Dirt Softens Soap: Anomalous Elasticity of Disordered Smectics

We show that a smectic in a disordered medium (e.g., aerogel) exhibits anomalous elasticity, with the compression modulus B(k) vanishing and the bend modulus K(k) diverging as k --> 0. In addition, the effective disorder develops long ranged correlations. These divergences are much stronger than those driven by thermal fluctuations in pure smectics, and are controlled by a zero temperature glassy fixed point, which we study in an $\epsilon=5-d$ expansion. We discuss the experimental implications of these theoretical predictions.