Topological charge and cooling scales in pure SU(2) lattice gauge theory

Using Monte Carlo simulations with overrelaxation, we have equilibrated lattices up to $\beta=2.928$, size $60^4$, for pure SU(2) lattice gauge theory with the Wilson action. We calculate topological charges with the standard cooling method and find that they become more reliable with increasing $\beta$ values and lattice sizes. Continuum limit estimates of the topological susceptibility $\chi$ are obtained of which we favor $\chi^{1/4}/T_c=0.643\,(12)$, where $T_c$ is the SU(2) deconfinement temperature. Differences between cooling length scales in different topological sectors turn out to be too small to be detectable within our statistical errors.


I. INTRODUCTION
Since Lüscher proposed the gradient flow method [1], the topic of scale setting has received increased attention.See, for instance, the review [2].In [3] Bonati and D'Elia suggested to replace the gradient flow by the computationally more efficient (standard) cooling flow [4] and supported this idea with numerical evidence for pure SU(3) lattice gauge theory (LGT).In a recent large statistics study of pure SU (2) LGT [5] we investigated the approach to the continuum limit for six gradient and six cooling scales.They are distinguished by the use of three different energy operators and two different ways of setting the initial scaling to agree with the deconfining scale on small lattices.We studied systematic errors of scale setting which, although only about 1% for our largest lattices at β ≈ 2.9 (2% at β ≈ 2.6), dominate the statistical errors.Quantitatively gradient and cooling scales worked equally well with differences between the six scales within the cooling and within the gradient group larger than differences between corresponding scales of the two groups.See [6] for a summary.
Using cooling we also calculated the topological charge Q on each of our configurations and showed that our charges of subsequent configurations are statistically independent.Here we supplement our previous publication by presenting details of our calculations of Q, and we consider the question whether there are noticeable differences in the scales when we restrict them to fixed topological sectors.
Although fixed topological sectors imply for local operators only a bias of order 1/V [7,8], getting trapped in one topological charge sector has often been a reason of concern.For instance, Lüscher and Schaefer [9] proposed to bypass the problem by imposing open boundary conditions in one of the lattice directions.Recently Lüscher [10] emphasized that master-field configurations on very large lattices would alleviate topological freezing.We find that the lattices used in our SU(2) investigation are so large that the 1/V effects due to topological freezing are swallowed by statistical errors.
In the next section we discuss our data for the topological charge and in section III we search for correlations of topological charge sectors with differences in the considered cooling scales.Summary and conclusions are given in the final section IV.

II. TOPOLOGICAL CHARGE ON THE LATTICE
The continuum equation of the topological charge, where * F is the dual field strength tensor, translates on the lattice to the discretization where the sum is over all lattice sites and Here ˜ = for positive indices while ˜ µνρσ = −˜ (−µ)νρσ for negative indices.
Measurements of this quantity on MC-generated lattice configurations suffer from lattice artifacts, which we suppressed by cooling.A SU(2) cooling step minimizes the action locally by replacing link variable U µ (x) by a function of the staple matrix U µ (x): After n c cooling sweeps one reaches metastable lattice configurations to which a topological charge Q nc is assigned.The obtained values still suffer from discretization errors, which can be absorbed by multiplicative normalization constants N L , replacing Q L by where we calculate the constant N L following the procedure most clearly explained in Ref. [3] and there attributed to [11].We minimize the equation arXiv:1710.09474v1 [hep-lat] 25 Oct 2017  where the sum is over all configurations for a fixed lattice size and β value.Table I gives an overview of the lattice sizes and β values for which we have calculated the topological charge distribution.For each parameter value we generated 128 configurations separated by a sufficiently large number of MCOR sweeps so that they are effectively statistically independent.Each MCOR update consists of one heat- - bath followed by two overrelaxation updates.For lattice sizes up to 52 4 the statistics is the one of Ref. [5].For the 60 4 lattice configurations are separated by 3 × 2 12  MCOR sweeps after 2 15 sweeps for equilibration.
On each lattice configuration we performed 2048 cooling sweeps and applied the minimization (6) with the summation using the Q 2048 charges of the final configurations.The multiplicative constants N L amount to corrections of about 10% and appear to approach slowly N L = 1 with increasing β values and lattice sizes.Subsequently, the 128 time series for the topological charge are plotted for each lattice.Examples of these plots are shown in Figs. 1 to 4 for increasing β values and lattice sizes.
As discussed in [12], when approaching the continuum limit the topological charge has to be defined at a fixed, large enough, number of n c cooling sweeps.For our first two figures a good choice does not exist, while for the last two figures n c = 1000 works well.Figs. 3 and 4 show that the removal of dislocations by this fixed number of sweeps becomes easier for increasing β, and metastable configurations are reached earlier and become more sta-ble.Some value n c < 1000 may suffice if one starts off at even larger β values.In the % stable column of Table I we report the stability of charge sectors under the next 1048 cooling sweeps.For increasing β from about β = 2.574 on we see up to statistical fluctuations a gradually improving trend.It is a bit subjective where to make the metastability cut from which on one is satisfied.If the choice is close to at least 90%, we have to require β > 2.75 and lattices large enough to accommodate physical instantons (their size increases proportionally to our scales to which the lattice sizes are already adjusted).

