suppSupplemental References

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

M.E.H. Bahri<EMAIL_ADDRESS>Department of Mechanical and Aerospace
Engineering, Princeton University, Princeton, NJ 08544, USA  S. Sarkar
<EMAIL_ADDRESS>Department of Physics, University of Michigan, Ann Arbor,
MI 48103, USA  D.A. Matoz-Fernandez<EMAIL_ADDRESS>Department of Theoretical
Physics, Complutense University of Madrid, 28040 Madrid, Spain  A. Košmrlj
<EMAIL_ADDRESS>Department of Mechanical and Aerospace Engineering,
Princeton University, Princeton, NJ 08544, USA  Princeton Materials Institute,
Princeton University, Princeton, NJ 08544, USA

(July 2023)

###### Abstract

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 Scheibner et
al. Scheibner  _et al._ (2020). Naturally this introduces active and non-
equilibrium effects. Passive equilibrium thermalized elastic membranes possess
effective (renormalized) Lamé coefficients that reduce with increasing system
size and a diverging effective bending rigidity Aronovitz and Lubensky (1988);
Le Doussal and Radzihovsky (1992). 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 Matoz-Fernandez  _et
al._ (2020). The novelty of the simulation method is that microscopic/discrete
and continuum in-plane elastic moduli are one-to-one and thus no coarse-
graining is needed.

Introduction. Elasticity has a long history and for much of it, has been
typically studied in the presence of general laws and symmetries Landau  _et
al._ (2012); Timoshenko and Woinowsky-Krieger (1959). With the development of
the Vicsek model and the Toner-Tu equations, the study of active systems has
transformatively shifted the field of non-equilibrium physics to include
active systems Vicsek  _et al._ (1995); Toner and Tu (1995). This shift has
flung open the door to the burgeoning field of active systems, where long-
standing interpretations of laws or symmetries, such as conservation of
energy, are being re-examined and cast in a new light. For an interesting
discussion of the history of active systems, which does go earlier than the
aforementioned papers, see Bowick  _et al._ (2022).

Such new considerations have also led to a vast new set of experimental
realizations and applications. One way to incorporate active effects is via
the inclusion of non-conservative forces. For example, a synthetically
engineered active elastic system was constructed out of polycrystals of
magnetic colloidal spinners Bililign  _et al._ (2022). Within the study, odd
elastic forces were found to organize the colloids into a polycrystalline
phase with motile dislocations whereby each grain has a tunable rotation rate
and size. Odd elastic forces are not conservative but are only present when
the system is deformed from a reference configuration Scheibner  _et al._
(2020). Another example includes robotic metabeams with piezoelectric
properties that can result in odd micropolar elasticity Chen  _et al._ (2021).
Such a system can convert mechanical and electrical energy into one another
via bending/shearing cycles as well as produce direction-dependent bending
rigidities. Additionally, work- generating limit cycles via active forces have
been exploited to produce robotic elements Brandenbourger  _et al._ (2021).
Within natural phenomena, active elasticity is also relevant for biological
systems. Analysis of already existing data of muscular hydraulics were found
to exhibit odd elastic behavior Shankar and Mahadevan (2022). In addition, new
phases of active chiral biological matter were obtained in the form of
spontaneously formed crystals consisting of starfish embryos Tan  _et al._
(2022). By examining the local shear-elongation angle in the vicinity of
defects, potential signatures of odd elasticity were also observed. In
addition, theoretically, active elastic surfaces have been generalized via
symmetry arguments and explored Salbreux and Jülicher (2017); Fossati  _et
al._ (2022), with particular emphasis on wave dynamics. One can also, for
example, model the mechanics of the actin cortex, a layer of cross-linked
actin that lies beneath the plasma membrane of animal cells Berthoumieux  _et
al._ (2014); Salbreux  _et al._ (2012). The activity of this mesh arises from
the myosin motors that exert contractile forces. More specifically amongst
animal cells, the mechanical behavior of human red blood cells is a
significant topic of interest. At these length scales thermal effects are
relevant to the fluctuations and mechanics of membranes Gov (2007); Nelson
_et al._ (2004). Active and non-equilibrium effects such as ATP concentration
softening the local shear modulus of the cytoskeleton have been confirmed,
leading to active height correlations that differ from thermal equilibrium
height correlations Gov (2007); Park  _et al._ (2010). This softening effect,
due to metabolic activity, can induce interesting curvature-mediated active
membrane motions Turlier  _et al._ (2016). Given the possibility that odd
elastic forces could be present in many biological systems such as the
aggregates of star-fish embryos and the cytoskeleton Salbreux and Jülicher
(2017); Tan  _et al._ (2022); Floyd  _et al._ (2022), it is worthwhile to
consider what the behavior of thermalized odd elastic membranes may be. This
paper aims to provide a first initial understanding of this topic.

We use a renormalization-group analysis along with Langevin-dynamic
simulations to investigate the long-range equal-time correlations of the
displacement fields of 2-D solvent-permeable isotropic odd elastic membranes
(which are necessarily chiral) at some non-zero temperature. These membranes
are characterized by Lamé coefficients ($\lambda,\mu$), a bending rigidity
($\kappa$) and two odd elastic constants ($A_{odd},K_{odd}$)Scheibner  _et
al._ (2020). Both odd elastic moduli break reflection symmetry, and more
specifically $K_{odd}$ breaks conservation of energy and couples pure and
simple shears whereas $A_{odd}$ breaks conservation of angular momentum and
couples dilation strains to torques. Finally, despite the fact that we
performed theoretical calculations for permeable membranes (Rouse dynamics),
we expect our scaling results to hold even in the case of impermeable
membranes (Zimm dynamics), where the membrane is not completely permeable to
the solvent and is modeled with Fourier vector dependent diffusivities.

Model and Theoretical Results We seek to describe a $D=2$ isotropic odd
elastic system embedded in $D+d_{c}$ dimensions (i.e. with $d_{c}$ co-
dimensions) at a temperature $T$ Scheibner  _et al._ (2020). In the absence of
thermal fluctuations and deformations, the reference configuration is that of
a flat undeformed elastic sheet. A schematic showing the elastic system under
non-zero $T$ conditions is shown in Fig. 1. Though such an active elastic
system cannot be described by a Hamiltonian, the constitutive equation for in-
plane stresses ($\sigma_{ij}$) and strains ($u_{ij}$) is still applicable
$\sigma_{ij}=\mathcal{C}_{ijkl}u_{kl}$ where

$\begin{split}\mathcal{C}_{ijkl}=&\lambda\delta_{ij}\delta_{kl}+\mu[\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}]\\\
&+K_{odd}\mathcal{E}_{ijkl}-A_{odd}\epsilon_{ij}\delta_{kl},\end{split}$ (1)

where we have assumed an orthonormal coordinate basis and with
$\mathcal{E}_{ijkl}=\frac{1}{2}[\epsilon_{ik}\delta_{jl}+\epsilon_{il}\delta_{jk}+\epsilon_{jk}\delta_{il}+\epsilon_{jl}\delta_{ik}]$.
$\epsilon$ is the 2-D Levi-Civita tensor
($\epsilon_{11}=\epsilon_{22}=0,\epsilon_{12}=-\epsilon_{21}=1$) and $\delta$
is the Kronecker delta. In addition, $\lambda,\mu$ are the 2-D Lamé
coefficients. We define $u_{i}(\mathbf{r},t)$ to be the displacement vector
along $i$-th axis (in-plane phonon) and
$f^{\alpha}(\mathbf{r},t);\alpha\in\\{1,...,d_{c}\\}$ to be the out-of-plane
displacement (flexural phonon). In the Monge gauge the strain tensor takes the
form:

$\begin{split}u_{ij}(\mathbf{r},t)=&\frac{1}{2}[\partial_{i}u_{j}(\mathbf{r},t)+\partial_{j}u_{i}(\mathbf{r},t)\\\
&+\partial_{i}f^{\alpha}(\mathbf{r},t)\partial_{j}f^{\alpha}(\mathbf{r},t)].\end{split}$
(2)

From these, a system of over-damped Langevin equations can be written down:

$\partial_{t}u_{j}(\mathbf{r},t)=\mathcal{D}\partial_{i}\\{\mathcal{C}_{ijkl}u_{kl}(\mathbf{r},t)\\}+\eta_{j}(\mathbf{r},t),$
(3)

where the Langevin noise satisfies $\langle\eta_{j}(\mathbf{r},t)\rangle=0$
and
$\langle\eta_{j}(\mathbf{r},t)\eta_{i}(\mathbf{r}^{\prime},t^{\prime})\rangle=2\mathcal{L}_{ij}k_{B}T\delta(\mathbf{r}-\mathbf{r}^{\prime})\delta(t-t^{\prime})$
and we assume $\mathcal{L}_{ij}=\mathcal{L}\delta_{ij}$. $\mathcal{D}$ is the
in-plane mobility and is equal to $\mathcal{L}$ if fluctuation-dissipation
holds. We may also write a Föppl-von-Kármán equation describing the dynamics
of flexural modes:

$\begin{split}\partial_{t}f^{\alpha}(\mathbf{r},t)=&\mathcal{D}_{f}[-\partial_{i}\partial_{j}\\{\mathcal{B}_{ijkl}\partial_{k}\partial_{l}f^{\alpha}(\mathbf{r},t)\\}\\\
&+\partial_{i}\\{\sigma_{ij}(\mathbf{r},t)\partial_{j}f^{\alpha}(\mathbf{r},t)\\}]+\eta_{f}^{\alpha}(\mathbf{r},t),\end{split}$
(4)

where
$\mathcal{B}_{ijkl}=\frac{\kappa}{2}\delta_{ij}\delta_{kl}+\frac{\kappa}{4}[\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}]$
is the bending rigidity tensor. $\mathcal{D}_{f}$ is the out-of-plane mobility
and is equal to $\mathcal{L}_{f}$ if fluctuation-dissipation relations hold.
One may fairly ask why an odd elastic component has not been incorporated into
this bending rigidity tensor but by noticing the symmetrized contraction in
$\mathcal{B}_{ijkl}\partial_{i}\partial_{j}\partial_{k}\partial_{l}$, no anti-
symmetric components will remain. Correspondingly the noise term satisfies
$\langle\eta_{f}^{\alpha}(\mathbf{r},t)\rangle=0$ and
$\langle\eta_{f}^{\alpha}(\mathbf{r},t)\eta_{f}^{\beta}(\mathbf{r}^{\prime},t^{\prime})\rangle=2\mathcal{L}_{f}k_{B}T\delta^{\alpha\beta}\delta(\mathbf{r}-\mathbf{r}^{\prime})\delta(t-t^{\prime})$
where $\delta^{\alpha\beta}$ is the Kronecker delta. We additionally assume
that
$\langle\eta_{f}^{\alpha}(\mathbf{r},t)\eta_{j}(\mathbf{r}^{\prime},t^{\prime})\rangle=0$.
To comprehend the effect of thermal fluctuations on the effective elastic
moduli, we use the Martin-Siggia-Rose-Janssen-DeDominicis formalism De
Dominicis (1978); Janssen (1979); Martin  _et al._ (1973); Täuber (2014); Frey
and Nelson (1991). The formalism adopts the path-integral formulation by
taking advantage of the form of the distribution of the thermal noises. Thus
the transition probability density is of the form:

$\begin{split}\mathcal{W}(\eta_{j},\eta_{f}^{\alpha})\propto&e^{-\frac{1}{4}\int
dt\int d^{d}\mathbf{r}(k_{B}T\mathcal{L})^{-1}\eta_{i}(\mathbf{r},t)^{2}}\\\
&e^{-\frac{1}{4}\int dt\int
d^{d}\mathbf{r}(k_{B}T\mathcal{L}_{f})^{-1}\eta_{f}^{\alpha}(\mathbf{r},t)^{2}}.\end{split}$
(5)

In this form, inserting in the Langevin equations, Eq. (4)and Eq. (3), renders
the expression into a complicated set of terms with high-degree non-
linearities. Thus, via an imaginary Hubbard-Stratonovich transformation
Altland and Simons (2010) we introduce the following response non-physical
variables and linearize the Langevin noises:

$\begin{split}\mathcal{W}(\eta_{j},\eta_{f}^{\alpha})\propto&\int\prod_{i}D[i\Upsilon_{i}]\prod_{\alpha}D[i\Phi^{\alpha}]\\\
&e^{\int dt\int
d^{d}\mathbf{r}[k_{B}T\mathcal{L}\Upsilon_{i}(\mathbf{r},t)^{2}-\Upsilon_{i}(\mathbf{r},t)\eta_{i}(\mathbf{r},t)]}\\\
&e^{\int dt\int
d^{d}\mathbf{r}[k_{B}T\mathcal{L}_{f}\Phi^{\alpha}(\mathbf{r},t)^{2}-\Phi^{\alpha}(\mathbf{r},t)\eta_{f}^{\alpha}(\mathbf{r},t)]},\end{split}$
(6)

where the integral represents a functional integration Tong . Assuming
periodic boundary conditions, one may then take a Fourier transform of the
MSRJD action seen in Eq. 6. Using the MSRJD action, and keeping terms only to
linear order, we may calculate the equal-time correlations (known as a
harmonic/Gaussian average) of our displacement fields to obtain:

$\langle
f^{\alpha}(\mathbf{q})f^{\beta}(-\mathbf{q})\rangle=\frac{k_{B}T\mathcal{L}_{f}\delta_{\alpha\beta}}{A\mathcal{D}_{f}\kappa
q^{4}},$ (7) $\begin{split}\langle
u_{i}(\mathbf{q})u_{j}(-\mathbf{q})\rangle=&\frac{k_{B}T\mathcal{L}}{A\mathcal{D}[\lambda+3\mu][\mu(\lambda+2\mu)+K_{odd}(K_{odd}-A_{odd})]q^{4}}\\\
&\\{q_{i}q_{j}[\mu(\lambda+3\mu)+K_{odd}(2K_{odd}-A_{odd})]\\\
&+[\epsilon_{ik}q_{j}q_{k}+\epsilon_{jl}q_{i}q_{l}][\mu
A_{odd}+(\lambda+\mu)K_{odd}]\\\
&+\epsilon_{ik}\epsilon_{jl}q_{k}q_{l}[(\lambda+2\mu)(\lambda+3\mu)+(K_{odd}-A_{odd})(2K_{odd}-A_{odd})]\\},\end{split}$
(8)

where $A$ is the area of our system. These calculations are performed in Sec.
III of the supplementary material. Active effects cannot be observed in the
out-of-plane correlations Eq. 7. One may ask if there is a correlation that is
otherwise zero in the absence of odd elastic moduli. The solution is in-plane
correlations of the type: $\langle
u_{1}(q_{1},0)u_{2}(-q_{1},0)\rangle=k_{B}T\mathcal{L}[A_{odd}\mu-
K_{odd}(\lambda+\mu)]/\\{\mathcal{D}q^{2}(\lambda+3\mu)(A_{odd}K_{odd}+K_{odd}^{2}+\mu(\lambda+\mu))\\}$.
This will be of use when extracting the scaling of the effective odd elastic
moduli from our simulation data.

Previous equilibrium studies, utilizing Boltzmann weights, established that
due to the geometric non-linearity of the strain tensor, transverse shear
modes generate a long-range coupling of Gaussian curvatures and thus the
existence of a low-temperature flat phase for such elastic surfaces was
established in Nelson and Peliti (1987). The long-range coupling is what
permits such 2-D systems to evade the Hohenberg-Mermin-Wagner theorem (which
states that continuous symmetries cannot be spontaneously broken in two and
less dimensions) Mermin and Wagner (1966); Mermin (1968); Hohenberg (1967);
Halperin (2019). Furthermore, via thermal fluctuations, the geometric non-
linearity of the strain tensor renormalizes the moduli of the theory when
fluctuations of the membrane become of the order of its thickness Nelson and
Peliti (1987). Further studies followed and established the presence of a
globally stable non-Gaussian fixed point named the Aronovitz-Lubensky fixed
point, as well as refined anomalous exponents for the moduli via a self-
consistent screening analysis (SCSA) Aronovitz and Lubensky (1988); Le Doussal
and Radzihovsky (1992). More specifically it was obtained that beyond the
thermal length scale ($\ell_{\text{th}}=\sqrt{16\pi^{3}\kappa^{2}/3k_{B}TY}$,
where $\kappa$ is the bending rigidity, $k_{B}$ is the Boltzmann constant, $T$
is the temperature and $Y$ is the 2-D Young’s modulus) the effective moduli of
the material renormalize with system size, $L$, as $\kappa^{R}(L)\sim
L^{\eta}$ ($\eta\approx 0.8$) whereas the Lamé coefficients scale as
$\lambda^{R}(L),\mu^{R}(L)\sim L^{-\eta_{u}}$ ($\eta_{u}\approx 0.4$).

One theoretical study utilizing our same model sans odd elastic moduli, has
explored the Langevin dynamics of elastic membranes with the addition of
hydrodynamic coupling when such membranes may or may not be permeable to the
surrounding solvent (Rouse and Zimm dynamics respectively) Frey and Nelson
(1991). While there are dynamical differences between the two cases, in the
study it was found that, regardless of whether the membranes are permeable or
not, the static correlations of the displacement fields are in agreement with
the equilibrium theory using the Boltzmann measure.

Importantly, however, the MSRJD action has one notable difference from the
analogous Boltzmann weight associated with an equilibrium elastic membrane
Hamiltonian such as in Aronovitz and Lubensky (1988); Nelson and Peliti
(1987); Nelson  _et al._ (2004); Guitter  _et al._ (1988). The MSRJD action
does not possess the quasi-rotational symmetry given in Guitter  _et al._
(1989). The absence of this symmetry is significant as this means that the
Ward identity, that would be derived from this symmetry, is also absent. Note
that this difference holds even in the absence of activity and is thus a
property of the Langevin dynamics. Thus, the form of the strain tensor is not
necessarily preserved through the renormalization group. As an example, if one
inserts Eq. 2 into Eq. 3, it may be noted that there are the linear and non-
linear terms both with coefficient $\mathcal{D}\mathcal{C}_{ijkl}$. Though we
may set these two coefficients to be equal at a microscopic length scale,
there is no symmetry that says that this necessarily need be the case at
larger length scales. In other words, the lack of this Ward identity may lead
to the breaking of fluctuation-dissipation relations at larger length scales.
Such a scenario, where the symmetry of the strain tensor is broken, has also
been treated in Le Doussal and Radzihovsky (2021). To achieve this, the
isotropy of the embedding space was broken via an external field to break the
form of the strain tensor explicitly. These perturbations will generically
result in a renormalization-group flow directed away from the Aronovitz-
Lubensky fixed point. More details are expanded upon in Sec. III of the
supplementary material. However, for the purposes of this study, where we
performed a performed 1-loop order $1/d_{c}$ renormalization group expansion,
if microscopic fluctuation-dissipation relations hold and the form of the
strain tensor is not violated microscopically, then they hold via
renormalization as well (one may think of this as an unstable but fixed
manifold).

We furthermore mention that by restricting to the Feynman diagrams we used
(via the $1/d_{c}$ analysis expanded upon in the supplementary material) in
our theoretical analysis, $A_{odd}$ and $K_{odd}$ do not generate each other.
Thus they may be considered as independent parameters when tuning bare moduli
such that either of the odd moduli is zero.

