Instantons and Monopoles

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by  Adriano Di Giacomo

Instantons and Monopoles
Adriano Di Giacomo
University of Pisa, Department of Physics and INFN,
Sezione di Pisa, Largo B. Pontecorvo 3, 56127 PISA, ITALY∗
Masayasu Hasegawa
Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics,
Dubna, Moscow Region, 141980, RUSSIAN Federation†
(Dated: January 27, 2015)
This study is part of a research program aimed to investigate the relations between instantons,
monopoles, and chiral symmetry breaking. Monopoles are important 3-dimensional topological
configurations existing in QCD, which are believed to produce colour confinement. Instantons are
5
4-dimensional topological configurations and are known tobe related to chiral symmetry breaking.
1
To study the relation between monopoles and instantons we generate configurations adding to the
0
2 vacuumstatestaticmonopole-antimonopolepairsofoppositechargesbyuseofamonopolecreation
operator. We observe that the monopole creation operator only adds long monopole loops to the
n
configurations. Wethencount thenumberoffermion zero modesusingOverlapfermions as atool.
a
As a result we find that each monopole-antimonopole pair of magnetic charge one adds one zero
J
mode of chirality 1, i.e. one instanton of topological charge 1.
6 ± ±
2
PACSnumbers: 11.15.Ha,12.38.Gc

]
t
a I. INTRODUCTION in the total number of positive and negative monopoles.
-l Magnetic charges are in any case shielded and the addi-
p Monopoles are 3-dimensional topological QCD config- tional monopoles are one-dimensional structures prop-
e
h urations, which are most probably responsible for colour agating in time: their effect on the vacuum vanishes
[ confinement,bycondensinginthevacuumandproducing at large volumes as V−34. Moreover the total mag-
dual superconductivity [1]. This mechanism has exten- netic charge being zero the operator which creates the
1 sively been studied on the Lattice during the years with monopoles has no infrared problems [8]. On the states
v
different approaches. constructed in this way we measure the change in the
7
1 Instantonsare4-dimensionalconfigurationsandarere- number of instantons.
5 lated to Chiral symmetry breaking as described e.g. by Asatooltocountinstantonsweusethezeromodesof
6 the instanton liquid model [2]. anOverlapDiracOperator. Indoingthatwerealisethat
0 There exist strong hints that the role of the two con- eachlatticeconfigurationcontainszeromodesofonlyone
.
1 figurations can be related. In Ref. [3] it is shown that sign,either+or ,andneverzeromodesofoppositesign
0 monopoles catalyse instantons, in the sense that the at the same time−: this contrasts with the naive expecta-
5 probability of having an instanton configuration in the tion that, zero modes being localisedand the correlation
1
presence of a monopole is of order 1. The analysis length finite, independent parts of the lattice could con-
:
v there was specifically addressed to grand-unified mod- tain instantons of opposite signs. We also would expect
Xi els in view of the proton decay, but it is equally valid in thatthenumberofinstantonsandofanti-instantons,i.e.
QCD. On the other hand the work of Ref. [4, 5] shows thenumberofzeromodes,beproportionaltothevolume.
r
a thatinstantons,andmoregenerallycaloronsaremadeof Wearguethat,forsomereasontobequantitativelystud-
monopoles. ied and understood we only observe the net topological
Asafirststeptowardsaquantitativeunderstandingof charge,while pairsof zeromodes ofopposite signescape
thisrelation,weaddstaticmonopole-antimonopolepairs detection. Similar views can be found in [9]. Maybe this
tothevacuumofanSU(3)puregaugetheory[6–8],with phenomenonisduetoourrathersmallvolumes,andwill
the aim of changing the number of monopoles without disappear at largervolumes. We check this argumentby
significant modifications of the vacuum state. Indeed, doing a careful study of the distributions of topological
if the vacuum is a dual superconductor adding a pair charges, a careful determination of the topological sus-
of magnetic charges does not modify the ground state, ceptibility,whichwecomparewithexistingresultsinthe
which is an eigenstate of the creation operators, except literature, to convince ourselves that the determination
ofthetopologicalchargeiscorrectandalsothatourcode
is correct [Section II].
∗ [email protected] The number of instantons can in any case be deter-
† UniversityofParma,DepartmentofPhysicsandINFN,Gruppo mined from the topologicalsusceptibility under the very
Collegato di Parma, Via G. P. Usberti 7/A, 43124 PARMA, general assumptions of translation invariance, existence
ITALY;[email protected] ofa finite correlationlength, andCP invariance[Section
2
III]. A. Lattice spacing
In Section IV we briefly recall the definition of the
operatorwhich creates a static pair of monopoles propa- To fix the scale, we determine the lattice spacing by
gating in time from t= to t=+ and sitting at a usingoftheanalyticinterpolationofRef.[17]. Asacheck
−∞ ∞
given spatial distance. we also measure the string tension and the q¯q potential
In Section V we check that the operator really cre- from Wilson loops. We use APE smearing [18] of the
ates monopoles by using the technique of Ref. [10, 11]. spacial link variables to suppress excited states and fit
We find that indeed it creates so called long monopoles, the function V(R) = σ R α/R+C to the potential
which are not short-range fluctuations but wrap the lat- V(R). The distance R b·etw−een quark and anti-quark is
tice along the time direction. improved to R using the Green function as defined in
I
FinallyinSectionVIwelookattheconfigurationswith Ref. [17, 19]. The Jackknife method is used to estimate
monopole pairs of different magnetic charges added, we the statistical errors.
