Phy5670/RPA

Random Phase Approximation for the Collective Modes of Nucleons

Polarization Propagator

To study excited states in meny-fermion systems, the limit of the two-particle (tp) propagator is used

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_{ph} (\alpha, \beta^{-1}; \gamma, \delta^{-1}; t-t') = \lim_{t_{\beta} \rightarrow t^{+}} \lim_{t_{\gamma} \rightarrow t'^{+}} G_{II} (\alpha t, \bar{\delta} t', \bar{\beta} t_{\beta}, \gamma t_{\gamma}) = -\frac{i}{\hbar} \langle \psi_{o}^{N}| T [a_{\bar{\beta}}^{H+}(t) a_{\alpha}^{H}(t) a_{\gamma}^{H+} (t') a_{\bar{\delta}}^{H} (t')] | \psi_{o}^{N} \rangle } (Eq. 1)

where "ph" means "particle-hole pairs". Substituting the explicit form of the Heisenberg operators and inserting a complete set of N-particle state one has

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_{ph} (\alpha, \beta^{-1}; \gamma, \delta^{-1}; t-t') = -\frac{i}{\hbar} \langle \psi_{o}^{N}| T [e^{iHt/\hbar}a_{\bar{\beta}}^{+}e^{-iHt/\hbar} e^{iHt/\hbar}a_{\alpha}e^{-iHt/\hbar} e^{iHt'/\hbar}a_{\gamma}^{+}e^{-iHt'/\hbar} e^{iHt'/\hbar}a_{\bar{\delta}}e^{-iHt'/\hbar}] | \psi_{o}^{N} \rangle }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = -\frac{i}{\hbar} \langle \psi_{o}^{N}| T [e^{iHt/\hbar}a_{\bar{\beta}} a_{\alpha}e^{-iHt/\hbar} e^{iHt'/\hbar}a_{\gamma}^{+} a_{\bar{\delta}}e^{-iHt'/\hbar}] | \psi_{o}^{N} \rangle }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = -\frac{i}{\hbar} \sum_{n}^{} \langle \psi_{o}^{N}| T [e^{iHt/\hbar}a_{\bar{\beta}} a_{\alpha}e^{-iHt/\hbar} | \psi_{n}^{N}\rangle \langle \psi_{n}^{N} | e^{iHt'/\hbar}a_{\gamma}^{+} a_{\bar{\delta}}e^{-iHt'/\hbar}] | \psi_{o}^{N} \rangle }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = -\frac{i}{\hbar} \langle \psi_{o}^{N}| T [e^{iHt/\hbar}a_{\bar{\beta}} a_{\alpha}e^{-iHt/\hbar} | \psi_{o}^{N}\rangle \langle \psi_{o}^{N} | e^{iHt'/\hbar}a_{\gamma}^{+} a_{\bar{\delta}}e^{-iHt'/\hbar}] | \psi_{o}^{N} \rangle -\frac{i}{\hbar} \sum_{n \neq 0}^{} \langle \psi_{o}^{N}| T [e^{iHt/\hbar}a_{\bar{\beta}} a_{\alpha}e^{-iHt/\hbar} | \psi_{n}^{N}\rangle \langle \psi_{n}^{N} | e^{iHt'/\hbar}a_{\gamma}^{+} a_{\bar{\delta}}e^{-iHt'/\hbar}] | \psi_{o}^{N} \rangle}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = -\frac{i}{\hbar} \langle \psi_{o}^{N} | a_{\bar{\beta}}^{+} a_{\alpha} | \psi_{o}^{N} \rangle \langle \psi_{o}^{N} | a_{\gamma}^{+} a_{\bar{\delta}} | \psi_{o}^{N} \rangle - \frac{i}{\hbar} [ \sum_{n \neq 0}^{} \theta (t-t') e^{i(E_{o}^{N} - E_{n}^{N})(t-t')/\hbar} \langle \psi_{o}^{N} | a_{\bar{\beta}}^{+} a_{\alpha} | \psi_{n}^{N} \rangle \langle \psi_{n}^{N} | a_{\gamma}^{+} a_{\bar{\delta}} | \psi_{o}^{N} \rangle + \sum_{n \neq 0}^{} \theta (t'-t) e^{i(E_{o}^{N} - E_{n}^{N})(t'-t)/\hbar} \langle \psi_{o}^{N} | a_{\gamma}^{+} a_{\bar{\delta}} | \psi_{n}^{N} \rangle \langle \psi_{n}^{N} | a_{\bar{\beta}}^{+} a_{\alpha} | \psi_{o}^{N} \rangle ] } (Eq. 2)

