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:

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

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

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

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