QRPA equation

The QRPA equations are the small-oscillations limit of the time-dependent Hartree-Fock-Bogoliubov approximation, see, e.g., [8,13]. In the canonical basis the most general equations take the form

$$\sum_{L<L^\prime} \left( \begin{array}{cc} A_{KK^\prime,LL^\... ...ime}\\ {Y}^k_{KK^\prime} \end{array}\right), \ \ K<K^\prime ,$$ (10)

$$A_{KK^\prime,LL^\prime} = E_{KL}\delta_{K^\prime L^\prime} -E_{K^\prime L}\delta_{K L^\prime} -E_{KL^\prime}\delta_{K^\prime L} +E_{K^\prime L^\prime}\delta_{KL}$$

$$-\bar{V}^{\rm ph}_{K \bar{L} \bar{K}^\prime L^\prime}u_{L^\prime}... ...V}^{\rm ph}_{K^\prime \bar{L} \bar{K} L^\prime}u_{L^\prime}v_L u_{K^\prime} v_K$$

$$+\bar{V}^{\rm ph}_{K \bar{L}^\prime \bar{K}^\prime L}u_{L}v_{L^\p... ...}^{\rm ph}_{K^\prime \bar{L}^\prime \bar{K} L}u_L v_{L^\prime} u_{K^\prime} v_K$$

$$-\bar{V}^{\rm pp}_{\bar{L} \bar{L}^\prime \bar{K}^\prime \bar{K}}... ... -\bar{V}^{\rm pp}_{K K^\prime {L}^\prime L}u_K u_{K^\prime} u_{L} u_{L^\prime}$$

$$-\bar{V}^{\rm 3p1h}_{\bar{L} \bar{L}^\prime {K} \bar{K}^\prime}v_... ..._{\bar{L} \bar{L}^\prime {K}^\prime \bar{K}}v_L v_{L^\prime} u_{K^\prime} v_{K}$$

$$-\bar{V}^{\rm 3p1h}_{{K} {K}^\prime \bar{L} {L}^\prime}u_{K}u_{K^... ... 3p1h}_{{K} {K}^\prime \bar{L}^\prime {L}}u_{K} u_{K^\prime} u_{L} v_{L^\prime}$$

$$-\bar{V}^{\rm 1p3h}_{\bar{L} {L}^\prime \bar{K}^\prime \bar{K}}u_... ...\bar{L}^\prime {L} \bar{K}^\prime \bar{K}}u_{L} v_{L^\prime} v_{K^\prime} v_{K}$$

$$-\bar{V}^{\rm 1p3h}_{{K} \bar{K}^\prime {L}^\prime {L}}u_{K}v_{K^... ...1p3h}_{{K}^\prime \bar{K} {L}^\prime {L}}u_{K^\prime} v_{K} u_{L} u_{L^\prime},$$ (11)

$$B_{KK^\prime,LL^\prime} = \bar{V}^{\rm ph}_{K^\prime L^\prime\bar{K} \bar{L}}u_{L^\prime}v_... ...\rm ph}_{K L^\prime \bar{K^\prime} \bar{L}}u_{L^\prime}v_{L} u_{K} v_{K^\prime}$$

$$-\bar{V}^{\rm ph}_{K^\prime L \bar{K} \bar{L}^\prime}u_{L}v_{L^\p... ...\rm ph}_{K L \bar{K}^\prime \bar{L}^\prime}u_{L}v_{L^\prime} u_{K} v_{K^\prime}$$

$$+\bar{V}^{\rm pp}_{K^\prime {K} \bar{L} \bar{L}^\prime}v_{L}v_{L^... ...m pp}_{L^\prime {L} \bar{K} \bar{K}^\prime}v_{K}v_{K^\prime} u_{L} u_{L^\prime}$$

$$+\bar{V}^{\rm 3p1h}_{K^\prime {K} {L^\prime} \bar{L}}u_{L^\prime}... ...\rm 3p1h}_{K^\prime {K} {L} \bar{L}^\prime}u_{L}v_{L^\prime} u_{K} u_{K^\prime}$$

