Conventions

The conventions of the package in one place, as a user meets them. The manual gives each one with its derivation: the sections are named in italics below.

Expressions

• IntegralOver[f, {x, y, ...}] is the integral, or sum, of f over the labels x, y, ...; a label stands for everything a field depends on (position, imaginary time, spin, any index) and must be a symbol. DeltaFunction[x, y] is the delta function, integrated automatically.
• Grassmann-odd factors are multiplied with **, in their order; Times would lose the order and the sign.
• FunctionalDerivative[F, s1, s2, ..., sn] is the left derivative δ/δs1 δ/δs2 … δ/δsn F: the rightmost acts first.
• Results are in canonical form: the integration variables are renamed z1, z2, ... and equal diagrams are added up, so SimplifyDiagrams[a - b] is 0 exactly when a and b are the same sum of diagrams (manual: Canonical form).

Fermions

The free propagator is G₀(x, y) = −⟨T ψ(x) ψ̄(y)⟩₀, with the generating functional and the fields

Needs["FunctionalDiagrams`"]
DeclareGrassmannSource[eta, etabar];
DeclareField[psi, etabar]; DeclareField[psibar, eta, -1];
Z0 = Exp[-IntegralOver[G0[x, y] etabar[x] ** eta[y], {x, y}]];
GaussianExpectationValue[Z0, {psi[x], psibar[y]}]

-G0[x, y]

so ψ ↔ δ/δη̄ and ψ̄ ↔ −δ/δη (the -1 of DeclareField), and the Green function is -PerturbativeCorrelationFunction[Z0, {psi[x], psibar[y]}, interaction, n]. G[x, y] is drawn as an arrow from y to x. The two-body interaction is S_int = ½ ∫ V(x, y) ψ̄(x) ψ̄(y) ψ(y) ψ(x), and the package expands exp(−S_int):

DeclareSymmetric[V]; DeclareInteractionLine[V]; DeclareDirectedPropagator[G0, G];
interaction =
  1/2 IntegralOver[V[x, y] psibar[x] ** psibar[y] ** psi[y] ** psi[x], {x, y}];

Closed fermion loops and exchange terms carry their signs by themselves (manual: Conventions for many-body problems).

Weights of the pictures

The number written in front of a picture, times the values of its elements, is the term. A propagator stands for itself; an interaction line (DeclareInteractionLine) for −V; a vertex for minus its coupling (DeclareCoupling); a local vertex (DeclareLocalVertex) for minus its kernel. The minus signs come from expanding exp(−S_int). So the Fock self-energy has the weight +1, the Hartree self-energy −1 (its closed loop), the tadpole of φ⁴ theory +½; a diagram of G or Σ with n_L closed fermion loops has the weight (−1)^n_L. Only the weights shown with the pictures follow this rule: the expressions keep their coefficients, and "Weights" -> "Coefficients" shows those. See Reading the weights and the manual, The weights.

Self-energy and vertices

• Φ[G] (LuttingerWardFunctional) is minus the sum of the two-particle-irreducible vacuum diagrams, with G₀ → G.
• Σ(x, y) = δΦ/δG(y, x): FunctionalDerivative[Phi, G[y, x]] (the shortcut SelfEnergy[Phi, G, {x, y}]).
• Γ⁰(x₁, x₂; x₃, x₄) = δΣ(x₁, x₂)/δG(x₄, x₃) = δ²Φ/δG(x₄, x₃)δG(x₂, x₁): FunctionalDerivative[Phi, G[x4, x3], G[x2, x1]] (the shortcut IrreducibleVertex). Fermions leave at x₁, x₃ and enter at x₂, x₄; the pictures put the legs counterclockwise from the top left: x₁ (out), x₂ (in), x₃ (out), x₄ (in). Every picture of a four-point function, boson or fermion, has its labels at the corners in the order given; "ExternalOrder" changes the order, and "ExternalOrder" -> "Uncrossed" exchanges the legs of one side of a diagram whose lines would cross (Particle–particle diagrams).
• The full vertex F: ⟨ψ(1′) ψ̄(2′) ψ̄(4′) ψ(3′)⟩_c = G(1′, 1) G(2, 2′) F(1, 2; 3, 4) G(4, 4′) G(3′, 3).
• A pair of propagators between two vertices is written L₀: in the particle–hole channel L₀(x₁, x₂; x₃, x₄) = G(x₁, x₄) G(x₃, x₂), so that F = Γ⁰ + Γ⁰L₀F.
• Real bosons ("Statistics" -> "Boson", G = ⟨φφ⟩ declared symmetric): Σ = −2 δΦ/δG(x, y), Γ⁰ = −2 δΣ/δG = 4 δ²Φ/δGδG, and a pair of lines in a channel carries −½. Complex bosons: "Statistics" -> "ComplexBoson".