where the definition of the time-ordering operator in terms of step functions is used also. The so-called polarization propagator is defined by Eq. (2) which includes the excited states only:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi (\alpha, \beta^{-1}; \gamma, \delta^{-1}; t-t') = G_{ph} (\alpha, \beta^{-1}; \gamma, \delta^{-1}; t-t') + \frac{i}{\hbar} \langle \psi_{o}^{N} | a_{\bar{\beta}}^{+} a_{\alpha} | \psi_{o}^{N} \rangle \langle \psi_{o}^{N} | a_{\gamma}^{+} a_{\bar{\delta}} | \psi_{o}^{N} \rangle }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = - \frac{i}{\hbar} [ \sum_{n \neq 0}^{} \theta (t-t') e^{i(E_{o}^{N} - E_{n}^{N})(t-t')/\hbar} \langle \psi_{o}^{N} | a_{\bar{\beta}}^{+} a_{\alpha} | \psi_{n}^{N} \rangle \langle \psi_{n}^{N} | a_{\gamma}^{+} a_{\bar{\delta}} | \psi_{o}^{N} \rangle + \sum_{n \neq 0}^{} \theta (t'-t) e^{i(E_{o}^{N} - E_{n}^{N})(t'-t)/\hbar} \langle \psi_{o}^{N} | a_{\gamma}^{+} a_{\bar{\delta}} | \psi_{n}^{N} \rangle \langle \psi_{n}^{N} | a_{\bar{\beta}}^{+} a_{\alpha} | \psi_{o}^{N} \rangle ]} (Eq. 3)

By employing the integral formulation of the step function, that is,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta (t-t_{o}) = \frac{-1}{2 \pi i} \int \frac{dE'}{E'+i \eta} e^{-iE'(t-t_{o})/\hbar} }

