UEG parameters

Mapping from ParaMC fields saved in short format to their corresponding types

Constructs a ParaMC object from its short string representation.

function KO_W(q, n, p::ParaMC)

KO interaction in momentum q and the Matsubara frequency index n Assume

\[r_q = v_q + f\]

then the KO interaction is

\[Rq = r_q / (1 - r_q Π0) - f\]

function KOdynamic_T(para::ParaMC)

Dynamic part of the interaction.

Assume

\[r_q = v_q + f\]

Then, the dynamic interaction is given by

\[δR_q/r_q = r_q Π₀/(1-r_q Π₀)\]

where Π₀ is the polarization of free electrons.

function KOdynamic_T(para::ParaMC)

Dynamic part of the interaction.

Assume

\[r_q = v_q + f\]

Then, the dynamic interaction is given by

\[d δR_q/df - 1 = 1/(1-r_q Π₀)^2 - 1\]

where Π₀ is the polarization of free electrons.

function counterKO_W(para::ParaMC; qgrid=para.qgrid, ngrid=[0,], order=para.order, proper=false)

calculate counter-terms of the KO interaction

Assume

\[r_q = v_q + f\]

and

\[Rq = r_q / (1 - r_q Π0) - f\]

Then, the counter-term is given by a power expansion of the form

\[(Rq ξ + f(ξ))/(1+(Rq ξ + f(ξ))Π0) - f(ξ)\]

where f(ξ) = f1 ξ + f2 ξ^2 + ...

Therefore, the counter-term is given by Order 1 (ξ^2)

\[(Rq+f1)^2 Π0\]

Order 2 (ξ^3)

\[(Rq+f1)^3 Π0^2 + (R_q+f1)f2 Π_0\]

function interactionDynamic(p::ParaMC, qd, τIn, τOut)

Dynamic part of the interaction.

Assume

\[r_q = v_q + f\]

Then, the dynamic interaction is given by

\[δR_q = (r_q)²Π₀/(1-r_q Π₀)\]

where Π₀ is the polarization of free electrons.

function interactionDynamic_df(p::ParaMC, qd, τIn, τOut)

Dynamic part of the interaction.

Assume

\[r_q = v_q + f\]

Then, the dynamic interaction is given by

\[d δR_q/df = (r_q)²Π₀/(1-r_q Π₀)\]

where Π₀ is the polarization of free electrons.

function interactionStatic(p::ParaMC, qd, τIn, τOut)

instant part of the renormalized interaction

Assume

\[r_q = v_q + f\]

Then, the instant interaction is given by

\[v_q = r_q - f\]

The current implementation involves one auxiliary time variable τOut for better sign cancellation.

To show the net result is v_q, one may perform a τOut integration explicitly, then

\[kostatic = r_q / (1-r_q Π₀) - f\]

where Π₀ is the polarization of free electrons. and,

\[dynamic = (r_q)²Π₀/(1-r_q Π₀)\]

so that,

\[kostatic - dynamic = v_q\]

function lindhard(x)

Dimensionless Linhard function so that lindhard(0) = 1 for both 2D and 3D

Arguments

• x : 2k/k_F

linear2D(data, xgrid, ygrid, x, y)

linear interpolation of data(x, y)

#Arguments:

• xgrid: one-dimensional grid of x
• ygrid: one-dimensional grid of y
• data: two-dimensional array of data
• x: x
• y: y

Hard-coded counterterm partitions for diagrams of max order order and minimal loop order offset, given in the form (nloop, nμ, n_λ). The default offset corresponds to partitions of the self-energy, where the minimal loop order is 1.