Figure 1: A triangular lattice, with periodic boundary conditions, is
implemented with fixed connectivity for Langevin simulations performed with a
Berendsen barostat. The color field in the schematic indicates the height
field with blue indicating lower portions of the elastic sheet and orange
indicating higher portions. The vertices associated with each triangular face
are assigned an ID $\\{1,2,3\\}$ in a counter-clockwise fashion. Reference
coordinates are referred to as $\\{\bar{\mathbf{r}}_{i}\\}$ where $i$ marks
the ID of the vertex for the triangular face. $\\{\mathbf{r}_{i}\\}$ marks the
current positions of the vertices. These coordinates may be used to calculate
the reference and current metrics and thereby the strain tensor for each
triangular face. The dual in-plane stresses for each triangular face were then
calculated via the constitutive relation whereas bending forces were
implemented using dihedral springs.

With these considerations we numerically integrate our renormalization group
equations and performed a stability analysis of the Aronovitz-Lubensky fixed
point explained further in the SI. Via our analyses, we find that in the
vicinity of the Aronovitz-Lubensky fixed point, $K_{odd}$ is an irrelevant
perturbation. Perturbations of non-zero $K_{odd}$, that preserve fluctuation-
dissipation relations, converge back to the Aronovitz-Lubensky fixed point
with exponent $K_{odd}^{R}(L)\sim L^{-2\eta_{u}}$. Thus
$\lambda^{R}(L),\mu^{R}(L)\sim L^{-\eta_{u}}$ and $\kappa^{R}(L)\sim L^{\eta}$
still hold. On the other hand, perturbations to non-zero $A_{odd}$, that
preserve fluctuation-dissipation relations, are marginal and thus expand the
Aronovitz-Lubensky fixed point to a higher-dimensional manifold. Thus,
$A_{odd}^{R}(L)\sim L^{-\eta_{u}}$, $\lambda^{R}(L),\mu^{R}(L)\sim
L^{-\eta_{u}}$ and $\kappa^{R}(L)\sim L^{\eta}$ still hold. In addition, in
the supplementary material, we also derive a new form for the thermal length
scale, $\ell_{\text{th}}$ that matches with the classical equilibrium form,
$\sqrt{16\pi^{3}\kappa^{2}/3k_{B}TY}$, in the absence of the odd-elastic
moduli and when fluctuation-dissipation relations hold. The new form of
$\ell_{\text{th}}$ can be found in Eq. (S105) in the supplementary material.
Thus, for systems sufficiently larger than the thermal length scale and
assuming fluctuation-dissipation relations, $K_{odd}^{R}$ can be ignored but
$A_{odd}^{R}$ cannot, with regard to the effective mechanics of the sheet.
Aside from the renormalization of equal-time correlations, from which we
derive the scaling of our elastic moduli, dynamic renormalization of the noise
variance does not occur Frey and Nelson (1991). Thus dynamic critical
exponents can be completely determined from our static renormalization
results, that is the scaling of our elastic moduli derived from equal-time
correlations.

Finally, as far as fluctuation-dissipation relations hold, we make a remark
that despite the fact that our results hold for completely permeable
membranes, known as Rouse dynamics, these results should hold even in the case
of Zimm dynamics where diffusivities are given a power dependence of the
Fourier vector magnitude such as in Frey and Nelson (1991). This is because
inserting a power of the Fourier vector magnitude into the diffusivities will
multiply all terms in our theoretical calculation (aside from the temporal
frequency, which is not relevant to equal-time correlations) and thus relative
ratios of these terms remains the same as in Rouse dynamics. This will just
shift the scaling of all the correlation functions by the same factor and thus
relative scaling between elastic moduli will not change.

However, the equal-time correlation results change if fluctuation-dissipation
relations do not hold microscopically, then we find the Aronovitz-Lubensky
fixed point unstable. Thus, the Aronovitz-Lubensky fixed point does not
characterize elastic membranes without fluctuation-dissipation relations. In
the absence of odd elastic moduli, the governing fixed point was obtained in
Le Doussal and Radzihovsky (2021). In the presence of odd elastic moduli, the
governing fixed point is yet to be found. For more on the complete theoretical
results please refer to Sec. III of the supplementary material.

Simulations Set-Up. To compare and contrast the theoretical results, we also
conducted Langevin simulations using following Ref. (Matoz-Fernandez  _et al._
, 2020). We introduced a new force implementations, explained below, yielding
a novel simulation technique for discrete membranes embedded in higher
dimensional spaces. Specifically, we simulated a 2-D elastic membrane embedded
in 3 dimensions so that $d_{c}=1$ under the influence of thermal noise.
Simulations were performed non-dimensionally so $k_{B}$, the mass of the
vertices and the lattice spacing were set to $1$. A Gaussian random noise
associated with a variance of the temperature, $T$, is produced by a random
number generator and acts as our Langevin noise. The temperature is a
particularly important parameter for our simulations because rather than
simulating a large system size, which can be computationally costly, we can
fix the system size, $L$, and instead change the temperature, which will
change the thermal length scale $\ell_{\text{th}}$ (thus allowing us to tune
$L/\ell_{\text{th}}$). Measuring equal-time correlation functions of the
displacements, we can then establish anomalous exponents of the effective
moduli. This is expanded upon further below. The specific physical system we
implemented was a triangular lattice with fixed connectivity under periodic
boundary conditions. For the integrator, we implemented a BAOAB-limit method
Leimkuhler and Matthews (2013), thus the simulation noise is not completely
memory-less, but the memory is short ranged in time. We used a Berendsen
barostat to tune the pressure of the system to zero; allowing us to tune the
system to its critical point (without a homogeneous stress). We used dihedral
springs to implement a bending rigidity for the elastic membrane Seung and
Nelson (1988). We set the continuum bending rigidity to $1$ as well. To
calculate in-plane forces, for each triangular face, the vertices that make up
the corners are assigned an ID $\\{1,2,3\\}$ with a consistent handedness
(counter-clockwise). Thus the same vertex can have a different ID with regards
to different faces. $\\{\bar{\mathbf{r}}_{i}\\}$ and $\\{\mathbf{r}_{i}\\}$
mark the reference and current positions of the vertices where $i$ is the ID
of the vertex within the triangular face in question. A schematic of this is
shown in Fig. 1, where the reference basis $\bar{\mathbf{e}}_{i}$ and current
basis $\mathbf{e}_{i}$ are defined in terms of the coordinates of the
vertices. We used the reference metric tensor,
$\bar{g}_{ij}(\mathbf{r},t)=\bar{\mathbf{e}}_{i}(\mathbf{r},t)\cdot\bar{\mathbf{e}}_{j}(\mathbf{r},t)$,
and current metric tensor,
$g_{ij}(\mathbf{r},t)=\mathbf{e}_{i}(\mathbf{r},t)\cdot\mathbf{e}_{j}(\mathbf{r},t)$,
to calculate the strain tensor
$u_{ij}(\mathbf{r},t)=[g_{ij}(\mathbf{r},t)-\bar{g}_{ij}(\mathbf{r},t)]/2$. We
will drop the $(\mathbf{r},t)$ argument now as it is implicit. As an aside, we
will use notation of differential geometry. For example, the dual of a general
tensor, tensor $J_{ij}$ is written down as $J^{ij}$ and is obtained via the
following contraction: $J^{ij}=g^{ik}g^{jl}J_{kl}$ where
$g^{ij}\equiv(g^{-1})_{ij}$ (meaning that the dual of the metric tensor is its
inverse; note that this property does not hold necessarily for other tensors)
Do Carmo (2016). Thus, in general, we use the current local metric tensor to
raise and lower indices on a general tensor. Returning to the stress tensor
calculation, we used the constitutive relation ($\sigma^{ij}=C^{ijkl}u_{kl}$)
via the generalized linear elastic tensor:

$\begin{split}\mathcal{C}^{ijkl}=&\lambda\bar{g}^{ij}\bar{g}^{kl}+\mu[\bar{g}^{ik}\bar{g}^{jl}+\bar{g}^{il}\bar{g}^{jk}]\\\
&+\frac{K^{odd}}{\sqrt{\det[\bar{g}]}}\mathcal{E}^{ijkl}-\frac{A_{odd}}{\sqrt{\det[\bar{g}]}}\epsilon^{ij}\bar{g}^{kl},\end{split}$
(9)

with
$\mathcal{E}^{ijkl}=\frac{1}{2}[\epsilon^{ik}\bar{g}^{jl}+\epsilon^{il}\bar{g}^{jk}+\epsilon^{jk}\bar{g}^{il}+\epsilon^{jl}\bar{g}^{ik}]$,
where $\epsilon^{ij}$ is again the 2-D Levi-Civita permutation symbol
satisfying $\epsilon^{11}=\epsilon^{22}=0,\epsilon^{12}=-\epsilon^{21}=1$.

Figure 2: Plots of the non-dimensionalized (a) flexural correlation function
($C_{ff}=k_{B}T/(A\kappa q_{\text{th}}^{4})$), (b) longitudinal in-plane
phonon correlation function
($C_{u_{1}u_{1}}=k_{B}T(2K_{odd}^{2}+\mu(\lambda+3\mu))/(A[\lambda+3\mu][K_{odd}^{2}+\mu(\lambda+2\mu)]q_{\text{th}}^{2})$)
and (c) semi-transverse in-plane phonon correlation function
($C_{u_{1}u_{2}}=k_{B}TK_{odd}(\lambda+\mu)/(A[\lambda+3\mu][K_{odd}^{2}+\mu(\lambda+2\mu)]q_{\text{th}}^{2})$).
The shading indicates the error bars showing the standard error of the mean.
For $q>q_{\text{th}}$, the harmonic correlations are obtained. The Aronovitz-
Lubensky anomalous exponents, $\eta\approx 0.8,\eta_{u}\approx 0.4$ are
observed in (a) and (b) for $q<q_{\text{th}}$. Instead in (c), no anomalous
exponents are observed and this is consistent with $K_{odd}^{R}(q)\sim
q^{2\eta_{u}}$ being irrelevant. Corresponding data sets can be found in Tab.
S1.

Once the stress tensor is calculated for a face, we calculated the normal
associated with the local basis of the triangular face:
$\hat{\mathbf{n}}_{face}=\mathbf{e}_{1}\times\mathbf{e}_{2}/\|\mathbf{e}_{1}\times\mathbf{e}_{2}\|$
and then the normal to an edge of the triangular face via an additional cross
product $\mathbf{n}_{i\rightarrow
j}=\hat{\mathbf{n}}_{face}\times\mathbf{r}_{i\rightarrow j}$ (note that this
normal is not of length $1$; in addition the symbol, $\hat{}$, marks a unit
vector). The normal vector can be decomposed in the current basis
$\\{\mathbf{e}_{1},\mathbf{e}_{2}\\}$ as follows: $(N_{i\rightarrow
j})_{k}=\mathbf{n}_{i\rightarrow j}\cdot\mathbf{e}_{k}$ for $k=\\{1,2\\}$. We
can then calculate the force on the edge as follows: $(T_{i\rightarrow
j})^{k}=(N_{i\rightarrow j})_{l}\sigma^{lk}$ and
$\mathbf{F}_{ij}=(T_{i\rightarrow j})^{k}\mathbf{e}_{k}$. To implement the
force on the edge, one can halve $\mathbf{F}_{ij}$ and apply each half on each
vertex $\\{i,j\\}$. Such an implementation is similar to that done in the
field of vertex models, the flat analogue bears the name (T)-scheme Lin  _et
al._ (2022); Tlili  _et al._ (2015, 2019). This method bears particular
advantages over the use of spring-mass systems. Firstly, via a coarse-graining
procedure one obtains that the continuum in-plane moduli,
$\mathcal{C}_{ijkl}$, are the exactly the same as those that we use in our
force implementation. Secondly, any stable elastic system, anisotropic or not
and odd or not, can be simulated very easily via this method. With springs, in
order to simulate $A_{odd}$ and $K_{odd}$ independently, for example, one
would require a unit cell that is larger than the triangular face and may
require a complicated microscopic force-displacement relation. Due to the fact
that the continuum description is being directly simulated in this new method,
these complications can be avoided. As for the time-step, $\Delta_{\tau}$, we
selected a small enough value such that all the inverse of the over-damped
frequencies associated with our system
($a^{4}/[8\pi^{3}\mathcal{D}_{f}\kappa],a^{2}/[2\pi\mathcal{D}\mathcal{C}_{ijkl}]$)
were larger. For our simulation procedure we had an
equilibration/thermalization period of $2\times 10^{7}\Delta_{\tau}$ after
which we began to record realizations of the system for $8\times
10^{8}\Delta_{\tau}$ at an interval of $10^{5}\Delta_{\tau}$ time steps. We
determined the thermalization length by performing autocorrelations of the
displacement fields to know over what period of time the system retains memory
of the initial configuration. In certain instances, particularly for
simulations at lower temperatures, achieving thermal equilibrium within the
system presents a challenge. Detailed elaboration on this topic can be found
in Sec. II of the supplementary material.

Given from our theoretical analysis that $A_{odd}$ and $K_{odd}$ do not
generate each other, we simulate odd elastic systems that satisfy $A_{odd}=0$
or $K_{odd}=0$ with regards to the bare moduli. In this way, we may examine
the effect of each modulus independently and further avoid potential non-
universal characteristics and instabilities Scheibner  _et al._ (2020). We
then measured the equal-time correlations of our displacement fields by
averaging over our snapshots of data. We utilize the form of the equal-time
correlations, Eq. (7) and Eq. (8), replacing the microscopic values
$\kappa,\lambda,\mu,A_{odd},K_{odd}$ with
$\kappa^{R}(q),\lambda^{R}(q),\mu^{R}(q),A_{odd}^{R}(q),K_{odd}^{R}(q)$ and
set $\mathcal{L}=\mathcal{D},\mathcal{L}_{f}=\mathcal{D}_{f}$. Since no
spatial symmetries are broken and we do not expect any anisotropies to develop
within the system. With these formulas we can then extrapolate the scaling of
the parameters
$\kappa^{R}(q),\lambda^{R}(q),\mu^{R}(q),A_{odd}^{R}(q),K_{odd}^{R}(q)$ from
the simulation data.

Results For $K_{odd}$. Correlation functions extracted from simulations with
$K_{odd}$ can be found in Fig. 2. One can find results consistent with
theoretical exponents extracted via the 1-loop $1/d_{c}$ expansion. That is,
on the one hand, for $q>q_{\text{th}}$, equal-time harmonic correlations are
obtained, that is, Eq. (7) and Eq. (8). Thus below the thermal length scale
($q>q_{\text{th}}$), the mechanical response is described by the bare moduli.
On the other hand, for $q<q_{\text{th}}$, $\langle
f(\mathbf{q})f(-\mathbf{q})\rangle\sim q^{-4+\eta}$ and thus effectively
$\kappa^{R}(q)/\kappa\sim q^{-\eta}$. The equal-time longitudinal-phonon
correlation scales as $\langle u_{1}(q_{1},0)u_{1}(-q_{1},0)\rangle\sim
q^{-2-\eta_{u}}$. On the other hand, the semi-transverse in-plane phonon
correlations never exhibit anomalous behavior, $\langle
u_{1}(q_{1},0)u_{2}(-q_{1},0)\rangle\sim q^{-2}$. With the ansatz that in the
vicinity of the Aronovitz-Lubensky fixed point, $\lambda^{R}(q),\mu^{R}(q)\sim
q^{\eta_{u}}$ and positing the correlation $K_{odd}^{R}(q)\sim q^{\alpha}$,
then $\langle u_{1}(q_{1},0)u_{2}(-q_{1},0)\rangle\sim
q^{-2+\alpha-2\eta_{u}}$. Thus these in-plane correlations could only be
explained by the scaling: $K_{odd}^{R}(q)\sim q^{2\eta_{u}}$, which is
consistent with our theoretical 1-loop $1/d_{c}$ expansion calculations. Such
a scaling is indicative that small perturbations in $K_{odd}$ to the
Aronovitz-Lubensky fixed point are irrelevant and thus
$K_{odd}^{R}(q)/\mu^{R}(q)$ scales to zero in the thermodynamic limit. The
significance of this is that breaking conservation of energy in the manner in
which $K_{odd}$ does is irrelevant (though $\mu^{R}(q),\lambda^{R}(q)$ do
scale to zero in the thermodynamic limit, they are examples of dangerously
irrelevant parameters. Instead $K_{odd}^{R}(q)$ is truly irrelevant).
Correlation functions extracted from simulations with $K_{odd}$ can be found
in Fig. 2. We remind the readers that these results hold when fluctuation-
dissipation relations hold.

Figure 3: Plots of the non-dimensionalized (a) flexural correlation function
($C_{ff}=k_{B}T/(A\kappa q_{\text{th}}^{4})$), (b) longitudinal in-plane
phonon correlation function
($C_{u_{1}u_{1}}=k_{B}T/(A(\lambda+2\mu)q_{\text{th}}^{2})$) and (c) semi-
transverse in-plane phonon correlation function
($C_{u_{1}u_{2}}=k_{B}TA_{odd}/(A[\lambda+3\mu][\lambda+2\mu]q_{\text{th}}^{2})$).
The shading indicates the error bars showing the standard error of the mean.
For $q>q_{\text{th}}$, the harmonic correlations are obtained. The Aronovitz-
Lubensky anomalous exponents, $\eta\approx 0.8,\eta_{u}\approx 0.4$ are
observed in (a) and (b) for $q<q_{\text{th}}$. In (c), the $\eta_{u}$
anomalous exponent is again observed and this is consistent with
$A_{odd}^{R}(q)\sim q^{\eta_{u}}$ being marginal. Corresponding data sets can
be found in Tab. S1.

Results For $A_{odd}$. Correlation functions extracted from simulations with
$A_{odd}$ can be found in Fig. 3. One can find results consistent with
theoretical exponents extracted via the 1-loop $1/d_{c}$ expansion. That is,
on the one hand, for $q>q_{\text{th}}$, harmonic correlations are obtained.
Whereas for $q<q_{\text{th}}$, $\langle f(\mathbf{q})f(-\mathbf{q})\rangle\sim
q^{-4+\eta}$ and $\langle u_{1}(q_{1},0)u_{1}(-q_{1},0)\rangle\sim
q^{-2-\eta_{u}}$ which exhibits the exponents associated with the Aronovitz-
Lubensky fixed point once again. The semi-transverse in-plane phonon
correlations exhibit the following scaling, $\langle
u_{1}(q_{1},0)u_{2}(-q_{1},0)\rangle\sim q^{-2-\eta_{u}}$, however, some
portions of the correlation functions are missing due to large fluctuations.
One can make sense of this by understanding that the microscopic value we took
for $A_{odd}$ is perturbative to $\lambda$ and $\mu$ and thus $\langle
u_{1}(q_{1},0)u_{2}(-q_{1},0)$, which is zero in the absence of odd elastic
moduli, experiences a greater amount of noise and error. Returning to the
scaling we have obtained: positing the following scaling $A^{R}_{odd}(q)\sim
q^{\alpha}$ then $\langle u_{1}(q_{1},0)u_{2}(-q_{1},0)\rangle\sim
q^{-2+\alpha-2\eta_{u}}$ and can thus only be explained by the scaling:
$A_{odd}^{R}(q)\sim q^{\eta_{u}}$. Such a scaling is indicative that small
perturbations in $A_{odd}$ to the Aronovitz-Lubensky fixed point are marginal
and thus $A_{odd}^{R}(q)/\mu_{odd}^{R}(q)\sim\text{constant}$ in the
thermodynamic limit. Thus breaking conservation of angular momentum, in the
manner that $A_{odd}$ does, cannot be ignored at the critical point and
thermal fluctuations cannot restore this manner in which the reflection
symmetry is broken. Thus the Aronovitz-Lubensky is no longer a fixed point but
becomes a higher-dimensional fixed manifold when fluctuation-dissipation
relations hold.

