The Vortex Structure of SU.pdf

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The Vortex Structure of SU(2) Calorons

Falk Bruckmann,1 Ernst-Michael Ilgenfritz,2, 3 Boris Martemyanov,4 and Bo Zhang1

1Institut fur Theoretische Physik, Universitat Regensburg, D-93040 Regensburg, Germany2Institut fur Physik, Humboldt-Universitat, D-12489 Berlin, Germany

3Institut fur Theoretische Physik, Universitat Heidelberg, D-69120 Heidelberg, Germany4Institute for Theoretical and Experimental Physics,B. Cheremushkinskaya 25, 117259 Moscow, Russia

We reveal the center vortex content of SU(2) calorons and ensembles of them. We use LaplacianCenter Gauge as well as Maximal Center Gauges to show that the vortex in a single caloron consistsof two parts. The first one connects the constituent dyons of the caloron (which are monopoles inLaplacian Abelian Gauge) and extends in time. The second part is predominantly spatial, enclosesone of the dyons and can be related to the twist in the caloron gauge field. This part depends stronglyon the caloron holonomy and degenerates to a plane when the holonomy is maximally nontrivial,i.e. when the asymptotic Polyakov loop is traceless. Correspondingly, we find the spatial vortices incaloron ensembles to percolate in this case. This finding fits perfectly in the confinement scenarioof vortices and shows that calorons are suitable to facilitate the vortex confinement mechanism.

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I. INTRODUCTION

To answer the question of what drives confinementand other nonperturbative effects in QCD, basically threesorts of topological excitations have been intensively ex-amined over the years: instantons, magnetic monopolesand center vortices. Instantons as solutions of the equa-tions of motion are special: they are the relevant ob-

jects in a semiclassical approach. While the generationof a chiral condensate is very natural via the (quasi)zero modes1, confinement remained unexplained in thismodel.

At finite temperature, where the classical solutions arecalled calorons [13], there has been quite some progressrecently, due to two effects. First of all, the asymptotic

Polyakov loop plays a key role determining the proper-ties of a new type of caloron solutions (for more detailson calorons see Sect. II and the reviews [4]). Under theconjecture that the asymptotic Polyakov loop is relatedto the average Polyakov loop, the order parameter of con-finement, calorons are sensitive to the phase of QCD un-der consideration.

Secondly, the new calorons with nontrivial holonomyconsist of N dyons/magnetic monopoles for the gaugegroup SU(N). In this way, contact is seemingly madeto the Dual Superconductor scenario. We stress that thedyon constituents of calorons appear in an unambiguousway as classical objects.

This is in contrast to Abelian monopoles and also cen-ter vortices, the other sorts of objects used to explainconfinement. In the sense, that they are widely accepted,they are not of semiclassical nature. They representgauge defects of codimension 3 and 2, respectively, whichremain after the respective gauge fixing and projection.

1 This mechanism is based on the index theorem and thus willwork for any object with topological charge.

Their interrelation and the fact that they are a prerequi-site for the occurrence of topological charge in general

has been quantitatively studied in the past [5].Monopoles are usually obtained by applying the Max-imal Abelian Gauge (MAG). Center vortices in latticeQCD can be defined through a center projection (there-fore called P-vortices) after the lattice gauge field hasbeen transformed into the Maximal Center Gauge (di-rectly by Direct Maximal Center Gauge [DMCG] or in-directly by Indirect Maximal Center Gauge [IMCG], withthe MAG as preconditioner) or into the Laplacian CenterGauge (LCG). We refer to Sect. III for the technicalities.

We would also like to mention another important de-velopment during recent years. Fermionic methods havebecome available to study topological structures with-out the necessity of smoothing. Singular, codimen-sion 1 sheets of sign-coherent topological charge havebeen found and proposed to be characteristic for gen-uine quantum configurations [6, 7] and potentially im-portant for the confinement property. The relation tothe other low-dimensional singular topological excita-tions is still not completely understood. In this scenario,(anti)selfdual objects like calorons typically appear astopological lumps after smearing [8], i.e. at a resolutionlength bigger than the lattice spacing [7].

The physical mechanisms assigned to calorons wouldbe based on their quantum weight [9], their moduli spacemetric [10], the particular features of their fermionic zeromodes [11] and the specific suppression of dyons accord-

ing to the action they acquire in different phases [12]. Thelast observation suggests an overall description of confine-ment [13] and deconfinement in terms of calorons dyonconstituents as independent degrees of freedom. The pro-posed generalized (approximative) moduli space metric,however, presents some difficulties [14] which are not yetovercome.

Since calorons unify instanton and monopoles, it isnatural to ask for the relation to center vortices. Infour dimensions the latter are two-dimensional world-

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sheets with Wilson loops taking values in the center ofthe gauge group if they are linked with the vortex sheet.Vortices that randomly penetrate a given Wilson loopvery naturally give rise to an area law. Since vortices areclosed surfaces, the necessary randomness can be facili-tated only by large vortices. This is further translatedinto the percolation of vortices, meaning that the sizeof the (largest) vortex clusters becomes comparable to

the extension of the space itself. This percolation hasbeen observed in lattice simulations of the confined phase[15], while in the deconfined phase the vortices align inthe timelike direction and the percolation mechanism re-mains working only for spatial Wilson loops [16, 17]. Thisparallels percolation properties of monopoles. Moreover,it conforms with the observation at high temperaturesthat the spatial Wilson loops keep a string tension incontrast to the correlators of Polyakov loops.

In this paper we will merge the caloron and vortexpicture focussing on two aspects: (i) to demonstrate howthe vortex content of individual calorons depends on theparameters of the caloron solution in particular the

holonomy and (ii) to obtain the vortices in correspond-ing caloron ensembles and analyze their percolation prop-erties. For simplicity we will restrict ourselves to gaugegroup SU(2). We will mainly use LCG, which has found acorrelation of vortices to instantons cores in [18, 19]. Werecall that LCG has been abandoned for finding vorticesin SU(2) Monte Carlo configurations because the vortexdensity did not possess a good continuum limit [20]. Thisobservation does not invalidate the application of LCGto smooth (semiclassical) field configurations. We alsocompare with results obtained by DMCG and IMCG. Inorder to enable the application of these gauge-fixing tech-niques we discretize calorons on a lattice, which is knownto reproduce continuum results very well.

In the combination of Laplacian Abelian Gauge (LAG)and LCG, magnetic monopole worldlines are known toreside on the vortex sheets, and we will confirm this forthe calorons constituent dyons (see [18] for some firstfindings).

In addition, we find another mainly spatial part ofthe vortex surfaces. It is strongly related to the twist ofthe dyons within the caloron. We explain this by ana-lytic arguments that yield good approximations for thelocations of these spatial vortices.

For a single caloron, both parts of the vortex systemtogether generate two intersection points needed to con-stitute the topological charge in the vortex language.2

In caloron ensembles we find the spatial vortex sur-faces to percolate only at low temperatures (where theholonomy is maximally non-trivial), while the space-timevortex surfaces are rather independent of the phase (i.e.the holonomy), both in agreement with physical expec-

2 Of course, for the caloron as a classical object the topologicalcharge density is continuously distributed.

tations and with observations in caloron gas simulationsthat have evaluated the QQ free energy on one hand andthe space like Wilson loops on the other [21].

From the results for individual calorons it is clear thatthe Polyakov loop, which we treat as an input parameterfor the caloron solution, is responsible for the percola-tion and hence the string tension in the confined phase.This lends support for the hypothesis that the holonomy

is important as the correct background for the classi-cal objects featuring in a semiclassical understanding offinite temperature QCD.

The paper is organised as follows. In the next two Sec-tions II and III we review the properties of calorons andvortices, including technicalities of how to discretize theformer and how to detect the latter. In Section IV we de-scribe the vortex content of single calorons. In Section Vwe demonstrate how the vortex content of caloron gaseschanges with the holonomy parameter. We conclude witha summary and a brief outlook. Part of our results havebeen published in [22].

II. CALORONS

A. Generalities

Calorons are instantons, i.e. selfdual3 Yang-Mills fieldsand therefore solutions of the equations of motion, atfinite temperature. In other words, their base space isR3S1 where the circle S1 has circumference = 1/kBTas usual.

As it turns out from the explicit solutions [13],calorons consist of localised lumps of topological chargedensity, which due to selfduality are lumps of action

density, too. For the gauge group SU(N) one can haveup to N lumps per unit topological charge. When wellseparated, these lumps are static4. Moreover, they pos-sess (quantised) magnetic charge equal to their electriccharge and hence are called dyons. Consequently, themoduli of calorons are the spatial locations of the dyons,which can take any value, plus phases [10].

Another important (superselection) parameter of thenew solutions by Kraan/van Baal and Lee/Lu [2, 3] isthe holonomy, the limit of the (untraced) Polyakov loopat spatial infinity,

P = lim

|x|Pexpi

0

A0dx0 . (1)Due to the magnetic neutrality of the dyons within acaloron, this limit is independent of the direction the

3 The results for antiselfdual calorons with negative topologicalcharge are completely analogous.

4 The gauge field, generically, can and will be time dependent, seeSect. II B.


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limit is taken. (In our convention the gauge fields arehermitean, we basically follow the notation of [3] but mul-tiply their antihermitean gauge fields by i and reinstate.)

