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Topological charge Q and its density
q(x) play an important role in the study of the non-trivial topological structure of QCD vacuum. Topological properties have important phenomenological implications, such asθ dependence and spontaneous chiral symmetry breaking. The confinement may also be related to nontrivial topological properties [1-3]. The topology of QCD gauge fields is a non-perturbative issue; therefore, the lattice method is a good choice to investigate it from first principles. Lattice QCD is powerful for studying the topological structure of the vacuum. There are many definitions of the topological charge for a lattice gauge field [4-6]. These definitions can be characterized either as gluonic or fermionic. In the fermionic definition, topological charge Q is the number of zero modes of the Dirac operator [7, 8]. In contrast, topological charge can be given by the field strength tensor (gluonic definition) on the lattice, and this definition approaches the fermionic definition in the limita→0 [9-11].The overlap Dirac operator is a solution of the Ginsparg-Wilson equation [12, 13], and the topological charge defined from the overlap fermion will be an exact integer. In the traditional method, the point source is used in the calculation of topological charge density [14, 15], which makes the computation on the large lattice almost impossible. To reduce the computational cost, the symmetric multi-probing source (SMP) method is introduced to calculate the topological charge density [16]. As the Wilson mass parameter m varies, the value of Q may change [17-20]. The topological charge density
q(x) has a strong correlation with the low-lying modes of the Dirac operator, which strongly influences how quarks propagate through the vacuum. Therefore, the topological charge densityq(x) is a useful probe of the gauge field. We visualize the topological charge density and view the detailed extra information [21]. In contrast, the topological charge cannot show the details of the QCD vacuum. Therefore, we will focus on the topological charge densityq(x) in the study of the topological properties of the QCD vacuum. We show an analysis of the topological chargeq(x) , obtained using the fermionic definition with different values of m and the gluonic definition with different Wilson flow time. Unlike in the case of Ref. [22], which studied just one time slice, we consider all time slices and show more details on the topological charge density with different topological charge. By analyzing the topological charge densityq(x) , we can obtain a great amount of information about the underlying topological structure. We show a comparison with the gluonic topological charge density that is calculated after the application of the Wilson flow method. This comparison is shown by calculating the matching parameterΞAB , which is defined later.ΞAB is used to measure the match of the topological charge density between the fermionic and gluonic definitions. The best match is found when the matching parameter is nearest to1 . The proper flow time of Wilson flow in the gluonic definition can also be obtained by analyzing the matching procedure. The behavior of the proper flow time toward the continuum limit is also discussed. -
The pure gauge lattice configurations were generated using a tadpole improved, plaquette plus rectangle gauge action through pseudo-heat-bath algorithm [23, 24]. This gauge action at tree-level
O(a2) -improved is defined asSG=5β3∑xμνν>μReTr[1−Pμν(x)]−β12u20∑xμνν>μReTr[1−Rμν(x)],
(1) where
Pμν is the plaquette term. The link productRμν(x) denotes the rectangular1×2 and2×1 loops. The mean linku0 is the tadpole improvement factor that largely corrects for the quantum renormalization of the coefficient for the rectangles relative to the plaquette.u0 is given byu0=(13ReTr⟨Pμν(x)⟩)1/4.
(2) In the fermionic definitions, we use the overlap operator to calculate topological charge density. The massless overlap Dirac operator is given by [20]
Dov=(1+DW√D†WDW),
(3) where
DW is the Wilson Dirac operator,DW=δa,bδα,βδi,j−κ4∑μ=1[(1−γμ)αβUμ(i)abδi,j−ˆμ+(1+γμ)αβU†μ(i−ˆμ)abδi,j+ˆμ],
(4) and
κ is the hopping parameter,κ=12(−m+4).