one can transform the polarization propagator, Eq. (3), into its Lehmann representation as following: (Let us calculate the first term in Eq. (3) first and let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau = t-t' } .)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi (\alpha, \beta^{-1}; \gamma, \delta^{-1}; E) = \int \pi (\alpha, \beta^{-1}; \gamma, \delta^{-1}; \tau) e^{iE \tau /\hbar} d \tau }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = - \frac{i}{\hbar} \sum_{n \neq 0}^{} \int \theta (\tau) e^{i(E_{o}^{N} - E_{n}^{N})\tau/\hbar} e^{iE \tau /\hbar} d \tau \langle \psi_{o}^{N} | a_{\bar{\beta}}^{+} a_{\alpha} | \psi_{n}^{N} \rangle \langle \psi_{n}^{N} | a_{\gamma}^{+} a_{\bar{\delta}} | \psi_{o}^{N} \rangle }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{1}{2 \pi \hbar} \sum_{n \neq 0}^{} \int \int \frac{dE'}{E'+i \eta} e^{-iE' \tau /\hbar} e^{i(E_{o}^{N} - E_{n}^{N})\tau/\hbar} e^{iE \tau /\hbar} d \tau \langle \psi_{o}^{N} | a_{\bar{\beta}}^{+} a_{\alpha} | \psi_{n}^{N} \rangle \langle \psi_{n}^{N} | a_{\gamma}^{+} a_{\bar{\delta}} | \psi_{o}^{N} \rangle }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{1}{2 \pi \hbar} \sum_{n \neq 0}^{} \int \int \frac{dE'}{E'+i \eta} e^{-i(E'-E-(E_{o}^{N}-E_{n}^{N})) \tau/\hbar} d \tau \langle \psi_{o}^{N} | a_{\bar{\beta}}^{+} a_{\alpha} | \psi_{n}^{N} \rangle \langle \psi_{n}^{N} | a_{\gamma}^{+} a_{\bar{\delta}} | \psi_{o}^{N} \rangle }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{1}{2 \pi \hbar} \sum_{n \neq 0}^{} \int \frac{dE'}{E'+i \eta} 2 \pi \hbar \delta(E'-E-(E_{o}^{N}-E_{n}^{N})) d \tau \langle \psi_{o}^{N} | a_{\bar{\beta}}^{+} a_{\alpha} | \psi_{n}^{N} \rangle \langle \psi_{n}^{N} | a_{\gamma}^{+} a_{\bar{\delta}} | \psi_{o}^{N} \rangle }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \sum_{n \neq 0}^{} \frac{\langle \psi_{o}^{N} | a_{\bar{\beta}}^{+} a_{\alpha} | \psi_{n}^{N} \rangle \langle \psi_{n}^{N} | a_{\gamma}^{+} a_{\bar{\delta}} | \psi_{o}^{N} \rangle}{E+(E_{o}^{N}-E_{n}^{N})+i \eta} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \sum_{n \neq 0}^{} \frac{\langle \psi_{o}^{N} | a_{\bar{\beta}}^{+} a_{\alpha} | \psi_{n}^{N} \rangle \langle \psi_{n}^{N} | a_{\gamma}^{+} a_{\bar{\delta}} | \psi_{o}^{N} \rangle}{E-(E_{n}^{N}-E_{0}^{N})+i \eta} }

Similarly, the second term in Eq. (3) cab be Fourier transformed into this form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - \sum_{n \neq 0}^{} \frac{\langle \psi_{o}^{N} | a_{\gamma}^{+} a_{\bar{\delta}} | \psi_{n}^{N} \rangle \langle \psi_{n}^{N} | a_{\bar{\beta}}^{+} a_{\alpha} | \psi_{o}^{N} \rangle}{E+(E_{n}^{N}-E_{0}^{N})-i \eta} }

Hence we obtain the polarization propagator in Lehmann representation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi (\alpha, \beta^{-1}; \gamma, \delta^{-1}; E) = \sum_{n \neq 0}^{} \frac{\langle \psi_{o}^{N} | a_{\bar{\beta}}^{+} a_{\alpha} | \psi_{n}^{N} \rangle \langle \psi_{n}^{N} | a_{\gamma}^{+} a_{\bar{\delta}} | \psi_{o}^{N} \rangle}{E-(E_{n}^{N}-E_{0}^{N})+i \eta} - \sum_{n \neq 0}^{} \frac{\langle \psi_{o}^{N} | a_{\gamma}^{+} a_{\bar{\delta}} | \psi_{n}^{N} \rangle \langle \psi_{n}^{N} | a_{\bar{\beta}}^{+} a_{\alpha} | \psi_{o}^{N} \rangle}{E+(E_{n}^{N}-E_{0}^{N})-i \eta} } (Eq. 4)

The polarization propagator incorporates the energy of excited states of N-particle system in its denominator, whereas its numerator contains the transition amplitudes connecting the ground state with those excited states.

