https://github.com/o-morikawa/wind3d
Discrete approximation to 3D winding number mapping T^3->U(N)
https://github.com/o-morikawa/wind3d
gradient-flow hep hep-lat julia julia-language three-dimensions torus winding-number-algorithm
Last synced: about 1 year ago
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Discrete approximation to 3D winding number mapping T^3->U(N)
- Host: GitHub
- URL: https://github.com/o-morikawa/wind3d
- Owner: o-morikawa
- License: mit
- Created: 2024-11-27T09:25:03.000Z (over 1 year ago)
- Default Branch: main
- Last Pushed: 2024-12-12T02:36:08.000Z (over 1 year ago)
- Last Synced: 2025-02-01T20:09:36.706Z (over 1 year ago)
- Topics: gradient-flow, hep, hep-lat, julia, julia-language, three-dimensions, torus, winding-number-algorithm
- Language: Mathematica
- Homepage: https://sites.google.com/view/o-morikawa
- Size: 34.6 MB
- Stars: 0
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# Wind3D
This provides the actual code and more datas in [arXiv:2412.03888 [hep-lat]](https://arxiv.org/abs/2412.03888).
We consider the winding number on numerical simulations.
We propose an effective approach to give an approximate integer for
```math
\int \tr(g^{-1}dg)^{3},
```
where $g$ is a smooth map: $X \to U(N)$.
To this end, we utilize a Julia repository, [o-morikawa/Gaugefields.jl](https://github.com/o-morikawa/Gaugefields.jl),
and formulate a gradient-flow method even on a course lattice.
We use the following terminologies:
- "map": Mapping function from $T^3$ to $SU(N)$ (See [hep-lat/0208057](https://arxiv.org/abs/hep-lat/0208057))
- "test": Random unitary matrices for $T^3\to U(N)$
Each directory contains
- src: main function which runs on [o-morikawa/Gaugefields.jl](https://github.com/o-morikawa/Gaugefields.jl)
- output: data and simple figures
- action: flow-time dependence of lattice action, which is a monotonically decreasing function
- det: flow-time dependence of determinant of configurations
- msearch: m (a param of "map" case) dependence of lattice action
- wind: flow-time dependence of winding number
- with "H": lattice improvement for $gd^3g$
- no "H": naive lattice definition of winding number
- m_nb: Mathematica notebooks for small lattice size
A parameter, $\eta$, is introduced for lattice improvement as [arXiv:2412.03888 [hep-lat]](https://arxiv.org/abs/2412.03888).
## Sample code
Use o-morikawa/Gaugefields.jl instead of the official repository, [Gaugefields.jl](https://github.com/akio-tomiya/Gaugefields.jl).
```julia
using Random
using Gaugefields
using LinearAlgebra
using Wilsonloop
function UN_test_3D(NX,NY,NT,NC,η;λ=1,rε=0.1)
Dim = 3
L = NX
n = 4
println("Run Configurations: n=$n")
eps = 0.01
flow_number = 6000
step = 10
println("L=$L, eps=$eps,flow=$(eps*flow_number)")
eta = η
randscale = λ
rand_eps = rε
println("eta: $eta, randscale: $randscale, rand_eps: $(rand_eps)")
for i = 1:n
Random.seed!(123)
if i == 1
println("Configuration 1: Cold start")
U = Initialize_3D_UN_Gaugefields(
NC,NX,NY,NT,
condition = "cold",
randomnumber="Random"
)
elseif i == 2
println("Configuration 2: Hot start")
U = Initialize_3D_UN_Gaugefields(
NC,NX,NY,NT,
condition = "hot",
randomnumber="Random",
randscale=randscale,
)
elseif i == 3
println("Configuration 3: Test mapping as T^3->SU(2)")
U = Initialize_3D_UN_Gaugefields(
NC,NX,NY,NT,
condition = "test_map",
m = -1, # -1, 1, 3, 5, 7
)
elseif i == 4
println("Configuration 4: Test mapping as T^3->SU(2) with Random noise")
U = Initialize_3D_UN_Gaugefields(
NC,NX,NY,NT,
condition = "test_map_rand",
m = -1,
randomnumber="Random",
reps = rand_eps,
)
end
println(typeof(U))
temps = Temporalfields(U, num=9)
println(typeof(temps))
println(winding_UN_3D(U,temps))
if i==1 || i==2
g = Gradientflow_eta_3D(U, eta, eps=eps)
else
g = Gradientflow_TA_eta_3D(U, eta, eps=eps)
end
flownumber = flow_number
W = winding_UN_3D(U,temps)
Wh =winding_UN_3D(U,1,temps)
S = calc_gdgaction_3D(U,eta,temps)
D = det_unitary(U)
println("flow time: 0")
println("W=$(W), Wh=$(Wh), S=$(S), Det=$(D)")
for iflow = 1:flownumber
flow!(U, g)
if iflow%step==0
W = winding_UN_3D(U,temps)
Wh =winding_UN_3D(U,1,temps)
S = calc_gdgaction_3D(U,eta,temps)
D = det_unitary(U)
println("flow time: $(iflow*eps)")
println("W=$(W), Wh=$(Wh), S=$(S), Det=$(D)")
end
end
end
end
function main()
NX = 20
NY = 20
NT = 20
NC = 2
η = 1
@time UN_test_3D(NX,NY,NT,NC,η,λ=5,rε=0.5)
end
main()
```