# Thermal buckling transition of crystalline membranes in a field

Pierre Le Doussal  Laboratoire de Physique de l’Ecole Normale Supérieure, ENS,
Université PSL, CNRS, Sorbonne Université, Université de Paris, 75005 Paris,
France<EMAIL_ADDRESS>Leo Radzihovsky  Department of Physics, University of
Colorado, Boulder, CO 80309<EMAIL_ADDRESS>
###### Abstract

Two dimensional crystalline membranes in isotropic embedding space exhibit a
flat phase with anomalous elasticity, relevant e.g., for graphene. Here we
study their thermal fluctuations in the absence of exact rotational invariance
in the embedding space. An example is provided by a membrane in an
orientational field, tuned to a critical buckling point by application of in-
plane stresses. Through a detailed analysis, we show that the transition is in
a new universality class. The self-consistent screening method predicts a
second order transition, with modified anomalous elasticity exponents at
criticality, while the RG suggests a weakly first order transition.

###### pacs:

64.60Fr,05.40,82.65Dp

Introduction and background. Experimental realization of freely suspended
graphene suspendGrapheneNature2007  and other exfoliated crystals, following
the 2004 pioneering works of Geim and Novoselov Geim2004 , launched extensive
research in electronic and mechanical properties of two-dimensional
crystalline membranesGeimMacDonald ; reviewRMPGraphene . This led to a
renaissance in the statistical mechanics of fluctuating elastic membranes,
first studied in the context of soft and biological matter three decades ago
NP ; AL ; CrumplingBucklingGuitter ; GDLP ; LRprl ; LRrapid ; GuitterMC ;
Jerusalem ; Bensimon ; RTtubule ; LRReview . Theoretical interest is also
motivated by the opportunity to explore the nontrivial and rich interplay
between field theory and geometry Jerusalem .

The most striking prediction is the existence of a low-temperature stable
“flat” phase of a tensionless crystalline membrane NP , that spontaneously
breaks rotational symmetry of the embedding space. This is in stark contrast
to canonical two-dimensional field theories for which the Hohenberg-Mermin-
Wagner theoremsHohenberg ; MerminWagner ; Coleman , preclude spontaneous
breaking of a continuous symmetry in two dimensions.

In such elastic membranes, in a spectacular phenomenon of order-from-disorder,
thermal fluctuations instead stiffen the long-wavelength ($k^{-1}$) bending
rigidity $\kappa_{0}\rightarrow\kappa_{0}k^{-\eta}$, $\eta>0$, via a universal
power-law “corrugation” effect, with membrane roughness scaling as
$h_{\text{r}ms}\sim L^{\zeta}$, with $\zeta=(4-D-\eta)/2$NP ; Jerusalem ,
where $D$ is membrane’s internal dimension, with $D=2$ for the physical case.
The resulting anomalous elasticity is characterized by universal exponents,
$\eta,\zeta$ and $\eta_{u}=4-D-2\eta$ determined exactly by the underlying
rotational invariance, with a scale dependent Young modulus $K_{0}\to
K_{0}q^{\eta_{u}}$. This was predicted, together with the values of the
exponents, by a variety of complementary methods NP ; AL ; GDLP ; LRprl ;
LRReview . It was verified in numerical simulations simulationsGraphene  and
continues to be explored experimentally experimentElasticModuli .

Most theoretical studies to-date have focused on stress-free fluctuating
membranes in an isotropic embedding environment NP ; AL ; GDLP ; LRprl ;
LRrapid ; RTtubule ; LRReview ; Gazit ; MirlinPoisson1 ; Mouhanna1 ;
MouhannaCrumpling ; MouhannaTwoLoopFlat , as appropriate for e.g., soft matter
realizations of a membrane in an isotropic fluid (though see interesting
generalizations for spherical shellsPouloseNelsonPNAS2012 ; Kosmrlj ).
However, many experiments on graphene and other solid-state membranes (even
some suspended ones) may be subjected to embedding space anisotropy and/or
external stresses due to the presence of a substrate GuineaPuddlesSubstrate ;
GuineaPinningSubstrate ; GuineaPLDSubstrate , clampingBowick2020Buckling ;
Bowick2017Clamped ; MorshedifardKosmrljBuckling2021 , or electric and magnetic
fieldsBoothGeimNanoLett2008 ; BleesMcEuenKirigamiNature2015 . Orientational
fields could also be imposed by suspending the membrane in a nematic
solventLehenyDiskNematic . This was realized in Barium hexaferrite platelets
by the Ljubljana groupnematicSheetsCopic ; nematicSheetsSmalyukh ;
grapheneNematicClark  showing that they form a ferromagnetic nematic, with
membranes’ normals aligning with the nematic director and manipulatable by an
external magnetic field. It is interesting to consider for instance the case
of an uniaxial easy axis field tending to order the membrane’s normal, and/or
the application of a boundary stress $\sigma$. In all previous theoretical
descriptions, the rotational invariance in the embedding space was assumed and
the response found to be controlled by the thermal tensionless membrane fixed
pointAL . The case of weak field or stresses is treated by simply introducing
a cutoff for the isotropic critical fluctuations, beyond a large scale
$\xi\sim(\kappa/\sigma)^{\nu}$, that diverges with a vanishing $\sigma$, where
$\nu$ is a universal exponent that we compute below. Such perturbations then
lead to an anomalous response, that in the context of tension predicts a non-
Hookean stress-strain relation $\varepsilon\sim\sigma^{\alpha}$, with
$\alpha=(D-2+\eta)/(2-\eta)=_{D=2}\eta/(2-\eta)$. CrumplingBucklingGuitter ;
GDLP ; ML ; RTtubule ; LRReview ; Mirlin ; MirlinPoisson1 ; MirlinPoisson2 .

Figure 1: A schematic illustration of a critical membrane tuned to a buckling
transition, subjected to an external in-plane isotropic stress
$\sigma_{ij}=\frac{1}{2}\sigma\delta_{ij}$, stabilized and balanced by an
external field $\vec{E}$, which tends to align the normals (blue vectors).

In this Letter we describe such experimental geometries, illustrated in Fig.1,
where the imposed stress and anisotropy lead to qualitatively richer and
universal buckling phenomenology. Generic buckling is a complex out-of-plane
instability of a sheet subjected to compression, that results in a strongly
distorted, non-perturbative state. Recently, there has been significant
interest and progress in the study of isotropic buckling, with focus on
effects of thermal fluctuations on the classical problem of Euler buckling,
stabilized only by finite size effects.MorshedifardKosmrljBuckling2021
Instead, here we focus on a gentler, continuous anisotropic form of this
transition, where the instability is controlled by a stabilizing external
field. Specifically, we consider an externally oriented membrane tuned to a
buckling transition by a compressional boundary stress applied within the
plane explicitly selected by the orientational field gentle . The compressive
stress can be tuned to a critical value, $\sigma_{c}$ to cancel out at
quadratic order the (embedding-space) rotational symmetry breaking fields. Our
key observation is that at this new buckling critical point (to which the
isotropic flat membrane critical pointAL  is unstable), although at harmonic
order the membrane appears to be rotationally invariant and stress-free, thus
exhibiting strong thermal fluctuations, it admits new important elastic
nonlinearities that are not rotationally invariant. These lead to a critical
membrane, tuned to the buckling point, that is, thus qualitatively distinct
from the conventional tensionless membranefootnote1 .

Results. Subjecting a crystalline membrane to a lateral compressive isotropic
boundary stress $\sigma$, tuned to a critical tensionless buckling point
$\sigma_{c}$ and stabilized by an orienting field, we find a new buckling
universality class, distinct from the isotropic tensionless membraneNP ; AL ;
GDLP ; LRprl ; LRReview . We propose a model based on symmetry arguments,
supported by more detailed considerations. We use two complementary approaches
to analyze the properties of the resulting critical state. The first is the
self-consistent screening approximation (SCSA) which was found to provide an
accurate description for the isotropic case LRprl ; LRReview . Thermal
fluctuations and elastic nonlinearities at the buckling transition lead to a
universal anomalous elasticity with exponent

$\displaystyle\eta^{\rm anis}$ $\displaystyle=$ $\displaystyle 0.754,$ (1)

characterizing the divergence of the effective length-scale dependent bending
rigidity $\kappa(k)\sim k^{-\eta}$. The in-plane elastic moduli remain finite
at the critical point, i.e., $\eta_{u}^{\rm anis}=0$ footnote2 . This is
despite the fact that the five eigen-couplings $w_{i}(q)\sim q^{4-D-2\eta}$
renormalize nontrivially, vanishing in the long wavelength limit. This is at
variance with the tensionless isotropic membrane for which SCSA predicts
universal exponents $\eta\approx 0.821$, $\eta_{u}\approx 0.358$ LRprl . The
corresponding roughness $h_{\text{rms}}\sim L^{\zeta}$ of the critically
buckled membrane is characterized by a universal roughness exponent

$\zeta^{\rm anis}=0.623,$ (2)

and it is thus rougher than a tensionless isotropic membrane, with a roughness
exponent $\zeta\approx 0.59$ LRprl .

We complement this SCSA calculation by an RG analysis in an expansion in
$\epsilon=4-D$. It confirms the instability of the standard anomalous
elasticity fixed point of the isotropic, tensionless membrane, under breaking
of the embedding space rotational symmetry. Let us recall that for the
isotropic membrane the elastic nonlinearities destabilize the harmonic theory
(i.e., the Gaussian fixed point) beyond the length scale $\xi^{\rm iso}_{\rm
NL}\sim(\frac{\kappa^{2}}{TK_{0}})^{\frac{1}{4-D}}$. If the anisotropy
perturbation is very weak, e.g., $w\sim\mu_{1,2},\lambda_{1,2}\ll K_{0}$ (see
below for definitions of these anisotropy parameters), the membrane still
experiences the standard isotropic anomalous elasticity up to scales $\xi^{\rm
iso}_{\rm NL}$, crossing over to the new anisotropic critical behavior beyond
the crossover length

$\xi^{\rm anis}_{NL}=\xi^{\rm iso}_{\rm
NL}\left(\frac{K_{0}}{w}\right)^{1/\rho}\ ,\quad\rho=\frac{\epsilon
d_{c}}{d_{c}+24}+O(\epsilon^{2}),$ (3)

where $\rho$ is the crossover exponent obtained from linearization of the RG
flow around the isotropic fixed point. If the anisotropy perturbation is
stronger, the thermal fluctuations and elastic nonlinearities directly
destabilize the harmonic theory at scales of order $\xi^{\rm iso}_{\rm NL}$.
Beyond these scales, the RG flows to a new stable buckling critical point,
which, within the $\epsilon$-expansion, is however accessible only for space
codimension $d_{c}=d-D>219$, analogous to the crumpling transition found by
Paczuski, et al.PKN . For the physical case, $d_{c}=1$, we interpret the
resulting runaway flows as a weakly first order transition, as for the
standard crumpling transition. We note that the SCSA is exact for the large
$d_{c}$ limit, and confirm that the two methods match in their common regime
of validity.