In SU(2) we diagonalise P,P = exp(2i3) (2)

with i the Pauli matrices. Note that = 0 o r 1/2

amount to trivial holonomies P = 12, whereas thecase = 1/4, i.e. trP = 0 is referred to as maximalnontrivial holonomy.

The constituent dyons have fractional topologicalcharges (masses) governed by the holonomy, namely2 and 2 1 2, cf. Fig. 1 upper panel, such that from the point of view of the topological density theconstituent dyons are identical in the case of maximalnontrivial holonomy = 1/4.

To be more concrete, the gauge field of a unit chargecaloron in the periodic gauge5 is given by

A3 =

1

2

3 log

2

,0

A1 iA2 = 1

2 (1 i2)( +

4i

,0) ,

(3)

where is the t Hooft tensor (we use the conventionin [3]) and and are (x0-periodic) combinations oftrigonometric and hyperbolic functions of x0 and x, re-spectively, see appendix A and [3]. They are given interms of the distances

r = |x z1| , s = |x z2| (4)from the following constituent dyon locations

z1 = (0, 0,

22/), z2 = (0, 0, 2

2/) , (5)

which we have put on the x3-axis with the center of massat the origin (which can always be achieved by space ro-tations and translations) and at a distance of d 2/to each other.

In case of large , the action consists of approximatelystatic lumps (of radius /4 and /4 in spatial di-rections) near z1 and z2. In the small limit the actionprofile approaches a single 4d instanton-like lump at theorigin. In Ref. [3] one can find more plots of the ac-tion density of SU(2) calorons with different sizes andholonomies.

In the far-field limit, away from both dyons the func-

tion behaves like

=4d

(r + s + d)2

re4r/e2ix0 + se4s/

[1 + O(emin(4r/,4s/))] , (6)

5 This gauge is in contrast to the nonperiodic algebraic gaugewhere A0 asymptotically vanishes and the holonomy is carriedby the transition function.

0

1

2

3

4

1

23

4

0

1

2

1

2

12

10

8

6

4

0

1

2

3

4

12

34

0

1

2

1

2

1.0

0.5

0.0

0.5

1.0

FIG. 1: Action density (top, shown in logarithmic scale andcut below e12) and Polyakov loop (bottom) in the (x1, x3)-plane (measured in units of ) at x0 = x2 = 0 for a caloronwith intermediate holonomy = 0.12 and size = 0.9 asdiscretized on a 8 482 80 lattice. The dyon locations arez1 = (0, 0,0.61) and z2 = (0, 0, 1.93).

and hence the off-diagonal part of A decays exponen-tially, while the Abelian part from

=r + s + d

r + s d + O(emin(4r/,4s/)) (7)

becomes a dipole field [3].

The Polyakov loop in the bulk plays a role similar toan exponentiated Higgs field in the gauge group: it is+12 and 12 in the vicinity of z1 and z2 6, respectively,cf. Fig. 1 lower panel. The existence of such points is of

topological origin [24]. Thus the Polyakov loop is a moresuitable pointer to the constituent dyon locations, whichagrees with the maxima of topological density for thelimiting case of well-separated dyons, but is valid even incase the two topological lumps merged into one for small.

6 On the line connecting the dyons the Polyakov loop can actuallybe computed exactly [23].


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B. The twist

A less-known feature of the caloron we want to de-scribe next is the Taubes twist. It basically means thatthe gauge field7 of one of the dyons is rotated by a timedependent gauge transformation (rotated in the direc-tion of the holonomy, here the third direction in colorspace) w.r.t. the gauge field of the other dyon when they

are combined into a caloron. This is the way the dyonsgenerate the unit topological charge [3].

The simplest way to reveal the twist is to consider thelimit of well-separated dyons, i.e. when their distance dis much larger than their radius /4 and /4. Let

us consider points near the first dyon, x = z1 + , where

the distance || is small compared to the separationd, but not necessarily compared to the dyon size. Thenthe relevant distances are obviously

r = , s = |(0, 0, d) + | = d 3 + O(2/d) . (8)In the appendix we derive the form of the functions and in this limit,

(x = z1 + ) 2dcoth(4/) 3 , (9)

(x = z1 + ) e2ix0/ 12d

sinh(4/). (10)

The large factors of 2d cancel in the expressions log and relevant for A, Eq. (3).

In the vicinity of the other dyon, x = z2 + , with

s = , r = d + 3 + O(2/d) , (11)we get very similar expressions with replaced by and3 by 3, but the time-dependent phase factor is absent,

(x = z2 + ) 2dcoth(4/) + 3

, (12)

(x = z2 + ) 12d

sinh(4/). (13)

This staticity of course also holds for A of this dyon andall quantities computed from it.

Plugging in those functions into the gauge field ofEq. (3) one can find that the corresponding gauge fieldcomponents are connected via a PT transformation, andthe exchange of and

(A1 iA2)(x0, z2 + ; ) = (A1 iA2)(x0, z1 ; )e2ix0/ (14)

A3(x0, z2 +; ) = A3(x0, z1 ; )

,0 , (15)

7 The twist can be formulated in a gauge-invariant way by fieldstrength correlators between points connected by Schwinger lines[25].

and a gauge transformation, namely

A(x0, z2 + ; ) = TA(x0, z1 ; ) (16)with the time-dependent twist gauge transformation

T(x0) = exp(i x0

3) . (17)

This gauge transformation is nonperiodic, T() = 12(but acts in the adjoint representation).The Polyakov loop values inside the dyons are obtainedfrom (x = z1,2 + ) = O(2) and

(x = z1 + ) =2d

/4 3 + O(2) , (18)

(x = z2 + ) =2d

/4 + 3 + O(2) , (19)

which results in

A0(z1) =

3 P(z1) = 12 , (20)A0(z2) = 0 P(z2) = +12 . (21)

Actually, the gauge field around z2 is that of a static

magnetic monopole with the Higgs field identified withA0 through dimensional reduction. Indeed, it vanishes atthe core according to (21) and approaches the vacuumexpectation value || = 2/ away from the core. Ac-cordingly, Di is identified with DiA0 = Fi0 = Ei, andthe Bogomolnyi equation with the selfduality equation.

The gauge field around z1 is that of a twistedmonopole, i.e. a monopole gauge rotated with T. Thecorresponding Higgs field is obtained from that of a staticmonopole by the same T, transforming in the adjoint rep-resentation. Therefore, the Higgs field of the twistedmonopole agrees with the gauge field A0 apart from theinhomogeneous term in Eq. (15). vanishes at the

core, too, and approaches the vacuum expectation value2/.

The electric and magnetic charges, as measured in the direction through the t Hooft field strength tensor, areequal and the same for both dyons. This is consistentwith the fact that selfdual configurations fulfilling theBPS bound must have positive magnetic charge.

These fields are in some unusual gauge: around thedyon cores the Higgs field has the hedgehog form a (xz1,2)a which is called the radial gauge. Far away fromthe dyons the Higgs field becomes diagonal up to ex-ponentially small corrections. Indeed, if one neglects theexponentially small s of (10) and (13) and replaces the

hyperbolic cotangent by 1 in the denominator of (9) and(12), this would be the so-called unitary gauge with diag-onal Higgs field and a Dirac string singularity (along theline connecting the dyons). Far away from the caloronsdyons, however, the hedgehog is not combed com-peletely and there is no need for a singularity8. In other

8 In contrast, the gauge field A4 written down in Sect. IIA of [9]is diagonal and A has a singularity at the x3-axis.


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words the covering of the color space happens in an ex-ponentially small but finite solid angle.

More precisely, the Higgs field approaches23/ and +23/ away from the static andtwisting dyon, respectively, for almost all directions.These values differ by 3/, and hence the correspond-ing A0s can be glued together (apart from a gauge singu-larity at the origin). Moreover, in A0 the leading far field

corrections to the asymptotic value, namely monopoleterms, are of opposite sign w.r.t. the fixed color direction3 and therefore do not induce a net winding number inthe asymptotic Polyakov loop. Hence the holonomy isindependent of the direction.

We remind the reader that this subsection has beendealing with the limit of well-separated dyons, i.e. allformulae are correct up to exponential corrections in /dand algebraic ones in /d.

C. Discretization

In order perform the necessary gauge transformationsor diagonalizations of the Laplace operator in numeri-cal form we translate the caloron solutions and latercaloron gas configurations into lattice configurations.For a space-time grid (with a temporal extent N0 = 8and spatial sizes of Ni = 48, . . . , 80, see specificationslater) we compute the links U(x) as path-ordered ex-ponentials of the gauge field A(x) (for single-caloronsolutions given by Eqn. (3)). Practically, the integral

U(x) = Pexp

ix+ax

A(y)dy

(22)

is decomposed into at least N = 20 subintervals, forwhich the exponential (22) is obtained by exponentiationof iA(y)a/N with A(y) evaluated in the midpoint ofthe subinterval. These exponential expressions are thenmultiplied in the required order (from x left to x + aright). A necessary condition for the validity of this ap-proximation is that a/N with characterizing thecaloron size or a typical caloron size in the multicaloronconfigurations.

