Symmetry conserving coupled cluster doubles wave function and the self-consistent odd particle number RPA

Abstract

Mixing single and triple fermions an exact annihilation operator of the Coupled Cluster Doubles wave function with good symmetry was found in Tohyama and Schuck (Phys Rev C 87:044316, 2013). Using these operators with the equation of motion method the so-called self-consistent odd particle number random phase approximation (odd-RPA) was set up. Together with the stationarity condition of the two body density matrix it is shown that the annihilation conditions allow to reduce the order of correlation functions contained in the matrix elements of the odd-RPA equations to a fully self consistent equation for the single particle occupation numbers. Excellent results for the latter and the ground state energies are obtained in an exactly solvable model from weak to strong couplings.

Appendices

Appendix A: Stationary condition

The stationary condition implies that the expectation value of the commutator $[H,K^\dag K^\dag ]$ must be zero with a general two-body Hamiltonian (12) where each elements are defined as,

$$\begin{aligned} E_{HF}= & {} \sum \limits _h \epsilon _{hh} +\frac{1}{2} \sum \limits _{hh'} {\bar{v}}_{hh'hh'} , \nonumber \\ H^{20}= & {} \sum \limits _{ph}\biggl ( \epsilon _{ph} +\sum \limits _{h'} {\bar{v}}_{ph'hh'}\biggl ) K^\dag _{ph} +\hbox {h.c.} \nonumber \\ H^{40}= & {} \frac{1}{4} \sum \limits _{pp',hh'} {\bar{v}}_{pp'hh'} K^\dag _{ph} K^\dag _{p'h'} +\hbox {h.c.}, \nonumber \\ H^{11}= & {} \sum \limits _{pp'} \biggl (\epsilon _{pp'}+\sum \limits _{h}{\bar{v}}_{php'h}\biggl ) S_{pp'}\nonumber \\&-\sum \limits _{hh'} \biggl (\epsilon _{h'h} +\sum \limits _{h_1} {\bar{v}}_{h'h_1hh_1} \biggl )S_{hh'} \nonumber \\ H^{31}= & {} \frac{1}{2} \sum \limits _{ph} K^\dag _{ph} \biggl (\sum \limits _{p'p_1} {\bar{v}}_{pp'hp_1} S_{p'p_1}\nonumber \\&-\sum \limits _{h'h_1} {\bar{v}}_{ph_1hh'} S_{h'h_1} \biggr ) + \hbox {h.c.}, \nonumber \\ H^{22}= & {} \sum \limits _{php'h'} {\bar{v}}_{ph'hp'} K^\dag _{ph} K _{h'p'}\nonumber \\&+\frac{1}{4} \sum \limits _{pp'p_1} {\bar{v}}_{pp_1p'p_1} S_{pp'} +\frac{1}{4} \sum \limits _{hh'h_1} {\bar{v}}_{hh_1h'h_1} S_{h'h} \nonumber \\&-\frac{1}{4} \sum \limits _{pp_1p'p_2} {\bar{v}}_{pp'p_1p_2} S_{pp_2} S_{p'p_1}\nonumber \\&-\frac{1}{4} \sum \limits _{hh'h_1h_2} {\bar{v}}_{hh'h_1h_2} S_{h_1h'} S_{h_2h} , \end{aligned}$$

(A1)

where the density matrix operators $S_{ij}$ are given by

$$\begin{aligned} S_{hh'}=\beta ^\dag _{h} \beta _{h'}, ~~~~~~~~~~~ S_{pp'}=\beta ^\dag _{p} \beta _{p'}. \end{aligned}$$

(A2)

This Hamilton operator is exactly the same as the normal ordered one given in [20], Eqs (E.24) and (E.25). For later convenience we write it here in a slightly non normal ordered form. Let us then calculate this commutator with the general Hamiltonian (12),

$$\begin{aligned}&\left[ H^{11},K^\dag _{ph}K^\dag _{p'h'}\right] = 2\sum \limits _{p_1} \varepsilon _{pp_1}K^\dag _{p_1h}K^\dag _{p'h'}\nonumber \\&\quad -2\sum \limits _{h_1} \varepsilon _{hh_1}K^\dag _{ph_1}K^\dag _{p'h'} \end{aligned}$$

(A3)

$$\begin{aligned}&\left[ H^{20},K^\dag _{ph}K^\dag _{p'h'}\right] = \varepsilon _{ph} K^\dag _{p'h'} +\varepsilon _{p'h'} K^\dag _{ph}\nonumber \\&\quad -\varepsilon _{ph'} K^\dag _{p'h} -\varepsilon _{p'h} K^\dag _{ph'}\nonumber \\&\quad -2K^\dag _{ph} \biggl ( \sum \limits _{h_1} \varepsilon _{p'h_1} S_{h'h_1} + \sum \limits _{p_1} \varepsilon _{p_1h'} S_{p'p_1}\biggr ) \end{aligned}$$

