Physics 880.05: Assignment #3

1. Directly solving for G⁰.
   1. One way to proceed is to substitute for G⁰ and also for the delta function and then to project out a given k. Remember that the non-interacting system is uniform.
   2. If you don't write these down by inspection, you're working too hard. Remember that the delta function is zero in each of those regions.
   3. Try integrating the equation from tau-tau' = -epsilon to +epsilon and then taking epsilon to zero. What survives?
2. The Beachball Diagram.
   1. Actually, I did most of this in class by mistake! So I guess the assignment is to write it up coherently.
   2. Here you need to do some contour integrals. Please come see me if you forget how to do these.
   3. First you need to identify the differences between one and three dimensions in the Feynman rules and in the non-interacting G⁰. Then just carry it out. You can compare the energy per particle rather than the energy density, if you prefer.
3. Vanishing Diagrams at Third Order.
   1. So here I'm just checking that you can do it correctly. I realize this is tedious, but if you label your diagrams clearly it should go quickly.
   2. Even if you want to do this by hand as well, I think it would be worthwhile to try out the notebook. Remember to save the package file with the name "deltasimplify.m" in the save directory as the Mathematica notebook "spinsums1.nb".
   3. Hint: Of the five third-order diagrams, three are zero. The explanation for one of them is similar to the second-order case, but you may need to work harder to explain the other two! (Do not assume they are equal to zero for the same reason!)
4. Three-body forces.
   1. Lowest-order means the smallest number of three-body and two-body vertices, with a least one three-body vertex. Hint: the diagram is analogous to the leading-order two-body diagram (the "bowtie" diagram).
   2. Apply the Feynman rules from the class notes in either momentum or coordinate space. There's not much difference! (So try both!)
5. Evaluation of the Grand Potential from the One-Particle Green's Function.
   • The goal here is not to derive the result (which is derived, more or less, in the class notes) but to verify that it works up to third order. That means identifying the combination of diagrams for the self-energy and for G that contribute at first, second, and third order, and to check that the correct overall factor is reproduced.
   • Note that there is an implicit trace over spin indices in the integral (see the class notes for a more explicit expression.)