The vacuum energy can be expanded around θ = 0 as"χ 2!
24E(θ) − E(0) = θ 1 + b2 θ + b4 θ + · · · ,2ere χ is the topological susceptibility, and b2i (i = 1, 2, 3, · · · ) are dimensionless conts describing the deviation of topological charge destribution from the Gaussian. Tantities can be determined on the lattice from configurations generated at θ = 0 as
#Q2 $θ=0χ=,V#Q4 $θ=0 − 3 #Q2 $2θ=0b2 = −,2
12 #Q $θ=0#Q6 $θ=0 − 15 #Q2 $θ=0 #Q4 $θ=0 + 30 #Q2 $3θ=0b4 =,2360 #Q $θ=0ere Q is the topological charge, whose precise definition is given in eqs. (10)-(14),· $θ=0 denotes an ensemble average over configurations generated with θ = 0. Accordthe large N analysis [2, 4], these quantities can be expressed, as a function of N , as−21/2βNSNTc(a Tc )2L σstrstatistics

1.750164.650.04624.980,100

1.850166.500.02373.571,040

1.975169.500.01112.430,490

1.975249.500.01113.6131,830TABLE I: The simulation parameters. Tc denotes the critical temperature. NTc is read fromRef. [33]. The uncertainties of NTc are below 1% and hence neglected in the following.1/2βNSNTc(a Tc )2L σstrstatistics

1.750164.650.04624.980,100

1.850166.500.02373.571,040

1.975169.500.01112.430,490

1.975249.500.01113.6131,830I: The simulation parameters. Tc denotes the critical temperature. NTc is read from. The uncertainties of NTc are below 1% and hence neglected in the following.

stance fluctuations, which we remove by introducing a smoothing technique. In the(new)mearing, new link variables UµUµ(new) = ProjXµ (x) =!

#$(old)are constructed from old ones Uµ(1 − ρ)U(old)(x) + ρ Xµ (x)µ",as(8)†Uν(old) (x)Uµ(old) (x + ν̂)Uν(old) (x + µ̂)

ν!=µ(old) †+Uν(x − ν̂)Uµ(old) (x − ν̂)Uν(old) (x − ν̂ + µ̂)%,(9)Proj acts as the projection back to an SU (2) element. This procedure minimizes0.8nAPE = 0201000.15

histogram of Q

histogram of Q0.20.1

0.6

0.40.050.200-15-10-50510nAPE = 02010015-15-10-5Q510

1510

15Q2.50.8histogram of QnAPE = 0201002
histogram of Q01.51

nAPE = 020100

0.60.40.20.500-15-10-50Q51015-15-10-505QFIG. 7: Histogram of Q for four ensembles at nAPE = 0, 20, 100.nAPE to nAPE + 1. Figure 5 shows the nAPE dependence of |"∆Q(nAPE )#|, where the symbolsre filled when its original value is negative. The results from four ensembles show expo-β=1.7501.8501.975NS=24, 1.975|〈 ΔQ(nAPE) 〉|1e+001e-02

1e-0401020nAPE3040FIG. 5: The decay rate of topological charge.

nential fall with approximately a common exponent for nAPE <∼ 10, while they take almost

io defined byfunction of nAPE for one particular configuration at β = 1.850. For nAPE >∼ 30, whenever!

#2Q changes, ln P shows a rapid increase" after slow decrease over many smearing steps. This2can be interpreted as that a local objectq(x,in topologicalnAPE ) charge density gradually shrinks and1 point withx a change of Q. This is precisely what happens whensuddenly disappears at some"P (nAPE ) =Nsite occurs q(x,the “falling through the lattice”[41]. nAPE )4,(15)xWe can explicitly see that the interpretationmakes sense. Figure 4 shows the topologicalcharge density, projected onto the z-t plane, of the same configuration as in Fig. 3. Betweendenotes the topological charge density q(x) in eq. 11) after nAPE steps ofnAPE = 50 and 60 and nAPE = 450 and 470, Q increases by unity, at the same time a negativearticipation ratio takes a value between 1/Nsite and 1. The maximal valueQ(nAPE), ln P(nAPE)ealized when q(x,2nAPE ) takes a flat distribution over the whole space-time.

nd, the possible minimumvalue, 1/Nsite , is attained0Q(nAPE) when the density formsln P(nAPE)x, nAPE ) = δ(x). -2Figure 3 shows the smearinghistory of Q and ln P as afor one particular-4 configuration at β = 1.850. For nAPE >∼ 30, whenevershows a rapid increaseafter slow decrease over many smearing steps. This-6d as that a local-8object in topological charge density gradually shrinks andars at some point-10 with a change of Q. This is precisely what happens when0100ugh the lattice” occurs [41].

