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ISRN Mathematical Physics
VolumeΒ 2012Β (2012), Article IDΒ 236783, 15 pages
Research Article

Evolving Center-Vortex Loops

Institut für Theoretische Physik, UniversitÀt Karlsruhe (TH), Kaiserstraße 12, 76131 Karlsruhe, Germany

Received 7 December 2011; Accepted 29 March 2012

Academic Editors: J. BičÑk and M. Ehrnstrâm

Copyright Β© 2012 Julian Moosmann and Ralf Hofmann. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider coarse-graining applied to nonselfintersecting planar centervortex loops as they emerge in the confining phase of an SU(2) Yang-Mills theory. Well-established properties of planar curve-shrinking predict that a suitably defined, geometric effective action exhibits (mean-field) critical behavior when the conformal limit of circular points is reached. This suggests the existence of an asymptotic mass gap. We demonstrate that the initially sharp mean center-of-mass position in a given ensemble of curves develops a variance under the flow as is the case for a position eigenstate in free-particle quantum mechanics under unitary time evolution. A possible application of these concepts is an approach to high-𝑇𝑐 superconductivity based (a) on the nonlocal nature of the electron (1 fold selfintersecting center-vortex loop) and (b) on planar curve-shrinking flow representing the decrease in thermal noise in a cooling cuprate.

1. Introduction

The problem of how a confining ground state emerges in 4D nonabelian Yang-Mills theory was addressed by ’t Hooft in terms of the definition of an order parameter that is dual to the Wilson loop [1]. The lattice-gauge-theory measurement as well as analytical model approaches to the ’t Hooft loop expectation was performed by several authors starting shortly thereafter [2–7]. The according confinement scenario is a condensation of magnetic center vortices which induces narrow electric flux lines in between color-electric sources granting a potential which rises sufficiently rapid with an increasing distance of separation. Due to the absence of magnetic center monopoles the associated center vortices form closed flux loops.

Four-dimensional SU(2) Yang-Mills theory occurs in three phases: a deconfining, a preconfining, and a confining one. While the former two phases possess propagating gauge fields, a complete decoupling thereof takes place at a Hagedorn transition towards the confining phase [8–12]. Namely, by the decay of a preconfining ground state, consisting of collapsing magnetic (w.r.t. the gauge fields in the defining SU(2) Yang-Mills Lagrangian) flux lines of finite core-size 𝑑, see also [13], into nonselfintersecting or selfintersecting center-vortex loops [1], the mass π‘šπ· of the dual gauge field diverges. This, in turn, implies 𝑑→0. As a consequence, center-vortex loops (CVLs) with nonvanishing selfintersection number 𝑁 become stable solitons in isolation. These solitons are classified according to their topology and center charge. That is, for 𝑑→0, the region of negative pressure 𝑃 is confined to the vanishing vortex core, and the soliton becomes a particle-like (𝑃=0) excitation whose stability is in addition assured topologically by its selfintersection(s).

The purpose of our paper is to investigate the sector with 𝑁=0 in some detail. Topologically, there is no reason for the stability of this sector’s excitations, and we will argue that on average, and as a consequence of a noisy environment a planar CVL with 𝑁=0 shrinks to nothingness within a finite β€œtime.” Here the role of β€œtime” is played by a variable measuring the decrease of externally provided resolving power applied to the system. By β€œplanar CVL” we mean an embedding of the 𝑁=0 soliton into a 2D flat and spatial plane. For an isolated SU(2) Yang-Mills theory the role of the environment is played by the sectors with 𝑁>0. If the SU(2) theory under consideration is part of a world subject to additional gauge symmetries, then a portion of such an environment arises from a mixing with these theories. In any case, a planar CVL at finite length 𝐿 is acquiring mass by frequent interactions with the environment after it was generated by a process subject to an inherent, finite resolution 𝑄0. At the same time, the CVL starts shrinking towards a circular point. The latter becomes unresolvable starting at some finite resolution π‘„βˆ—. That is, all properties that are related to the existence of extended lines of center flux, observable for 𝑄0β‰₯𝑄>π‘„βˆ—, do not occur for π‘„β‰€π‘„βˆ—, and the CVL vanishes from the spectrum of confining SU(2) Quantum Yang-Mills theory. Since CVLs with 𝑁>0 have a finite mass (positive-integer multiples of the Yang-Mills scale Ξ›), we β€œobserve” a gap in the mass spectrum of the theory when probing the system with resolution π‘„β‰€π‘„βˆ—.