Model of anisotropic membrane buckling. The coordinates of the atoms in the
$d$-dimensional embedding space are denoted $\vec{r}({\bf
x})\in\mathbb{R}^{d}$, with the atoms labeled by their position ${\bf
x}\in\mathbb{R}^{D}$ in the internal space. For graphene $D=2$, and atoms span
a triangular lattice, described here in the continuum limit. The deformations
with respect to the flat sheet are described by $D$ phonon fields
$u_{\alpha}({\bf x})$, and $d_{c}=d-D$ height fields
$\vec{h}\in\mathbb{R}^{d_{c}}$ (orthogonal to the $\vec{e}_{\alpha}$) as
$\vec{r}({\bf x})=(x_{\alpha}+u_{\alpha}({\bf
x}))\vec{e}_{\alpha}+\vec{h}({\bf x})$, where the $\vec{e}_{\alpha}$ are a set
of $D$ orthonormal vectors. While the physical case corresponds to $d=3$ and
$d_{c}=1$, it is useful to study the theory for a general $d_{c}$. The
nonlinear strain tensor measures the deformation of the induced metric
relative to the preferred flat metric,
$u_{\alpha\beta}=\frac{1}{2}(\partial_{\alpha}\vec{r}\cdot\partial_{\beta}\vec{r}-\delta_{\alpha\beta})\simeq\frac{1}{2}(\partial_{\alpha}u_{\beta}+\partial_{\beta}u_{\alpha}+\partial_{\alpha}\vec{h}\cdot\partial_{\beta}\vec{h})$
to the accuracy needed here, with the $O((\partial u)^{2})$ phonon
nonlinearities irrelevant and therefore neglected (see below). The tensor
$u_{\alpha\beta}$ encodes full rotational invariance in the embedding space,
its approximate form being invariant under infinitesimal rotations by
$\theta$, i.e., the $O(\theta^{2})$ term vanishes under the (apparent)
distortion $u_{1}=x_{1}(\cos\theta-1)$, $h_{1}=x_{1}\sin\theta$, which
corresponds to a rigid rotation, with the corresponding vanishing of the exact
strain tensor.

Here we build on the model of a rotationally invariant tensionless membrane.
Its Hamiltonian is the sum of curvature energy and in-plane stretching energy

${\cal F}_{1}[\vec{h},u_{\alpha}]=\int
d^{D}x\,\left[\frac{\kappa}{2}(\partial^{2}\vec{h})^{2}+\tau
u_{\alpha\alpha}+\mu(u_{\alpha\beta})^{2}+\frac{\lambda}{2}(u_{\alpha\alpha})^{2}\right]$
(4)

where $\kappa$ is the bending modulus, $\lambda,\mu$ the in-plane Lamé elastic
constants. The parameter $\tau$ controls the preferred extension of the
membrane in the $\vec{e}_{\alpha}$ plane.

Based on symmetry considerations, complemented by a model-building derivation
(presented at the end of the paper), external orientational and boundary
stresses introduce new relevant elastic nonlinearities, with five new
independent couplings, that by symmetry lead to a modified effective
Hamiltonian ${\cal F}={\cal F}_{1}+{\cal F}_{2}$, where ${\cal F}_{2}$ breaks
rotational invariance in the embedding space,

$\displaystyle{\cal F}_{2}[\vec{h},u_{\alpha}]=\int
d^{D}x\,\bigg{(}\frac{\gamma}{2}(\partial_{\alpha}\vec{h})^{2}$ (5)
$\displaystyle+\frac{\lambda_{1}}{2}\partial_{\alpha}u_{\alpha}(\partial_{\beta}\vec{h})^{2}+\frac{\lambda_{2}}{8}[(\partial_{\alpha}\vec{h})^{2}]^{2}$
$\displaystyle+\mu_{1}\partial_{\alpha}u_{\beta}(\partial_{\alpha}\vec{h}\cdot\partial_{\beta}\vec{h})+\frac{\mu_{2}}{4}[\partial_{\alpha}\vec{h}\cdot\partial_{\beta}\vec{h}]^{2}\bigg{)},$

retaining in-plane isotropy and the $h\to-h$ invariance as a feature of our
geometry, preserving the equivalence between the two sides of the membrane.

We now study the membrane with parameters tuned to the thermal buckling
critical point defined by the renormalized $\gamma_{R}=0$. Integrating over
the in-plane phonon modes $u_{\alpha}$ and, rescaling for convenience all
elastic constants by $1/d_{c}$, we obtain an effective Hamiltonian for the
height field,

$\displaystyle{\cal F}[\vec{h}]=\int
d^{D}x\,[\frac{\kappa}{2}(\partial^{2}\vec{h})^{2}+\frac{\gamma}{2}(\partial_{\alpha}\vec{h})^{2}]+\frac{1}{4d_{c}}\int
d^{D}x\,d^{D}y$ $\displaystyle\,\,\,\,\times\partial_{\alpha}\vec{h}({\bf
x})\cdot\partial_{\beta}\vec{h}({\bf x})\,R_{\alpha\beta,\gamma\delta}({\bf
x}-{\bf y})\,\partial_{\gamma}\vec{h}({\bf
y})\cdot\partial_{\delta}\vec{h}({\bf y}),$ (6)

with a non-local quartic tensorial interaction, which in Fourier space is
given byfootnoteFisherShankaNelsonRG

$R_{\alpha\beta,\gamma\delta}({\bf
q})=\sum_{i=1}^{5}w_{i}\,(W_{i})_{\alpha\beta,\gamma\delta}({\bf q}).$ (7)

The $W_{i}$ are five projectors in the space of four index tensors, equal to
bilinear combinations of longitudinal $P^{L}_{\alpha\beta}({\bf
q})=q_{\alpha}q_{\beta}/q^{2}$ and transverse $P^{T}({\bf
q})=\delta_{\alpha\beta}-P^{L}_{\alpha\beta}({\bf q})$ projectors on the wave
vector ${\bf q}$. The five ”bare couplings” $w_{i}$ are given in the
Supplementary Material (SM) SM  in terms of the bare elastic moduli in (4) and
(5), together with the basis tensors $W_{i}$. footnote3  The important
features are the following. When rotational symmetry breaking is absent,
$\gamma=0$, $\mu_{1}=\mu_{2}=\lambda_{1}=\lambda_{2}=0$, the couplings
$w_{2},w_{4},w_{5}$ vanish and

$w_{1}=\mu\quad,\quad w_{3}=\mu+(D-1)\frac{\mu\lambda}{\lambda+2\mu},$ (8)

leading to (the ${\bf q}$ dependence suppressed)

$R_{\alpha\beta,\gamma\delta}=(w_{3}-w_{1})P_{\alpha\beta}^{T}P_{\gamma\delta}^{T}+w_{1}\frac{1}{2}(P_{\alpha\gamma}^{T}P_{\beta\delta}^{T}+P_{\alpha\delta}^{T}P_{\beta\gamma}^{T}),$
(9)

which is the usual quartic coupling associated to ${\cal F}_{1}$. When
$\lambda_{1}$ and $\lambda_{2}$ are turned on, while $\mu_{1}=\mu_{2}=0$, all
$w_{i}$ are nonzero except $w_{2}=0$. Finally, when all couplings in ${\cal
F}_{2}$ are nonzero, all $w_{i}$ are nonzero.

SCSA analysis. The form (Thermal buckling transition of crystalline membranes
in a field) is suitable to apply the SCSA method, which is exact in the limit
of large $d_{c}$. The calculation is performed in the SM SM  and parallels the
one in Section IV. A of LRReview . Consider the two point correlation of the
height field in Fourier space, $\langle h^{i}({\bf k})h^{j}({\bf
k}^{\prime})\rangle={\cal G}(k)(2\pi)^{d}\delta^{d}({\bf k}+{\bf
k}^{\prime})\delta_{ij}$. If we neglect the quartic nonlinearities in (Thermal
buckling transition of crystalline membranes in a field) we find ${\cal
G}(k)=G(k)=1/(\gamma k^{2}+\kappa k^{4})$. The nonlinearities lead to a
nonzero self-energy $\Sigma(k)={\cal G}(k)^{-1}-\gamma k^{2}-\kappa k^{4}$.
Together with the renormalized interaction tensor, $\tilde{R}({\bf q})$, it
satisfies the SCSA equations

$\displaystyle\Sigma(k)=\frac{2}{d_{c}}\int_{q}k_{\alpha}(k_{\beta}-q_{\beta})(k_{\gamma}-q_{\gamma})k_{\delta}\tilde{R}_{\alpha\beta,\gamma\delta}({\bf
q}){\cal G}({\bf k}-{\bf q})$ (10) $\displaystyle\tilde{R}({\bf q})=R({\bf
q})-R({\bf q})\Pi({\bf q})\tilde{R}({\bf q})$ (11)

where $\Pi({\bf q})$ encodes the screening of the in-plane elasticity by out
of plane fluctuations

$\Pi_{\alpha\beta,\gamma\delta}({\bf
q})=\frac{1}{4}\int_{p}v_{\alpha\beta}({\bf q},{\bf q}-{\bf
p})v_{\gamma\delta}({\bf q},{\bf q}-{\bf p}){\cal G}({\bf p}){\cal G}({\bf
q}-{\bf p})$ (12)

and $v_{\alpha\beta}({\bf p},{\bf
p}^{\prime})=p_{\alpha}p^{\prime}_{\beta}+p^{\prime}_{\alpha}p_{\beta}$. One
can decompose $\Pi({\bf q})=\sum_{i=1}^{5}\pi_{i}(q)W_{i}({\bf q})$ and
$\tilde{R}({\bf q})=\sum_{i=1}^{5}\tilde{w}_{i}(q)W_{i}({\bf q})$, where
$\tilde{w}_{i}(q)$ are the momentum dependent renormalized couplings. Looking
for a solution which behaves at small $k$ as ${\cal G}(k)\simeq
Z_{\kappa}^{-1}/k^{4-\eta}$, and evaluating the integrals $\pi_{i}(q)$ SM  one
finds that they diverge at small $q$ as $\pi_{i}(q)\simeq
Z_{\kappa}^{-2}a_{i}q^{-(4-D-2\eta)}$ where $a_{i}=a_{i}(\eta,D)$. From (11)
we find that the renormalized couplings are softened at small $q$ as
$\tilde{w}_{i}(q)\propto Z_{\kappa}^{2}c_{i}q^{\eta_{u}}$, with
$\eta_{u}=4-D-2\eta$ and $c_{i}=1/a_{i}$ for $i=1,2$ and

$\displaystyle\begin{pmatrix}c_{3}&c_{4}\\\
c_{4}&c_{5}\end{pmatrix}\simeq\begin{pmatrix}a_{3}&a_{4}\\\
a_{4}&a_{5}\end{pmatrix}^{-1}$ (13)

Inserting this into the self-energy equation (Thermal buckling transition of
crystalline membranes in a field) and performing the integrals we find that
the factors of $Z_{\kappa}$ cancel and the self-consistent equation, which
implicitly determines $\eta$ as a function of $D,d_{c}$ is given by

$\frac{d_{c}}{2}=\sum_{i=1,2}{b_{i}\over
a_{i}}+{b_{3}a_{5}-b_{4}a_{4}+b_{5}a_{3}\over{a_{3}a_{5}-a_{4}^{2}}},$ (14)

where $b_{i}=b_{i}(\eta,D)$ are self-energy integrals, given with the
$a_{i}(\eta,D)$ in the SM. Note that here we have considered the case where
all bare couplings $w_{i}$ are nonzero. For a physical membrane, $D=2$, (14)
reduces to finding the root of a cubic equation

$\displaystyle d_{c}=\frac{24(\eta-1)^{2}(2\eta+1)}{(\eta-4)\eta(2\eta-3)}.$
(15)

For $d_{c}=1$ we obtain our main result (1). For large $d_{c}$ we find
$\eta=2/d_{c}+O(1/d_{c}^{2})$. The roughness of a size $L$ membrane is
characterized by $h_{\rm rms}=\langle h^{2}\rangle^{1/2}\simeq L^{\zeta}$
where $\zeta=(4-D-\eta)/2$. Hence for $d_{c}=1$ we find $\zeta=0.623$.

