A New Perspective on Thermally Fluctuating 2D Elastic Membranes: Introducing Odd Elastic Moduli and Non-Equilibrium Effects

Non-equilibrium and active effects in mesoscopic scale systems have heralded a new era of scientific inquiries, whether concerning meta-materials or biological systems such as bacteria and cellular components. At mesoscopic scales, experimental and theoretical treatments of membranes, and other quasi-two-dimensional elastic surfaces cannot generically ignore Brownian motion and other thermal effects. In this paper we aim to study the behavior of thermally fluctuating 2-D elastic membranes possessing odd elastic moduli embedded in higher dimensions. We implement an isotropic generalization of the elastic tensor that includes odd elastic moduli, $K_{odd}$ and $A_{odd}$, that break conservation of energy and angular momentum respectively, due to cite{scheibner2020odd}. Naturally this introduces active and non-equilibrium effects. Passive equilibrium thermalized elastic membranes possess effective (renormalized) Lam\'e coefficients that reduce with increasing system size and a diverging effective bending rigidity. Introducing two odd elastic moduli means that deformations from a reference state can induce chiral forces that cannot be derived from a Hamiltonian. Thus, the behavior of odd elastic membranes must instead be investigated with Langevin equations. If fluctuation-dissipation relations hold, we calculate via the renormalization group that at long length scales, active effects due to $K_{odd}$ can be effectively ignored whereas $A_{odd}$ cannot. To validate these findings, we developed an advanced force implementation methodology, inspired by the $(T)$-scheme prevalent in vertex models. This contributed to a new method for the simulation of elastic membranes in higher dimensions, as detailed recently in \cite{matoz2020wrinkle}. The novelty of the method is that microscopic/discrete and continuum in-plane elastic moduli are one-to-one and thus no coarse-graining is needed.