$U(1)$ Defects on Domain Lines

Based on recent experimental results, we give field-theoretic description of $U(1)$ defects localized on the domain lines on thin films. We describe topology of our model and solve this model in the adiabatic approximation. It turns out that such a model naturally provides periodic structure observed in experiment. The effective theory turns out to be the sine-Gordon model, but unlike the previous theoretical considerations we argue that in this case it is favorable for sine-Gordon kinks to merge into one defect with a uniform winding. We consider a system of adjacent domain lines and anti-lines and explain the experimental fact that the appearance of defects on a domain line prevents defect creation on the adjacent anti-lines. We also quantize the model and investigate possible effects of finite transverse dimension of the film.