Random Phase Approximation

First, let us consider the non-interacting limit of the polarization propagator, which can be obtained from Eq. (3) by replacing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H } by the non-interacting Hamiltonian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{o} } and replacing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi_{o}^{N} \rangle } by the non-interacting ground state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\phi_{o}^{N} \rangle } ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi^{(o)} (\alpha, \beta^{-1}; \gamma, \delta^{-1}; t-t') = G_{ph}^{(o)} (\alpha, \beta^{-1}; \gamma, \delta^{-1}; t-t') + \frac{i}{\hbar} \langle \phi_{o}^{N} | a_{\bar{\beta}}^{+} a_{\alpha} | \phi_{o}^{N} \rangle \langle \phi_{o}^{N} | a_{\gamma}^{+} a_{\bar{\delta}} | \phi_{o}^{N} \rangle }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = - \frac{i}{\hbar} [ \theta (t-t') \theta (\alpha - F) \theta (F - \beta) \delta_{\alpha, \gamma} \delta_{\beta, \delta} e^{-i(E_{\alpha} - E_{\beta})(t-t')/\hbar} + \theta (t'-t) \theta (F - \alpha) \theta (\beta - F) \delta_{\alpha, \gamma} \delta_{\beta, \delta} e^{-i(E_{\beta} - E_{\alpha})(t'-t)/\hbar}] } (Eq. 5)

The first term corresponds to the independent propagation of a particle with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha (\gamma) } from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t' } to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t } , and a hole with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta (\delta) } from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t } to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t' } . The second term exchanges the role of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t' } , as well as that of the quantum numbers, and corresponds to the independent hole-particle (hp) propagation. See Fig. 1. Using the integral representation of the step functions in Eq. (5), the Fourier transform of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi^{o} } is obtained as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi^{(o)} (\alpha, \beta^{-1}; \gamma, \delta^{-1}; E) = \delta_{\alpha, \gamma} \delta_{\beta, \delta} [\frac{\theta(\alpha - F) \theta(F - \beta)}{E-(E_{\alpha}-E_{\beta})+i \eta} - \frac{\theta(F - \alpha) \theta(\beta - F)}{E+(E_{\beta}-E_{\alpha})-i \eta}] } (Eq. 6)

The poles of the non-interacting polarization propagator correspond to ph states obtained by removing a particle from an occupied level and placing it in an empty level of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{o} } , as shown in Fig. 2. The numerator shows that the ph pair added to the non-interaction ground state propagates without changing its quantum numbers.

Higher-order contributions are terms that represent correlations between the initial and final ph states, and terms that dress the non-interacting single-particle (sp) propagators. Both types of corrections are included in the first-order contribution given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi^{(1)} (\alpha, \beta^{-1}; \gamma, \delta^{-1}; t-t') = (-\frac{i}{\hbar})^2 \int_{-\infty}^{\infty} dt_{1} \frac{1}{4} \sum_{\kappa \lambda \mu \nu}^{} \langle \kappa \lambda |V| \mu \nu \rangle \langle \phi_{o}^{N}| T [ a_{\kappa}^{+}(t_{1}) a_{\lambda}^{+}(t_{1}) a_{\nu}(t_{1}) a_{\mu}(t_{1}) a_{\bar{\beta}}^{+}(t) a_{\alpha}(t) a_{\gamma}^{+}(t') a_{\bar{\delta}}(t')] | \phi_{o}^{N} \rangle } (Eq. 7)

in the time formulation. A scheme to calculate excited states using non-interacting sp propagators can be obtained by keeping only those terms in Eq. (7) which link the propagators. Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V } is symmetric, one has

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi^{(1)} (\alpha, \beta^{-1}; \gamma, \delta^{-1}; t-t') \rightarrow (i \hbar)^2 \int_{-\infty}^{\infty} dt_{1} \sum_{\kappa \lambda \mu \nu}^{} \langle \kappa \lambda |V| \mu \nu \rangle G^{(o)}(\alpha, \kappa;t-t_{1}) G^{(o)}(\mu, \bar{\beta}; t_{1}-t) G^{(o)}(\nu, \gamma; t_{1}-t') G^{(o)}(\bar{\delta}, \lambda; t'-t_{1}) } (Eq. 8)