(A4)

with

$$\begin{aligned}&\varepsilon _{pp'}=t_{pp'}+\sum \limits _{h_1}{\bar{v}}_{ph_1p'h_1},\nonumber \\&\quad \varepsilon _{h'h} = t_{h'h}+\sum \limits _{h_1}{\bar{v}}_{h'h_1hh_1},\nonumber \\&\quad \varepsilon _{ph}= t_{ph}+\sum \limits _{h_1}{\bar{v}}_{ph_1hh_1}. \end{aligned}$$

(A5)

$$\begin{aligned}&4\left[ H^{40},K^\dag _{ph}K^\dag _{p'h'}\right] \nonumber \\&\quad = -\sum \limits _{p_1h_1}\left( {\bar{v}}_{p_1h_1p'h} K^0_{ph',h_1p_1} + {\bar{v}}_{p_1h_1ph'} K^0_{p'h,h_1p_1}\right) \nonumber \\&\qquad +\frac{1}{2}\sum \limits _{p_1p_2} {\bar{v}}_{p_2p_1pp_1} K^\dag _{p_2h} K^\dag _{p'h'} \nonumber \\&\qquad +\sum \limits _{p_1p_2h_1h_2} {\bar{v}}_{p_1h_1p_2h_2}\biggl (K^0_{ph,h_1p_1} K^0_{p'h',h_2p_2}\nonumber \\&\qquad + K^0_{p'h',h_1p_1} K^0_{ph,h_2p_2}\biggr )+\frac{1}{2}\sum \limits _{h_1h_2} {\bar{v}}_{hh_1h_2h_1} K^\dag _{ph_2} K^\dag _{p'h'} \nonumber \\&\qquad +2\sum \limits _{p_1p_2h_1h_2} {\bar{v}}_{p_1h_1p_2h_2}\biggl (K^\dag _{ph}K_{h_1p_1}K^0_{p'h',h_2p_2}\nonumber \\&\qquad +K^\dag _{p'h'}K_{h_1p_1}K^0_{ph,h_2p_2}\biggr ) \nonumber \\&\qquad +\sum \limits _{p_1h_1}\biggl ({\bar{v}}_{p'h_1p_1h'} +{\bar{v}}_{p_1h'p'h_1}\biggr ) K^\dag _{ph}K_{h_1p_1}\nonumber \\&\qquad +\sum \limits _{p_1h_1}\biggl ({\bar{v}}_{p_1hph_1} +{\bar{v}}_{ph_1p_1h}\biggr ) K^\dag _{p'h'}K_{h_1p_1} \nonumber \\&\qquad -\sum \limits _{p_1h_1}\biggl ({\bar{v}}_{p'h_1p_1h}+{\bar{v}}_{p_1h_1p'h}\biggr ) K^\dag _{ph'}K_{h_1p_1}\nonumber \\&\qquad -\sum \limits _{p_1h_1}\biggl ({\bar{v}}_{p_1h_1ph'} +{\bar{v}}_{ph'p_1h_1}\biggr ) K^\dag _{p'h}K_{h_1p_1} \end{aligned}$$

(A6)

