Kernels and Lipkin parameters up to sixth order

To calculate kernels $N_m(\phi)$ from derivatives of the overlap kernel $I(\phi)$, in Eq. (5), we use the Onishi theorem [2]:

$$\langle\Phi\vert\Phi(\phi)\rangle = \frac{\det^{1/2}(1+e^{2i\phi}CC^+)} {\det^{1/2}(1+ CC^+)}e^{-iN_0\phi} \, ,$$ (17)

where $C$ is the Thouless matrix. In the canonical basis of the Bogoliubov transformation, the above overlap is given as

$$\langle\Phi\vert\Phi(\phi)\rangle = \prod_{\mu>0} (u_{\mu}^2+ v_{\mu}^2 e^{2i\phi} )e^{-iN_0\phi} \, ,$$ (18)

where the label $\mu>0$ is associated with one of the pair-conjugated canonical states $\{\mu,\bar{\mu}\}$. Then the density matrix in canonical basis can be obtained as

$$\rho_{\mu\nu}(\phi) = \frac{ \langle\Phi\vert a^+_{\nu}a_{\mu} \vert\Phi(\phi)\rangle} ... ...{e^{2i\phi} v_{\mu}^2 } {u_{\mu}^2+ v_{\mu}^2 e^{2i\phi} } \delta_{\mu\nu} \, .$$ (19)

We then have the reduced kernels $n_m(\phi)=N_m(\phi)/I(\phi)$ given as

$$n_1(\phi) =\!\! {\rm Tr}\rho(\phi)-N_0 \equiv R_0(\phi) \, ,$$ (20)

$$n_2(\phi) =\!\! R^2_0(\phi)-i\frac{d}{d\phi}R_0(\phi) \equiv R^2_0(\phi)+R_1(\phi) \, ,$$ (21)

$$n_3(\phi) =\!\! R^3_0(\phi)+3R_0(\phi)R_1(\phi)+R_2(\phi) \, ,$$ (22)

$$n_4(\phi) =\!\! R^4_0(\phi)+6R^2_0(\phi)R_1(\phi)+4R_0(\phi)R_2(\phi)$$

$$+3R^2_1(\phi)+R_3(\phi) \, ,$$ (23)

$$n_5(\phi) = R^5_0(\phi)+10R^3_0(\phi)R_1(\phi)+10R^2_0(\phi)R_2(\phi)$$

$$+15R_0(\phi)R^2_1(\phi) +5R_0(\phi)R_3(\phi)$$

$$+10R_1(\phi)R_2(\phi)+R_4(\phi) \, ,$$ (24)

$$n_6(\phi) = R^6_0(\phi)+15R^4_0(\phi)R_1(\phi)+20R^3_0(\phi)R_2(\phi)$$

$$+45R^2_0(\phi)R^2_1(\phi) +15R^2_0(\phi)R_3(\phi)$$

$$+60R_0(\phi)R_1(\phi)R_2(\phi)+6R_0(\phi)R_4(\phi)$$

$$+15R^3_1(\phi)+15R_1(\phi)R_3(\phi)+10R^2_2(\phi)$$

$$+R_5(\phi) \, ,$$ (25)

where we used the definition

$$R_n (\phi) = (-i)^n\frac{d^n}{d\phi^n}\left({\rm Tr}\rho(\phi) - N_0\right) \, .$$ (26)

Derivatives of the density matrix $\rho(\phi)$ (19) can be calculated as,

$$-i\frac{d}{d\phi} \rho(\phi) = 2 \rho(\phi)(1-\rho(\phi)) \, ,$$ (27)

$$(-i)^2\frac{d^2}{d\phi^2} \rho(\phi) = 4 \rho(\phi)(1-\rho(\phi))(1-2\rho(\phi)) \, ,$$ (28)

$$(-i)^3\frac{d^3}{d\phi^3} \rho(\phi) = 8 \rho(\phi)(1-\rho(\phi))$$

$$- 48 \rho^2(\phi)(1-\rho(\phi))^2 \, ,$$ (29)

$$(-i)^4\frac{d^4}{d\phi^4} \rho(\phi) = 16 \rho(\phi)(1-\rho(\phi))^2$$

$$- 16 \rho^2(\phi)(1-\rho(\phi))$$

$$-192\rho^2(\phi)(1-\rho(\phi))^3$$

$$+192\rho^3(\phi)(1-\rho(\phi))^2 \, ,$$ (30)