The reader may object that the restriction of the shrinking dynamics to a 2D plane does not justify the conclusion about the aforementioned occurrence of a mass gap since CVLs evolve in 3D. However, according to the work of [14, 15] and for low-energy SU(2) Yang-Mills theory the assumption of planar, closed, and smooth curves accurately describing the late-β€œtime” behavior of the evolution of any given center-vortex loop with 𝑁=0 is valid because the CVL becomes planar after a finite β€œtime” under the curve shrinking equation. Thus, to discuss the approach of the round-point limit (mass-gap property) we may pick planar initial conditions and consider the subsequent evolution in the plane. The formation of a singularity (a loop pinching off to form a cusp) is a planar phenomenon [15] which, however, is excluded to occur in low-energy, that is, weakly resolved SU(2) Yang-Mills theory because the energy required to form and resolve such a singularity is not available.

Notice that by embedding an 𝑁=0 CVL of an isolated SU(2) Yang-Mills theory into a flat 2D surface at π‘šπ·<∞, 𝑑>0, a hypothetical observer measuring a positive (negative) curvature along a given sector of the vortex line experiences more (less) negative pressure in the intermediate vicinity of this curve sector leading to this sector’s motion towards (away from) the observer, see Figure 1. The speed of this motion is a monotonic function of curvature. On average, this shrinks the CVL. Alternatively, one may globally consider the limit π‘šπ·β†’βˆž, 𝑑→0, 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. One possibility to describe this situation is by the following flow equation in the (dimensionless) parameter πœπœ•πœ1𝐱=πœŽπœ•2𝑠𝐱,(1.1) where 𝑠 is arc length, 𝐱 is a point on the planar CVL, and 𝜎 is a string tension effectively expressing the distortions induced by the environment. After rescaling, βˆšΜ‚π‘₯β‰‘πœŽπ±, βˆšπœ‰=πœŽπ‘ , (1.1) assumes the following form:πœ•πœΜ‚π‘₯(𝑒,𝜏)=πœ•2πœ‰Μ‚π‘₯=π‘˜(𝑒,𝜏)𝐧(𝑒,𝜏),(1.2) where 𝑒 is a (dimensionless) curve parameter (π‘‘πœ‰=|πœ•π‘’Μ‚π‘₯|𝑑𝑒), 𝐧 the (inward-pointing) Euclidean unit normal, and π‘˜ the scalar curvature, defined as||πœ•π‘˜β‰‘2πœ‰||=||||1Μ‚π‘₯||πœ•π‘’||πœ•Μ‚π‘₯𝑒1||πœ•π‘’||πœ•Μ‚π‘₯𝑒||||Μ‚π‘₯,(1.3)√|𝐯|≑𝐯⋅𝐯, and 𝐯⋅𝐰 denotes the Euclidean scalar product of the vectors 𝐯 and 𝐰 lying in the plane. In the following, we resort to a slight abuse of notation by using the same symbol Μ‚π‘₯ for the functional dependence on 𝑒 or πœ‰.

Figure 1: Highly space-resolved snapshot of an 𝑁=0 CVL curve-sector. The pressure 𝑃𝑖 in the region pointed to by the normal vector 𝐧 is more negative than the pressure 𝑃𝑒, thus leading to a motion of the sector along 𝐧.

It is worth mentioning that (1.2) expresses a special case of the local condition that the rate of decrease of the (dimensionless) curve length ∫𝐿(𝜏)=0𝐿(𝜏)βˆ«π‘‘πœ‰=02πœ‹π‘‘π‘’|πœ•π‘’Μ‚π‘₯(𝑒,𝜏)| is maximal w.r.t. a variation of the direction of the velocity πœ•πœΜ‚π‘₯ of a given point on the curve at fixed |πœ•πœΜ‚π‘₯| [16]:𝑑𝐿(𝜏)ξ€œπ‘‘πœ=βˆ’0𝐿(𝜏)π‘‘πœ‰π‘˜π§β‹…πœ•πœΜ‚π‘₯.(1.4) The 1D heat equation (1.2) is well understood mathematically [17, 18]. The evolution of a particular initial configuration of a planar CVL (𝜏=0) into the round-point limit (𝜏=𝑇) is depicted in Figure 2.

Figure 2: Plot of the evolution of a planar CVL under (1.2). The thick central line depicts the graph of the CVL’s β€œcenter of mass.” The flow is started at 𝜏=0 and ends at 𝜏=𝑇.

