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When we calculate the Cooper instability with the perturbation theory, we will obtain an effective interaction constant $g^{\prime}=\frac{g}{1-g\nu\ln(\frac{w_D}{T})}$, where $g>0$. The Cooper instability appears in the condition that $T=w_D e^{-\frac{1}{g\nu}}$. When we plot the $g^{\prime}$ function, we will get a figure like this: enter image description here.

From this figure we can know that on the right of the critical temperature, $g^{\prime}>0$, and on the left of the critical temperature, $g^{\prime}<0$. At the first beginning ,we introduce the attractive interaction which means the interaction is always attractive. However $g^{\prime}$ can change sign which means the interaction can be attractive or repulsive. I'm very confused about this, can someone explain this?

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  • $\begingroup$ pg 79 in these notes (physik.tu-dresden.de/~timm/personal/teaching/thsup_w11/…) has a plot similar to yours. The plot there shows the effective interaction between electrons in jellium that arises from competition between attractive phonon exchange and repulsive photon exchange (Coulomb interaction). I'm not sure how you derived the expressions in your post but could it be that they regard the same problem? $\endgroup$ – wcc Mar 28 at 15:00
  • $\begingroup$ That expression can be found in pg 270 in Condensed Matter Field Theory by Altland& Simons. And I think these are two different problems. Before this expression was obtained, we have assumed that there is only the attractive interaction between electrons or the net interaction is attracitve. $\endgroup$ – Morning Mar 28 at 15:51
  • $\begingroup$ okay, then what makes the interaction effective instead of bare? $\endgroup$ – wcc Mar 28 at 19:23

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