$$+\bar{V}^{\rm 3p1h}_{L^\prime {L} {K^\prime} \bar{K}}u_{K^\prime}... ...\rm 3p1h}_{L^\prime {L} {K} \bar{K}^\prime}u_{K}v_{K^\prime} u_{L} u_{L^\prime}$$

$$+\bar{V}^{\rm 1p3h}_{K^\prime \bar{K} \bar{L} \bar{L}^\prime}v_{L... ...}_{K \bar{K}^\prime \bar{L} \bar{L}^\prime}v_{L}v_{L^\prime} u_{K} v_{K^\prime}$$

$$+\bar{V}^{\rm 1p3h}_{L^\prime \bar{L} \bar{K} \bar{K}^\prime}v_{K... ..._{L \bar{L}^\prime \bar{K} \bar{K}^\prime}v_{K}v_{K^\prime} u_{L} v_{L^\prime},$$ (12)

$$\bar{V}^{\rm ph}_{KL K^\prime L^\prime} = \frac{ \delta^2 E... ...ppa^\ast] }{ \delta\rho_{K^\prime K}\delta\rho_{L^\prime L} },$$ (13)

$$\bar{V}^{\rm pp}_{K^\prime K L^\prime L} = \frac{ \delta^2 ... ...{ \delta\kappa^\ast_{K^\prime K} \delta\kappa_{L^\prime L} },$$ (14)

$$\bar{V}^{\rm 3p1h}_{K^\prime K L^\prime L} = \frac{ \delta^... ...\prime} } ={\bar{V}^{\rm 1p3h\ \ast}}_{L L^\prime K^\prime K},$$ (15)

where $K$ and $L$ are single-particle indices for the canonical basis, and the states are assumed to be ordered. The symbol $\bar{K}$ refers to the conjugate partner of $K$, $u_K$ and $v_K$ come from the BCS transformation associated with the canonical basis, and the $E_{KL}$ are the one-quasiparticle matrix elements of the HFB Hamiltonian (cf. Eq. (4.14b) of Ref. [60]). $X^k_{LL^\prime}$ and $Y^k_{LL^\prime}$ are the forward and backward amplitudes of the QRPA solution $k$, and $E_k$ is the corresponding excitation energy. $E[\rho,\kappa,\kappa^\ast]$ is the energy functional (see App. B for an explicit definition) and $\rho$ and $\kappa$ are the density matrix and pairing tensor. After taking the functional derivatives, we replace $\rho$ and $\kappa$ by their HFB solutions, in complete analogy with an ordinary Taylor-series expansion.

To write the equations in coupled form, we introduce the notation

$$K \equiv (n_\mu l_\mu j_\mu m_\mu) \equiv (\mu m_\mu), \ \ L \equiv (\nu m_\nu), \ \$$ (16)

where $(nljm)$ denote spherical quantum numbers. Using (i) rotational, time-reversal, and parity symmetries of the HFB state, (ii) the conjugate single-particle state²

$$\vert\overline{K}\rangle = \vert\overline{\mu m_\mu}\rangle = (-)^{j_\mu-m_\mu}\vert\mu\: -m_\mu\rangle ,$$ (17)

and (iii) the relations

$$X^k_{KK^\prime} = \langle j_\mu m_\mu j_{\mu^\prime} m_{\mu^\prim... ...array}{cc}\sqrt{2}, &\mu=\mu^\prime,\\ 1, &\mbox{otherwise},\end{array}\right.$$ (18)

$$Y^k_{KK^\prime} = (-)^{j_\mu-m_\mu}(-)^{j_{\mu^\prime}-m_{\mu^\pr... ...array}{cc}\sqrt{2}, &\mu=\mu^\prime,\\ 1, &\mbox{otherwise},\end{array}\right.$$ (19)

with $J_k$ the angular momentum of the state $k$ and the factor $\sqrt{2}$ for convenience [12], one can rewrite the QRPA equation as

$$\sum_{\nu\leq\nu^\prime} \left( \begin{array}{cc} A_{[\mu\mu... ...mu\mu^\prime]J_k} \end{array}\right),\ \ \mu \leq \mu^\prime ,$$ (20)