Still this might be not sufficient to ensure that the po-tential A(y) is reasonably constant within the subinter-val of all links and give a converged result. In particular,the gauge field (3) is singular at the origin and has big

gradients near the line connecting the dyons, as visu-alised in Fig. 2 of [26]. Hence we dynamically adjust thenumber of subintervals N for every link, ensuring thatfurther increasing N would leave unchanged all entries ofthe resulting link matrix U(x).

The lattice field constructed this way is not strictlyperiodic in the three spatial directions, but this is notimportant for the lattices at hand with Ni N0. Theaction is already very close to 82, the maximal deviationoccurs for large calorons ( 0.9) and is about 15 %.

Lateron, we will make heavy use of the lowest Lapla-cian eigenmodes in the LCG. When computing thesemodes in the caloron backgrounds we enforce spatial pe-riodicity by hand. In Maximal Center gauges we alsoconsider the caloron gauge field as spatially periodic.

D. Caloron ensembles

The caloron gas configurations considered later in thispaper have been created along the lines of Ref. [21]. Thefour-dimensional center of mass locations of the caloronsare sampled randomly as well as the spatial orientationof the dipole axis connecting the two dyons and theangle of a global U(1) rotation around the axis 3 incolor space. The caloron size is sampled from a suitablesize distribution D(, T).

The superposition is performed in the socalled alge-braic gauge with the same holonomy parameter takenfor all calorons and anticalorons9. Finally, the additivesuperposition is gauge-rotated into the periodic gauge.

Then the field A(x) is periodic in Euclidean time andpossesses the required asymptotic holonomy. We haveapplied cooling to the superpositions in order to ensurespatial periodicity of the gauge field.

In Sect. V we will compare sequences of randomcaloron gas configurations which differ in nothing elsethan the global holonomy parameter .

III. CENTER VORTICES

To detect center vortices, we will mainly use theLaplacian Center Gauge (LCG) procedure, which canbe viewed as a generalization of Direct Maximal CenterGauge (DMCG) with the advantage to avoid the Gribovproblem of the latter [27]. We will compare our resultsto vortices from the maximal center gauges DMCG andIMCG in Sect. IV E.

In LCG one has to compute the two lowest eigenvectorsof minus the gauge covariant Laplacian operator in theadjoint representation10,

[UA]0,1 = 0,10,1 (23)abxy[U

A] =1

a2

UA (x)abx+a,y

+ UA (x )baxa,y 2abxy(24)

a, b = 1, 2, 3 ,

9 Superposing (anti)calorons with different holonomies would cre-ate jumps ofA0 in the transition regions.

10 We use with a subindex for the eigenmodes of the Laplacian,not to be confused with the auxiliary function involved in thecaloron gauge field, Eq. (3).


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which we do by virtue of the ARPACK package [28].For the vortex detection, the lowest mode 0 is rotated

to the third color direction, i.e. diagonalised,

V0 = |0| 3 . (25)The remaining Abelian freedom of rotations around thethird axis, V vV with v = exp(i3) is fixed (up tocenter elements) by demanding for 1 a particular form

with vanishing second component and positive first com-ponent, respectively,

(vV1)a=2 = 0 , (vV1)

a=1 > 0 . (26)

Defects of this gauge fixing procedure appear when0 and 1 are collinear, because then the Abelian free-dom parametrised by v remains unfixed. In [27] it wasshown, that the points x, where 0(x) and 1(x) arecollinear, define the generically two-dimensional vortexsurface, as the Wilson loops in perpendicular planes takecenter elements. This includes points x, where 0 van-ishes, 0(x) = 0, which define monopole worldlines in theLaplacian Abelian Gauge (LAG) [29].

We detect the center vortices in LCG with the help ofa topological argument: after having diagonalised 0 byvirtue ofV, Eq. (25), the question whether 0 and 1 arecollinear amounts to V1 being diagonal too, i.e. havingzero first and second component. We therefore inspecteach plaquette, take all four corners and consider theprojections ofV1 taken in these points onto the (1, 2)-plane, see Fig. 2.

By assuming continuity11 of the field V1 (more pre-cisely, its (1, 2)-projection) between the lattice sites ofthis plaquette, we can easily assign a winding number toit. By normalization of the two-dimensional arrows thisis actually a discretization of a mapping from a circle in

coordinate space to a circle in color space. In the contin-uum this could give rise to any integer winding number,while with four discretization points the winding numbercan only take values {1, 0, 1}. This winding numbercan easily be computed by adding the angles betweenthe two-dimensional vectors on neighbouring sites.

A nontrivial winding number around the plaquette im-plies that the (1, 2)-components of

V1 have a zeropoint inside the plaquette, which in turn means that thetwo eigenvectors are there collinear in color space. Inthis case we can declare the midpoint of that plaquettebelonging to the vortex surface. The vortex surface is atwo-dimensional closed surface formed by the plaquettes

of the dual lattice. The plaquettes of the dual lattice areorthogonal to and shifted by a/2 in all directions relativeto the plaquettes of the original lattice.

At face value the above procedure is plagued by pointswhere the lowest eigenvector 0 is close to the negative

11 The continuity assumption underlies all attempts to measuretopological objects on lattices. For semiclassical objects it iscertainly justified.

V1

V0

FIG. 2: The topological argument to detect vortices on agiven plaquette: The transverse components of the first ex-cited mode 1 to the direction of the lowest mode 0 (afterboth have been gauge transformed by V) are plotted for thefour sites of a plaquette. The configuration shown here hasa nonvanishing winding number, which implies that the twoeigenvectors are collinear in color space somewhere inside theplaquette.

3-direction. Such situations are inevitable when 0 hasa hedgehog behavior around one of its zeroes, i.e. formonopoles in the LAG. Then the diagonalising gauge

transformation V changes drastically in space. The cor-responding transformed first excited mode V1 may giveartificial winding numbers and thus unphysical vorticesif we insist on the continuity assumption in this case.

Actually, to detect vortices, the lowest eigenvector canbe fixed to any color direction [27], i.e. to different di-rections on different plaquettes. Using this we rotate 0plaquette by plaquette to the direction of the average 0over the four corners of the plaquette. This gauge ro-tation is in most cases a small rotation. Afterwards weinspect 1s color components perpendicular to the av-erage direction (this can be done by inspecting V1 inthe (1, 2)-plane after diagonalising the four-site aver-aged lowest eigenvector, the resulting gauge transforma-tion now changes only mildly throughout the four sitesof the plaquette).

Note that the winding number changes sign under0 0, but not under 1 1 (both changes ofsign do not change the fact that these fields are eigen-modes of the Laplacian). Hence the global signs of 0,1 and also the signs of the winding numbers are am-biguous.

IV. VORTICES IN INDIVIDUAL CALORONS

The following results are obtained for single caloronsdiscretized on space-time lattices with N0 = 8 (meaningthat our lattice spacing is a = /8) and N1 = N2 =48, N3 = 80 or N1 = N2 = N3 = 64 points.

For the LCG vortices we have to take an ambiguity intoaccount, namely the dependence of the adjoint Laplacianspectrum on the lattice discretization, in particular theratio N3/N1,2. From experience we can summarize thatthe lowest adjoint eigenmode 0 is rather independent ofthat aspect ratio. The first excited mode 1 dependson it in the following way, cf. Fig. 3: for large N3/N1,2


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1

2

8

2

4

8

1

12

8

FIG. 3: The lowest 30 eigenvalues of the adjoint Laplacian op-erator for a caloron with = 0.12 and = 0.7discretized on848280 (top left) and 864264 (bottom left). Two-folddegeneracies are plotted as bold lines (and some eigenvalueshave been slightly shifted to be distinguishable at this reso-lution). For comparison we plotted in the right panels thefree spectra on the same lattices marking their degeneracies

by numbers. The lowest singlets on the rhs. always belong tothe eigenvalue = 0.

the first excited mode 1 is a singlet, whereas for inter-mediate and small N3/N1,2 it is a doublet.

This ambiguity reflects the fact that we are forcingstates of a continuous spectrum into a finite volume,which like waves in a potential well are then sen-

sitive to the periodic boundary conditions12. Localisedbound states, on the contary, should not depend muchon the discretization.

Indeed, the absolute values and degeneracies of theeigenvalues can be understood by mimicking the caloronwith constant links,

U0 = exp(2i3/N0) , Ui = 12 , (27)

that reproduce the holonomy (and have zero action). ForLaplacian modes in the fundamental representation thisapproximation was shown to be useful in [31].

