Center-Vortex Loops with One Self-Intersection

Abstract

We investigate the 2D behavior of one-fold self-intersecting, topologically stabilized center-vortex loops in the confining phase of an SU(2) Yang-Mills theory. This coarse-graining is described by curve-shrinking evolution of center-vortex loops immersed in a flat 2D plane driving the renormalization-group flow of an effective “action.” We observe that the system evolves into a highly ordered state at finite noise level, and we speculate that this feature is connected with 2D planar high Tc superconductivity in FeAs systems.

1. Introduction

The idea of a nontrivial ground state being responsible for the emergence of “elementary” particles is a rather old one: already Lord Kelvin proposed that atoms and molecules should be considered knotted lines of vortices representing distortions in a universal medium or ground state — the ether. As we know now, the physics of atoms and molecules is described in terms of a much more efficient and elegant framework, quantum mechanics. The agent responsible for the chemical bond — Lord Kelvin’s electron — is considered a spinning point particle in quantum mechanics, and this yields an excellent description of atomic physics, collider physics, and in the bulk of condensed matter physics.

There are, however, theoretical discrepancies with the concept of the electron being a point particle, and there are exceptional experimental situations pointing to the limitations of this concept to describe reality. As for the former, we have the old problem of a diverging classical self-energy not resolved in quantum electrodynamics where the electron mass is introduced as a free parameter whose running with resolution needs an experimental boundary condition. On the other hand, the two-dimensional dynamics of strongly correlated electrons in condensed matter physics signals the relevance of nonlocal effects possibly related to the nontrivial anatomy of the electron becoming relevant in collective phenomena. Also, recent high-temperature plasma experiments indicate unexpected explosive behavior not unlikely related to the mechanism for lepton emergence.

Early papers, mostly investigating on a 4D Euclidean lattice the role of center vortices in forming a confining ground state at low temperatures and resolution, are based on the definition of a dual order parameter for quark confinement. Recent developments in understanding the confining phase of an SU(2) Yang-Mills theory suggest that Lord Kelvin’s ideas may actually be realized in Nature. Those authors construct a plausible effective low-energy action for the 4D SU(2) Yang-Mills theory with solutions to the associated field equations representing closed confining strings knotted into stable solitons. In the thermodynamic approach the emergence of magnetic center-vortex loops (CVLs) is related to discontinuous phase changes of a complex order parameter for confinement across the downward Hagedorn transition and the fact that no magnetic charges exist where these flux lines could end. It was also discussed there how the locations of topologically stabilized self-intersection represent isolated, spinning magnetic charges. Notice that with respect to the electromagnetic U(1) of the Standard Model there is a dual interpretation of magnetic charges emerging in an SU(2) Yang-Mills theory.

In our previous article we have investigated the sector with N = 0 self-intersections by considering a resolution-dependent ensemble average. The corresponding weight-functional is defined purely in terms of the planar curves’ geometry. The resolution dependence of this geometry, in turn, is determined by a curve-shrinking equation (heat equation). The validity of this description of spatial coarse-graining is motivated by considerations relating local curvature with the direction and speed of “motion” of the associated line-segment. The requirement that the partition function over a given ensemble of planar curves is invariant under a change of the resolution then yields the renormalization-group evolution of the weight-functional which is written as the exponential of an “action.” Here the term “action” is slightly misleading since we do not aim at describing the time-evolution of the system by demanding stationarity of the “action” under curve variation. To do the latter, a model, which relates resolution and time being a macroscopic concept associated with the measuring apparatus, needs to be introduced. We thus regard resolution over time as the more fundamental quantity to describe certain subatomic systems.

Our observation is that the effective “action” exhibits a transition towards dilational invariance after a finite, critical decrease of resolution. On average, CVLs with N = 0 are shrunk to circular points for a resolution less than the critical value which de facto removes them from the spectrum and thus generates an asymptotic mass gap. CVLs with N greater than zero are massive. Knowing the evolution of the weight-functional, one is in a position to compute the resolution dependence of “observables” as ensemble averages of the associated nonlocal or local “operators.” As for the evolution of the initially sharp center-of-mass position, we observe a spread of the variance with decreasing resolution saturating at a finite value. This is similar to the unitary free-particle evolution of a position eigenstate in quantum mechanics.