$$A_{[\mu\mu^\prime]J_k,[\nu\nu^\prime]J_k} = \frac{1}{\sqrt{1+\delta_{\mu\mu^\prime}}} \frac{1}{\sqrt{1+\delta... ...E_{\mu^\prime\nu}\:\delta_{\mu\nu^\prime}(-)^{j_\mu+j_{\mu^\prime}-J_k} \right.$$

$$-E_{\mu\nu^\prime}\:\delta_{\mu^\prime\nu}(-)^{j_\mu+j_{\mu^\prime}-J_k} +E_{\mu^\prime\nu^\prime}\:\delta_{\mu\nu}$$

$$+G(\mu\mu^\prime\nu\nu^\prime;J_k) (u_{\mu^\prime} u_\mu u_\nu u_{\nu^\prime} +v_{\nu} v_{\nu^\prime} v_{\mu^\prime} v_{\mu})$$

$$+F(\mu\mu^\prime\nu\nu^\prime;J_k) (u_{\mu} v_{\nu^\prime} u_\nu v_{\mu^\prime} +u_{\mu^\prime} v_{\nu} u_{\nu^\prime} v_{\mu})$$

$$-(-)^{ j_{\nu^\prime}+j_\nu-J_k }F(\mu\mu^\prime\nu^\prime\nu;J_k... ...} u_{\nu^\prime} v_{\mu^\prime} +u_{\mu^\prime} v_{\nu^\prime} u_{\nu} v_{\mu})$$

$$-H(\mu\mu'\nu\nu';J_k)( v_\nu v_{\nu'} u_\mu v_{\mu'} +u_{\mu'} v_\mu u_\nu u_{\nu'} )$$

$$+(-)^{j_\mu+j_{\mu'}-J_k} H(\mu'\mu\nu\nu';J_k) (v_\nu v_{\nu'} u_{\mu'} v_\mu + u_\mu v_{\mu'} u_\nu u_{\nu'} )$$

$$-H^\ast(\nu\nu'\mu\mu';J_k)( u_\mu u_{\mu'} u_{\nu'} v_\nu + u_\nu v_{\nu'} v_{\mu'} v_\mu)$$

$$+(-)^{j_\nu+j_{\nu'}-J_k}H^\ast(\nu'\nu\mu\mu';J_k) (u_\mu u_{\mu'} u_\nu v_{\nu'} + u_{\nu'} v_\nu v_{\mu'} v_\mu) \left.\right\},$$ (21)

$$B_{[\mu\mu^\prime]J_k,[\bar{\nu}\bar{\nu}^\prime]J_k} = \frac{1}{\sqrt{1+\delta_{\mu\mu^\prime}}} \frac{1}{\sqrt{1+\delta... ...\mu v_\nu v_{\nu^\prime} +u_{\nu} u_{\nu^\prime} v_{\mu^\prime} v_{\mu})\right.$$

$$-(-)^{ j_\nu+j_{\nu^\prime}-J_k }F(\mu\mu^\prime\nu^\prime\nu;J_k... ...} v_{\mu^\prime} v_{\nu^\prime} +u_{\mu^\prime} u_{\nu^\prime} v_{\nu} v_{\mu})$$

$$+(-)^{ j_\nu+j_{\nu^\prime}+j_\mu+j_{\mu^\prime} }F(\mu^\prime\mu... ... u_{\nu} v_{\nu^\prime} v_{\mu} +u_{\mu} u_{\nu^\prime} v_{\mu^\prime} v_{\nu})$$

$$+H(\mu\mu'\nu\nu';J_k)(v_\nu v_{\nu'} u_{\mu'} v_\mu + u_\mu v_{\mu'} u_\nu u_{\nu'})$$

$$-(-)^{j_\mu+j_{\mu'}-J_k} H(\mu'\mu\nu\nu';J_k) (v_\nu v_{\nu'} u_\mu v_{\mu'} + u_{\mu'} v_\mu u_\nu u_{\nu'})$$

$$-H^\ast(\nu\nu'\mu\mu';J_k) (v_\mu v_{\mu'} u_{\nu'} v_\nu + u_\nu v_{\nu'} u_\mu u_{\mu'})$$