measure the topological charge and from it we compute The lattice spacing is then computed in two different
the number of instantons. The result is that a pair of ways (1) a(1): from α and Sommer scale r = 0.5 [fm],
0
monopolesofcharges 1producesoneinstantonoranti- and (2) a(2): from the String tension √σ = 440 [MeV]
±
instanton. A pair with magnetic chargemc produces mc [Table I]. The results are reasonably consistent with the
instantons or anti-instantons. analytic interpolation. Therefore, we take the lattice
In Section VII we draw some conclusions. spacing from the analytic interpolation [17], and use the
Some of the results have already been reported at the Sommer scale r =0.5 [fm].
0
LATTICE 2014 Conference [12] and at the Conference
Confinement 2014 [13].
B. Simulation details
II. OVERLAP FERMIONS
WegenerateconfigurationsusingtheWilsongaugeac-
tion and periodic boundary conditions. The number of
Wilson fermions explicitly break Chiral symmetry. iterationsfor the thermalizationis (2.0 104), and the
Therefore we use Overlap fermions which preserve Chi- configurations are sampled after O(5.0 ×103) iterations
ral symmetry on the Lattice. The Ginsparg-Wilson re- O ×
between them. The simulation parameters are listed in
lation [14] describes Chiral symmetry in Lattice gauge
Table II.
theory,
We construct the Overlap Dirac operator D from the
massless Wilson fermions D which are computed with
γ D+Dγ =aDRγ D. (1) W
5 5 5
the gauge links of the configurations. An anti-periodic
a is the lattice spacing, D the Overlap Dirac operator, boundary condition in the temporal direction is used for
and R a parameter. The right hand side of this relation the Wilson fermions.
is non zero, because of the Nielsen-Ninomiya [15] no go In more detail, we carry out the numerical compu-
theorem. MultiplyingbothsidesofEq. (1)bytheinverse tations by the technique of Ref. [20–22]. The massless
of the OverlapDirac operator, D−1, gives Overlap Dirac operator D(ρ) is
γ D−1+D−1γ =aRγ , (2)
5 5 5 ρ D (ρ)
W
D(ρ)= 1+ . (7)
showing that Chiral symmetry breaking of the propaga- a( DW(ρ)†DW(ρ))
tor D−1 is an operator of (a) vanishing in the contin-
uum limit. The exact formOof the Overlap Dirac opera- DW(ρ)iscomputedfrommpasslessWilsonDiracoperator
torisdefinedinRef.[16]intermsofthemasslessWilson DW, Eq. (4), as follows:
Dirac operator D .
W ρ
D (ρ)=D (8)
W W
1 A − a
D = 1+ (3)
Ra(cid:18) √A†A(cid:19) ρ is a (negative) mass parameter, 0 < ρ < 2. We set
A= M +aD , M is a parameter, 0<M <2. D ρ=1.4 in this study. Using the Hermitian Wilson Dirac
0 W 0 0 W
is the−massless Wilson Dirac operator defined as follows: operatorHW(ρ)=γ5DW(ρ),the signfunctionisdefined
as in Ref. [16]
1
D = γ ( ∗ + ) a ∗ (4)
W 2 µ ∇µ ∇µ − ∇µ∇µ HW(ρ)
ǫ(H (ρ)) . (9)
W
(cid:8) (cid:9) ≡ H (ρ)†H (ρ)
W W
1
[ ψ](n)= U ψ(n+µˆ) ψ(n) (5)
∇µ a{ n,µ − } Therefore, the massless Opverlap Dirac operator is com-
puted by the sign function
1
[ ∗ψ](n)= ψ(n) U† ψ(n µˆ) (6) ρ
∇µ a − n−µˆ,µ − D(ρ)= 1+γ5ǫ(HW(ρ)) . (10)
a{ }
n o
3
TABLE I. Determination of the lattice spacing. The lattice spacing a(0) is computed by the analytic function of [17]. The
lattice spacing a(1) is computed by our simulations of the q¯q potential (n is thenumberof smearing steps and α is the weight
factor of smearing, FR is therange of thefit) and a(2) from our determination of the string tension.
β a(0) [fm] a(1) [fm] a(2) [fm] V (n, α) FR (RI/a) χ2/ndf Nconf.
6.00 9.315 10−2 9.39(9) 10−2 9.93(8) 10−2 184 (20, 0.5) 0.9 - 7.0 4.3/4.0 440
× × ×
TABLEII.Thistablesummarisestheresultsofoursimulationsforthenumberofzeromodesandthetopologicalsusceptibility.
χ2/ndf refers to thefit of thefunction PQ= √e−22πhQhQQ222ii to thedistributions of the topological charges.
β a/r0 V V/r04 NZero hQ2i hQ2ir04/V χ2/ndf Nconf.