The present work interprets the shrinking of closed and 2D flat-embedded (planar) curves as a Wilsonian renormalization group evolution governed by an effective action, which is defined purely in geometric terms. In the presence of an environment represented by the parameter 𝜎, this action possesses a natural decomposition into a conformal and a nonconformal factor. One of our goals is to show that the transition to the conformal limit of vanishing mean curve length really is a critical phenomenon characterized by a mean-field exponent if a suitable parameterization of the effective action is used. To see this, various initial conditions are chosen to generate an ensemble whose partition function is invariant under curve shrinking. A (second-order) phase transition is characterized by the critical behavior of the coefficient associated with the nonconformal factor in the effective action. That is, in the presence of an environment the (nearly massless) 𝑁=0 sector in confining SU(2) Yang-Mills dynamics, generated during the Hagedorn transition, practically disappears after a finite β€œtime” leading to an asymptotic mass gap. We also believe that 𝑁=0 CVLs play the role of Majorana neutrinos in physics: their disappearance after a finite time and the absence of antiparticles would be manifestations of lepton-number violation [19–21] forbidden in the present standard model of particle physics, see [22] for a discussion of experimental signatures in this context.

In Section 2, we provide information on proven properties of curve-shrinking evolution. The philosophy underlying the statistics of geometric fluctuations in the 𝑁=0 sector is elucidated in Section 3. In Section 4, we present our results for the renormalization-group flow of the effective action, give an interpretation, and compute the evolution of a local quantity. Finally, in Section 5 we summarize our work and give an outlook to the 𝑁=1 case which we suspect to be relevant to high 𝑇𝑐 superconductivity.

2. Prerequisites on Mathematical Results

In this section, we provide knowledge on the properties of the shrinking of embedded (nonselfintersecting) curves in a plane [17, 18]. It is important to stress that only for curve-shrinking in a plane are the following results valid. To restrict the motion of a CVL to a plane, however, is justified when physics close to the round-point limit is considered [14, 15], see Section 1.

The properties of the 𝜏-evolution of smooth, embedded, and closed curves Μ‚π‘₯(𝑒,𝜏) subject to (1.2) were investigated in [17] for the purely convex case and in [18] for the general case. The main result of [18] is that an embedded curve with finitely many points of inflection remains embedded and smooth when evolving under (1.2) and that such a curve flows to a circular point for πœβ†—π‘‡, where 0<𝑇<∞. That is, asymptotically the curve converges (w.r.t. the 𝐢∞-norm) to a shrinking circle: limπœβ†’π‘‡πΏ(𝜏)=0 and limπœβ†’π‘‡π΄(𝜏)=0, 𝐴 being the (dimensionless) area enclosed by the curve, such that the isoperimetric ratio 𝐿2(𝜏)/𝐴(𝜏) approaches the value 4πœ‹ from above. For later use, we present the following two identities, see Lemmas 3.1.2 and 3.1.7 in [17]:πœ•πœξ€œπΏ=βˆ’πΏ0π‘‘πœ‰π‘˜2ξ€œ=βˆ’02πœ‹||πœ•π‘‘π‘’π‘’||π‘˜Μ‚π‘₯2,πœ•(2.1)𝜏𝐴=βˆ’2πœ‹.(2.2) Setting 𝐴(𝜏=0)≑𝐴0, the solution to (2.2) is𝐴(𝜏)𝐴0=1βˆ’2πœ‹πœπ΄0.(2.3) By virtue of (2.3) the critical value 𝑇 is related to 𝐴0 as𝐴𝑇=02πœ‹.(2.4)

3. Geometric Partition Function

We now wish to interpret curve-shrinking as a Wilsonian renormalization-group flow taking place in the 𝑁=0 planar CVL sector. A partition function, defined as a statistical average (according to a suitably defined weight) over 𝑁=0 CVLs, is to be left invariant under a decrease of the resolution determined by the flow parameter 𝜏. Notice that, physically, 𝜏 is interpreted as a strictly monotonic decreasing (dimensionless) function of a ratio 𝑄/𝑄0 where 𝑄 (𝑄0) are mass scales associated with an actual (initial) resolution applied to the system.

