Wilsonloop.jl

Abstract

In Lattice Quantum Chromo-Dynamics (QCD), the gauge action is constructed by gauge invariant objects, Wilson loops, in discretized spacetime. Wilsonloop.jl helps us to treat with the Wilson loops and generic Wilson lines in any Nc and dimensions.

This is a package for lattice QCD codes.

This package is used in LatticeQCD.jl and a code in a project JuliaQCD.

What this package can do

• From a symbolic definition of Wilson lines, this returns SU(Nc)-valued Wilson lines as objects
• Constructing all staples from given symbolic Wilson lines
• Constructing derivatives of given symbolic Wilson lines (auto-grad for SU(Nc) variables)

How to install

add Wilsonloop

Notation warning

In Julia, adjoint represents hermitian conjugate, and we follow this terminology.

For example Adjoint_GLink means hermitian conjugate of a gauge link, not the link in the adjoint representation.

Please do not confuse with a link in the adjoint representation in conventional lattice QCD context. We do not support links in adjoint representation.

Basic idea

This package defines Wilsonline{Dim} type.

mutable struct Wilsonline{Dim}
        glinks::Array{GLink{Dim},1}
end

This is a array of GLink{Dim}.

The GLink{Dim} is defined as

abstract type Gaugelink{Dim} end

struct GLink{Dim} <: Gaugelink{Dim}
    direction::Int8
    position::NTuple{Dim,Int64}
    isdag::Bool
end

GLink{Dim} has a direction of a bond on the lattice and relative position $U_{\mu}(n)$. The direction and position are obtained by get_direction(a) and get_position(a), respectively. For example if we want to have 2nd link of the Wilson loop w, just do get_position(w[2]).

How to use

Plaquette and its staple

We can easily generate a plaquette.

println("plaq")
plaq = make_plaq()
display(plaq)

The output is

plaq
1-st loop
L"$U_{1}(n)U_{2}(n+e_{1})U^{\dagger}_{1}(n+e_{2})U^{\dagger}_{2}(n)$"	
2-nd loop
L"$U_{1}(n)U_{3}(n+e_{1})U^{\dagger}_{1}(n+e_{3})U^{\dagger}_{3}(n)$"	
3-rd loop
L"$U_{1}(n)U_{4}(n+e_{1})U^{\dagger}_{1}(n+e_{4})U^{\dagger}_{4}(n)$"	
4-th loop
L"$U_{2}(n)U_{3}(n+e_{2})U^{\dagger}_{2}(n+e_{3})U^{\dagger}_{3}(n)$"	
5-th loop
L"$U_{2}(n)U_{4}(n+e_{2})U^{\dagger}_{2}(n+e_{4})U^{\dagger}_{4}(n)$"	
6-th loop
L"$U_{3}(n)U_{4}(n+e_{3})U^{\dagger}_{3}(n+e_{4})U^{\dagger}_{4}(n)$"

If we want to consider 2D system, we can do make_plaq(Dim=2).

The staple of the plaquette is given as

    for μ=1:4
        println("μ = $μ")
        staples = make_plaq_staple(μ)
        display(staples)
    end

