[NOTE: This is an extended version by Okuto Morikawa, in order to implement higher-form gauge fields (i.e., 't Hooft twisted boundary condition/flux).]
[NOTE: O.M. also provides a memory-safer code set than the original one.]
This is a package for lattice QCD codes. Treating gauge fields (links), gauge actions with MPI and autograd.
This package will be used in LatticeQCD.jl. See also the orginal package in Gaugefields.jl.
O.M. would like to thank Yuki Nagai and Akio Tomiya (the contributors of the original package). O.M. is also grateful to Hiroshi Suzuki.
This package has following functionarities
- SU(Nc) (Nc > 1) gauge fields in 2 or 4 dimensions with arbitrary actions.
- Z(Nc) 2-form gauge fields in 4 dimensions, which are given as 't Hooft flux.
- U(1) gauge fields in 2 dimensions with arbitrary actions.
- Configuration generation
- Heatbath
- quenched Hybrid Monte Carlo
- quenched Hybrid Monte Carlo being subject to 't Hooft twisted b.c.
- with external (non-dynamical) Z(Nc) 2-form gauge fields
- quenched Hybrid Monte Carlo for SU(Nc)/Z(Nc) gauge theory
- with dynamical Z(Nc) 2-form gauge fields
- Gradient flow via RK3
- Yang-Mills gradient flow
- Yang-Mills gradient flow being subject to 't Hooft twisted b.c.
- Gradient flow for SU(Nc)/Z(Nc) gauge theory
- I/O: ILDG and Bridge++ formats are supported (c-lime will be installed implicitly with CLIME_jll )
- MPI parallel computation (experimental. See documents.)
- quenched HMC with MPI being subject to 't Hooft twisted b.c.
The implementation of higher-form gauge fields is based on arXiv:2303.10977 [hep-lat].
Dynamical fermions will be supported with LatticeDiracOperators.jl.
In addition, this supports followings
- Autograd for functions with SU(Nc) variables
- Stout smearing (exp projecting smearing)
- Stout force via backpropagation
Autograd can be worked for general Wilson lines except for ones have overlaps.
add Wilsonloop
add https://github.com/o-morikawa/Gaugefields.jl.git
This is a non-official package in Julia, and you are recommended to use it as a develop (dev) package if there's a possibility that you use the original Gaugefields.jl package or modify it.
To install the oringinal package, in Julia REPL in the package mode,
add Gaugefields.jl
Download the code locally, then in Julia REPL in the package mode,
dev /<your full path>/Gaugefields
When you use this package in Julia REPL, in the package mode,
activate Gaugefields
or, when in command line,
julia --project="Gaugefields" test.jl
Please see the orginal docs in Gaugefields.jl. Basically, you can use this package in a same way as the original code if the argument of any function, (..., U, ...), is rewritten by (..., U, B, ...).
ILDG format is one of standard formats for LatticeQCD configurations.
We can read ILDG format like:
NX = 4
NY = 4
NZ = 4
NT = 4
Nwing = 1
Dim = 4
U = Initialize_Gaugefields(NC,Nwing,NX,NY,NZ,NT,condition = "cold")
ildg = ILDG(filename)
i = 1
L = [NX,NY,NZ,NT]
load_gaugefield!(U,i,ildg,L,NC)Then, we can calculate the plaquette:
temp1 = similar(U[1])
temp2 = similar(U[1])
comb = 6
factor = 1/(comb*U[1].NV*U[1].NC)
@time plaq_t = calculate_Plaquette(U,B,temp1,temp2)*factor
println("plaq_t = $plaq_t")
poly = calculate_Polyakov_loop(U,temp1,temp2)
println("polyakov loop = $(real(poly)) $(imag(poly))")We can write a configuration as the ILDG format like
filename = "hoge.ildg"
save_binarydata(U,filename)Gaugefields.jl also supports a text format for Bridge++.
using Gaugefields
filename = "testconf.txt"
load_BridgeText!(filename,U,L,NC)filename = "testconf.txt"
save_textdata(U,filename)SU(N) gauge fields possess Z(N) center symmetry, which is called 1-form global symmetry, a type of generalized symmetry. To gauge the 1-form center symmetry, we can define the Z(N) 2-form gauge fields in four dimensions, B, as
NX = 4
NY = 4
NZ = 4
NT = 4
Nwing = 0
NC = 3
flux=[1,0,0,0,0,1] # FLUX=[Z12,Z13,Z14,Z23,Z24,Z34]
println("Flux is ", flux)
U1 = Initialize_Gaugefields(NC,Nwing,NX,NY,NZ,NT,condition = "cold")
B1 = Initialize_Bfields(NC,flux,Nwing,NX,NY,NZ,NY,condition = "tflux")
println("Initial conf of B at [1,2][2,2,:,:,NZ,NT]")
display(B1[1,2][2,2,:,:,NZ,NT])We can do the HMC simulations. The example code is as follows.
