Phy5670/RPA

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.