The output is

μ = 1
1-st loop
L"$U_{2}(n+e_{1})U^{\dagger}_{1}(n+e_{2})U^{\dagger}_{2}(n)$"	
2-nd loop
L"$U^{\dagger}_{2}(n+e_{1}-e_{2})U^{\dagger}_{1}(n-e_{2})U_{2}(n-e_{2})$"	
3-rd loop
L"$U_{3}(n+e_{1})U^{\dagger}_{1}(n+e_{3})U^{\dagger}_{3}(n)$"	
4-th loop
L"$U^{\dagger}_{3}(n+e_{1}-e_{3})U^{\dagger}_{1}(n-e_{3})U_{3}(n-e_{3})$"	
5-th loop
L"$U_{4}(n+e_{1})U^{\dagger}_{1}(n+e_{4})U^{\dagger}_{4}(n)$"	
6-th loop
L"$U^{\dagger}_{4}(n+e_{1}-e_{4})U^{\dagger}_{1}(n-e_{4})U_{4}(n-e_{4})$"	
μ = 2
1-st loop
L"$U^{\dagger}_{1}(n-e_{1}+e_{2})U^{\dagger}_{2}(n-e_{1})U_{1}(n-e_{1})$"	
2-nd loop
L"$U_{1}(n+e_{2})U^{\dagger}_{2}(n+e_{1})U^{\dagger}_{1}(n)$"	
3-rd loop
L"$U_{3}(n+e_{2})U^{\dagger}_{2}(n+e_{3})U^{\dagger}_{3}(n)$"	
4-th loop
L"$U^{\dagger}_{3}(n+e_{2}-e_{3})U^{\dagger}_{2}(n-e_{3})U_{3}(n-e_{3})$"	
5-th loop
L"$U_{4}(n+e_{2})U^{\dagger}_{2}(n+e_{4})U^{\dagger}_{4}(n)$"	
6-th loop
L"$U^{\dagger}_{4}(n+e_{2}-e_{4})U^{\dagger}_{2}(n-e_{4})U_{4}(n-e_{4})$"	
μ = 3
1-st loop
L"$U^{\dagger}_{1}(n-e_{1}+e_{3})U^{\dagger}_{3}(n-e_{1})U_{1}(n-e_{1})$"	
2-nd loop
L"$U_{1}(n+e_{3})U^{\dagger}_{3}(n+e_{1})U^{\dagger}_{1}(n)$"	
3-rd loop
L"$U^{\dagger}_{2}(n-e_{2}+e_{3})U^{\dagger}_{3}(n-e_{2})U_{2}(n-e_{2})$"	
4-th loop
L"$U_{2}(n+e_{3})U^{\dagger}_{3}(n+e_{2})U^{\dagger}_{2}(n)$"	
5-th loop
L"$U_{4}(n+e_{3})U^{\dagger}_{3}(n+e_{4})U^{\dagger}_{4}(n)$"	
6-th loop
L"$U^{\dagger}_{4}(n+e_{3}-e_{4})U^{\dagger}_{3}(n-e_{4})U_{4}(n-e_{4})$"	
μ = 4
1-st loop
L"$U^{\dagger}_{1}(n-e_{1}+e_{4})U^{\dagger}_{4}(n-e_{1})U_{1}(n-e_{1})$"	
2-nd loop
L"$U_{1}(n+e_{4})U^{\dagger}_{4}(n+e_{1})U^{\dagger}_{1}(n)$"	
3-rd loop
L"$U^{\dagger}_{2}(n-e_{2}+e_{4})U^{\dagger}_{4}(n-e_{2})U_{2}(n-e_{2})$"	
4-th loop
L"$U_{2}(n+e_{4})U^{\dagger}_{4}(n+e_{2})U^{\dagger}_{2}(n)$"	
5-th loop
L"$U^{\dagger}_{3}(n-e_{3}+e_{4})U^{\dagger}_{4}(n-e_{3})U_{3}(n-e_{3})$"	
6-th loop
L"$U_{3}(n+e_{4})U^{\dagger}_{4}(n+e_{3})U^{\dagger}_{3}(n)$"	
1-st loop
L"$U^{\dagger}_{1}(n-e_{1})U_{4}(n-e_{1})U_{1}(n-e_{1}+e_{4})$"	
2-nd loop
L"$U_{1}(n)U_{4}(n+e_{1})U^{\dagger}_{1}(n+e_{4})$"	
3-rd loop
L"$U^{\dagger}_{2}(n-e_{2})U_{4}(n-e_{2})U_{2}(n-e_{2}+e_{4})$"	
4-th loop
L"$U_{2}(n)U_{4}(n+e_{2})U^{\dagger}_{2}(n+e_{4})$"	
5-th loop
L"$U^{\dagger}_{3}(n-e_{3})U_{4}(n-e_{3})U_{3}(n-e_{3}+e_{4})$"	
6-th loop
L"$U_{3}(n)U_{4}(n+e_{3})U^{\dagger}_{3}(n+e_{4})$"

Input loops

The arbitrary Wilson loop is constructed as

loop = [(1,+1),(2,+1),(1,-1),(2,-1)]
println(loop)
w = Wilsonline(loop)
println("P: ")
show(w)

Its adjoint is calculated as

println("P^+: ")
show(w')

Its staple is calculated as

println("staple")
for μ=1:4
    println("μ = $μ")
    V1 = make_staple(w,μ)
    V2 = make_staple(w',μ)
    show(V1)
    show(V2)
end

Derivatives

The derivative of the lines $dw/dU_{\mu}$ is calculated as

println("derive w")
for μ=1:4
    dU = derive_U(w,μ)
    for i=1:length(dU)
        show(dU[i])
    end
end

Note that the derivative is a rank-4 tensor.