as shown in Fig. 3. The physical meaning of this expression is that a ph pair is added at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t' } with quantum number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma \delta^{-1} } , and then propagates to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_{1} } . At this momentan interaction changes the prapagation to a ph pair with quantum number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \beta^{-1} } which ends at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t } when the pair is removed. The Fourier transform of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi^{(1)} } is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi^{(1)} (\alpha, \beta^{-1}; \gamma, \delta^{-1}; E) = \pi^{(o)} (\alpha, \beta^{-1}; E) \langle \alpha \beta^{-1} | V_{ph} | \gamma \delta^{-1} \rangle \pi^{(o)} (\gamma, \delta^{-1}; E) } (Eq. 9)

which is shown in Fig. 4. Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi^{0}(\alpha, \beta^{-1};E) } is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi^{(1)} (\alpha, \beta^{-1}; \gamma, \delta^{-1}; E) = \int \frac{dE'}{2 \pi i} G^{(o)}(\alpha, \gamma; E+E')G^{(o)}(\bar{\gamma}, \bar{\beta}; E') = \delta_{\alpha, \gamma} \delta_{\beta, \delta} \pi^{(o)}(\alpha, \beta^{-1}; E) }

Equation (9) can be cast into the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi^{(o)} (\alpha, \beta^{-1}; E) \langle \alpha \beta^{-1} | V_{ph} | \gamma \delta^{-1} \rangle \pi^{(o)} (\gamma, \delta^{-1}; E) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \pi^{(o)} (\alpha, \beta^{-1}; E) \sum_{\epsilon \theta} \langle \alpha \beta^{-1} | V_{ph} | \epsilon \theta^{-1} \rangle \pi^{(o)} (\epsilon, \theta^{-1}; \gamma, \delta^{-1}; E) }

$\rightarrow \pi ^{(o)}(\alpha ,\beta ^{-1};E)\sum _{\epsilon \theta }\langle \alpha \beta ^{-1}|V_{ph}|\epsilon \theta ^{-1}\rangle \pi ^{RPA}(\epsilon ,\theta ^{-1};\gamma ,\delta ^{-1};E)$

When this expression is included to the non-interacting polarization propagator, one obtains the corresponding approximation to the exact polarization propagator, in which the ph interaction $V_{ph}$ is iterated to all orders with the non-interacting polarization propagator. Schematically,

$\pi ^{RPA}=\pi ^{(o)}+\pi ^{(o)}V_{ph}\pi ^{(o)}+\pi ^{(o)}V_{ph}\pi ^{(o)}V_{ph}\pi ^{(o)}+\pi ^{(o)}V_{ph}\pi ^{(o)}V_{ph}\pi ^{(o)}V_{ph}\pi ^{(o)}+...$

$=\pi ^{(o)}+\pi ^{(o)}V_{ph}[\pi ^{(o)}+\pi ^{(o)}V_{ph}\pi ^{(o)}+\pi ^{(o)}V_{ph}\pi ^{(o)}V_{ph}\pi ^{(o)}+...]$

$=\pi ^{(o)}+\pi ^{(o)}V_{ph}\pi ^{RPA}$

Explicitly,

$\pi ^{RPA}(\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};E)=\pi ^{(o)}(\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};E)+\pi ^{(o)}(\alpha ,\beta ^{-1};E)\sum _{\epsilon \theta }\langle \alpha \beta ^{-1}|V_{ph}|\epsilon \theta ^{-1}\rangle \pi ^{RPA}(\epsilon ,\theta ^{-1};\gamma ,\delta ^{-1};E)$ (Eq. 10)

or

$\pi ^{RPA}(\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};E)=\pi ^{(o)}(\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};E)+\sum _{\epsilon \theta \zeta \xi }\pi ^{(o)}(\alpha ,\beta ^{-1};\zeta ,\xi ^{-1};E)\langle \alpha \beta ^{-1}|V_{ph}|\epsilon \theta ^{-1}\rangle \pi ^{RPA}(\epsilon ,\theta ^{-1};\gamma ,\delta ^{-1};E)$ (Eq. 11)

As shown in Fig. 5, successively replacing $\pi ^{RPA}$ generates higher order terms, the bubbles or rings emerge when the direct part of the interaction is used to connect the unperturbed ph propagation. Each bubble represents the sum of a forward- and backward-going term corresponding to the first or second term in Eq. (6), respectively. The existence of both terms implies interplay between the components when Eq. (11) is solved. It generates the possibility of intermediate states in which many ph states are present at the same time. It has been argued that these Pauli exchange terms add up with random phases and might therefore be rather small, hence the name random phase approximation (RPA).