The purpose of the present paper is to extend the earlier procedure to the case of N = 1. We now have a singled-out point on the curve: the location of the self-intersection where practically the entire mass of the soliton resides. Setting the Yang-Mills scale Λ of the SU(2) theory equal to the electron mass, 511 keV, which in turn determines the mass of the intersection point, we interpret this soliton as an electron or a positron. In the presence of a static electric or magnetic background field it is physically possible to lift the two-fold degeneracy with respect to the two possible directions of center-flux: the soliton exhibits a two-fold spin degeneracy. Notice that as long as both wings of center flux are of finite size the position of the intersection point can be shifted at almost no cost of energy. In particular, if the inner angle between in- and outgoing center-flux at the intersection is sufficiently small, then a motion of points on the vortex line directed perpendicular to the bisecting line of the angle easily generates a velocity of the intersection point which exceeds the speed of light, see Figure 1. Recall that the path-integral formulation of quantum mechanics admits such superluminal motion in the sense that the according trajectories sizably contribute to transition amplitudes.

Figure 1. Points on the center flux lines moving oppositely on a line perpendicular to the bisecting line of the angle α with velocity modulus v1. For sufficiently small α the velocity modulus v2 of the intersection point is superluminal: v2 equals v1 times the cotangent of α divided by two.

The paper is organized as follows. In Section 2 we discuss the physics associated with the emergence of topologically stabilized CVLs with intersection number N = 1, and how their spatial 2D coarse-graining is captured by a curve-shrinking flow. Some mathematical results on the properties of this flow for immersed curves, which are relevant for our subsequent numerical analysis, are briefly discussed. Also, we repeat our earlier discussion of how the renormalization-group flow of an effective “action” is driven by the curve-shrinking evolution of the members of a given ensemble of curves. In Section 3 we explain our numerical analysis concerning the computation of the effective “action,” the variance of the location of the self-intersection, and the entropy associated with a given ensemble. Finally, in Section 4 we summarize our results and interpret them in view of certain 2D layered, quasimetallic systems exhibiting high-Tc superconductivity.

2. Conceptual Framework

2.1. Self-Intersecting Center-Vortex Loops

The transition from the non-self-intersecting to the self-intersecting CVL sector is by twisting of non-self-intersecting curves. The emergence of a localized antimonopole in the process is due to its capture by oppositely directed center fluxes in the intersection core (eye of the storm). By a rotation of the left half-plane in Figure 2(a) by an angle of π, see Figure 2(b), each wing of the CVLs forms a closed flux loop by itself thereby introducing equally directed center fluxes at the intersection point. This does not allow for an isolation of a single, spinning antimonopole in the core of the intersection and thus is topologically equivalent to the untwisted case, Figure 2(a). However, another rotation of the left-most half-plane in Figure 2(c) introduces an intermediate loop which by shrinking is capable of isolating a spinning antimonopole due to oppositely directed center fluxes. Notice that in the last stage of such a shrinking process, at short distances between the cores of the flux lines, propagating dual gauge modes are available. On large distances these modes are infinitely massive which is characteristic of the confining phase: there is repulsion due to Biot-Savart which needs to be overcome. This necessitates an investment of energy manifesting itself in terms of the mass of the isolated antimonopole (eye of the storm). Alternatively, the emergence of an isolated antimonopole is possible by a simple pinching of the untwisted curve, again having to overcome local repulsion in the final stage of this process.

Figure 2. Topological transition from the N = 0 sector, panels a, b and c, to the N = 1 sector, panel d, by twisting and subsequent capture of a magnetic antimonopole in the core of the final intersection. Arrows indicate the direction of center flux.

For the analysis performed in the present work we solely regard the situation depicted in Figure 2(d) and thus no longer need to discuss the direction of center flux within a given curve segment. This is not relevant for the process of a spatial coarse-graining microscopically described by the same curve-shrinking flow as applied to the sector with N = 0.