In this free-field configuration the eigenmodes arewaves proportional to

exp(2inx/Na) with inte-

ger n. At nontrivial holonomies and on our lattices withN0 N1,2 N3 one can easily convince oneself, that thelowest part of the spectrum is formed by modes in thethird color direction, 3, which do not depend on x0,n0 = 0. The eigenvalues are then given by trigonomet-

ric functions of 2ni/Ni, which for large Ni can be wellapproximated by

1a2

i

2

niNi

2(lowest ) . (28)

In other words, a wave in the ith direction contributesn2i quanta of (2/Ni)

2 to the eigenvalue. The lowesteigenvalue in this approximation is always zero. This fitsour numerical findings quite well, see Fig. 3.

In the asymmetric case, N3 = 80, N1,2 = 48 obvi-

ously the cheapest excitation is a wave along the x3-axis (connecting the dyons), i.e. n3 = 1. This givesa doublet, which in the presence of the caloron is splitinto two lines, see Fig. 3 top, the first excited mode isthus a singlet (the next modes are those with nontrivialn1 = 1 or n2 = 1 forming an approximate quartetand so on).

In the symmetric case, Ni = 64, on the other hand,excitations along all xi give equal energy contribution.For the excited modes this gives a sextet, which is againsplit by the caloron, see Fig. 3 bottom. It turns out thatthe first excited mode remains two-fold degenerate. Theeigenmodes are close to combinations of waves with non-

trivial n1 = 1 and with nontrivial n2 = 1, reflectingthe calorons axial symmetry around the x3-axis.

This finally explains the different spectra and differentshape of the eigenmodes on the different lattices.

12 A similar effect has been observed in Fig. 1 of [30], where theadjoint modes in the background of an instanton over the four-sphere have been shown to depend on the radius of the sphere.


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A. The lowest eigenvector and the LAG monopoles

As it turns out, away from the dyons the lowest mode0 becomes diagonal

13 and constant, for normalisabil-ity reasons it is then approximately (0, 0, 1/

Vol)T with

Vol = N0N1N2N3.Near each dyon core we find a zero of the third com-

ponent of the lowest mode, a=30 , see Fig. 4 top panel.

Together with the first and second component being verysmall on the whole x3-axis, we expect zeroes in the mod-ulus |0| at the constituent dyons, which means that thedyons are LAG-monopoles, cf. Fig. 3 in [18] and Fig. 10in [31].

Such zeroes can be unambiguously detected by a wind-ing number on lattice cubes similar to that of Sect. III.As a result we find almost static LAG-monopole world-

0 1 2 3 41234

0.0010

0.0005

0.0005

0.0010

0 1 2 3 41234

4

3

2

1

1

FIG. 4: Top: The third component of the lowest mode, a=30 ,along the x3-axis (in units ofat x0 = /2) for a caloron withintermediate holonomy = 0.1 and size = 1.0 discretizedon a 848280 lattice. The dyons have x3-locations 0.63and 2.51. Note that for that lattice 1/

Vol = 0.00082, a

value that is indeed taken on by the lowest mode far away

from the dyons. The other components

a=1,2

0 are found tobe of order 108 [not shown]. Bottom: the gauge field Aa=30(in units of inverse ), which is related to the Higgs field used to explain the behaviour of the lowest mode around thedyons (see text). Note that A30 takes the value / near thetwisting dyon.

13 The third direction in color space is distinguished by our (gauge)choice of the holonomy, Eqn. (2).

lines for large calorons at the locations of their dyons,while monopole loops around the caloron center of massare seen for small calorons (with 0.5, where the ac-tion density is strongly time-dependent as well), see Fig.5. Note that these locations are part of the LCG vortexsurface by definition. Similar monopole worldlines havebeen obtained in the MAG [8, 32]. Adjoint fermionic zeromodes, on the other hand, detect the constituent dyons

by maxima [33].The lowest mode also reflects the twist of the caloron:

the first and second component of 0 near the dyon coreare either static or rotate once with time x0 evolvingfrom 0 to . Fig. 6 shows this for the lowest modeas well as for the first excited mode. Our results areessentially equal to Fig. 9 of [31], just with a resolutionof N0 = 8 (instead of N0 = 4) more clearly revealing thesine- and cosine-like behaviours.

In order to understand the behaviour found for thelowest adjoint mode 0, we propose to compare it to theHiggs field discussed in Sect. II B. For the static dyon

one has from time-independence D0 = 0 and from theequation of motion Di(Di) = DiFi0 = 0. Therefore of a single static dyon is a zero mode of the adjointLaplacian = D2. For the twisting dyon the sameequations apply due to the transformation properties of (under T) and the latter is again a zero mode of theLaplacian. These zero modes approach a constant (thevev) asymptotically, so they are normalizable like a planewave.

Around each dyon core, the lowest adjoint mode 0 be-haves similar to of that dyon: it vanishes at the dyoncore, becomes constant and dominated by the third com-ponent away from the dyons, it reveals the Taubes twist(around the twisting dyon) and is in the same gauge as .Since the latter are zero modes of in the backgroundof isolated dyons, a combination of them is a naturalcandidate to be the lowest mode of that (non-negative)operator in the caloron background.

The lowest adjoint mode 0 for calorons with well-

-3 -2 -1 0 1 2 3

-0.4-0.2

0.20.4

-3 -2 -1 0 1 2 3

-0.4-0.2

0.20.4

FIG. 5: Zeroes of the lowest adjoint mode, i.e. monopolesin Laplacian Abelian Gauge, in the (x0, x3)-plane (both inunits of , x3 horizontally, at x1 = x2 = 0) for calorons ofholonomy = 0.1 and sizes = 0.5 (upper panel, z1 =(0, 0,0.16), z2 = (0, 0, 0.63)) and = 0.9 (lower panel,z1 = (0, 0,0.51), z2 = (0, 0, 2.04)). At the origin a closedmonopole wordline of minimal size occurs, which we ascribeto the gauge singularity in the caloron gauge field.


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1

4

1

2

3

41

0.0002

0.0001

0.0000

0.0001

0.0002

1

4

1

2

3

41

0.0002

0.0001

0.0000

0.0001

0.0002

FIG. 6: The twist of the caloron gauge field reflected in thebehaviour of the adjoint Laplacian modes. Shown are the

individual color components of the lowest mode (top) andthe first excited mode (bottom) as a function of time x0 inthe vicinity of the twisting dyon at z1. The dashed curvedepicts the (almost constant) third color component. Thecorresponding plots in the vicinity of the static dyon wouldsimply show static lines.

separated dyons is therefore best described in the fol-lowing way, cf. Fig. 4: Around the static dyon at z2one has 0 = A0, where the proportionality con-stant of course disappears from the eigenvalue equation(23), but is approximately given by the normalization:

|0

| (0, 0, 1/

Vol)T. Around the twisting dyon at

z1, one has to compensate for the inhomogeneous term0 = A0 + 3 (/) (cf. eqn. (15)). The propor-tionality constant there turns out to be negative, suchthat the lowest mode is able to interpolate between theseshapes with a rather mild variation throughout the re-maining space, see Fig. 4 upper panel.

B. Dyon charge induced vortex

In the following we present and discuss one part ofthe calorons vortex that is caused by the magneticcharge of constituent dyons. Our findings are summa-rized schematically in Figs. 7 and 9.

The ambiguity of the first excited mode 1 of the ad-joint Laplacian influences this part of the vortex mostsuch that we have to discuss the singlet and doubletcases separately. We find that for the singlet 1, e.g.for N3/N1,2 = 80/48, the vortex consists of the whole(x0, x3)-plane at x1 = x2 = 0 only, see Figs. 7 and 8.Hence this part of the vortex is space-time like. It in-cludes the LAG-monopole worldlines, which are eithertwo open (straight) lines or form one closed loop in that

x0

x3

/2

/2

0twisting dyon static dyon

x0

x3

/2

/2

0

FIG. 7: The dyon charge induced part of the vortex in case

the first excited mode is a singlet: for a large caloron (top)and for a small caloron (bottom), shown schematically in theplane x1 = x2 = 0.

plane. In other words, the space-time vortex connectsthe dyons once through the center of mass of the caloronand once through the periodic spatial boundary of thelattice.

The magnetic flux (measured through the windingnumber as described in Sect. III) at every time slicepoints into the x3-direction. Its sign changes at the

-4

-2

0

2

4

-2

0

2

-2

0

2

-4

-2

0

2

4

FIG. 8: The dyon charge induced part of the vortex fromthe singlet first excited mode as measured in a caloron withholonomy = 0.25 and = 0.6 in a time slice. The outcomeis identical to the x3-axis and the same for all time slices. Thedots denote points on the vortex (lines along x0) where theflux changes, i.e. the LAG-monopoles.


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x0

x30twisting dyon static dyon

/2

/2

x1

x0

x3

/2

/2

x1

FIG. 9: The dyon charge induced part of the vortex from thedoublet first excited mode for a large caloron (top) and for asmall (bottom) caloron schematically at x2 = 0.

dyons as indicated by arrows14 in Fig. 7. The flux isalways pointing towards the twisting dyon.

