Fluctuating nematic elastomer membranes

We study the flat phase of nematic elastomer membranes with rotational symmetry spontaneously broken by an in-plane nematic order. Such a state is characterized by a vanishing elastic modulus for simple shear and soft transverse phonons. At harmonic level, the in-plane orientational (nematic) order is stable to thermal fluctuations that lead to short-range in-plane translational (phonon) correlations. To treat thermal fluctuations and relevant elastic nonlinearities, we introduce two generalizations of two-dimensional membranes in a three-dimensional space to arbitrary D-dimensional membranes embedded in a d-dimensional space and analyze their anomalous elasticities in an expansion about $D=4.\mathrm{}$ We find a stable fixed point that controls long-scale properties of nematic elastomer membranes. It is characterized by singular in-plane elastic moduli that vanish as a power law ${\eta }_{\lambda }=4\mathrm{}-D$ of a relevant inverse length scale (e.g., wave vector) and a finite bending rigidity. Our predictions are asymptotically exact near four dimensions.