One can define renormalized amplitude ratio as

$\displaystyle\lim_{q\to
0}\frac{\tilde{w}_{i}(q)}{\tilde{w}_{j}(q)}=\frac{c_{i}}{c_{j}}$ (16)

for any pair $(i,j)$ such that the bare couplings $w_{i},w_{j}$ are nonzero.
Near $D=4$ we find that these renormalized couplings take values such that the
interaction energy becomes
$\frac{v_{1}}{2}[(\partial_{\alpha}\vec{h})^{2}]^{2}+v_{2}(\partial_{\alpha}\vec{h}\cdot\partial_{\beta}\vec{h})^{2}$,
i.e., local in the fields $\partial_{\alpha}\vec{h}$. This property however
does not hold for $D<4$, e.g., one finds $c_{2}/c_{1}=(D+\eta-2)/(2-\eta)$
instead of unity for $D=4$, $\eta=0$. Thus the critical point requires a fully
non-local five coupling description. The $c_{i}$ are given in SM . In the
physical case of $D=2$ and $d_{c}$=1 we find

$c_{i}=\\{\frac{1}{2},0.302,0.338,-0.029,0.173\\},$ (17)

and the universal $\lambda/\mu=-0.978$ and the Poisson ratio (not to be
confused with external stress),

$\sigma^{\rm anis}=-0.968,$ (18)

to be contrasted with $\sigma=-1/3$ for an isotropic tensionless membraneLRprl
; LRReview

There are other fixed points that lie in the invariant subspaces of the SCSA
equations. The rotationally invariant membrane corresponds to bare couplings
$w_{2}=w_{4}=w_{5}=0$. The corresponding renormalized couplings also vanish,
which amounts to $b_{2}=b_{4}=b_{5}=0$ in (14), leading to

$\frac{d_{c}}{2}=\frac{b_{1}}{a_{1}}+{b_{3}\over{a_{3}}},$ (19)

which is precisely the SCSA equation for the anomalous flat phase of the
isotropic membrane, leading for $D=2$ to
$\eta=\frac{4}{d_{c}+\sqrt{16-2d_{c}+d_{c}^{2}}}$, and $\eta\simeq 0.821$,
$\zeta=0.590$ for $d_{c}=1$ LRprl ; LRReview . Near $D=4$ one recovers
$\eta=\frac{12}{d_{c}+24}\epsilon+O(\epsilon^{2})$ from the Aronovitz-
Lubensky’s $\epsilon$-expansionAL . Another fixed manifold is $w_{2}=0$,
corresponding to a choice of bare couplings so that
$(\mu+\mu_{1})^{2}=\mu(\mu+\mu_{2})$, which includes the choice
$\mu_{1}=\mu_{2}=0$, leading to $\tilde{w}_{2}(q)=0$ and

$\frac{d_{c}}{2}={b_{1}\over
a_{1}}+{b_{3}a_{5}-b_{4}a_{4}+b_{5}a_{3}\over{a_{3}a_{5}-a_{4}^{2}}}.$ (20)

This leads to yet another fixed point with slightly different exponents. For
$D=2$ and $d_{c}=1$ we find $\eta=0.854$ and $\zeta=0.573$. Near $D=4$ we find
$\eta=\frac{18}{d_{c}+36}\epsilon+O(\epsilon^{2})$. Universal amplitude ratios
have $c_{2}=0$.

RG analysis. As a nontrivial check and for further insight, we have
complemented this SCSA calculation and results using an RG analysis,
controlled by an $\epsilon=4-D$ expansion near $D=4$. We have calculated the
one-loop corrections to the Hamiltonian (Thermal buckling transition of
crystalline membranes in a field) and obtained the RG equations for the five
dimensionless couplings
$\hat{w}_{i}=w_{i}/\kappa^{2}C_{4}\Lambda_{\ell}^{-\epsilon}$ of the form
$\partial_{\ell}\hat{w}_{i}=\epsilon\hat{w}_{i}+a_{ijk}\hat{w}_{j}\hat{w}_{k}$,
where the $a_{ijk}$ and details of the calculation are given in SM . The
anomalous dimension of the out-of-plane height field $h$ defines the exponent
$\eta$ given by

$\displaystyle\eta=\frac{1}{12}\left(10\hat{w}_{1}-18\hat{w}_{2}+5\hat{w}_{3}+3\hat{w}_{5}-6\hat{w}_{44}\right),$
(21)

with $\hat{w}_{44}=\sqrt{3}\hat{w}_{4}$, and evaluated at the fixed point of
interest $\hat{w}_{i}^{*}$ (see below). The anomalous dimension of the phonon
field is given by

$\eta_{u}=\frac{1}{12}(\hat{w}_{1}-\hat{w}_{2}).$ (22)

The isotropic membrane corresponds to the space
$\hat{w}_{2}=\hat{w}_{4}=\hat{w}_{5}=0$, which is preserved by the RG flow and
along which

$\displaystyle\partial_{\ell}\hat{w}_{1}=-\frac{1}{12}\hat{w}_{1}\left((d+20)\hat{w}_{1}+10\hat{w}_{3}\right),$
(23)
$\displaystyle\partial_{\ell}\hat{w}_{3}=-\frac{5}{24}\hat{w}_{3}\left((d+4)\hat{w}_{3}+8\hat{w}_{1}\right).$
(24)

The isotropic membrane fixed point is
$\hat{w}^{*}_{1}=\frac{12\epsilon}{d+24}$,
$\hat{w}^{*}_{3}=\frac{24\epsilon}{5(d+24)}$, corresponding to
$\hat{\mu}^{*}=\frac{12\epsilon}{24+d}$,
$\hat{\lambda}^{*}=\frac{-4\epsilon}{24+d}$ AL . Diagonalizing the RG flow for
$\hat{w}_{i}=\hat{w}_{i}^{*}+\delta\hat{w}_{i}$ around this fixed point in the
larger space of five couplings shows that, in addition to the two negative
eigenvalues $-1$ and $-\frac{d_{c}}{d_{c}+24}$ within the plane
$\delta\hat{w}_{1,3}$ of the isotropic membrane, (i) there is a marginal
direction mixing $\delta\hat{w}_{1,3,4}$ (eigenvalue $0$) (ii) there are two
unstable directions with eigenvalues $\frac{d_{c}}{d_{c}+24}$ with $\delta
w_{2,5}$ nonzero (in the large $d_{c}$ limit this eigenspace is purely along
$\delta w_{2,5}$). Hence, consistent with the SCSA findings, the isotropic
membrane fixed point is unstable to anisotropy of the orientational field and
external boundary stress.

To determine where the general flow goes we searched for attractive fixed
points of the RG equations. We found one such fixed point in the subspace of
couplings $\hat{w}_{i}$ at which, the interaction energy is fully local in the
gradients $\partial_{\alpha}\vec{h}$ and parameterized by two couplings
$v_{1},v_{2}$ as defined above. This subspace is preserved by the RG and also
arises in the study of the crumpling transition. In fact the RG flow within
this subspace is identical to the one obtained in PKN  with $d$ replaced by
$d_{c}$. It admits a stable FP for $d_{c}>219$. Here we demonstrated that this
FP is fully attractive in the space of the five couplings. Hence the RG
approach is consistent, around $D=4$, with the SCSA (which is exact for large
$d_{c}$ and any $D$), predicting a new fixed point for membrane in anisotropic
embedding space. For the physical membrane $D=2$ and $d_{c}=1$, while the SCSA
predicts this new ”anisotropic buckling transition” to be continuous, the RG,
if extrapolated from $D=4$, suggests a weakly first order transition, as
argued for the crumpling transition PKN ; Mouhanna1 ; MouhannaCrumpling .

To reach the new anisotropic buckling critical point requires tuning
$\gamma=\gamma_{c}$, so that $\gamma_{R}=0$. Slightly away from criticality
the correlation length is long but finite, $\xi\sim|\delta\gamma|^{-\nu}$,
diverging with a vanishing $\delta\gamma=\gamma-\gamma_{c}$. Linearizing the
RG flow around the fixed point yields $\delta\gamma(L)\sim\delta\gamma
L^{\theta}$, where
$\theta=-\frac{\epsilon}{d_{c}}(1-\frac{66}{d_{c}}+O(\frac{1}{d_{c}^{2}}))$,
see the SM SM . By balancing
$\kappa(\xi)\xi^{-4}\sim\delta\gamma(\xi)\xi^{-2}$ and using that
$\kappa(\xi)\sim\xi^{\eta}$ we obtain the correlation length exponent as
$\nu=1/(2+\theta-\eta)$.

Model development. Until now we argued for the model (4,5) based on symmetry
considerations. Here, as illustrated in Fig.1, we develop an explicit model of
a membrane undergoing buckling in the absence of rotational invariance in the
embedding space. We consider an elastic membrane in an external field
$\vec{E}$ (taken along the z-axis) that aligns the membrane’s normal $\hat{n}$
along the field. We thus expect the energy density to be a monotonic function
of $\hat{n}\cdot\vec{E}$, namely of the small tilt angle $\theta$,

$\displaystyle{\cal H}_{\text{o}rient}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{1}}{2}\theta^{2}+\frac{\tilde{\alpha}_{2}}{4}\theta^{4}+\ldots,$
(25)

with $\alpha_{1}>0$, $\tilde{\alpha}_{2}>0$. Combining this orientational
field energy with the Hamiltonian for an elastic membraneJerusalem ; LRReview
, subjected to an in-plane compressional boundary stress $\sigma>0$, isotropic
in the membrane’s xy plane, and using that, to lowest order
$\theta\sim|\partial_{\alpha}h|$, we obtain,

$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle\frac{\kappa}{2}(\partial^{2}h)+\mu
u_{\alpha\beta}^{2}+\frac{\lambda}{2}u_{\alpha\alpha}^{2}+\sigma\partial_{\alpha}u_{\alpha}$
(26)
$\displaystyle+\frac{\alpha_{1}}{2}(\partial_{\alpha}h)^{2}+\frac{\alpha_{2}}{4}(\partial_{\alpha}h)^{4}+\ldots.$

We note that the external stress, $\sigma$ is an in-plane boundary term, that
induces a stress-dependent inward displacement of the membrane’s edges.
Observing that $\sigma\partial_{\alpha}u_{\alpha}=\sigma
u_{\alpha\alpha}-\frac{1}{2}\sigma(\partial_{\alpha}h)^{2}$, the rotationally
invariant strain component $\sigma u_{\alpha\alpha}$ can be accommodated by
simply changing the preferred extension of the membrane without breaking the
embedding space rotational symmetry (i.e., it amounts to a redefinition of the
parameter $\tau$ in (4), which determines the preferred membrane’s projected
area footnote5 ). The negative in-plane strain $\partial_{\alpha}u_{\alpha}$
induced by positive $\sigma$ can be relieved by a membrane tilt,
$(\partial_{\alpha}h)^{2}>0$, stress-free in the actual plane of the membrane.
The lowering of the energy associated with the membrane tilt is then given by
the second term, i.e., ${\cal
H}_{\sigma}=-\frac{1}{2}\sigma(\partial_{\alpha}h)^{2}$, which, neglecting
bending energy and boundary conditions, is unbounded, since tilt is
unconstrained in the absence of the orientational field. Putting these
ingredients together and rescaling xy coordinate system, we obtain the
Hamiltonian governing a buckling transition of a membrane in an orientational
field,

$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle\frac{\kappa}{2}(\partial^{2}h)+\mu
u_{\alpha\beta}^{2}+\frac{\lambda}{2}u_{\alpha\alpha}^{2}$ (27)
$\displaystyle+\frac{\gamma}{2}(\partial_{\alpha}h)^{2}+\frac{\alpha_{2}}{4}(\partial_{\alpha}h)^{4}+\ldots,$

where $\gamma=\alpha_{1}-\sigma$ is the critical parameter which can be tuned
to $\gamma_{c}$ to reach the buckling transition (with $\gamma_{c}=0$ at
$T=0$), studied in here. As detailed in SM, we can estimate the buckling
stress $\sigma_{c}$ based on a model of homeotropic alignment of a membrane in
a nematic solvent (using typical values of Frank elastic
constants)LehenyDiskNematic  and a model of a ferroelectric membrane aligned
by an electric field. These give $\sigma_{c}\sim 1-10eV/\mu m^{2}$, with the
thermal fluctuation corrections to $\gamma$ that we show in SM to be
subdominant.