$$\begin{aligned}&4\left[ H^{22},K^\dag _{ph}K^\dag _{p'h'}\right] \nonumber \\&\quad = -4\sum \limits _{p_1h_1}\biggl ({\bar{v}}_{p_1hh_1p'} K^\dag _{ph'} + {\bar{v}}_{p_1h'h_1p} K^\dag _{p'h}\biggr ) K^\dag _{p_1h_1} \nonumber \\&\qquad +4\sum \limits _{p_1p_2h_1h_2} {\bar{v}}_{p_1h_2h_1p_2} K^\dag _{p_1h_1}\biggl (K^\dag _{ph} K^0_{p'h',h_2p_2}\nonumber \\&\qquad + K^\dag _{p'h'} K^0_{ph,h_2p_2}\biggr ) \nonumber \\&\qquad -\sum \limits _{p_1p_2p_3}\biggl [ ({\bar{v}}_{p_2p_3pp_1}+{\bar{v}}_{p_2p_3p_1p})K^\dag _{p'h'}K^\dag _{p_3h}S_{p_2p_1}\nonumber \\&\qquad + ({\bar{v}}_{p_3p_2p'p_1}+{\bar{v}}_{p_2p_3p_1p'})K^\dag _{ph}K^\dag _{p_2h}S_{p_3p_1} \biggr ] \nonumber \\&\qquad -\sum \limits _{p_1p_2}\biggl [ \biggl ({\bar{v}}_{p_1p_2p'p} + {\bar{v}}_{p_2p_1pp'} \biggr ) K^\dag _{p_1h}K^\dag _{p_2h'}\nonumber \\&\qquad +{\bar{v}}_{p_1p_2p'p_2} K^\dag _{ph}K^\dag _{p_1h'} + {\bar{v}}_{p_1p_2pp_2} K^\dag _{p'h'}K^\dag _{p_1h} \biggr ] \nonumber \\&\qquad -\sum \limits _{h_1h_2h_3}\biggl [ \biggl ({\bar{v}}_{h_3hh_1h_2}+{\bar{v}}_{hh_3h_1h_2}\biggr ) K^\dag _{p'h'}K^\dag _{ph_1}S_{h_2h_3}\nonumber \\&\qquad + \biggl ({\bar{v}}_{h'h_3h_2h_1}+{\bar{v}}_{h_3h'h_1h_2}\biggr ) K^\dag _{ph}K^\dag _{p'h_1}S_{h_2h_3}\biggr ] \nonumber \\&\qquad -\sum \limits _{h_1h_2}\biggl [ \biggl ({\bar{v}}_{h'hh_1h_2} + {\bar{v}}_{hh'h_2h_1}\biggr ) K^\dag _{ph_1}K^\dag _{p'h_2}\nonumber \\&\qquad +{\bar{v}}_{h'h_2h_1h_2} K^\dag _{ph}K^\dag _{p'h_1} +{\bar{v}}_{hh_2h_1h_2} K^\dag _{p'h'}K^\dag _{ph_1}\biggr ] \end{aligned}$$

(A7)

$$\begin{aligned}&2\left[ H^{31},K^\dag _{ph}K^\dag _{p'h'}\right] =K^\dag _{ph} \sum \limits _{p_1h_1} K^\dag _{p_1h_1} \biggl (\sum \limits _{p_2} {\bar{v}}_{p_1p_2h_1p'} K^\dag _{p_2h'}\nonumber \\&\quad - \sum \limits _{h_2} {\bar{v}}_{p_1h'h_1h_2} K^\dag _{p'h_2}\biggr ) \nonumber \\&\quad + K^\dag _{p'h'} \sum \limits _{p_1h_1} K^\dag _{p_1h_1}\biggl (\sum \limits _{p_2} {\bar{v}}_{p_1p_2h_1p} K^\dag _{p_2h}\nonumber \\&\quad - \sum \limits _{h_2} {\bar{v}}_{p_1hh_1h_2} K^\dag _{ph_2}\biggr ) \nonumber \\&\quad + K^\dag _{ph} \sum \limits _{p_1h_1}\biggl (\sum \limits _{p_2} {\bar{v}}_{p_1p'h_1p_2} K^\dag _{p_2h'}\nonumber \\&\quad - \sum \limits _{h_2} {\bar{v}}_{p_1h_2h_1h'} K^\dag _{p'h_2}\biggr ) K_{h_1p_1} \nonumber \\&\quad + K^\dag _{p'h'} \sum \limits _{p_1h_1}\biggl (\sum \limits _{p_2} {\bar{v}}_{p_1ph_1p_2} K^\dag _{p_2h} - \sum \limits _{h_2} {\bar{v}}_{p_1h_2h_1h} K^\dag _{ph_2}) K_{h_1p_1} \nonumber \\&\quad +\biggl (K^\dag _{ph}+K^\dag _{p'h'}\biggl ) \sum \limits _{p_1h_1} K_{h_1p_1}\biggl (\sum \limits _{p_2p_3} {\bar{v}}_{p_1p_2h_1p_3} S_{p_3p_2}\nonumber \\&\quad - \sum \limits _{h_2h_3} {\bar{v}}_{p_1h_3h_1h_2} S_{h_3h_2}\biggr ) \nonumber \\&\quad + \biggl (\sum \limits _{p_2p_3} {\bar{v}}_{p'p_2hp_3} S_{p_3p_2} - \sum \limits _{h_2h_3} {\bar{v}}_{p'h_3hh_2} S_{h_3h_2}\biggr )K^\dag _{ph'}\nonumber \\&\quad +\biggl (\sum \limits _{p_2p_3} {\bar{v}}_{pp_2h'p_3} S_{p_3p_2} - \sum \limits _{h_2h_3} {\bar{v}}_{ph_3h'h_2} S_{h_3h_2}\biggr )K^\dag _{p'h} \nonumber \\&\quad +\sum \limits _{p_1h_1}\biggl (\sum \limits _{p_2p_3} {\bar{v}}_{p_1p_2h_1p_3} S_{p_3p_2}\nonumber \\&\quad - \sum \limits _{h_2h_3} {\bar{v}}_{p_1h_3h_1h_2} S_{h_3h_2}\biggr )\biggl ( K^\dag _{p'h'}K^0_{ph,h_1p_1}+K^\dag _{ph}K^0_{p'h',h_1p_1}\biggr ) \end{aligned}$$

