Skyrmions and disoriented domains in 2d-O(3) model

Gottfried Holzwarth

Attempts to describe the cooling of a hot hadronic gas through effective chiral field theories, (in the simplest case an O(4) model (3 pseudoscalar plus one scalar field)), start out from a random ensemble of field configurations (high temperature, short correlation length) and follow the evolution of each individual field configuration in real time. During the cooling process formation of topological defects and of disoriented ordered domains (DCC) takes place, with baryon-antibaryon production and anomalous pion multiplicities as potential corresponding experimental signatures.

In the effective action the dominant next-to-leading order term is the Skyrme term. Its strength determines the size of localized structures that appear as static solutions of the field equations in nontrivial topological sectors: skyrmions and bags. The Skyrme-Witten conjecture identifies the (integer) winding number B with baryon number.

I discuss in this talk the simpler case of the 2d-O(3) model. It has a similar topological structure, but here the Skyrme term alone is not sufficient to provide the defects with a finite size. In condensed matter applications the presence of the Zeeman coupling to an external magnetic field helps to set the scale. However, due to its explicitly violating the O(3)-symmetry this stabilizing mechanism prevents the formation of disoriented domains.

Another possibility to fix the size of defects, without breaking the symmetry is to include a potential for the modulus of the O(3)-field. The cooling process then is monitored by the change of this potential in time, in relation to typical relaxation times of the fluctuating field configuration. The simplest situation is realized in a sudden quench where the hot initial configuration is exposed at t=0 to the zero temperature potential and evolves without dynamical feedback towards an ordered low-energy configuration. In order to study the mutual interplay in the formation of defects and DCC's

we distinguish two types of evolutions: B-conserving and B-violating. As long as the modulus field does not take the value of zero at any point in space and time, conservation of B is topologically protected. During the field evolution the modulus will frequently pass through zero and defects will unwind to approach the lowest-energy topologically trivial configuration. By imposing a constraint on B (which is a global property of a configuration) we force the evolution to leave localized defects embedded in otherwise smooth fields. The size of these defects is determined by the coupling constants in the effective action, their density (depending on their size) emerges from the evolution.

While B is given by the integral over the winding density tex2html_wrap_inline24, the number of defects (particles plus antiparticles) may be defined as the integral over the absolute value of tex2html_wrap_inline24. Generally this is not an integer, but for configurations with well-developed separate defects it is close to an integer. The equal-time correlation functions reflect in their short- and long-range behaviour both the size of the emerging defects and the growth of disoriented domains during the evolution.

To implement the B-constraint into lattice simulations we have to provide a way to calculate B for a given configuration. We do this through a 'geodesic' approach which maps the elementary spatial lattice triangle onto the smaller one of the two spherical triangles which are cut out by the three image points on the tex2html_wrap_inline32 sphere on which the O(3)-vectors with unit length live. In the continuum limit this converges smoothly to the standard differential form of the winding density. For random configurations it implies that on one lattice unit cell we can have either zero or one defect with well-defined probability, but not more (similar to the Kibble mechanism in cosmology). The total winding B obtained in this way for some initial configuration then is preserved in each update step of the evolution which finally leads to a smooth configuration with that value of B.

The structure of the emerging defects is characterized by bag formation: large gradients in the angular variables (which imply large winding density) is preferably tied to small values of the modulus field (to reduce local energy) i.e. the charged matter gets concentrated inside localized bags. Depending on the coupling constants cluster formation may set in, where defects with several units of winding density sit inside a common bag.

To get an idea what sort of phenomenology we have to confront we demonstrate in an event-by-event analysis typical features of different evolutions.

All B-violating evolutions lead to a very small number of defects, (independent of the initial value of B), which finally will also unwind on much larger time scales (depending on the Metropolis algorithm). Once the particle number has dropped to a few units, the size of aligned domains in the embedding field approaches the lattice size.

In B-conserving evolutions (which are of interest due to their possible implications for the hadronic 3d-O(4) case) defects can disappear only via annihilation processes which happen preferably for overlapping defects. Therefore the annihilation rate is sensitive to their size. If their size is of the order of the lattice unit then the evolution leaves a large number of them embedded in the smooth background (even for B=0 configurations) and the size of aligned domains saturates at a finite value (inversely proportional to the sqrt of the defect density). This result is not very sensitive to the actual value of B, as long as B is much smaller than the initial number of randomly produced defects.

If the defect size is increased by about one order of magnitude then annihilation dominates the B-violating unwinding such that for B=0 configurations both types of evolutions proceed in a quite similar pattern: both types eventually lead to fully aligned configurations with all defects eliminated.

The result of B-conserving evolutions then, however, depends sensitively on the actual value of B, because in the end the number of defects will be equal to B (with all antiparticles eliminated) and, correspondingly, the size of aligned domains then reflects B. or the number of clusters in which the total B is split up.

Evidently, already this most simple case of a sudden quench is of remarkable complexity, and indicates that descriptions in terms of spatially constant mean fields may not be adequate. Dynamical feedback from a time-dependent potential is naturally included. Depending on the ratio of the corresponding time scale to the typical relaxation time it leads to reheating and particle production. Sampling the initial configurations according to an appropriate statistical ensemble we obtain in this way the final particle density (and its variance) as function of quench time. Information about the spectrum of elementary field modes may be obtained from analysing the time-dependent configurations in momentum space.