Conclusion. To summarize, in this Letter, in contrast to previous works on
tensionless crystalline membranes, we studied a thermal elastic sheet tuned by
an external boundary stress to a critical point of a buckling transition,
stabilized by an orientational field. We find that this breaking of embedding
rotational symmetry has profound effects, and leads to a new class of
anomalous elasticity, that we have explored in detail here using the SCSA and
RG analyses. With much recent interest in elastic sheets, most notably
graphene and other van der Waals monolayers, we hope that our predictions will
stimulate further experiments to probe the rich universal phenomenology
predicted here for an elastic membrane tuned to a buckling transition in an
anisotropic environment. We also expect that ideas explored here can be
extended to a richer class of anomalously elastic media.LRelastomer ;
footnoteFutureLR

Note Added: We have recently became aware of an ongoing work by S. Shankar and
D. R. Nelson on a membrane subjected to a boundary stress or strain, which, in
contrast to our work only breaks embedding rotational symmetry at the
boundary.

Acknowledgments. We thank John Toner, David Nelson and Suraj Shankar for
enlightening discussions. LR also acknowledges support by the NSF grants MRSEC
DMR-1420736, Simons Investigator Fellowship, and thanks École Normale
Supérieure for hospitality. PLD acknowledge support from ANR under the grant
ANR-17-CE30-0027-01 RaMaTraF. Both authors thank KITP for hospitality. This
research was supported in part by the National Science Foundation under Grant
No. NSF PHY-1748958.

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Supplementary Material for Thermal buckling transition of crystalline
membranes

We give the principal details of the calculations described in the main text
of the Letter.

A. Projectors and tensor multiplication

Here we consider four index tensors, such as
$R_{\alpha\beta,\gamma\delta}({\bf q})$ introduced in the text, which are
symmetric in $\alpha\leftrightarrow\beta$, in $\gamma\leftrightarrow\delta$
and in $(\alpha,\beta)\leftrightarrow(\gamma,\delta)$. The product of such
tensors is defined as $(T\cdot
T^{\prime})_{\alpha\beta,\gamma\delta}=T_{\alpha\beta,\gamma^{\prime}\delta^{\prime}}T^{\prime}_{\gamma^{\prime}\delta^{\prime},\gamma\delta}$,
the identity being $I_{\alpha\beta,\gamma\delta}={1\over
2}(\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma})$.
We recall the definition LRReview  of the five ”projectors” $W_{i}$,
$i=1,\ldots,5$, which span the space of such four index tensors

$\displaystyle(W_{3})_{\alpha\beta,\gamma\delta}({\bf
q})={1\over{D-1}}P^{T}_{\alpha\beta}P^{T}_{\gamma\delta}\;,\;\;\;(W_{5})_{\alpha\beta,\gamma\delta}({\bf
q})=P^{L}_{\alpha\beta}P^{L}_{\gamma\delta}\;,$ (28)
$\displaystyle(W_{4})_{\alpha\beta,\gamma\delta}({\bf
q})=(W_{4a})_{\alpha\beta,\gamma\delta}({\bf
q})+(W_{4b})_{\alpha\beta,\gamma\delta}({\bf q})\;,$ (29)
$\displaystyle(W_{4a})_{\alpha\beta,\gamma\delta}({\bf
q})={1\over{\sqrt{D-1}}}P^{T}_{\alpha\beta}P^{L}_{\gamma\delta}\;,\quad(W_{4b})_{\alpha\beta,\gamma\delta}({\bf
q})={1\over{\sqrt{D-1}}}P^{L}_{\alpha\beta}P^{T}_{\gamma\delta}\;,$ (30)
$\displaystyle(W_{2})_{\alpha\beta,\gamma\delta}({\bf q})={1\over
2}(P^{T}_{\alpha\gamma}P^{L}_{\beta\delta}+P^{T}_{\alpha\delta}P^{L}_{\beta\gamma}+P^{L}_{\alpha\gamma}P^{T}_{\beta\delta}+P^{L}_{\alpha\delta}P^{T}_{\beta\gamma})\;,$
(31) $\displaystyle W_{1}({\bf q})={1\over
2}(\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma})-W_{3}({\bf
q})-W_{5}({\bf q})-W_{2}({\bf q})\;,$ (32) $\displaystyle W_{1}({\bf
q})+W_{3}({\bf q})={1\over 2}(P^{T}_{\alpha\gamma}({\bf
q})P^{T}_{\beta\delta}({\bf q})+P^{T}_{\alpha\delta}({\bf
q})P^{T}_{\beta\gamma}({\bf q}))$ (33)

where $P^{T}_{\alpha\beta}=\delta_{\alpha\beta}-q_{\alpha}q_{\beta}/q^{2}$ and
$P^{L}_{\alpha\beta}=q_{\alpha}q_{\beta}/q^{2}$ are the standard transverse
and longitudinal projection operators associated to ${\bf q}$. The first two
projectors $W_{1},W_{2}$ are mutually orthogonal and orthogonal to the other
three. Note that while $R$, being symmetric, can be expressed in terms of the
symmetric tensors $W_{i}$, $i=1,..5$, we will need at some intermediate stages
of the calculations some products (such as $\Pi*R$ see below), which are not
symmetric. Hence we introduced $W_{4}^{a}$ and $W_{4}^{b}$, which together
with $W_{i}$, $i=1,2,3$ and $W_{5}$ make the representation complete under
tensor multiplication. The rules for the tensor multiplication
$T^{\prime\prime}=T^{\prime}*T$ of the tensors
$T=\sum_{i=1}^{3}w_{i}W_{i}+w_{4a}W_{4a}+w_{4b}W_{4b}+w_{5}W_{5}$ and
$T^{\prime}=\sum_{i=1}^{3}w^{\prime}_{i}W_{i}+w^{\prime}_{4a}W_{4a}+w^{\prime}_{4b}W_{4b}+w^{\prime}_{5}W_{5}$
are

$\displaystyle
w_{1}^{\prime\prime}=w^{\prime}_{1}w_{1}\;,\;\;\;w_{2}^{\prime\prime}=w^{\prime}_{2}w_{2}\;,\;\;\;\begin{pmatrix}w_{3}^{\prime\prime}&w_{4a}^{\prime\prime}\\\
w_{4b}^{\prime\prime}&w_{5}^{\prime\prime}\end{pmatrix}=\begin{pmatrix}w^{\prime}_{3}&w_{4a}^{\prime}\\\
w_{4b}^{\prime}&w^{\prime}_{5}\end{pmatrix}\begin{pmatrix}w_{3}&w_{4a}\\\
w_{4b}&w_{5}\end{pmatrix}\;,$ (34)

with
$T^{\prime\prime}=\sum_{i=1}^{3}w^{\prime\prime}_{i}W_{i}+w^{\prime\prime}_{4a}W_{4a}+w^{\prime\prime}_{4b}W_{4b}+w^{\prime\prime}_{5}W_{5}$.

B. Integration over in-plane deformations

The integration over the phonon fields $u_{\alpha}(x)$ of the Gibbs measure
$\sim e^{-{\cal F}[\vec{h},u_{\alpha}]/T}$, with ${\cal F}={\cal F}_{1}+{\cal
F}_{2}$ given by (4) and (5) leads to the Gibbs measure $\sim e^{-{\cal
F}[\vec{h}]/T}$ for the height fields with an effective Hamiltonian of the
form (Thermal buckling transition of crystalline membranes in a field) in the
text (we set $\tau=0$). To perform it we use a method slightly different from
the one in e.g. LRReview  Section III B. Let us introduce the elastic matrix

$C^{\mu,\lambda}_{\alpha\beta,\gamma\delta}=\lambda\delta_{\alpha\beta}\delta_{\gamma\delta}+\mu(\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma})$
(35)

and denote
$\tilde{u}_{\alpha\beta}=\frac{1}{2}(\partial_{\alpha}u_{\beta}+\partial_{\beta}u_{\alpha})$
and
$A_{\alpha\beta}=\frac{1}{2}\partial_{\alpha}\vec{h}\cdot\partial_{\beta}\vec{h}$.
We then rewrite the model ${\cal F}={\cal F}_{1}+{\cal F}_{2}$ as

$\displaystyle{\cal F}[u,\vec{h}]=\int
d^{D}x\,\,\bigg{[}\frac{\kappa}{2}(\nabla^{2}h)^{2}+\frac{1}{2}C^{\mu,\lambda}_{\alpha\beta,\gamma\delta}\tilde{u}_{\alpha\beta}\tilde{u}_{\gamma\delta}+\tilde{u}_{\alpha\beta}C^{\mu+\mu_{1},\lambda+\lambda_{1}}_{\alpha\beta,\gamma\delta}A_{\gamma\delta}+\frac{1}{2}C^{\mu+\mu_{2},\lambda+\lambda_{2}}_{\alpha\beta,\gamma\delta}A_{\alpha\beta}A_{\gamma\delta}\bigg{]}$
(36)

We must treat separately the contributions of the in plane strains which are
uniform (i.e. with zero momentum), and those with nonzero wavevector, i.e. the
phonons.

###  .1 B.1 Phonon integration: nonzero wavevector

.

We recall the phonon field propagator

$\langle u_{\alpha}({\bf q})u_{\beta}({\bf
q}^{\prime})\rangle=T(2\pi)^{D}\delta^{D}({\bf q}+{\bf
q}^{\prime})\left(\frac{P_{\alpha\beta}^{T}({\bf q})}{\mu
q^{2}}+\frac{P_{\alpha\beta}^{L}({\bf q})}{(2\mu+\lambda)q^{2}}\right)$ (37)

from which the in-plane strain correlator at nonzero wavevector is obtained as

$\langle\tilde{u}_{\alpha\beta}({\bf q})\tilde{u}_{\gamma\delta}({\bf
q}^{\prime})\rangle=T(2\pi)^{D}\delta^{D}({\bf q}+{\bf
q}^{\prime})D_{\alpha\beta,\gamma\delta}({\bf q})$ (38)

with, for ${\bf q}\neq 0$,

$D_{\alpha\beta,\gamma\delta}({\bf
q})=\frac{1}{4}\left[\hat{q}_{\alpha}\hat{q}_{\gamma}\left(\frac{P_{\beta\delta}^{T}({\bf
q})}{\mu}+\frac{P_{\beta\delta}^{L}({\bf q})}{2\mu+\lambda}\right)+\text{3
permutations}\right]$ (39)

This tensor has a simple expression in terms of the projectors (suppressing
the indices and the $q$ dependence)

$D=\frac{1}{2\mu}W_{2}+\frac{1}{2\mu+\lambda}W_{5}$ (40)

The integration over the phonon field in (36) using (37) is then a simple
quadratic Gaussian integral leading to the form given in Eq. (Thermal buckling
transition of crystalline membranes in a field) in the main text

${\cal F}_{\rm eff}(\vec{h})=\frac{1}{d_{c}}\int_{q\neq
0}R_{\alpha\beta,\gamma\delta}({\bf q})A_{\alpha\beta}(-{\bf
q})A_{\gamma\delta}({\bf q})$ (41)

where the interaction tensor is

$R=\frac{1}{2}C^{\mu+\mu_{2},\lambda+\lambda_{2}}-\frac{1}{2}C^{\mu+\mu_{1},\lambda+\lambda_{1}}\cdot
D\,\cdot C^{\mu+\mu_{1},\lambda+\lambda_{1}}$ (42)

Thanks to the projectors its explicit calculation is easy. One decomposes

$C^{\mu,\lambda}=2\mu(W_{1}+W_{2}+W_{3}+W_{5})+\lambda[(D-1)W_{3}+\sqrt{D-1}W_{4}+W_{5}]$
(43)

and uses the above multiplication rules for the $W_{i}$’s. One obtains

$R_{\alpha\beta,\gamma\delta}({\bf q})=\sum_{i=1}^{5}w_{i}W_{i}({\bf q})$ (44)

in terms of the five ”elastic constants” $w_{i}$

$\displaystyle w_{1}=\mu+\mu_{2}$ (45) $\displaystyle
w_{2}=\mu+\mu_{2}-\frac{(\mu+\mu_{1})^{2}}{\mu}$ $\displaystyle
w_{3}=\mu+\mu_{2}+\frac{(D-1)}{2}(\lambda+\lambda_{2}-\frac{(\lambda+\lambda_{1})^{2}}{\lambda+2\mu})$
$\displaystyle
w_{44}=\frac{1}{2}(D-1)\left(\lambda+\lambda_{2}-\frac{(\lambda+\lambda_{1})(\lambda+\lambda_{1}+2\mu+2\mu_{1})}{\lambda+2\mu}\right)\quad,\quad
w_{44}=\sqrt{D-1}w_{4}$ $\displaystyle
w_{5}=\frac{1}{2}\left(\lambda+\lambda_{2}+2\mu+2\mu_{2}-\frac{(\lambda+\lambda_{1}+2\mu+2\mu_{1})^{2}}{\lambda+2\mu}\right)$

Note that this is true under the condition that the phonon propagator is
positive definite i.e.