Independently of the flux one can investigate thealignment between the lowest and first excited mode. Itchanges from parallel15 to antiparallel near the staticdyon, because the lowest mode 0 vanishes (i.e. the dyonis a LAG-monopole) [27]. In addition we find two otherimportant facts not mentioned in [27]: the alignmentdoes not change at the twisting dyon since both modes0 and 1 vanish there and it changes at some otherlocations outside of the calorons dyons because 1 hasanother zero there [not shown].

For the doublet excited mode, i.e. at smaller N3/N1,2 =64/64, the dyon charge induced vortex is slightly differ-ent: again it connects the dyons, but now (for a fixedtime) via two lines in the interior of the caloron, pass-ing near the center of mass, see Figs. 9 and 10. Theselines exist for all times for which the monopole worldlineexists, that is for all times if the caloron is large and for

some subinterval of x0 if the caloron is small (and themonopole worldline is a closed loop existing during the

14 We have fixed the ambiguity in the winding number described inSect. III by fixing the asymptotic behaviour of the lowest mode.

15 In itself, calling 0 and 1 parallel is ambiguous as that changeswhen one of these eigenfunctions is multiplied by -1. The tran-sition from parallel to antiparallel or vice versa, however, is anunambiguous statement.

subinterval).

-4

-2

0

2

-4

-2

0

2

-4

-2

0

2

-4

-2

0

2

4

-2

0

2

FIG. 10: The dyon charge induced part of the vortex fromthe doublet first excited state as measured in a caloron withholonomy = 0.25 and = 0.6 (same as in Fig. 8) at afixed time slice. Like in Fig. 8 the dots denote points on thevortex where the flux changes. The x3-axis has been addedto guide the eye, it is not part of the vortex surface here.

These two vortex surfaces spread away from the x3-axiswhich connects the dyons. The axial symmetry aroundthis axis is seemingly broken. However, using other linearcombinations of the doublet in the role of the first excitedmode (keeping the lowest one) in the procedure of centerprojection, the vortex surface is rotated around the x3-axis. The situation is very similar to the breaking ofspherical symmetry in the hydrogen atom by choosing astate of particular quantum number m out of a multipletwith fixed angular momentum l.

The magnetic flux flips at the dyons, just like in thecase with singlet 1.

Notice that these vortices are predominantly space-time like, but have parts that are purely spatial, in par-ticular for small calorons, namely at minimal and maxi-mal x0 of the dyon charge induced vortex surface (and atother locations in addition, when the smooth continuumsurface is approximated by plaquettes).

C. Twist-induced vortex

In this section we will discuss the second part of theLCG vortex surfaces we found for individual calorons.We start again by discussing the singlet case. The twist-induced vortex in the singlet case appears at a fixed timeslice and hence is a purely spatial vortex. In contrast tothe space-time part, this vortex surface does not containthe monopole/dyon worldlines. Hence it is not obviousthat this part of the vortex structure is caused by them.


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-4

-2

0

2

4

-2

0

2

-2

0

2

-4

-2

0

2

4

-4

-2

0

2

4

-2

0

2

-2

0

2

-4

-2

0

2

4

-4

-2

0

2

4

-2

0

2

-2

0

2

-4

-2

0

2

4

-4

-2

0

2

4

-2

0

2

-2

0

2

-4

-2

0

2

4

-4

-2

0

2

4

-2

0

2

-2

0

2

-4

-2

0

2

4

-4

-2

0

2

4

-2

0

2

-2

0

2

-4

-2

0

2

4

-4

-2

0

2

4

-2

0

2

-2

0

2

-4

-2

0

2

4

2 0 2 4

3

2

1

1

2

3

FIG. 11: Twist-induced part of the vortex (bubble) from singlet first excited modes for calorons of size = 0.6 andholonomies from left to right: = 0.1, 0.12, 0.16 (upper row) = 0.2, 0.25, 0.3 (middle row) and = 0.34 (lower row, leftpanel). The plot in the lower right panel summarises the results for = 0.1, 0.12, 0.16, 0.2, 0.25, 0.3, 0.34, at x1 = 0, i.e. thebubbles are cut to circles. The plane near the boundary in the = 0.25 picture is an artifact caused by periodic boundaryconditions.

The properties of this spatial vortex depend strongly

on the holonomy, which will be very important for thepercolation of vortices in caloron ensembles in Sect. V.

In short, our finding is that the twist-induced part ofthe vortex is a closed surface around the twisting dyon aslong as the holonomy parameter is < 1/4, and becomesa closed surface around the static dyon for > 1/4, wewill refer to these surfaces as bubbles. For maximalnontrivial holonomy = 1/4 the vortex is the x3 = 0plane, i.e. the midplane perpendicular to the axis con-necting the dyons, we will refer to it as degenerate bub-

ble.

The bubble depends on the holonomy as shown inFig. 11. For two complementary holonomies = 0 and = 12 0 the bubbles are of same shape just reflectedat the origin, thus one of them encloses the static dyonand another encloses the twisting dyon. This is to beexpected from the symmetry of the underlying calorons.In the limit of 1/4 the bubbles grow to become aflat plane which enables to turn over to the other dyon.

In our = 0.25 data we find another piece of the vortexnear the boundary of the lattice, see Fig. 11. It is an


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-4

-2

0

2

4

-2

0

2

-2

0

2

-4

-2

0

2

4

-4

-2

0

2

4

-2

0

2

-2

0

2

-4

-2

0

2

4

-4

-2

0

2

4

-2

0

2

-2

0

2

-4

-2

0

2

4

5 4 3 2 1

3

2

1

1

2

3

FIG. 12: Spatial part of the vortex (bubble) for calorons of fixed intermediate holonomy = 0.12 and sizes from left to right: = 0.6, 0.7, 0.9. The panel on the very right shows a summary of the bubbles for = 0.6, 0.7, 0.8, 0.9 at x1 = 0.That the bubble for = 0.9 is much bigger than that for = 0.7, 0.8 is probably a finite volume effect. For small sizes (and also in the limiting cases of holonomy close to the trivial values 0 and 1/2) we have met difficulties in resolving thecorresponding small bubbles in the lattice dicretization.

artefact of the finite periodic volume. Likewise, very large

bubbles in our results have deformations since they comeclose to the boundary of the lattice. The intermediatebubbles shown in these figures are generally free fromdiscretization artefacts and can easily be extrapolated(at least qualitatively) to these limits.

The size of the bubble also depends on the size param-eter of the caloron, i.e. the distance between the dyons,as shown in Fig. 12.

The time-coordinate of LCG bubbles in large caloronsis always consistent with x0 = 0.5. For small calorons,on the other hand, x0 = 0 is the exclusive time slice: theaction density peaks there and the LAG monopoles arecircling around it (cf. Fig. 5). However, the bubbles ofsmall calorons are too small to be detected.

In the case of the first excited mode being a doublet,similar bubbles have been found. They also enclose oneof the dyons and degenerate to the midplane for =1/4. Their sizes, however, may be different and theyare distributed over several time slices. Considering thecollection of all time slices, these fragments add up to fullbubbles.

1. Analytic considerations

In the following we present two analytic arguments relying on the twist that support the existence of thebubbles (playing the role of spatial vortices) and help toestimate their sizes.

The first one is specific for vortices in LCG. As we havedemonstrated in Sect. IV, the lowest mode 0 twists nearthe twisting dyon and is static near the static dyon. Thesame holds for the first excited mode 1, see Fig. 6.

Then a topological argument shows that they have tobe (anti)parallel somewhere inbetween, cf. Fig. 13. As0, 1 and the diagonalising gauge transformation V

are static around the static dyon, so is V1 and its pro-

jection along the third direction (see the right part ofFig. 13). We assume that this projection is nonzero, oth-erwise the two states are obviously (anti)parallel and thepoint would belong to the vortex already16.

In the twisting region called S, the two lowest modes

3

FIG. 13: Behaviour of the nondiagonal elements of V tran-formed first excited mode V1 in the twisting region (left) andin the static region (right) with time x0 evolving upwards.The lattice sites inbetween are indicated only at x0 = 0. On

the entire discretised rectangle the field has winding number 1,meaning it contains the twist-induced vortex. More precisely,it is the plaquette marked with filled circles that containsthe winding (in analogy to Fig. 2) and thus the vortex (in allother plaquettes the field performs a partial winding but thenwinds back).

16 in particular to the space-time part since then the two modes are(anti)parallel for all x0


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13

behave like (suppressing arguments x)

0(x0) = T(x0)0(x0 = 0)T(x0)

1(x0) = T(x0)1(x0 = 0)T(x0) (29)

with the twisting transformation/rotation from Eq. (17).The time dependence of the diagonalising V can be de-duced easily17,

V(x0) = T(x0)V(0)T(x0) , (30)

such that

V1(x0) = T(x0)V1(0)T

(x0) . (31)

Again we assume that the two modes are not(anti)parallel at x0 = 0. Then

V1(0) has a nonvanishingcomponent perpendicular to 3. According to Eq. (31)this component then rotates in time x0 around the thirddirection (left part of Fig. 13). This immediately impliesthat there is a space-time plaquette (in the sense ofFig. 2, marked in Fig. 13 with filled circles) that contains

a point where the two modes are collinear. Notice thesimilarity of Figs. 13 and 2.

