Replica Symmetry Breaking in Renormalization: Application to the Randomly Pinned Planar Flux Array

The randomly pinned planar flux line array is supposed to show a phase transition to a vortex glass phase at low temperatures. This transition has been examined by using a mapping onto a 2D XY-model with random an\-iso\-tropy but without vortices and applying a renormalization group treatment to the replicated Hamiltonian based on the mapping to a Coulomb gas of vector charges. This renormalization group approach is extended by deriving renormalization group flow equations which take into account the possibility of a one-step replica symmetry breaking. It is shown that the renormalization group flow is unstable with respect to replica asymmetric perturbations and new fixed points with a broken replica symmetry are obtained. Approaching these fixed points the system can optimize its free energy contributions from fluctuations on large length scales; an optimal block size parameter $m$ can be found. Correlation functions for the case of a broken replica symmetry can be calculated. We obtain both correlations diverging as $\ln{r}$ and $\ln^2{r}$ depending on the choice of $m$.Comment: 14 pages, LaTeX, 1 uuencoded PostScript figure (accepted at 15 Nov 94 for publication in March 95 issue of J. Phys. I France

Duality mapping and unbinding transitions of semiflexible and directed polymers

Directed polymers (strings) and semiflexible polymers (filaments) are one-dimensional objects governed by tension and bending energy, respectively. They undergo unbinding transitions in the presence of a short-range attractive potential. Using transfer matrix methods we establish a duality mapping for filaments and strings between the restricted partition sums in the absence and the presence of a short-range attraction. This allows us to obtain exact results for the critical exponents related to the unbinding transition, the transition point and transition order.Comment: 7 pages, eq. (20) corrected in this submissio

Secondary polygonal instability of buckled spherical shells

When a spherical elastic capsule is deflated, it first buckles axisymmetrically and subsequently loses its axisymmetry in a secondary instability, where the dimple acquires a polygonal shape. We explain this secondary polygonal buckling in terms of wrinkles developing at the inner side of the dimple edge in response to compressive hoop stress. Analyzing the axisymmetric buckled shape, we find a compressive hoop stress with parabolic stress profile at the dimple edge. We further show that there exists a critical value for this hoop stress, where it becomes favorable for the membrane to buckle out of its axisymmetric shape, thus releasing the compression. The instability mechanism is analogous to the formation of wrinkles under compressive stress. A simplified stability analysis allows us to quantify the critical stress for secondary buckling. Applying this secondary buckling criterion to the axisymmetric shapes, we can determine the critical volume for secondary buckling. Our analytical result is in close agreement with existing numerical data