(5) In the overlap formalism,
κ has to be in the range(κc,0.25) forDov to describe a single massless Dirac fermion, andκc is the critical value ofκ at which the pion mass extrapolates to zero in the simulation with ordinary Wilson fermions. We call m in Eq. (5) the Wilson mass parameter. In this work, we choose parameterκ as the input parameter.The overlap topological charge density can be calculated as follows:
qov(x)=12Trc,d(γ5Dov(x))=Trc,d(˜Dov(x)),
(6) where the trace is over the color and Dirac indices. It is well known that the traditional way of computing
qov(x) with a point source is almost impossible for a large lattice volume.To avoid the high computational effort in the calculation of the
q(x) with a point source, we apply the SMP method to calculateq(x) [16, 25],qsmp(x)=∑α,aψ(x,α,a)(˜Dov(x))ϕP(S(x,P),α,a)=∑α,aψ(x,α,a)(˜Dov(x))ψ(x,α,a)+y≠x∑y∈S(x,P)ψ(x,α,a)(˜Dov(x))ψ(y,α,a)≈∑α,aψ(x,α,a)(˜Dov(x))ψ(x,α,a),
(7) where x is the seed site at site
(x1,x2,x3,x4) , and y represents the other lattice sites belonging to the setS(x,P) .ϕP(S(x,P),α,a) is the SMP source vector, andψ is the normalized point source vector.S(x,P) represents the sites with the same color of x obtained by the symmetric coloring schemeP(nsd,nsd,nsd,ntd,mode) .ns andnt are the spatial and temporal sizes of the lattice, d is the minimal distance of the coloring scheme,mode=0,1,2 corresponds to the Normal, Split, and Combinedmode for scheme P, and the number of SMP sources that cover all lattice sites are12d4 ,24d4 , and6d4 , respectively. The term in the third line of Eq. (7) is the summation of space-time off-diagonal elements of˜Dov(x) . Because of the space-time locality ofDov , this term can be regarded as the error in the calculation of topological charge density. If we choose the proper scheme P of the SMP source, we can neglect the error term and obtain the last line in Eq. (7).However, the number of normalized point sources is
12NL , andNL=NxNyNzNt is the lattice volume. This shows that the SMP method is much cheaper than the point source method in the calculation of topological charge density with the fermionic definition, especially for a large lattice volume. The topological charge using the SMP method is denoted asQsmp , given byQsmp=∑xqsmp(x).
(8) Gradient flow is a non-perturbative smoothing procedure, which has been proven to have well-defined numerical and perturbative properties. The gradient flow is defined as the solution of the evolution equations [6, 26-28]
˙Vμ(x,τ)=−g20[∂x,μSG(V(τ))]Vμ(x,τ),Vμ(x,0)=Uμ(x),
(9) where
τ is the dimensionless gradient flow time (Wilson flow time in this work), andg20∂x,μSG(U) is given byg20∂x,μSG(U)=2i∑aTaImTr[TaΩμ]=12(Ωμ(x)−Ω†μ)−16Tr(Ωμ(x)−Ω†μ),
(10) where
Ta(a=1,2,⋯,8) are the Hermitian generators of theSU(3) group.Ωμ=Uμ(x)X†μ(x) , andXμ(x) represents the so-called staples.In practice, the gradient flow moves the gauge configuration along the steepest descent direction in the configuration space, such as along the gradient of the action. The chosen sign in the evolution equations leads to a minimization of the action, which is as expected. We use the third order Runge-Kutta method to obtain the solution of the flow in Eq. (9). The gluonic definition of topological charge density in Euclidean spacetime is defined as
q(x)=132π2ϵμνρσTr[FμνFρσ],
(11) with
Fμν the gluonic field strength tensor. The topological charge of a gauge field is the four-dimensional integral over space-time of the topological charge density,Q=∫d4xq(x).
(12) The most common definition of the topological charge density in lattice discretization is the clover definition, given by
qclovL(x)=132π2ϵμνρσTr[CclovμνCclovρσ],
(13) where
Cclovμν is the usual clover leaf.The field strength tensor
Fμν used in this work is three-loopO(a4) -improved and defined as [29]FImpμν=2718C(1,1)−27180C(2,2)+190C(3,3),
(14) where
C(m,n) denotes the threem×n loops used to construct the clover term, andC(1,1) is the clover leaf mentioned above.The
q(x) calculated after Wilson flow to the gauge configuration is denoted asqwf(x) . The topological charge obtained by Wilson flow is denoted asQwf , given byQwf=∑xqwf(x).
(15) To fairly compare the two definitions for the topological charge density with the varied Wilson mass parameter, we calculate the matching parameter
ΞAB , given by [30]ΞAB=χ2ABχAAχBB,
(16) with
χAB=1V∑x(qA(x)−ˉqA)(qB(x)−ˉqB),
(17) where
ˉq denotes the mean value ofq(x) , and in this work,qA(x)≡qsmp(x) ,qB(x)≡qwf(x) . When theΞAB is nearest to1 , the best match is found [22]. When the best match is reached, the flow timeτ is called the proper flow time of Wilson flow, denoted asτpr . In this work, the step length for numerical integration of Wilson flow isδτ=0.005 .We also calculate the factor
Zcalc , defined asZ_{{\rm calc}}\equiv\frac{\displaystyle\sum_{x}\left\big|q_{{\rm ov}}\left(x\right)\right\big|}{\displaystyle\sum_{x}\left\big|q_{{\rm wf}}\left(x\right)\right\big|}.