This argument applies to all pairs of points with onepoint in the twisting region S and one point in the staticregion (its complement) S: on any line connecting thetwo there exists a point which belongs to the vortex. Thisresults in a closed surface at the boundary between SandS (see below). The time-coordinate of this surface is notdetermined by these considerations.

Our argument can be extended to vortices beyondLCG. For that aim we mimic the caloron gauge fieldby A0 = 0, Ai = 0 in the static region S and A0 =3(/), Ai = 0 in the twisting region S (cf. Eq. (15))[22]. In this simplified gauge field vortices can be locateddirectly by the definition that 1 Wilson loops are linkedwith them. Obviously rectangular Wilson loops connect-ing (0, x1), (, x1), (, x2), (0, x2) and (0, x1) are 12if and only if x1 belongs to S and x2 belongs to S (orvice versa). This again predicts spatial vortices at theboundary between the twisting and the static region.

Actually, this argument is exact if one chooses for thepoints x1,2 the dyon locations z1,2: the path ordered ex-ponentials at fixed x1,2 are the Polyakov loops 12 andthe remaining spatial parts are inverse to each other be-cause of periodicity and cancel. Hence there should al-ways be a spatial vortex between the two dyons.

Thus the twist in the gauge field of the caloron itselfgives rise to a spatial vortex. This vortex extends in thetwo spatial directions perpendicular to lines connectingS and S, just like a bubble.

Note that the two arguments above do not workpurely within the twisting region or purely within the

17 The first factor is necessary, otherwise V is singular around thenorth pole and nonperiodic.

static region.

It remains to be specified where the boundary betweenthe twisting region S and its complement S is. To thatend one should consider the competing terms twistingvs. static in the relevant function , see Eq. (A.3). Ac-tually its derivatives enter the off-diagonal gauge fields,see Eq. (3). In the periodic gauge we have used so far,

there is an additional term proportional to itself. Todecide whether the static or the twisting part dominates(at a given point) it is better to go over to the algebraicgauge, where this term is absent and where must be re-placed by = exp(4ix0/) [3]. The two competingterms become

e4ix0/sinh(4r/)

r fstatic (32)

e4ix0/sinh(4s/)

s ftwist (33)

with given in Eqn. (A.1). Note that the time de-pendence of these functions still differs by a factor

exp(2ix0/).We finally define the twisting region S as where the

gradient of ftwist dominates

|ftwist|2 |fstatic|2 . (34)

4 2 0 2 4

3

2

1

0

1

2

3

4 2 0 2 4

3

2

1

0

1

2

3

FIG. 14: The bubbles measured for calorons with holonomy = 0.12, size = 0.6 (top) and = 0.9 (bottom) respec-tively as a function of x2 (vertically) and x3 (horizontally) atx1 = 0 compared to the boundary of the twisting region S,the smooth curve computed from the equality in Eq. (34).


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and its boundary where the equality holds.In two particular cases this can be determined ana-

lytically. For the case of maximally nontrivial holonomy = = 1/4, the two functions fstatic,twist only differby the arguments r vs. s. Then the boundary of S isobviously r = s, which gives the midplane between thedyons. This indeed amounts to our numerical finding,the degenerate bubble for = 1/4.

In the large caloron limit and if we further as-sume the solutions of the equality in Eq. (34) to obeyr/, s/ 1, it is enough to compare in fstatic,twistthe exponentially large terms in sinh and . This yieldsfor the boundary of S the equation r = s, which canbe worked out to give

x21 + x22 + (x3 d)2 = (d)2 ,

2

(35)

Thus, the boundary of the twisting region S is a spherewith midpoint (0, 0, d) and radius ||d. This sphere al-ways touches the origin, is centered at negative and pos-itive x3 for < 1/4 and > 1/4, respectively, and againdegenerates to the midplane of the dyons for maximallynontrivial holonomy = = 1/4.

In Fig. 14 we compare the boundary of S obtainedfrom the equality in Eq. (34) to the numerically obtainedbubbles in LCG for two different values of the caloronparameter . The graphs agree qualitatively.

One could also think of characterising the locations xof the twist-induced vortex by a fixed value of the tracedPolyakov loop, say tr P(x) = 0. This also encloses oneof the dyons and becomes the midplane for = 1/4. Inthe large separation limit, however, this surface is thatof a single dyon of fixed size set by and (just like thetopological density). It does not grow with the separation

d, which however seems to be the case for the measuredvortices as well as for the boundary of Susing the far fieldlimit, Eq. (35). Hence the local Polyakov loop seems nota perfect pointer to the spatial vortex.

D. Intersection and topological charge

To a good approximation the dyon charge induced vor-tex extends in space and time connecting the dyons twice,whereas the twist induced vortex is purely spatial aroundone of the dyons. This results in two intersection pointsgenerating topological charge as we will describe now.

The notion of topological charge also exists for (sin-gular) vortex sheets. In order to illustrate that let uschoose a local coordinate system and denote the two di-rections perpendicular to the vortex sheet, in which aWilson loop is 1, by and . The Wilson loop canbe generated by a circular Abelian gauge field decayingwith the inverse distance, which generates a gauge fieldF (the magnetic field, say B3 F12 for a static vortexin the x3-direction, is tangential to the vortex, respec-tively). The corresponding flux is via an Abelian Stokes

Theorem connected to the Wilson loop and is nothing butthe winding number used in LCG to detect the vortex.

In order to generate topological charge proportionalto FF, the vortex thus needs to extend in alldirections. This is made more precise by the geomet-ric objects called writhe and self-intersection. The re-lation to the topological charge including example con-figurations has been worked out for vortices consisting

of hypercubes in [34] and for smooth vortices in [35].The result is that a (self)intersection point where twobranches of the vortex meet such that the combined tan-gential space is four-dimensional contributes 1/2 tothe topological charge. The contribution of the writheis related to gradients of the vortex tangential and nor-mal space w.r.t. the two coordinates parametrising thevortex. Two trivial examples are important for vorticesin a caloron: a two-dimensional plane as well as a two-dimensional sphere embedded in four-dimensional spacehave no writhe. Since the two parts of our vortex are ofthese topologies, we immediately conclude that the topo-logical charge of vortices in calorons comes exclusivelyfrom intersection points.

We first discuss the position of the intersection pointsin the singlet case, cf. Fig. 15 top panel. The twist in-duced bubble occurs at a fixed time slice and so does anyintersection point. The dyon induced vortex consists oftwo static straight lines from one of the dyon to anotherand therefore intersects the bubble twice on the x3-axis.

There are two exceptions to this fact: for smallcalorons (small ) the dyon induced vortex exists in sometime slices only and the number of intersection pointsdepends on whether the time-coordinate of the bubble is

x3

x1

x2

x3

x1

x2

FIG. 15: The intersection of the spatial bubble (at fixed x0 =/2) for < 1/4 with the space-time part of the vortex fromthe singlet (top) and the doublet (bottom) excited mode (inthe doublet case the bubble is distributed over several timeslices).


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15

within that time-interval (but the bubble is usually toosmall to detect when is small).

The case of maximal nontrivial holonomy is particularbecause for the corresponding degenerate bubble there isonly one intersection point at the center of mass of thecaloron, the other one moved to x3 as 1/40in infinite volume.

Concerning the sign of the contributions, it is essential

that the relative sign of the vortex flux is determined:the magnetic flux of the dyon induced vortex flips atthe dyons and hence is of opposite sign at the intersec-tion points on the bubble. One can depict the flux onthe bubble by an electric field normal to the bubble (i.e.hedgehog-like). It follows that in LCG the contributionsof the intersection points to the topological charge of thevortex are both +1/2.

The vortex in the caloron is thus an example for ageneral statement, that a non-orientable vortex surfaceis needed for a nonvanishing total topological charge. Inour case the two branches of the dyon charge inducedvortex have been glued together at the dyons in a non-orientable way: the magnetic fluxes start or end at thedyons as LAG-monopoles (this construction is impossiblefor the bubble as the dyons are not located on them).Thus, vortices without monopoles on them possess trivialtotal topological charge.

In the doublet case with its fragmented bubbles thereare still two intersection points (cf. Fig. 15 bottom panel)which again contribute topological charges of +1/2 each.

To sum up this section we have demonstrated that thevortex has unit topological charge like the caloron back-ground gauge field. This result is not completely triv-ial as there is to our knowledge no general proof thatthe topological charge from the gauge field persists forits vortex skeleton after center projection (P-vortices).

Moreover, the topological density of the caloron is notmaximal at the two points where the topological densityof the vortex is concentrated and the total topologicalcharge of the caloron is split into fractions of 2 and 2whereas that of the vortex always comes in equal frac-tions 1/2 from two intersection points, close to the staticdyon if < 1/4 and close to the twisting dyon if > 1/4.