To devise a geometric ansatz for the effective action 𝑆=𝑆[Μ‚π‘₯(𝜏)], which is a functional of the curve Μ‚π‘₯ representable in terms of integrals over local densities in πœ‰ (reparametrization invariance), the following reflection on symmetries is in order. (i) scaling symmetry Μ‚π‘₯β†’πœ†Μ‚π‘₯,πœ†βˆˆπ‘+: For both, πœ†β†’βˆž and πœ†β†’0, implying πœ†πΏβ†’βˆž and πœ†πΏβ†’0 at fixed 𝐿, the action 𝑆 should be invariant under further finite rescalings (decoupling of the fixed length scale πœŽβˆ’1/2), (ii) Euclidean point symmetry of the plane (rotations, translations and reflections about a given axis): Sufficient but not necessary for this is a representation of 𝑆 in terms of integrals over scalar densities w.r.t. these symmetries. That is, the action density should be expressible as a series involving products of Euclidean scalar products of (πœ•π‘›/πœ•πœ‰π‘›)Μ‚π‘₯, π‘›βˆˆπ+, or constancy. However, an exceptional scalar integral over a nonscalar density can be devised. Consider the area 𝐴, calculated as||||1𝐴=2ξ€œ02πœ‹||||.π‘‘πœ‰Μ‚π‘₯⋅𝐧(3.1) The density Μ‚π‘₯⋅𝐧 in (3.1) is not a scalar under translations.

We now resort to a factorization ansatz as𝑆=𝐹𝑐×𝐹nc,(3.2) where in addition to Euclidean point symmetry 𝐹𝑐 (𝐹nc) is (is not) invariant under Μ‚π‘₯β†’πœ†Μ‚π‘₯. In principle, infinitely many operators can be defined to contribute to 𝐹𝑐. Since the evolution generates circles for πœβ†—π‘‡ higher derivatives of π‘˜ w.r.t. πœ‰ rapidly converge to zero [17]. We expect this to be true also for Euclidean scalar products involving higher derivatives (πœ•π‘›/πœ•πœ‰π‘›)Μ‚π‘₯. To yield conformally invariant expressions, such integrals need to be multiplied by powers of √𝐴 and/or 𝐿 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 𝐹𝑐 equal to the isoperimetric ratio:𝐹𝑐(𝜏)≑𝐿(𝜏)2𝐴(𝜏).(3.3) We conceive the nonconformal factor 𝐹nc in 𝑆 as a formal expansion in inverse powers of 𝐿. 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 (1.2), we allow for an explicit 𝜏 dependence of the coefficient 𝑐 of the lowest nontrivial power 1/𝐿. In principle, this sums up the contribution to 𝐹nc of certain higher-power operators which do not exhibit an explicit 𝜏 dependence. Hence, we make the following ansatz𝐹nc(𝜏)=1+𝑐(𝜏)𝐿(𝜏).(3.4) The initial value 𝑐(𝜏=0) is determined from a physical boundary condition such as the mean length 𝐿 at 𝜏=0 which determines the mean mass π‘š of a CVL as π‘š=𝜎𝐿.

For latter use, we investigate the behavior of 𝐹nc(𝜏) as πœβ†—π‘‡ for an ensemble consisting of a single curve only and require the independence of the β€œpartition function” under changes in 𝜏. Integrating (2.1) in the vicinity of 𝜏=𝑇 under the boundary condition that 𝐿(𝜏=𝑇)=0, we have√𝐿(𝜏)=√8πœ‹π‘‡βˆ’πœ.(3.5) Since 𝐹𝑐(πœβ†—π‘‡)=4πœ‹ independence of the β€œpartition function” under the flow in 𝜏 implies thatβˆšπ‘(𝜏)βˆπ‘‡βˆ’πœ.(3.6) That is, 𝐹nc approaches constancy for πœβ†—π‘‡ which brings us back to the conformal limit by a finite renormalization of the conformal part βˆ«πΉπ‘ of the action. In this parameterization of 𝑆, 𝑐(𝜏) can thus be regarded as an order parameter for conformal symmetry with mean-field critical exponent.

4. Renormalization-Group Flow

4.1. Effective Action

Let us now numerically investigate the effective action 𝑆[Μ‚π‘₯(𝜏)] resulting from a partition function 𝑍 w.r.t. a nontrivial ensemble 𝐸. The latter is defined as the average𝑍=𝑖expβˆ’π‘†Μ‚π‘₯𝑖(𝜏)ξ€»ξ€Έ(4.1) over the ensemble 𝐸={Μ‚π‘₯1,…}. Let us denote by 𝐸𝑀 an ensemble consisting of 𝑀 curves where 𝐸𝑀 is obtained from πΈπ‘€βˆ’1 by adding a new curve Μ‚π‘₯𝑀(𝑒,𝜏). In Figure 3, eight initial curves are depicted which in this way generate the ensembles 𝐸𝑀 for 𝑀=1,…,8.