Conclusion. We have extracted the behavior of thermally fluctuating elastic
membranes in the presence of non-equilibrium odd elastic moduli. For
perturbative values of the odd elastic moduli, we observe that
$K_{odd}^{R}(q)\sim q^{2\eta_{u}}$ is an irrelevant parameter which can be
ignored in the thermodynamic limit whereas $A_{odd}^{R}(q)\sim q^{\eta_{u}}$
is marginal and cannot be ignored at any scale. We emphasize once again that
this study should be considered as an investigation in the case where these
odd elastic moduli are perturbations. Potentially for very large odd elastic
moduli, other behaviors and potential non-linear instabilities are possible
(for which the stress may not be able to be tuned to zero for given system
sizes). In addition, further exploration is merited on the basis that one can
also microscopically break the fluctuation-dissipation relations and our
eigen-value analysis of the Aronovitz-Lubensky fixed point yields unstable
eigen-vectors. This analysis, in the absence of odd elastic moduli, has been
obtained in Le Doussal and Radzihovsky (2021), where the same problem with
broken fluctuation-dissipation relations can be mapped to an equilibrium
problem that breaks the isotropy of the embedding space.

Obtaining a globally stable fixed point in the full phase space where
fluctuation-dissipation relations do not hold and in the presence of odd
elastic moduli is unresolved at the current time. Despite this, an important
point to note is that active components making up an elastic network, such as
actin or myosin in biological systems, are not to be necessarily associated
with the breaking of fluctuation-dissipation relations. The origin of the non-
equilibrium nature of biological systems, such as the cytoskeleton, would not
originate from odd active behavior necessarily. Instead, it may come from non-
Gaussian noise distributions as well as chemical reactions. For example, in
the context of the KPZ equation, multiplicative noise was even considered to
probe non-equilibrium properties Tu  _et al._ (1997). On the other hand, in
the context of chemical reactions, Park  _et al._ (2010) found that metabolic
activity of ATP can help facilitate non-equilibrium fluctuations of membranes.
To model this, one may conduct simulations and repeat a similar analysis with
a concentration parameter analagous to what is done in Turlier  _et al._
(2016). In addition, our results are valid for permeable membranes, further
can be done to explore membrane-solvent interactions as was done in Frey and
Nelson (1991). We expect that our equal-time correlation results should hold
even in the case of Zimm dynamics. Purely hydrodynamic solvent interactions
are expected to modify the dynamical scaling but not the static Frey and
Nelson (1991), though this remains to be verified via simulations. Lastly,
many more open directions can be investigated by incorporating other forms of
non-equilibrium features and activity such as non-Gaussian noises and odd
visco-elastic components for example Tu  _et al._ (1997); Floyd  _et al._
(2022). Thus, various open problems remain to fully explore the non-
equilibrium inducing aspects of our system to properly model biological and
potential synthetic systems at mesoscopic scales.

Acknowledgements. The authors thanks Uwe Taüber for a useful discussion in
development of the ideas with regards to fluctuation-dissipation relations,
and John Toner for a discussion on mass renormalization. The authors would
also like to acknowledge the following funding source: NSF Career award
DMR-1752100. DMF was supported by the Comunidad de Madrid and the Complutense
University of Madrid (Spain) through the Atracción de Talento program
2022-T1/TIC-24007.

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Supplemental Material

###### Contents

  1. S-1 Choice Of Over-Damped Equations
    1. S-1.1 Stability of Fully Inertial System
      1. S-1.1.1  $J<-\mu(B+\mu)$
      2. S-1.1.2  $-\mu(B+\mu)<J<(B/2)^{2}$
      3. S-1.1.3 $J>(B/2)^{2}$
    2. S-1.2 Inertial Term Irrelevance
    3. S-1.3 Potential Non-Steady State Behaviors
  2. S-2 Details of Simulations
    1. S-2.1 Implementation of Dihedral Forces
    2. S-2.2 Explicit In-Plane Constitutive Relations
    3. S-2.3 Barostat, Integrators and Simulation Procedure
    4. S-2.4 Procedure
    5. S-2.5 Data Tables
  3. S-3 Field Theoretic and Renormalization Group Analysis
    1. S-3.1 Fourier Transform of MSRJD Action and Propagators
    2. S-3.2 Reformulating MSRJD Action
    3. S-3.3 Scaling of Harmonic Theory
    4. S-3.4 Absence of Ward Identity
    5. S-3.5  Renormalization of Over-Damped Odd Elastic Membranes
      1. S-3.5.1  Feynman Diagrams and Renormalization Group Equations
      2. S-3.5.2 Stability Analysis
      3. S-3.5.3 Derivation of Thermal Length Scale Assuming Fluctuation-Dissipation Relations
      4. S-3.5.4 Numerical Renormalization With Fluctuation-Dissipation Relations

##  S-1 Choice Of Over-Damped Equations

We will illustrate why we opt for over-damped Langevin equations for examining
our model system, as described in sub-sections S-1.1, S-1.2, and S-1.3. The
choice is driven by three key reasons:

  1. 1.

The presence of the inertial term does not cause instability that might lead
to substantial differences from the over-damped scenario.

  2. 2.

At large length scales, the inertial term becomes irrelevant in the context of
field theory physics.

  3. 3.

Adopting the over-damped condition simplifies simulations, as it enables us to
circumvent potential non-steady state behaviors.

Regarding our theoretical analysis, it’s crucial to note that we will conduct
all calculations based on an orthonormal basis, contrary to the non-orthogonal
basis employed for the simulations. This approach will streamline the
application of the Fourier transformation, a critical step in our forthcoming
analysis.

###  S-1.1 Stability of Fully Inertial System

We first show that inclusion of the intertial term will not provide us with a
qualitative difference from the over-damped version of the equations via some
instability. Thus, we begin with the fully inertial athermal (sans thermal
noise) equations, which are the following:

$\begin{split}\rho\partial_{t}^{2}u_{j}(\mathbf{r},t)+\partial_{t}u_{j}(\mathbf{r},t)&=\mathcal{D}\partial_{i}\sigma_{ij}(\mathbf{r},t)=\mathcal{D}\mathcal{C}_{ijkl}\partial_{i}u_{kl}(\mathbf{r},t)\\\
\rho\partial_{t}^{2}f^{\alpha}(\mathbf{r},t)+\partial_{t}f^{\alpha}(\mathbf{r},t)&=\mathcal{D}_{f}[-\partial^{i}\partial^{j}M_{ij}^{\alpha}(\mathbf{r},t)+\partial_{i}[\sigma_{ij}(\mathbf{r},t)\partial_{j}f^{\alpha}(\mathbf{r},t)]]\\\
&=\mathcal{D}_{f}[-\partial_{i}\partial_{j}\\{\mathcal{B}_{ijkl}\partial_{k}\partial_{l}f^{\alpha}(\mathbf{r},t)\\}+\partial_{i}\\{\sigma_{ij}(\mathbf{r},t)\partial_{j}f^{\alpha}(\mathbf{r},t)\\}]\end{split}$
(S1)

where $\rho$ is the density. Roman indices vary through the number of in-plane
dimensions, which in this case $D=2$, whereas Greek indices vary through the
co-dimension $d_{c}=d-D$ of the membrane (with $d$ being the total dimension
of the embedding space). $\mathcal{D}$ and $\mathcal{D}_{f}$ are the in-plane
and out-of-plane mobilities of the membrane respectively. The strain tensor
$u_{ij}$, moment tensor $M_{ij}$, and stress tensor $\sigma_{ij}$ are the
following:

$\begin{split}&u_{ij}(\mathbf{r},t)=\frac{1}{2}[\partial_{i}u_{j}(\mathbf{r},t)+\partial_{j}u_{i}(\mathbf{r},t)+\partial_{i}f^{\alpha}(\mathbf{r},t)\partial_{j}f^{\alpha}(\mathbf{r},t)]\\\
&M_{ij}^{\alpha}(\mathbf{r},t)=\mathcal{B}_{ijkl}\partial_{k}\partial_{l}f^{\alpha}(\mathbf{r},t)\\\
&\sigma_{ij}(\mathbf{r},t)=\mathcal{C}_{ijkl}u_{kl}(\mathbf{r},t)\end{split}$
(S2)

We shall also use an isotropic bending rigidity
$\mathcal{B}_{ijkl}=\frac{\kappa}{2}\delta_{ij}\delta_{kl}+\frac{\kappa}{4}[\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}]$
so that
$\mathcal{B}_{ijkl}\partial_{i}\partial_{j}\partial_{k}\partial_{l}=\kappa\Delta^{2}$.
Under periodic boundary conditions, the Gaussian bending rigidity may be
ignored by means of the Gauss-Bonnet theorem. To perform a stability analysis
we neglect non-linear terms and arrive at the linearized form in Fourier space
(via the defined Fourier transform
$g(\mathbf{r},t)=\frac{1}{A\tau}\sum_{\mathbf{q},\omega}g(\mathbf{q},\omega)e^{i(\mathbf{q}\cdot\mathbf{r}-\omega
t)}$ with $A$ being the area of the system and $\tau$ the length of time):

$\begin{split}&-\rho\omega^{2}u_{j}(\mathbf{q},\omega)-i\omega
u_{j}(\mathbf{q},\omega)=-\mathcal{D}\mathcal{C}_{ijkl}q_{i}q_{k}u_{l}(\mathbf{q},\omega)\\\
&-\rho\omega^{2}f^{\alpha}(\mathbf{q},\omega)-i\omega
f^{\alpha}(\mathbf{q},\omega)=\mathcal{D}_{f}[-\kappa
q^{4}f^{\alpha}(\mathbf{q},\omega)-\mathcal{C}_{ijkl}u^{o}_{kl}q_{i}q_{j}f^{\alpha}(\mathbf{q},\omega)]\end{split}$
(S3)

where $u^{o}_{kl}$ is the homogeneous strain tensor defined via the Fourier
transform of the strain tensor:

$\begin{split}u_{ij}(\mathbf{r},t)=&u_{ij}^{o}+\frac{i}{2}\sum_{\mathbf{q},\omega}[q_{i}u_{j}(\mathbf{q},\omega)+q_{j}u_{i}(\mathbf{q},\omega)]e^{i(\mathbf{q}\cdot\mathbf{r}-\omega
t)}\\\
&-\frac{1}{2}\sum_{\mathbf{q}_{1},\mathbf{q}_{2},\omega_{1},\omega_{2}}\mathbf{q}_{1,i}\mathbf{q}_{2,j}f^{\alpha}(\mathbf{q}_{1},\omega_{1})f^{\alpha}(\mathbf{q}_{2},\omega_{2})e^{i(\\{\mathbf{q}_{1}+\mathbf{q}_{2}\\}\cdot\mathbf{r}-\\{\omega_{1}+\omega_{2}\\}t)}\end{split}$
(S4)

Note that in addition via the constitutive relation:
$\sigma_{ij}^{o}=\mathcal{C}_{ijkl}u^{o}_{kl}$ can be reformulated as the
homogeneous stress tensor. Thus we may rewritten Eq. LABEL:eq:LangevinLinear
we may rewrite the equations as:

$\begin{split}&-\rho\omega^{2}u_{j}(\mathbf{q},\omega)-i\omega
u_{j}(\mathbf{q},\omega)=-\mathcal{D}\mathcal{C}_{ijkl}q_{i}q_{k}u_{l}(\mathbf{q},\omega)\\\
&-\rho\omega^{2}f^{\alpha}(\mathbf{q},\omega)-i\omega
f^{\alpha}(\mathbf{q},\omega)=\mathcal{D}_{f}[-\kappa
q^{4}-\sigma_{ij}^{o}q^{i}q^{j}]f^{\alpha}(\mathbf{q},\omega)\end{split}$ (S5)

In addition, assuming isotropy we can further rewrite
$\sigma_{ij}^{o}q_{i}q_{j}=\sigma^{o}q^{2}$ and so one obtains:

$\begin{split}&-\rho\omega^{2}u_{j}(\mathbf{q},\omega)-i\omega
u_{j}(\mathbf{q},\omega)=-\mathcal{D}\mathcal{C}_{ijkl}q_{i}q_{k}u_{l}(\mathbf{q},\omega)\\\
&-\rho\omega^{2}f^{\alpha}(\mathbf{q},\omega)-i\omega
f^{\alpha}(\mathbf{q},\omega)=\mathcal{D}_{f}[-\kappa
q^{4}-\sigma^{o}q^{2}]f^{\alpha}(\mathbf{q},\omega)\end{split}$ (S6)

From here the stability condition for the flexural field, $f^{\alpha}$, is
clear. We require simply that the system have a positive bending rigidity
($\kappa>0$) and that the elastic sheet not be buckled, i.e. $\kappa
q^{4}+\sigma^{o}q^{2}>0$. This leaves us with the stability analysis of the
linearized in-plane phonon equation. To do this we can follow an analysis
similar to that given by Scheibner  _et al._ (2020). The strain tensor is
taken to be the same as Eq. (1) in the main text:

$\begin{split}\mathcal{C}_{ijkl}=\lambda\delta_{ij}\delta_{kl}+\mu[\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}]+K_{odd}\mathcal{E}_{ijkl}-A_{odd}\epsilon_{ij}\delta_{kl}\end{split}$
(S7)

with with
$\mathcal{E}_{ijkl}=\frac{1}{2}[\epsilon_{ik}\delta_{jl}+\epsilon_{il}\delta_{jk}+\epsilon_{jk}\delta_{il}+\epsilon_{jl}\delta_{ik}]$.
The symbol $\epsilon_{il}$ is the permutation Levi-Civita tensor
($\epsilon_{11}=\epsilon_{22}=0,\epsilon_{12}=-\epsilon_{21}=1$). We begin by
reformulating the equations:

$(-\rho\omega^{2}-i\omega)\begin{bmatrix}u_{1}\\\
u_{2}\end{bmatrix}=\mathcal{D}q^{2}\left[\begin{array}[]{ccccc}B+\mu&K_{odd}\\\
-K_{odd}-A_{odd}&\mu\\\ \end{array}\right]\begin{bmatrix}u_{\parallel}\\\
u_{\perp}\end{bmatrix}$ (S8)

where $B=\lambda+\mu$ and $u_{\parallel}=\hat{q}_{i}u_{i}$ and
$u_{\perp}=\epsilon_{ij}\hat{q}_{i}u_{j}$ (where the symbol $\hat{}$ marks a
unit vector). By doing the eigenvalue analysis of the matrix we obtain:

$(-\rho\omega^{2}-i\omega)=-\mathcal{D}\bigg{[}\frac{B}{2}+\mu\pm\sqrt{\bigg{(}\frac{B}{2}\bigg{)}^{2}-A_{odd}K_{odd}-K_{odd}^{2}}\bigg{]}q^{2}$
(S9)

For simplicity let us redefine $J=A_{odd}K_{odd}+K_{odd}^{2}$. We can then
solve this quadratic equation to obtain:

$\omega=-\frac{i}{2\rho}\pm\frac{1}{2\rho}\sqrt{-1+\rho\mathcal{D}\bigg{[}\frac{B}{2}+\mu\pm\sqrt{\bigg{(}\frac{B}{2}\bigg{)}^{2}-J}\bigg{]}q^{2}}$
(S10)

We do not want $\omega$ to have a positive imaginary branch, otherwise this
leads to a real positive exponential solution which leads to a blow-up of the
equations. Thus when the second term in the above equation becomes more
"positive" than the first, then we have obtained an instability. We find 3
regimes which are as follows:

####  S-1.1.1  $J<-\mu(B+\mu)$

If this condition is satisfied then the following inequality is implied:

$\sqrt{\bigg{(}\frac{B}{2}\bigg{)}^{2}-J}>\frac{B}{2}+\mu$ (S11)

and thus the following eigen-value:

$i\omega=\frac{1}{2\rho}\pm\frac{i}{2\rho}\sqrt{-1+\rho\mathcal{D}\bigg{[}\frac{B}{2}+\mu-\sqrt{\bigg{(}\frac{B}{2}\bigg{)}^{2}-J}\bigg{]}q^{2}}$
(S12)

will induce a solution that blows up for all $q$. Hence, one must require that
$J>-\mu(B+\mu)$ in order to have a stable odd elastic system.

####  S-1.1.2  $-\mu(B+\mu)<J<(B/2)^{2}$

In this case a careful analysis of all 4 eigenvalues shows that there is no
possibility for $\omega$ to have a positive imaginary branch since:

$|\mathrm{Im}\bigg{(}\sqrt{-1+\rho\mathcal{D}\bigg{[}\frac{B}{2}+\mu\pm\sqrt{\bigg{(}\frac{B}{2}\bigg{)}^{2}-J}\bigg{]}q^{2}}\bigg{)}|<1$
(S13)

Thus we have a stable system for all $q$.

