Historical research guide · 1089 · 2012
Center-Vortex Loops with One Self-Intersection
Julian Moosmann · Ralf Hofmann
In one minute
Julian Moosmann and Ralf Hofmann take a picture of the electron that goes back to Lord Kelvin — not a point, but a knotted line of flux in the ground state — and make it calculable. In the confining phase of an SU(2) Yang–Mills theory the ground state is threaded by closed magnetic center-vortex loops. Twist one so that it crosses itself once and a single magnetic charge is trapped at the crossing; set the theory’s energy scale to the electron mass and almost the whole mass of the object sits at that crossing. The authors then ask what such a loop looks like as you view it more coarsely. Coarse-graining is modelled as curve shrinking, the loop evolving like a curve under a heat equation, and demanding that the statistical weight of an ensemble of loops be unchanged by that flow gives the flow of an effective action. The result is a surprise: the ensemble orders itself. Entropy falls to zero at finite resolution, and they connect that to two-dimensional high-temperature superconductivity in iron-arsenide layers.
Connection
Chapter 2 asks what the vacuum actually is, and this paper is one of the few places where a concrete, calculable answer is offered — a ground state with topology in it, whose knots are the particles rather than something added to it. For chapter 13 it supplies the unified move in miniature: one structure in the ground state, viewed at different resolutions, gives a lepton at one scale and a spontaneously ordered, dissipation-free two-dimensional electron system at another.
From the source
In the confining phase of an SU(2) Yang–Mills theory the ground state carries closed magnetic center-vortex loops; twisting a loop until it crosses itself once captures an isolated, spinning magnetic antimonopole in the core of the intersection, and this configuration is topologically distinct from the non-self-intersecting sector because its rotation number cannot be smoothly deformed to zero.Section 2.1, Self-Intersecting Center-Vortex Loops; Figure 2, panels a to d
Setting the Yang–Mills scale of the SU(2) theory equal to the electron mass of 511 keV fixes the mass of the intersection point, where practically the entire mass of the soliton resides; the authors then read the one-fold self-intersecting loop as an electron or a positron, with a two-fold degeneracy in the direction of center flux that a static electric or magnetic background lifts as a two-fold spin degeneracy.Section 1, Introduction
Because both wings of center flux have finite size, the position of the intersection point can be shifted at almost no cost in energy; when the inner angle between incoming and outgoing center flux is small enough, motion of points on the vortex line perpendicular to the bisector of that angle carries the intersection point faster than light, at the velocity of those points times the cotangent of half the angle — a motion the authors note the path-integral formulation of quantum mechanics already admits, with such trajectories contributing sizeably to transition amplitudes.Section 1, Introduction; Figure 1
Spatial coarse-graining of a center-vortex loop is modelled as a curve-shrinking flow in a dimensionless resolution parameter, each point of the curve moving along the inward normal at a rate set by the local curvature; requiring the partition function over an ensemble of such curves to be invariant under that flow yields the renormalization-group evolution of an effective action, written as an isoperimetric conformal factor times a nonconformal factor expanded in inverse powers of length or area.Sections 2.2 to 2.4, Equations 2.1 to 2.13
Numerically evolving ensembles of up to sixteen self-intersecting curves, all normalised to the same initial area, the variance of the location of the self-intersection first grows, reaches a maximum and then falls to zero at a finite value of the flow parameter — in contrast with the non-self-intersecting sector, where the variance of the centre of mass saturates at a finite value.Section 3.3; Figures 9 and 10
The entropy of the ensemble approaches zero continuously at finite resolution, so that one member of the ensemble is singled out with weight approaching unity: order emerges spontaneously as resolution decreases, no energy supplied by the environment can be dissipated through the monopole in the intersection core below a critical resolution, and the authors propose this as the connection to two-dimensional high-transition-temperature superconductivity in the FeAs systems.Section 3.4 and Section 4; Figure 11
The way in
https://doi.org/10.5402/2012/601749How to cite it
Julian Moosmann, Ralf Hofmann (2012) Center-Vortex Loops with One Self-Intersection doi:10.5402/2012/601749