$\mu>0\quad,\quad 2\mu+\lambda>0$ (46)

Note also that the interaction $R_{\alpha\beta,\gamma\delta}(q)$ given above
is understood to explicitly exclude the zero-mode $q=0$, which we address
below. The stability of the zero-mode requires $\mu>0$ and $2\mu+D\lambda>0$,
which is a more stringent condition.

Note that when $\mu_{1}=\mu_{2}=0$ one has $w_{2}=0$. When in addition
$\lambda_{1}=\lambda_{2}=0$ one has $w_{2}=w_{4}=w_{5}=0$ and

$w_{1}=\mu\quad,\quad w_{3}=\mu+(D-1)\frac{\mu\lambda}{\lambda+2\mu}$ (47)

as given in the text. Note that in general there are 5 couplings $w_{i}$ and 6
original couplings. Inversion thus determines only the following five ratio as

$\displaystyle\mu+\mu_{2}=w_{1}$ (48)
$\displaystyle\frac{(\mu+\mu_{1})^{2}}{\mu}=w_{1}-w_{2}$
$\displaystyle\frac{(\lambda+\lambda_{1})^{2}}{\mu}=\frac{4\left(w_{1}-w_{2}\right)\left(w_{1}-w_{3}+w_{44}\right){}^{2}}{\left(D\left(w_{1}-w_{5}\right)-w_{3}+w_{5}+2w_{44}\right){}^{2}}$
$\displaystyle\lambda+\lambda_{2}=\frac{2\left(w_{44}^{2}-(D-1)\left(w_{1}-w_{3}\right)\left(w_{1}-w_{5}\right)\right)}{(D-1)\left(D\left(w_{1}-w_{5}\right)-w_{3}+w_{5}+2w_{44}\right)}$
$\displaystyle\frac{\lambda}{\mu}=\frac{2(D-1)\left(w_{1}-w_{2}\right)}{D\left(w_{1}-w_{5}\right)-w_{3}+w_{5}+2w_{44}}-2$

Here $\mu+\mu_{2}$ and $\lambda+\lambda_{2}$ are the two $h^{4}$ vertex
couplings in the original $u,h$ theory (before integrating phonons) and and
$\frac{(\lambda+\lambda_{1})^{2}}{\mu}$ and $\frac{(\mu+\mu_{1})^{2}}{\mu}$
are the natural $uhh$ vertex couplings combination appearing in perturbation
theory. Finally $\lambda/\mu$ is a ratio of elastic constants. Hence the
overall elastic constant scale, $\mu$, remains undetermined and must be
calculated separately from the $u,h$ theory. Note that combining the above
equations, one also obtains the following ratio

$\frac{\lambda+\lambda_{1}}{\mu+\mu_{1}}=\frac{2(w_{3}-w_{1}-w_{44})}{D\left(w_{1}-w_{5}\right)-w_{3}+w_{5}+2w_{44}}$
(49)

Finally, in the case $\mu_{2}=\mu_{1}=0$ one has $w_{2}=0$ and one can invert
the above relations for all remaining couplings

$\displaystyle\mu=w_{1}\quad,\quad\lambda=-\frac{2w_{1}\left(-Dw_{5}+w_{1}-w_{3}+w_{5}+2w_{44}\right)}{D\left(w_{1}-w_{5}\right)-w_{3}+w_{5}+2w_{44}}$
(50)
$\displaystyle\lambda_{1}=\frac{2w_{1}\left(-Dw_{5}+w_{5}+w_{44}\right)}{D\left(w_{1}-w_{5}\right)-w_{3}+w_{5}+2w_{44}}\quad,\quad\lambda_{2}=\frac{2w_{44}\left(2(D-1)w_{1}+w_{44}\right)-2(D-1)w_{5}\left((D-2)w_{1}+w_{3}\right)}{(D-1)\left(D\left(w_{1}-w_{5}\right)-w_{3}+w_{5}+2w_{44}\right)}$
(51)

consistent with the above result.

One can also ask about necessary conditions for the quartic form in the
effective stretching energy (Thermal buckling transition of crystalline
membranes in a field) to be positive definite. Positivity of the quartic form

$k_{1}^{\alpha}({\bf q}-{\bf k}_{1})^{\alpha}R_{\alpha\beta,\gamma\delta}({\bf
q})k_{3}^{\beta}({\bf q}-{\bf k}_{3})^{\delta}$ (52)

for any choice of ${\bf k}_{1},{\bf k}_{3},{\bf q}$ implies for instance: (i)
choosing all ${\bf k}_{i}$ aligned with $q$

$w_{5}>0$ (53)

(ii) choosing ${\bf k}_{3}={\bf k}_{1}$ and considering various limits we also
find

$w_{2}>0\quad,\quad(D-2)w_{1}+w_{3}>0$ (54)

Finally, note that one must have $w_{1}\geq w_{2}$ for the above equations
(48) to make sense.

###  .2 B.2 zero-mode

We must treat separately the uniform part of the nonlinear strain tensor,
$u_{\alpha\beta}({\bf q}=0)$. It is the sum of the uniform part of the in-
plane strain tensor, which we denote $\tilde{u}^{0}_{\alpha\beta}$ and of
$A^{0}_{\alpha\beta}=\frac{1}{2}[(\partial_{\alpha}h)(\partial_{\beta}h)]({\bf
q}=0)$. The energy per unit volume associated to this zero-mode is

$f(\tilde{u}^{0},A^{0})=\mu(\tilde{u}^{0}_{\alpha\beta}+A^{0}_{\alpha\beta})^{2}+\frac{\lambda}{2}(\tilde{u}^{0}_{\alpha\alpha}+A^{0}_{\alpha\alpha})^{2}+\lambda_{1}\tilde{u}^{0}_{\alpha\alpha}A^{0}_{\alpha\alpha}+2\mu_{1}\tilde{u}^{0}_{\alpha\beta}A^{0}_{\alpha\beta}+\mu_{2}(A^{0}_{\alpha\beta})^{2}+\frac{\lambda_{2}}{2}(A^{0}_{\alpha\alpha})^{2},$
(55)

which can be rewritten as

$f(\tilde{u}^{0},A^{0})=\frac{1}{2}C^{\mu,\lambda}_{\alpha\beta,\gamma\delta}\tilde{u}^{0}_{\alpha\beta}\tilde{u}^{0}_{\gamma\delta}+\tilde{u}^{0}_{\alpha\beta}C^{\mu+\mu_{1},\lambda+\lambda_{1}}_{\alpha\beta,\gamma\delta}A^{0}_{\gamma\delta}+(\mu+\mu_{2})(A^{0}_{\alpha\beta})^{2}+\frac{\lambda+\lambda_{2}}{2}(A^{0}_{\alpha\alpha})^{2}.$
(56)

Minimizing the energy over the $D(D+1)/2$ independent components of the in-
plane strain tensor $\tilde{u}^{0}_{\alpha\beta}$ (or integrating the Gibbs
measure, which is equivalent since the energy is quadratic in the
$\tilde{u}^{0}_{\alpha\beta}$) we obtain the minimum

$[\tilde{u}^{0}_{\rm
min}]_{\alpha\beta}=-[C^{\mu,\lambda}]^{-1}_{\alpha\beta,\gamma^{\prime}\delta^{\prime}}C^{\mu+\mu_{1},\lambda+\lambda_{1}}_{\gamma^{\prime}\delta^{\prime},\gamma,\delta}A^{0}_{\gamma,\delta}=-\frac{\mu+\mu_{1}}{\mu}A^{0}_{\alpha\beta}+\frac{\lambda\mu_{1}-\lambda_{1}\mu}{\mu(2\mu+D\lambda)}\delta_{\alpha\beta}A^{0}_{\gamma\gamma}\
,$ (57)

where we have used that

$\displaystyle[C^{\mu,\lambda}]^{-1}_{\alpha\beta,\gamma\delta}=\frac{-\lambda}{2\mu(2\mu+D\lambda)}\delta_{\alpha\beta}\delta_{\gamma\delta}+\frac{1}{4\mu}(\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma}).$
(58)

Plugging back this minimum into the energy we find

$\displaystyle f_{\rm
eff}[h]=f_{0}(u_{\rm\min}^{0},A^{0})=\frac{1}{2}\bar{C}_{\alpha\beta,\gamma\delta}A^{0}_{\alpha\beta}A^{0}_{\gamma\delta},$
(59)

where

$\bar{C}=C^{\mu+\mu_{2},\lambda+\lambda_{2}}_{\alpha\beta,\gamma\delta}-C^{\mu+\mu_{1},\lambda+\lambda_{1}}\cdot[C^{\mu,\lambda}]^{-1}\cdot
C^{\mu+\mu_{1},\lambda+\lambda_{1}}.$ (60)

Upon explicit calculation the final result is

$f_{\rm
eff}[h]=(\mu_{2}-2\mu_{1}-\frac{\mu_{1}^{2}}{\mu})(A^{0}_{\alpha\beta})^{2}+\frac{1}{2}(\lambda_{2}-\frac{D\lambda_{1}\left(2\lambda+\lambda_{1}\right)\mu-2\lambda\mu_{1}^{2}+4\lambda_{1}\mu\left(\mu+\mu_{1}\right)}{\mu(D\lambda+2\mu)})(A^{0}_{\alpha\alpha})^{2}.$
(61)

Note that it vanishes when the new terms breaking rotational symmetry are
absent i.e. when $\mu_{1}=\mu_{2}=\lambda_{1}=\lambda_{2}=0$. These zero-mode
terms are thus generated only by the bulk anisotropy since we are working in
the fixed stress setting and freely integrate over the zero-mode of the in-
plane strain. We leave their study for the future footnoteFutureLR .