Manual: Self-energy, Luttinger–Ward functional and vertices.

Channels

For fermions, with the legs above:

channel	name in the package	pairs of legs
particle–hole (ph)	"ParticleHole"	(x₁ x₂ | x₃ x₄)
transverse particle–hole (ph̄)	"TransverseParticleHole"	(x₁ x₄ | x₃ x₂)
particle–particle (pp)	"ParticleParticle"	(x₁ x₃ | x₂ x₄)

Γ⁰ is irreducible in ph: Γ⁰ = F − R_ph = Γ_irr + R_ph̄ + R_pp, with Γ_irr the fully irreducible vertex (the bare vertex up to third order). For real bosons and superfields the channels are "s" (a b | c d), "t" (a c | b d) and "u" (a d | b c), with a, b, c, d at the corners counterclockwise from the top left: the diagrams of t join opposite corners and are drawn with two legs crossing, unless "Uncrossed" exchanges them. The particle–particle kernel of fermions is a derivative with respect to the anomalous propagators, so it needs the superfield below. Manual: Two-particle channels, Bethe–Salpeter ladders and the parquet; Particle–particle diagrams.

The vertex of the density and Hedin's equations

• Λ(x₁, x₂; z) = −δG⁻¹(x₁, x₂)/δU(z) for a potential U coupled to ρ = ψ̄ψ (PerturbativeDensityVertex): ⟨ψ(x₁) ψ̄(x₂) ρ(z)⟩_c = L₀Λ, and Λ = γ + Γ⁰L₀Λ = γ + FL₀γ with γ = δ(x₁, z) δ(x₂, z); Σ = Σ_H − VGΛ.
• Hedin: W = V − V⟨ρρ⟩_cV, P = −(proper part of ⟨ρρ⟩_c) = L₀Λ̃, Σ = Σ_H − GWΛ̃, with Λ̃ the proper vertex.
• Hubbard–Stratonovich (HubbardStratonovich): ⟨φφ⟩₀ = V and the coupling −i∫φρ.

Manual: Exact equations.

Superfields

"Statistics" -> "Superfield": the Nambu field 𝛙 = (ψ, ψ̄) is one Grassmann field with one source. Its propagator 𝒢 = −⟨T𝛙𝛙⟩ is antisymmetric (DeclareAntisymmetric), with the blocks G, 𝒢_ψψ = −⟨Tψψ⟩ and 𝒢_ψ̄ψ̄ = −⟨Tψ̄ψ̄⟩; derivatives with respect to it are antisymmetrised. 𝚺 = −2 δΦ/δ𝒢, with the gap functions Δ = 𝚺_ψ̄ψ̄ and Δ̄ = 𝚺_ψψ; Γ⁰ = 4 δ²Φ/δ𝒢δ𝒢, whose channels s, t, u are ph, pp and ph̄. SuperfieldComponents projects on ψ and ψ̄. In the pictures 𝒢_ψψ has two arrow heads pointing to its ends ("DivergingArrows") and 𝒢_ψ̄ψ̄ two pointing to its middle ("ConvergingArrows"). Manual: Superfields.

Effective actions

The effective actions are Legendre transforms of −log Z. The 1PI action is the transform with respect to the sources of the fields, a functional of the means φ̄ = ⟨φ⟩ (for fermions the Grassmann-odd Ψ = ⟨ψ⟩ and Ψ̄ = ⟨ψ̄⟩); the 2PI action is the transform with respect to a two-point source as well, a functional of the means and of the propagator G. In both, S is the classical action, written whole, and S'' its second derivative with respect to the means.

The 1PI action (EffectiveAction1PI):