2.2. Euclidean Curve Shrinking Flow

The restriction of evolution of a CVL with self-intersection number N = 1 to the plane is an essential constraint on generality if we aim at a fundamental description of the effectively quantum mechanical behavior of a charged lepton (electron, muon, tau-lepton). An electron bound inside a hydrogen atom certainly “moves” in 3D and quantum mechanics describing it as a spinning, relativistic point particle is in accurate agreement with experiment. The quantum mechanically incompletely understood condensed-matter physics of strongly correlated 2D electrons, however, is a possibly fertile application field of our restriction of curve evolution to the plane, see Section 4.

Notice that by immersing an SU(2) CVL with finite core size d and finite mass of the dual gauge field into a flat 2D surface, a hypothetic observer measuring a positive (negative) curvature of a segment of the vortex line experiences a more (less) negative pressure in the intermediate vicinity of this curve segment leading to its motion towards (away from) the observer, see Figure 3.

Figure 3. Highly space-resolved snapshot of a CVL segment. The pressure in the region pointed to by the normal vector is more negative than the pressure outside, thus leading to a motion of the segment along the normal.

The inward directed speed of a point in the core of the vortex will be a monotonic function of the curvature at this point. On average, this shrinks the CVL. Alternatively, one may globally consider the limit of infinite dual gauge mass and vanishing core size, that is, the confining phase of an SU(2) Yang-Mills theory, but now take into account the effects of an environment which locally relaxes this limit by collisions and thus also induces curve shrinking.

(Equations 2.1 to 2.3 of the article set up the flow. The derivative of a point on the loop with respect to the dimensionless flow parameter equals the second derivative of that point with respect to arc length, divided by a string tension that effectively expresses the distortions induced by the environment. After rescaling lengths by the square root of the string tension, the flow reads: the derivative of the rescaled point with respect to the flow parameter equals its second derivative with respect to rescaled arc length, which is the scalar curvature times the inward-pointing Euclidean unit normal. The scalar curvature is defined in the usual reparametrisation-invariant way. The exact typeset forms are in the published article.)

We now consider curves with one self-intersection, that is, N = 1, in the sense of the stable situation of Figure 2(d). Since the direction of center flux is inessential for the shrinking process we may actually treat this situation in a way as depicted in Figure 2(b), where the curve is defined to be a smooth immersion into the plane with exactly one double point and a total rotation number zero, the integral of the curvature over the loop vanishing. Here the dimensionless curve length is given by the smooth integration over the curve parameter of the modulus of the derivative of the curve. Notice that this is topologically distinct from the case of Figure 2(d) where one encounters a nonvanishing rotation number which is not smoothly deformable to zero.

In the N = 0 case a smooth, embedded curve shrinks to a circular point under the flow at a finite critical value of the flow parameter. That is, the isoperimetric ratio approaches four pi from above. The curve in the situation of Figure 2(b) separates the plane into three disjoint areas two of which are finite and denoted by A1 and A2. We understand by the critical value of the flow parameter the finite value where either A1 or A2 or both vanish. This corresponds to a singularity encountered and thus terminates the flow.

Recall that in the N = 0 case the rate of area change is a constant, minus two pi. This is no longer true for N = 1. However, we have that the difference of A1 and A2 is constant along the flow. Also, for N = 1 we have, in comparison to the N = 0 case, the more relaxed constraint that the rate of change of the sum of A1 and A2 lies between minus four pi and minus two pi.

In contrast to the N = 0 case the isoperimetric ratio for the N = 1 case is bounded up to the critical value of the flow parameter if and only if A1 and A2 differ. Notice that the case of equal areas physically is extremely fine-tuned.

2.3. Effective “Action”

We now wish to interpret curve-shrinking as a Wilsonian renormalization-group flow taking place in the N = 1 CVL sector in the sense defined in Section 2.2. A partition function, defined as a statistical average according to a suitably defined weight over N = 1 CVLs, is to be left invariant under a decrease of the resolution determined by the flow parameter. Notice that, physically, the flow parameter is interpreted as a strictly monotonic decreasing dimensionless function of a ratio of mass scales associated with an actual and an initial resolution applied to the system. The role of the resolution scale can also be played by the finite temperature of a reservoir coupled to the system.