using Random
using Gaugefields
using Wilsonloop
using LinearAlgebra
function calc_action(gauge_action,U,B,p)
NC = U[1].NC
Sg = -evaluate_GaugeAction(gauge_action,U,B)/NC
Sp = p*p/2
S = Sp + Sg
return real(S)
end
function MDstep!(gauge_action,U,B,p,MDsteps,Dim,Uold,temp1,temp2)
Δτ = 1.0/MDsteps
gauss_distribution!(p)
Sold = calc_action(gauge_action,U,B,p)
substitute_U!(Uold,U)
for itrj=1:MDsteps
U_update!(U,p,0.5,Δτ,Dim,gauge_action)
P_update!(U,B,p,1.0,Δτ,Dim,gauge_action,temp1,temp2)
U_update!(U,p,0.5,Δτ,Dim,gauge_action)
end
Snew = calc_action(gauge_action,U,B,p)
println("Sold = $Sold, Snew = $Snew")
println("Snew - Sold = $(Snew-Sold)")
ratio = min(1,exp(-Snew+Sold)) # bug is fixed
if rand() > ratio
substitute_U!(U,Uold)
return false
else
return true
end
end
function U_update!(U,p,ϵ,Δτ,Dim,gauge_action)
temps = get_temporary_gaugefields(gauge_action)
temp1 = temps[1]
temp2 = temps[2]
expU = temps[3]
W = temps[4]
for μ=1:Dim
exptU!(expU,ϵ*Δτ,p[μ],[temp1,temp2])
mul!(W,expU,U[μ])
substitute_U!(U[μ],W)
end
end
function P_update!(U,B,p,ϵ,Δτ,Dim,gauge_action,temp1,temp2) # p -> p +factor*U*dSdUμ
NC = U[1].NC
temp = temp1
dSdUμ = temp2
factor = -ϵ*Δτ/(NC)
for μ=1:Dim
calc_dSdUμ!(dSdUμ,gauge_action,μ,U,B)
mul!(temp,U[μ],dSdUμ) # U*dSdUμ
Traceless_antihermitian_add!(p[μ],factor,temp)
end
end
function HMC_test_4D_tHooft(NX,NY,NZ,NT,NC,Flux,β)
Dim = 4
Nwing = 0
flux = Flux
println("Flux : ", flux)
Random.seed!(123)
U = Initialize_Gaugefields(NC,Nwing,NX,NY,NZ,NT,condition = "cold",randomnumber="Reproducible")
B = Initialize_Bfields(NC,flux,Nwing,NX,NY,NZ,NT,condition = "tflux")
temp1 = similar(U[1])
temp2 = similar(U[1])
if Dim == 4
comb = 6 #4*3/2
elseif Dim == 3
comb = 3
elseif Dim == 2
comb = 1
else
error("dimension $Dim is not supported")
end
factor = 1/(comb*U[1].NV*U[1].NC)
@time plaq_t = calculate_Plaquette(U,B,temp1,temp2)*factor
println("0 plaq_t = $plaq_t")
poly = calculate_Polyakov_loop(U,temp1,temp2)
println("0 polyakov loop = $(real(poly)) $(imag(poly))")
gauge_action = GaugeAction(U,B)
plaqloop = make_loops_fromname("plaquette")
append!(plaqloop,plaqloop')
β = β/2
push!(gauge_action,β,plaqloop)
#show(gauge_action)
p = initialize_TA_Gaugefields(U) #This is a traceless-antihermitian gauge fields. This has NC^2-1 real coefficients.
Uold = similar(U)
substitute_U!(Uold,U)
MDsteps = 50
temp1 = similar(U[1])
temp2 = similar(U[1])
comb = 6
factor = 1/(comb*U[1].NV*U[1].NC)
numaccepted = 0
numtrj = 100
for itrj = 1:numtrj
t = @timed begin
accepted = MDstep!(gauge_action,U,B,p,MDsteps,Dim,Uold,temp1,temp2)
end
if get_myrank(U) == 0
# println("elapsed time for MDsteps: $(t.time) [s]")
end
numaccepted += ifelse(accepted,1,0)
#plaq_t = calculate_Plaquette(U,B,temp1,temp2)*factor
#println("$itrj plaq_t = $plaq_t")
if itrj % 10 == 0
@time plaq_t = calculate_Plaquette(U,B,temp1,temp2)*factor
println("$itrj plaq_t = $plaq_t")
poly = calculate_Polyakov_loop(U,temp1,temp2)
println("$itrj polyakov loop = $(real(poly)) $(imag(poly))")
println("acceptance ratio ",numaccepted/itrj)
end
end
return plaq_t,numaccepted/numtrj
end
function main()
β = 5.7
NX = 4
NY = 4
NZ = 4
NT = 4
NC = 3
Flux = [0,0,1,1,0,0]
#HMC_test_4D(NX,NY,NZ,NT,NC,β)
HMC_test_4D_tHooft(NX,NY,NZ,NT,NC,Flux,β)
end
main()HMC simulations with dynamical B fields are as follows:
using Random
using Gaugefields
using Wilsonloop
using LinearAlgebra
function calc_action(gauge_action,U,B,p)
NC = U[1].NC
Sg = -evaluate_GaugeAction(gauge_action,U,B)/NC
Sp = p*p/2
S = Sp + Sg
return real(S)
end
function MDstep!(gauge_action,U,B,flux,p,MDsteps,Dim,Uold,Bold,flux_old,temp1,temp2)
Δτ = 1.0/MDsteps
gauss_distribution!(p)
Sold = calc_action(gauge_action,U,B,p)
substitute_U!(Uold,U)
substitute_U!(Bold,B)
flux_old[:] = flux[:]
Flux_update!(B,flux)
for itrj=1:MDsteps
U_update!(U,p,0.5,Δτ,Dim,gauge_action)
P_update!(U,B,p,1.0,Δτ,Dim,gauge_action,temp1,temp2)
U_update!(U,p,0.5,Δτ,Dim,gauge_action)
end