####  S-1.1.3 $J>(B/2)^{2}$

In this case we can rewrite the eigenvalues as:

$\omega=-\frac{i}{2\rho}\pm\frac{1}{2\rho}\sqrt{-1+\rho\mathcal{D}\bigg{[}\frac{B}{2}+\mu\pm
i\sqrt{J-\bigg{(}\frac{B}{2}\bigg{)}^{2}}\bigg{]}q^{2}}$ (S14)

By rewriting:

$Re^{-i\theta_{\pm}}\equiv 1-\rho\mathcal{D}\bigg{[}\frac{B}{2}+\mu\pm
i\sqrt{J-\bigg{(}\frac{B}{2}\bigg{)}^{2}}\bigg{]}q^{2}$ (S15)

And taking the principal branch cut in the complex plane to be the non-
positive x-axis (so that $-\pi<\theta_{\pm}<\pi$), we can take the square root
and write:

$\omega=-\frac{i}{2\rho}\pm\frac{i}{2\rho}\sqrt{R}e^{-i\theta_{\pm}/2}$ (S16)

And thus:

$\mathrm{Im}(\omega)=-\frac{1}{2\rho}\pm\frac{1}{2\rho}\sqrt{R}\cos{\theta_{\pm}/2}$
(S17)

Since $-\pi<\theta_{\pm}<\pi$, then $\cos{\frac{\theta_{\pm}}{2}}>0$.
Therefore we need only examine:

$\mathrm{Im}(\omega)=-\frac{1}{2\rho}+\frac{1}{2\rho}\sqrt{R}\cos{\theta_{\pm}/2}$
(S18)

Once $\sqrt{R}\cos{(\theta_{\pm}/2)}>1$ we have an instability. This is
equivalent to checking when $R(\cos{(\theta_{\pm}/2)})^{2}>1$. Via the half-
angle formula this becomes:

$R(1+\cos{\theta_{\pm}})>2$ (S19)

Thus we obtain:

$\sqrt{(1-\rho
q^{2}\mathcal{D}\bigg{[}\frac{B}{2}+\mu\bigg{]})^{2}+\rho^{2}\mathcal{D}q^{4}\bigg{[}J-\bigg{(}\frac{B}{2}\bigg{)}^{2}\bigg{]}}+1-\rho\mathcal{D}q^{2}\bigg{[}\frac{B}{2}+\mu\bigg{]}>2$
(S20)

which leads us to:

$\sqrt{(1-\rho\mathcal{D}q^{2}\bigg{[}\frac{B}{2}+\mu\bigg{]})^{2}+\rho^{2}\mathcal{D}^{2}q^{4}\bigg{[}J-\bigg{(}\frac{B}{2}\bigg{)}^{2}\bigg{]}}>1+\rho\mathcal{D}q^{2}\bigg{[}\frac{B}{2}+\mu\bigg{]}>0$
(S21)

By squaring the inequality once more and solving we see that we obtain
instabilities when:

$\rho^{2}\mathcal{D}^{2}q^{4}\bigg{[}J-\bigg{(}\frac{B}{2}\bigg{)}^{2}\bigg{]}>4\rho\mathcal{D}q^{2}\bigg{[}\frac{B}{2}+\mu\bigg{]}$
(S22)

resulting in:

$q>\sqrt{\frac{4}{\rho\mathcal{D}}\frac{\bigg{[}\frac{B}{2}+\mu\bigg{]}}{\bigg{[}J-\bigg{(}\frac{B}{2}\bigg{)}^{2}\bigg{]}}}\equiv
q_{c}$ (S23)

Note the difference in the direction of this inequality is different from that
obtained in the stability analysis of the visco-elastic equations in Scheibner
_et al._ (2020), where the dissipative term has two extra spatial derivatives,
$\eta^{ijkl}\partial_{t}\partial_{i}\partial_{k}u_{l}$. Hence we see that in
this regime, if a system has a small enough physical length scale (i.e. the
lattice spacing of discrete model is small enough) then we have instabilities.
If we were to explore over-damping via the zero-inertia Brownian method
($\rho=0$), we see that $q_{c}=\infty$ and thus we should not be able to
observe this instability. In the Langevin method where we have under-damping,
if $q_{c}<2\pi/a$ where $a$ is the lattice spacing of our discrete system,
then we should once again not see this instability. Otherwise, the under-
damped Langevin case should show this instability.

###  S-1.2 Inertial Term Irrelevance

Further justifying the over-damped case, we show the irrelevance of the
inertial term near the Gaussian fixed point of the corresponding MSRJD action.
To do this a power-counting analysis Tong  is done in Section S-3.3. From this
one can see that $\partial_{t}^{2}f,\partial_{t}^{2}u_{j}$ scale to zero in
the low frequency limit (since $\zeta_{t}=4$). Thus high-frequency phenomena
will be unimportant to the long-wavelength limiting behavior of the theory.

###  S-1.3 Potential Non-Steady State Behaviors

As a final note, in the case where over-damping is not assumed, it is well
known that one must consider an active heat flow Falasco  _et al._ (2016);
Löwen (2020). This active heat flow is a form of work done by non-equilibrium
or active forces that are not derived from a free energy. However, in the
over-damped limit, one may disregard such terms. This will help us to simplify
simulations that we performed where we use barostats and thermostats which in
general should contain an active heat flow term.

In addition, other variables such as the pressure may also not be treated as
state functions Solon  _et al._ (2015). Thus we also consider Rouse dynamics
so that we do not couple hydrodynamics to the membrane fluctuations Frey and
Nelson (1991).

##  S-2 Details of Simulations

###  S-2.1 Implementation of Dihedral Forces

The bending forces were determined directly from a set of dihedral springs
between the faces of the triangles. The elastic bending energy of such a
system can be formulated as:

$\text{E}_{\text{bend}}=\hat{\kappa}\sum_{\langle
IJ\rangle}[1-\hat{n}_{I}\cdot\hat{n}_{J}]$ (S24)

where $\hat{\kappa}$ is the discrete dihedral spring stiffness and
$\hat{n}_{I}$ is the normal to face $I$. We can relate the microscopic spring
stiffness, $\hat{\kappa}$, to the coarse-grained bending rigidity, $\kappa$
Seung and Nelson (1988):

$\kappa=\frac{\sqrt{3}}{2}\hat{\kappa}$ (S25)

###  S-2.2 Explicit In-Plane Constitutive Relations

For our simulations we used the implementation of the constitutive relation to
enact forces explained in the main body of the text. We also added in a non-
linear area potential (which we call a bulk strain term) which helped to
prevent the areas of faces of the triangles from becoming too small and thus
aids with the stabilization of our numerical simulations. We show here the
derivation of explicit constitutive relations with respect to each of these
contributions to the in-plane stresses:

We begin by defining the strain tensor for the each triangular face as
follows:

$u_{ij}=\frac{1}{2}[g_{ij}-\bar{g}_{ij}]$ (S26)

where $\\{g,\bar{g}\\}$ are the current and reference metric tensors
respectively. We use the following elastic modulus tensor in our constitutive
relation:

$\begin{split}\mathcal{C}^{ijkl}=&\lambda\bar{g}^{ij}\bar{g}^{kl}+\mu[\bar{g}^{ik}\bar{g}^{jl}+\bar{g}^{il}\bar{g}^{jk}]+\frac{K^{odd}}{\sqrt{\det[\bar{g}]}}\mathcal{E}^{ijkl}-\frac{A_{odd}}{\sqrt{\det[\bar{g}]}}\epsilon^{ij}\bar{g}^{kl}\end{split}$
(S27)

with
$\mathcal{E}^{ijkl}=\frac{1}{2}[\epsilon^{ik}\bar{g}^{jl}+\epsilon^{il}\bar{g}^{jk}+\epsilon^{jk}\bar{g}^{il}+\epsilon^{jl}\bar{g}^{ik}]$,
where $\epsilon^{ij}$ is again the 2-D Levi-Civita permutation symbol
satisfying $\epsilon^{11}=\epsilon^{22}=0,\epsilon^{12}=-\epsilon^{21}=1$.

Then, in the linear elastic limit where deformations are small, one can write
down the contribution of $A_{odd}$ to the constitutive relation:

$\begin{split}\sigma_{A}^{ij}=-\frac{A_{odd}}{\sqrt{|\bar{g}|}}\epsilon^{ij}\mathrm{Tr}(u)\end{split}$
(S28)

where $\mathrm{Tr}$ is the trace and $\epsilon$ is again the 2-D Levi-Civita
tensor ($\epsilon^{11}=\epsilon^{22}=0,\epsilon^{12}=-\epsilon^{21}=1$). So
now:

$\begin{split}\sigma_{A}^{11}&=0\\\
\sigma_{A}^{12}&=-\frac{A_{odd}}{\sqrt{|\bar{g}|}}\mathrm{Tr}(u)\\\
\sigma_{A}^{22}&=0\\\ \sigma_{A}^{21}&=-\sigma_{A}^{12}\\\ \end{split}$ (S29)

Next, we move the contribution to $K$,

$\begin{split}\sigma_{K}^{ij}=\frac{K_{odd}}{2\sqrt{|\bar{g}|}}\left(2\epsilon^{ik}u_{k}^{j}+2\epsilon^{jk}u_{k}^{i}\right)\\\
\end{split}$ (S30)

Thus we obtain:

$\begin{split}\sigma_{K}^{11}&=\frac{K_{odd}}{\sqrt{|\bar{g}|}}\left(2u_{2}^{1}\right)\\\
\sigma_{K}^{12}&=\frac{K_{odd}}{\sqrt{|\bar{g}|}}\left(u_{2}^{2}-u_{1}^{1}\right)\\\
\sigma_{K}^{21}&=\sigma_{K}^{12}\\\
\sigma_{K}^{22}&=-\frac{K_{odd}}{\sqrt{|\bar{g}|}}\left(2u_{1}^{2}\right)\\\
\end{split}$ (S31)

We can write the same for the normal elastic parameters $\lambda,\mu$:

$\begin{split}\sigma_{\lambda,\mu}^{ij}=\lambda\bar{g}^{ij}\mathrm{Tr}(u)+2\mu\bar{g}^{ik}u_{k}^{j}\\\
\end{split}$ (S32)

$\begin{split}\sigma_{\lambda,\mu}^{11}&=\lambda\bar{g}^{11}\mathrm{Tr}(u)+2\mu(\bar{g}^{11}u_{1}^{1}+\bar{g}^{12}u_{2}^{1})\\\
\sigma_{\lambda,\mu}^{12}&=\lambda\bar{g}^{12}\mathrm{Tr}(u)+2\mu(\bar{g}^{11}u_{1}^{2}+\bar{g}^{12}u_{2}^{2})\\\
\sigma_{\lambda,\mu}^{21}&=\lambda\bar{g}^{21}\mathrm{Tr}(u)+2\mu(\bar{g}^{21}u_{1}^{2}+\bar{g}^{22}u_{2}^{1})\\\
\sigma_{\lambda,\mu}^{22}&=\lambda\bar{g}^{22}\mathrm{Tr}(u)+2\mu(\bar{g}^{21}u_{1}^{2}+\bar{g}^{22}u_{2}^{2})\\\
\end{split}$ (S33)

Thus we have established stress-strain relations in terms of the metric tensor
with respect to $A_{odd},K_{odd},\lambda,\mu$. We now move on, to showing the
implementation of the area potential.

##### Area Potential

For reasons concerned with the stability of the triangular lattice under large
deformations in the presence of odd elastic parameters, we also added in an
area potential (to prevent small areas of the triangular faces and ultimately
two edges crossing each other; which can cause numerical divergences by means
of the calculation of our normal to the face:
$\hat{n}_{face}=\mathbf{e}_{1}\times\mathbf{e}_{2}/\|\mathbf{e}_{1}\times\mathbf{e}_{2}\|$
where $\\{\mathbf{e}_{i}\\}$ is the current basis derived from the edges of a
triangular face as was discussed in the main text). To add such a term we have
to potentially implement the following energy term:

$\mathcal{F}=C_{1}\log\frac{\det g}{\det\bar{g}}+C_{2}\bigg{(}\log\frac{\det
g}{\det\bar{g}}\bigg{)}^{2}$ (S34)

where $g$ and $\bar{g}$ are the current and reference metrics respectively.
$\det g$ and $\det\bar{g}$ are intuitively allowing us to compare the local
change in area. Thus the idea of this potential is to create a restoring force
that does not allow $\det g$ to become too small. Of course we want to avoid
using the energy term as we don’t have any conservation of energy in our
system. Thus we are instead interested in deriving a stress strain relation:

$\sigma^{ij}=\frac{\delta\mathcal{F}}{\delta u_{ij}}$ (S35)

In order to do this we must express the current metric tensor in terms of the
reference basis and the strains. The strain tensor takes the following form:

$u_{ij}=\frac{1}{2}[g_{ij}-\bar{g}_{ij}]$ (S36)

We use the same notation as in the main text: $\\{\mathbf{e}_{i}\\}$ is our
current basis and $\\{\bar{\mathbf{e}}_{i}\\}$ is our reference basis.
Meanwhile the current and reference metric tensors are expressed respectively
as $g_{ij}=\mathbf{e}_{i}\cdot\mathbf{e}_{j}$ and
$\bar{g}_{ij}=\bar{\mathbf{e}}_{i}\cdot\bar{\mathbf{e}}_{j}$. With this, we
can write:

$\begin{split}\det
g=&(\mathbf{e}_{1}\cdot\mathbf{e}_{1})(\mathbf{e}_{2}\cdot\mathbf{e}_{2})-(\mathbf{e}_{1}\cdot\mathbf{e}_{2})^{2}\\\
=&[2u_{11}+\bar{\mathbf{e}}_{1}\cdot\bar{\mathbf{e}}_{1}][2u_{22}+\bar{\mathbf{e}}_{2}\cdot\bar{\mathbf{e}}_{2}]-[2u_{12}+\bar{\mathbf{e}}_{1}\cdot\bar{\mathbf{e}}_{2}]^{2}\end{split}$
(S37)

Using this we can rewrite and define:

$\mathcal{G}=\log\frac{\det
g}{\det\bar{g}}=\log\frac{[2u_{11}+\bar{\mathbf{e}}_{1}\cdot\bar{\mathbf{e}}_{1}][2u_{22}+\bar{\mathbf{e}}_{2}\cdot\bar{\mathbf{e}}_{2}]-[2u_{12}+\bar{\mathbf{e}}_{1}\cdot\bar{\mathbf{e}}_{2}]^{2}}{(\bar{\mathbf{e}}_{1}\cdot\bar{\mathbf{e}}_{1})(\bar{\mathbf{e}}_{2}\cdot\bar{\mathbf{e}}_{2})-(\bar{\mathbf{e}}_{1}\cdot\bar{\mathbf{e}}_{2})^{2}}$
(S38)

Differentiating with respect to the strain we obtain:

$\frac{\delta\mathcal{G}}{\delta
u_{11}}=\frac{2[2u_{22}+\bar{\mathbf{e}}_{2}\cdot\bar{\mathbf{e}}_{2}]}{\det
g}$ (S39) $\frac{\delta\mathcal{G}}{\delta
u_{22}}=\frac{2[2u_{11}+\bar{\mathbf{e}}_{1}\cdot\bar{\mathbf{e}}_{1}]}{\det
g}$ (S40) $\frac{\delta\mathcal{G}}{\delta
u_{12}}=-\frac{2[2u_{12}+\bar{\mathbf{e}}_{1}\cdot\bar{\mathbf{e}}_{2}]}{\det
g}$ (S41)

This gives us:

$\sigma^{ij}=\frac{\delta\mathcal{F}}{\delta
u_{ij}}=C_{1}\frac{\delta\mathcal{G}}{\delta
u_{ij}}+2C_{2}\mathcal{G}\frac{\delta\mathcal{G}}{\delta u_{ij}}$ (S42)

And thus, having obtained the stress tensor we can use the rest of the
implementation in the main text to determine the necessary forces to apply on
the vertices. By evaluating these expressions at $u_{ij}=0$, we immediately
obtain that $C_{1}=0$ so that the stress tensor is zero in the absence of
strain. Thus we are left with:

$\sigma^{ij}=\frac{\delta\mathcal{F}}{\delta
u_{ij}}=2C_{2}\mathcal{G}\frac{\delta\mathcal{G}}{\delta u_{ij}}$ (S43)

To obtain the linear elastic response from the area potential in terms of
$C_{2}$, we can differentiate again with respect to $u_{kl}$ and then set
$u_{ij}=0$. This in turn gives us the respective contribution,
$\mathcal{C}^{ijkl}_{C_{2}}$, to the elastic modulus tensor:

$\mathcal{C}^{ijkl}_{C_{2}}=\frac{\delta\sigma^{ij}}{\delta
u_{kl}}=2C_{2}\bigg{[}\mathcal{G}\frac{\delta^{2}\mathcal{G}}{\delta
u_{ij}\delta u_{kl}}+\frac{\delta\mathcal{G}}{\delta
u_{ij}}\frac{\delta\mathcal{G}}{\delta
u_{kl}}\bigg{]}|_{u=0}=2C_{2}\frac{\delta\mathcal{G}}{\delta
u_{ij}}\frac{\delta\mathcal{G}}{\delta u_{kl}}|_{u=0}$ (S44)

We thus obtain that:

$\mathcal{C}^{1111}_{C_{2}}=2C_{2}\bigg{[}\frac{2\bar{\mathbf{e}}_{2}\cdot\bar{\mathbf{e}}_{2}}{\det\bar{g}}\bigg{]}^{2}$
(S45)
$\mathcal{C}^{2222}_{C_{2}}=2C_{2}\bigg{[}\frac{2\bar{\mathbf{e}}_{1}\cdot\bar{\mathbf{e}}_{1}}{\det\bar{g}}\bigg{]}^{2}$
(S46)
$\mathcal{C}^{1122}_{C_{2}}=\mathcal{C}^{2211}_{C_{2}}=2C_{2}\bigg{[}\frac{2\bar{\mathbf{e}}_{2}\cdot\bar{\mathbf{e}}_{2}}{\det\bar{g}}\bigg{]}\bigg{[}\frac{2\bar{\mathbf{e}}_{1}\cdot\bar{\mathbf{e}}_{1}}{\det\bar{g}}\bigg{]}$
(S47)
$\mathcal{C}^{1212}_{C_{2}}=\mathcal{C}^{2112}_{C_{2}}=\mathcal{C}^{2121}_{C_{2}}=\mathcal{C}^{1221}_{C_{2}}=2C_{2}\bigg{[}\frac{2\bar{\mathbf{e}}_{1}\cdot\bar{\mathbf{e}}_{2}}{\det\bar{g}}\bigg{]}^{2}$
(S48)
$\begin{split}&\mathcal{C}^{1112}_{C_{2}}=\mathcal{C}^{1121}_{C_{2}}=\mathcal{C}^{1211}_{C_{2}}=\mathcal{C}^{2111}_{C_{2}}=2C_{2}\bigg{[}\frac{2\bar{\mathbf{e}}_{1}\cdot\bar{\mathbf{e}}_{2}}{\det\bar{g}}\bigg{]}\bigg{[}\frac{2\bar{\mathbf{e}}_{2}\cdot\bar{\mathbf{e}}_{2}}{\det\bar{g}}\bigg{]}\end{split}$
(S49)
$\begin{split}&\mathcal{C}^{2221}_{C_{2}}=\mathcal{C}^{2212}_{C_{2}}=\mathcal{C}^{2122}_{C_{2}}=\mathcal{C}^{1222}_{C_{2}}=2C_{2}\bigg{[}\frac{2\bar{\mathbf{e}}_{1}\cdot\bar{\mathbf{e}}_{2}}{\det\bar{g}}\bigg{]}\bigg{[}\frac{2\bar{\mathbf{e}}_{1}\cdot\bar{\mathbf{e}}_{1}}{\det\bar{g}}\bigg{]}\end{split}$
(S50)

###  S-2.3 Barostat, Integrators and Simulation Procedure

It is known that there are certain cases, such as active brownian fluids with
alignment interactions, for which pressure is not an equation of state Solon
_et al._ (2015); Falasco  _et al._ (2016). Thus, we begin by remarking on the
fact that active systems have to be simulated differently from equilibrium
systems due to active heat flow as seen in Falasco  _et al._ (2016). For a
generic active matter system with $N$ particles in $d$ dimensions experiencing
a damping $\gamma$, the equation of state takes the form:

$\frac{1}{\gamma}\sum_{i=1}^{Nd}\langle
F^{nc}_{i}\frac{dr_{i}}{dt}\rangle+dNk_{B}T=\int_{V}d\mathbf{r}Tr(\sigma)-\sum_{i=1}^{Nd}\langle
F^{c}_{i}r_{i}\rangle-\sum_{i=1}^{Nd}\langle F^{nc}_{i}r_{i}\rangle$ (S51)

where $F^{nc}$ are the non-conservative forces, $F^{c}$ are conservative
forces and $r_{i}$ is the position of the i-th degree of freedom. The brackets
$\langle\cdot\rangle$ represent ensemble averages. The first term on the
right-hand side of the equation is the pressure (this formula is how we
calculate the pressure in our simulations as well). The second and third terms
on the right-hand side represent the work being done by conservative and non-
conservative forces respectively. In the absence of non-conservative forces,
this equation simplifies to the usual equilibrium equation of state. The left
most term is the active heat flow and represents the energy change by means of
non-conservative forces Falasco  _et al._ (2016). However, with over-damped
systems ($\gamma\rightarrow\infty$), this term can be ignored and velocity
distributions take the values in thermal equilibrium (i.e.
$p_{i}^{2}/2m=k_{B}T$ as by the virial theorem). This enables simpler
equations of motion and allows us to treat the pressure as a steady-state
variable.

