Dirac documentation

docs/make.jl

@@ -10,5 +10,6 @@ makedocs(sitename="LatticeGPU", modules=[LatticeGPU], doctest=true,
              "Space-time" => "space.md",
              "Groups and algebras" => "groups.md",
              "Fields" => "fields.md"
+             "Dirac" => "dirac.md"
              ], 
          repo = "https://igit.ific.uv.es/alramos/latticegpu.jl")

docs/src/dirac.md

+
+# Dirac operator
+
+The module `Dirac` has the necessary stuctures and function 
+to simulate non-dynamical 4-dimensional Wilson fermions.
+
+There are two main data structures in this module, the structure [`DiracParam`](@ref)
+
+```@docs
+DiracParam
+```
+
+and the workspace [`DiracWorkspace`](@ref)
+
+```@docs
+DiracWorkspace
+```
+
+The workspace stores four fermion fields, namely `.sr`, `.sp`, `.sAp` and `.st`, used
+for different purposes. If the representation is either `SU2fund` of `SU3fund`, an extra
+field with values in `U2alg`/`U3alg` is created to store the clover, used for the improvent.
+
+## Functions
+
+The functions [`Dw!`](@ref), [`g5Dw!`](@ref) and [`DwdagDw!`](@ref) are all related to the 
+Wilson-Dirac operator.
+
+The action of the Dirac operator `Dw!` is the following:
+
+```math
+D\psi (\vec{x} = x_1,x_2,x_3,x_4) = (4 + m_0)psi(\vec{x})
+```
+```math
+    - \frac{1}{2}\sum_{\mu = 1}^4 \theta (\mu) (1-\gamma_\mu) U_\mu(\vec{x}) \psi(\vec{x} + \hat{\mu}) 
+```
+```math
+    + \theta^* (\mu) (1 + \gamma_\mu) U^{-1}_\mu(\vec{x} - \hat{\mu}) \psi(\vec{x} - \hat{\mu})
+```
+
+where $$m_0$$ and $$\theta$$ are respectively the values `.m0` and `.th` of [`DiracParam`](@ref).
+Note that $$|\theta(i)|=1$$ is not built into the code, so it should be imposed explicitly. 
+
+Additionally, if |`dpar.csw`| > 1.0E-10, the clover term is assumed to be stored in `ymws.csw`, which
+can be done via the [`Csw`](@ref) function. In this case we have an extra term in `Dw!`:
+
+```math
+    \frac{i}{2}C_{sw} \sum_{\pi = 1}^6 F^{cl}_\pi \sigma_\pi \psi(\vec{x})
+```
+where the $$\sigma$$ matrices are those described in the `Spinors` module and the index $$\pi$$ runs
+as specified in `lp.plidx`.
+
+If the boudary conditions, defined in `lp`, are either `BC_SF_ORBI,D` or `BC_SF_AFWB`, the 
+improvement term 
+
+```math
+    (c_t -1) (\delta_{x_4,a} \psi(\vec{x}) + \delta_{x_4,T-a} \psi(\vec{x}))
+```
+is added. Since the time-slice $$t=T$$ is not stored, this accounts to modifying the second
+and last time-slice. 
+
+Note that the Dirac operator for SF boundary conditions assumes that the value of the field 
+in the first time-slice is zero. To enforce this, we have the function
+
+```@docs
+SF_bndfix!
+```
+
+The function [`Csw`](@ref) is used to store the clover in `dws.csw`. It is computed 
+according to the expression
+
+```math
+F_{\mu,\nu} = \frac{1}{8} (Q_{\mu \nu} - Q_{\nu \mu})
+```
+
+where 
+```math
+Q_{\mu\nu} = U_\mu(\vec{x})U_{\nu}(x+\mu)U_{\mu}^{-1}(\vec{x}+\nu)U_{\nu}(\vec{x}) +
++ U_{\nu}^{-1}(\vec{x}-\nu) U_\mu (\vec{x}-\nu) U_{\nu}(\vec{x} +\mu - \nu) U^{-1}_{\mu}(\vec{x}) +
++ U^{-1}_{\mu}(x-\mu)U_\nu^{-1}(\vec{x} - \mu - \nu)U_\mu(\vec{x} - \mu - \nu)U_\nu^{-1}(x-\nu) +
++ U_{\nu}(\vec{x})U_{\mu}^{-1}(\vec{x} + \nu - \mu)U^{-1}_{\nu}(\vec{x} - \mu)U_\mu(\vec{x}-\mu)
+
+```
+
+The correspondence between the tensor field and the GPU-Array is the following:
+```math
+F[b,1,r] \to F_{41}(b,r) ,\quad F[b,2,r] \to F_{42}(b,r) ,\quad F[b,3,r] \to F_{43}(b,r) 
+```
+```math
+F[b,4,r] \to F_{31}(b,r) ,\quad F[b,5,r] \to F_{32}(b,r) ,\quad F[b,6,r] \to F_{21}(b,r) 
+```
+where $$(b,r)$$ labels the lattice points as explained in the module `Space`
+
+The function [`pfrandomize!`](@ref), userfull for stochastic sources, is also present. 
+
+The generic interface of these functions reads
+
+```@docs
+Dw!
+g5Dw!
+DwdagDw!
+Csw!
+pfrandomize!
+```