Figure 3: Initial curves contributing to the ensembles 𝐸𝑀, see text. Points locate the respective positions of the centers of mass.

We are interested in a situation where all curves in 𝐸𝑀 shrink to a point at the same value 𝜏=𝑇. Because of (2.3) and (2.4), we thus demand that at 𝜏=0 all curves in 𝐸𝑀 initially have the same area 𝐴0. The effective action 𝑆 in (3.2) (when associated with the ensemble 𝐸𝑀 we will denote it as 𝑆𝑀) is determined by the function 𝑐𝑀(𝜏), compared with (3.4), whose flow follows from the requirement of 𝜏-independence of 𝑍𝑀:π‘‘π‘π‘‘πœπ‘€=0.(4.2) This is an implicit, first-order ordinary differential equation for 𝑐(𝜏) which needs to be supplemented with an initial condition 𝑐0,𝑀=𝑐𝑀(𝜏=0). A natural initial condition is to demand that the quantity:𝐿𝑀1(𝜏=0)≑𝑍𝑀(𝜏=0)𝑀𝑖=1𝐿̂π‘₯𝑖(𝜏=0)expβˆ’π‘†π‘€ξ€ΊΜ‚π‘₯𝑖(𝜏=0)ξ€»ξ€Έ(4.3) coincides with the arithmetic mean 𝐿𝑀(𝜏=0) defined as𝐿𝑀1(𝜏=0)≑𝑀𝑀𝑖=1𝐿̂π‘₯𝑖(𝜏=0).(4.4) From 𝐿𝑀𝐿(𝜏=0)=𝑀(𝜏=0) a value for 𝑐0,𝑀 follows. We also have considered a modified factor 𝐹nc(𝜏)=1+(𝑐(𝜏)/𝐴(𝜏)) in (3.2). In this case the choice of initial condition 𝐿𝑀𝐿(𝜏=0)=𝑀(𝜏=0) leads to 𝐹nc(𝜏)≑0. While the geometric effective action thus is profoundly different for such a modification of 𝐹nc(𝜏) physical results such as the evolution of the variance of center-of-mass position agree remarkably well, see Section 4.2. That is, the geometric effective action itself is not a physical object. Rather, going from one ansatz for 𝑆𝑀 to another describes a particular way of redistributing the weight in the ensemble which seems to have no significant impact on the physics. This is in contrast to quantum field theory and conventional statistical mechanics, where the action in principle is related to the physical properties of a given member of the ensemble.

For the curves depicted in Figure 3, we make the convention that 𝐴0≑2πœ‹Γ—100. It then follows that 𝑇=100 by (2.4). The dependence 𝑐2𝑀(𝜏) is plotted in Figure 4. According to Figure 4, it seems that the larger the ensemble the closer 𝑐2𝑀(𝜏) to the evolution of a single circle of initial radius βˆšπ‘…=𝐴0/πœ‹. That is, for growing 𝑀 the function 𝑐2𝑀(𝜏) approaches the form:𝑐2as,𝑀(𝜏)=π‘˜π‘€(π‘‡βˆ’πœ),(4.5) where the slope π‘˜π‘€ depends on the strength of deviation from circles of the representatives in the ensemble 𝐸𝑀 at 𝜏=0, that is, on the variance Δ𝐿𝑀 at a given value 𝐴0. Physically speaking, the value 𝜏=0 is associated with a certain initial resolution of the measuring devise (the strictly monotonic function 𝜏(𝑄), 𝑄 being a physical scale such as energy or momentum transfer, expresses the characteristics of the measuring device and the measuring process), the value of 𝐴0 describes the strength of noise associated with the environment (𝐴0 determines how fast the conformal limit of circular points is reached), and the values of 𝑐0,𝑀 and π‘˜π‘€, see (4.5), are associated with the conditions at which the to-be-coarse-grained system is prepared. Notice that this interpretation is valid for the action 𝑆𝑀=(𝐿(𝜏)2/𝐴(𝜏))(1+𝑐𝑀(𝜏)/𝐿(𝜏)) only.