Snew = calc_action(gauge_action,U,B,p)
println("Sold = $Sold, Snew = $Snew")
println("Snew - Sold = $(Snew-Sold)")
ratio = min(1,exp(-Snew+Sold)) # bug is fixed
if rand() > ratio
substitute_U!(U,Uold)
substitute_U!(B,Bold)
flux[:] = flux_old[:]
return false
else
return true
end
end
function Flux_update!(B,flux)
NC = B[1,2].NC
NDW = B[1,2].NDW
NX = B[1,2].NX
NY = B[1,2].NY
NZ = B[1,2].NZ
NT = B[1,2].NT
i = rand(1:6)
flux[i] += rand(-1:1)
flux[i] %= NC
flux[i] += (flux[i] < 0) ? NC : 0
# flux[:] = rand(0:NC-1,6)
B = Initialize_Bfields(NC,flux,NDW,NX,NY,NZ,NT,condition = "tflux")
end
function U_update!(U,p,ϵ,Δτ,Dim,gauge_action)
temps = get_temporary_gaugefields(gauge_action)
temp1 = temps[1]
temp2 = temps[2]
expU = temps[3]
W = temps[4]
for μ=1:Dim
exptU!(expU,ϵ*Δτ,p[μ],[temp1,temp2])
mul!(W,expU,U[μ])
substitute_U!(U[μ],W)
end
end
function P_update!(U,B,p,ϵ,Δτ,Dim,gauge_action,temp1,temp2) # p -> p +factor*U*dSdUμ
NC = U[1].NC
temp = temp1
dSdUμ = temp2
factor = -ϵ*Δτ/(NC)
for μ=1:Dim
calc_dSdUμ!(dSdUμ,gauge_action,μ,U,B)
mul!(temp,U[μ],dSdUμ) # U*dSdUμ
Traceless_antihermitian_add!(p[μ],factor,temp)
end
end
function HMC_test_4D_dynamicalB(NX,NY,NZ,NT,NC,β)
Dim = 4
Nwing = 0
Random.seed!(123)
flux = [1,1,1,1,2,0]
U = Initialize_Gaugefields(NC,Nwing,NX,NY,NZ,NT,condition = "cold",randomnumber="Reproducible")
B = Initialize_Bfields(NC,flux,Nwing,NX,NY,NZ,NT,condition = "tflux")
L = [NX,NY,NZ,NT]
filename = "test/confs/U_beta6.0_L8_F111120_4000.txt"
load_BridgeText!(filename,U,L,NC)
temp1 = similar(U[1])
temp2 = similar(U[1])
if Dim == 4
comb = 6 #4*3/2
elseif Dim == 3
comb = 3
elseif Dim == 2
comb = 1
else
error("dimension $Dim is not supported")
end
factor = 1/(comb*U[1].NV*U[1].NC)
@time plaq_t = calculate_Plaquette(U,B,temp1,temp2)*factor
println("0 plaq_t = $plaq_t")
poly = calculate_Polyakov_loop(U,temp1,temp2)
println("0 polyakov loop = $(real(poly)) $(imag(poly))")
gauge_action = GaugeAction(U,B)
plaqloop = make_loops_fromname("plaquette")
append!(plaqloop,plaqloop')
β = β/2
push!(gauge_action,β,plaqloop)
#show(gauge_action)
p = initialize_TA_Gaugefields(U) #This is a traceless-antihermitian gauge fields. This has NC^2-1 real coefficients.
Uold = similar(U)
substitute_U!(Uold,U)
Bold = similar(B)
substitute_U!(Bold,B)
flux_old = zeros(Int, 6)
MDsteps = 50
temp1 = similar(U[1])
temp2 = similar(U[1])
comb = 6
factor = 1/(comb*U[1].NV*U[1].NC)
numaccepted = 0
numtrj = 100
for itrj = 1:numtrj
t = @timed begin
accepted = MDstep!(
gauge_action,
U,
B,
flux,
p,
MDsteps,
Dim,
Uold,
Bold,
flux_old,
temp1,
temp2
)
end
if get_myrank(U) == 0
println("Flux : ", flux)
# println("elapsed time for MDsteps: $(t.time) [s]")
end
numaccepted += ifelse(accepted,1,0)
#plaq_t = calculate_Plaquette(U,B,temp1,temp2)*factor
#println("$itrj plaq_t = $plaq_t")
if itrj % 10 == 0
@time plaq_t = calculate_Plaquette(U,B,temp1,temp2)*factor
println("$itrj plaq_t = $plaq_t")
poly = calculate_Polyakov_loop(U,temp1,temp2)
println("$itrj polyakov loop = $(real(poly)) $(imag(poly))")
println("acceptance ratio ",numaccepted/itrj)
end
end
return plaq_t,numaccepted/numtrj
end
function main()
β = 6.0
NX = 8
NY = 8
NZ = 8
NT = 8
NC = 3
HMC_test_4D_dynamicalB(NX,NY,NZ,NT,NC,β)
end
main()We can use Lüscher's gradient flow.
NX = 4
NY = 4
NZ = 4
NT = 4
Nwing = 0
NC = 3
flux=[1,0,0,0,0,1] # FLUX=[Z12,Z13,Z14,Z23,Z24,Z34]
U = Initialize_Gaugefields(NC,Nwing,NX,NY,NZ,NT,condition = "hot")
B = Initialize_Bfields(NC,flux,Nwing,NX,NY,NZ,NY,condition = "tflux")
temp1 = similar(U[1])
temp2 = similar(U[1])
comb = 6
factor = 1/(comb*U[1].NV*U[1].NC)
g = Gradientflow(U, B)
for itrj=1:100
flow!(U,B,g)
@time plaq_t = calculate_Plaquette(U,B,temp1,temp2)*factor
println("$itrj plaq_t = $plaq_t")
poly = calculate_Polyakov_loop(U,temp1,temp2)
println("$itrj polyakov loop = $(real(poly)) $(imag(poly))")
end
We can do the gradient flow with general terms with the use of Wilsonloop.jl, which is shown below.