$$+(-)^{j_\nu + j_{\nu'}-J_k}H^\ast(\nu'\nu\mu\mu';J_k) (v_\mu v_{\mu'} u_\nu v_{\nu'} +u_{\nu'} v_\nu u_\mu u_{\mu'}) \left.\right\},$$ (22)

$$G(\mu\mu^\prime\nu\nu^\prime;J_k) = \sum_{ m_\mu m_{\mu^\prime}m_\nu m_{\nu^\prime} } \langle j_\mu m... ...e} m_{\nu^\prime} \vert J_k M_k\rangle \bar{V}^{\rm pp}_{K K^\prime L L^\prime}$$

$$\equiv \langle [\mu\mu^\prime]J_k \vert \bar{V}^{\rm pp} \vert[\nu\nu^\prime]J_k\rangle ,$$ (23)

$$F(\mu\mu^\prime\nu\nu^\prime;J_k) = \sum_{ m_\mu m_{\mu^\prime}m_\nu m_{\nu^\prime} } \langle j_\mu m... ...^\prime} \vert J_k M_k\rangle \bar{V}^{\rm ph}_{K\bar{L^\prime}\bar{K^\prime}L}$$

$$= \sum_{J^\prime} (-)^{j_{ \mu^\prime}+j_\nu +J^\prime } \left\{ \b... ...e [\mu\nu^\prime]J_k \vert \bar{V}^{\rm ph} \vert [\mu^\prime \nu]J_k \rangle ,$$ (24)

$$H(\mu\mu'\nu\nu';J_k) = \sum_{m_\mu m_{\mu'} m_\nu m_{\nu'}} \langle j_\mu m_\mu j_{\mu'}... ...\nu'} m_{\nu'} \vert JM \rangle \bar{V}^{\rm 3p1h}_{\bar{L}\bar{L}' K \bar{K}'}$$

$$= \sum_{J^\prime} (-)^{j_{ \mu}+j_\nu + 1 + l_{\nu^\prime} -J_k - J... ... [\mu\nu]J_k \vert \bar{V}^{\rm 3p1h} \vert [\mu^\prime \nu^\prime]J_k \rangle.$$

We have represented the second derivatives of the energy functional $E[\rho,\kappa,\kappa^\ast]$ as unsymmetrized matrix elements of effective interactions $\bar{V}^{\rm pp}$, $\bar{V}^{\rm ph}$, and $\bar{V}^{\rm 3p1h}$. These effective interactions are given in App. B. Although the ``matrix elements'' are unsymmetrized, the underlying two-quasiparticle states are of course antisymmetric. As a consequence, $A_{[\mu\mu^\prime]J_k,[\nu\nu^\prime]J_k}= B_{[\mu\mu^\prime]J_k,[ \bar{\nu} \bar{\nu}^\prime ]J_k}=0$ if $J_k$ is odd and either $\mu=\mu^\prime$ or $\nu=\nu^\prime$.

The nuclear energy functional $E[\rho,\kappa,\kappa^\ast]$ is usually separated into particle-hole (ph) and pairing pieces (again, see App. B for explicit expressions). If the pairing functional, which we will call $E_{\rm pair}[\rho,\kappa,\kappa^\ast]$, depends on $\rho$ then the derivatives of $E_{\rm pair}[\rho,\kappa,\kappa^\ast]$ with respect to $\rho_{KK'}$ are called pairing-rearrangement terms [13]. In the QRPA, two kinds of pairing-rearrangement terms can arise in general. One has particle-hole character and is included in $\bar{V}^{\rm ph}_{KLK'L'}$; the other affects 3-particle-1-hole (3p1h) and 1-particle-3-hole (1p3h) configurations and is represented by $\bar{V}^{\rm 3p1h}_{K'KL'L}$ and $\bar{V}^{\rm 1p3h}_{K'KL'L}$. If the $\rho$-dependence of $E_{\rm pair}[\rho,\kappa,\kappa^\ast]$ is linear, then the ph-type pairing-rearrangement term does not appear. Furthermore, the 3p1h and 1p3h pairing-rearrangement terms arise only for $J^{\pi}=0^+$ modes if the HFB state has $J=0$. Most existing work uses a pairing functional that is linear in $\rho$, and so needs no pairing-rearrangement terms in $J^{\pi} \neq 0^+$ channels.