Figure 4: The square of the coefficient 𝑐𝑀(𝜏) entering the effective action of (3.2) by virtue of (3.4) for various ensemble sizes. Notice the early onset of the linear drop of 𝑐2𝑀(𝜏) and the saturation in 𝑀 for 𝑀β‰₯5. The slope of 𝑐2𝑀(𝜏) near 𝜏=𝑇 does not depend on 𝑐20,𝑀≑𝑐2𝑀(𝜏=0) and thus not on the initial choice of 𝐿.
4.2. Statistical Uncertainty of Center of Mass Position

We are now in a position to compute the flow of a more local β€œobservable,” namely, the mean β€œcenter-of-mass” (COM) position in a given ensemble and the statistical variance of the COM position. The COM position Μ‚π‘₯COM of a given curve Μ‚π‘₯(πœ‰,𝜏) is defined asΜ‚π‘₯COM1(𝜏)=ξ€œπΏ(𝜏)0𝐿(𝜏)π‘‘πœ‰Μ‚π‘₯(πœ‰,𝜏).(4.6) We will below present only results on the statistical variance of the COM position.

Let as assume that at 𝜏=0, the ensembles 𝐸𝑀 are modified such that a translation is applied to each representative letting its COM position coincide with the origin. Recall that such a modification πΈπ‘€β†’πΈξ…žπ‘€ does not alter the (effective) action (Euclidean point symmetry). That is, at 𝜏=0 the statistical variance in the position of the COM is prepared to be nil, physically corresponding to an infinite resolution applied to the system by the measuring device.

The mean COM position Μ‚π‘₯COM over ensemble πΈξ…žπ‘€ is defined asΜ‚π‘₯COM1(𝜏)≑𝑍𝑀𝑀𝑖=1Μ‚π‘₯COM,𝑖(𝜏)expβˆ’π‘†π‘€ξ€ΊΜ‚π‘₯𝑖(𝜏)ξ€»ξ€Έ.(4.7) The scalar statistical deviation Δ𝑀,COM of Μ‚π‘₯COM over the ensemble πΈξ…žπ‘€ is defined asΔ𝑀,COM(𝜏)≑var𝑀,COM;π‘₯(𝜏)+var𝑀,COM;𝑦(𝜏),(4.8) where var𝑀,COM;π‘₯≑1𝑍𝑀𝑀𝑖=1ξ€·π‘₯COM,𝑖(𝜏)βˆ’π‘₯COMξ€Έ(𝜏)2ξ€·expβˆ’π‘†π‘€ξ€ΊΜ‚π‘₯𝑖(𝜏)ξ€»ξ€Έ=βˆ’π‘₯2COM1(𝜏)+𝑍𝑀𝑀𝑖=1π‘₯2COM,𝑖(𝜏)expβˆ’π‘†π‘€ξ€ΊΜ‚π‘₯𝑖,(𝜏)ξ€»ξ€Έ(4.9) and similarly for the coordinate 𝑦. In Figure 5, plots of Δ𝑀,COM(𝜏) are shown when Δ𝑀,COM(𝜏) is evaluated over the ensembles πΈξ…ž3,…,πΈξ…ž8 with the action:𝑆𝑀=𝐿(𝜏)2𝐴𝑐(𝜏)1+𝑀(𝜏)𝐿(𝜏),(4.10) and subject to the initial condition 𝐿𝑀𝐿(𝜏=0)=M(𝜏=0). In Figure 5(b) the according plots of Δ𝑀,COM(𝜏) are depicted as obtained with the action𝑆𝑀=𝐿(𝜏)2𝐴𝑐(𝜏)1+𝑀(𝜏)𝐴(𝜏),(4.11) and subject to the initial condition 𝐿𝑀𝐿(𝜏=0)=𝑀(𝜏=0). In this case, one has 𝑐𝑀(𝜏)=βˆ’π΄(𝜏) leading to equal weights for each curve in πΈξ…žπ‘€.

