Week 45: Many-body perturbation theory

Week 45, November 4-8, 2024

Topics to be covered

1. Thursday:
   1. Continuation of examples from week 45
   2. Many-body perturbation theory, discussion of diagrammatic representation
   3. Diagram rules
   4. Video of lecture at https://youtu.be/4L6Yhl9SIUI
   5. Whiteboard notes at https://github.com/ManyBodyPhysics/FYS4480/blob/master/doc/HandwrittenNotes/2024/NotesNovember7.pdf
2. Friday:
   >
   1. Diagram examples
   2. Discussion of second midterm
   3. Video of lecture at URL
   4. Whiteboard notes at https://github.com/ManyBodyPhysics/FYS4480/blob/master/doc/HandwrittenNotes/2024/NotesNovember8.pdf
3. Second midterm at https://github.com/ManyBodyPhysics/FYS4480/blob/master/doc/Exercises/2024/SecondMidterm.pdf
4. Lecture Material: Whiteboard notes (see above) and Shavitt and Bartlett chapters 5-7

Diagram rules, topological distinct diagrams

Draw all topologically distinct diagrams by linking up particle and hole lines with various interaction vertices. Two diagrams can be made topologically equivalent by deformation of fermion lines under the restriction that the ordering of the vertices is not changed and particle lines and hole lines remain particle and\ hole lines.

For the explicit evaluation of a diagram: Sum freely over all internal indices and label all lines.

Diagram rules: matrix elements

Extract matrix elements for the one-body operators (if present) as \( \langle \mathrm{out} |\hat{f}| \mathrm{in}\rangle \) and for the two-body operator (if \ present) as \( \bra{\mathrm{left\hspace{0.1cm}out, right\hspace{0.1cm}out}}|\hat{v}|\ket{\mathrm{left\hspace{0.1cm}in, right\hspace{0.1cm}in}} \).

Diagram rules: phase factors

Calculate the phase factor: \( (-1)^{\mathrm{holelines} + \mathrm{loops}} \)

Diagram rules: equivalent pairs

Multiply by a factor of \( \frac{1}{2} \) for each equivalent pair of lines (particle lines or hole lines) that begin at the same interaction vertex and end at the same (yet different from the first) interaction vertex.

Diagram rules: energy denominators

For each interval between successive interaction vertices with minimum one single-particle state above the Fermi level with \( n \) hole states and \( m \) particle stat\ es there is a factor !

bt \[ \frac{1}{\sum_i^n\epsilon_i-\sum_a^m\epsilon_a}. \]