Thus, for the simulations we used a Berendsen barostat to tune the box
pressure, $\int_{V}d\mathbf{r}Tr(\sigma)/[dV]$, to zero and to tune the
temperature of the system, we utilized a Gaussian distributed noise-force
whose variance dictated the magnitude of the temperature. For the integrator,
we implemented a BAOAB-limit method Leimkuhler and Matthews (2013), thus the
noise is not completely memory-less, but the memory is short ranged in time.
We selected the BAOAB-limit method as it reduces error while being very
efficient in terms of the descriptive equations of motion. Specifically the
BAOAB-limit differential equation is of the form:

$\mathbf{r}_{i}([n+1]\Delta_{\tau})=\mathbf{r}_{i}(n\Delta_{\tau})+\frac{\Delta_{\tau}\mathbf{\Gamma}}{2}[\mathbf{F}_{face}+\mathbf{F}_{dih}]+\sqrt{\frac{k_{B}T\Delta_{\tau}\mathbf{\Gamma}}{4}}(\mathbf{R}_{n,i}+\mathbf{R}_{n+1,i})$
(S52)

where $\mathbf{\Gamma}$ is the a diagonal diffusivity matrix with components
$(\mathcal{D},\mathcal{D},\mathcal{D}_{f})$, $\Delta_{\tau}$ is the time step,
$r_{i}(n\Delta_{\tau})$ is the position of the $i$-th atom at time-step n,
$k_{B}T$ sets the temperature (we take $k_{B}=1$), $\mathbf{R}_{n,i}$ is a
random number generator of variance 1 and mean 0 for time step $n$ operating
on vertex $i$. $\mathbf{F}_{dih}$ is the force originating from dihedral
degrees of freedom whereas $\mathbf{F}_{face}$ are the forces from the in-
plane degrees of freedom of the faces. These results are merely an
implementation of forces discussed in the main text. As one can see from this
equation, the noise that is used is not independent between adjacent time
steps but the correlation is short-ranged nonetheless.

In our simulations we took the matricial diffusivity matrix, $\mathbf{\Gamma}$
in Eq. (S52) to be the identity matrix and thus the fluctuation-dissipation is
assumed at the microscopic scale.

###  S-2.4 Procedure

We simulated $A_{odd}$ and $K_{odd}$ separately in an otherwise normal elastic
system. We took a variety of values of each of the parameters while varying
the temperature. Varying the temperature allows us to access different
effective length scales (by effectively changing $L/\ell_{\text{th}}$ without
changing the linear dimension of the system $L$; look further below for our
derivation of the new form of $\ell_{\text{th}}$), rather than doing a
computationally costly simulation with a large system size. We allowed each
elastic system to “thermally equilibrate” for around $2\cdot 10^{7}$ time
steps, after which we would begin recording instantaneous snapshots of the
configurations of the system. The time step was determined by the limiting
factors given by the over-damped frequencies of the system:

$\begin{split}\Delta_{\tau,\kappa}=a^{4}/[8\pi^{3}\mathcal{D}_{f}\kappa],\Delta_{\tau,\mathcal{C}_{ijkl}}=a^{2}/[2\pi\mathcal{D}\mathcal{C}_{ijkl}]\end{split}$
(S53)

where $a$ is the lattice spacing. Thus
$\Delta_{\tau}\leq\text{Min}\\{\Delta_{\tau,\kappa},\Delta_{\tau,\mathcal{C}_{ijkl}}\\}$.

Each simulation was thermally equilibrated with $2\times 10^{7}$ time steps.
This length of simulation steps was determined by performing auto-correlations
of the displacement fields to know the amount of time it takes for the system
to lose memor of the initial configuration. For low temperatures, however,
thermal equilibration is very difficult as the auto-correlation times grow.
After thermal equilibration steps were taken, the simulation ran through
$8\times 10^{8}$ time steps, recording snapshots every $10^{5}$ time steps.
This gave us a total of $8\times 10^{3}$ snapshots of data. For each
simulation we measured the auto-correlation time for the amplitude peak to
decay by an order of magnitude and used this time-interval to sample fully
independent realizations of membrane configurations out of the total data set.
With this reduced independent data set, we calculated averaged equal-time
correlations. In computation time, this amounted to about 80 hours.

###  S-2.5 Data Tables

Here we show the data tables for the plots we generated in Fig. 2 and Fig. 3
of the main text.

Table S1:  Data Sets for Fig. 2
---
$L/a$ |  $\kappa/k_{B}T$ |  $K_{odd}/(\lambda+2\mu)$ |  $C_{2}/(\lambda+2\mu)$
$35$ |  $1$ |  $4.76$ |  $.048$
$35$ |  $10$ |  $4.76$ |  $.048$
$35$ |  $1e3$ |  $4.76$ |  $.048$
$35$ |  $1e5$ |  $4.76$ |  $.048$
$35$ |  $1$ |  $.73$ |  $.007$
$35$ |  $10$ |  $.73$ |  $.007$
$35$ |  $1e3$ |  $.73$ |  $.007$
$35$ |  $1e5$ |  $.73$ |  $.007$
$35$ |  $1$ |  $.076$ |  $8e-4$
$35$ |  $10$ |  $.076$ |  $8e-4$
$35$ |  $1e3$ |  $.076$ |  $8e-4$
$35$ |  $1e5$ |  $.076$ |  $8e-4$
Table S2:  Data Sets for Fig. 3
---
$L/a$ |  $\kappa/k_{B}T$ |  $A_{odd}/(\lambda+2\mu)$ |  $C_{2}/(\lambda+2\mu)$
$35$ |  $1$ |  $.1$ |  $.048$
$35$ |  $10$ |  $.1$ |  $.048$
$35$ |  $100$ |  $.1$ |  $.048$
$35$ |  $1e3$ |  $.1$ |  $.048$
$35$ |  $1e4$ |  $.1$ |  $.048$
$35$ |  $1e5$ |  $.1$ |  $.048$
$35$ |  $1e6$ |  $.1$ |  $.048$

##  S-3 Field Theoretic and Renormalization Group Analysis

###  S-3.1 Fourier Transform of MSRJD Action and Propagators

We can take Eq. (6) of the main text and begin by taking the Fourier transform
of the MSRJD action (assuming once again that we are operating in an
orthonormal basis), $\mathcal{A}_{MSRJD}$:

$\begin{split}&\mathcal{W}(\eta_{j},\eta_{f}^{\alpha},\Upsilon_{i},\Phi^{\alpha})\propto
e^{-\mathcal{A}_{MSRJD}}\equiv e^{\int d\omega
A\cdot\tau\sum_{\mathbf{q}}[k_{B}T\mathcal{L}|\Upsilon_{i}(\mathbf{q},\omega)|^{2}-\Upsilon_{i}(\mathbf{q},\omega)\eta_{i}(-\mathbf{q},-\omega)+k_{B}T\mathcal{L}_{f}|\Phi^{\alpha}(\mathbf{q},\omega)|^{2}-\Phi^{\alpha}(\mathbf{q},\omega)\eta_{f}^{\alpha}(-\mathbf{q},-\omega)]}\end{split}$
(S54)

where $A$ is the area of the system and $\tau$ is the length of time. For
completeness, the corresponding Fourier equations of the noises are the
following:

$\eta_{j}(\mathbf{q},\omega)=\mathcal{D}\mathcal{C}_{ijkl}q_{i}[q_{k}u_{l}(\mathbf{q},\omega)-\frac{i}{2}\sum_{\mathbf{p},\gamma}p_{k}(p_{l}-q_{l})f^{\alpha}(\mathbf{p},\gamma)f^{\alpha}(\mathbf{q}-\mathbf{p},\omega-\gamma)]-i\omega
u_{j}(\mathbf{q},\omega)$ (S55)
$\begin{split}&\eta_{f}^{\alpha}(\mathbf{q},\omega)=(\mathcal{D}_{f}[\kappa
q^{4}+\mathcal{C}_{ijkl}u^{o}_{kl}q_{i}q_{j}]-i\omega)f^{\alpha}(\mathbf{q},\omega)\\\
&+\mathcal{D}_{f}\mathcal{C}_{ijkl}q_{i}\bigg{[}\sum_{(\mathbf{p},\gamma)\neq\mathbf{0}}\bigg{(}ip_{k}u_{l}(\mathbf{p},\gamma)-\frac{1}{2}\sum_{\mathbf{z},\xi}(p_{k}-z_{k})z_{l}f^{\beta}(\mathbf{p}-\mathbf{z},\gamma-\xi)f^{\beta}(\mathbf{z},\xi)\bigg{)}(q_{j}-p_{j})f^{\alpha}(\mathbf{q}-\mathbf{p},\omega-\gamma)\bigg{]}\end{split}$
(S56)

where we have used the isotropic bending rigidity, $\kappa$ in place of the
bending rigidity tensor and we can further assume isotropy of the homogeneous
strain term: $\mathcal{C}_{ijkl}u^{o}_{kl}q_{i}q_{j}=\sigma^{o}q^{2}$. The
MSRJD action, $\mathcal{A}_{MSRJD}$, possesses the following harmonic terms in
matrix form:

$\begin{split}&\mathcal{A}_{MSRJD}^{harm.}=\frac{1}{2}\begin{bmatrix}\Upsilon_{j}(\mathbf{q},\omega)\\\
u_{j}(\mathbf{q},\omega)\end{bmatrix}^{T}\begin{bmatrix}-2k_{B}T\mathcal{L}\delta_{jl}&i\omega\delta_{jl}+\mathcal{D}\mathcal{C}_{ijkl}q_{i}q_{k}\\\
-i\omega\delta_{jl}+\mathcal{D}\mathcal{C}_{ilkj}q_{i}q_{k}&0\end{bmatrix}\begin{bmatrix}\Upsilon_{l}(-\mathbf{q},-\omega)\\\
u_{l}(-\mathbf{q},-\omega)\end{bmatrix}\\\
&+\frac{\delta_{\alpha\beta}}{2}\begin{bmatrix}\Phi^{\alpha}(\mathbf{q},\omega)\\\
f^{\alpha}(\mathbf{q},\omega)\end{bmatrix}^{T}\begin{bmatrix}-2k_{B}T\mathcal{L}_{f}&i\omega+\mathcal{D}_{f}\kappa
q^{4}+\mathcal{D}_{f}\sigma^{o}q^{2}\\\ -i\omega+\mathcal{D}_{f}\kappa
q^{4}+\mathcal{D}_{f}\sigma^{o}q^{2}&0\end{bmatrix}\begin{bmatrix}\Phi^{\beta}(-\mathbf{q},-\omega)\\\
f^{\beta}(-\mathbf{q},-\omega)\end{bmatrix}\end{split}$ (S57)

From here we can obtain the propagators of the linear theory:

$\langle
f^{\alpha}(\mathbf{q},\omega)f^{\beta}(-\mathbf{q},-\omega)\rangle=\frac{2k_{B}T\mathcal{L}_{f}\delta_{\alpha\beta}}{A\tau\\{\mathcal{D}_{f}^{2}[\kappa
q^{4}+\sigma^{o}q^{2}]^{2}+\omega^{2}\\}}$ (S58)

This corresponds to a static propagator, which is obtained by integrating over
all $\omega$, thus applying the Cauchy residue theorem Barabási  _et al._
(1995); Täuber (2014):

$\langle
f^{\alpha}(\mathbf{q})f^{\beta}(-\mathbf{q})\rangle=\frac{k_{B}T\mathcal{L}_{f}\delta_{\alpha\beta}}{A\mathcal{D}_{f}[\kappa
q^{4}+\sigma^{o}q^{2}]}$ (S59)

Analogously we can write the in-plane propagator explicitly as:

$\begin{split}&\langle
u_{h}(\mathbf{q},\omega)u_{v}(-\mathbf{q},-\omega)\rangle=\frac{2k_{B}T\mathcal{L}\epsilon_{hl}\epsilon_{vp}}{A\tau\text{Det}}[-i\omega\delta_{jl}+q^{2}\mathcal{D}\mathcal{C}_{ijkl}P^{L}_{ik}(\mathbf{q})][i\omega\delta_{jp}+q^{2}\mathcal{D}\mathcal{C}_{mjrp}P^{L}_{rp}(\mathbf{q})]\end{split}$
(S60)

where Det is the determinant of the matrix that couples the harmonic terms
$u_{j}$ of the action (the determinant of
$[-i\omega\delta_{jl}+q^{2}\mathcal{D}\mathcal{C}_{ijkl}P^{L}_{ik}(\mathbf{q})][i\omega\delta_{jp}+q^{2}\mathcal{D}\mathcal{C}_{mjrp}P^{L}_{rp}(\mathbf{q})]$).
In addition $P^{L}_{ij}(\mathbf{q})=q_{i}q_{j}/q^{2}$ is the linear projection
operator and $\epsilon$ is the 2-D Levi-Civita tensor. To obtain the static
propagators one must apply the Cauchy residue formula once again. The lower-
half-plane residues of the determinant denominator can be found as well in
Scheibner  _et al._ (2020) and are:

$\omega_{res}=-i\frac{\mathcal{D}q^{2}}{2}[\lambda+3\mu\pm\sqrt{(\lambda+\mu)^{2}-4K_{odd}(A_{odd}+K_{odd})}]$
(S61)

Applying the residue formula results in the following static propagator:

$\begin{split}&\langle
u_{h}(\mathbf{q})u_{v}(-\mathbf{q})\rangle=\frac{k_{B}T\mathcal{L}}{A\mathcal{D}[\lambda+3\mu][\mu(\lambda+2\mu)+K_{odd}(K_{odd}-A_{odd})]q^{4}}\\\
&\\{[\mu(\lambda+3\mu)+K_{odd}(2K_{odd}-A_{odd})]q_{h}q_{v}\\\
&+[\epsilon_{hk}q_{v}q_{k}+\epsilon_{vl}q_{h}q_{l}][A_{odd}\mu+K_{odd}(\lambda+\mu)]\\\
&+\epsilon_{hk}\epsilon_{vl}q_{k}q_{l}[(K_{odd}-A_{odd})(2K_{odd}-A_{odd})+(\lambda+2\mu)(\lambda+3\mu)]\\}\end{split}$
(S62)

###  S-3.2 Reformulating MSRJD Action

We can further simplify these equations via two observations. Firstly, we can
scale out $\kappa$ via the following transformation:

$\begin{split}&\\{\Phi^{\alpha},f^{\alpha},\mathcal{C}_{ijkl},u_{j},\Upsilon_{j},\mathcal{D},\mathcal{L},\mathcal{D}_{f},\mathcal{L}_{f},\sigma^{o}\\}\rightarrow\\{\sqrt{\kappa}\Phi^{\alpha},\frac{1}{\sqrt{\kappa}}f^{\alpha},\mathcal{C}_{ijkl}\kappa^{2},u_{j}/\kappa,\kappa\Upsilon_{j},\frac{1}{\kappa^{2}}\mathcal{D},\frac{1}{\kappa^{2}}\mathcal{L},\mathcal{D}_{f}/\kappa,\mathcal{L}_{f}/\kappa,\sigma^{o}\kappa\\}\end{split}$
(S63)

The second simplifying step can be taken by recognizing that the absolute
value of the critical temperature of the theory,
$\mathcal{L}_{f}/\mathcal{D}_{f}$, is not important Täuber (2014). Thus one
can rescale all the order parameters and diffusivities such that
$\mathcal{L}_{f}$ disappears from the equations:

$\begin{split}&\\{\Phi^{\alpha},f^{\alpha},\mathcal{C}_{ijkl},u_{j},\Upsilon_{j},\mathcal{D},\mathcal{L}\\}=\\{\sqrt{\frac{\mathcal{D}_{f}}{\mathcal{L}_{f}}}\tilde{\Phi}^{\alpha},\sqrt{\frac{\mathcal{L}_{f}}{\mathcal{D}_{f}}}\tilde{f}^{\alpha},\frac{\mathcal{D}_{f}}{\mathcal{L}_{f}}\tilde{\mathcal{C}}_{ijkl},\frac{\mathcal{L}_{f}}{\mathcal{D}_{f}}\tilde{u}_{j},\frac{\mathcal{D}_{f}}{\mathcal{L}_{f}}\tilde{\Upsilon}_{j},\frac{\mathcal{L}_{f}}{\mathcal{D}_{f}}\tilde{\mathcal{D}},\bigg{(}\frac{\mathcal{L}_{f}}{\mathcal{D}_{f}}\bigg{)}^{2}\tilde{\mathcal{L}}\\}\end{split}$
(S64)

Hence, the harmonic theory adjusts as follows:

$\begin{split}&\mathcal{A}_{MSRJD}^{harm.}=\frac{1}{2}\begin{bmatrix}\tilde{\Upsilon}_{j}(\mathbf{q},\omega)\\\
\tilde{u}_{j}(\mathbf{q},\omega)\end{bmatrix}^{T}\begin{bmatrix}-2k_{B}T\tilde{\mathcal{L}}\delta_{jl}&i\omega\delta_{jl}+\tilde{\mathcal{D}}\tilde{\mathcal{C}}_{ijkl}q_{i}q_{k}\\\
-i\omega\delta_{jl}+\tilde{\mathcal{D}}\tilde{\mathcal{C}}_{ilkj}q_{i}q_{k}&0\end{bmatrix}\begin{bmatrix}\tilde{\Upsilon}_{l}(-\mathbf{q},-\omega)\\\
\tilde{u}_{l}(-\mathbf{q},-\omega)\end{bmatrix}\\\
&+\frac{\delta_{\alpha\beta}}{2}\begin{bmatrix}\tilde{\Phi}^{\alpha}(\mathbf{q},\omega)\\\
\tilde{f}^{\alpha}(\mathbf{q},\omega)\end{bmatrix}^{T}\begin{bmatrix}-2k_{B}T\mathcal{D}_{f}&i\omega+\mathcal{D}_{f}q^{4}+\mathcal{D}_{f}\sigma^{o}q^{2}\\\
-i\omega+\mathcal{D}_{f}q^{4}+\mathcal{D}_{f}\sigma^{o}q^{2}&0\end{bmatrix}\begin{bmatrix}\tilde{\Phi}^{\beta}(-\mathbf{q},-\omega)\\\
\tilde{f}^{\beta}(-\mathbf{q},-\omega)\end{bmatrix}\end{split}$ (S65)

Despite having introduced these extra Hubbard-Stratonovich variables, a key
simplifying feature of the MSRJD approach can be seen from the form of these
matrices: Feynman diagrams which contain the contraction of two response
variables together will be null Täuber (2014); Altland and Simons (2010).
Thus, such Feynman diagrams need not be considered. But before obtaining
information about the renormalization factors and the Feynman diagrams, we
first review the scaling of the harmonic theory.

