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QFT Propagator Poles

Complex p0p_0 plane for evaluating a Matsubara frequency sum over a product of two propagators, one of them shifted by the external frequency q0q_0. The Matsubara frequencies iωni\omega_n lie on the imaginary axis. The two zigzag branch cuts at Im⁡p0=0\operatorname{Im} p_0 = 0 and Im⁡p0=−q0\operatorname{Im} p_0 = -q_0 split the plane into regions (I), (II) and (III), and the dots α1,21,2\alpha_{1,2}^{1,2} mark the poles of the two propagators in each region. Closing the contour region by region turns the frequency sum into residues at these poles plus integrals along the cuts.


QFT Propagator Poles

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Code

qft-propagator-poles.typ (54 lines) 
#import "@preview/cetz:0.5.3": canvas, decorations, draw
#import draw: circle, content, line

#set page(width: auto, height: auto, margin: 8pt, fill: none)
#set text(size: 12pt)

#canvas({
  let (xrange, yrange, cut-x, q0) = (6, 4, 2, 3)
  let zigzag-style = (amplitude: 0.1, segment-length: 0.2, stroke: 0.75pt)

  // Real axis turning into a branch cut at cut-x, and a second cut shifted down by q0.
  line((-1, 0), (cut-x, 0), stroke: 0.75pt)
  decorations.zigzag(line((cut-x, 0), (xrange, 0)), ..zigzag-style, name: "x-axis-right")
  content((rel: (-0.3, 0.3), to: "x-axis-right.end"), $"Re"(p_0)$)
  decorations.zigzag(line((cut-x, -q0), (xrange, -q0)), ..zigzag-style)
  for y-pos in (0, -q0) {
    content((xrange, y-pos), [branch cut], anchor: "north-west", padding: (left: 6pt, top: 4pt))
  }

  line(
    (0, -yrange - 1),
    (0, 2),
    mark: (end: "stealth", fill: black),
    stroke: 0.75pt,
    name: "y-axis",
  )
  content((rel: (0.8, -0.2), to: "y-axis.end"), $"Im"(p_0)$)

  content((cut-x, -q0 / 2), [#math.underbrace(box(width: 7.5em))], name: "q0-brace", angle: -90deg)
  content((rel: (-0.5, 0), to: "q0-brace"), $q_0$)

  // Matsubara frequencies
  for n in range(-yrange, 2).filter(n => n != 0) {
    circle((0, n), radius: 0.04, fill: black, name: "matsubara")
    content("matsubara", $i omega_#n$, anchor: "west", padding: 0.2)
  }
  content((-0.2, -2.5), align(right)[Matsubara\ frequencies], anchor: "east")
  circle((0, 0), radius: 0.03, fill: black)
  content((0.2, 0.1), $0$)

  // Poles one unit above and below each cut
  for (y-pos, superscript) in ((1, 1), (-1, 1), (1 - q0, 2), (-1 - q0, 2)) {
    for (x-pos, subscript) in ((cut-x + 1, 2), (cut-x + 3, 1)) {
      circle((x-pos, y-pos), radius: 0.06, fill: black)
      content((x-pos, y-pos), $alpha_#subscript^#superscript$, anchor: "south", padding: 0.1)
    }
  }
  content((cut-x + 3.3, 1), [propagator poles], anchor: "west")

  // Region labels
  for (y-pos, label) in ((1.5, [(I)]), (-q0 / 2, [(II)]), (-q0 - 1.5, [(III)])) {
    content((cut-x + 2, y-pos), text(fill: rgb("#00008B"), label))
  }
})