To devise a geometric ansatz for the effective “action,” which is a functional of the curve representable in terms of integrals over local densities (reparametrization invariance), the following reflection on symmetries is in order. First, scaling symmetry: for a rescaling of the curve towards infinite length at fixed shape, the “action” should be invariant under further finite rescalings, which is the decoupling of the fixed length scales set by the string tension and by the Yang-Mills scale. Second, Euclidean point symmetry of the plane — rotations, translations, and reflections about a given axis. Sufficient but not necessary for this is a representation of the “action” in terms of integrals over scalar densities with respect to these symmetries. That is, the “action” density should be expressible as a series involving products of Euclidean scalar products of derivatives of the curve with respect to arc length, or constancy. However, an exceptional scalar integral over a nonscalar density can be devised: the area, calculated as one half the modulus of the integral over the curve of the point vector dotted into the normal. That density is not a scalar under translations.

We now resort to a factorization ansatz in which the “action” is a conformal factor times a nonconformal factor, the latter not being invariant under rescaling of the curve. In principle, infinitely many operators can be defined to contribute to the conformal factor. Since the evolution homogenizes the curvature except for a small vicinity of the intersection point, higher derivatives of the curvature with respect to arc length should not be of importance. We expect this to be true also for Euclidean scalar products involving higher derivatives of the curve. To yield conformally invariant expressions such integrals need to be multiplied by powers of the square root of the area and/or the length, or the inverse of integrals involving lower derivatives. At this stage, we are not capable of constraining the expansion in derivatives by additional physical or mathematical arguments. To be pragmatic, we simply set the conformal factor equal to the isoperimetric ratio, the squared length divided by the area.

We conceive the nonconformal factor in the “action” as a formal Taylor expansion in inverse powers of the length or of the total area, due to the property of conformal invariance in the limit of large length and area.

Since we regard the renormalization-group evolution of the effective “action” as induced by the flow of an ensemble of curves, where the evolution of each member is dictated by the curve-shrinking equation, we allow for an explicit flow-parameter dependence of the coefficient of the lowest nontrivial inverse power of length or of area. In principle, this sums up the contribution to the nonconformal factor of certain higher-power operators which do not exhibit an explicit flow-parameter dependence. Hence the nonconformal factor is taken as one plus a flow-dependent coefficient divided by the length. The initial value of that coefficient is determined from a physical boundary condition such as the mean length at the start of the flow.

2.4. Geometric Partition Function

Let us now numerically investigate the effective “action” resulting from a partition function with respect to a nontrivial ensemble. The latter is defined as the average of the exponential of minus the “action” over the ensemble of curves. Let us denote by an ensemble of M curves the set obtained from the ensemble of M minus one curves by adding a new curve. The effective “action” associated with the ensemble of M curves is determined by the coefficient function of that ensemble, whose flow follows from the requirement that the partition function be independent of the flow parameter.

This is an implicit, first-order ordinary differential equation for the coefficient which needs to be supplemented with an initial condition. A natural initial condition is to demand that the weighted mean length at the start of the flow coincides with the algebraic mean length at the start of the flow; from this a value for the initial coefficient follows.

We also have considered a modified nonconformal factor in which the coefficient is divided by the area rather than by the length. While the ansatz for the geometric effective “action” thus is profoundly different for such a modification, physical results such as the evolution of the variance of the intersection agree remarkably well, see Section 3.

3. Results of Simulation

3.1. Preparation of Ensembles

Similar as in our previous work we normalize all curves to have the same initial area, the sum of the two enclosed areas, and, since we are now interested in the position of the intersection where the antimonopole is localized, we have applied a translation to each curve in the ensembles such that the location of the intersections initially coincides with the origin.

Since the critical value of the flow parameter varies from curve to curve, we order the members of the maximal-size ensemble of sixteen curves into subensembles such that the critical values are in decreasing order. The types of ensembles obtained in this way are referred to as T-ordered. We also have performed all simulations with smaller ensembles whose members are picked randomly from the maximal-size ensemble and have obtained strikingly similar results for ensemble averages of “observables,” for the evolution up to the smallest critical value present in the subensemble.