The coefficient of the action can be complex. The complex conjugate of the action defined here is added automatically to make the total action hermitian.
The code is
using Random
using Test
using Gaugefields
using Wilsonloop
function gradientflow_test_4D(NX,NY,NZ,NT,NC)
Dim = 4
Nwing = 0
flux = [0,0,1,1,0,0]
Random.seed!(123)
U = Initialize_Gaugefields(NC,Nwing,NX,NY,NZ,NT,condition = "hot",randomnumber="Reproducible")
B = Initialize_Bfields(NC,flux,Nwing,NX,NY,NZ,NT,condition = "tflux")
temp1 = similar(U[1])
temp2 = similar(U[1])
if Dim == 4
comb = 6 #4*3/2
elseif Dim == 3
comb = 3
elseif Dim == 2
comb = 1
else
error("dimension $Dim is not supported")
end
factor = 1/(comb*U[1].NV*U[1].NC)
@time plaq_t = calculate_Plaquette(U,B,temp1,temp2)*factor
println("0 plaq_t = $plaq_t")
poly = calculate_Polyakov_loop(U,temp1,temp2)
println("0 polyakov loop = $(real(poly)) $(imag(poly))")
#Plaquette term
loops_p = Wilsonline{Dim}[]
for μ=1:Dim
for ν=μ:Dim
if ν == μ
continue
end
loop1 = Wilsonline([(μ,1),(ν,1),(μ,-1),(ν,-1)],Dim = Dim)
push!(loops_p,loop1)
end
end
#Rectangular term
loops = Wilsonline{Dim}[]
for μ=1:Dim
for ν=μ:Dim
if ν == μ
continue
end
loop1 = Wilsonline([(μ,1),(ν,2),(μ,-1),(ν,-2)],Dim = Dim)
push!(loops,loop1)
loop1 = Wilsonline([(μ,2),(ν,1),(μ,-2),(ν,-1)],Dim = Dim)
push!(loops,loop1)
end
end
listloops = [loops_p,loops]
listvalues = [1+im,0.1]
g = Gradientflow_general(U,B,listloops,listvalues,eps = 0.1)
for itrj=1:10
flow!(U,B,g)
if itrj % 10 == 0
@time plaq_t = calculate_Plaquette(U,B,temp1,temp2)*factor
println("$itrj plaq_t = $plaq_t")
poly = calculate_Polyakov_loop(U,temp1,temp2)
println("$itrj polyakov loop = $(real(poly)) $(imag(poly))")
end
end
return plaq_t
end
const eps = 0.1
println("4D system")
@testset "4D" begin
NX = 4
NY = 4
NZ = 4
NT = 4
Nwing = 1
@testset "NC=2" begin
β = 2.3
NC = 2
println("NC = $NC")
@time plaq_t = gradientflow_test_4D(NX,NY,NZ,NT,NC)
end
@testset "NC=3" begin
β = 5.7
NC = 3
println("NC = $NC")
@time plaq_t = gradientflow_test_4D(NX,NY,NZ,NT,NC)
end
@testset "NC=4" begin
β = 5.7
NC = 4
println("NC = $NC")
val = 0.7301232810349298
@time plaq_t =gradientflow_test_4D(NX,NY,NZ,NT,NC)
end
endHere, we show the HMC with MPI. the REPL and Jupyternotebook can not be used when one wants to use MPI. At first, in Julia REPL in the package mode,
add MPI
Then,
using MPI
MPI.install_mpiexecjl()and
export PATH="/<your home path>/.julia/bin/:$PATH"
The command is like:
mpiexecjl --project="Gaugefields" -np 2 julia mpi_sample.jl 1 1 1 2 true
1 1 1 2 means PEX PEY PEZ PET. In this case, the time-direction is diveded by 2.
The sample code is written as
using Random
using Gaugefields
using Wilsonloop
using LinearAlgebra
using MPI
if length(ARGS) < 5
error("USAGE: ","""
mpirun -np 2 exe.jl 1 1 1 2 true
""")
end
const pes = Tuple(parse.(Int64,ARGS[1:4]))
const mpi = parse(Bool,ARGS[5])
function calc_action(gauge_action,U,B,p)
NC = U[1].NC
Sg = -evaluate_GaugeAction(gauge_action,U,B)/NC
Sp = p*p/2
S = Sp + Sg
return real(S)
end
function MDstep!(gauge_action,U,B,p,MDsteps,Dim,Uold,temp1,temp2)
Δτ = 1.0/MDsteps
gauss_distribution!(p)
Sold = calc_action(gauge_action,U,B,p)
substitute_U!(Uold,U)
for itrj=1:MDsteps
U_update!(U,p,0.5,Δτ,Dim,gauge_action)
P_update!(U,B,p,1.0,Δτ,Dim,gauge_action,temp1,temp2)
U_update!(U,p,0.5,Δτ,Dim,gauge_action)
end
Snew = calc_action(gauge_action,U,B,p)
if get_myrank(U) == 0
println("Sold = $Sold, Snew = $Snew")
println("Snew - Sold = $(Snew-Sold)")
end
ratio = min(1,exp(-Snew+Sold)) # bug is fixed
r = rand()
if mpi
r = MPI.bcast(r, 0, MPI.COMM_WORLD)
end
#ratio = min(1,exp(-Snew+Sold))
if r > ratio
substitute_U!(U,Uold)
return false
else
return true
end
end
function U_update!(U,p,ϵ,Δτ,Dim,gauge_action)
temps = get_temporary_gaugefields(gauge_action)
temp1 = temps[1]
temp2 = temps[2]
expU = temps[3]
W = temps[4]
for μ=1:Dim
exptU!(expU,ϵ*Δτ,p[μ],[temp1,temp2])
mul!(W,expU,U[μ])
