Table Of ContentCalorons, instantons and constituent monopoles in SU(3) lattice gauge theory
Christof Gattringer
Institut fu¨r Theoretische Physik, Universit¨at Regensburg, D-93040 Regensburg, Germany
(October 1, 2002)
We analyze the zero-modes of the Dirac operator in quenched SU(3) gauge configurations at non-
zero temperature and compare periodic and anti-periodic temporal boundary conditions for the
fermions. Itisdemonstratedthatforthedifferentboundaryconditionsoftenthemodesarelocalized
atdifferentspace-timepointsandhavedifferentsizes. Ourobservationsareconsistentwithpatterns
expectedforKraan-vanBaalsolutionsoftheclassicalYang-Millsequations. Thesesolutionsconsist
of constituent monopoles and the zero-modes are localized on different constituents for different
boundary conditions. Our findings indicate that the excitations of the QCD vacuum are more
structured than simple instanton-likelumps.
3 PACS numbers: 11.15.Ha
0
0
2
Understanding the relevant excitations of the QCD can be expressed as P∞ = exp(i2πdiag(µ1,µ2,µ3)) with
n vacuum is a major scientific goaland lattice calculations µ1+µ2+µ3 =0 and µ1 ≤µ2 ≤µ3 ≤µ4 ≡1+µ1. The
a cancontributevaluablenon-perturbativeinformation. A actiondensitycanbe writtendowninarelativelysimple
J
prominentcandidate forsuchrelevantexcitationsarein- form[5]anddepends in additionto the phasesµ1,µ2,µ3
9 stantons which are particularly successful in providing also on three 3-vectors ~y1,~y2,~y3. As remarked the KvB
a mechanism for chiral symmetry breaking [1]. Lattice solutions can be seen to be superpositions of constituent
2
calculations have helped to corroborate these ideas. Re- monopoles and the 3 corresponding lumps are in space
v
1 cently several studies of the zero-modes and near zero- locatedat~y1,~y2,~y3. Them-thconstituentmonopolecan
0 modes of the Dirac operator [2–4] which provide a pow- be assigned a mass 8π2(µm+1 −µm) (both the masses
0 erfulandnaturalfilterretainingtheinfraredfluctuations and the monopole positions ~y are in units of the inverse
i
0 of the underlying gauge field were presented. It was es- temperature). Ageneralchoicefortheparametersofthe
1 tablishedthatthevacuumhasalumpystructureandthe KvB solution will give rise to three lumps at different
2
lumps are locally chiral [3] and (anti) self-dual [4]. positions with different width.
0
However, it is generally believed, that instantons are Thezero-modeψ inthe backgroundofaKvBsolution
/
t not responsible for confinement. On the other hand all has been computed [9] for arbitrary temporal boundary
a
l lattice calculations show that the deconfinement transi- conditionsψ(t+1/T,~x)=exp(i2πz)ψ(t,~x)withz =1/2
-
p tionandthe restorationofchiralsymmetrycoincideata correspondingtoanti-periodicboundaryconditions(b.c.)
