Defect formation in hot Abelian Higgs model

A. Rajantie
University of Sussex

Defect formation is believed to be relatively well understood in second-order symmetry breaking phase transitions. Zurek [1] has given a simple estimate for the produced defect density, based on the critical slowing down of the system near a second-order transition. If the temperature decreases rapidly enough, the fields fall out of equilibrium before reaching the critical point, and the correlation length of the order parameter freezes to a value
where tex2html_wrap_inline72 is the time scale related to the the temperature ``quench'', and tex2html_wrap_inline74 and tex2html_wrap_inline76 are the critical exponents related to the diverging of the correlation length and the relaxation time, respectively:
This freeze-out correlation length gives the average distance between the defects. In cosmology, one also gets a lower bound for the defect density from the fact that the correlation length cannot be larger than the horizon size.

In cosmology, however, the phase transitions are more likely to be related to a ``breakdown'' of local gauge symmetries, i.e. the Higgs mechanism, than a true spontaneous symmetry breakdown. In that case, the direction of the order parameter has no physical meaning and there is no correlation length that would be directly related to the final defect density. Therefore, Zurek's arguments, or even the causality argument, do not necessarily have to apply.

In the absence of any convincing analytical arguments, the only possible way to understand defect formation is to carry out real-time non-equilibrium simulations numerically. With M. Hindmarsh, we chose to study the Abelian Higgs model in the hard thermal loop approximation [2]. The equations of motion can be formulated in a local way by introducing two extra fields tex2html_wrap_inline80 and tex2html_wrap_inline82, which depend, in addition to the space-time coordinates, on an internal ``coordinate'' tex2html_wrap_inline84.

We took the initial conditions from a thermal ensemble in the Coulomb (symmetric) phase, and solved numerically the equations of motion, decreasing the bare mass parameter smoothly during the simulation so that eventually the system underwent a phase transition to the Higgs phase. The rate of change of tex2html_wrap_inline86 was parameterized by the quench time-scale tex2html_wrap_inline72. In the lattice we used, one dimension was much shorter than the others, but still longer than the correlation lengths. In the final state, we counted the number of string defects passing through the system in this short direction. This procedure was repeated for many different quench rates and initial configurations.

Figure 1:   Number of ``long'' strings, i.e. those that stretch through the short dimension of the lattice, in the final state as a function of the quench time-scale tex2html_wrap_inline72.

Our results show that the dependence of the final defect density on tex2html_wrap_inline72 is indeed consistent with a power law, but with a fairly small exponent tex2html_wrap_inline94 (see Fig. 1). In comparison, the value for a global theory, i.e. one without gauge fields, is much longer , tex2html_wrap_inline96[3]. This shows that the presence of the gauge field changes essentially this process. It remains an open question, whether there is a simple physical picture, like the Zurek picture for global theories, that can explain the density of defects formed in the transition.