5.79 0.2795 124 126.5 2.32(12) 8.3(7) 0.066(6) 4.7/14.0 200
5.81 0.2659 104 49.96 1.44(4) 3.50(17) 0.070(3) 8.5/11.0 844
144 191.9 2.80(13) 12.2(1.1) 0.064(5) 12.4/17.0 249
164 327.4 3.63(17) 20.9(1.8) 0.064(6) 19.6/19.0 275
5.85 0.2484 124 78.95 1.87(10) 5.3(5) 0.068(6) 3.9/11.0 200
164 249.5 3.30(13) 17.0(1.3) 0.068(5) 22.0/19.0 344
5.86 0.2395 144 126.5 2.32(9) 8.3(7) 0.066(6) 16.7/15.0 338
5.90 0.2216 124 49.96 1.40(4) 3.32(16) 0.067(3) 10.1/11.0 835
164 157.9 2.48(11) 10.3(9) 0.065(5) 18.1/19.0 320
5.93 0.2129 144 78.95 1.95(9) 5.8(5) 0.073(6) 15.0/12.0 278
5.99 0.1899 144 49.96 1.39(4) 3.17(15) 0.063(3) 8.8/11.0 862
6.00 0.1863 124 24.98 0.88(4) 1.47(11) 0.059(4) 6.9/8.0 430
144 46.28 1.35(5) 3.17(19) 0.069(4) 10.2/11.0 592
123 24 49.96 1.34(4) 2.98(15) 0.060(3) 11.7/12.0 790
×
164 78.95 1.81(7) 5.4(4) 0.068(5) 7.0/14.0 405
184 126.5 2.42(11) 8.8(7) 0.069(5) 16.7/14.0 258
6.07 0.1662 164 49.96 1.33(4) 3.04(15) 0.061(3) 4.1/12.0 895
Next, we numerically compute the sign function by the imaginary axis as shown in Figure 1. The improved
using the minmax approximation by Chebyshev polyno- massless Overlap Dirac operator Dimp(ρ) is defined as:
mials of Ref. [20, 22, 23]. We solve eigenvalue problems
byusingthe Arnoldimethod(subroutinesofARPACK), a −1
Dimp(ρ)= 1 D(ρ) D(ρ) (11)
and save (80) pairs of the low-lying eigenvalues and
− 2ρ
eigenvectoOrs of the Overlap Dirac operator. (cid:18) (cid:19)
Ifn+isthenumberofexactzeromodesofpluschirality The spectral density ρ(λ, V) is defined as
and n of minus chirality in the spectrum the massless
−
OverlapDiracoperator,thetopologicalchargeisdefined 1
ρ(λ, V)= δ(λ λ¯) . (12)
as Q = n+ n−. The average square of the topological V h − i
charge Q2−, and the topological susceptibility Q2 /V Xλ
h i h i
are computed from the topological charges. All of our λ¯ = Im(λ ). We show a typical spectral density
results are listed in Table II. imp
ρ(λ, V) of the non zero modes in Figure 2.
C. Eigenvalues and Spectral density D. The number of zero modes, the topological
charge, and the topological susceptibility
The eigenvalues λ of the OverlapDirac operator lie in
the complex plane on a circle of centre (ρ,0) and radius In our simulations, we never observe zero modes of
ρ as shown in Figure 1. In our case, ρ = 1.4. We con- opposite chirality in the same configuration. The zero
sider insteadthe eigenvalues λ ofthe improvedmass- modes have either all + chirality or all - chirality in
imp
less Overlap Dirac operator Dimp(ρ) [24]. They lie on each configuration: as a consequence the number N
obs
4
V = 184, β = 6.00
25
1.5 0.42
20
1
λ
λ
0.5 imp 0.14 15
λ λ 〉2 Q
Im0 Im 〈 10
-0.5 -0.14
5
-1
0
0 50 100 150 200 250 300 350
-1.5 -0.42
0 0.5 1 Re1λ.5 2 2.5 0 Reλ 0.05 V/r40
FIG. 4. The average square of the topological charges Q2
FIG. 1. An example of distribution of eigenvalues λ, and
versus the physical volume V/r4. The linear line is Qh2 =i
improved eigenvalues λimp. Inthisconfiguration thereis one A V/r4+B, A=6.8(2) 10−20, B= 0.20(13). h i
zero mode. The right figure is an enlargement of the region · 0 × −
λ 0.
≈
of observed zero modes is N = n n = Q and
×106 N2 =Q2. By definition Q=obsn | n+ −. −| | |
14 V = 184, β = 6.00 oTbshe average N = N +−is p−lotted as a func-
Zero obs
h i
12 tion of the volume V in Figure 3 and is proportional
to √V: fitting a function N = A V/r4 + B to
3] V10 χth2e/nddafta=g1i5v.e1s/1A5.=0. 4T.h9e(3fi)t×tinZ1ge0rr−oa2n,gBepin=ph−·y0s.i1ca80l(5v)o,luamnde
e 8
M units V/r4 is from24 to 330. Q2 is consequently linear
λ) [ 6 in V [Figu0re 4]. The slope is thhe itopological susceptibil-
ρ(
ity whichis determined to be Q2 r4/V =6.8(2) 10−2.
