu(\varepsilon) \end{array} \end{aligned} $$

(31)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \widehat{a_{\nu m}(\varepsilon) a_{\nu' \bar{m}'}(\varepsilon)} &\displaystyle =&\displaystyle \delta_{\nu \nu'} \, \delta_{m,- m'} \, (-)^{j-m} \,u_\nu(\varepsilon) \, v_\nu(\varepsilon)\, g_\nu(\varepsilon) \end{array} \end{aligned} $$

(32)

where we have substituted δ(ε − ε) by g _ν(ε).

The diagonal part of the BCS Hamiltonian reads,

$$\displaystyle \begin{aligned} \begin{array}{rcl} H_{sp} - \lambda \hat{N} &\displaystyle =&\displaystyle \sum_{jm} (\varepsilon_j - \lambda) v^2_j + \sum_{\nu m} \int d\varepsilon (\varepsilon - \lambda) v^2_\nu(\varepsilon) g(\varepsilon) \\ &\displaystyle &\displaystyle + \sum_{jm} (\varepsilon_j - \lambda) (u^2_j-v^2_j) \alpha^+_{jm} \alpha_{jm} \\ &\displaystyle &\displaystyle + \sum_{\nu m} \int d\varepsilon (\varepsilon - \lambda) \left[ u^2_\nu(\varepsilon)-v^2_\nu(\varepsilon)\right] \alpha^+_{\nu m}(\varepsilon) \alpha_{\nu m}(\varepsilon) \\ &\displaystyle &\displaystyle + \sum_{jm} (\varepsilon_j - \lambda) (-)^{j-m} u_j v_j \left( \alpha^+_{jm} \alpha^+_{j\bar{m}} + hc \right) \\ &\displaystyle &\displaystyle + \sum_{\nu m} \int d\varepsilon (\varepsilon - \lambda) (-)^{j_{\nu}-m} u_\nu(\varepsilon) v_\nu(\varepsilon) \left[ \alpha^+_{\nu m}(\varepsilon) \alpha^+_{\nu \bar{m}}(\varepsilon) + hc \right] \qquad {} \end{array} \end{aligned} $$

(33)

For the pairing interaction we have

$$\displaystyle \begin{aligned} V_P= -\textit{GP}_d^+ P_d -\textit{GP}_d^+ P_c -\textit{GP}_c^+ P_d -\textit{GP}_c^+ P_c \end{aligned} $$

(34)

where $P_d^+ = \sum _j A^+_j$ and $P_c^+ = \int _0^\infty d\varepsilon \;A^+(\varepsilon )$. Thus,

$$\displaystyle \begin{aligned} \begin{array}{rcl} -\textit{GP}_d^+ P_d &\displaystyle =&\displaystyle - \frac{\varDelta^2_d}{G} + 2 \varDelta_d \sum_{jm} u_j v_j \alpha^+_{jm} \alpha_{jm} \\ &\displaystyle &\displaystyle - \frac{\varDelta_d}{2} \sum_{jm} (-)^{j-m} (u^2_j-v^2_j) \left( \alpha^+_{jm} \alpha^+_{j\bar{m}} + hc \right) \\ &\displaystyle &\displaystyle - G (\mathrm{terms}\; \mathrm{with}\; \mathrm{four}\; \alpha) \end{array} \end{aligned} $$

(35)

$$\displaystyle \begin{aligned} \begin{array}{rcl} -\textit{GP}_c^+ P_c &\displaystyle =&\displaystyle - \frac{\varDelta^2_c}{G} + 2 \varDelta_c \sum_{\nu m} \int d\varepsilon u_\nu(\varepsilon) v_\nu(\varepsilon) \alpha^+_{\nu m}(\varepsilon) \alpha_{\nu m}(\varepsilon) \\ &\displaystyle &\displaystyle - \frac{\varDelta_c}{2} \sum_{\nu m} \int d\varepsilon (-)^{j_\nu-m} \left[ u^2_\nu(\varepsilon)-v^2_\nu(\varepsilon) \right] \left[ \alpha^+_{\nu m}(\varepsilon) \alpha^+_{\nu \bar{m}}(\varepsilon) + hc \right] \\ &\displaystyle &\displaystyle - G (\mathrm{terms}\; \mathrm{with}\; \mathrm{four}\; \alpha) \end{array} \end{aligned} $$

