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Non-equilibrium and active effects in mesoscopic scale systems have heraldeda new era of scientific inquiries, whether concerning meta-materials orbiological systems such as bacteria and cellular components. At mesoscopicscales, experimental and theoretical treatments of membranes, and otherquasi-two-dimensional elastic surfaces cannot generically ignore Brownianmotion and other thermal effects. In this paper we aim to study the behavior ofthermally fluctuating 2-D elastic membranes possessing odd elastic moduliembedded in higher dimensions. We implement an isotropic generalization of theelastic tensor that includes odd elastic moduli, $K_{odd}$ and $A_{odd}$, thatbreak conservation of energy and angular momentum respectively, due tocite{scheibner2020odd}. Naturally this introduces active and non-equilibriumeffects. Passive equilibrium thermalized elastic membranes possess effective(renormalized) Lam\'e coefficients that reduce with increasing system size anda diverging effective bending rigidity. Introducing two odd elastic modulimeans that deformations from a reference state can induce chiral forces thatcannot be derived from a Hamiltonian. Thus, the behavior of odd elasticmembranes must instead be investigated with Langevin equations. Iffluctuation-dissipation relations hold, we calculate via the renormalizationgroup that at long length scales, active effects due to $K_{odd}$ can beeffectively ignored whereas $A_{odd}$ cannot. To validate these findings, wedeveloped an advanced force implementation methodology, inspired by the$(T)$-scheme prevalent in vertex models. This contributed to a new method forthe simulation of elastic membranes in higher dimensions, as detailed recentlyin \cite{matoz2020wrinkle}. The novelty of the method is thatmicroscopic/discrete and continuum in-plane elastic moduli are one-to-one andthus no coarse-graining is needed.

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