(18) Because the topological charge of gluonic definition is not always an integer,
Zcalc is needed in the visualization of the matching procedure. The matching parameterΞAB is independent of the value ofZcalc .We will analyze
q(x) of all time slices on lattices of164 ,243×48 and324 at the inverse coupling,β=4.50 ,β=4.80 , andβ=5.0 , corresponding to the lattice spacinga=0.1289 ,0.0845 and0.0655fm , respectively. In this work, we calculate the topological charge density of two configurations for lattice volumes164 and244×48 , and one configuration for324 . We show the visualization of the topological charge density and apply the matching procedure to obtain the proper flow time of Wilson flow in the calculation of topological charge density. -
Before we proceed with the analysis, we first demonstrate that the SMP method with proper d is a good method to calculate the topological charge density with an overlap Dirac operator. When the source in the calculation of topological charge is a point source,
qps(x) is an exact result ofqov(x) . It is reasonable to useqps(x) as a benchmark for comparison. Due to the high computational cost, we only make a comparison of the point source and the SMP source in Eq. (6) on a123×24 lattice withβ=4.8 ,κ=0.21 . We showqps(x) obtained using the point source method andqsmp(x) using the SMP method withd=6 in Fig. 1. To show the visualization more clearly, we use a cutoff method, shown as the color map. The same cutoff procedure is used in other visualized figures. It shows thatqsmp(x) is highly matched withqps(x) . The matching parameterΞAB for these two methods is0.9997 , and we can barely see the difference with the naked eye. This shows that the error caused by the space-time off-diagonal elements ofDov is indeed very small when the distance parameter d of SMP source is proper. Thus, the SMP method is a good choice to calculate the topological charge density while the parameter d is large enough. It is expected that a better match corresponds to a larger distance d [16].Figure 1. (color online)
qps(x) andqsmp(x) on lattice123×24 by the point source and SMP source for thet=12 slice; other time slices are similar. To the naked eye, they are almost the same. The parametersκ=0.21 andβ=4.80 are the same for both setups. In the figures, we use a color cutoff method shown as the color map. (left): Point source, (right): SMP source.To obtain more precise topological charge density, we choose the distance parameter
d=8 in the SMP method on164 ,243×48 , and324 ensembles. The results are summarized in Tables 1-5. We only show the visualization of topological charge densityq(x) for lattice volume243×48 atβ=4.80 as an example. The results for other lattice ensembles are similar. In these calculations ofq(x) withd=8 by the SMP method, we find that the topological chargeQsmp is very close to an integer and it is not always the same for the same ensemble with differentκ values. This is acceptable as zero crossings in the spectral flow of the˜Dov(x) occur for differentκ on the lattice [6, 18]. From the tables, it also shows thatQsmp is very sensitive toκ on the coarsest lattice.Qsmp changes a little on the finer lattice, and is stable versus the change ofκ on the finest lattice. This sensitivity is probably the effect of finite lattice space a. All results indicate that even though the topological chargeQsmp from overlap fermions is always an integer, its value is not unique, and it depends on the Wilson mass parameter m.κ τpr Zcalc ΞAB Qsmp Qwf 0.17 0.360 0.5121 0.6853 5.0010 3.5420 0.18 0.365 0.7351 0.7259 5.0015 3.5299 0.19 0.350 0.8656 0.7309 6.0005 3.5847 0.21 0.330 1.0045 0.7267 3.9990 3.6378 0.23 0.320 1.0073 0.7030 1.9963 3.6796 Table 1. Proper time of Wilson flow
τpr needed to match the SMP topological charge density with differentκ atβ=4.50 and lattice volume164 forconf.1 .Qsmp is obtained by the SMP method, andQwf is the result of Wilson flow withτpr. κ τpr Zcalc ΞAB Qsmp Qwf 0.17 0.365 0.5230 0.7006 −6.0091 −7.2945 0.18 0.360 0.7235 0.7339 −6.0052 −7.3027 0.19 0.350 0.8680 0.7471 −5.0026 −7.3181 0.21 0.330 1.0077 0.7322 −3.9997 −7.3458 0.23 0.320 1.0121 0.7096 −5.9964 −7.3581 Table 2. Proper time of Wilson flow
τpr needed to match the SMP topological charge density with differentκ atβ=4.50 and lattice volume164 forconf.2 .Qsmp is obtained by the SMP method, andQwf is the result of Wilson flow withτpr. κ τpr Zcalc ΞAB Qsmp Qwf 0.17 0.365 0.6811 0.7653 8.0077 8.5997 0.18 0.345 0.8364 0.7670 7.0078 8.6863 0.19 0.325 0.9275 0.7580 7.0095 8.7954 0.21 0.305 1.0215 0.7324 9.0169 8.9322 0.23 0.295 0.9811 0.6972 9.0279 9.0130 Table 3. Proper time of Wilson flow
