3.8 KiB
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
DiracParam
and the workspace DiracWorkspace
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!, g5Dw! and DwdagDw! are all related to the
Wilson-Dirac operator.
The action of the Dirac operator Dw! is the following:
D_w\psi (\vec{x} = x_1,x_2,x_3,x_4) = (4 + m_0 + i \mu \gamma_5)\psi(\vec{x}) -
- \frac{1}{2}\sum_{\mu = 1}^4 \theta (\mu) (1-\gamma_\mu) U_\mu(\vec{x}) \psi(\vec{x} + \hat{\mu}) + \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.
Note that |\theta(\mu)|=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! function. In this case we have the Sheikholeslami–Wohlert (SW) term
in Dw!:
\delta D_w^{sw} = \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
\delta D_w^{SF} = (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
SF_bndfix!
Note that this is not enforced in the Dirac operators, so if the field so does not satisfy SF
boundary conditions, it will not (in general) satisfy them after applying Dw!
or g5Dw!. This function is called for the function DwdagDw!, so in this case
so will always be a proper SF field after calling this function.
The function Csw! is used to store the clover in dws.csw. It is computed
according to the expression
F_{\mu,\nu} = \frac{1}{8} (Q_{\mu \nu} - Q_{\nu \mu})
where
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:
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)
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!, userfull for stochastic sources, is also present. It
randomizes a fermion field either in all the space or in a specifit time-slice.
The generic interface of these functions reads
Dw!
g5Dw!
DwdagDw!
Csw!
pfrandomize!
mtwmdpar