Anomalous elasticity of nematic and critically soft elastomers

Uniaxial elastomers are characterized by five elastic constants. If their elastic modulus C5 describing the energy of shear strains in planes containing the anisotropy axis vanishes, they are said to be soft. In spatial dimensions d less than or equal to 3, soft elastomers exhibit anomalous elasticity with certain length-scale-dependent bending moduli that diverge and shear moduli that vanish at large length scales. Using renormalized field theory at d=3 and to first order in epsilon=3-d, we calculate critical exponents and other properties characterizing the anomalous elasticity of two soft systems: (i) nematic elastomers in which softness is a manifestation of a Goldstone mode induced by the spontaneous symmetry breaking associated with a transition from an isotropic state to a nematic state, and (ii) a particular version of what we call a critically soft elastomer in which C(5)=0 corresponds to a critical point terminating the stability regime of a uniaxial elastomer with C5>0.