###  S-3.3 Scaling of Harmonic Theory

We briefly discuss the scaling of the harmonic theory. Since we have two order
parameters with different dispersion relations, we are first confronted with
how to rescale frequencies. To resolve this we must go to the propagators in
the linear theory:

$\langle\tilde{\Phi}^{\alpha}(\mathbf{q},\omega)\tilde{f}^{\alpha}(-\mathbf{q},-\omega)\rangle\sim\frac{k_{B}T\mathcal{D}_{f}}{A\tau[-i\omega+\mathcal{D}_{f}(q^{4}+\sigma^{o}q^{2})]}$
(S66)

where we will make the assumptions of vanishing stress,
$\sigma^{o}q^{2}<<\kappa q^{4}$, and

$\langle\tilde{\Upsilon}_{j}(\mathbf{q},\omega)\tilde{u}_{l}(-\mathbf{q},-\omega)\rangle\sim\frac{k_{B}T\tilde{\mathcal{L}}}{A\tau[-i\omega\delta_{jl}+\tilde{\mathcal{D}}\tilde{\mathcal{C}}_{ijkl}q_{i}q_{k}]}$
(S67)

From this we see that in the small stress limit, the residues scale as
$\omega\sim q^{4}$ in the flexural response propagator and $\omega\sim q^{2}$
for the in-plane phonon response propagator Frey and Nelson (1991);
Radzihovsky and Toner (1995); Täuber (2014). Thus in the small frequency
limit, the flexural modes are much slower and thus in-plane phonons should be
considered as “fast” variables in the field theoretic sense. Thus, the terms
that establish the harmonic theory in $\mathcal{A}_{MSRJD}$ are:

$\tilde{\Phi}^{\alpha}\partial_{t}\tilde{f}^{\alpha},\tilde{\Phi}^{\alpha}\nabla^{4}\tilde{f}^{\alpha},\tilde{\Phi}^{\alpha
2}$ (S68)

This makes the coefficients of these terms automatically scale-invariant. We
now perform a power counting procedure to obtain the scaling of the theory and
render the action, $\mathcal{A}_{MSRJD}$, massless Kardar (2007). Thus if we
assign scale powers in a momentum-shell sense Kardar (2007):

$\\{\mathbf{r}\rightarrow b\mathbf{r}\\},\\{t\rightarrow
b^{\zeta_{t}}t\\},\\{\tilde{f}^{\alpha}\rightarrow
b^{\zeta_{f}}\tilde{f}^{\alpha}\\},\\{\tilde{\Phi}^{\alpha}\rightarrow
b^{\zeta_{\Phi}}\tilde{\Phi}^{\alpha}\\}$ (S69)

then we obtain that:

$\zeta_{t}=4,\zeta_{f}=\frac{-D+\zeta_{t}}{2},\zeta_{\Phi}=\frac{-D-\zeta_{t}}{2}$
(S70)

Though we are taking $D=2$ (the number of in-plane dimensions), we leave $D$
un-inserted to show the general scaling. If we also assign the following
rescaling factors:

$\\{\tilde{u}_{i}\rightarrow
b^{\zeta_{u}}\tilde{u}_{i}\\},\\{\tilde{\Upsilon}_{i}\rightarrow
b^{\zeta_{\Upsilon}}\tilde{\Upsilon}_{i}\\}$ (S71)

then via the form of the strain tensor we also obtain:

$\zeta_{u}=2\zeta_{f}-1=-D+\zeta_{t}-1$ (S72)

and thus via observing that $\tilde{\Upsilon}_{i}\partial_{t}\tilde{u}_{i}$
must also be scale invariant we obtain:

$\zeta_{\Upsilon}=D+1$ (S73)

The linear scaling of the theory thus results in the following mass
dimensions:

$\\{\tilde{\mathcal{C}}_{ijkl}\rightarrow
b^{4-D}\tilde{\mathcal{C}}_{ijkl}\\},\\{\tilde{\mathcal{L}},\tilde{\mathcal{D}}\rightarrow
b^{D-2}\tilde{\mathcal{L}},\tilde{\mathcal{D}}\\},\\{\sigma^{o}\rightarrow
b^{2}\sigma^{o}\\}$ (S74)

This establishes that the upper critical dimension for
$\tilde{\mathcal{C}}_{ijkl}$ is $4$. With this scaling, one can see that the
terms $\tilde{\Upsilon}_{i}\partial_{t}^{2}\tilde{u}_{i}$ and
$\tilde{\Phi}^{\alpha}\partial_{t}^{2}\tilde{f}^{\alpha}$ would be irrelevant,
which both scale with exponent $-4$.

###  S-3.4 Absence of Ward Identity

An important tool in the renormalization group are the use of symmetries of
the action. In the case of the free energy associated with equilibrium
fluctuating elastic membranes, Guitter et al. Guitter  _et al._ (1989)
established a symmetry of both the strain tensor:

$\begin{split}&f^{\alpha}(\mathbf{r})\rightarrow
f^{\alpha}(\mathbf{r})+A_{i}^{\alpha}r_{i}\\\ &u_{i}(\mathbf{r})\rightarrow
u_{i}(\mathbf{r})-A_{i}^{\alpha}f^{\alpha}(\mathbf{r})-\frac{1}{2}A_{i}^{\alpha}A_{j}^{\alpha}r_{j}\end{split}$
(S75)

which helps to establish the Ward identity associated with the effective
action, $\Gamma$ Peskin (2018):

$\int d^{D}\mathbf{r}\bigg{[}r_{i}\frac{\delta\Gamma}{\delta
f^{\alpha}}-f^{\alpha}\frac{\delta\Gamma}{\delta u_{i}}\bigg{]}=0$ (S76)

This Ward identity spares extra calculations as it enforces that the
coefficient of the $\partial_{i}u_{j}\partial_{k}u_{l}$ vertex will
renormalize exactly as
$\partial_{i}u_{j}\partial_{k}f^{\beta}\partial_{l}f^{\beta}$ and
$\partial_{i}f^{\alpha}\partial_{j}f^{\alpha}\partial_{k}f^{\beta}\partial_{l}f^{\beta}$.
Thus one need only renormalize one of these vertices to obtain the
renormalization of the in-plane elastic constants, $C_{ijkl}$.

Because the symmetry Guitter  _et al._ (1989) obtained is a symmetry of the
strain tensor, as soon as one incorporates components that prevent an elastic
action from being entirely formulated in terms of $u_{ij}$ and $\nabla^{2}f$,
the symmetry and its associated Ward identity no longer hold. Indeed the only
dynamic term that could be added to a free energy should be of the form
$(\partial_{t}u_{ij})^{2}$, which would lead to non-physical forces and
equations that we wouldn’t derive kinematically. One need not the formalism of
the Ward identity to observe this either. From Eq. (3) and Eq. (4) of the main
text, since the kinematic condition is not necessarily satisfied,
$\partial_{i}\sigma_{ij}=\mathcal{C}_{ijkl}\partial_{i}u_{kl}\neq 0$, Eq. S75
is no longer a symmetry of the dynamic equations. This holds whether we are
working with the odd elastic equations or not. Thus Frey and Nelson (1991) is
also missing the Ward identity. In the case of equilibrium Langevin elastic
membranes, Feynman diagrams will retain the same effective structure as in the
case of Guitter  _et al._ (1989) and thus they derive the matching results.
However, for odd elastic systems where there is no analogue and the structure
of elastic tensors has been generalized, this is no longer a guarantee. Thus,
without a Ward identity, further care will be required because many more
Feynman diagrams must be calculated.

One immediate consequence of the lack of the Ward identity means that
renormalization may not preserve equality between the coefficients of vertices
in Eq. (3) and Eq. (4) of the main text, potentially resulting in breaking any
microscopic fluctuation-dissipation. Thus these equations must be generalized
into the following form:

$\partial_{t}\tilde{u}_{j}(\mathbf{r},t)=\tilde{\mathcal{C}}_{ijkl}^{u,\mathcal{D}}\partial_{i}\partial_{k}\tilde{u}_{l}(\mathbf{r},t)+\frac{1}{2}\tilde{\mathcal{C}}_{ijkl}^{f,\mathcal{D}}\partial_{i}(\partial_{k}\tilde{f}^{\alpha}(\mathbf{r},t)\partial_{l}\tilde{f}^{\alpha}(\mathbf{r},t))+\eta_{j}(\mathbf{r},t)$
(S77)
$\begin{split}\partial_{t}\tilde{f}^{\alpha}(\mathbf{r},t)&=[-\mathcal{D}_{f}\Delta^{2}+\tilde{\sigma}^{o}\Delta]\tilde{f}^{\alpha}(\mathbf{r},t)+\tilde{\mathcal{C}}_{ijkl}^{u,\mathcal{D}_{f}}\partial_{i}[\partial_{k}\tilde{u}_{l}(\mathbf{r},t)\partial_{j}\tilde{f}^{\alpha}(\mathbf{r},t)]+\tilde{\mathcal{C}}_{ijkl}^{f,\mathcal{D}_{f}}\partial_{i}[\frac{1}{2}\partial_{k}\tilde{f}^{\beta}(\mathbf{r},t)\partial_{l}\tilde{f}^{\beta}(\mathbf{r},t)\partial_{j}\tilde{f}^{\alpha}(\mathbf{r},t)]+\eta_{f}^{\alpha}(\mathbf{r},t)\end{split}$
(S78)

where one can observe that the diffusivities and elastic tensor have been
combined and $\tilde{\sigma}^{o}=\mathcal{D}_{f}\sigma^{o}$. Consequently the
MSRJD action must also be generalized accordingly. We re-state them in their
final form before commencing the renormalization scheme:

$\begin{split}\eta_{j}(\mathbf{q},\omega)=&q_{i}[\tilde{\mathcal{C}}_{ijkl}^{u,\mathcal{D}}q_{k}\tilde{u}_{l}(\mathbf{q},\omega)-\frac{i}{2}\tilde{\mathcal{C}}_{ijkl}^{f,\mathcal{D}}\sum_{\mathbf{p},\gamma}p_{k}(p_{l}-q_{l})\tilde{f}^{\alpha}(\mathbf{p},\gamma)\tilde{f}^{\alpha}(\mathbf{q}-\mathbf{p},\omega-\gamma)]-i\omega\tilde{u}_{j}(\mathbf{q},\omega)\end{split}$
(S79)

$\begin{split}&\eta_{f}^{\alpha}(\mathbf{q},\omega)=([\mathcal{D}_{f}q^{4}+\tilde{\sigma}^{o}q^{2}]-i\omega)\tilde{f}^{\alpha}(\mathbf{q},\omega)\\\
&+q_{i}\bigg{[}\sum_{(\mathbf{p},\gamma)\neq\mathbf{0}}\bigg{(}i\tilde{\mathcal{C}}_{ijkl}^{u,\mathcal{D}_{f}}p_{k}u_{l}(\mathbf{p},\gamma)-\frac{\tilde{\mathcal{C}}_{ijkl}^{f,\mathcal{D}_{f}}}{2}\sum_{\mathbf{z},\xi}(p_{k}-z_{k})z_{l}\tilde{f}^{\beta}(\mathbf{p}-\mathbf{z},\gamma-\xi)\tilde{f}^{\beta}(\mathbf{z},\xi)\bigg{)}(q_{j}-p_{j})\tilde{f}^{\alpha}(\mathbf{q}-\mathbf{p},\omega-\gamma)\bigg{]}\end{split}$
(S80)

$\begin{split}&\mathcal{W}(\eta_{j},\eta_{f}^{\alpha},\Upsilon_{i},\tilde{\Phi}^{\alpha})\propto
e^{-\mathcal{A}_{MSRJD}}=e^{\int d\omega
A\cdot\tau\sum_{\mathbf{q}}[k_{B}T\tilde{\mathcal{L}}|\tilde{\Upsilon}_{i}(\mathbf{q},\omega)|^{2}-\tilde{\Upsilon}_{i}(\mathbf{q},\omega)\eta_{i}(-\mathbf{q},-\omega)+k_{B}T\mathcal{D}_{f}|\tilde{\Phi}^{\alpha}(\mathbf{q},\omega)|^{2}-\tilde{\Phi}^{\alpha}(\mathbf{q},\omega)\eta_{f}^{\alpha}(-\mathbf{q},-\omega)]}\end{split}$
(S81)

with the harmonic portion of the action taking the form:

$\begin{split}&\mathcal{A}_{MSRJD}^{harm.}=\frac{1}{2}\begin{bmatrix}\tilde{\Upsilon}_{j}(\mathbf{q},\omega)\\\
\tilde{u}_{j}(\mathbf{q},\omega)\end{bmatrix}^{T}\begin{bmatrix}-2k_{B}T\tilde{\mathcal{L}}\delta_{jl}&i\omega\delta_{jl}+\tilde{\mathcal{C}}_{ijkl}^{u,\mathcal{D}}q_{i}q_{k}\\\
-i\omega\delta_{jl}+\tilde{\mathcal{C}}_{ijkl}^{u,\mathcal{D}}q_{i}q_{k}&0\end{bmatrix}\begin{bmatrix}\tilde{\Upsilon}_{l}(-\mathbf{q},-\omega)\\\
\tilde{u}_{l}(-\mathbf{q},-\omega)\end{bmatrix}\\\
&+\frac{\delta_{\alpha\beta}}{2}\begin{bmatrix}\tilde{\Phi}^{\alpha}(\mathbf{q},\omega)\\\
\tilde{f}^{\alpha}(\mathbf{q},\omega)\end{bmatrix}^{T}\begin{bmatrix}-2k_{B}T\mathcal{D}_{f}&i\omega+\mathcal{D}_{f}q^{4}+\tilde{\sigma}^{o}q^{2}\\\
-i\omega+\mathcal{D}_{f}q^{4}+\tilde{\sigma}^{o}q^{2}&0\end{bmatrix}\begin{bmatrix}\tilde{\Phi}^{\beta}(-\mathbf{q},-\omega)\\\
\tilde{f}^{\beta}(-\mathbf{q},-\omega)\end{bmatrix}\end{split}$ (S82)

with additional anharmonic terms:

$\begin{split}&\mathcal{A}_{MSRJD}^{anharm.}=-\tilde{\Upsilon}_{i}(\mathbf{q},\omega)\bigg{[}\frac{iq_{i}}{2}\tilde{\mathcal{C}}^{f,\mathcal{D}}_{ijkl}\sum_{\mathbf{p},\gamma}p_{k}(p_{l}+q_{l})\tilde{f}^{\alpha}(\mathbf{p},\gamma)\tilde{f}^{\alpha}(-\mathbf{q}-\mathbf{p},-\omega-\gamma)\bigg{]}\\\
&+\tilde{\Phi}^{\alpha}(\mathbf{q},\omega)q_{i}\bigg{[}\sum_{(\mathbf{p},\gamma)\neq\mathbf{0}}\bigg{(}i\tilde{\mathcal{C}}_{ijkl}^{u,\mathcal{D}_{f}}p_{k}u_{l}(\mathbf{p},\gamma)-\frac{\tilde{\mathcal{C}}_{ijkl}^{f,\mathcal{D}_{f}}}{2}\sum_{\mathbf{z},\xi}(p_{k}-z_{k})z_{l}\tilde{f}^{\beta}(\mathbf{p}-\mathbf{z},\gamma-\xi)\tilde{f}^{\beta}(\mathbf{z},\xi)\bigg{)}(-q_{j}-p_{j})\tilde{f}^{\alpha}(-\mathbf{q}-\mathbf{p},-\omega-\gamma)\bigg{]}\end{split}$
(S83)

These non-linear/anharmonic terms can be found in the next section in Fig. S1
as diagrams $(a),(b),(c)$. From here forwards we will take $k_{B}T=1$ as was
done in Frey and Nelson (1991), due to the fact that there are many parameters
that are present in the theory. Setting $k_{B}T=1$ will also not change the
asymptotic results of the scaling of the moduli.

###  S-3.5  Renormalization of Over-Damped Odd Elastic Membranes

####  S-3.5.1  Feynman Diagrams and Renormalization Group Equations

With the set up of the equations complete, we can commence the process of
renormalization. Each term can be renormalized by the contraction of an-
harmonicities Taylor-expanded from the MSRJD action. We intend to show the
calculations diagrammatically with an attached Mathematica code which performs
the accompanying analytic calculations. We begin by representing an-harmonic
terms diagrammatically in isolation. This is done in Fig. S1 (a),(b) and (c).
By further integrating out the harmonic in-plane field using Eq.
(LABEL:eq:MeasureOddFinal) and Eq. (S82), one can also obtain effective
vertices of the flexural field shown in (d) and (e). This is analytically done
in the attached Mathematica code, but is too complicated to put in closed form
in text. The effective flexural vertices will aid us by allowing us to
calculate less Feynman diagrams in total.

Figure S1: The Feynman diagrams corresponding to the linear terms in equations
Eq. (S79) and Eq. (S80) are shown in (a), (b) and (c). If one integrates out
the in-plane order parameters $\tilde{u}_{j},\tilde{\Upsilon}_{j}$, then one
obtains in effective flexural vertices shown in (d) and (e).

We perform a 1-loop perturbative momentum-shell renormalization group scheme
to leading order in $d_{c}$. The 1-particle-irreducible diagrams included in
such a scheme are given in Fig. S2 Peskin (2018); Täuber (2014); Barabási  _et
al._ (1995); Frey and Nelson (1991). Other diagrams in a 1-loop scheme are
ignored as they are lower in order $d_{c}$, such as that found in Fig. S3(c).
However, such diagrams would have to be hypothetically included in an
$\epsilon$-expansion analysis, as all 1-loop diagrams are of order $\epsilon$.
In the case when fluctuation-dissipation relations hold and in the absence of
odd elasticity, these extra diagrams vanish (thus leading to the exact same
analysis as presented by Frey and Nelson (1991)). This is not the case if
those two conditions do not hold. The contributions of these extra diagrams in
the non-odd-elastic case has been recently done by Le Doussal and Radzihovsky
(2021), where breaking the isotropy of the embedding space is effectively
equivalent to our case in which fluctuation-dissipation relations do not hold.
We also note here that these other ignored Feynman diagrams can also allow
$A_{odd}$ and $K_{odd}$ to generate one another. Thus further investigation is
merited in exploring these other potential Feynman diagrams that we have
ignored here. Other diagrams such as Fig. S3(a,b) can be ignored on the
grounds that they produce only higher-order-wavelength contributions to the
original vertices and thus are respectively irrelevant.

