Phy5670/RPA

Polarization Propagator

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

$G_{ph}(\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};t-t')=\lim _{t_{\beta }\rightarrow t^{+}}\lim _{t_{\gamma }\rightarrow t'^{+}}G_{II}(\alpha t,{\bar {\delta }}t',{\bar {\beta }}t_{\beta },\gamma t_{\gamma })=-{\frac {i}{\hbar }}\langle \psi _{o}^{N}|T[a_{\bar {\beta }}^{H+}(t)a_{\alpha }^{H}(t)a_{\gamma }^{H+}(t')a_{\bar {\delta }}^{H}(t')]|\psi _{o}^{N}\rangle$ (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

$G_{ph}(\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};t-t')=-{\frac {i}{\hbar }}\langle \psi _{o}^{N}|T[e^{iHt/\hbar }a_{\bar {\beta }}^{+}e^{-iHt/\hbar }e^{iHt/\hbar }a_{\alpha }e^{-iHt/\hbar }e^{iHt'/\hbar }a_{\gamma }^{+}e^{-iHt'/\hbar }e^{iHt'/\hbar }a_{\bar {\delta }}e^{-iHt'/\hbar }]|\psi _{o}^{N}\rangle$

$=-{\frac {i}{\hbar }}\langle \psi _{o}^{N}|T[e^{iHt/\hbar }a_{\bar {\beta }}a_{\alpha }e^{-iHt/\hbar }e^{iHt'/\hbar }a_{\gamma }^{+}a_{\bar {\delta }}e^{-iHt'/\hbar }]|\psi _{o}^{N}\rangle$

$=-{\frac {i}{\hbar }}\sum _{n}^{}\langle \psi _{o}^{N}|T[e^{iHt/\hbar }a_{\bar {\beta }}a_{\alpha }e^{-iHt/\hbar }|\psi _{n}^{N}\rangle \langle \psi _{n}^{N}|e^{iHt'/\hbar }a_{\gamma }^{+}a_{\bar {\delta }}e^{-iHt'/\hbar }]|\psi _{o}^{N}\rangle$

$=-{\frac {i}{\hbar }}\langle \psi _{o}^{N}|T[e^{iHt/\hbar }a_{\bar {\beta }}a_{\alpha }e^{-iHt/\hbar }|\psi _{o}^{N}\rangle \langle \psi _{o}^{N}|e^{iHt'/\hbar }a_{\gamma }^{+}a_{\bar {\delta }}e^{-iHt'/\hbar }]|\psi _{o}^{N}\rangle -{\frac {i}{\hbar }}\sum _{n\neq 0}^{}\langle \psi _{o}^{N}|T[e^{iHt/\hbar }a_{\bar {\beta }}a_{\alpha }e^{-iHt/\hbar }|\psi _{n}^{N}\rangle \langle \psi _{n}^{N}|e^{iHt'/\hbar }a_{\gamma }^{+}a_{\bar {\delta }}e^{-iHt'/\hbar }]|\psi _{o}^{N}\rangle$

$=-{\frac {i}{\hbar }}\langle \psi _{o}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\psi _{o}^{N}\rangle \langle \psi _{o}^{N}|a_{\gamma }^{+}a_{\bar {\delta }}|\psi _{o}^{N}\rangle -{\frac {i}{\hbar }}[\sum _{n\neq 0}^{}\theta (t-t')e^{i(E_{o}^{N}-E_{n}^{N})(t-t')/\hbar }\langle \psi _{o}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\psi _{n}^{N}\rangle \langle \psi _{n}^{N}|a_{\gamma }^{+}a_{\bar {\delta }}|\psi _{o}^{N}\rangle +\sum _{n\neq 0}^{}\theta (t'-t)e^{i(E_{o}^{N}-E_{n}^{N})(t'-t)/\hbar }\langle \psi _{o}^{N}|a_{\gamma }^{+}a_{\bar {\delta }}|\psi _{n}^{N}\rangle \langle \psi _{n}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\psi _{o}^{N}\rangle ]$ (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:

$\pi (\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};t-t')=G_{ph}(\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};t-t')+{\frac {i}{\hbar }}\langle \psi _{o}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\psi _{o}^{N}\rangle \langle \psi _{o}^{N}|a_{\gamma }^{+}a_{\bar {\delta }}|\psi _{o}^{N}\rangle$

$=-{\frac {i}{\hbar }}[\sum _{n\neq 0}^{}\theta (t-t')e^{i(E_{o}^{N}-E_{n}^{N})(t-t')/\hbar }\langle \psi _{o}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\psi _{n}^{N}\rangle \langle \psi _{n}^{N}|a_{\gamma }^{+}a_{\bar {\delta }}|\psi _{o}^{N}\rangle +\sum _{n\neq 0}^{}\theta (t'-t)e^{i(E_{o}^{N}-E_{n}^{N})(t'-t)/\hbar }\langle \psi _{o}^{N}|a_{\gamma }^{+}a_{\bar {\delta }}|\psi _{n}^{N}\rangle \langle \psi _{n}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\psi _{o}^{N}\rangle ]$ (Eq. 3)

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

$\theta (t-t_{o})={\frac {-1}{2\pi i}}\int {\frac {dE'}{E'+i\eta }}e^{-iE'(t-t_{o})/\hbar }$

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 $\tau =t-t'$.)

$\pi (\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};E)=\int \pi (\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};\tau )e^{iE\tau /\hbar }d\tau$

$=-{\frac {i}{\hbar }}\sum _{n\neq 0}^{}\int \theta (\tau )e^{i(E_{o}^{N}-E_{n}^{N})\tau /\hbar }e^{iE\tau /\hbar }d\tau \langle \psi _{o}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\psi _{n}^{N}\rangle \langle \psi _{n}^{N}|a_{\gamma }^{+}a_{\bar {\delta }}|\psi _{o}^{N}\rangle$

$={\frac {1}{2\pi \hbar }}\sum _{n\neq 0}^{}\int \int {\frac {dE'}{E'+i\eta }}e^{-iE'\tau /\hbar }e^{i(E_{o}^{N}-E_{n}^{N})\tau /\hbar }e^{iE\tau /\hbar }d\tau \langle \psi _{o}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\psi _{n}^{N}\rangle \langle \psi _{n}^{N}|a_{\gamma }^{+}a_{\bar {\delta }}|\psi _{o}^{N}\rangle$

$={\frac {1}{2\pi \hbar }}\sum _{n\neq 0}^{}\int \int {\frac {dE'}{E'+i\eta }}e^{-i(E'-E-(E_{o}^{N}-E_{n}^{N}))\tau /\hbar }d\tau \langle \psi _{o}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\psi _{n}^{N}\rangle \langle \psi _{n}^{N}|a_{\gamma }^{+}a_{\bar {\delta }}|\psi _{o}^{N}\rangle$

$={\frac {1}{2\pi \hbar }}\sum _{n\neq 0}^{}\int {\frac {dE'}{E'+i\eta }}2\pi \hbar \delta (E'-E-(E_{o}^{N}-E_{n}^{N}))d\tau \langle \psi _{o}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\psi _{n}^{N}\rangle \langle \psi _{n}^{N}|a_{\gamma }^{+}a_{\bar {\delta }}|\psi _{o}^{N}\rangle$

$=\sum _{n\neq 0}^{}{\frac {\langle \psi _{o}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\psi _{n}^{N}\rangle \langle \psi _{n}^{N}|a_{\gamma }^{+}a_{\bar {\delta }}|\psi _{o}^{N}\rangle }{E+(E_{o}^{N}-E_{n}^{N})+i\eta }}$

$=\sum _{n\neq 0}^{}{\frac {\langle \psi _{o}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\psi _{n}^{N}\rangle \langle \psi _{n}^{N}|a_{\gamma }^{+}a_{\bar {\delta }}|\psi _{o}^{N}\rangle }{E-(E_{n}^{N}-E_{0}^{N})+i\eta }}$