substitute_U!(U[μ],W)
end
end
function P_update!(U,B,p,ϵ,Δτ,Dim,gauge_action,temp1,temp2) # p -> p +factor*U*dSdUμ
NC = U[1].NC
temp = temp1
dSdUμ = temp2
factor = -ϵ*Δτ/(NC)
for μ=1:Dim
calc_dSdUμ!(dSdUμ,gauge_action,μ,U,B)
mul!(temp,U[μ],dSdUμ) # U*dSdUμ
Traceless_antihermitian_add!(p[μ],factor,temp)
end
end
function HMC_test_4D_tHooft(NX,NY,NZ,NT,NC,Flux,β)
Dim = 4
Nwing = 0
flux = Flux
Random.seed!(123)
if mpi
PEs = pes#(1,1,1,2)
U = Initialize_Gaugefields(NC,Nwing,NX,NY,NZ,NT,condition = "hot",mpi=true,PEs = PEs,mpiinit = false)
B = Initialize_Bfields(NC,flux,Nwing,NX,NY,NZ,NT,condition = "tflux",mpi=true,PEs = PEs,mpiinit = false)
else
U = Initialize_Gaugefields(NC,Nwing,NX,NY,NZ,NT,condition = "hot")
B = Initialize_Bfields(NC,flux,Nwing,NX,NY,NZ,NT,condition = "tflux")
end
if get_myrank(U) == 0
println("Flux : ", flux)
end
if get_myrank(U) == 0
println(typeof(U))
end
temp1 = similar(U[1])
temp2 = similar(U[1])
if Dim == 4
comb = 6 #4*3/2
elseif Dim == 3
comb = 3
elseif Dim == 2
comb = 1
else
error("dimension $Dim is not supported")
end
factor = 1/(comb*U[1].NV*U[1].NC)
@time plaq_t = calculate_Plaquette(U,B,temp1,temp2)*factor
if get_myrank(U) == 0
println("0 plaq_t = $plaq_t")
end
poly = calculate_Polyakov_loop(U,temp1,temp2)
if get_myrank(U) == 0
println("0 polyakov loop = $(real(poly)) $(imag(poly))")
end
gauge_action = GaugeAction(U,B)
plaqloop = make_loops_fromname("plaquette")
append!(plaqloop,plaqloop')
β = β/2
push!(gauge_action,β,plaqloop)
#show(gauge_action)
p = initialize_TA_Gaugefields(U) #This is a traceless-antihermitian gauge fields. This has NC^2-1 real coefficients.
Uold = similar(U)
substitute_U!(Uold,U)
MDsteps = 50
temp1 = similar(U[1])
temp2 = similar(U[1])
comb = 6
factor = 1/(comb*U[1].NV*U[1].NC)
numaccepted = 0
numtrj = 100
for itrj = 1:numtrj
t = @timed begin
accepted = MDstep!(gauge_action,U,B,p,MDsteps,Dim,Uold,temp1,temp2)
end
if get_myrank(U) == 0
println("elapsed time for MDsteps: $(t.time) [s]")
end
numaccepted += ifelse(accepted,1,0)
#plaq_t = calculate_Plaquette(U,B,temp1,temp2)*factor
#println("$itrj plaq_t = $plaq_t")
if itrj % 10 == 0
plaq_t = calculate_Plaquette(U,B,temp1,temp2)*factor
if get_myrank(U) == 0
println("$itrj plaq_t = $plaq_t")
end
poly = calculate_Polyakov_loop(U,temp1,temp2)
if get_myrank(U) == 0
println("$itrj polyakov loop = $(real(poly)) $(imag(poly))")
println("acceptance ratio ",numaccepted/itrj)
end
end
end
return plaq_t,numaccepted/numtrj
end
function main()
β = 5.7
NX = 4
NY = 4
NZ = 4
NT = 4
NC = 3
Flux = [0,0,1,1,0,0]
#HMC_test_4D(NX,NY,NZ,NT,NC,β)
HMC_test_4D_tHooft(NX,NY,NZ,NT,NC,Flux,β)
end
main()We can access the gauge field defined on the bond between two neigbohr points.
In 4D system, the gauge field is like u[ic,jc,ix,iy,iz,it].
There are four directions in 4D system. Gaugefields.jl uses the array like:
NX = 4
NY = 4
NZ = 4
NT = 4
Nwing = 1
Dim = 4
U = Initialize_Gaugefields(NC,Nwing,NX,NY,NZ,NT,condition = "cold")
In the later exaples, we use, mu=1 and u=U[mu] as an example.
If you want to get the hermitian conjugate of the gauge fields, you can do like
u'This is evaluated with the lazy evaluation.
So there is no memory copy.
This returms
If you want to shift the gauge fields, you can do like
shifted_u = shift_U(u, shift)This is also evaluated with the lazy evaluation.
Here shift is shift=(1,0,0,0) for example.
If you want to calculate the matrix-matrix multiplicaetion on each lattice site, you can do like
As a mathematical expression, for matrix-valued fields A(n), B(n),
we define "matrix-field matrix-field product" as,
for all site index n.
In our package, this is expressed as,
mul!(C,A,B)which means C = A*B on each lattice site.
Here A, B, C are same type of u.
If you want to calculate the trace of the gauge field, you can do like
tr(A)It is useful to evaluation actions. This trace operation summing up all indecis, spacetime and color.