RPA in Finite Systems and the Schematic Model

To solve Eq. (10) we assume that $\pi ^{RPA}$ also has a Lehmann representation, like the non-interacting and the exact propagator. For RPA, define

$X_{\alpha \beta }^{n}=\langle \psi _{n}^{N}|a_{\alpha }^{+}a_{\bar {\beta }}|\psi _{n}^{N}\rangle ^{*}$

$Y_{\alpha \beta }^{n}=\langle \psi _{n}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\psi _{n}^{N}\rangle ^{*}=-X_{{\bar {\beta }}{\bar {\alpha }}}^{n}$

and

$\epsilon _{n}^{\pi }=E_{n}^{N}-E_{o}^{N}$

Then $\pi ^{RPA}$ in Lehmann representation is

$\pi ^{RPA}(\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};E)=\sum _{n\neq 0}^{}{\frac {X_{\alpha \beta }^{n}X_{\gamma \delta }^{n*}}{E-\epsilon _{n}^{\pi }+i\eta }}-\sum _{n\neq 0}^{}{\frac {Y_{\alpha \beta }^{n*}Y_{\gamma \delta }^{n}}{E+\epsilon _{n}^{\pi }-i\eta }}$ (Eq. 12)

In case of a finite system, we consider bound states and the summation in Eq. (12) involves some discrete states. Taking the limit

$\lim _{E\rightarrow \epsilon _{n}^{\pi }}(E-\epsilon _{n}^{\pi })[\pi ^{RPA}=\pi ^{(o)}+\pi ^{(o)}V_{ph}\pi ^{RPA}]$

one has

$X_{\alpha \beta }^{n}X_{\gamma \delta }^{n*}=0+\pi ^{(o)}(\alpha ,\beta ^{-1};\epsilon _{n}^{\pi })\sum _{\epsilon \theta }^{}\langle \alpha \beta ^{-1}|V_{ph}|\epsilon \theta ^{-1}\rangle X_{\epsilon \theta }^{n}X_{\gamma \delta }^{n*}$

$X_{\alpha \beta }^{n}=\pi ^{(o)}(\alpha ,\beta ^{-1};\epsilon _{n}^{\pi })\sum _{\epsilon \theta }^{}\langle \alpha \beta ^{-1}|V_{ph}|\epsilon \theta ^{-1}\rangle X_{\epsilon \theta }^{n}$ (Eq. 13)

Using Eq. (6) the equation above for $\alpha >F>\beta$ can be written as

$[\epsilon _{n}^{\pi }-(\epsilon _{\alpha }-\epsilon _{\beta })]X_{\alpha \beta }^{n}=\sum _{\epsilon \theta }^{}\langle \alpha \beta ^{-1}|V_{ph}|\epsilon \theta ^{-1}\rangle X_{\epsilon \theta }^{n}$ (Eq. 14)

whereas for $\alpha <F<\beta$ one has

$[\epsilon _{n}^{\pi }+(\epsilon _{\beta }-\epsilon _{\alpha })]X_{\alpha \beta }^{n}=-\sum _{\epsilon \theta }^{}\langle \alpha \beta ^{-1}|V_{ph}|\epsilon \theta ^{-1}\rangle X_{\epsilon \theta }^{n}$ (Eq. 15)