200

300
nAPE

400500600itly see that the interpretation makes sense. Figure 4 shows the topologicalFIG. 3: An example of Q(nAPE ) and ln P (nAPE ) as a function of nAPE at β = 1.850.

rojected onto the z-t plane, of the same configuration as in Fig. 3. Between0 and n9 by unity, at the same time a negative= 450 and 470, Q increasesnAPE = 50nAPE = 600.2120.180400

481216t20

240.1zz12

0.280-0.14-0.1-0.2028-0.204

8nAPE = 1001216t202428

nAPE = 2000.2120.180400

481216t20

240.1zz12

0.280-0.14-0.1-0.2028-0.204

8nAPE = 4501216t2024

28

nAPE = 4700.2120.1804004

81216t2024280.180-0.14-0.1-0.20z

z120.2-0.204

81216
t202428Gaussian shape is seen in all ensembles.
15β=1.75010

1055QQ1500-5-5-10-10-1510000-15100001050011000configuration number15

β=1.9751010550-5-10-10-1510000-1510000configuration number1100011000

NS=24, β=1.9750-51050010500configuration numberQQ15

β=1.85010500configuration numberFIG. 6: Monte Carlo history of Q at four ensembles.

110000.8nAPE = 0201000.15

histogram of Q

histogram of Q

0.20.1

0.6

0.40.050.200-15-10-50510nAPE = 02010015-15-10-5Q510

1510

15Q2.50.8

histogram of QnAPE = 0201002
histogram of Q01.51

nAPE = 020100

0.60.40.20.500-15-10-50Q51015-15-10-505QFIG. 7: Histogram of Q for four ensembles at nAPE = 0, 20, 100.

32.5β=1.750214

a χ1.20.8β=1.850

0.60.25β=1.975NS=24, 1.9750.2010

20nAPE30

400.1β=1.750
0-0.1
0.1b2

β=1.8500-0.10.1β=1.975NS=24, 1.975

0-0.1010

20nAPE3040alter the limit for the constant fit by much. The final results thus obtained areχ= 0.200(39) ,4Tcχ1/4= 0.674(31) ,Tcb2 = −0.049(20) ,e the errors are summed in quadrature.

n Refs. [14–16], the topological susceptibility χ is calculated in SU(N ) gauge theal values of N to study the large N behavior. In Refs. [14, 39, 45–47], χ is estim0.30.115

constant
linearconstantlinearb2

χ / Tc400.2-0.10.1-0.200.010.020.03(aTc)20.04

0.0500.010.020.03(aTc)20.040.05FIG. 10: The continuum limit of χ/Tc4 and b2 . The solid lines in both plots are the results from aconstant fit using only two finer lattices, and the dashed lines are those from a linear fit using all0.1χ/σ20.080.06LuciniDel DebbioBonatiprevious SU(2)this workph fit0.040.0200

0.10.221/N0

0.30.1χ/σ20.080.06LuciniDel DebbioBonatiprevious SU(2)this workph fit0.040.0200

0.10.221/N00.3b20

DIGADel Debbio
Bonatithis workph fit-0.05-0.10

0.10.20.321/N2 and b . Each data point is slightly shifted horizontally toe N dependence of χ/σstr2r to see. The horizontal dashed line in the b2 plot represents the dilute instanton gason (DIGA).1/Nb20

DIGADel DebbioBonatithis workph fit-0.05-0.10

0.10.20.321/N2 and b . Each data point is slightly shifted horizontally toe N dependence of χ/σstr2r to see. The horizontal dashed line in the b2 plot represents the dilute instanton gasn (DIGA).