e singlecriticaltemperaturewhichsuggeststhatthereisan and z = 0 giving the periodic case. The remarkable
h underlyingmechanismlinkingthetwophenomena. Inre- feature of the zero-mode is that it traces only one of
:
v cent years Kraanand van Baal[5] have constructed new the monopoles. In particular it is located on the m-th
Xi solutions for the classical SU(N) Yang Mills equations monopole when z ∈ {µm,µm+1}. At least for well sepa-
with compactified time, i.e. at finite temperature (KvB ratedmonopolesthe width ofthe lump seenby the zero-
r solutions). The KvB solutions generalize the caloronso- modedependsonthesizeoftheunderlyinggluoniclump
a
lution[6]andalumpoftopologicalcharge1isbuiltofN and also the quantity σ = min[z −µm,µm+1−z]. The
constituent monopoles, a feature which might provide a lump is narrow for large σ and broad for small σ. This
linktotheconfinementproblem. NotethatforKvBsolu- is an important feature in the deconfined phase where
tionsthemonopolelumpsappearinmultiplesofN.Asin- due to the Z3 symmetry of the action the Polyakovloop
gle lump is not an independent object and only N lumps can have three different phases 0,±2π/3. At sufficiently
togetherhavetopologicalcharge±1. Strongevidencefor hightemperaturesconfigurationswithrealPolyakovloop
SU(2) KvB solutions in cooled lattice gauge configura- have µ ∼0,i=1,2,3. The anti-periodic zero-mode has
i
tions were given for twisted [7] and periodic boundary z = 1/2, is located on monopole 3 and has σ ∼ 1/2
conditions [8]. The zero-mode for the KvB solution [9] andthus is quite localized. The periodic zero-modewith
shows characteristic features which should allow to use z = 0 can be located on any of the monopoles. It has
the eigenvectors of the Dirac operator as a filter retain- σ ∼ 0 and is very broad. For configurations with phase
ing infrared fluctuations and search for traces of KvB 2π/3 of the Polyakov loop (the case with phase −2π/3
solutions also in thermalized configurations. is similar) one has (µ1,µ2,µ3) ∼ (−2/3,1/3,1/3). The
Beforepresentingourlatticecalculationswebrieflydis- anti-periodic mode has σ ∼1/6 while the periodic mode
cuss a few properties of SU(3) KvB solutions and their hasσ ∼1/3andboth ofthemarelocalizedonmonopole
zero-mode. KvB solutions [5] are characterized by the 1. This implies that for complex Polyakovloop the roles
Polyakovloopatspatialinfinitywhichinasuitablegauge are reversed and here the periodic mode is more local-
1
size β statistics a [fm] T [MeV] T/Tc
6×203 8.20 70 0.115 287 0.96 0.006 0.002
6×203 8.45 64 + 25 0.094 350 1.17
0.004
0.001
0.002
TABLEI. Parameters of ourcalculations.
0 20 0 20
16 16
ized. However, the difference in localization is not as 5 1y0 15 204 81z2 5 1y0 15 204 81z2
pronounced since the values σ ∼ 1/6 and 1/3 differ less
FIG. 1. Scalar density of the zero-mode for anti-periodic
than σ ∼1/2 and 0 obtained for real Polyakovloop. temporal b.c. (l.h.s.) and periodic temporal b.c. (r.h.s.). We
Tosummarize,ifKvBsolutionsarepresentinthermal- show y-z slices for a configuration below Tc.
ized SU(3) gauge configurationsthen one should find in-
stanceswherethezero-modeswithanti-periodicb.c.and
periodic b.c. are located at different positions and the figurations. We will refer to these subensembles as the
lumpstheyseecandifferinsize. Inthedeconfinedphase complex respectively the real sector.
the differences in size should show a particular pattern In order to analyze the localization of the zero modes
when comparing configurations with real and complex we form the scalar density ρ(x) = Pc,d|ψ(x)c,d|2 from
Polyakovloop. These are simple stable signatures which the zero-modes ψ(x) by summing over the color and
c,d
distinguish KvB solutions from instantons or calorons. Dirac indices c,d. The resulting density ρ is gauge in-
InournumericalanalysisweusequenchedSU(3)gauge variant. InFig.1 weshowy-z slicesofthe scalardensity
configurations generated with the Lu¨scher-Weisz action and compare the anti-periodic case (l.h.s.) to the peri-
[10] on 6 × 203 lattices. We adjust the inverse gauge odiccase(r.h.s.). Theplotsareforaconfigurationinthe
coupling β such that one ensemble is below the critical confining phase.
temperature and the other one in the deconfined high Fig. 1 clearly demonstrates that the zero-modes with
temperature phase. The lattice spacing a and the val- periodic and anti-periodic b.c. are located at different
ues for the temperature T and Tc for the Lu¨scher-Weisz space-time points. For anti-periodic b.c. ρ has its maxi-
action listed in Table I were determined in [11]. mumat(t,x,y,z)=(2,9,13,13)whileforperiodicb.c.it
We calculate the eigenvalues and eigenvectors of the is located at (1,9,6,7). The figure shows the y-z slices
chirallyimprovedDiracoperator[12]usingtheimplicitly through the maxima, i.e. the t = 2,x = 9 slice for anti-
restartedArnoldimethod[13]. Thisisdoneforboth,pe- periodic b.c., respectively the t = 1,x = 9 slice for the
riodic and anti-periodic temporal b.c. for the fermions periodic case. The maxima are separated by 9.3 lattice
(the spatial b.c. are always periodic). The chirally im- spacings which corresponds to 1.1 fm. It is also obvious
proved Dirac operator is not exactly chiral but a good from the figure that the two lumps have different height
approximation of an exactly chiral lattice Dirac opera- and width. This is not what is expected for the zero-
tor. The topologicalzero-modes of the continuum mani- mode of a classical caloron but agrees well with the be-
festthemselvesassmallrealmodes. Itcanbeshown[14] havior predicted for the KvB solutions. When scanning
that although the eigenvalues are not exactly zero they through our ensemble we have detected many configura-
give riseto the correcttopologicalsusceptibility through tions showing such behavior.
