Tattered Membranes and Constrained Magnets

We formulate a generalized D-dimensional field theory, parametrized by an O(d)×O(D) tensor field with an energetic longitudinal constraint, describing a new class of fluctuating "tattered" membranes, exhibiting a nonzero density of topological connectivity defects-slits, cracks, and faults at an effective medium level. For infinite-coupling constraint, the model reproduces the elastic membrane, with its rich anomalous elasticity realized by, e.g., graphene. Two additional fixed points emerge within the critical manifold: (i) globally attractive, "isotropic" O(d)×O(D), and (ii) "transverse," which in D=2 is the exact "dual" of the elastic membrane. Their properties are obtained in general D, d from the renormalization group and the self-consistent screening analyses. They correspond to critical points of an interesting class of constrained spin models.