(36)

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle -G (P_d^+ P_c + P_c^+ P_d) = -2 \frac{\varDelta_d \varDelta_c}{G} + 2 \varDelta_c \sum_{jm} u_j v_j \alpha^+_{jm} \alpha_{jm} \\ &\displaystyle &\displaystyle \quad + 2 \varDelta_d \sum_{\nu m} \int d\varepsilon u_\nu(\varepsilon) v_\nu(\varepsilon) \alpha^+_{\nu m}(\varepsilon) \alpha_{\nu m}(\varepsilon) \\ &\displaystyle &\displaystyle \quad - \frac{\varDelta_d}{2} \sum_{\nu m} \int d\varepsilon (-)^{j_\nu-m} \left[ u^2_\nu(\varepsilon)-v^2_\nu(\varepsilon)) \right] \left[ \alpha^+_{\nu m}(\varepsilon) \alpha^+_{\nu \bar{m}}(\varepsilon) + hc \right] \\ &\displaystyle &\displaystyle \quad - \frac{\varDelta_c}{2} \sum_{jm} (-)^{j-m} (u^2_j-v^2_j) \left( \alpha^+_{jm} \alpha^+_{j\bar{m}} + hc \right) \\ &\displaystyle &\displaystyle \quad - G (\mathrm{terms}\; \mathrm{with}\; \mathrm{four}\; \alpha) \end{array} \end{aligned} $$

(37)

where we have introduced the discrete Δ _d and the continuum Δ _c gaps,

$$\displaystyle \begin{aligned} \begin{array}{rcl} \varDelta_d &\displaystyle =&\displaystyle \frac{G}{2} \sum_{jm} u_j v_j \end{array} \end{aligned} $$

(38)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \varDelta_c &\displaystyle =&\displaystyle \frac{G}{2} \sum_{\nu m} \int d\varepsilon u_\nu(\varepsilon) v_\nu(\varepsilon) g_\nu(\varepsilon) \end{array} \end{aligned} $$

(39)

Then, the pairing interaction reads

$$\displaystyle \begin{aligned} \begin{array}{rcl} V_P &\displaystyle =&\displaystyle - \frac{\varDelta^2}{G} + 2 \varDelta \sum_{jm} u_j v_j \alpha^+_{jm} \alpha_{jm} + 2 \varDelta \sum_{\nu m} \int d\varepsilon u_\nu(\varepsilon) v_\nu(\varepsilon) \alpha^+_{\nu m}(\varepsilon) \alpha_{\nu m}(\varepsilon) \\ &\displaystyle &\displaystyle - \frac{\varDelta}{2} \sum_{jm} (-)^{j-m} (u^2_j-v^2_j) \left( \alpha^+_{jm} \alpha^+_{j\bar{m}} + hc \right) \\ &\displaystyle &\displaystyle - \frac{\varDelta}{2} \sum_{\nu m} \int d\varepsilon (-)^{j_\nu-m} \left[ u^2_\nu(\varepsilon)-v^2_\nu(\varepsilon)) \right] \left[ \alpha^+_{\nu m}(\varepsilon) \alpha^+_{\nu \bar{m}}(\varepsilon) + hc \right] \\ &\displaystyle &\displaystyle - G (\mathrm{terms}\; \mathrm{with}\; \mathrm{four}\; \alpha) {} \end{array} \end{aligned} $$

(40)

Notice that the unknown discrete and continuum gap do not appear separately but in the combination Δ = Δ _d + Δ _c.