III. SCALES IN TOPOLOGICAL SECTORS
As observables we use three definitions [5] of the energy density: E 0 (t), E 1 (t) and E 4 (t).E 0 (t) is the Wilson action up to a constant factor, E 1 (t) is the sum of the squared Pauli matrices of the plaquette variables, and E 4 (t) is Lüscher's [1] energy density which averages over the four plaquettes attached to each site n in a fixed µν, µ = ν plane.The functions are used to set up three cooling scales by choosing appropriate fixed target values y 0 i and performing cooling steps (4) until As a function of β, the observable then scales like a length.
There is some ambiguity in the choice of target values.In [5] they are chosen so that either (superscripts 01) initial estimates of the scales s 01 0 and s 01 1 (they give almost identical values) agree with the deconfinement scaling from β ≈ 2.3 on a 4 × 8 3 to β ≈ 2.44 on a 6 × 12 3 lattice, or so that (superscripts 02) s 02 4 agrees.This leads to two possible values per energy observable, i.e., a total of six targets: For our lattices up to size 52 4 the resulting values are found in Tables 7 and 8 of [5] and for our new 60 4 lattice they are given in Table III.Following the convention of [5] we label these scales L 7 − L 12 .
For the five lattices with β ≥ 2.71 we calculated the scales on the topological sectors

and performed
Student difference test between them.No statistically significant discrepancies are encountered.In particular there are none when comparing the Q 1000 < 0 with the Q 1000 > 0 scales.To increase the statistics by a factor two in the average for the |Q 1000 | = 0 sectors, we combined them into |Q 1000 | = 1 and |Q 1000 | ≥ 2. Together with the scales for Q 1000 = 0 their values are listed in Table IV.
Comparing each of the scales with itself in the other two |Q 1000 | sectors by means of Student difference tests, the q-values compiled in Table V are obtained.The results for the pairs (L 7 , L 8 ) and (L 10 , L 11 ) are almost identical because the fluctuations of the operators E 0 and E 1 are strongly correlated [5].So, their q values are combined in the following, and the subsequent analysis is done with the remaining q values.
When the compared data are statistically independent, rely on the same estimator, and are drawn from a Gaussian distribution, the Student different tests return uniformly distributed random numbers q in the range 0 < q < 1.A visual check of the histogram of our q values  supports immediately that they are indeed uniformly distributed in (0,1).In particular, their mean value comes out to be q = 0.508 (40) in agreement with the expected 0.5.If there are still some residual correlations between our q values, this would have decreased the error bar, because the number of independent q would have been counted too high, while each of them still fluctuates like a uniformly distributed random number in the interval (0,1).So, we find convincing evidence that the 1/V bias expected for our scales due to topological freezing disappears within our statistical noise.

IV. SUMMARY AND CONCLUSIONS
Using cooling we calculated the topological charge of pure SU (2) LGT for larger lattices and β values than it was done in previous literature.For the first time they appear to be large enough to yield stable topological sectors.See Figs. 3 and 4.
Within our statistical fluctuations we find no observable correlations between cooling scales (7) and topological charge sectors.Our number of statistically independent configurations is of a typical size as used for scale setting, e.g., [1,9].So, our results support that the problem of topological freezing becomes only serious when a much larger precision is targeted.

TABLE I :
Lattice size used to generate data at each β value.
. Our previous Table I gives the largest value of |Q 1000 | and compares it with the largest value of |Q 2048 |.

TABLE IV :
Cooling scales one topological sectors.

TABLE V :
Student difference tests for the 60 4 lattice.