τpr needed to match the SMP topological charge density with differentκ atβ=4.80 and lattice volume243×48 forconf.1 .Qsmp is obtained by the SMP method, andQwf is the result of Wilson flow withτpr. κ τpr Zcalc ΞAB Qsmp Qwf 0.17 0.365 0.6802 0.7661 4.9938 4.3224 0.18 0.345 0.8359 0.7656 4.9932 4.3289 0.19 0.325 0.9274 0.7574 3.9942 4.3365 0.21 0.305 1.0211 0.7309 3.9949 4.3431 0.23 0.290 0.9596 0.6953 3.9967 4.3454 Table 4. Proper time of Wilson flow
τpr needed to match the SMP topological charge density with variedκ atβ=4.80 and lattice volume243×48 forconf.2 .Qsmp is obtained by the SMP method, andQwf is the result of Wilson flow withτpr. κ τpr Zcalc ΞAB Qsmp Qwf 0.17 0.355 0.7271 0.7678 3.0099 2.8066 0.18 0.335 0.8724 0.7627 3.0059 2.7340 0.19 0.315 0.9507 0.7515 3.0069 2.6380 0.21 0.295 1.0233 0.7222 3.0127 2.5120 0.23 0.285 0.9636 0.6818 3.0240 2.4352 Table 5. Proper time of Wilson flow
τpr needed to match the SMP topological charge density with variedκ atβ=5.0 and lattice volume324 .Qsmp is obtained by the SMP method, andQwf is the result of Wilson flow withτpr. The SMP topological charge density for three choices
κ compared with the proper flow time of Wilson flowτpr for time slicet=24 as an example is shown in Fig. 2, andqwf(x) is renormalized usingZcalc . Other time slices have a similar property. This shows that more Wilson flow time forqwf(x) is needed to match the topological charge densityqsmp(x) with a smallerκ or larger mass m. This phenomenon may be a result of the smallerκ showing sparser small eigenvalues ofDov , which has a similar effect of smoothing the configurations. However, the detailed reasons need further study. This indicates that the overlap Dirac operator is less sensitive to small objects asκ is decreased, and these objects can be removed by Wilson flow smoothing.Figure 2. (color online) Best matched topological charge density
qwf(x) (right) calculated by the Wilson flow method compared with the overlapqsmp(x) for the time slicet=24 , whereqwf(x) is renormalized usingZcalc .τpr is the proper flow time of Wilson flow fixed by a matching procedure. A color cutoff method is used shown as the color map in the figures.τpr ,Zcalc ,ΞAB ,Qsmp andQwf of different ensembles for five differentκ are shown in Tables 1-5, respectively. When calculatingZcalc , the flow time of Wilson flow isτpr . As the parameterκ increases, there is a monotonically decreasing trend in the proper flow time of Wilson flow in all tables. This shows that the matching procedure is effective and the proper flow time of Wilson flow is almost equal at the sameκ for different configurations of the same lattice ensembles. When the parameterκ is fixed,ΞAB andZcalc are independent of topological charge Q and very close for different configurations with the same lattice volume. The value ofΞAB is approximately in the range[0.68,0.77] . We can see that the proper flow time of Wilson flowτpr is different for differentκ . However, it is reasonable to choose the average value of the proper flow time of Wilson flow of differentκ as the properˉτpr . This is approximatelyˉτpr=0.345 ,ˉτpr=0.327 ,ˉτpr=0.317 , or the proper flow radius of Wilson flow√8τ≈0.214 ,0.137 and0.104fm for the lattice ensemble164 atβ=4.5 ,243×48 atβ=4.8 , and324 atβ=5.0 , respectively. This indicates that we can chooseˉτpr for gluonicqwf(x) to match with fermionicqsmp(x) .All results show that
Qwf deviates largely from an integer for the proper flow time of Wilson flow fixed by the matching procedure. This phenomenon may be due to the fact that the topological charge is the global property of topological structure. However, the matching parameterΞAB only shows the local matching property of topological charge density. Otherwise, the topological charge is the effect of the infrared properties, and the ultraviolet fluctuations lead to unphysical results as well as to non-integer topological charge values. Wilson flow is indeed a smoothing scheme of gauge fields. However, Wilson flow will modify the gauge field at the same time, which does not satisfy that the topological charge is conserved. In contrast,FImpμν in Eq. (14) is applicable under the classical expansion with respect to lattice spacing a, and it may be affected by some quantum fluctuations even though the smoothing is performed.However, the topological charge is not the main quantity of interest, and the physically relevant observable is the topological susceptibility. The results show that to make the topological charge close to an integer, larger Wilson flow time is required. It is noted that too large a Wilson flow time may wipe out the negative core of the topological charge density correlator [31]. However, the flow time for the topological susceptibility to reach a plateau is smaller than the flow time for the topological charges reach some integers [28]. Nonetheless, the topological susceptibility using the SMP method is too expensive to calculated. In future work, we may try to consider it.