4 h i 0 ×
This value is consistent with the determination from the
2 widths of the distributions of topological charges and
withthedeterminationsofothergroups,asdiscussedbe-
0
0 100 200 300 400 500 low.
λ [MeV]
However our counting of zero modes contrasts with
the expectation that the number of zero modes be pro-
FIG. 2. Spectral density of non zero modes (ρ(λ)).
portional to the volume. Indeed, if zero modes are asso-
ciatedtolocalizedconfigurationsliketheinstantons,and
the vacuumis made ofindependent regionsofsize of the
4 orderofthe correlationlength, the number ofinstantons
is proportional to the volume. In addition n = n
3.5 h +i h −i
by invariance under CP.
3
Iftheseargumentsarecorrectthe onlywayoutisthat
2.5 pairs of instantons of opposite chirality somehow escape
ero detection by our way of counting zero modes, at least at
NZ 2 the rather modest volumes of our simulations. This ef-
1.5 fectpreservesanyhowthe valueofthe topologicalcharge
andwouldnotaffectthedeterminationofthetopological
1
susceptibility. We shall use this assumption in the next
0.5 Section to extract from our data the number density of
0 instantons. Acarefulstudyofthisphenomenon,however,
0 50 100 150 200 250 300 350
shouldbedone,toputonsaferbasisthedeterminationof
V/r4
0 thetopologicalsusceptibility. Acomplementarystrategy
couldbetoimprovenumericallythedeterminationbased
FIG. 3. The number of observed zero modes NZero versus on the bosonic sector of the theory [25], which does not
the physical volume V/r04. The continuous curve is NZero = rely on the counting of zero modes, and cross-check the
pA V/r4+B, A=4.9(3), B= 0.18(5).
· 0 − two methods.
5
0.12
70 β = 6.00
β = 6.00, JHEP 11 (2003) 023
0.1
60 β = 6.00, Phys. Rev. Lett. 94, (2005) 032003
50 0.08
V
NQ3400 〉42 / r Q00.06
〈 0.04
20
10 0.02
0 0
-10 -5 0 5 10 0 20 40 60 80 100 120 140 160 180 200 220
Q V/r4
0
FIG. 5. A distribution of topological charges Q. The lattice FIG. 6. The topological susceptibility Q2 r4/V at β =6.00
is V =164, β=6.00. compared to results of other groups [26h, 27i].0
Asanticipatedabovethetopologicalsusceptibilitycan 0.12
beextractedfromthedistributionoftopologicalcharges.
A typical distribution of the topological charges is that 0.1
in Figure 5: a Gaussian function [26]
0.08
V
PQ = e−2π2hQQQ22i2 (13) 〉42 / r Q00.06
h i 〈 0.04
p
fits the data with χ2/ndf = 7.0/14.0, and topological
0.02
susceptibility Q2 r4/V = 6.7(5) 10−2. This value is
consistent withh thie0value Q2 r4/×V = 6.8(5) 10−2 di-
h i 0 × 0
rectly computed from the number of zero modes. More- 0 50 100 150 200 250 300 350
over, seventeen distributions of the topological charges V/r4
0
are computed from our seventeen different lattices, and
we fit the Gauss function to all distributions. All the re- FIG.7. Thetopological susceptibilities Q2 r4/V inoursim-
sulting values of χ2/ndf are in the range from 0.3 to 1.3 ulationsasshowninTableIIversusthephhysiic0alvolumeV/r4.
0
[Table II]. The topologicalchargesall have the Gaussian A dotted line marks thephysical volume V/r4=50.00.
distribution.
In Figure 6, we compare the topological susceptibility
at the same β = 6.00 with results of other groups. The
lap fermions in our simulations are properly computed.
figure shows that our results are consistent with them.
Moreover, we check that the finite lattice volume does Our result: χ=(1.86(6) 102 [MeV])4 (14)
notaffectthetopologicalsusceptibilityuptoL=2.1[fm] ×
as indicated in Figure 7. Ref [27]: χ=(1.91(5) 102 [MeV])4 (15)
×
Ref [28]: χ=(1.88(12)(5) 102 [MeV])4 (16)
×
The theoretical expectation [29, 30]:
F
E. Topological susceptibility in the continuum limit χ= 6π(m2η+m2η′ −2m2K)|exp ≃(1.80×102 [MeV])4
(17)
Last, we fix the physical volume at the value V/r4 =
0
50.00 marked in Figure 7, and extrapolate the five data
points of the topological susceptibility to the continuum III. INSTANTONS
limit using a linear expression Q2 r4/V = c + c a2.
h i 0 0 1
The results are consistent with other groups. We are We want to compute the number of instantons from
thus confident that eigenvalues and eigenvectors of over- the number of zero modes, but, as anticipated, there are
6
problemstodetermineitsincesomeofthemescapedetec- When Q 1, N 1 at Q2 fixed
tion. In any translation invariant model indeed, e.g. the ≫ I ≫ NI
instantonliquidmodel,thenumberofinstantonslinearly −Q2
e 2NI
increases with the physical volume. The number of our P(Q) (23)
≃ √2N π
zero modes instead clearly increases as the square root I
ofthe physicalvolume [Figure 3]. We never observezero Finally, the number of instantons is determined as N =
I
modesn+andn−ofoppositechiralityinthesameconfig- hQ2i=hNZ2eroi.
uration: ournumberofzeromodesalwayscoincideswith
the topologicalcharge. The distributions of the topolog-
ical charges determined in this way are Gaussian with B. The density of instantons
0.3<χ2/ndf<1.3, and agree with other groups results.