Figure 5: (a) plots of Δ𝑀,COM(𝜏) for 𝑀=3,…,8 when evaluated with the action 𝑆𝑀=(𝐿(𝜏)2/𝐴(𝜏))(1+(𝑐𝑀(𝜏)/𝐿(𝜏))). Notice the rapid generation of an uncertainty in the COM position under the flow and its saturation when approaching the conformal limit πœβ†—π‘‡. There is also a saturation of this limiting value with a growing ensemble size. (b) plots of Δ𝑀,COM(𝜏) for 𝑀=3,…,8 when evaluated with the action 𝑆𝑀=(𝐿(𝜏)2/𝐴(𝜏))(1+𝑐𝑀(𝜏)/𝐴(𝜏)). Notice the qualitative agreement with the results displayed in (a).
4.3. Quantum Mechanical versus Statistical Uncertainty

In view of the results obtained in Section 4.2, we would say that an ensemble of evolving planar CVLs in the 𝑁=0 sector qualitatively resembles the quantum mechanics of a free-point particle (it is of no relevance at this point whether this particle carries spin or not) of mass π‘š in 1D. Namely, an initially localized square of the wave function πœ“ with |πœ“(𝜏=0,π‘₯)|2∝exp[βˆ’(π‘₯2/π‘Ž20)], where Ξ”π‘₯(𝜏=0)=π‘Ž0, according to unitary time evolution in quantum mechanics evolves as |πœ“(𝜏,π‘₯)|2=|exp[βˆ’π‘–(𝐻𝜏/ℏ)]πœ“(𝜏=0,π‘₯)|2∝exp[βˆ’((π‘₯βˆ’(𝑝/π‘š)𝜏)2/π‘Ž2(𝜏))], where 𝐻=𝑝2/2π‘š is the free-particle Hamiltonian, 𝑝 is the spatial momentum, and π‘Ž(𝜏)β‰‘π‘Ž01+(β„πœ/π‘šπ‘Ž20)2. In agreement with Heisenberg’s uncertainty relation one has during the process that Δπ‘₯Δ𝑝=(ℏ/2)1+(πœβ„/π‘šπ‘Ž20)2β‰₯ℏ/2. Time evolution in quantum mechanics and the process of coarse-graining in a statistical system describing planar CVLs share the property that in both systems the evolution generates out of a small initial position uncertainty (corresponding to a large initial resolution Δ𝑝) a larger position uncertainty in the course of the evolution. Possibly, future development will show that interference effects in quantum mechanics can be traced back to the nonlocal nature of the degrees of freedom (CVLs) entering a statistical partition function.

5. Summary, Conclusions, and Outlook

5.1. Summary of Present Work

In this exploratory paper an attempt has been undertaken to interpret the effects of an environment on 2D planar center-vortex loops, as they emerge in the confining phase of an SU(2) Yang-Mills theory, in terms of a Wilsonian renormalization-group flow carried by purely geometric entities. Our (mainly numerical) analysis uses established mathematics on the shrinking of embedded curves in the plane. In the case of nonintersecting CVLs (𝑁=0) the role of the environment is played by the entirety of all sectors with 𝑁>0 and, possibly, an explicit environment. In a particular parametrization of the effective action we observe critical behavior as the limit of circular points is approached during the evolution. That is, planar 𝑁=0 CVLs on average disappear from the spectrum for resolving powers smaller than a critical, finite value. Using this formalism to compute the evolution of the mean values of local observables, such as the center-of-mass position, a behavior is generated that qualitatively resembles the associated unitary evolution in quantum mechanics. We also have found evidence that this situation is practically not altered when changing the ansatz for the effective action.

5.2. Outlook on Strongly Correlated Electrons in a Plane

Let us conclude this article with a somewhat speculative outlook on planar 𝑁=1 CVLs. Setting the Yang-Mills scale (mass of the intersection in the CVL) of the associated SU(2) theory equal to the electron mass, this soliton is interpreted as an electron or positron, see [8, 9, 23, 24] for a more detailed discussion on the viability of such an assignment and Figure 6 for a display. Important for our purpose are the facts that the two-fold directional degeneracy of the center flux represents the two-fold degeneracy of the spin projection and that a large class of curve deformations (shifts of the intersection point) leaves the mass of the soliton invariant in the free case.

Figure 6: An isolated CVL with 𝑁=1, which when associated with the confining phase of an SU(2) Yang-Mills theory of scale Ξ›βˆΌπ‘šπ‘’=511keV, is interpreted as an electron or positron. The arrows depict one out of two possible directions of center flux responsible for the magnetic moment (spin), the intersection point is the location of mass and electric charge.