E. Results from Maximal Center Gauges

We have performed complementary studies of vorticesboth in the Direct and in the Indirect Maximal Center

Gauges (DMCG [36] and IMCG [37], respectively). TheDMCG in SU(2) is defined by the maximization of thefunctional

FDMCG[U] =,x

(tr gU(x))2 , (36)

with respect to gauge transformations g(x) SU(2).U(x) are the lattice links and

gU(x) = g(x)U(x)g(x+

) the gauge transformed ones. Maximization of (36)minimizes the distance to center elements and fixes the

gauge up to a Z(2) gauge transformations. The corre-sponding, projected Z(2) links are defined as

Z(x) = sign (trgU(x)) . (37)

The Gribov copy problem is known to spoil gauges withmaximizations such as DMCG. In practice we also ap-plied random SU(2) gauge transformations before max-

imizing FDMCG and selected the configuration with thelargest value of that functional.The IMCG goes an indirect way. At first, one fixes the

maximal Abelian gauge (MAG [38]) by maximizing thefunctional

FMAG[U] =,x

trgU(x)3(

gU(x))3

, (38)

with respect to gauge transformations g SU(2). TheMAG minimizes the off-diagonal elements of the linksand fixes the gauge up to U(1). Therefore, the followingprojection to a U(1) gauge field through the phase of thediagonal elements of the links, (x) = arg (gU(x))11,is not unique. Exploiting the remaining U(1) gaugefreedom, which amounts to a shift (x) (x) =(x) + (x) + (x + ), one maximize the IMCG func-tional

FIMCG[U] =,x

(cos((x)))2 , (39)

that serves the same purpose as FDMCG in (36). Finally,the projected Z(2) gauge links are defined as

Z(x) = sign(cos((x))) . (40)

Finally, the Z(2) links are used to form Z(2) plaque-ttes. All dual plaquettes of the negative Z(2) plaquettesform closed two dimensional surfaces the vortexsurfaces.

In the background of calorons we tried to confirm bothdyon charge induced and twist-induced vortices seen inLCG. In DMCG, the dyon charge induced vortices areobserved and the twist induced part splits into severalparts in adjacent time slices. Choosing the best amongrandom gauge copies, the dyon charge induced part dis-appears and the twist-induced vortex bubble occurs atfixed time slice. In both cases, the bubble is much smallerthan that found in LCG.

In IMCG, the situation for dyon charge induced vor-tices is rather stable: we find always a space-time vor-tex connecting the dyons or better to say the Abelianmonopoles representing the dyons in MAG [8, 32]. Sim-ilar to the LCG doublet case two vortex lines pass nearthe center of mass of the caloron. This structure propa-gates either statically or nonstatically in time, dependingon the distance between dyons in the caloron.

Straight lines through the center of mass and throughthe outer space, as found in the LCG singlet case, were


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never observed. One can convince oneself, that it is ac-tually impossible to get such vortex structures from Z(2)link configurations.

The situation with twist-induced vortices is unsta-ble, as a rule they do not appear in IMCG. The rea-son for this could be partially understood in IMCGconsiderations. Let us consider two gauge equivalentAbelian configurations that generate local Polyakov loops12 tr P(x) = cos(a(x)) and how their temporal links con-tribute to the Abelian gauge functional FIMCG given inEq. (39). In the quasi-temporal gauge when all sub-sequent temporal links are the same, they give a con-tribution equal to N0 cos(a(x)/N0)

2 to the correspond-ing part of the functional. When, on the other hand,all but one temporal links are trivial, the contributionis equal to N0 1 + cos(a(x))2. For cos(a(x)) = 1and for sufficiently large N0 we have N0 cos(a(x)/N0)

2 N0 a(x)2/N0 > N0 sin(a(x))2. This means that af-ter maximizing the functional (39) and projecting ontoZ(2), we get all temporal links trivial in all points xwhere the Polyakov loop is not equal to 1. So, thetwist-induced vortex shrinks to one point where the (un-traced) Polyakov loop is equal to 12.

Maximization of the functional (39) is equivalent tothe minimization of the functional

F[U] =,x

(sin((x)))2 . (41)

If we would replace it by the functional

F[U] =,x

(sin((x)))

2 . (42)

the situation with the Z(2) projected Polyakov loop

would change because N0

sin(a(x)/N0)2

>

sin(a(x))2

on points where cos(a(x)) < 0 and now we would have12 temporal links in some time slice and trivial tem-poral links in all other time slices in the spatial regionwhere the Polyakov loop is negative as well as trivialtemporal Z(2) links in all time slices in the region wherethe Polyakov loop is positive. Numerical studies supportthe appearance of twist-induced vortex on the boundarywhere initial Polyakov loop changes the sign from nega-tive to positive.

One may conclude from this section that the Gri-bov copy problems of DMCG and IMCG persist for thesmooth caloron backgrounds. The basic features of thevortices can be reproduced, but for clarity we stick to the

vortices obtained in the Laplacian Center gauge.

V. VORTICES IN CALORON ENSEMBLES

In this section we present the vortex content of ensem-bles of calorons. The generation of the latter has beendescribed in Sect. II D. We superposed 6 calorons and 6anticalorons with an average size of = 0.6 on a 8643lattice.

The most important feature of these ensembles is theirholonomy P = exp(2i3). Under the conjecturementioned in the introduction we will use 12 tr P =cos(2) as equivalent to the order parameter 12 tr P.In particular, caloron ensembles with maximally nontriv-ial holonomy = 1/4 mimic the confined phase with12 tr P = 0.

For each of the holonomy parameters =

{0.0625, 0.0125, 0.01875, 0.25} we considered onecaloron ensemble with otherwise equal parameters.Again we computed the lowest adjoint modes in thesebackgrounds and used the routines based on windingnumbers to detect the LCG vortex content.

In Figs. 16 and 17 we show the space-time part respec-tively the purely spatial part of the corresponding vor-tices as a function of the holonomy . One can clearly seethat with holonomy approaching the confining value 1/4the vortices grow in size, especially the spatial vorticesstart to percolate, which will be quantified below.

Fig. 18 shows another view on this property. In theseplots we have fixed one of the spatial coordinates to

a particular value, such that vortices become line-likeor remain surfaces (and may appear to be non-closed,when they actually close through other slices than thefixed one). These plots should be compared to Fig. 7of Ref. [17], which however does not show a particularvortex configuration, but the authors interpretation ofmeasurements (in addition, the authors of [17] seem tohave overlooked that vortices cut at fixed spatial coordi-nate still have surface-like parts, i.e. dual plaquettes).

We find that vortices in the deconfined phase tendto align in the time-like direction, while in the confinedphase vortices percolate in the spatial directions. We re-mind the reader that we distinguish the different phasesby the values of the holonomy,

0,

1/2 vs.

= 1/4, and not by different temperatures, which wouldlead to different caloron density and size distribution.

We start our interpretation of these results by the factthat the caloron background is dilute in the sense that thetopological density is well approximated by the sum overthe constituent dyons of individual calorons (of course,the long-range Aa=3 components still interact with the

short-range Aa=1,2 components inside other dyon cores).Therefore it is permissible and helpful to interpret thevortices in caloron ensembles as approximate recombi-nation of vortices from individual calorons presented inthe previous sections.

Indeed, the space-time vortices resemble the dyon-induced vortices which are mostly space-time like andtherefore line-like at fixed x0. The spatial vortices, onthe other hand, resemble the twist-induced bubbles inindividual calorons.

Following the recombination interpretation, the bub-bles should become larger and larger when the holon-omy approaches the confining holonomy = 1/4, wherethey degenerate to flat planes. This is indeed the case incaloron ensembles: towards = 1/4 the individual vor-tices merge to from one big vortex, see also the schematic


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FIG. 16: The space-time part of vortices in caloron ensembles in a fixed time slice. Only the holonomy varies from left to right: = {0.0625, 0.0125, 0.01875, 0.25} (from deconfined phase to confined phase). The upper row shows the entire vortex contentin each caloron ensemble, the lower row shows the corresponding biggest vortex cluster.

FIG. 17: The spatial part of vortices in the caloron ensembles of Fig. 16 (with the same values of the holonomy ) summed

over all time slices. Again the upper row shows the entire vortex content and the lower row the corresponding biggest vortexcluster.

plot Fig. 7 in [22]. In other words, the maximal nontrivialholonomy has the effect of forcing the spatiall vortices topercolate. Consequently the vortices yield exponentiallydecaying Polyakov loop correlators, the equivalent of theWilson loop area law at finite temperature (see below).

Note also that there is no similar scenario for the

space-time vortices. The dyon-induced vortices in eachcaloron are either always as large as the lattice (in thesinglet case) or are always confined to the interior of thecaloron (in the doublet case). This is consistent withthe physical picture, that spatial Wilson loops do notchange much across the phase transition.