Figure S2: The Feynman diagrams corresponding to a 1-loop renormalization
group to leading order in $d_{c}$ are shown. Diagrams in (a) renormalize
$\mathcal{D}_{f}$ and $\tilde{\sigma}^{o}$, (b)
$\tilde{\mathcal{C}}_{ijkl}^{u,\mathcal{D}}$, (c)
$\tilde{\mathcal{C}}_{ijkl}^{f,\mathcal{D}}$, (d)
$\tilde{\mathcal{C}}_{ijkl}^{u,\mathcal{D}_{f}}$, (e)
$\tilde{\mathcal{C}}_{ijkl}^{f,\mathcal{D}_{f}}$.  Figure S3: Examples of
numerous Feynman diagrams that have been ignored are shown. (a) renormalizes
$(\tilde{\Phi}^{\alpha})^{2}$ to non-zero orders of the wave-vector and thus
would be irrelevant with respect to its scalar coefficient. (b) is ignored on
basis of the same argument with respect to $\tilde{\Upsilon}_{j}^{2}$. (c)
Diagrams of this type have been ignored as they contribute to
$\tilde{\mathcal{C}}_{ijkl}^{f,\mathcal{D}}$ to a lower order in $d_{c}$ than
the diagram found in Fig. S2(c).

Considering these Feynman diagrams, we can write down the renormalization
factors in order to calculate our renormalization group equations. To remove
the $UV$ divergences due to an-harmonic Feynman diagrams and reformulate the
theory in terms of renormalized variables, we make the following ansatz of how
to rescale the theory:

$\begin{split}&\tilde{u}_{i}^{R}=Z^{-1}\tilde{u}_{i},\tilde{f}^{\alpha,R}=Z_{f}^{-1/2}\tilde{f}^{\alpha},\\\
&\tilde{\Upsilon}^{R}=Z_{\Upsilon}^{-1}\tilde{\Upsilon},\tilde{\Phi}^{\alpha,R}=Z_{\Phi}^{-1/2}\tilde{\Phi}^{\alpha}\end{split}$
(S84) $\mathcal{D}_{f}^{R}=Z_{\mathcal{D}_{f}}D_{f}$ (S85)
$\tilde{\mathcal{L}}^{R}=Z_{L}\tilde{\mathcal{L}}$ (S86)
$\tilde{\mathcal{C}}_{ijkl}^{u,\mathcal{D},R}=Z_{ijkl}^{u,\mathcal{D}}\tilde{\mathcal{C}}_{ijkl}^{u,\mathcal{D}}$
(S87)
$\tilde{\mathcal{C}}_{ijkl}^{f,\mathcal{D},R}=Z_{ijkl}^{f,\mathcal{D}}\tilde{\mathcal{C}}_{ijkl}^{f,\mathcal{D}}$
(S88)
$\tilde{\mathcal{C}}_{ijkl}^{u,\mathcal{D}_{f},R}=Z_{\mathcal{D}_{f}}^{-1}Z_{ijkl}^{u,\mathcal{D}_{f}}\tilde{\mathcal{C}}_{ijkl}^{u,\mathcal{D}_{f}}$
(S89)
$\tilde{\mathcal{C}}_{ijkl}^{f,\mathcal{D}_{f},R}=Z_{\mathcal{D}_{f}}^{-1}Z_{ijkl}^{f,\mathcal{D}_{f}}\tilde{\mathcal{C}}_{ijkl}^{f,\mathcal{D}_{f}}$
(S90) $\tilde{\sigma}^{R}=Z_{\sigma}\tilde{\sigma}^{o}$ (S91)

where we have used our scale transformations in Eq.
LABEL:eq:DfScaleTransformation to account for how the diffusivity,
$\mathcal{D}_{f}$, has been absorbed into the tensors
$\tilde{\mathcal{C}}_{ijkl}^{u,\mathcal{D}_{f}},\tilde{\mathcal{C}}_{ijkl}^{f,\mathcal{D}_{f}}$
and the stress $\tilde{\sigma}^{o}$. From this ansatz, similar as to Frey and
Nelson (1991), we can obtain certain reductions in the number of independent
renormalization factors. We obtain symmetries of
$\Gamma_{M,\tilde{M},N,\tilde{N}}$ which is the effective vertex function with
$M$ $f$ fields, $\tilde{M}$ $\Phi$ fields, $N$ $u$ fields and $\tilde{N}$
$\Upsilon$ fields as in Frey and Nelson (1991); Peskin (2018). Since:

$\begin{split}&\partial_{\omega}\Gamma_{1100}(\mathbf{q}=0,\omega)\sim
i\sqrt{Z_{f}Z_{\Phi}}\\\
&\partial_{\omega}\Gamma_{0011}(\mathbf{q}=0,\omega)\sim
iZ_{u}Z_{\Upsilon}\end{split}$ (S92)

and since all an-harmonic vertices vanish when $\mathbf{q}=0$, we obtain that
$Z=1/Z_{\Upsilon}$ and $Z_{f}=1/Z_{\Phi}$. Furthermore:

$\Gamma_{0200}(\mathbf{q}=0,\omega)\sim Z_{\mathcal{D}_{f}}Z_{\Phi}$ (S93)

which also implies that $Z_{\mathcal{D}_{f}}=1/Z_{\Phi}$. Thus we have
established that $Z_{f}=Z_{\mathcal{D}_{f}}$. However, unlike Frey and Nelson
(1991), we cannot use any Ward identity as we have shown that it is not valid
for a dynamical action and thus we cannot establish that $Z=Z_{f}$. This is an
important point because without the Ward identity, we cannot establish further
simplifications of the following renormalization factors
$Z_{ijkl}^{u,\mathcal{D}},Z_{ijkl}^{f,\mathcal{D}},Z_{ijkl}^{u,\mathcal{D}_{f}},Z_{ijkl}^{f,\mathcal{D}_{f}}$.
Thus we must treat these renormalization factors as independent. With
renormalization factors established, we can write down the following
renormalization group ODEs for the following parameters with the use of our
evaluated Feynman diagrams:

$\displaystyle\beta_{\tilde{\mu}^{u,\mathcal{D},R}}=2\tilde{\mu}^{u,\mathcal{D},R}-d_{c}\mathcal{D}_{f}^{R}\Lambda^{2}\frac{\tilde{\mu}^{f,\mathcal{D},R}\tilde{\mu}^{u,\mathcal{D}_{f},R}-\tilde{K}_{odd}^{f,\mathcal{D},R}\tilde{K}_{odd}^{u,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\tilde{\sigma}^{R}]^{2}}$
(S94a)
$\displaystyle\beta_{\tilde{\lambda}^{u,\mathcal{D},R}}=2\tilde{\lambda}^{u,\mathcal{D},R}-d_{c}\mathcal{D}_{f}^{R}\Lambda^{2}\frac{2\tilde{\lambda}^{f,\mathcal{D},R}(\tilde{\lambda}^{u,\mathcal{D}_{f},R}+\tilde{\mu}^{u,\mathcal{D}_{f},R})+\tilde{\mu}^{f,\mathcal{D},R}(2\tilde{\lambda}^{u,\mathcal{D}_{f},R}+\tilde{\mu}^{u,\mathcal{D}_{f},R})-\tilde{K}_{odd}^{f,\mathcal{D},R}\tilde{K}_{odd}^{u,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\tilde{\sigma}^{R}]^{2}}$
$\displaystyle\beta_{\tilde{K}_{odd}^{u,\mathcal{D},R}}=2\tilde{K}_{odd}^{u,\mathcal{D},R}-d_{c}\mathcal{D}_{f}^{R}\Lambda^{2}\frac{\tilde{K}_{odd}^{f,\mathcal{D},R}\tilde{\mu}^{u,\mathcal{D}_{f},R}+\tilde{\mu}^{f,\mathcal{D},R}\tilde{K}_{odd}^{u,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\tilde{\sigma}^{R}]^{2}}$
$\displaystyle\beta_{\tilde{A}_{odd}^{u,\mathcal{D},R}}=2\tilde{A}_{odd}^{u,\mathcal{D},R}-d_{c}\mathcal{D}_{f}^{R}\Lambda^{2}\frac{\tilde{A}_{odd}^{f,\mathcal{D},R}(\tilde{\mu}^{u,\mathcal{D}_{f},R}+\tilde{\lambda}^{u,\mathcal{D}_{f},R})}{4\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\tilde{\sigma}^{R}]^{2}}$
$\displaystyle\beta_{\tilde{\mu}^{u,\mathcal{D}_{f},R}}=2\tilde{\mu}^{u,\mathcal{D}_{f},R}-d_{c}\mathcal{D}_{f}^{R}\Lambda^{2}\frac{\tilde{\mu}^{f,\mathcal{D}_{f},R}\tilde{\mu}^{u,\mathcal{D}_{f},R}-\tilde{K}_{odd}^{f,\mathcal{D}_{f},R}\tilde{K}_{odd}^{u,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\tilde{\sigma}^{R}]^{2}}-\frac{\tilde{\mu}^{u,\mathcal{D}_{f},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\tilde{\lambda}^{u,\mathcal{D}_{f},R}}=2\tilde{\lambda}^{u,\mathcal{D}_{f},R}-d_{c}\mathcal{D}_{f}^{R}\Lambda^{2}\frac{2\tilde{\lambda}^{f,\mathcal{D}_{f},R}(\tilde{\lambda}^{u,\mathcal{D}_{f},R}+\tilde{\mu}^{u,\mathcal{D}_{f},R})+\tilde{\mu}^{f,\mathcal{D}_{f},R}(2\tilde{\lambda}^{u,\mathcal{D}_{f},R}+\tilde{\mu}^{u,\mathcal{D}_{f},R})}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\tilde{\sigma}^{R}]^{2}}$
(S94b)
$\displaystyle+d_{c}\mathcal{D}_{f}^{R}\Lambda^{2}\frac{\tilde{K}_{odd}^{f,\mathcal{D}_{f},R}\tilde{K}_{odd}^{u,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\tilde{\sigma}^{R}]^{2}}-\frac{\tilde{\lambda}^{u,\mathcal{D}_{f},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\tilde{K}_{odd}^{u,\mathcal{D}_{f},R}}=2\tilde{K}_{odd}^{u,\mathcal{D}_{f},R}-d_{c}\mathcal{D}_{f}^{R}\Lambda^{2}\frac{\tilde{K}_{odd}^{f,\mathcal{D}_{f},R}\tilde{\mu}^{u,\mathcal{D}_{f},R}+\tilde{\mu}^{f,\mathcal{D}_{f},R}\tilde{K}_{odd}^{u,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\tilde{\sigma}^{R}]^{2}}-\frac{\tilde{K}_{odd}^{u,\mathcal{D}_{f},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\tilde{A}_{odd}^{u,\mathcal{D}_{f},R}}=2\tilde{A}_{odd}^{u,\mathcal{D}_{f},R}-d_{c}\mathcal{D}_{f}^{R}\Lambda^{2}\frac{\tilde{A}_{odd}^{f,\mathcal{D}_{f},R}(\tilde{\mu}^{u,\mathcal{D}_{f},R}+\tilde{\lambda}^{u,\mathcal{D}_{f},R})}{4\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\tilde{\sigma}^{R}]^{2}}-\frac{\tilde{A}_{odd}^{u,\mathcal{D}_{f},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\tilde{\mu}^{f,\mathcal{D},R}}=2\tilde{\mu}^{f,\mathcal{D},R}-d_{c}\mathcal{D}_{f}^{R}\Lambda^{2}\frac{\tilde{\mu}^{f,\mathcal{D},R}\tilde{\mu}^{f,\mathcal{D}_{f},R}-\tilde{K}_{odd}^{f,\mathcal{D},R}\tilde{K}_{odd}^{f,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\tilde{\sigma}^{R}]^{2}}$
$\displaystyle\beta_{\tilde{\lambda}^{f,\mathcal{D},R}}=2\tilde{\lambda}^{f,\mathcal{D},R}-d_{c}\mathcal{D}_{f}^{R}\Lambda^{2}\frac{2\tilde{\lambda}^{f,\mathcal{D},R}(\tilde{\lambda}^{f,\mathcal{D}_{f},R}+\tilde{\mu}^{f,\mathcal{D}_{f},R})+\tilde{\mu}^{f,\mathcal{D},R}(2\tilde{\lambda}^{f,\mathcal{D}_{f},R}+\tilde{\mu}^{f,\mathcal{D}_{f},R})-\tilde{K}_{odd}^{f,\mathcal{D},R}\tilde{K}_{odd}^{f,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\tilde{\sigma}^{R}]^{2}}$
$\displaystyle\beta_{\tilde{K}_{odd}^{f,\mathcal{D},R}}=2\tilde{K}_{odd}^{f,\mathcal{D},R}-d_{c}\mathcal{D}_{f}^{R}\Lambda^{2}\frac{\tilde{K}_{odd}^{f,\mathcal{D},R}\tilde{\mu}^{f,\mathcal{D}_{f},R}+\tilde{\mu}^{f,\mathcal{D},R}\tilde{K}_{odd}^{f,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\tilde{\sigma}^{R}]^{2}}$
$\displaystyle\beta_{\tilde{A}_{odd}^{f,\mathcal{D},R}}=2\tilde{A}_{odd}^{f,\mathcal{D},R}-d_{c}\mathcal{D}_{f}^{R}\Lambda^{2}\frac{\tilde{A}_{odd}^{f,\mathcal{D},R}(\tilde{\mu}^{f,\mathcal{D}_{f},R}+\tilde{\lambda}^{f,\mathcal{D}_{f},R})}{4\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\tilde{\sigma}^{R}]^{2}}$
$\displaystyle\beta_{\tilde{\mu}^{f,\mathcal{D}_{f},R}}=2\tilde{\mu}^{f,\mathcal{D}_{f},R}-d_{c}\mathcal{D}_{f}^{R}\Lambda^{2}\frac{\tilde{\mu}^{f,\mathcal{D}_{f},R}\tilde{\mu}^{f,\mathcal{D}_{f},R}-\tilde{K}_{odd}^{f,\mathcal{D}_{f},R}\tilde{K}_{odd}^{f,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\tilde{\sigma}^{R}]^{2}}-\frac{\tilde{\mu}^{f,\mathcal{D}_{f},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\tilde{\lambda}^{f,\mathcal{D}_{f},R}}=2\tilde{\lambda}^{f,\mathcal{D}_{f},R}-d_{c}\mathcal{D}_{f}^{R}\Lambda^{2}\frac{2\tilde{\lambda}^{f,\mathcal{D}_{f},R}(\tilde{\lambda}^{f,\mathcal{D}_{f},R}+\tilde{\mu}^{f,\mathcal{D}_{f},R})+\tilde{\mu}^{f,\mathcal{D}_{f},R}(2\tilde{\lambda}^{f,\mathcal{D}_{f},R}+\tilde{\mu}^{f,\mathcal{D}_{f},R})}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\tilde{\sigma}^{R}]^{2}}$
$\displaystyle+d_{c}\mathcal{D}_{f}^{R}\Lambda^{2}\frac{\tilde{K}_{odd}^{f,\mathcal{D}_{f},R}\tilde{K}_{odd}^{f,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\tilde{\sigma}^{R}]^{2}}-\frac{\tilde{\lambda}^{f,\mathcal{D}_{f},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\tilde{K}_{odd}^{f,\mathcal{D}_{f},R}}=2\tilde{K}_{odd}^{f,\mathcal{D}_{f},R}-d_{c}\mathcal{D}_{f}^{R}\Lambda^{2}\frac{\tilde{K}_{odd}^{f,\mathcal{D}_{f},R}\tilde{\mu}^{f,\mathcal{D}_{f},R}+\tilde{\mu}^{f,\mathcal{D}_{f},R}\tilde{K}_{odd}^{f,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\tilde{\sigma}^{R}]^{2}}-\frac{\tilde{K}_{odd}^{f,\mathcal{D}_{f},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\tilde{A}_{odd}^{f,\mathcal{D}_{f},R}}=2\tilde{A}_{odd}^{f,\mathcal{D}_{f},R}-d_{c}\mathcal{D}_{f}^{R}\Lambda^{2}\frac{\tilde{A}_{odd}^{f,\mathcal{D}_{f},R}(\tilde{\mu}^{f,\mathcal{D}_{f},R}+\tilde{\lambda}^{f,\mathcal{D}_{f},R})}{4\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\tilde{\sigma}^{R}]^{2}}-\frac{\tilde{A}_{odd}^{f,\mathcal{D}_{f},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\tilde{\sigma}^{R}}=2\tilde{\sigma}^{R}+\frac{\partial
logZ_{\sigma}}{\partial b}$
$\displaystyle\beta_{\tilde{\mathcal{L}}^{R}}=2\frac{\tilde{\mathcal{L}}^{R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\mathcal{D}_{f}^{R}}=\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b},$

where we have used that $Z_{\mathcal{L}}=1$ since no Feynman diagrams
contribute to zero-th order in the wave vectors to $\Upsilon_{j}^{2}$.
Furthermore $\Lambda$ is the UV cutoff, in other words the Fourier wave-vector
corresponding to the microscopic length scale of the theory (for example, the
lattice spacing of the system). The parameter $b=e^{s}$ where $s$ is the re-
scaling parameter Tong ; Peskin (2018); Kardar (2007). Note as well that the
stress does in fact renormalize, thus from a zero-stress state, a non-zero
stress can be generated in the presence of odd elastic moduli or a mis-match
of diffusivities. More on this will be mentioned in Sec. S-3.5.4.