All that suggests that we observe the net number of We obtain the instanton density by fitting a linear
zero modes. At least at our rather small physical lattice function N = 2ρ r4 V/r4 + B to Q2 as shown in
volumes for some reason pairs of zero modes of opposite Figure 4. AIll of daita0 p·oint0s are incluhdediin the fit and
signseemtoappearasnon-zeromodes. Thiscanexplain χ2/ndf = 11.9/15.0. The slope is 2ρ r4 = 6.8(2) 10−2
whyweobtainthecorrecttopologicalchargeanyway. To and the intercept is B = 0.20(13)i.0The inter×cept is
estimate the density of instantons we can use a model compatible with zero. Fina−lly, the instanton density is
based on the reasonable assumption that the instantons evaluated as
of both Chiralities are uniformly distributed in space-
time and independent. ρi =8.3(3) 10−4 [GeV4]. (24)
×
This resultis consistentwiththe instantonliquidmodel,
Ref. [2].
A. The number of instantons
Letusdenotethenumberofinstantonsofpositivechi-
IV. THE MONOPOLE CREATION OPERATOR
rality in a volume V by n , the number of instantons of
+
negative chirality by n . Of course
− In this section, we briefly recall the definition of
N the monopole creation operator and the method to
I
n+ = n− = =ρiV, (18) count monopoles. In order to understand the relation
h i h i 2
between instantons and monopoles we add monopole-
ρi isthedensityofinstantons. BecauseofCP invariance antimonopole pairs of opposite charges in the configu-
instantons and anti-instantons have the same distribu- rations using the monopole creation operator, and mea-
tion. The instantons are independent and the distribu- sure the variation of the number of instantons. We first
tionPoisson-likeO( 1 ): indeedwecanviewaspace-time check that the monopoles are successfully added in the
Vst
of volume Vst as coveredby non overlapping hypercubes configurations by counting the additional monopoles.
of size V, which are all equivalent by translation invari- The monopole creation operator is defined in [6–8].
ance, independent from each other if their size is bigger Specifically, in this study, we use the monopole cre-
than the correlation length, and can all be used at the ation operator defined in [8], [Eq. (41) et seq.], which,
same time to determine the distribution, which is then acting on the vacuum state, produces a static pair of
Poisson-like. monopole-anti-monopole of opposite charge propagating
P(n+)= 1 NI n+e−N2I (19) from t=−∞ to t=+∞. The operator is defined as
n ! 2 µ¯=exp( β∆S). (25)
+ (cid:18) (cid:19) −
∆S isdefinedbymodifyingthenormalactionatthetime
P(n−)= n1! N2I n−e−N2I (20) tfobllyowresp:lacing the usual plaquette Πµν(n) by Π¯µν(n), as
− (cid:18) (cid:19)
The resulting distribution for Q=n+−n− is S+∆S ≡ Re(1−Π¯µν(n)) (26)
n,µ<ν
∞ X
P(Q)= P(n )P(n +Q)=exp( N )I (N ). Π¯ (n)=Π (n)inallsites nwithn =t;atn =t the
− − I Q I µν µν 0 0
− 6
nX−=0 (21) sifipeadceΠ-¯spa(ct,e~nc)om=pΠone(ntt,s~n)i,jwh=ile1Π−¯ 3(ta,r~ne)aigsaainmuondmifioedd-
ij ij i0
Here I (x) is the modified Bessel function plaquette with inserted matrices M (~n) and M (~n)†,
Q i i
IQ(x)= ∞ (x)n−(x)n−+Qexp( NI) 1 Π¯i0(t,~n)= Tr1[I]Tr[Ui(t,~n)Mi†(~n+ˆi)
2 2 − n !(n +Q)!
− −
Xk=0 U (t,~n+ˆi)M (~n+ˆi)U†(t+1,~n)U†(t,~n)] (27)
(22) × 0 i i 0
7
configurations varying the number of added monopole
TABLEIII.ThedefinitionofDistanceN andthelocation of
charges from zero to four. The distance between the
themonopole and theanti-monopole in thefour-dimensional
monopole and the anti-monopole is defined in Table III.
space (x,y,z,t). V =L3 T.
× Next, the configurations are iteratively transformed to
N Monopole Anti-monopole the Maximally Abelian gauge using the Simulated An-
Odd (cid:0)L+N+2,L+N+2,L+1,T(cid:1) (cid:0)L−N+2,L−N+2,L−1,T(cid:1) nealing algorithm. To remove the effects of the Gribov
2 2 2 2 2 2 2 2
Even (cid:0)L+N+1,L+N+1,L+1,T(cid:1) (cid:0)L−N+1,L−N+1,L−1,T(cid:1) copies, 20 iterations are carried out in our simulations.