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

$-\sum _{n\neq 0}^{}{\frac {\langle \psi _{o}^{N}|a_{\gamma }^{+}a_{\bar {\delta }}|\psi _{n}^{N}\rangle \langle \psi _{n}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\psi _{o}^{N}\rangle }{E+(E_{n}^{N}-E_{0}^{N})-i\eta }}$

Hence we obtain the polarization propagator in Lehmann representation

$\pi (\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};E)=\sum _{n\neq 0}^{}{\frac {\langle \psi _{o}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\psi _{n}^{N}\rangle \langle \psi _{n}^{N}|a_{\gamma }^{+}a_{\bar {\delta }}|\psi _{o}^{N}\rangle }{E-(E_{n}^{N}-E_{0}^{N})+i\eta }}-\sum _{n\neq 0}^{}{\frac {\langle \psi _{o}^{N}|a_{\gamma }^{+}a_{\bar {\delta }}|\psi _{n}^{N}\rangle \langle \psi _{n}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\psi _{o}^{N}\rangle }{E+(E_{n}^{N}-E_{0}^{N})-i\eta }}$ (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 $H$ by the non-interacting Hamiltonian $H_{o}$ and replacing $|\psi _{o}^{N}\rangle$ by the non-interacting ground state $|\phi _{o}^{N}\rangle$,

$\pi ^{(o)}(\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};t-t')=G_{ph}^{(o)}(\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};t-t')+{\frac {i}{\hbar }}\langle \phi _{o}^{N}|a_{\bar {\beta }}^{+}a_{\alpha }|\phi _{o}^{N}\rangle \langle \phi _{o}^{N}|a_{\gamma }^{+}a_{\bar {\delta }}|\phi _{o}^{N}\rangle$

$=-{\frac {i}{\hbar }}[\theta (t-t')\theta (\alpha -F)\theta (F-\beta )\delta _{\alpha ,\gamma }\delta _{\beta ,\delta }e^{-i(E_{\alpha }-E_{\beta })(t-t')/\hbar }+\theta (t'-t)\theta (F-\alpha )\theta (\beta -F)\delta _{\alpha ,\gamma }\delta _{\beta ,\delta }e^{-i(E_{\beta }-E_{\alpha })(t'-t)/\hbar }]$ (Eq. 5)

The first term corresponds to the independent propagation of a particle with $\alpha (\gamma )$ from $t'$ to $t$, and a hole with $\beta (\delta )$ from $t$ to $t'$. The second term exchanges the role of $t$ and $t'$, 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 $\pi ^{o}$ is obtained as

$\pi ^{(o)}(\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};E)=\delta _{\alpha ,\gamma }\delta _{\beta ,\delta }[{\frac {\theta (\alpha -F)\theta (F-\beta )}{E-(E_{\alpha }-E_{\beta })+i\eta }}-{\frac {\theta (F-\alpha )\theta (\beta -F)}{E+(E_{\beta }-E_{\alpha })-i\eta }}]$ (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 $H_{o}$, 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

$\pi ^{(1)}(\alpha ,\beta ^{-1};\gamma ,\delta ^{-1};t-t')=(-{\frac {i}{\hbar }})^{2}\int _{-\infty }^{\infty }dt_{1}{\frac {1}{4}}\sum _{\kappa \lambda \mu \nu }^{}\langle \kappa \lambda |V|\mu \nu \rangle \langle \phi _{o}^{N}|T[a_{\kappa }^{+}(t_{1})a_{\lambda }^{+}(t_{1})a_{\nu }(t_{1})a_{\mu }(t_{1})a_{\bar {\beta }}^{+}(t)a_{\alpha }(t)a_{\gamma }^{+}(t')a_{\bar {\delta }}(t')]|\phi _{o}^{N}\rangle$ (Eq. 7)

in the time formulation.