The output is

L"$I  \otimes U_{2}(n+e_{1})U^{\dagger}_{1}(n+e_{2})U^{\dagger}_{2}(n)\delta_{m,n}$"	
L"$U_{1}(n-e_{1}) \otimes U^{\dagger}_{1}(n-e_{1}+e_{2})U^{\dagger}_{2}(n-e_{1})\delta_{m,n+e_{1}}$"

The derivatives are usually used for making the smearing of the gauge fields (Stout smearing can be used in Gaugefields.jl).

Examples

Long lines and its staple

mu = 1
nu = 2
rho = 3
loops = [(mu,2),(nu,1),(rho,3),(mu,-2),(rho,-3),(nu,-1)]
w = Wilsonline(loops)

L"$U_{1}(n)U_{1}(n+e_{1})U_{2}(n+2e_{1})U_{3}(n+2e_{1}+e_{2})U_{3}(n+2e_{1}+e_{2}+e_{3})U_{3}(n+2e_{1}+e_{2}+2e_{3})U^{\dagger}_{1}(n+e_{1}+e_{2}+3e_{3})U^{\dagger}_{1}(n+e_{2}+3e_{3})U^{\dagger}_{3}(n+e_{2}+2e_{3})U^{\dagger}_{3}(n+e_{2}+e_{3})U^{\dagger}_{3}(n+e_{2})U^{\dagger}_{2}(n)$"

Its staple:

staple = make_staple(w,mu)

1-st loop
L"$U_{1}(n+e_{1})U_{2}(n+2e_{1})U_{3}(n+2e_{1}+e_{2})U_{3}(n+2e_{1}+e_{2}+e_{3})U_{3}(n+2e_{1}+e_{2}+2e_{3})U^{\dagger}_{1}(n+e_{1}+e_{2}+3e_{3})U^{\dagger}_{1}(n+e_{2}+3e_{3})U^{\dagger}_{3}(n+e_{2}+2e_{3})U^{\dagger}_{3}(n+e_{2}+e_{3})U^{\dagger}_{3}(n+e_{2})U^{\dagger}_{2}(n)$"	

2-nd loop
L"$U_{2}(n+e_{1})U_{3}(n+e_{1}+e_{2})U_{3}(n+e_{1}+e_{2}+e_{3})U_{3}(n+e_{1}+e_{2}+2e_{3})U^{\dagger}_{1}(n+e_{2}+3e_{3})U^{\dagger}_{1}(n-e_{1}+e_{2}+3e_{3})U^{\dagger}_{3}(n-e_{1}+e_{2}+2e_{3})U^{\dagger}_{3}(n-e_{1}+e_{2}+e_{3})U^{\dagger}_{3}(n-e_{1}+e_{2})U^{\dagger}_{2}(n-e_{1})U_{1}(n-e_{1})$"	

Derivative of Wilson line

The derivative of the staple

dev = derive_U(staple[1],nu)

L"$U_{1}(n-e_{1}) \otimes U_{3}(n+e_{2})U_{3}(n+e_{2}+e_{3})U_{3}(n+e_{2}+2e_{3})U^{\dagger}_{1}(n-e_{1}+e_{2}+3e_{3})U^{\dagger}_{1}(n-2e_{1}+e_{2}+3e_{3})U^{\dagger}_{3}(n-2e_{1}+e_{2}+2e_{3})U^{\dagger}_{3}(n-2e_{1}+e_{2}+e_{3})U^{\dagger}_{3}(n-2e_{1}+e_{2})U^{\dagger}_{2}(n-2e_{1})\delta_{m,n+2e_{1}}$"

The derivative of the Wilson loops with respect to a link is a rank-4 tensor (ref), which is expressed as

, where A and B are matrices. We can get the A and B matrices, expressed by Wilsonline{Dim} type :

devl = get_leftlinks(dev[1])
devr = get_rightlinks(dev[1])

The derivative of the action

The action is usually expressed as

The derivative of the action is

Therefore, the staple V is important to get the derivative.

Note that we define the derivative as

Acknowledgment

If you write a paper using this package, please refer this code.

BibTeX citation is following

@article{Nagai:2024yaf,
    author = "Nagai, Yuki and Tomiya, Akio",
    title = "{JuliaQCD: Portable lattice QCD package in Julia language}",
    eprint = "2409.03030",
    archivePrefix = "arXiv",
    primaryClass = "hep-lat",
    month = "9",
    year = "2024"
}

and the paper is arXiv:2409.03030.