####  S-3.5.2 Stability Analysis

With Eq. (S94), we may perform a stability analysis of the Aronovitz-Lubensky
fixed point Aronovitz and Lubensky (1988). We are interested in having only
physically relevant parameters. Thus we further reduce the equations so that
the only parameters considered are:

$\begin{split}&\\{\tilde{\mathcal{C}}_{ijkl}^{u,\mathcal{D},R}/\tilde{\mathcal{L}}^{R},\tilde{\mathcal{C}}_{ijkl}^{f,\mathcal{D},R}/\tilde{\mathcal{L}}^{R},\tilde{\mathcal{C}}_{ijkl}^{u,\mathcal{D}_{f},R}/\mathcal{D}_{f}^{R},\tilde{\mathcal{C}}_{ijkl}^{f,\mathcal{D}_{f},R}/\mathcal{D}_{f}^{R},\tilde{\sigma}^{R}/\mathcal{D}_{f}^{R}\\}=\\{\hat{\tilde{\mathcal{C}}}_{ijkl}^{u,\mathcal{D},R},\hat{\tilde{\mathcal{C}}}_{ijkl}^{f,\mathcal{D},R},\hat{\tilde{\mathcal{C}}}_{ijkl}^{u,\mathcal{D}_{f},R},\hat{\tilde{\mathcal{C}}}_{ijkl}^{f,\mathcal{D}_{f},R},\hat{\tilde{\sigma}}^{R}\\}\end{split}$
(S95)

The equations derived are then:

$\displaystyle\beta_{\hat{\tilde{\mu}}^{u,\mathcal{D},R}}=2\hat{\tilde{\mu}}^{u,\mathcal{D},R}-d_{c}(\mathcal{D}_{f}^{R})^{2}\Lambda^{2}\frac{\hat{\tilde{\mu}}^{f,\mathcal{D},R}\hat{\tilde{\mu}}^{u,\mathcal{D}_{f},R}-\hat{\tilde{K}}_{odd}^{f,\mathcal{D},R}\hat{\tilde{K}}_{odd}^{u,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}-2\frac{\hat{\tilde{\lambda}}^{u,\mathcal{D},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\hat{\tilde{\lambda}}^{u,\mathcal{D},R}}=2\hat{\tilde{\lambda}}^{u,\mathcal{D},R}-d_{c}(\mathcal{D}_{f}^{R})^{2}\Lambda^{2}\frac{2\hat{\tilde{\lambda}}^{f,\mathcal{D},R}(\hat{\tilde{\lambda}}^{u,\mathcal{D}_{f},R}+\hat{\tilde{\mu}}^{u,\mathcal{D}_{f},R})+\hat{\tilde{\mu}}^{f,\mathcal{D},R}(2\hat{\tilde{\lambda}}^{u,\mathcal{D}_{f},R}+\hat{\tilde{\mu}}^{u,\mathcal{D}_{f},R})}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}$
$\displaystyle+d_{c}(\mathcal{D}_{f}^{R})^{2}\Lambda^{2}\frac{\hat{\tilde{K}}_{odd}^{f,\mathcal{D},R}\hat{\tilde{K}}_{odd}^{u,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}-2\frac{\hat{\tilde{\lambda}}^{u,\mathcal{D},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\hat{\tilde{K}}_{odd}^{u,\mathcal{D},R}}=2\hat{\tilde{K}}_{odd}^{u,\mathcal{D},R}-d_{c}(\mathcal{D}_{f}^{R})^{2}\Lambda^{2}\frac{\hat{\tilde{K}}_{odd}^{f,\mathcal{D},R}\hat{\tilde{\mu}}^{u,\mathcal{D}_{f},R}+\hat{\tilde{\mu}}^{f,\mathcal{D},R}\hat{\tilde{K}}_{odd}^{u,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}-2\frac{\hat{\tilde{K}}_{odd}^{u,\mathcal{D},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\hat{\tilde{A}}_{odd}^{u,\mathcal{D},R}}=2\hat{\tilde{A}}_{odd}^{u,\mathcal{D},R}-d_{c}(\mathcal{D}_{f}^{R})^{2}\Lambda^{2}\frac{\hat{\tilde{A}}_{odd}^{f,\mathcal{D},R}(\hat{\tilde{\mu}}^{u,\mathcal{D}_{f},R}+\hat{\tilde{\lambda}}^{u,\mathcal{D}_{f},R})}{4\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}-2\frac{\hat{\tilde{A}}_{odd}^{u,\mathcal{D},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\hat{\tilde{\mu}}^{u,\mathcal{D}_{f},R}}=2\hat{\tilde{\mu}}^{u,\mathcal{D}_{f},R}-d_{c}(\mathcal{D}_{f}^{R})^{2}\Lambda^{2}\frac{\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R}\hat{\tilde{\mu}}^{u,\mathcal{D}_{f},R}-\hat{\tilde{K}}_{odd}^{f,\mathcal{D}_{f},R}\hat{\tilde{K}}_{odd}^{u,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}-2\frac{\hat{\tilde{\mu}}^{u,\mathcal{D}_{f},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\hat{\tilde{\lambda}}^{u,\mathcal{D}_{f},R}}=2\hat{\tilde{\lambda}}^{u,\mathcal{D}_{f},R}-d_{c}(\mathcal{D}_{f}^{R})^{2}\Lambda^{2}\frac{2\hat{\tilde{\lambda}}^{f,\mathcal{D}_{f},R}(\hat{\tilde{\lambda}}^{u,\mathcal{D}_{f},R}+\hat{\tilde{\mu}}^{u,\mathcal{D}_{f},R})+\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R}(2\hat{\tilde{\lambda}}^{u,\mathcal{D}_{f},R}+\hat{\tilde{\mu}}^{u,\mathcal{D}_{f},R})}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}$
$\displaystyle+\frac{\hat{\tilde{K}}_{odd}^{f,\mathcal{D}_{f},R}\hat{\tilde{K}}_{odd}^{u,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}-2\frac{\hat{\tilde{\lambda}}^{u,\mathcal{D}_{f},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\hat{\tilde{K}}_{odd}^{u,\mathcal{D}_{f},R}}=2\hat{\tilde{K}}_{odd}^{u,\mathcal{D}_{f},R}-d_{c}(\mathcal{D}_{f}^{R})^{2}\Lambda^{2}\frac{\hat{\tilde{K}}_{odd}^{f,\mathcal{D}_{f},R}\hat{\tilde{\mu}}^{u,\mathcal{D}_{f},R}+\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R}\hat{\tilde{K}}_{odd}^{u,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}-2\frac{\hat{\tilde{K}}_{odd}^{u,\mathcal{D}_{f},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\hat{\tilde{A}}_{odd}^{u,\mathcal{D}_{f},R}}=2\hat{\tilde{A}}_{odd}^{u,\mathcal{D}_{f},R}-d_{c}(\mathcal{D}_{f}^{R})^{2}\Lambda^{2}\frac{\hat{\tilde{A}}_{odd}^{f,\mathcal{D}_{f},R}(\hat{\tilde{\mu}}^{u,\mathcal{D}_{f},R}+\hat{\tilde{\lambda}}^{u,\mathcal{D}_{f},R})}{4\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}-2\frac{\hat{\tilde{A}}_{odd}^{u,\mathcal{D}_{f},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\hat{\tilde{\mu}}^{f,\mathcal{D},R}}=2\hat{\tilde{\mu}}^{f,\mathcal{D},R}-d_{c}(\mathcal{D}_{f}^{R})^{2}\Lambda^{2}\frac{\hat{\tilde{\mu}}^{f,\mathcal{D},R}\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R}-\hat{\tilde{K}}_{odd}^{f,\mathcal{D},R}\hat{\tilde{K}}_{odd}^{f,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}-2\frac{\hat{\tilde{\mu}}^{f,\mathcal{D},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\hat{\tilde{\lambda}}^{f,\mathcal{D},R}}=2\hat{\tilde{\lambda}}^{f,\mathcal{D},R}-d_{c}(\mathcal{D}_{f}^{R})^{2}\Lambda^{2}\frac{2\hat{\tilde{\lambda}}^{f,\mathcal{D},R}(\hat{\tilde{\lambda}}^{f,\mathcal{D}_{f},R}+\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R})+\hat{\tilde{\mu}}^{f,\mathcal{D},R}(2\hat{\tilde{\lambda}}^{f,\mathcal{D}_{f},R}+\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R})}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}$
$\displaystyle+d_{c}(\mathcal{D}_{f}^{R})^{2}\Lambda^{2}\frac{\hat{\tilde{K}}_{odd}^{f,\mathcal{D},R}\hat{\tilde{K}}_{odd}^{f,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}-2\frac{\hat{\tilde{\lambda}}^{f,\mathcal{D},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\hat{\tilde{K}}_{odd}^{f,\mathcal{D},R}}=2\hat{\tilde{K}}_{odd}^{f,\mathcal{D},R}-d_{c}(\mathcal{D}_{f}^{R})^{2}\Lambda^{2}\frac{\hat{\tilde{K}}_{odd}^{f,\mathcal{D},R}\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R}+\hat{\tilde{\mu}}^{f,\mathcal{D},R}\hat{\tilde{K}}_{odd}^{f,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}-2\frac{\hat{\tilde{K}}_{odd}^{f,\mathcal{D},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\hat{\tilde{A}}_{odd}^{f,\mathcal{D},R}}=2\hat{\tilde{A}}_{odd}^{f,\mathcal{D},R}-d_{c}(\mathcal{D}_{f}^{R})^{2}\Lambda^{2}\frac{\hat{\tilde{A}}_{odd}^{f,\mathcal{D},R}(\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R}+\hat{\tilde{\lambda}}^{f,\mathcal{D}_{f},R})}{4\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}-2\frac{\hat{\tilde{A}}_{odd}^{f,\mathcal{D},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R}}=2\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R}-d_{c}(\mathcal{D}_{f}^{R})^{2}\Lambda^{2}\frac{\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R}\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R}-\hat{\tilde{K}}_{odd}^{f,\mathcal{D}_{f},R}\hat{\tilde{K}}_{odd}^{f,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}-2\frac{\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\hat{\tilde{\lambda}}^{f,\mathcal{D}_{f},R}}=2\hat{\tilde{\lambda}}^{f,\mathcal{D}_{f}^{R}}-d_{c}(\mathcal{D}_{f}^{R})^{2}\Lambda^{2}\frac{2\hat{\tilde{\lambda}}^{f,\mathcal{D}_{f},R}(\hat{\tilde{\lambda}}^{f,\mathcal{D}_{f},R}+\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R})+\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R}(2\hat{\tilde{\lambda}}^{f,\mathcal{D}_{f},R}+\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R})}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}$
$\displaystyle+d_{c}(\mathcal{D}_{f}^{R})^{2}\Lambda^{2}\frac{\hat{\tilde{K}}_{odd}^{f,\mathcal{D}_{f},R}\hat{\tilde{K}}_{odd}^{f,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}-2\frac{\hat{\tilde{\lambda}}^{f,\mathcal{D}_{f},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\hat{\tilde{K}}_{odd}^{f,\mathcal{D}_{f},R}}=2\hat{\tilde{K}}_{odd}^{f,\mathcal{D}_{f},R}-d_{c}(\mathcal{D}_{f}^{R})^{2}\Lambda^{2}\frac{\hat{\tilde{K}}_{odd}^{f,\mathcal{D}_{f},R}\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R}+\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R}\hat{\tilde{K}}_{odd}^{f,\mathcal{D}_{f},R}}{8\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}-2\frac{\hat{\tilde{K}}_{odd}^{f,\mathcal{D}_{f},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\hat{\tilde{A}}_{odd}^{f,\mathcal{D}_{f},R}}=2\hat{\tilde{A}}_{odd}^{f,\mathcal{D}_{f},R}-d_{c}(\mathcal{D}_{f}^{R})^{2}\Lambda^{2}\frac{\hat{\tilde{A}}_{odd}^{f,\mathcal{D}_{f},R}(\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R}+\hat{\tilde{\lambda}}^{f,\mathcal{D}_{f},R})}{4\pi[\mathcal{D}_{f}^{R}\Lambda^{2}+\hat{\tilde{\sigma}}^{R}]^{2}}-2\frac{\hat{\tilde{A}}_{odd}^{f,\mathcal{D}_{f},R}}{\mathcal{D}_{f}^{R}}\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$
$\displaystyle\beta_{\hat{\tilde{\sigma}}^{R}}=2\hat{\tilde{\sigma}}^{R}+\frac{\partial
logZ_{\sigma}}{\partial b}$

Formulated in terms of these variables, one in principle should solve for the
fixed point/manifold. However, as we know $\frac{\partial
logZ_{\mathcal{D}_{f}}}{\partial b}$ is a complicated expression and thus
renders obtainment of these solutions analytically intractable. In addition we
note that, $\frac{\partial logZ_{\sigma}}{\partial b}$ is also quite
complicated as an expression, however it indicates that stress is generated
when fluctuation dissipation is broken or when odd elastic parameters are
present. Despite the complexity of these expressions, if we define the
Aronovitz-Lubensky fixed point as:

$\\{\hat{\tilde{\mu}}^{\gamma,\delta,R},\hat{\tilde{\lambda}}^{\gamma,\delta,R},\hat{\tilde{K}}_{odd}^{\gamma,\delta,R},\hat{\tilde{A}}_{odd}^{\gamma,\delta,R},\hat{\tilde{\sigma}}^{R}\\}=\\{\frac{16\pi\Lambda^{2}}{4+d_{c}},\frac{-8\pi\Lambda^{2}}{4+d_{c}},0,0,0\\}$
(S97)

where $\gamma\in\\{u,f\\},\delta\in\\{\mathcal{D},\mathcal{D}_{f}\\}$, one can
check that it is indeed a fixed point of the equations and we can analyze its
stability. Performing a stability analysis on these 16 variables gives the
following eigen-values:

$\begin{split}&\text{Eigen-
values}=\\{0,0,0,0,0,0,0,0,\frac{-2d_{c}}{4+d_{c}},\frac{-2d_{c}}{4+d_{c}},-2,\frac{2d_{c}}{4+d_{c}},\frac{2d_{c}}{4+d_{c}},\frac{2d_{c}}{4+d_{c}},\frac{2d_{c}}{4+d_{c}},\frac{2d_{c}}{4+d_{c}},\frac{2(2+d_{c})}{4+d_{c}}\\}\end{split}$
(S98)

The eight zero eigen-values indicate that the Aronovitz-Lubensky fixed point
potentially belongs to a higher-dimensional fixed manifold, although perhaps
not a stable one. Four of these eight zero eigen-values are perturbations
purely in
$\\{\hat{\tilde{\mu}}^{\gamma,\delta,R},\hat{\tilde{\lambda}}^{\gamma,\delta,R}\\}$
and three of them are somewhat analytically complicated, however they indicate
that there are potentially a broader set of fixed points where fluctuation-
dissipation may not necessarily hold Le Doussal and Radzihovsky (2021). The
one simple eigen-vector that is a pure perturbation in
$\\{\hat{\tilde{\mu}}^{\gamma,\delta,R},\hat{\tilde{\lambda}}^{\gamma,\delta,R}\\}$
is:

$\begin{split}&[\delta\hat{\tilde{A}}_{odd}^{u,\mathcal{D},R},\delta\hat{\tilde{A}}_{odd}^{f,\mathcal{D},R},\delta\hat{\tilde{A}}_{odd}^{u,\mathcal{D}_{f},R},\delta\hat{\tilde{A}}_{odd}^{f,\mathcal{D}_{f},R},\delta\hat{\tilde{K}}_{odd}^{u,\mathcal{D},R},\delta\hat{\tilde{K}}_{odd}^{f,\mathcal{D},R},\delta\hat{\tilde{K}}_{odd}^{u,\mathcal{D}_{f},R},\delta\hat{\tilde{K}}_{odd}^{f,\mathcal{D}_{f},R},\\\
&\delta\hat{\tilde{\lambda}}^{u,\mathcal{D},R},\delta\hat{\tilde{\lambda}}^{f,\mathcal{D},R},\delta\hat{\tilde{\lambda}}^{u,\mathcal{D}_{f},R},\delta\hat{\tilde{\lambda}}^{f,\mathcal{D}_{f},R},\delta\hat{\tilde{\mu}}^{u,\mathcal{D},R},\delta\hat{\tilde{\mu}}^{f,\mathcal{D},R},\delta\hat{\tilde{\mu}}^{u,\mathcal{D}_{f},R},\delta\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R},\delta\hat{\tilde{\sigma}}^{R}]\\\
&=[0,0,0,0,0,0,0,0,-\frac{-4-3d_{c}}{16},-\frac{-3(4+d_{c})}{32},-\frac{-3(4+d_{c})}{32},-1/2,-1,0,0,1,0]\end{split}$
(S99)

The other 4 zero eigen-values are perturbations in
$\\{\hat{\tilde{K}}_{odd}^{\gamma,\delta,R},\hat{\tilde{A}}_{odd}^{\gamma,\delta,R}\\}$
and include:

$\begin{split}&[\delta\hat{\tilde{A}}_{odd}^{u,\mathcal{D},R},\delta\hat{\tilde{A}}_{odd}^{f,\mathcal{D},R},\delta\hat{\tilde{A}}_{odd}^{u,\mathcal{D}_{f},R},\delta\hat{\tilde{A}}_{odd}^{f,\mathcal{D}_{f},R},\delta\hat{\tilde{K}}_{odd}^{u,\mathcal{D},R},\delta\hat{\tilde{K}}_{odd}^{f,\mathcal{D},R},\delta\hat{\tilde{K}}_{odd}^{u,\mathcal{D}_{f},R},\delta\hat{\tilde{K}}_{odd}^{f,\mathcal{D}_{f},R},\\\
&\delta\hat{\tilde{\lambda}}^{u,\mathcal{D},R},\delta\hat{\tilde{\lambda}}^{f,\mathcal{D},R},\delta\hat{\tilde{\lambda}}^{u,\mathcal{D}_{f},R},\delta\hat{\tilde{\lambda}}^{f,\mathcal{D}_{f},R},\delta\hat{\tilde{\mu}}^{u,\mathcal{D},R},\delta\hat{\tilde{\mu}}^{f,\mathcal{D},R},\delta\hat{\tilde{\mu}}^{u,\mathcal{D}_{f},R},\delta\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R},\delta\hat{\tilde{\sigma}}^{R}]\\\
&=[1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],\\\
&[0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0],\\\
&[0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0],\\\
&[0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0]\end{split}$ (S100)

These four marginal perturbations indicate that perturbations in
$\hat{A}_{odd}^{\gamma,\delta,R}$ that preserve fluctuation dissipation are
marginal. Furthermore the two eigen-vectors corresponding to the eigen-value
$\frac{-2d_{c}}{4+d_{c}}$ are:

$\begin{split}&[\delta\hat{\tilde{A}}_{odd}^{u,\mathcal{D},R},\delta\hat{\tilde{A}}_{odd}^{f,\mathcal{D},R},\delta\hat{\tilde{A}}_{odd}^{u,\mathcal{D}_{f},R},\delta\hat{\tilde{A}}_{odd}^{f,\mathcal{D}_{f},R},\delta\hat{\tilde{K}}_{odd}^{u,\mathcal{D},R},\delta\hat{\tilde{K}}_{odd}^{f,\mathcal{D},R},\delta\hat{\tilde{K}}_{odd}^{u,\mathcal{D}_{f},R},\delta\hat{\tilde{K}}_{odd}^{f,\mathcal{D}_{f},R},\\\
&\delta\hat{\tilde{\lambda}}^{u,\mathcal{D},R},\delta\hat{\tilde{\lambda}}^{f,\mathcal{D},R},\delta\hat{\tilde{\lambda}}^{u,\mathcal{D}_{f},R},\delta\hat{\tilde{\lambda}}^{f,\mathcal{D}_{f},R},\delta\hat{\tilde{\mu}}^{u,\mathcal{D},R},\delta\hat{\tilde{\mu}}^{f,\mathcal{D},R},\delta\hat{\tilde{\mu}}^{u,\mathcal{D}_{f},R},\delta\hat{\tilde{\mu}}^{f,\mathcal{D}_{f},R},\delta\hat{\tilde{\sigma}}^{R}]\\\
&=[0,0,0,0,0,0,0,0,-5/4,-5/4,-5/4,-5/4,1,1,1,1,0],\\\
&[0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0]\end{split}$ (S101)

and the eigen-value $-2$ corresponds to the eigen-vector:

$\begin{split}&[\delta\hat{\tilde{A}}_{odd}^{u,\mathcal{D},R},\delta\hat{\tilde{A}}_{odd}^{f,\mathcal{D},R},\delta\hat{\tilde{A}}_{odd}^{u,\mathcal{D}_{f},R},\delta\hat{\tilde{A}}_{odd}^{f,\mathcal{D}_{f},R},\delta\hat{\tilde{K}}_{odd}^{u,\mathcal{D},R},\delta\hat{\tilde{K}}_{odd}^{f,\mathcal{D},R},\delta\hat{\tilde{K}}_{odd}^{u,\mathcal{D}_{f},R},\delta\hat{\tilde{K}}_{odd}^{f,\mathcal{D}_{f},R},\\\