2 2 2 2 2 2 2 2 Abelian link variables, holding U(1) U(1) symme-
×
try, are derived by Abelian projection from non-Abelian
link variables. The monopole current is defined for each
Tr[I]isthe traceofthe identity,andthe matrixM (~n)is
i
colour direction as follows:
the discretised versionof the classical field configuration
bekaAdd0i(e~nd−inx~tkh)eplorocadtuicoends ~xatk,snitaem~nelbyy the monopoles to kµi(n)= 21ǫµνρσ∂νniρσ(n+ν) (32)
P
M (~n)=exp(i A0(~n x~ )), (i=x, y, z). (28)
i i − k The colour index can be i = 1, 2, and 3. niρσ(n+ν) is
Xk the Dirac string [32], and the monopole current satisfies
The form used for the monopole fields in a spherical co- the conservation law,
ordinate system (r, θ, φ) centred at the monopole is
Wu-Yang: ki(n)= 1ǫ ∂ ni (n+ν)=0. (33)
µ 2 µνρσ ν ρσ
(i) nz−z ≧0 Xi Xi
A0x 2mgcrsinφ(s1in+θcosθ)λ3 The monopole density is defined as follows:
AA0y0z=−2mgcrcosφ0(s1in+θcosθ)λ3 (29) ρmr03 = 121V |kµi(n)|r03. (34)
    i n,µ
XX
(ii) n z <0
z − First,wemeasurethemonopoledensityasshowninFig-
A0x −2mgcrsinφ(s1in−θcosθ)λ3
AA0y0z= 2mgcrcosφ(s10in−θcosθ)λ3  (30) 4
  Normal Conf.
The electric chargeis   3.5 mmc == 01
c
3 mc = 2
6 m = 3
g = β : (gauge coupling constant) (31) 2.5 mcc = 4
r
30
We give the monopoles magnetic charges rm 2
mc = 0,1,2,3,4. One monopole has charge +mc and ρ
1.5
the other has charge m . The total magnetic charge
c
−
is zero. mc = 0 is the reference configuration with no 1
monopoles added.
0.5
The monopole of charge +m and anti-monopole of
c
charge −mc are placed at time slice t=T/2: the choice 00 1 2 3 4 5 6 7 8
is irrelevant since boundary conditions are periodic in Distance
time.
ThelocationsofthemonopolesinthelatticeV =L3× FIG. 8. The monopole density ρmr03 versus the distance
T arechoseninthe(x,y,z,t)spaceasinTableIII,where between the monopole and anti-monopole. The lattice is
also the distance N between the monopoles is defined. V =144,β=6.00.
While Monte Carlo simulations are carried out, the
pairofmonopolesmakeslongmonopoleloopsinthecon-
figurations. ure 8, varying the distance between the monopole and
anti-monopole from one to eight, moreover, varying the
monopole charges m from zero to four. The monopole
c
V. DETECTING THE ADDITIONAL densities go to plateaus when the monopole and anti-
MONOPOLES monopole are adequately separated. On this plateau we
fix the distance D between the monopole and the anti-
Toverifythatthe monopolesaresuccessfullyaddedto monopole, D = 6 and D=8, and measure the monopole
the configurations, we detect the monopoles in the con- density. The monopole density does not depend on the
figurations,as done in Ref. [10, 11, 31]. We generate the distance between them as indicated in Table IV. The
8
TABLEIV.Themonopoledensityρmr03. D6andD8indicate 107
Distance = 6 and 8 respectively. N indicate the number
conf.
of configurations each distance. 106 Histgram of Monopole loops
V = 323×64, β = 5.29, κ = 0.13632
β V mc ρmr03 (D6) ρmr03 (D8) Nconf. 105
6.00 144 0 0.893(12) 0.893(12) 80
1 1.026(12) 1.032(13) 80 a) 104
2 1.658(14) 1.660(13) 80 H(L/103
3 2.281(15) 2.247(17) 80
4 2.991(14) 3.024(15) 80 102
6.00 184 0 0.917(7) 0.916(9) 80
10
1 0.981(7) 0.993(8) 80
2 1.307(7) 1.305(8) 80 1
0 10000 20000 30000 40000
3 1.619(8) 1.621(9) 80
L/a
4 1.998(9) 2.056(8) 80
FIG. 10. The histogram of the length of the monopole loops
L/a. Thenormalconfigurationswithtwoflavorsofdynamical
Wilsonfermions,V/r4=1.565(13) 103, r /a=6.05(5),are
3.5 0 × 0
used.
