Quenched disorder in polymerized membranes: from anomalous elasticity to a glassy phase

Abstract

Polymerized membranes are paradigmatic fluctuating systems in which geometry, elasticity, and statistical fluctuations combine to produce anomalous long-distance behavior. In the disorder-free flat phase, bending rigidity increases at long wavelengths while in-plane elastic moduli soften. Quenched random metric and curvature introduce competing thermal and disorder-induced fluctuations. We review perturbative and nonperturbative field-theoretical approaches to these systems, from pure membranes to their disordered counterparts. Early one-loop studies identified a disorder-free fixed point, $P_4$, and a zero-temperature disorder-dominated fixed point, $P_5$. A subsequent nonperturbative renormalization-group analysis revealed a third fixed point, $P_c$, at finite temperature and disorder, governing a continuous wrinkling transition between the thermally dominated flat phase and a disorder-dominated glassy phase. Higher-order perturbative calculations also recover a $P_c$-like fixed point, with anomalous dimensions remarkably close to the reexpanded nonperturbative predictions. This scenario connects naturally with the wrinkling transition observed in partially polymerized lipid membranes. The three experimental scaling regimes of height fluctuations can be associated with $P_4$, $P_c$, and $P_5$, with theoretical exponents showing good quantitative agreement with measurements, particularly at criticality. Finally, we discuss the status of the glassy scenario, the unresolved discrepancy between perturbative and nonperturbative fixed-point stability predictions, and possible extensions to defective graphene and related two-dimensional crystalline materials.