This package and Wilsonloop.jl enable you to perform several calculations. Here we demonstrate them.
Some of them will be simplified in LatticeQCD.jl.
We develop Wilsonloop.jl, which is useful to calculate Wilson loops. If you want to use this, please install like
add Wilsonloop.jl
For example, if you want to calculate the following quantity:
which is Z(Nc) 1-form gauge invariant [arXiv:2303.10977 [hep-lat]].
You can use Wilsonloop.jl as follows
using Wilsonloop
loop = [(1,1),(2,1),(1,-1),(2,-1)]
w = Wilsonline(loop)The output is L"$U_{1}(n)U_{2}(n+e_{1})U^{\dagger}_{1}(n+e_{2})U^{\dagger}_{2}(n)$".
Then, you can evaluate this loop with the use of the Gaugefields.jl like:
using LinearAlgebra
NX = 4
NY = 4
NZ = 4
NT = 4
NC = 3
Nwing = 0
Dim = 4
U = Initialize_Gaugefields(NC,Nwing,NX,NY,NZ,NT,condition = "cold")
flux=[1,0,0,0,0,1] # FLUX=[Z12,Z13,Z14,Z23,Z24,Z34]
B = Initialize_Bfields(NC,flux,Nwing,NX,NY,NZ,NY,condition = "tflux")
temp1 = similar(U[1])
temp2 = similar(U[1])
temp3 = similar(U[1])
temp4 = similar(U[1])
V = similar(U[1])
evaluate_gaugelinks!(V,w,U,B,[temp1,temp2,temp3,temp4])
println(tr(V))For example, if you want to calculate the clover operators, you can define like:
function make_cloverloop(μ,ν,Dim)
loops = Wilsonline{Dim}[]
loop_righttop = Wilsonline([(μ,1),(ν,1),(μ,-1),(ν,-1)],Dim = Dim) # Pmunu
push!(loops,loop_righttop)
loop_rightbottom = Wilsonline([(ν,-1),(μ,1),(ν,1),(μ,-1)],Dim = Dim) # Qmunu
push!(loops,loop_rightbottom)
loop_leftbottom= Wilsonline([(μ,-1),(ν,-1),(μ,1),(ν,1)],Dim = Dim) # Rmunu
push!(loops,loop_leftbottom)
loop_lefttop = Wilsonline([(ν,1),(μ,-1),(ν,-1),(μ,1)],Dim = Dim) # Smunu
push!(loops,loop_lefttop)
return loops
endWe can calculate actions from this packages with fixed gauge fields U. We introduce the concenpt "Scalar-valued neural network", which is S(U) -> V, where U and V are gauge fields.
using Gaugefields
using LinearAlgebra
function test1()
NX = 4
NY = 4
NZ = 4
NT = 4
Nwing = 0
Dim = 4
NC = 3
flux=[1,0,0,0,0,1]
U = Initialize_Gaugefields(NC,Nwing,NX,NY,NZ,NT,condition = "cold")
B = Initialize_Bfields(NC,flux,Nwing,NX,NY,NZ,NY,condition = "tflux")
gauge_action = GaugeAction(U,B) #empty network
plaqloop = make_loops_fromname("plaquette") #This is a plaquette loops.
append!(plaqloop,plaqloop') #We need hermitian conjugate loops for making the action real.
β = 1 #This is a coefficient.
push!(gauge_action,β,plaqloop)
show(gauge_action)
Uout = evaluate_GaugeAction_untraced(gauge_action,U,B)
println(tr(Uout))
end
test1()The output is
----------------------------------------------
Structure of the actions for Gaugefields
num. of terms: 1
-------------------------------
1-st term:
coefficient: 1.0
-------------------------
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)$"
7-th loop
L"$U_{2}(n)U_{1}(n+e_{2})U^{\dagger}_{2}(n+e_{1})U^{\dagger}_{1}(n)$"
8-th loop
L"$U_{3}(n)U_{1}(n+e_{3})U^{\dagger}_{3}(n+e_{1})U^{\dagger}_{1}(n)$"
9-th loop
L"$U_{4}(n)U_{1}(n+e_{4})U^{\dagger}_{4}(n+e_{1})U^{\dagger}_{1}(n)$"
10-th loop
L"$U_{3}(n)U_{2}(n+e_{3})U^{\dagger}_{3}(n+e_{2})U^{\dagger}_{2}(n)$"
11-th loop
L"$U_{4}(n)U_{2}(n+e_{4})U^{\dagger}_{4}(n+e_{2})U^{\dagger}_{2}(n)$"
12-th loop
L"$U_{4}(n)U_{3}(n+e_{4})U^{\dagger}_{4}(n+e_{3})U^{\dagger}_{3}(n)$"
-------------------------
----------------------------------------------
8928.0 + 0.0im
function calculate_topological_charge_plaq(U::Array{T,1}, B::Array{T,2}, temp_UμνTA, temps) where {T}
UμνTA = temp_UμνTA
numofloops = calc_UμνTA!(UμνTA, "plaq", U, B, temps)
Q = calc_Q(UμνTA, numofloops, U)
return Q
end
function calculate_topological_charge_clover(U::Array{T,1}, B::Array{T,2}, temp_UμνTA, temps) where {T}
UμνTA = temp_UμνTA
numofloops = calc_UμνTA!(UμνTA, "clover", U, B, temps)
Q = calc_Q(UμνTA, numofloops, U)
return Q
end
function calculate_topological_charge_improved(
U::Array{T,1},
B::Array{T,2},
temp_UμνTA,
Qclover,
temps,
) where {T}
UμνTA = temp_UμνTA
numofloops = calc_UμνTA!(UμνTA, "rect", U, B, temps)
Qrect = 2 * calc_Q(UμνTA, numofloops, U)
c1 = -1 / 12
c0 = 5 / 3
Q = c0 * Qclover + c1 * Qrect
return Q
end
function calc_UμνTA!(
temp_UμνTA,
name::String,
U::Array{T,1},
B::Array{T,2},
