Gauge Smearing

The fundamental principle behind gauge smearing is to take information about neighbouring links and use this information to augment a given link. In this sense, low frequency information is propagated though the lattice and high frequency information is suppressed. A common object used in gauge smearing is the 1x1 link staple,

          x+nu-->--x+mu+nu
            |        |
            ^        v
            |        |
C_{mu}(x) = x       x+mu      + ...

which is defined using the following formula:

C_{mu}(x) = Sum_{mu != nu} rho_{mu, nu}[U_{nu}(x)U_{mu}(x+nu)U^dag_{nu}(x+mu) +
                                        U^dag_{nu}(x-nu)U_{mu}(x-nu)U_{nu}(x+mu-nu)]

where rho defines the weight of each staple. Typically, all elements of rho are either zero or one (include, or do not include).

In a typical smearing calculation, a new link U^prime_{mu}(x) is constructed using the lattice, then the entire lattice links are replaced with the new links. This process is performed some O(50) times.

APE gauge smearing

When APE smearing, one produces the new links iteratively using the following formula:

U^tilde_{mu}(x) = (1 - alpha) * U_{mu}(x) + alpha/6 * C_{mu}(x)

Notice that each new link is NOT a member of SU(3). As a result each new link is projected down to SU(3) before the next iteration,

U^prime_{mu}(x) = P_{su3}[U^tilde_{mu}(x)]

To call the APE smearing routine in your application, use the QUDA interface function:

void performAPEnStep(unsigned int nSteps, double alpha);

which defines the alpha parameter in APE smearing, and the number of steps to perform.

STOUT gauge smearing

Wilson and Symanzik Flow