By adding Eqs. (33) and (40), we get the expression of the BCS Hamiltonian $H_{BCS}=H_{sp}+V_P-\lambda \hat {N}$ in terms of the quasi-particle operators. By eliminating the “dangerous” terms [29] that contain $\alpha ^+_{nm} \alpha ^+_{n\bar {m}} + hc$ with n = j and n = ν(ε), we get the following two equations:

$$\displaystyle \begin{aligned} \begin{array}{rcl} - \frac{\varDelta}{2} (u^2_j-v^2_j) + (\varepsilon_j - \lambda) u_j v_j &\displaystyle =&\displaystyle 0 {} \end{array} \end{aligned} $$

(41)

$$\displaystyle \begin{aligned} \begin{array}{rcl} - \frac{\varDelta}{2} \left[ u^2_\nu(\varepsilon)-v^2_\nu(\varepsilon) \right] + (\varepsilon - \lambda) u_\nu(\varepsilon) v_\nu(\varepsilon) &\displaystyle =&\displaystyle 0 {} \end{array} \end{aligned} $$

(42)

which, together with the normalization conditions $u^2_j+v^2_j=1$ and $u^2_\nu (\varepsilon )+v^2_\nu (\varepsilon )=1$, yield

$$\displaystyle \begin{aligned} \begin{array}{rcl} v^2_j &\displaystyle =&\displaystyle \frac{1}{2} \left[ 1 - \frac{(\varepsilon_j - \lambda)} {\sqrt{(\varepsilon_j - \lambda)^2 + \varDelta^2}} \right] {} \end{array} \end{aligned} $$

(43)

$$\displaystyle \begin{aligned} \begin{array}{rcl} v_\nu^2(\varepsilon) &\displaystyle =&\displaystyle \frac{1}{2} \left[ 1 - \frac{(\varepsilon - \lambda)} {\sqrt{(\varepsilon - \lambda)^2 + \varDelta^2}} \right] {} \end{array} \end{aligned} $$

(44)

Eq. (44) shows that the occupation probability in the continuum does not depend on the quantum number ν, i.e., v _ν(ε) = v(ε). The coefficients $u^2_j$ and u ²(ε) are obtained from the normalization condition. Substituting Eqs. (43) and (44) into Eqs. (41) and (42) we get

$$\displaystyle \begin{aligned} \begin{array}{rcl} u_j v_j &\displaystyle =&\displaystyle \frac{\varDelta}{2 \sqrt{(\varepsilon_j - \lambda)^2 + \varDelta^2}} \end{array} \end{aligned} $$

(45)

$$\displaystyle \begin{aligned} \begin{array}{rcl} u(\varepsilon) v(\varepsilon) &\displaystyle =&\displaystyle \frac{\varDelta}{2 \sqrt{(\varepsilon - \lambda)^2 + \varDelta^2}} \end{array} \end{aligned} $$

(46)

which can be used to calculate the gap parameter,

$$\displaystyle \begin{aligned} \begin{array}{rcl} \varDelta &\displaystyle =&\displaystyle \varDelta_d + \varDelta_c \end{array} \end{aligned} $$

(47)

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle =&\displaystyle \frac{G}{2} \sum_{jm} u_j v_j + \frac{G}{2} \sum_{\nu m} \int_0^\infty d\varepsilon\; u(\varepsilon) v(\varepsilon) g_\nu(\varepsilon) \end{array} \end{aligned} $$

(48)

The above equation reduces to the so-called gap equation,

$$\displaystyle \begin{aligned} \frac{4}{G}= \sum_{jm} \frac{1}{E_j} + \sum_{\nu m} \int_0^\infty d\varepsilon\; \frac{g_\nu(\varepsilon)}{E(\varepsilon)} \end{aligned} $$

(49)

where we have introduced the quasi-particle energies, $E_j=\sqrt {(\varepsilon _j - \lambda )^2 + \varDelta ^2}$ and $E(\varepsilon )=\sqrt {(\varepsilon - \lambda )^2 + \varDelta ^2}$.