Normal Conf., V = 144
3 V = 144
Normal Conf., V = 184
2.5 V = 184 104 104
Normal Conf. mc = 2
ρ3 r0m1.25 H(L/a) H(L/a) 10
10
1 0 1000 L/a 2000 3000 0 1000 L/a 2000 3000
0.5
104 m = 0 104 m = 3
c c
0 0 1 2mc 3 4 H(L/a) H(L/a)
10 10
FIG. 9. The monopole density ρmr03 versus the magnetic
charge mc. Distance = 8. The data points of the normal 0 1000 L/a 2000 3000 0 1000 L/a 2000 3000
configurations are slightly shifted horizontally from mc = 0
to distinguish them. 104 m = 1 104 m = 4
c c
a) a)
L/ L/
monopoledensitiescomputedfromnormalconfigurations H( H(

of β = 6.00, V = 144 and 184 are ρ r3 = 0.940(11) 10 10
m 0
and ρ r3 = 0.919(7) respectively. The monopole den-
m 0 0 1000 L/a 2000 3000 0 1000 L/a 2000 3000
sity increases with monopole chargesas shown in Figure
9. However, if the physical volume becomes larger, the
FIG. 11. The histogram of the length of the monopole loops
slope becomes lower. As the physical volume becomes
L/a. Monopoles with charges mc ranging from 0 to 4 are
larger, the number density of the monopoles is rarefied. added to the normal configurations. The lattice is V = 144,
We add the number of monopoles in the configurations β=6.00
by increasing the monopole charges.
Next, we measure the lattice spacing in the same way
as in Section IIA. The results are listed in Table V and
madeoftheshortmonopoleloops,whichareshort-range
showthat the additionalmonopolesdo notchangeitap-
fluctuations. The large (infrared) clusters, which perco-
preciably.
latethroughthelatticeandwraparoundtheboundaries,
aremadeofthelongestmonopoleloopineachcolordirec-
tion. Thewayhowtocomputenumericallythemonopole
A. The monopole clusters world line in four dimension is explained in [35]. If the
physical lattice volume is large enough, the small clus-
Themonopolesareknowntoformtwoclusters[11,33, ters and the large clusters are separated as in Figure
34] in MA gauge. The small (ultraviolet) clusters are 10. InquenchedSU(2)study [33,36],the longmonopole
9
TABLE V. Determination of the lattice spacing. The lattice spacing a(1) and a(2) are computed by our simulations. n is the
numberof smearing steps and α is the weight factor of smearing.
β V Distance mc a(1) [fm] a(2) [fm] (n, α) FR (RI/a) χ2/ndf Nconf.
6.00 184 6 0 0.092(3) 0.0967(13) (30, 0.5) 1.8 - 8.0 4.0/4.0 500
1 0.0951(8) 0.1005(7) (25, 0.5) 0.9 - 8.0 5.2/5.0 500
2 0.097(3) 0.1021(17) (20, 0.5) 1.8 - 8.0 3.8/4.0 500
3 0.097(4) 0.101(2) (25, 0.5) 1.8 - 7.0 2.9/3.0 500
4 0.1010(11) 0.1068(10) (25, 0.5) 0.9 - 6.0 3.2/3.0 500
8 0 0.092(3) 0.0968(14) (30, 0.5) 1.8 - 8.0 4.0/4.0 500
1 0.095(4) 0.100(2) (25, 0.5) 1.8 - 6.0 2.0/2.0 500
2 0.0980(8) 0.1036(7) (25, 0.5) 0.9 - 9.0 6.1/6.0 500
3 0.099(4) 0.104(2) (15, 0.5) 1.8 - 7.0 3.0/3.0 500
4 0.1009(10) 0.1062(9) (20, 0.5) 0.9 - 8.0 3.3/5.0 500
loops only exist in confinement phase, and dominate the VI. THE RELATIONS BETWEEN ZERO
string tension. The long monopole loops are therefore MODES, INSTANTONS, AND MONOPOLES
considered to play the important role of producing color
confinement. Going to the Maximally Abelian gauge is A. Simulation details
essential to divide monopole into clusters.
We create a histogram of the length of the monopole
loopswhenonepairofmonopoleswithchargesfromzero
tofourareaddedasshowninFigure11. Toclarifywhich
cluster increases with the monopole charges, we deduct Normal Conf.
2.5
the sum of the longest loops from the sum of all loops. Distance = 8
Prediction
We define the remainder of the subtraction as the sum
2
of short loops. The averages of the sums of the long
loops and short loops divided by the total number of
configurations are determined. The results are shown in ero1.5
Z
N
Figure 12. We conclude that the monopoles added are
long monopoles wrapping the lattice, as expected. The 1
length of the long loops is proportional to the charge: a
chargem isequivalenttom monopolepairofcharge1. 0.5
c c
0
0 1 2 3 4
m
c
8000 Normal Conf., Long loops FIG. 13. The number of zero modes NZero versus monopole
Long loops
charges mc. The data point computed from the normal con-
Normal Conf., Short loops
Short loops figurations is slightly shifted from mc =0.
6000
We generate configurations with one monopole-anti-
a
L/4000 monopolepairaddedwithdifferentmagneticchargesmc.
The distances between the monopole and anti-monopole
are fixed at 6, and 8: slightly changing the distance be-
2000 tween them, allows to check finite lattice volume effects.