###  .3 B.3 Stability

Here we note that we can rewrite

$\displaystyle{\cal F}_{1}+{\cal F}_{2}=\int
d^{D}x\,\left[\frac{\kappa}{2}(\partial^{2}\vec{h})^{2}+\tau
u_{\alpha\alpha}+\frac{\gamma}{2}(\partial_{\alpha}\vec{h})^{2}+f_{\rm
el}\right].$ (62)

Using the traceless tensors and the traces as

$\displaystyle f_{\rm
el}=\mu\left(\tilde{u}_{\alpha\beta}-\frac{1}{D}\delta_{\alpha\beta}\tilde{u}_{\gamma\gamma}+\frac{\mu+\mu_{1}}{\mu}(A_{\alpha\beta}-\frac{1}{D}\delta_{\alpha\beta}A_{\gamma\gamma})\right)^{2}+\frac{2\mu+D\lambda}{2D}\left(\tilde{u}_{\alpha\alpha}+\frac{2(\mu+\mu_{1})+D(\lambda+\lambda_{1})}{2\mu+D\lambda}A_{\alpha\alpha}\right)^{2}$
(63)
$\displaystyle+\hat{\mu}_{2}\left(A_{\alpha\beta}-\frac{1}{D}\delta_{\alpha\beta}A_{\gamma\gamma}\right)^{2}+B_{2}A_{\alpha\alpha}^{2},$
(64)

with

$\displaystyle\hat{\mu}_{2}=\mu+\mu_{2}-\frac{(\mu+\mu_{1})^{2}}{\mu}$ (65)
$\displaystyle
B_{2}=\frac{1}{2D}\left(2(\mu+\mu_{2})+D(\lambda+\lambda_{2})-\frac{(2(\mu+\mu_{1})+D(\lambda+\lambda_{1}))^{2}}{2\mu+D\lambda}\right),$
(66)

where we recall that
$A_{\alpha\beta}=\frac{1}{2}\partial_{\alpha}\vec{h}\cdot\partial_{\beta}\vec{h}$.
Let us set $\tau=0$. Note that $\hat{\mu}_{2}=w_{2}$ as defined in (45). Hence
we see that, since the traceless part and the trace are independent, for
$w_{2}>0$ and $B_{2}>0$ the last two square terms imply that at the minimum
energy (which is zero) one must have $A_{\alpha\beta}=0$, and, in turn from
the two first squares, $\tilde{u}_{\alpha\beta}=0$. Hence in that case
$u_{\alpha}=0$, $\vec{h}=0$ is indeed the stable ground state. We note that,
in contrast, in the rotationally invariant case (i.e., setting
$\mu_{1}=\mu_{2}=\lambda_{1}=\lambda_{2}$), the same reasoning leads to the
zero energy minimum condition,
$u_{\alpha\beta}=\tilde{u}_{\alpha\beta}+A_{\alpha\beta}=0$, instead of the
above anisotropic condition of $\tilde{u}_{\alpha\beta}$ and $A_{\alpha\beta}$
vanishing separately. This is expected since in isotropic embedding space,
rotations of the membrane do not change its energy.

C. SCSA analysis

Below we present the details of the SCSA analysis that was outlined in the
main text, following closely the calculation in Ref.LRReview, .

###  .4 C.1. SCSA equations

The SCSA is given by the pair of coupled equations (Thermal buckling
transition of crystalline membranes in a field) and (11) given in the text for
the self-energy $\Sigma(k)={\cal G}(k)^{-1}-\kappa k^{4}$ and for the
renormalized interaction $\tilde{R}_{\alpha\beta,\gamma\delta}({\bf q})$. The
equation (11) involves tensor multiplication. We can thus decompose $\Pi({\bf
q})=\sum_{i=1}^{5}\pi_{i}(q)W_{i}({\bf q})$ and
$\tilde{R}(q)=\sum_{i=1}^{5}\tilde{w}_{i}(q)W_{i}({\bf q})$, as indicated in
the text, where $\tilde{w}_{i}(q)$ are the momentum dependent renormalized
couplings and $\pi_{i}(q)$ are polarization integrals calculated below. The
rules for the tensor multiplication were given in the previous section. Since
the tensors $R_{\alpha\beta,\gamma\delta}({\bf q})$,
$\tilde{R}_{\alpha\beta,\gamma\delta}({\bf q})$ and
$\Pi_{\alpha\beta,\gamma\delta}({\bf q})$ are symmetric in
$\alpha\leftrightarrow\beta$, in $\gamma\leftrightarrow\delta$ and in
$(\alpha,\beta)\leftrightarrow(\gamma,\delta)$, they can be parameterized in
terms of five couplings (i.e., with $w_{4a}=w_{4b}=w_{4}$).

We can now solve the equation (11) and find the renormalized couplings
$\tilde{w}_{i}(q)$ as

$\displaystyle\tilde{w}_{1}(q)={w_{1}\over{1+w_{1}\pi_{1}(q)}}\;,\;\;\;\tilde{w}_{2}(q)={w_{2}\over{1+w_{2}\pi_{2}(q)}}\;,$
(67) $\displaystyle\begin{pmatrix}\tilde{w}_{3}(q)&\tilde{w}_{4}(q)\\\
\tilde{w}_{4}(q)&\tilde{w}_{5}(q)\end{pmatrix}=\begin{pmatrix}w_{3}&w_{4}\\\
w_{4}&w_{5}\end{pmatrix}\left(\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+\begin{pmatrix}\pi_{3}(q)&\pi_{4}(q)\\\
\pi_{4}(q)&\pi_{5}(q)\end{pmatrix}\begin{pmatrix}w_{3}&w_{4}\\\
w_{4}&w_{5}\end{pmatrix}\right)^{-1}\;\;.$ (68)

These can be substituted into (Thermal buckling transition of crystalline
membranes in a field) to express the self-energy as

$\displaystyle\Sigma(k)={2\over d_{c}}\sum_{i=1,5}\int_{\bf
q}\tilde{w}_{i}(q){\cal G}({\bf k}-{\bf
q})k_{\alpha}(k_{\beta}-q_{\beta})(W_{i})_{\alpha\beta,\gamma\delta}({\bf
q})k_{\gamma}(k_{\delta}-q_{\delta})\;,$ (69)

The above equations form a closed set of SCSA equations for the five
renormalized elastic coupling constants $\tilde{w}_{i}(q)$, together with the
self energy $\Sigma(k)$. The complete Dyson equation for the self-energy
contains an additional UV divergent ”tadpole” diagram contribution, which
scales as $k^{2}$. The integral in (69) also contains a component that scales
as $k^{2}$ at small $k$. Both contributions have been substracted by tuning
the bare coefficient $\gamma$ in order to sit at the critical point.

To solve the above SCSA equations at the critical point, we look for a
solution with the long wavelength form ${\cal G}(k)\simeq
Z^{-1}_{\kappa}/k^{4-\eta}$. The $\pi_{i}(q)$ integrals have been calculated
in the Appendix B of LRReview . They diverge for small $q$ as:

$\displaystyle\pi_{i}(q)\simeq
Z_{\kappa}^{-2}a_{i}(\eta,D)q^{-(4-D-2\eta)}\;.$ (70)

For the amplitudes $a_{i}(\eta,D)$ we find

$\displaystyle
a_{1}=2A\;,\;\;a_{2}=A{2(2-\eta)\over{D+\eta-2}}\;,\;\;a_{3}=A(D+1)\;,\;\;a_{4}=A\sqrt{D-1}(D+2\eta-3)\;,$
(71) $\displaystyle
a_{5}={A\over{D-2+\eta}}(-22+31D-10D^{2}+D^{3}+43\eta-32D\eta+5D^{2}\eta-24\eta^{2}+8D\eta^{2}+4\eta^{3})\;,$

with

$A=A(\eta,D)={\Gamma(2-\eta-D/2)\Gamma(D/2+\eta/2)\Gamma(D/2+\eta/2)\over
4(4\pi)^{D/2}\Gamma(2-\eta/2)\Gamma(2-\eta/2)\Gamma(D+\eta)}\;.$ (72)

To compute the self-energy we define the amplitudes $b_{i}(\eta,D)$ through:

$\displaystyle\int_{\bf q}q^{4-D-2\eta}|{\bf k}-{\bf
q}|^{-(4-\eta)}k_{\alpha}(k_{\beta}-q_{\beta})(W_{i})_{\alpha\beta,\gamma\delta}({\bf
q})k_{\gamma}(k_{\delta}-q_{\delta})=b_{i}(\eta,D)k^{4-\eta}\;.$ (73)

The explicit calculation in the Appendix B of LRReview  gives

$\displaystyle
b_{1}=B(D-2)(D+1)\;,\;\;b_{2}=-B{(D-1)(D^{2}-4+2\eta)\over{D-2+\eta}}\;,\;\;b_{3}=B(D+1)\;,$
(74) $\displaystyle
b_{4}=2B\sqrt{D-1}(2\eta-3)\;,\;\;b_{5}={B\over{D-2+\eta}}(-22+15D-2D^{2}+43\eta-16D\eta-24\eta^{2}+4D\eta^{2}+4\eta^{3})\;,$

where

$B=B(\eta,D)={\Gamma(\eta/2)\Gamma(D/2+\eta/2)\Gamma(2-\eta)\over
4(4\pi)^{D/2}\Gamma(2-\eta/2)\Gamma(D/2+\eta)\Gamma(D/2+2-\eta/2)}\;.$ (75)

###  .5 C.2. Anisotropic fixed point

Let us first search for a solution to the SCSA equations when all the bare
couplings $w_{i}$ are nonzero. This corresponds to the ”anisotropic fixed
point” discussed in the text. In the limit ${\bf q}\to 0$ we find from (68)

$\displaystyle\tilde{w}_{1}(q)\simeq\frac{1}{\pi_{1}(q)}\;,\quad\tilde{w}_{2}(q)\simeq\frac{1}{\pi_{2}(q)}\;,\quad,\quad\begin{pmatrix}\tilde{w}_{3}(q)&\tilde{w}_{4}(q)\;,\\\
\tilde{w}_{4}(q)&\tilde{w}_{5}(q)\end{pmatrix}\simeq\begin{pmatrix}\pi_{3}(q)&\pi_{4}(q)\\\
\pi_{4}(q)&\pi_{5}(q)\end{pmatrix}^{-1}\;.$ (76)

independent of the bare values, as long as they are nonzero. Substituting
Eqs.(68),(72),(75) into (69) we see that factors of $Z_{\kappa}$ cancel and we
find the self-consistent equation:

$\frac{d_{c}}{2}=\sum_{i=1,2}{b_{i}(\eta,D)\over
a_{i}(\eta,D)}+{b_{3}(\eta,D)a_{5}(\eta,D)-b_{4}(\eta,D)a_{4}(\eta,D)+b_{5}(\eta,D)a_{3}(\eta,D)\over{a_{3}(\eta,D)a_{5}(\eta,D)-a_{4}(\eta,D)^{2}}}\;.$
(77)

Putting everything together, after considerable simplifications, the equation
determining the exponent $\eta=\eta^{\rm anis}(D,d_{c})$ is found to be:

$\displaystyle d_{c}={{D(D+1)(D-4+\eta)(D-4+2\eta)(2D-3+2\eta)\Gamma[{1\over
2}\eta]\Gamma[2-\eta]\Gamma[\eta+D]\Gamma[2-{1\over
2}\eta]}\over{2(2-\eta)(5-D-2\eta)(D+\eta-1)\Gamma[{1\over 2}D+{1\over
2}\eta]\Gamma[2-\eta-{1\over 2}D]\Gamma[\eta+{1\over 2}D]\Gamma[{1\over
2}D+2-{1\over 2}\eta]}}\;,$ (78)

which in $D=2$ reduces to

$\displaystyle d_{c}=\frac{24(\eta-1)^{2}(2\eta+1)}{(\eta-4)\eta(2\eta-3)}\;.$
(79)

as given in the main text, leading to $\eta^{\rm
anis}(D=2,d_{c}=1)=0.753645..$. In the limit of large $d_{c}$, the solution of
(78) behaves as

$\displaystyle\eta^{\rm
anis}(D,d_{c})\simeq\frac{C(D)}{d_{c}}+O(1/d_{c}^{2})\quad,\quad
C(D)=\frac{(D-4)^{2}(2D-3)\Gamma(D+2)}{2(5-D)(D-1)\Gamma\left(2-\frac{D}{2}\right)\Gamma\left(\frac{D}{2}+2\right)\Gamma\left(\frac{D}{2}\right)^{2}}\;,$

with $C(2)=2$. As discussed in LRReview  the leading coefficient $C(D)$ in the
$1/d_{c}$ expansion is an exact result, while the higher orders are specific
to the SCSA.