the index theorem. Although the small real eigenvalues Having found instances of configurations resembling
do not vanish exactly we will refer to the correspond- KvB solutions we now turn to a more quantitative anal-
ing modes as zero-modes in the following. From a larger ysis. One interesting observable was already addressed,
setofconfigurationsweselectthoseconfigurationswhich the distance between the maximum of ρ and the maxi-
A
have a single zero-mode for both b.c., i.e. are in the sec- mum of ρ . The subscript A refers to ρ computed from
P
tor with topological charge Q = ±1. The restriction to the zeromode withanti-periodicb.c, while the subscript
the |Q| = 1 sector is necessary to avoid mixing between P is used for ρ computed from the periodic zero-mode.
multiple zero-modes that occurs in higher sectors. For A related observable is the overlap of the support of
our configurations with |Q| = 1 we find that for both, the two lumps. The support is defined as the function
periodic andanti-periodic boundaryconditions the zero- θ(x) which has θ(x)=1 for the lattice points x carrying
modeisalwaysdominatedbyasinglelump. Theposition the lump in ρ and θ(x) = 0 for all other points. An
ofthelump,however,canbedifferentforthetwochoices open question is how to identify the lump. For a narrow
of the boundary conditions. lumpthesupportissmallwhileabroadlumphasamuch
The ensemble in the high temperature phase has a larger support.
non-vanishing Polyakov loop. We will use this ensem- We will base our definition of the support on the so-
ble to study the influence of the expectation value of called inverse participation ratio I =V P ρ(x)2, where
x
the Polyakov loop on the zero-modes. We divide the V is the total number of points in our lattice. Note that
deconfining ensemble into sub-ensembles with complex weusenormalizedeigenvectorssuchthatP ρ(x)=1. It
x
respectively real Polyakov loop containing 64 (25) con- iseasytoseethatforanentirelylocalizedstate(ρ(x)=1
2
20 20
all sectors
48 real sector
s15 s15
nt nt
u u
o10 o10
c c
f f
o o
# 5 # 5
0 0
0 5 10 0.0 0.5 1.0 0 5 10 0.0 0.5 1.0
distance relative overlap distance relative overlap
FIG.2. Histograms for thedistance of the maxima (l.h.s.)
FIG.3. Sameas Fig. 2 butfor theensemble above Tc.
and the relative overlap of the lumps for different boundary
conditions (r.h.s.). Thedata are for theensemble below Tc.
(r.h.s. plot in Fig. 2) one comes to the same conclusion.
Only for a small fraction of the configurations one finds
for only one lattice point and 0 else) the inverse partici-
that the lumps seen by anti-periodic and periodic zero-
pation ratio is I =V, while for a completely delocalized
modes coincide and for many cases they do not overlap
state (ρ(x) = 1/V for all lattice points) I = 1. The
at all. A comparison of the two plots in Fig. 2 shows
inverse participation ratio is a convenient measure for
that although many pairs of lumps may have the same
thelocalization,assuminglargevaluesforlocalizedstates
position of their maximum they still differ in size such
and small values for spread out states.
that their relative overlapis smaller than 1.
