matsubara-contour-5.typ

#import "@preview/cetz:0.3.4": canvas, draw, decorations
#import draw: line, content, circle, arc, set-style, bezier

#set page(width: auto, height: auto, margin: 8pt)

// Constants
#let x-range = 3.5
#let y-range = 3
#let main-radius = y-range + 0.75
#let y-offset = 0.25
#let dot-radius = 0.05
#let small-dot-radius = 0.03
#let zigzag-start = 0.5

// Styles
#let arrow-style = (mark: (end: "stealth", scale: 0.5, fill: black))
#let dark-blue = blue.darken(20%)
#let marc-style = (symbol: "stealth", fill: dark-blue, scale: 0.5, shorten-to: none)
#let contour-style = (paint: dark-blue, thickness: 0.8pt)
#let zigzag-style = (amplitude: 0.15, segment-length: 0.25)

// Helper function to draw a semicircle with arrows
#let draw-semicircle(center-x, start-angle, end-angle, name) = {
  arc(
    (center-x, 0),
    radius: main-radius,
    start: start-angle,
    stop: end-angle,
    stroke: contour-style,
    mark: (
      end: (
        (pos: 25%, ..marc-style),
        (pos: 50%, ..marc-style),
        (pos: 75%, ..marc-style),
      ),
    ),
    name: name,
    anchor: "origin",
  )
}

// Helper function to draw a vertical line with arrows
#let draw-vertical-line(x, start-y, end-y, name) = {
  line(
    (x, start-y),
    (x, end-y),
    stroke: contour-style,
    mark: (
      end: (
        (pos: 25%, ..marc-style),
        (pos: 50%, ..marc-style),
        (pos: 75%, ..marc-style),
      ),
    ),
    name: name,
  )
}

// Helper function to draw zigzag line
#let draw-zigzag(start, end, segments: 6, amplitude: 0.1, name: none) = {
  let dx = end.at(0) - start.at(0)
  let dy = end.at(1) - start.at(1)
  let length = calc.sqrt(dx * dx + dy * dy)
  let segment-length = length / segments

  let points = ()

  for i in range(segments + 1) {
    let t = i / segments
    let x = start.at(0) + dx * t
    let y = start.at(1) + dy * t

    // Add zigzag effect
    if i > 0 and i < segments {
      if calc.rem(i, 2) == 1 {
        y += amplitude
      } else {
        y -= amplitude
      }
    }

    points.push((x, y))
  }

  // Draw the zigzag line as a series of connected lines
  for i in range(segments) {
    line(
      points.at(i),
      points.at(i + 1),
      stroke: (thickness: 0.8pt),
      name: if name != none and i == 0 { name } else { none },
    )
  }
}

#canvas({
  // Calculate the exact point where the x-axis intersects the right arc
  // For a circle centered at (y-offset, 0) with radius main-radius,
  // the x-coordinate where y=0 intersects is y-offset + main-radius
  let arc_intersection = y-offset + main-radius

  // Draw x-axis with zigzag on both sides
  // Center straight segment
  line((-zigzag-start, 0), (zigzag-start, 0), stroke: 0.8pt, name: "x-axis-center")

  // Left side zigzag
  decorations.zigzag(
    line((-x-range - 0.4, 0), (-zigzag-start, 0)),
    ..zigzag-style,
    stroke: 0.8pt,
    name: "x-axis-left",
  )

  // Right side zigzag with straight end segment
  // First the zigzag part - ending a bit earlier
  decorations.zigzag(
    line((zigzag-start, 0), (arc_intersection - 0.4, 0)),
    ..zigzag-style,
    stroke: 0.8pt,
    name: "x-axis-right-zigzag",
  )

  // Then the straight arrow part that touches the arc - extending to exactly touch the arc
  line(
    (arc_intersection - 0.4, 0),
    (arc_intersection, 0),
    stroke: 0.8pt,
    ..arrow-style,
    name: "x-axis-right-arrow",
  )

  content("x-axis-right-arrow.end", $"Re"(p_0)$, anchor: "south-east", padding: 2pt)

  // Draw y-axis
  line((0, -y-range - 0.7), (0, y-range + 0.7), ..arrow-style, name: "y-axis")
  content("y-axis.97%", $"Im"(p_0)$, anchor: "north-east", padding: (right: 8pt))

  // Draw Matsubara frequencies
  for n in range(-y-range, y-range + 1) {
    if n != 0 {
      circle((0, n), radius: small-dot-radius, fill: black, name: "freq-" + str(n))
      content(
        "freq-" + str(n),
        $i omega_#text(size: 0.7em)[#n]$,
        anchor: "west",
        padding: (left: 10pt),
      )
    }
  }

  // Draw origin
  circle((0, 0), radius: small-dot-radius, fill: black, name: "origin")
  content("origin", [0], anchor: "north-east", padding: 2pt)

  // Draw split contour
  // Right half (C₁): vertical line down + right semicircle
  draw-vertical-line(y-offset, main-radius, -main-radius, "right-line")
  draw-semicircle(y-offset, -90deg, 90deg, "right-arc")

  // Left half (C₂): vertical line up + left semicircle
  draw-vertical-line(-y-offset, -main-radius, main-radius, "left-line")
  draw-semicircle(-y-offset, 90deg, 270deg, "left-arc")

  // Add contour labels
  content("right-line.end", text(fill: dark-blue)[$C_1$], anchor: "south-west", padding: 4pt)
  content("left-line.start", text(fill: dark-blue)[$C_2$], anchor: "south-east", padding: 4pt)

  // Draw poles
  circle((x-range / 2, y-range / 4), radius: dot-radius, fill: black, name: "pole-e")
  content("pole-e", $E$, anchor: "west", padding: 2pt)

  circle((-x-range / 2, -y-range / 4), radius: dot-radius, fill: black, name: "pole-minus-e")
  content("pole-minus-e", $-E$, anchor: "east", padding: 2pt)
})