Supported in part by the National Natural Science Foundation of China (NSFC) (11335001, 11275169)
We evaluate the topological charge density of SU(3) gauge fields on a lattice by calculating the trace of the overlap Dirac matrix employing the symmetric multiprobing (SMP) method in 3 modes. Since the topological charge
Article funded by SCOAP^{3}
Topological charge
where the trace is over color indices, and
where the trace is over color and Dirac indices,
where the hopping parameter
in which the Zolotarev series expansion [
Fortunately,
In this work, we employ the symmetric multiprobing(SMP) method to evaluate the topological charge density by calculating
The overlap Dirac operator
where the trace is over the Dirac and color indices,
which is equivalent to solving the following linear equation:
where
This equation has a high computational cost when solving for a large matrix
We know from above that calculations of the trace of
then Eq. (5) can be rewritten as:
where
are spacetime offdiagonal elements of
where
(color online) The locality of
The approximate topological charge density
We see that the approximate density of both sites can be easily calculated from
We first mark all sites in the lattice with a color label, called the coloring algorithm. The sites with the same color form a subset of all lattice sites. The multiprobing source can be constructed from a subset as follows: choose a color and Dirac index, construct point sources on sites in the subset, and add all these point source vectors.
In order to reduce the calculation cost, we would like to reduce the number of MP sources by combining more point sources in one MP source. However, the errors introduced by MP sources depend exponentially on the distance between the sites in the subset, as shown in Eq. (12). To control the systematic errors, we should include in one MP source the sites that as far as possible from each other. Therefore, an appropriate coloring algorithm is important to optimize the balance of the systematic errors and calculation cost.
We chose a symmetric coloring algorithm and the definition of Euclid distance in this work, which marks the sites with the same color in each direction with a fixed distance. There are other algorithms, like GCA (Greedy Coloring Algorithm) [
(color online) Example of the Split Mode on a
(color online) Examples of the Split Mode on a
Furthermore, we can divided a subset into two subsets by marking half of the sites with a new color, where each site has a different color with respect to its neighbors, see examples in
We can also combine two subsets into a new subset which includes two sites with equal spacetime offsets
(color online) Example of the Combined Mode on a
(color online) Example of the Combined Mode on a
Let us consider a lattice
where
The topological charge density on any site
where the second term sums the corresponding offdiagonal components of
The lattice sites can be divided into
where Eq. (12) is used, and the sum over the color and Dirac indices results in a factor of 12. In practice, the errors are far smaller than the upper bound. We define the minimal distance
and we believe that
For the case of a spatial symmetric lattice
Therefore, the choice of
We calculate the topological charge density and the total topological charge for pure gauge lattice configurations with Iwasaki gauge action [
where
First, we calculate the topological charge density exactly with point sources on a
(color online) The topological charge density in the
A series of gauge field configurations were then generated with different lattice spacing
Lattice setup and the multiprobing scheme. Symmetric schemes
lattice 

No. of config.  SMP scheme 



0.1  20 

2,

0.133  20  

0.1  20 

2,


0.1  20 

2,


0.1  20 

2,

The number of sources
lattice 




192, 384, 972, 1944, 1536, 3072, 15552 


192, 384, 1536, 3072, 6144, 49152 


192, 384, 972, 1944, 1536, 3072 


192, 384, 1536, 3072 

Since
(color online) The series of
(color online) The average absolute error of
Considering the same lattice with different lattice spacing
(color online) Comparison of the average absolute error of
(color online) Comparison of the average absolute error of
(color online) The errors versus the efficiency for all lattices. A larger lattice has a higher efficiency.
All results of
The average absolute systematic error






12^{4}  0.1  (6, 0)  192  0.471(71)  16^{4}  0.1  (8, 1)  384  0.384(79)  
(6, 1)  384  0.220(37)  (4, 2)  1536  0.0117(21)  
(4, 0)  972  0.141(24)  (4, 0)  3072  0.098(15)  
(4, 1)  1944  0.051(10)  (4, 1)  6144  0.022(3)  
(3, 2)  1536  0.060(10)  (2, 0)  49152  0.003(1)  
(3, 0)  3072  0.049(7)  24^{4}  0.1  (12, 0)  192  1.767(244)  
(2, 0)  15552  0.010(2)  (12, 1)  384  1.029(153)  
0.133  (6, 0)  192  0.467(94)  (8, 0)  972  0.406(81)  
(6, 1)  384  0.301(65)  (8, 1)  1944  0.253(41)  
(4, 0)  972  0.181(29)  (6, 2)  1536  0.250(36)  
(4, 1)  1944  0.059(12)  (6, 0)  3072  0.186(27)  
(3, 2)  1536  0.082(16)  32^{4}  0.1  (16, 0)  192  2.335(576)  
(3, 0)  3072  0.065(11)  (16, 1)  384  1.026(320)  
(2, 0)  15552  0.010(2)  (8, 2)  1536  0.242(63)  
16^{4}  0.1  (8, 0)  192  0.592(101)  (8, 0)  3072  0.133(29) 
There is another method to obtain the topological charge by calculating a few low modes of the chiral Dirac operator, counting the number of zero modes and applying the index theorem. However, to estimate the topological charge density
The topological charge
We developed a symmetric scheme with 3 modes of the coloring algorithm and evaluated the topological charge with the SMP method. The results of the average absolute systematic error were presented, as well as the efficiency of the SMP method compared to the point source method. Potential applications of the SMP method are expected in calculations of the trace of the inverse of any large sparse matrix with locality.