In Fig. 3,
ΞAB versus the flow time of Wilson flow of one configuration for lattices of164 ,243×48 , and324 are shown.ΞAB for other configurations have a similar trend. We see that as the flow time of Wilson flowτ increases,ΞAB reaches a maximum value and then decreases. When the parameterκ is fixed, we can observe that as the lattice spacing decreases, the proper flow time of Wilson flow almost uniformly decreases, as expected. It also shows that as the lattice spacing a decreases,ΞAB tends to increase.Figure 3. (color online)
ΞAB versusτ of one configuration for lattices of164 ,243×48 and324 . From left to right, the first figure isΞAB versusτ for lattice volume164 , the second figure for lattice volume243×48 , and the last one for lattice volume324 . Asκ is reduced, the proper flow time of Wilson flow is increased. As the lattice spacing decreases,ΞAB tends to increase. -
We have analyzed the topological charge density
q(x) of all time slices using direct visualizations. We find that the SMP method is a good choice to study the topological charge density in the fermionic definition, and the SMP method is much cheaper than the traditional point source method, especially for a large lattice volume. The results show that the topological charge density depends on the Wilson mass parameter m in the fermionic definition. By comparing theqsmp(x) with the gluonic definition ofqwf(x) , a correlation between m andτ is revealed. Smaller values ofκ remove non-trivial topological charge fluctuations, which are similar to Wilson flow with a larger flow time. The detailed reasons are worthy of further study. By analyzing the topological charge densityq(x) , we find that the proper flow time of Wilson flow for the gluonic definition of topological charge density can be obtained by the comparison ofqsmp(x) withqwf(x) . We also observe that the proper flow time of Wilson flow decreases with the decrease in lattice spacing a, which is consistent with expectations. Furthermore, as the lattice spacing a decreases,ΞAB tends to increase.We also find that the topological charge obtained by Wilson flow at the proper flow time is far from an integer, and it is different from that of the fermionic definition. Although the topological charge density of the fermionic definition and that of the gluonic definition have the best match, the topological charge from the fermionic and gluonic definition are very different. The reason for this phenomenon may be that the matching parameter
ΞAB only shows the local matching property of topological charge density, but the topological charge is the global property of topological structure. In contrast, Wilson flow can indeed smooth the gauge field. However, Wilson flow also changes the configurations, which does not guarantee the conservation of topological charge. Otherwise, in the gluonic definition of topological charge density, we used the improved field strength tensor corrected in the classical expansion with respect to lattice spacing, which may be affected by the quantum fluctuations even though the Wilson flow is used. The detailed reasons need further study.In the SMP method, we had known that the error is dependent on the off-diagonal components of the overlap operator. With a larger distance parameter, it has smaller errors in the SMP method. We can try to choose a larger distance parameter to decrease the error in future work. Otherwise, we could try to improve the field strength tensor to reduce the effect of classical expansion with respect to the lattice spacing in the gluonic definition of topological charge density.
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We are grateful to Heng-tong Ding for careful reading of the manuscript and useful suggestions. Numerical simulations have been performed on the Tianhe-2 supercomputer at the National Supercomputer Center in Guangzhou (NSCC-GZ), China.
Dependence of overlap topological charge density on Wilson mass parameter
- Received Date: 2021-03-07
- Available Online: 2021-07-15
Abstract: In this paper, we analyze the dependence of the topological charge density from the overlap operator on the Wilson mass parameter in the overlap kernel by the symmetric multi-probing source (SMP) method. We observe that non-trivial topological objects are removed as the Wilson mass is increased. A comparison of topological charge density calculated by the SMP method using the fermionic definition with that of the gluonic definition by the Wilson flow method is shown. A matching procedure for these two methods is used. We find that there is a best match for topological charge density between the gluonic definition with varied Wilson flow time and the fermionic definition with varied Wilson mass. By using the matching procedure, the proper flow time of Wilson flow in the calculation of topological charge density can be estimated. As the lattice spacing a decreases, the proper flow time also decreases, as expected.