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-3

0

3

0

1

-3

0

3

-3

0

-3

0

3

0

1

-3

0

3

-3

0

FIG. 18: The vortex content of a caloron ensemble in thedeconfined phase (upper panel, mimicked by holonomy =0.0625 close to trivial) and in the confined phase (lower panel,maximal nontrivial holonomy = 0.25) in a lattice slice atfixed spatial coordinate. The short direction is x0 while theother directions are the remaining spatial ones, all given inunits of .

In order to quantify the percolation we measured twoobservables. The first one concerns the spatial and space-time extensions of the largest vortices. We have includedplots of them in the second row of Figs. 16 and 17, re-

spectively.For the spatial extension we superpose the purely spa-

tial vortex plaquettes of all time slices in one 3d lattice,whereas for the space-time extension we remove all purelyspatial vortex plaquettes. Then we pick the largest con-nected cluster in the remaining vortex structure.

Table I shows the extension of largest spatial andspace-time vortex clusters for different holonomy pa-rameters of otherwise identical caloron ensembles. Thespatial-spatial vortex cluster extension changes drasti-

holonomy parameter 0.0625 0.125 0.1875 0.25

space-time extension 47 56 56 56

spatial extension 20 35 56 56

TABLE I: Extensions of the largest vortex cluster (see text) inthe caloron ensembles of Figs.16 and 17. Note that the largestextension on a 8 643 lattice is

p(8/2)2 + 3 (32/2)2 = 55.6.

cally with the holonomy parameter whereas the space-time one almost keeps to percolate.

The second row in Table I is related to confinementgenerated by vortices. If center vortices penetrate a Wil-son loop with extensions T and L randomly, the proba-bility to find n vortices penetrating the area A = T L isgiven by the Poisson distribution

Pr(n; T, L) =(pA)n

n!epA (43)

where p is the density of (spatial) vortices. The indexr characterizes the perfect randomness of this distribu-tion, in order to distinguish it from an arbitrary empiricaldistribution. The average Wilson loop in such a centerconfiguration is given by an alternating sum

W(T, L) =n

(1)nP(n; T, L) (44)

Obviously, for the Poisson distribution Pr one obtains anarea law, logW T L, with a string tension = 2p.

We explore space-time Wilson loops

W(L, )

, which

amount to Polyakov loop correlators at distance L andprobe confinement, as well as purely spatial Wislon loopsas a function of their area, W(L, L) W(A = LL).As Fig. 19 clearly shows, the values of logW(L, ) showa confining linear behaviour (like from random vortices)reaching larger and larger distances L when the holonomyapproaches the confinement value = 1/4. At the sametime the corresponding string tensions also grow by anorder of magnitude.

One can also explain the deviation from the confiningbehaviour in holonomies far from 1/4 . The probabilityP(2; L, ) for two vortices penetrating the Wilson loop isfound much larger than in the case of Poisson distribution

[not shown]. This comes from small bubbles, which verylikely penetrate a given rectangular twice.

The behaviour of logW(A) on the other hand changesonly slightly for different holonomies, see Fig. 20. Thecorresponding slopes (string tensions) vary by a factorof approximately 2. For holonomy = 0.25 we find anexponential decay stronger than proportional to the area.A quantitative analysis of this effect needs to include suit-able caloron densities and size distributions around thecritical temperature.


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5 10 15 20 25 30

0.02

0.04

0.06

0.08

5 10 15 20 25 30

0.05

0.1

0.15

0.2

0.25

0.3

5 10 15 20 25 30

0.2

0.4

0.6

0.8

1

5 10 15 20 25 30

0.25

0.5

0.75

1

1.25

1.5

FIG. 19: The observable

log

W(L, )

as a function ofL in units of lattice spacing a = /8 from vortices in caloron ensembles

with holonomies as in Figs. 16 and 17 (from left to right: = {0.0625, 0.0125, 0.01875, 0.25}). Note the very different scales.Percolating vortices, i.e. random penetrations, would give a linear behavior. The line shown here is a linear extrapolation ofthe first data point. The caloron vortices follow this line for higher and higher distances L when the holonomy approaches theconfinement value 1/4, thus inducing confinement in the Polyakov loop correlator (see text).

200 400 600 800 1000

0.5

1

1.5

2

200 400 600 800 1000

1

2

3

4

5

FIG. 20: The observable logW(A) as a function of thearea A in lattice units a2 = 2/64 from vortices in caloronensembles with holonomy parameters = 0.0625 (left) and = 0.25 (right). Percolating vortices, i.e. random penetra-tions, would give a linear behavior. The line shown here is alinear extrapolation of the first data point.

VI. SUMMARY AND OUTLOOK

In this paper we have extracted the vortex contentof SU(2) calorons and ensembles made of them, mainlywith the help of the Laplacian Center Gauge, andstudied the properties of the emerging vortices. Ourmain results are

(1) The constituent dyons of calorons induce zeroesof the lowest adjoint mode and therefore appear asmonopoles in the Laplacian Abelian Gauge. The cor-responding worldlines are either two static lines or oneclosed loop for large and small calorons, respectively.

(2) One part of the calorons vortex surface containsthe dyon/monopole worldlines. The vortex changesits flux there (hence the surface should be viewed asnon-orientable). These are general properties of LCGvortices. The specific shapes of these dyon-induced

vortices depend on the caloron size as well as on thelattice extensions. These vortices are predominantlyspace-time like.

(3) Another part of the vortex surface consists of abubble around one of the dyons, depending on theholonomy. The bubble degenerates into the midplane ofthe dyon molecule in the case of maximal nontrivialholonomy = 1/4. This part is predominantly spatial.We have argued that it is induced by the relative twist

between different dyons in the caloron.

(4) Both parts of the vortex together reproduce theunit topological charge of the caloron by 2 intersectionpoints with contributions 1/2.

(5) In dilute caloron ensembles that differ only in

the holonomy mimicking confined and deconfined phase the vortices can be described to a good approximationby recombination of vortices from individual calorons.The spatial vortices in ensembles with deconfiningholonomies 0 form small bubbles. With theholonomy approaching the confinement value = 1/4,the spatial bubbles grow and merge with each other, i.e.they percolate in spatial directions. We have quantifiedthis by the extension of the largest cluster on one handand by the quark-antiquark potential revealed by thePolyakov loop correlator on the other.

In particular the last finding is in agreement with the(de)confinement mechanism based on the percolation of

center vortices.We postpone a more detailed analysis of this mech-

anism as well as working out the picture for the morephysical case of gauge group SU(3) to a later paper.

Acknowledgements

We are grateful to Michael Muller-Preussker, StefanOlejnik and Pierre van Baal for useful comments and toPhilippe de Forcrand for discussions in an early stage ofthe project. We also thank Philipp Gerhold for makinghis code for the generation of caloron ensembles available.

FB and BZ have been supported by DFG (BR 2872/4-1)and BM by RFBR grant 09-02-00338-a.

Appendix

A. Caloron gauge fields

Here we give the functions necessary for the gaugefields of calorons [3] including and their form in the


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limits described in Sect. II. The first set of two auxiliary dimensionless functions are

= cos(2x0/) + cosh(4r/) cosh(4s/) + r2 + s2 + d2

2rssinh(4r/) sinh(4s/)

+d sinh(4r/)

r cosh(4s/) + cosh(4r/)

sinh(4s/)

s

(A.1)

= cos(2x0/) + cosh(4r/) cosh(4s/) + r2 + s2 d2

2rssinh(4r/) sinh(4s/) . (A.2)

We remind the reader that r = |xz1| and s = |xz2| are the distances to the dyon locations and d = |z1z2| = 2/is the distance between the dyon locations, the size of the caloron. The next set of two auxiliary functions enteringEqn. (3) are

=

, =

1

d

sinh(4r/)

r+ e2ix0/

sinh(4s/)

s

(A.3)

For the twist we have analyzed the limit of large sized in Sect. II. For points x = z1 + near the locationof the first dyon, r = || is small and s = d3 +O(2/d)is large. Hence the argument 4s/ is much larger than1 (unless trivial holonomy = 0) and the hyperbolicfunctions can be replaced by exponential functions withexponentially small corrections. On the other hand, nomanipulations are made in all functions with argument4r/, such that we get the exact expressions in termsof the distance r,

= e4s/d

rsinh(4r/) (A.4)

= 12

e4s/

cosh(4r/) 3r

sinh(4r/)

The exponentially large prefactors cancel in the functions and :

(x = z1 + ) =2d

coth(4/) 3 (A.5)

(x = z1 + ) = e2ix0/

1

2d

sinh(4/)(A.6)

where we have replaced r by ||.For points x = z2 + near the location of the second

dyon, s = || is small and r = d + 3 + O(2/d) is largeleading to

= e4r/ dsinh(4s/)

s(A.7)

=1

2e4r/

cosh(4s/) +

3s

sinh(4s/)

and

(x = z2 + ) =2d

coth(4/) + 3(A.8)

(x = z2 + ) =1

2d

sinh(4/)(A.9)

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