The maximal-size ensemble at the start of the flow is depicted in Figure 4 with the universal choice of an initial area of two hundred pi. The curves in Figure 4 are arranged in a T-ordered way. The largest critical value is 65 and the smallest is 43. In Figure 5 the evolution of an initial curve under curve shrinking is shown from two viewpoints. The flow is started at zero and stopped at a value shortly below the critical one. In Figure 6 the flow of the intersection points, corresponding to the initial curves depicted in Figure 4, is shown.

The search for solutions to the second-order partial differential equation subject to periodic boundary conditions in the curve parameter, and for the initial conditions depicted in Figure 4, was performed numerically using the method of lines. That is, the partial differential equation was discretized on a uniform grid in the parameter yielding a semidiscrete problem in terms of a system of ordinary differential equations in the flow parameter which was solved using Mathematica. Figure 5 indicates why this technique is called the numerical method of lines. As one can also see from Figure 5, a set of discrete points on the curve, although remaining equidistant in the curve parameter, may evolve under the flow such that the spatial distances between next-neighbour points fall below the numerical precision. Numerically, the flow then encounters a singularity, not to be confused with the earlier mentioned nonfictitious singularities. To recognize such a situation automatically, the constancy of the difference of the two areas was exploited: the evolution was stopped as soon as a sizable deviation occurred from what that constancy predicts. The configuration obtained at this point in the flow was fitted in such a way that a new discretization yielded well-separated points to restart the method of lines. The same constancy was also used as an indicator for the final singularity where A1 or A2 or both vanish.

Figure 4. Initial curves contributing to the maximal-size ensemble of sixteen. The intersection points coincide with the origin, and all curves have the same area, two hundred pi. By definition this ensemble is T-ordered.

Figure 5. Plot of the evolution of an N = 1 CVL, curve 12 of Figure 4, under the curve-shrinking flow. The thick central line indicates the trajectory of the intersection point which coincides with the origin at the start of the flow.

Figure 6. Flow of the intersection points for the initial curves depicted in Figure 4.

3.2. Renormalization-Group Invariance of Partition Function

For all ensembles, the flow-parameter dependence of the coefficient in the nonconformal factor roughly behaves like a square root of the difference between a weakly ensemble-dependent minimal resolution and the flow parameter. For the modified “action,” in which the coefficient is divided by the area, the coefficient is well approximated by a linear function of that same difference. Again, the minimal resolution is weakly ensemble-dependent. For T-ordered ensembles, the results for the two forms of the “action” are shown in Figures 7 and 8, respectively. The results for randomly assembled ensembles do not differ sizably from those presented in Figures 7 and 8.

Figure 7. The squares of the coefficients entering the ansatz for the effective “action,” specializing to the nonconformal factor with the coefficient divided by the length, for T-ordered ensembles up to sixteen curves.

Figure 8. The coefficient entering the ansatz for the effective “action,” specializing to the nonconformal factor with the coefficient divided by the area, for T-ordered ensembles up to sixteen curves.

3.3. Variance of Location of Self-Intersection

The mean intersection over the ensemble is defined as the weighted average of the locations of self-intersection of the member curves, the weight being the exponential of minus the “action” divided by the partition function. The scalar statistical deviation of the intersection location over the ensemble is defined as the sum of the variances of its two Cartesian coordinates, each variance being the weighted average of the squared coordinate minus the square of the weighted average.

In Figure 9 plots of the deviation are shown when evaluated over the ensembles of one through sixteen curves subject to the “action” given by the isoperimetric ratio times one plus the coefficient divided by the length, and the initial condition equating the weighted and algebraic mean lengths. In Figure 10 the according plots are depicted as obtained with the “action” in which the coefficient is divided by the area, subject to the same initial condition. Relaxing the constraint of T-ordering does not entail a qualitative change of the results.