We note that the above equation (78) is the same as the one obtained for the
crumpling transition, replacing $d$ by $d_{c}$. Hence, studying our new fixed
point amounts, formally, to studying the crumpling transition fixed point in
embedding space dimension $d=1$ instead of $d=3$. Not surprisingly then, the
leading term in the large $d_{c}$ expansion above then coincides with the one
in the $1/d$ expansion for the crumpling transition of Ref. AL, . We can also
expand our SCSA prediction in $\epsilon=4-D$, finding

$\displaystyle\eta^{\rm
anis}(D,d_{c})\simeq\frac{25}{3d_{c}}(4-D)^{3}+O((4-D)^{3}),$ (80)

consistent with the vanishing of the leading order $O(\epsilon)$ of $\eta^{\rm
anis}(D,d_{c})$ found below in the section on the RG calculation.

This new ”anisotropic” membrane fixed point is characterized by several
universal amplitude ratio. As discussed in the text, from (76) we obtain

$\displaystyle\tilde{w}_{i}(q)\simeq Z_{\kappa}^{2}c_{i}q^{4-D-2\eta}/A$ (81)
$\displaystyle c_{i}=1/a_{i}\quad\text{for}\quad
i=1,2\quad,\quad\begin{pmatrix}c_{3}&c_{4}\\\
c_{4}&c_{5}\end{pmatrix}\simeq\begin{pmatrix}a_{3}&a_{4}\\\
a_{4}&a_{5}\end{pmatrix}^{-1}$

Inserting the $\tilde{w}_{i}(q)$ into (48), we obtain the renormalized
couplings of the $u,h$ theory. More precisely we obtain the $h^{4}$ couplings
$\tilde{\mu}(q)+\tilde{\mu}_{2}(q)$,
$\tilde{\lambda}(q)+\tilde{\lambda}_{2}(q)$, and the $uh^{2}$ couplings
$(\tilde{\mu}(q)+\tilde{\mu}_{1}(q))^{2}/\tilde{\mu}(q)$ and
$(\tilde{\lambda}(q)+\tilde{\lambda}_{1}(q))^{2}/\tilde{\lambda}(q)$. These
four couplings thus vanish as $q^{4-D-2\eta}$ at small $q$. In addition we
obtain the ratio $\tilde{\lambda}(q)/\tilde{\mu}(q)$ which has a finite limit
at small $q$. The determination of $\tilde{\mu}(q)$ however requires an
additional calculation (see below), with the result that $\tilde{\mu}(q)\sim
q^{\eta_{u}}$ where $\eta_{u}$ is now an independent exponent (at variance
with the rotationally invariant case where one has $\eta_{u}=4-D-2\eta$). For
this anisotropic fixed point, $\eta_{u}=0$, i.e. $\tilde{\mu}(0)$ is finite.
Hence we find that $\tilde{\mu}_{2}(q)\to-\tilde{\mu}(0)$ at small $q$, so
that the $h^{4}$ coupling can vanish at small $q$ as
$\tilde{\mu}(q)+\tilde{\mu}_{2}(q)\sim q^{4-D-2\eta}$, and similarly for
$\tilde{\lambda}_{2}(q)$. A similar property holds for the $uh^{2}$ couplings.

Let us now determine the amplitude ratio, which are universal at the fixed
point. From (81) we obtain the amplitude ratio in the long wavelength limit as

$\displaystyle\lim_{q\to
0}\frac{\tilde{w}_{i}(q)}{\tilde{w}_{j}(q)}=\frac{c_{i}}{c_{j}}$ (82)

for any pair $(i,j)$, with, using (71) and (81) we obtain

$\displaystyle c_{1}=\frac{1}{2}\quad,\quad
c_{2}=-\frac{D+\eta-2}{2(\eta-2)}\quad,\quad
c_{3}=\frac{1}{4}\left(\frac{1-D}{(D+3)(D+\eta-1)}+\frac{D}{\eta-2}-\frac{8}{(D+3)(D+2\eta-5)}+2\right)$
(83) $\displaystyle
c_{4}=-\frac{\sqrt{D-1}(D+\eta-2)(D+2\eta-3)}{4(\eta-2)(D+\eta-1)(D+2\eta-5)}\quad,\quad
c_{5}=\frac{(D+1)(D+\eta-2)}{4(\eta-2)(D+\eta-1)(D+2\eta-5)}$ (84)

Note that these values of the $c_{i}$ assume that all bare $w_{i}$ are nonzero
hence they are valid only at the anisotropic fixed point. Inserting the value
of $\eta$ for $D=2$ and $d_{c}$=1 we find, at the anisotropic fixed point

$c_{i}(D=2,d_{c}=1)=\left\\{\frac{1}{2},0.30234,0.338287,-0.0292957,0.173248\right\\}$
(85)

From this, using (48), we find $\lim_{q\to
0}\tilde{\lambda}(q)/\tilde{\mu}(q)=-0.978449$ and the Poisson ratio
$\sigma_{R}(q)=\frac{\tilde{\lambda}(q)}{2\tilde{\mu}(q)+(D-1)\tilde{\lambda}(q)}=-0.957808$

Note that for $D=2$ and large $d_{c}$ we find, up to $O(1/d_{c}^{2})$ terms

$c_{i}=\left\\{\frac{1}{2},\frac{1}{2d_{c}},\frac{1}{3}+\frac{1}{36d_{c}},\frac{1}{12d_{c}},\frac{1}{4d_{c}}\right\\}$
(86)

Hence for $D=2$, the anisotropic membrane fixed point converges as
$d_{c}\to+\infty$ to the one of the isotropic membrane since $\tilde{w}_{2}$,
$\tilde{w}_{4}$, $\tilde{w}_{5}$ are parametrically smaller in that limit than
$\tilde{w}_{1}$ and $\tilde{w}_{3}$ (which span the couplings of the isotropic
membrane). However, from (83) we can state that these two fixed points are
different at infinite $d_{c}$ for $D>2$. In this limit one can simply set
$\eta\to 0$ in (83) and one finds for the anisotropic fixed point

$\lim_{d_{c}\to+\infty}c_{i}=\left\\{\frac{1}{2},\frac{D-2}{4},\frac{D^{2}-9D+22}{40-8D},\frac{(D-3)(D-2)}{8(D-5)\sqrt{D-1}},\frac{-D^{2}+D+2}{8\left(D^{2}-6D+5\right)}\right\\}$
(87)

while for the isotropic one $c_{i}=\\{\frac{1}{2},0,\frac{1}{D+1},0,0\\}$, see
below. Hence, for $d_{c}=+\infty$, the anisotropic fixed point leaves the
isotropic subspace as $D$ increases from $D=2$ to $D=4$.

As mentionned in the text, there is an interesting subspace of couplings which
corresponds to a purely local interaction between the gradient fields
$\partial_{\alpha}\vec{h}$

$R_{\alpha\beta,\gamma\delta}=\frac{\mu_{0}}{2}(\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma})+\frac{\lambda_{0}}{2}\delta_{\alpha\beta}\delta_{\gamma\delta}.$
(88)

for some constants denoted $\mu_{0}$ and $\lambda_{0}$ (these are denoted
$4v_{2}$ and $4v_{1}$ respectively in the main text). It is realized by the
choice

$\displaystyle
w_{1}=w_{2}=\mu_{0},\;\;w_{3}=\frac{1}{2}(D-1)\lambda_{0}+\mu_{0},\;\;w_{4}=\frac{1}{2}\sqrt{D-1}\lambda_{0},\;\;w_{5}=\frac{1}{2}\lambda_{0}+\mu_{0}\;.$
(89)

Note that the two eigenvalues of the matrix formed by the $w_{i}$, $i=3,4,5$,
are then $\mu_{0}$, and $\mu_{0}+\frac{1}{2}D\lambda$. Replacing $d_{c}$ by
$d$ this is also the subspace corresponding to the bare action of the Landau
theory for the crumpling transition PKN .

This subspace is preserved by the one-loop RG in an expansion in $D=4$, as we
will see in the next section. However, for any fixed $D<4$, it is not
preserved by the RG flow in the large $d_{c}$ limit (hence it is also not
preserved by the SCSA). In $D=4$ at large $d_{c}$ it is indeed preserved
(consistent with the RG), since in that case one has

$\displaystyle\lim_{d_{c}\to+\infty}c_{i}=\left\\{\frac{1}{2},\frac{1}{2},\frac{1}{4},-\frac{1}{4\sqrt{3}},\frac{5}{12}\right\\}$
(90)

which indeed belongs to the subspace (89). However, from the above discussion,
we expect the two-loop corrections in the RG to fail to preserve this
subspace. This indicates that the study of the RG of the crumpling transition
to higher order in $\epsilon$ will be qualitatively different from the one
given in PKN , a subject we leave for future investigation.

###  .6 C 3. RG flow associated to the SCSA equations

It is instructive to recast the SCSA equations into an RG flow. We start with
large $d_{c}$, and discuss general $d_{c}$ below. The SCSA equations allow one
to obtain the exact RG beta function to leading order in $1/d_{c}$ in any
dimension $D$. Indeed, taking a derivative $\partial_{\ell}=-q\partial_{q}$ on
both sides of equations (67) we obtain,

$\partial_{\ell}\tilde{w}_{i}(q)=-\frac{w_{i}^{2}}{(1+w_{i}\pi_{1}(q))^{2}}(-q\partial_{q})\pi_{i}(q)=-\tilde{w}_{i}(q)^{2}(-q\partial_{q}\pi_{i}(q))\simeq-\tilde{w}_{i}(q)^{2}\kappa^{-2}\epsilon
q^{-\epsilon}a_{i}(0,D)$ (91)

where we have used (70) setting $\eta\to 0$, i.e., using the bare propagator
with $Z_{\kappa}=\kappa$. The natural dimensionless coupling for the RG is

$\hat{w}_{i}:=\tilde{w}_{i}(q)\kappa^{-2}q^{-\epsilon}$ (92)

In terms of these couplings we obtain the RG equation for $d_{c}=+\infty$,
exact for any $\epsilon=4-D$,

$\displaystyle\partial_{\ell}\hat{w}_{i}=\epsilon\hat{w}_{i}-\epsilon{\color[rgb]{1.00,0.00,0.00}a_{i}(0,D)}\hat{w}_{i}^{2}\quad,\quad
i=1,2$ (93)
$\displaystyle\partial_{\ell}\begin{pmatrix}\hat{w}_{3}&\hat{w}_{4}\\\
\hat{w}_{4}&\hat{w}_{5}\end{pmatrix}=\epsilon\begin{pmatrix}\hat{w}_{3}&\hat{w}_{4}\\\
\hat{w}_{4}&\hat{w}_{5}\end{pmatrix}-\epsilon\begin{pmatrix}\hat{w}_{3}&\hat{w}_{4}\\\
\hat{w}_{4}&\hat{w}_{5}\end{pmatrix}\begin{pmatrix}a_{3}(0,D)&a_{4}(0,D)\\\
a_{4}(0,D)&a_{5}(0,D)\end{pmatrix}\begin{pmatrix}\hat{w}_{3}&\hat{w}_{4}\\\
\hat{w}_{4}&\hat{w}_{5}\end{pmatrix}$ (94)

The fixed point of these RG equations which describes the anisotropic membrane
for $d_{c}=+\infty$ is then $\hat{w}_{i}=\hat{w}_{i}^{*}$ with

$\hat{w}_{i}^{*}=\frac{1}{a_{i}(0,D)}\quad,\quad
i=1,2\quad,\quad\begin{pmatrix}\hat{w}^{*}_{3}&\hat{w}^{*}_{4}\\\
\hat{w}^{*}_{4}&\hat{w}^{*}_{5}\end{pmatrix}=\begin{pmatrix}a_{3}(0,D)&a_{4}(0,D)\\\
a_{4}(0,D)&a_{5}(0,D)\end{pmatrix}^{-1}$ (95)

is consistent with the above analysis (81). The calculation of the exponent
$\eta$ to leading order $O(1/d_{c})$ is then as follows. If one calculates
$-k\partial_{k}(\Sigma(k)/k^{4})$ from (69) one obtains a convergent integral.
Replacing $\tilde{w}_{i}(q)=\kappa^{2}q^{\epsilon}\hat{w}_{i}$ in (69) and
using (73) we can write the RG function $\eta=\eta(\hat{w})$ as

$\eta=-k\partial_{k}(\Sigma(k)/k^{4})={2\over
d_{c}}\sum_{i=1,5}\tilde{b}_{i}(D)\hat{w}_{i}\quad,\quad\tilde{b}_{i}(D)=\lim_{\eta\to
0}\eta b_{i}(\eta,D),$ (96)

where in the r.h.s we used $\eta$ as a regulator to obtain the needed (finite)
integral. One can then easily check that at the fixed point (95) the exponent
$\eta=\eta(\hat{w}^{*})$ recovers the result $\eta\simeq C(D)/d_{c}$ predicted
by the self-consistent equation (77).