$$(-i)^5\frac{d^5}{d\phi^5} \rho(\phi) = 32 \rho(\phi)(1-\rho(\phi))^3$$

$$-128 \rho^2(\phi)(1-\rho(\phi))^2$$

$$+32\rho^3(\phi)(1-\rho(\phi))$$

$$+2304\rho^3(\phi)(1-\rho(\phi))^3$$

$$-768\rho^2(\phi)(1-\rho(\phi))^4$$

$$-768\rho^4(\phi)(1-\rho(\phi))^2 \, ,$$ (31)

which gives

$$R_0 (\phi) = {\rm Tr}\rho(\phi) - N_0 \, ,$$ (32)

$$R_1 (\phi) = 2 {\rm Tr}\rho(\phi)(1-\rho(\phi)) \, ,$$ (33)

$$R_2 (\phi) = 4 {\rm Tr}\rho(\phi)(1-\rho(\phi))(1-2\rho(\phi)) \, ,$$ (34)

$$R_3 (\phi) = 8 {\rm Tr}\rho(\phi)(1-\rho(\phi))$$

$$-48{\rm Tr}\rho^2(\phi)(1-\rho(\phi))^2 \, ,$$ (35)

$$R_4 (\phi) = 16 {\rm Tr}\rho(\phi)(1-\rho(\phi))^2$$

$$-16 {\rm Tr}\rho^2(\phi)(1-\rho(\phi))$$

$$-192{\rm Tr}\rho^2(\phi)(1-\rho(\phi))^3$$

$$~ +192{\rm Tr}\rho^3(\phi)(1-\rho(\phi))^2 \, ,$$ (36)

$$R_5 (\phi) = 32 {\rm Tr}\rho(\phi)(1-\rho(\phi))^3$$

$$-128 {\rm Tr}\rho^2(\phi)(1-\rho(\phi))^2$$

$$+32 {\rm Tr}\rho^3(\phi)(1-\rho(\phi))$$

$$+2304{\rm Tr}\rho^3(\phi)(1-\rho(\phi))^3$$

$$-768{\rm Tr}\rho^2(\phi)(1-\rho(\phi))^4$$

$$~ -768{\rm Tr}\rho^4(\phi)(1-\rho(\phi))^2 \, .$$ (37)

Up to the sixth order, the average Routhian of Eq. (7) to be minimized reads

$$E_{N_0} = \langle\Phi\vert\hat{H}-k_1(\hat{N}-N_0)-k_2(\hat{N}-N_0)^2$$

$$-k_3(\hat{N}-N_0)^3-k_4(\hat{N}-N_0)^4$$

$$-k_5(\hat{N}-N_0)^5-k_6(\hat{N}-N_0)^6\vert\Phi\rangle \, .$$ (38)

Average values of powers of the particle-number operator are given in Eqs. (20)-(25) taken at $\phi=0$. Moreover, in all terms with $m\geq2$, one can set $R_0\equiv0$. This gives

$$E_{N_0} = \langle\Phi\vert\hat{H}\vert\Phi\rangle-k_1({\rm Tr}\rho-N_0)-2 k_2{\rm Tr}\rho(1-\rho)$$

$$-4 k_3{\rm Tr}\rho(1-\rho)(1-2\rho)-12k_4({\rm Tr}\rho(1-\rho))^2$$

$$-8k_4{\rm Tr}\rho(1-\rho)+48 k_4 {\rm Tr}\rho^2(1-\rho)^2$$

$$-80k_5({\rm Tr}\rho(1-\rho))({\rm Tr}\rho(1-\rho)(1-2\rho))$$

$$-16k_5 {\rm Tr}\rho(1-\rho)^2+16k_5 {\rm Tr}\rho^2(1-\rho)$$

$$+192k_5{\rm Tr}\rho^2(1-\rho)^3-192k_5{\rm Tr}\rho^3(1-\rho)^2$$

$$-120k_6({\rm Tr}\rho(1-\rho))^3-240k_6({\rm Tr}\rho(1-\rho))^2$$

$$+1440k_6({\rm Tr}\rho(1-\rho))({\rm Tr}\rho^2(1-\rho)^2)$$

$$-160k_6({\rm Tr}\rho(1-\rho)(1-2\rho))^2$$

$$-32k_6{\rm Tr}\rho(1-\rho)^3+128k_6{\rm Tr}\rho^2(1-\rho)^2$$

$$-32k_6{\rm Tr}\rho^3(1-\rho)-2304k_6{\rm Tr}\rho^3(1-\rho)^3$$

$$+768k_6{\rm Tr}\rho^2(1-\rho)^4+768k_6{\rm Tr}\rho^4(1-\rho)^2 \, .$$

Hence, the corresponding mean-field Routhians (to be used in the HFB equations) reads ¹