Now, we further assume that the ph interaction is separable, that is,

$\langle \alpha \beta ^{-1}|V_{ph}|\epsilon \theta ^{-1}\rangle =\lambda Q_{\alpha \beta }Q_{\epsilon \theta }^{*}$ (Eq. 16)

where $\lambda$ is the coupling constant. Substituting this into Eq. (14) yields

$[\epsilon _{n}^{\pi }-(\epsilon _{\alpha }-\epsilon _{\beta })]X_{\alpha \beta }^{n}=\lambda Q_{\alpha \beta }\sum _{\epsilon \theta }^{}Q_{\epsilon \theta }^{*}X_{\epsilon \theta }^{n}$

for $\alpha >F>\beta$. This implies that

$X_{\alpha \beta }^{n}=-C{\frac {Q_{\alpha \beta }}{\epsilon _{n}^{\pi }-(\epsilon _{\alpha }-\epsilon _{\beta })}}$ (Eq. 19)

Inserting Eqs. (17) and (19) into Eq. (18) leads to

$X_{\alpha \beta }^{n}=C{\frac {Q_{\alpha \beta }}{\epsilon _{n}^{\pi }-(\epsilon _{\alpha }-\epsilon _{\beta })}}$

where

$C=\lambda \sum _{\epsilon \theta }^{}Q_{\epsilon \theta }^{*}X_{\epsilon \theta }^{n}$ (Eq. 18)

Similarly, for $\alpha <F<\beta$ we have

$C=\lambda \sum _{\epsilon >F>\theta }^{}{\frac {C|Q_{\epsilon \theta }|^{2}}{\epsilon _{n}^{\pi }-(\epsilon _{\epsilon }-\epsilon _{\theta })}}-\lambda \sum _{\epsilon <F<\theta }^{}{\frac {C|Q_{\epsilon \theta }|^{2}}{\epsilon _{n}^{\pi }-(\epsilon _{\epsilon }-\epsilon _{\theta })}}$

${\frac {1}{\lambda }}=\sum _{\epsilon >F>\theta }^{}{\frac {|Q_{\epsilon \theta }|^{2}}{\epsilon _{n}^{\pi }-(\epsilon _{\epsilon }-\epsilon _{\theta })}}-\sum _{\epsilon <F<\theta }^{}{\frac {|Q_{\epsilon \theta }|^{2}}{\epsilon _{n}^{\pi }-(\epsilon _{\epsilon }-\epsilon _{\theta })}}$ (Eq. 20)

This is an eigenvalue equation which can be solved schematically. For simplicity, let us ignore the second term (hp) and consider the first term (ph) only, that is,

${\frac {1}{\lambda }}=\sum _{\epsilon >F>\theta }^{}{\frac {|Q_{\epsilon \theta }|^{2}}{\epsilon _{n}^{\pi }-(\epsilon _{\epsilon }-\epsilon _{\theta })}}$ (Eq. 21)

As shown in Fig. 7, the curve is the right side of Eq. (21) as a function of $\epsilon _{n}^{\pi }$, whose poles are indicated by the vertical asymptotes. For repulsive interaction ($\lambda >0$), the energies are bound between the poles, except for the largest one. If the interaction is very small, all the energies are close to those for non-interacting case, as expected. On the other hand, if the interaction is very large all the energies deviate from the non-interacting ones. In particular, the highest shifts upward a lot. The states with energies bound between the poles are close to the eigenstates for the non-interacting Hamiltonian with tiny admixture with each other. However, the state of the highest energy is a coherent superposition of all the non-interacting eigenstates, in which each non-interacting eigenstate contributes almost the same amount. Similar situation happens for attractive interaction ($\lambda <0$). In this case, when the coupling constant is very large, there is a state which mixes all the non-interacting eigenstates almost equally and its energy is lowered greatly. Such coherent states correspond to the collective modes of the system. Examples are the first octupole states appearing in O-16, Ca-40, and Pb-208 at low excitation energies and the giant dipole resonance.

Reference

[1] W. Dickhoff and D. Van Neck, Many-Body Theory Exposed, 2nd ed, World Scientific

[2] P. Ring and P. Schuck, The Nuclear Many-Body Problem, Study ed, Springer