The results presented in Figures 9 and 10 are unexpected since in the N = 0 sector the variance of the “center-of-mass” saturates rapidly to finite values. In contrast, for the N = 1 sector the variance of the location of the self-intersection initially increases, reaches a maximum, and decreases to zero at a finite value of the flow parameter. This is readily confirmed by the evaluation of the entropy, see Section 3.4.

Figure 9. Plots of the deviation of the intersection location for the T-ordered ensembles with one through sixteen members. We have employed the ansatz for the “action” given by the isoperimetric ratio times one plus the coefficient divided by the length.

Figure 10. Plots of the deviation of the intersection location for the T-ordered ensembles with one through sixteen members. We have employed the ansatz for the “action” given by the isoperimetric ratio times one plus the coefficient divided by the area.

3.4. Evolution of Entropy

Let us now evaluate the flow of the entropy, defined as the logarithm of the partition function plus the weighted ensemble average of the “action.” In Figure 11 plots are shown for the entropy when evaluated with the “action” given by the isoperimetric ratio times one plus the coefficient divided by the length, for T-ordered ensembles of size one through sixteen. These graphs look very much alike to the ones generated using the “action” in which the coefficient is divided by the area. Notice the continuous approach to zero at finite values of the flow parameter. This implies that order emerges spontaneously in the system with decreasing resolution: starting at a finite value of the flow parameter, a particular member of the ensemble is singled out by its weight approaching unity. Judging from our results for the N = 0 sector, this behavior is highly unexpected. Therefore the nontrivial topology of N = 1 induces qualitative differences to the coarse-graining process.

Figure 11. Flow of the entropies for T-ordered ensembles of size one through sixteen when evaluated with the “action” given by the isoperimetric ratio times one plus the coefficient divided by the length. The situation does not change qualitatively if the “action” with the coefficient divided by the area is used.

4. Summary, Interpretation of Results, and Conclusion

In this paper we have investigated the spatial coarse-graining of CVLs, immersed in a flat 2D plane, of an SU(2) Yang-Mills theory being in its confining phase. The focus was on the sector with one topologically stabilized self-intersection — the existence of an isolated magnetic charge at its location, N = 1. We have analysed this coarse-graining process in terms of curve ensembles generated by evolving an initial situation under the curve-shrinking flow. The idea here is to suppose that curve shrinking in the flow parameter represents an exact coarse-graining of a given initial state and to reconstruct the associated ensemble-weight of the statistical approach — the exponential of an effective “action” — by demanding invariance of the corresponding partition function under the flow, that is, renormalization-group evolution. Notice that the flow parameter is related to a physical resolution, applied to probing the system, in a strictly monotonic decreasing manner. This resolution may be associated with a local momentum transfer exerted by an observer or a globally defined temperature of an environment. The functional dependence of the flow parameter on these physical parameters depends on the given experimental situation. It is, however, reasonable to assume that finite values of the flow parameter universally correspond to finite values of these physical parameters.

In Sections 3.3 and 3.4 we have obtained the unexpected result that a statistical ensemble of renormalization-group evolved curves spontaneously orders itself in the sense that, starting from finite values of the flow parameter, only a particular member of the ensemble survives the process of 2D spatial coarse-graining. That is, the entropy attributed to the ensemble is practically zero for sufficiently large values of the flow parameter. For the location of self-intersection — the charge of an electron — this means that no dissipation of energy, provided by the environment, can be mediated by the monopole situated within the core of the intersection if the resolution falls below a critical, finite value. This result must drastically depend on the two-dimensionality of space and the fact that we consider the sector with N = 1.

The recently discovered, unconventional FeAs systems do not appear to exhibit an explicit, strong correlation between the electrons contained in their theoretically suggested, 2D-superconducting layers. If the two-dimensional behavior of noninteracting electrons, subject to an environment represented by the flow parameter, indeed is described by the coarse-graining process investigated in the present work, then the sudden decrease of entropy that we observe at a finite value of the flow parameter should ultimately be connected to this particular kind of high-Tc superconductivity. Here the flow parameter is a monotonically decreasing function of temperature.

Acknowledgment

The authors would like to thank Francesco Giacosa and Markus Schwarz for useful conversations.