ThesimulationparametersandthedataarelistedinTa-
ble VI. The Overlap Dirac operator is constructed from
0 the gauge links of the configurations. The eigenvalue
0 1 2 3 4
mc problems are solved by the Arnoldi method, and (60)
O
pairsofthe low-lyingeigenvaluesandeigenvectorsofthe
FIG. 12. The average of the long and short monopole loops Overlap Dirac operator are saved. We then count the
divided by the number of configurations. The lattice is V = number of zero modes, and calculate the average square
144, β = 6.00. The data points of the normal configurations
of topologicalcharges,i.e. the number of instantons. Fi-
are slightly shifted from mc =0 for graphical reasons. nally,wecomparethe simulationresultswiththe predic-
tions based on the hypothesis that the added monopoles
10
eratorofthepair,andisthereforeleftunchanged. Inany
10 case, as a dual superconductor the vacuum shields the
9 Normal Conf. monopoles, so that they are one-dimensional structures
8 Distance = 8 and do not influence the external space time: O(V−43).
Prediction For these reasonswe expect that the distribution of zero
7
modes Eq. (19), (20) is unaffected by the additional
6 monopoles especially at large space-time volumes V, ex-
〉
2
Q 5 cept for the addition of some instantons.
〈 4 If each monopole-antimonopole pair of charge mc = 1
producesqinstantonsoranti-instantonswithequalprob-
3
ability (CP invariance), and the original distribution is
2
not modified, the value of Q2 becomes
1 h i
(Q+∆Q)2 = Q2 +2 Q ∆Q + (∆Q)2 (35)
0 h i h i h ih iI h iI
0 1 2 3 4
mc Here means average on the configuration, I
h···i h···i
the average on the charge distribution of the additional
FIG.14. Theaveragesquareof topological charges Q2 ver- M m q instantons. The middle term in the right-
h i ≡ c×
susmonopolechargesmc. Thedatapointcomputedfromthe hand side of Eq. (35) vanishes since Q = 0. To define
normal configurations is slightly shifted from mc =0. I we assume thateachchiralchahrgeicanbe 1 with
h···i ±
equalprobability. Theprobabilitytohavek positiveand
M k negative instantons is
TABLE VI. Charges of the added monopole, together with −
numberofzeromodesNZero,andtheaveragesquareoftopo- P(k,M)= 1 M , (k =0...M). (36)
logical charges Q2 . The lattice is β=6.00, V =144. 2M k
h i (cid:18) (cid:19)
Distance mc NZero Q2 Nconf. Corresponding to this partition ∆Q=2k M so that
h i −
6 1 1.50(6) 3.6(3) 400
M
2 1.78(7) 5.4(4) 401 (∆Q)2 = P(k,M)(2k M)2 =M. (37)
I
3 2.03(8) 6.7(5) 423 h i −
0
X
4 2.10(8) 7.2(5) 440
The sum is easily computed as shown in Appendix A.
8 0 1.39(5) 3.2(2) 501
Recalling that M = m q, the prediction follows that, in
c
1 1.55(6) 4.0(3) 502
presence of m pairs of monopoles
c
2 1.85(7) 5.6(4) 510
(Q+∆Q)2 = Q2 +m q (38)
3 2.09(7) 6.8(4) 500 c
h i h i
4 2.16(8) 7.5(5) 505 Comparingwiththedata[Figure14]itfollowsthatq =1.
Fitofalinearfunction Q2 =A m +B totheresultsof
c
h i ·
the average square of the topological charges computed
do not perturb the vacuum and specifically the distribu- from simulations gives a slope A compatible with 1, and
tion of instantons, but can only change the total topo- intercept B compatible with Q2 = 3.17(19) (the value
h i
logical charge. from normal configurations) [Table VIII].
Note: We do not do smearing, cooling, or MA gauge The observedbehavioursupports the assumptionthat
fixing in simulations. The number of zero modes, Dis- monopoles do not alter the vacuum. Each pair of
tance 6 m = 0, coincide within errors with Distance 8. monopolesaddsoneinstantonoroneanti-instantonwith
c
We take this as a check of volume independence. equalprobability. Havingfixedonceandforallthatq =1
In Figure 13 and Figure 14 we show the number of we can explicitly compute the topological charge distri-
zero modes and the average square of the topological bution, under our basic assumption, and compare with
chargesrespectively as functions of the monopole charge data. Since our monopole configurationis CP invariant,
m . m = 0 is the ordinary case with no monopoles the distribution has to be even in Q. In the simple case
c c
added. The results are compared with the predictions mc =1thedistributionisthesumwithequalweight 21 of
developed below. two”unperturbed”distributionsP0(Q)centredatQ=1
and Q = 1, P(Q) = 1[P (Q+1)+P (Q 1)]. P (Q)
− 2 0 0 − 0
is well approximated by Eq. (23). In the case of general
m
B. Predictions c
P(Q,m )=
c
The creation operator Eq. (25) acting on the vacuum 1 mc m
produces a pair of static monopoles propagating in time = c [P (Q+m 2k)+P (Q m +2k)]
from to+ . Ifthevacuumisadualsuperconductor 2mc+1 k=0(cid:18) k (cid:19) 0 c− 0 − c
−∞ ∞ X
itisacoherentstateandaneigenstateofthecreationop- (39)