• A real boson: Γ[φ̄] = S[φ̄] + ½ Tr ln S''[φ̄] + Γ₁[φ̄]. S''[φ̄] = D⁻¹[φ̄] = D₀⁻¹ + S_int''[φ̄] is the inverse propagator in the background, and Γ₁, defined by this equation, is minus the connected 1PI vacuum diagrams with two or more loops, with the lines D[φ̄] = S''[φ̄]⁻¹ and the vertices of S_int[φ̄ + φ] of degree three or more in φ.
• Fermions: Γ[Ψ̄, Ψ] = S[Ψ̄, Ψ] − ½ Tr ln(−S''[Ψ̄, Ψ]) + Γ₁[Ψ̄, Ψ], with S'' the Nambu matrix of the second derivatives with respect to (Ψ, Ψ̄): its normal block is D⁻¹[Ψ̄, Ψ](x, y) = δ²S/δΨ̄(x)δΨ(y), its anomalous blocks are δ²S/δΨ̄(x)δΨ̄(y) = V(x, y)Ψ(x)Ψ(y) and δ²S/δΨ(x)δΨ(y) = V(x, y)Ψ̄(x)Ψ̄(y), and the lines of Γ₁ are S''⁻¹. A superfield has the same form, Γ[𝚿] = S[𝚿] − ½ Tr ln(−S''[𝚿]) + Γ₁[𝚿]. The package computes these two with bare lines G₀, the vertices of degree two kept as vertices: grouped by loops, their diagrams are S_int, the rings of the one loop, and Γ₁ with its lines expanded.
• At zero means the derivatives are the vertex functions: for a boson the second is D₀⁻¹ − Σ and the fourth Γ⁽⁴⁾; for fermions δ²Γ/δΨ̄(a)δΨ(b) = G₀⁻¹(a, b) − Σ(a, b) (the rightmost derivative first), and the fourth derivative, with respect to Ψ̄(x₁), Ψ(x₂), Ψ̄(x₃), Ψ(x₄), is the full vertex F(x₁, x₂; x₃, x₄).

The 2PI action (EffectiveAction2PI):

• At zero means: Γ[G] = Φ[G] − Tr ln(−G⁻¹) − Tr(G₀⁻¹G − 1) for fermions, Φ[G] + ½ Tr ln G⁻¹ + ½ Tr(D₀⁻¹G − 1) for a real boson, with Φ the Luttinger–Ward functional. It is stationary where G⁻¹ = G₀⁻¹ − Σ[G], the Dyson equation, and equal to −log Z there.
• With the means (option "MeanFields"), the action of Cornwall, Jackiw and Tomboulis: Γ[φ̄, G] = S[φ̄] + ½ Tr ln G⁻¹ + ½ Tr(D⁻¹[φ̄]G − 1) + Γ₂[φ̄, G], and for fermions Γ[Ψ̄, Ψ, G] = S[Ψ̄, Ψ] − Tr ln(−G⁻¹) − Tr(D⁻¹[Ψ̄, Ψ]G − 1) + Γ₂[Ψ̄, Ψ, G]. D⁻¹ is that of the 1PI action, for fermions D⁻¹[Ψ̄, Ψ] = G₀⁻¹ − Σ_HF[−ΨΨ̄]: Σ_HF[−ΨΨ̄], the first-order self-energy with −Ψ(x)Ψ̄(y) in place of G(x, y), is the Hartree and Fock potentials of the means. Γ₂ is minus the 2PI vacuum diagrams with the lines G and the vertices of degree three or more; at zero means it is Φ. The Dyson equation is G⁻¹ = D⁻¹ − Σ, with Σ the self-energy of Γ₂.
• For a boson and for a superfield, at G = S''⁻¹, Γ₂ is the part of Γ₁ that is 2PI in these lines.

Manual: Effective actions and operator traces.

Matsubara frequencies

G₀(k, iω) = 1/(iω − ξ_k). The density correlation function ⟨ρρ⟩_c of the package is minus the Lindhard function χ₀, and its boson propagator ⟨φφ⟩₀ is minus the 𝒟₀ = −⟨Tφφ⟩₀ of the textbooks. Each closed fermion loop gets the factor "SpinDegeneracy" on evaluation. Manual: Evaluation: Matsubara sums, and Evaluating diagrams.

Dimensional regularization

Euclidean, a line is 1/(q² + m²) (ScalarPropagator), d = 4 − 2ε, every loop μ^(2ε) (e^γE/4π)^ε ∫d^dk/(2π)^d: minimal subtraction is then the MS-bar scheme, with no γE or ln 4π in the results. φ⁴ theory with λ/4!; Γ⁽⁴⁾ = −(amputated 1PI part of the connected four-point function); β = μ dλ/dμ, γ_φ = ½ μ d ln Z_φ/dμ, γ_m² = μ d ln m²/dμ. Manual: Evaluation: dimensional regularization.

QED at one loop

Minkowski, metric (+, −, −, −), Peskin and Schroeder except for the sign of the charge: e > 0 everywhere and the electron has the charge −e, so S_int = +e∫ψ̄γ^μψA_μ and the vertex is +ieγ^μ; α = e²/4π. No γ5 (GammaMatrix[5] is refused). The package expands exp(−X), so the interaction passed is X = −iS_int, and the 1PI functions are −iΣ, iΠ^μν and +ieΓ^μ. With the MS-bar measure above, the 2/ε of Peskin and Schroeder is our 1/ε. Manual: Evaluation: QED at one loop.

Names

Names of propagators, fields and sources must be symbols that are not built into the Wolfram Language: C, D, E, I, K, N, O, Gamma, Beta, GreenFunction are (the package warns with FunctionalDiagrams::builtin). See the FAQ.