temps,
) where {NC,Dim,T<:AbstractGaugefields{NC,Dim}}
loops_μν, numofloops = calc_loopset_μν_name(name, Dim)
calc_UμνTA!(temp_UμνTA, loops_μν, U, B, temps)
return numofloops
end
function calc_UμνTA!(
temp_UμνTA,
loops_μν,
U::Array{T,1},
B::Array{T,2},
temps,
) where {NC,Dim,T<:AbstractGaugefields{NC,Dim}}
UμνTA = temp_UμνTA
for μ = 1:Dim
for ν = 1:Dim
if ν == μ
continue
end
evaluate_gaugelinks!(temps[1], loops_μν[μ, ν], U, B, temps[2:6])
Traceless_antihermitian!(UμνTA[μ, ν], temps[1])
end
end
return
end
#=
implementation of topological charge is based on
https://arxiv.org/abs/1509.04259
=#
function calc_Q(UμνTA, numofloops, U::Array{<:AbstractGaugefields{NC,Dim},1}) where {NC,Dim}
Q = 0.0
if Dim == 4
ε(μ, ν, ρ, σ) = epsilon_tensor(μ, ν, ρ, σ)
else
error("Dimension $Dim is not supported")
end
for μ = 1:Dim
for ν = 1:Dim
if ν == μ
continue
end
Uμν = UμνTA[μ, ν]
for ρ = 1:Dim
for σ = 1:Dim
if ρ == σ
continue
end
Uρσ = UμνTA[ρ, σ]
s = tr(Uμν, Uρσ)
Q += ε(μ, ν, ρ, σ) * s / numofloops^2
end
end
end
end
return -Q / (32 * (π^2))
end
#topological charge
function epsilon_tensor(mu::Int, nu::Int, rho::Int, sigma::Int)
sign = 1 # (3) 1710.09474 extended epsilon tensor
if mu < 0
sign *= -1
mu = -mu
end
if nu < 0
sign *= -1
nu = -nu
end
if rho < 0
sign *= -1
rh = -rho
end
if sigma < 0
sign *= -1
sigma = -sigma
end
epsilon = zeros(Int, 4, 4, 4, 4)
epsilon[1, 2, 3, 4] = 1
epsilon[1, 2, 4, 3] = -1
epsilon[1, 3, 2, 4] = -1
epsilon[1, 3, 4, 2] = 1
epsilon[1, 4, 2, 3] = 1
epsilon[1, 4, 3, 2] = -1
epsilon[2, 1, 3, 4] = -1
epsilon[2, 1, 4, 3] = 1
epsilon[2, 3, 1, 4] = 1
epsilon[2, 3, 4, 1] = -1
epsilon[2, 4, 1, 3] = -1
epsilon[2, 4, 3, 1] = 1
epsilon[3, 1, 2, 4] = 1
epsilon[3, 1, 4, 2] = -1
epsilon[3, 2, 1, 4] = -1
epsilon[3, 2, 4, 1] = 1
epsilon[3, 4, 1, 2] = 1
epsilon[3, 4, 2, 1] = -1
epsilon[4, 1, 2, 3] = -1
epsilon[4, 1, 3, 2] = 1
epsilon[4, 2, 1, 3] = 1
epsilon[4, 2, 3, 1] = -1
epsilon[4, 3, 1, 2] = -1
epsilon[4, 3, 2, 1] = 1
return epsilon[mu, nu, rho, sigma] * sign
end
function calc_loopset_μν_name(name, Dim)
loops_μν = Array{Vector{Wilsonline{Dim}},2}(undef, Dim, Dim)
if name == "plaq"
numofloops = 1
for μ = 1:Dim
for ν = 1:Dim
loops_μν[μ, ν] = Wilsonline{Dim}[]
if ν == μ
continue
end
plaq = make_plaq(μ, ν, Dim = Dim)
push!(loops_μν[μ, ν], plaq)
end
end
elseif name == "clover"
numofloops = 4
for μ = 1:Dim
for ν = 1:Dim
loops_μν[μ, ν] = Wilsonline{Dim}[]
if ν == μ
continue
end
loops_μν[μ, ν] = make_cloverloops_topo(μ, ν, Dim = Dim)
end
end
elseif name == "rect"
numofloops = 8
for μ = 1:4
for ν = 1:4
if ν == μ
continue
end
loops = Wilsonline{Dim}[]
loop_righttop = Wilsonline([(μ, 2), (ν, 1), (μ, -2), (ν, -1)])
loop_lefttop = Wilsonline([(ν, 1), (μ, -2), (ν, -1), (μ, 2)])
loop_rightbottom = Wilsonline([(ν, -1), (μ, 2), (ν, 1), (μ, -2)])
loop_leftbottom = Wilsonline([(μ, -2), (ν, -1), (μ, 2), (ν, 1)])
push!(loops, loop_righttop)
push!(loops, loop_lefttop)
push!(loops, loop_rightbottom)
push!(loops, loop_leftbottom)
loop_righttop = Wilsonline([(μ, 1), (ν, 2), (μ, -1), (ν, -2)])
loop_lefttop = Wilsonline([(ν, 2), (μ, -1), (ν, -2), (μ, 1)])
loop_rightbottom = Wilsonline([(ν, -2), (μ, 1), (ν, 2), (μ, -1)])
loop_leftbottom = Wilsonline([(μ, -1), (ν, -2), (μ, 1), (ν, 2)])
push!(loops, loop_righttop)
push!(loops, loop_lefttop)
push!(loops, loop_rightbottom)
push!(loops, loop_leftbottom)
loops_μν[μ, ν] = loops
end
end
else
error("$name is not supported")
end
return loops_μν, numofloops
end
function make_cloverloops_topo(μ, ν; Dim = 4)
loops = Wilsonline{Dim}[]
loop_righttop = Wilsonline([(μ, 1), (ν, 1), (μ, -1), (ν, -1)])
loop_lefttop = Wilsonline([(ν, 1), (μ, -1), (ν, -1), (μ, 1)])
loop_rightbottom = Wilsonline([(ν, -1), (μ, 1), (ν, 1), (μ, -1)])
loop_leftbottom = Wilsonline([(μ, -1), (ν, -1), (μ, 1), (ν, 1)])
push!(loops, loop_righttop)
push!(loops, loop_lefttop)
push!(loops, loop_rightbottom)
push!(loops, loop_leftbottom)
return loops
endWe can calculate the topological charge as
Qplaq = calculate_topological_charge_plaq(U,B,temp_UμνTA,temps[1:6]),
Qclover = calculate_topological_charge_clover(U,B,temp_UμνTA,temps[1:6]),
Qimproved= calculate_topological_charge_improved(U,B,temp_UμνTA,Qclover,temps[1:6]).