(A8)

with

$$\begin{aligned} K^0_{p_2h_2,h_1p_1}= & {} \left[ K_{h_1p_1},K^\dag _{p_2h_2}\right] \nonumber \\= & {} \delta _{p_1p_2} \delta _{h_1h_2} -\delta _{p_1p_2} S_{h_1h_2} -\delta _{h_1h_2} S_{p_1p_2} \end{aligned}$$

(A9)

Summing the mean values of the different commutators in the ground state $\vert \hbox {Z}\rangle$ for $\langle [H,K^{\dag }_{ph}K^{\dag }_{p'h'}]\rangle =0$ yields a relation between $\langle S_{kk'}S_{ll'}\rangle$ and $\langle S_{nn'}\rangle$.

Appendix B: Calculation of correlation functions in the Lipkin model

All correlation functions can be expressed in terms of z, $\langle J_0\rangle$ and $\langle J^2_0\rangle$. We also have

$$\begin{aligned} n_0= & {} \tfrac{N}{2}-\langle J_0\rangle , \quad n_1 = \tfrac{N}{2}+\langle J_0\rangle \end{aligned}$$

(B1)

From the first Eq. (31), we find

$$\begin{aligned} z\langle J_{+}^{2}\rangle =\tfrac{N}{2}+\langle J_{0}\rangle \end{aligned}$$

(B2)

Multiplying (31) by $J^2_+$, gives

$$\begin{aligned} \langle J^2_+ J_{0}\rangle = -\tfrac{N}{2} \langle J^2_+ \rangle +z\langle J_{+}^{4}\rangle \end{aligned}$$

(B3)

which yields

$$\begin{aligned} 2z\langle J_{+}^{4}\rangle = N \langle J^2_+\rangle +2\langle J^2_+ J_{0}\rangle \end{aligned}$$

(B4)

Multiplying the second Eq. (32) by $J_+$, we can write

$$\begin{aligned} \langle J_+ J_{-} \rangle = z(N-1)\langle J^2_{+}\rangle -z^{2} \langle J_{+}^{4}\rangle \end{aligned}$$

(B5)

which leads to

$$\begin{aligned} z^{2}\langle J_{+}^{4}\rangle= & {} (N-1)z \langle J^2_{+}\rangle -\langle J_+ J_{-}\rangle \end{aligned}$$

(B6)

Then, with (B4) and (B6), we find

$$\begin{aligned} z\langle J^2_+ J_{0}\rangle =(\tfrac{N}{2} -1)z\langle J^2_{+}\rangle - \langle J_+ J_{-}\rangle \end{aligned}$$

(B7)

We multiply (31) from the left by $J_0$,

$$\begin{aligned} J^2_0|Z\rangle= & {} -\tfrac{N}{2}J_0|Z\rangle +z J_0J^2_+|Z\rangle \nonumber \\= & {} -\tfrac{N}{2}J_0|Z\rangle +2z J^2_+|Z\rangle +z J^2_+J_0|Z\rangle \end{aligned}$$

(B8)

Replacing all terms of (B8) by there mean values, we find the Casimir relation

$$\begin{aligned} J_{+}J_{-}|Z\rangle =\left( \tfrac{1}{4}N(N+2)-J^2_0 + J_{0}\right) |Z\rangle \end{aligned}$$

(B9)

In summary, the correlation functions are given by

$$\begin{aligned} 2z\langle J^2_+\rangle= & {} N +2\langle J_0\rangle \nonumber \\ \langle J_{+}J_{-}\rangle= & {} \frac{1}{4}N(N+2)-\langle J^2_0 \rangle + \langle J_0\rangle \nonumber \\ z\langle J^2_+ J_0\rangle= & {} \frac{1}{2}(N-2)z\langle J^2_+\rangle -\langle J_{+}J_{-}\rangle \nonumber \\ z\langle J_+J_-J_0\rangle= & {} \frac{N}{2}(N+2)z +\frac{1}{2}(N-6)z\langle J_{+}J_{-}\rangle \nonumber \\&~ -(N-4)z\langle J_0\rangle -\langle J^2_+\rangle \end{aligned}$$

(B10)

what allows to solve the equation for $\langle J_0\rangle$.