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I have been working through lecture notes of a condensed matter field theory class, and unfortunately I haven't found references for some of the specialized calculations though the way they're done makes them seem common knowledge in the cond-mat community. I have started this thread because it turns out I have some very basic doubts about the interpretations of advanced and retarded Green's functions which haven't been answered by successive re-readings of the standard textbooks. I will be grateful if someone can answer some of these queries.

1. To begin with, I want to write down the transverse susceptibility through a particle hole propagator in the ladder approximation. Suppose [itex]U[/itex] denotes the intra-site (or intra-orbital, depending on the context -- I am using a Hubbard model like system) energy term, then the transverse susceptibility [itex]\chi^{-+}[/itex] is given in the Fourier domain as

[tex]\chi^{-+}(Q,\Omega) = \frac{\chi_{0}(Q,\Omega)}{1-U\chi_{0}(Q,\Omega)}[/tex]

where

[tex]\chi_{0}(Q,\Omega)=i\sum_{k_1}\int_{-\infty}^{\infty}\frac{d\omega_1}{2\pi}G_{\uparrow}^{0}(k_1,\omega_1)G_{\downarrow}^{0}(k_1-Q,\omega_1-\Omega)[/tex]

is just i times the bare particle hole propagator (w/o self energy corrections) without any interaction lines (the first term in the ladder).

2. Now, the exact forms of the Green's functions will depend on the band structure of the system being considered. In the most general case, we can write

[tex]G_{\uparrow}^{0}(k_1,\omega_1) = \frac{\theta(k_1-k_F^{\uparrow})}{\omega_1-E_{k_1}^{\uparrow}+i\eta} + \frac{\theta(k_F^{\uparrow}-k_1)}{\omega_1-E_{k_1}^{\uparrow}-i\eta}[/tex]

and

[tex]G_{\downarrow}^{0}(k_1-Q,\omega_1-\Omega) = \frac{\theta(|k_1-Q|-k_F^{\downarrow})}{\omega_1-\Omega-E_{k_1-Q}^{\downarrow}+i\eta} + \frac{\theta(k_F^{\downarrow}-|k_1-Q|)}{\omega_1-\Omega-E_{k_1}^{\downarrow}-i\eta}[/tex]

Here [itex]E_{k}^{\uparrow,\downarrow}[/itex] denote the band energies for up-spin and down-spin. I consider retarded and advanced contributions from both the up-spin and down-spin bands.

But I have a problem with theinterpretation of the individual terms, especially in the down-spin Green's function in the form that it has been written. The variables [itex]k_1, \omega_1[/itex] correspond to internal lines, and are integrated over.With respect to [itex]\omega_1[/itex], the first term in [itex]G_{\downarrow}^{0}(k_1-Q,\omega_1-\Omega)[/itex] is retarded, whereas with respect to [itex]\Omega[/itex] it is advanced.

Would you call the first term advanced or retarded? What's the convention here?

3. Substituting the full form of the Green's functions into the expression for [itex]\chi^0[/itex] we find that only terms (1x4 + 2x3) contribute as the contour integral (over [itex]\omega_1[/itex]) of a product of two purely retarded or purely advanced terms (wrt [itex]\omega_1[/itex]). So I am left with

[tex]\chi_{0}(Q,\Omega) = \sum_{k_1}\left[\frac{\theta(k_1-k_F^{\uparrow})\theta(k_F^{\downarrow}-|k_1-Q|)}{E_{k_1}^{\uparrow}-E_{k_1-Q}^{\downarrow}-\omega-i\eta} + \frac{\theta(|k_1-Q|-k_{F}^{\downarrow})\theta(k_{F}^{\uparrow}-k_1)}{E_{k_1-Q}^{\downarrow}-E_{k_1}^{\uparrow}+\omega-i\eta}\right][/tex]

3. Suppose I consider asaturated ferromagnet, i.e. one in which he up-spin band is partially below the Fermi level, but the down-spin band is completely above the Fermi level. At T = 0 K, all states below the Fermi level will be filled with particles, so, the down spin electron density is zero. Incidentally now, there is no real value of [itex]k_{F}^{\downarrow}[/itex] as the down spin band lies completely above the Fermi Level, and there is no intersection with it.

For such a system, the down spin Green's function should only get a contribution from the hole states, as there can't be any particle (electron) states which are down spin. So, which term would you pick? There is some ambiguity with respect to which one is an electronic contribution and which one is a hole -- there is after all no Fermi wave vector for the down-spin band.

In addition to answers and explanations for the questions I've posed, I would appreciate suggestions and also references for such calculations. I haven't found them in Atland & Simons or Fetter & Walecka.

Eagerly awaiting a response! Thanks in advance.

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# Transverse Susceptibility, and doubts about retarded/advanced Green's Functions

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