Using I we now define a number N = V/(32I) and
In order to further study the evidence for KvB so-
use this to determine the supportbysetting θ(x)=1 for
lutions we now turn to the ensemble in the high-
thoseN latticepointsxwhichcarrytheN largestvalues
temperature phase. For these configurations the theory
ofρ andθ(x)=0forallotherpoints. Thenormalization
is deconfined and the Polyakovloop has a non-vanishing
ofN waschosensuchthatifthelumpwasa4-DGaussian
expectation value. As discussed above the phase of the
the support would consist of the 4-volume carrying the
Polyakovloop plays an important role for the properties
points inside the half-width of the Gaussian. Note that
of the KvB solutions and one expects significant differ-
N is the size of the support, i.e. N =P θ(x).
x ences between the complex and the real sector.
With our definition of the support θ for both the
Letus beginourdiscussionofthe deconfinedphaseby
anti-periodic (θ ) and periodic (θ ) lumps we can de-
A P againlookingatthe histogramsforthe distancebetween
fine the overlap function C of the two lumps as C(x) =
the peaks of ρ and their overlap. Fig. 3 is equivalent to
θ (x)θ (x). From this we compute the relative overlap
P A Fig.2but now showsdata for the ensemble aboveT . In
c
R=2/(N +N )P C(x). The relative overlap R is a
A P x additiontoshowingthedataforthefullensemblewealso
real number ranging from 0.0 to 1.0. If the lump in ρ
A displaytheresultsforthesubensembleintherealsector.
and the lump in ρ sit on top of eachother (at least the
P These data are represented by thinner, black bars.
core up to about the half-width) and have the same size
When inspecting the histograms for the distance be-
then R = 1.0. If the two lumps are entirely separated
tween the peaks (l.h.s. plot) we find that again a certain
one has R=0.0. Values in between give the ratio of the
amountofconfigurationsshowsratherlargedistancesbut
common volume of the two supports to the average size
the fraction of these configurations is smaller. 12% of
of the two supports. Note that even if the two lumps sit
the ensemble show distances of at least 7 lattice spac-
ontopofeachother,buthavedifferentsizesN ,N one
A P ings (again corresponding to ∼0.7 fm). Note that these
has R<1.0.
configurations are entirely dominated by the real sector.
In Fig.2 we showhistogramsfor the distance between
This is exactly as pointed out in our summary of the
the maxima in ρ and ρ (l.h.s.) and histograms for
A P KvBzero-modeswherewediscussedthatinthe complex
the relative overlap (r.h.s). The distances were rounded
sector the two zero-modes sit on the same lump.
to the nearest integer while the relative overlap R was
Another effect is evident from the two plots in Fig. 3.
rounded to the nearest multiple of 0.1. The data shown
Although more than half of the configurations have the
are for the ensemble below T . When looking at the
c centerofthe twolumps atexactly the same lattice point
l.h.s.plotonefindsthatformanyconfigurationsthemax-
wedonotseemanyconfigurationswherethe lumpshave
ima of the two lumps sit on top of each other. However,
a large overlap. The distribution is instead peaked at
the distribution shows also that for 50% of the configu-
an overlap of 0.5. The reason for this behavior is the
rations the two lumps are separated by at least 6 lattice
difference in size between the two lumps. One ofthem is
spacings which corresponds to 0.7 fm, almost twice the
muchnarrowerthantheotheroneandthustheirrelative
typical instanton size. This indicates that for a large
overlapis considerably smaller than 1.
fraction of the ensemble the zero-modes with different
We directly study the phenomenon of the large dif-
b.c. see different lumps.
ference of the sizes of the two lumps by comparing the
Also from the histograms for the relative overlap
inverse participation ratio I of the zero-mode for anti-
3
andasubstructureresemblingtheconstituentmonopoles
200
of Kraan - van Baal solutions is evident.
I
P
150
Acknowldegements: I would like to thank Meinulf
G¨ockeler,MichaelMu¨ller-Preussker,PaulRakow,Stefan
100
Schaefer, Andreas Scha¨fer, Christian Weiss and in par-
real sector ticular Pierre van Baal for discussions and the Austrian
50 complex sector Academy of Science for support (APART 654). The cal-
culationsweredoneontheHitachiSR8000attheLeibniz
0 Rechenzentrumin Munich and I thank the LRZ staff for
0 50 100 150 200 training and support.
I
A
FIG. 4. Comparison of the inverse participation ratio of
the zero mode for periodic (IP) and anti-periodic (IA) b.c.
The data are from the ensemble above Tc.
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