$$h' = h-k_1-\left(2 k_2+24k_4{\rm Tr}\rho(1-\rho)+8k_4\right)(1-2\rho)$$

$$-4 k_3\left((1-2\rho)^2-2\rho(1-\rho)\right)$$

$$+96 k_4\rho(1-\rho)(1-2\rho)$$

$$-80 k_5({\rm Tr}\rho(1-\rho)(1-2\rho))(1-2\rho)$$

$$-80 k_5({\rm Tr}\rho(1-\rho))(1-2\rho)^2$$

$$+160k_5({\rm Tr}\rho(1-\rho))\rho(1-\rho)$$

$$-16k_5(1-\rho)^2+64k_5\rho(1-\rho)$$

$$-16k_5\rho^2+384k_5\rho(1-\rho)^3$$

$$-1152k_5\rho^2(1-\rho)^2+384k_5\rho^3(1-\rho)$$

$$-32k_6-360k_6({\rm Tr}\rho(1-\rho))^2$$

$$-320k_6({\rm Tr}\rho(1-\rho)(1-2\rho))$$

$$-480k_6({\rm Tr}\rho(1-\rho))+1440k_6({\rm Tr}\rho^2(1-\rho)^2)$$

$$+1984k_6\rho+720k_6({\rm Tr}\rho(1-\rho))^2\rho$$

$$+1920k_6({\rm Tr}\rho(1-\rho)(1-2\rho))\rho$$

$$+3840k_6({\rm Tr}\rho(1-\rho))\rho$$

$$-2880k_6({\rm Tr}\rho^2(1-\rho)^2)\rho-17280k_6\rho^2$$

$$-8640k_6({\rm Tr}\rho(1-\rho))\rho^2$$

$$-1920k_6({\rm Tr}\rho(1-\rho)(1-2\rho))\rho^2$$

$$+49920k_6\rho^3+5760k_6({\rm Tr}\rho(1-\rho))\rho^3$$

$$-57600k_6\rho^4+23040k_6\rho^5 \, .$$ (39)

Although the expression for the mean-field Routhian looks complicated, it does not require too much of the computational resources. Additional terms appearing within the Lipkin method are simply composed by powers of density matrices, and the manipulations of these terms can be easily performed numerically. The particle-particle mean fields remain the same, so that the HFB equations are modified only through Eq. (40).

Lipkin parameters $k_{m}$ for $m=1,\ldots,M$ can be determined from Eq. (9) by requiring that it is fulfilled at gauge angle $\phi=\phi_0=0$ and also at all $M$ other nonzero values of the gauge angle $\phi_i$. This gives

$$C+\sum_m k_m n_m(\phi_i)=h(\phi_i) \, ,$$ (40)

where $C$ is the flattened Routhian. Then, at sixth order, Lipkin parameters $k_m$ can be easily obtained by inverting the matrix built of coefficients $n_m(\phi_i)$ as

$$\left(\begin{array}{c} C\\ k_1\\ k_2\\ k_3\\ k_4\\ k_5\\ k_... ..._2)\\ h(\phi_3)\\ h(\phi_4)\\ h(\phi_5)\\ h(\phi_6) \end{array}\right) \, .$$

At lower orders, or when neglecting odd orders, a smaller number of the gauge angle points can be used.

In fact, the value of $C$ obtained from the first row of Eq. (42) does not appear in the mean-field Routhian (40) and can be ignored. Anyhow, at convergence it is calculated from Eq. (39). Moreover, during the iteration of the HFB equation, parameter $k_1\equiv\lambda$ is treated as a Lagrange multiplier, determined so as to adjust the average particle number, for which, at convergence, one has ${\rm Tr}\rho = N_0$ and thus $k_1$ has no influence on the value of the right-hand side of Eq. (39). It means that in the linear equation (41), the term related to $k_1$ can be simply moved from the left hand side to the right, and the dimension of the matrix in Eq. (42) can be further reduced correspondingly.