It is a remarkable fact that a high level of mathematical understanding exists for the behavior of curve-shrinking in a 2D plane [17, 18] even in the case of one selfintersection [25]. Incidentally, 2D quantum systems exist in nature which exhibit unconventional behavior. Specifically, the phenomenon of high 𝑇𝑐 superconductivity appears to be strongly related to the two-dimensionality of electron dynamics as it is enabled by rare-earth doping of cuprate materials [26]. Apparently, the Coulomb repulsion between the electrons moving in the would-be valence band within the cuprate planes of high 𝑇𝑐 superconductors (Mott insulators) is effectively screened by the interplane environment also providing for the very existence of these electrons by doping. The question then is how the long-range order of electronic spins, which at given (optimal) doping and at a sufficiently low temperature leads to superconductivity, emerges within the cuprate planes. As it seems, quantum Monte Carlo simulations of a transformed Hubbard model (𝑑-𝐽 model) subject to Gutzwiller projection yield quantitative explanations of a number of experimental results related to the existence of the pseudogap phase (Nernst effect, nonlinear diamagnetic susceptibility), see [27, 28] and references therein.

We would here like to offer a sketch of an alternative approach to high 𝑇𝑐 superconductivity being well aware of our ignorance on the details of present-day research in this field. The key idea is already encoded in Figure 6. Namely, according to confining SU(2) Yang-Mills theory the electron is a nonlocal object with the physics of its charge localization being only loosely related to the physics of its magnetic moment (spin): the magnetic moment, carried by the core of the flux line, microscopically manifested by (oppositely) moving (opposite) electric charges, receives contributions from vortex sectors that are spatially far separated (on the scale of the diameter of the intersection point) from the location of the isolated electric charge. This suggests that in certain physical circumstances, where the ordering effect of interacting vortex lines becomes important, the postulate of a spinning point particle fails to describe reality.

Concerning the strong correlations in 2D electron dynamics responsible for high 𝑇𝑐 superconductivity, we imagine a situation as depicted in Figure 7. Each electron’s spin in the plane interacts with the spin of its neighbors as follows. Equally directed electric fluxes (dually interpreted center fluxes of SU(2)𝑒) attract one another, and there is attraction for four out of six vortex sectors defined by the neighboring electrons (Ampere’s law), while two vortex sectors experience repulsion (that the electric charges of confining SU(2)𝑒 are seen by the photon is a consequence of the mixing between the corresponding two gauge groups: SU(2)𝑒 and SU(2)CMB [8, 9, 24]). Notice that for a given electron two of the adjacent electrons exhibit equal spin projection, while four of the adjacent electrons have opposite spins. This is in agreement with the observation that high 𝑇𝑐 superconductivity is an effect not related to 𝑠-wave Cooper-pair condensation.

Figure 7: An array of strongly correlated electrons in the plane possibly representing the superconducting state in a cuprate. The equally directed center flux in adjacent vortex sections provides for an attractive force (Ampere’s law) at intermediate distances. For a given electron, out of six neighboring vortex sectors there are four sectors with attraction and two sectors with repulsion. At short distance, there is repulsion since an overlap of CVL sections, leading to new intersection points each of mass βˆΌπ‘šπ‘’, is topologically forbidden. Thus there is a typical equilibrium configuration contributing to long-range order in the 2D system. If the externally provided resolution (temperature) falls below a critical value, then statistical fluctuations of the position of an intersection point relative to another one (the location of the electronic charge) will vanish. That is, electrons no longer can disperse energy provided by the heat bath (phonons; spin fluctuations) and thus provide a 2D material free of electric resistivity.

An overlap of vortex sectors, hypothetically leading to the creation of extra intersection points, is topologically forbidden. That is, the fluctuations in the energy density of the system are far too weak to create an intersection point of mass π‘šπ‘’=511keV. Therefore, at very small spatial separation repulsion must occur between adjacent vortex sectors. At a sufficiently low temperature and an optimal screening of Coulomb repulsion by the interplane environment (doping), this would lead to a typical equilibrium configuration as depicted in Figure 7 where the intersection points (electronic charge) do not move relative to one another. A local demolition of this highly ordered state would cost a finite amount of energy manifested in terms of the (gigantic) gaps measured experimentally in the cuprate systems. Applying an electric field vector with a component parallel to the plane would set into resistivity-free motion the thus locked electrons. For a macroscopic analogue, imagine a stiff table cloth being pulled over the table’s surface in a friction-free fashion. The occurrence of the pseudogap phase would possibly be explained by local defects in the fabric of Figure 7 due to insufficient Coulomb screening and/or too large thermal noise (macroscopic vorticity; liquid of point-like defects in 2D).


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


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