We can easily calculate the matrix derivative of the actions. The matrix derivative is defined as
We can calculate this like
dSdUμ = calc_dSdUμ(gauge_action,μ,U,B)or
calc_dSdUμ!(dSdUμ,gauge_action,μ,U,B)With the use of the matrix derivative, we can do the Hybrid Monte Carlo method. The simple code is as follows.
using Gaugefields
using LinearAlgebra
function MDtest!(gauge_action,U,B,Dim)
p = initialize_TA_Gaugefields(U) #This is a traceless-antihermitian gauge fields. This has NC^2-1 real coefficients.
Uold = similar(U)
substitute_U!(Uold,U)
MDsteps = 50
temp1 = similar(U[1])
temp2 = similar(U[1])
comb = 6
factor = 1/(comb*U[1].NV*U[1].NC)
numaccepted = 0
numtrj = 100
for itrj = 1:numtrj
accepted = MDstep!(gauge_action,U,B,p,MDsteps,Dim,Uold)
numaccepted += ifelse(accepted,1,0)
plaq_t = calculate_Plaquette(U,B,temp1,temp2)*factor
println("$itrj plaq_t = $plaq_t")
println("acceptance ratio ",numaccepted/itrj)
end
endWe define the functions as
function calc_action(gauge_action,U,B,p)
NC = U[1].NC
Sg = -evaluate_GaugeAction(gauge_action,U,B)/NC
Sp = p*p/2
S = Sp + Sg
return real(S)
end
function MDstep!(gauge_action,U,B,p,MDsteps,Dim,Uold)
Δτ = 1.0/MDsteps
gauss_distribution!(p)
Sold = calc_action(gauge_action,U,B,p)
substitute_U!(Uold,U)
for itrj=1:MDsteps
U_update!(U,p,0.5,Δτ,Dim,gauge_action)
P_update!(U,B,p,1.0,Δτ,Dim,gauge_action)
U_update!(U,p,0.5,Δτ,Dim,gauge_action)
end
Snew = calc_action(gauge_action,U,B,p)
println("Sold = $Sold, Snew = $Snew")
println("Snew - Sold = $(Snew-Sold)")
ratio = min(1,exp(-Snew+Sold))
if rand() > ratio
substitute_U!(U,Uold)
return false
else
return true
end
end
function U_update!(U,p,ϵ,Δτ,Dim,gauge_action)
temps = get_temporary_gaugefields(gauge_action)
temp1 = temps[1]
temp2 = temps[2]
expU = temps[3]
W = temps[4]
for μ=1:Dim
exptU!(expU,ϵ*Δτ,p[μ],[temp1,temp2])
mul!(W,expU,U[μ])
substitute_U!(U[μ],W)
end
end
function P_update!(U,B,p,ϵ,Δτ,Dim,gauge_action) # p -> p +factor*U*dSdUμ
NC = U[1].NC
temp = gauge_action._temp_U[end]
dSdUμ = similar(U[1])
factor = -ϵ*Δτ/(NC)
for μ=1:Dim
calc_dSdUμ!(dSdUμ,gauge_action,μ,U,B)
mul!(temp,U[μ],dSdUμ) # U*dSdUμ
Traceless_antihermitian_add!(p[μ],factor,temp)
end
endThen, we can do the HMC:
function test1()
NX = 4
NY = 4
NZ = 4
NT = 4
Nwing = 0
Dim = 4
NC = 3
flux=[1,0,0,0,0,1]
U = Initialize_Gaugefields(NC,Nwing,NX,NY,NZ,NT,condition = "cold")
B = Initialize_Bfields(NC,flux,Nwing,NX,NY,NZ,NY,condition = "tflux")
gauge_action = GaugeAction(U,B)
plaqloop = make_loops_fromname("plaquette")
append!(plaqloop,plaqloop') # add hermitian conjugate
β = 5.7/2 # real part; re[p] = (p+p')/2
push!(gauge_action,β,plaqloop)
show(gauge_action)
MDtest!(gauge_action,U,B,Dim)
end
test1()