In the above RG equations (93) we have neglected the renormalization of
$\kappa$ which is subdominant in $1/d_{c}$. We can now take it into account
and define accordingly the running RG couplings as
$\hat{w}_{i}:=\tilde{w}_{i}(q)\kappa^{-2}q^{2\eta-\epsilon}=\tilde{w}_{i}(q)\tilde{\kappa}(q)^{-2}q^{-\epsilon}$.
This allows to write the SCSA equations as RG flow equations for any $d_{c}$
as follows

$\displaystyle\partial_{\ell}\hat{w}_{i}=(\epsilon-2\eta)\hat{w}_{i}-(\epsilon-2\eta)a_{i}(D,\eta)\hat{w}_{i}^{2}\quad,\quad
i=1,2$ (97)
$\displaystyle\partial_{\ell}\begin{pmatrix}\hat{w}_{3}&\hat{w}_{4}\\\
\hat{w}_{4}&\hat{w}_{5}\end{pmatrix}=(\epsilon-2\eta)\begin{pmatrix}\hat{w}_{3}&\hat{w}_{4}\\\
\hat{w}_{4}&\hat{w}_{5}\end{pmatrix}-(\epsilon-2\eta)\begin{pmatrix}\hat{w}_{3}&\hat{w}_{4}\\\
\hat{w}_{4}&\hat{w}_{5}\end{pmatrix}\begin{pmatrix}a_{3}(D,\eta)&a_{4}(D,\eta)\\\
a_{4}(D,\eta)&a_{5}(D,\eta)\end{pmatrix}\begin{pmatrix}\hat{w}_{3}&\hat{w}_{4}\\\
\hat{w}_{4}&\hat{w}_{5}\end{pmatrix},$ (98)

where the $\eta$ RG function, $\eta=\eta(\hat{w})$, is defined as

$\displaystyle\eta:=-k\partial_{k}\tilde{\kappa}(k)=-k\partial_{k}(\Sigma(k)/k^{4})={2\over
d_{c}}\eta\sum_{i=1,5}b_{i}(\eta,D)\hat{w}_{i}.$ (99)

The fixed point of these RG equations, corresponding to all bare $w_{i}$ being
nonzero, i.e., the anisotropic membrane fixed point, is given by

$\hat{w}_{i}^{*}=\frac{1}{a_{i}(D,\eta^{*})}\quad,\quad
i=1,2\quad,\quad\begin{pmatrix}\hat{w}^{*}_{3}&\hat{w}^{*}_{4}\\\
\hat{w}^{*}_{4}&\hat{w}^{*}_{5}\end{pmatrix}=\begin{pmatrix}a_{3}(D,\eta^{*})&a_{4}(D,\eta^{*})\\\
a_{4}(D,\eta^{*})&a_{5}(D,\eta^{*})\end{pmatrix}^{-1},$ (100)

where $\eta^{*}$ is determined by (99) at the fixed point. Equivalence with
the full SCSA equation (77) is then immediately follows.

Other fixed points

As discussed in the main text, there are a number of other subspaces which are
preserved by renormalization within the SCSA method (hence also at large
$d_{c}$). These can be labeled as $S_{i_{1},\dots i_{n}}$, with $1\leq
i_{1}<i_{2}<\dots<i_{n}\leq 5$, where the only nonzero bare couplings $w_{i}$
are $w_{i_{1}},\dots w_{i_{n}}$. For those with $w_{4}=0$, i.e.,
$i_{1},\dots,i_{n}\in\\{1,2,3,5\\}$, there are four with $n=1$, five with
$n=2$ (that is $S_{12},S_{13},S_{15},S_{23},S_{25}$) together with $S_{123}$
and $S_{125}$ (note that $w_{4}=0$ is not preserved unless one has also
$w_{3}=0$ or $w_{5}=0$). Then, one has $S_{345},S_{1345},S_{2345},S_{12345}$
with $w_{4}\neq 0$. In each of these subspaces there is a fixed point denoted
by $P_{i_{1},\dots i_{n}}$. It is obtained from (100) by setting to zero the
$\hat{w}_{i}^{*}$ not in the set $\\{i_{1},\dots,i_{n}\\}$ (disregarding their
corresponding equation, except $a_{4}$ which must be set to zero when
$\hat{w}_{4}=0$). Their associated SCSA equation is obtained as
$\frac{d_{c}}{2}=\sum_{i=1}^{5}b_{i}(\eta,D)\hat{w}_{i}^{*}$. Let us give some
examples.

  1. 1.

The fixed point $P_{13}$ describes the isotropic flat phase. Its exponent
$\eta$ is determined by
$\frac{d_{c}}{2}=\frac{b_{1}}{a_{1}}+{b_{3}\over{a_{3}}}$ i.e., Eq. (19) in
the main text, leading to the well known value
$\eta=4/(1+\sqrt{15})=0.820852..$, $\zeta=\frac{1}{7}(8-\sqrt{15})=0.589574..$
for the exponents describing out-of-plane fluctuations of the physical
membrane $D=2,d_{c}=1$. The amplitudes are $c_{1}=A/a_{1}$ and $c_{3}=A/a_{3}$
which gives $c_{i}=\\{\frac{1}{2},0,\frac{1}{D+1},0,0\\}$, leading to
$\lim_{q\to 0}\tilde{\lambda}(q)/\tilde{\mu}(q)=-\frac{2}{D+2}$ and to the
universal Poisson ratio, $\sigma_{R}=-1/3$ within the SCSA.

  2. 2.

The fixed point $S_{1345}$ has $\hat{w}_{2}^{*}=0$ and describes the case
where $\mu_{2}=\mu_{1}=0$. The exponent $\eta$ is determined by (20) in the
text, which for $D=2$ gives

$\frac{d_{c}}{2}=\frac{93}{5(\eta-4)}+\frac{6}{\eta-2}+\frac{1}{\eta}+\frac{16}{15-10\eta}+8.$
(101)

For $d_{c}=1$ one finds $\eta=0.853967$, and $\zeta=0.573016$. The amplitudes
$c_{i}$ are then given by (83), where one sets $c_{2}=0$. For $D=2$, $d_{c}=1$
inserting the above value of $\eta$ one finds

$c_{i}=\left\\{\frac{1}{2},0,0.346325,-0.0550542,0.233302\right\\}.$ (102)

Note that the manifold $\hat{w}_{2}=0$ is however not preserved within the
$\epsilon$-expansion (see analysis in section below).

Remark. One bonus of these RG equations, as compared to the original self-
consistent equations, is that one can determine the direction of the RG flow,
the Hessian around each fixed point, and the various crossovers in the flow.
For instance, to determine the Hessian around a fixed point $\hat{w}_{i}^{*}$,
with associated exponent $\eta^{*}$, we need the variation of $\eta$.
Variation of (99) around the fixed point gives
$\delta\eta=-\frac{\sum_{j=1}^{5}b_{j}\delta\hat{w}_{j}}{\sum_{k=1}^{5}b^{\prime}_{k}\hat{w}^{*}_{k}}$,
where we have denoted $a_{i}\equiv a_{i}(D,\eta^{*})$, $b_{i}\equiv
b_{i}(D,\eta^{*})$,
$a^{\prime}_{i}\equiv\partial_{\eta}b_{i}(D,\eta)|_{\eta=\eta^{*}}$,
$b^{\prime}_{i}\equiv\partial_{\eta}b_{i}(D,\eta)|_{\eta=\eta^{*}}$. Using
this, one can obtain the Hessian, and the flow around the fixed point. We
defer this study to the futurefootnoteFutureLR .

D. Renormalization group calculation for the $h^{4}$ theory

Here we present the details of the one-loop RG calculation for the quartic
model $h^{4}$ defined in Eq. (Thermal buckling transition of crystalline
membranes in a field) of the main text. The power-counting is the same as in
the standard $\phi^{4}$ $O(N)$ model with quartic nonlinearities which are
relevant for $D<D_{uc}=4$. This allows us to control the RG analysis by an
expansion in $\epsilon=4-D$ around $D=4$ WilsonFisher ; ZinnJustin ; CL  Here
we will simply display the calculation using the momentum shell RG, i.e
introducing a running UV cutoff $\Lambda_{\ell}=\Lambda e^{-\ell}$ and
integrating the internal momentum in the shell
$\Lambda_{\ell}e^{-d\ell}<q<\Lambda_{\ell}$. However, we have checked all of
our formula also using dimensional regularization for $D<4$ with the external
momentum as an IR cutoff.

In the critical theory there are two types of relevant one-loop corrections,
the correction to the $h^{4}$ vertex $\delta R$ and the correction to the
bending rigidity $\delta\kappa$. Away from criticality one also needs to
calculate the correction to $\gamma$.

###  .7 D 1. Correction to the quartic interaction

Having constructed the generic vertex $R_{\alpha\beta,\gamma\delta}({\bf q})$,
the analysis of the diagrams is then quite similar to that of the $O(N)$
modelWilsonFisher ; ZinnJustin ; CL . There are three distinct channels
contributing to the renormalization of $R_{\alpha\beta,\gamma\delta}({\bf
q})$, with only one of them taken into account in the large $d_{c}$ and SCSA
analysis. The corrections to the quartic coupling can be written as the sum

$\delta R=\delta R^{(1)}+\delta R^{(2)}+\delta R^{(3)}$ (103)

depicted by the three diagrams in Fig.2.

The contribution from the first (vacuum polarization) diagram, proportional to
$d_{c}$, is given by the following integral

$\displaystyle\delta R_{\alpha\beta,\gamma\delta}^{(1)}({\bf q})$
$\displaystyle=$
$\displaystyle-\frac{Td_{c}}{\kappa^{2}}R_{\alpha\beta,\gamma^{\prime}\delta^{\prime}}({\bf
q})R_{\gamma^{\prime\prime}\delta^{\prime\prime},\gamma\delta}({\bf
q}){\int_{\Lambda_{\ell}e^{-d\ell}}^{\Lambda_{\ell}}{d^{D}p\over(2\pi)^{D}}}\frac{p_{\gamma^{\prime}}(q_{\delta^{\prime}}-p_{\delta^{\prime}})p_{\gamma^{\prime\prime}}(q_{\delta^{\prime\prime}}-p_{\delta^{\prime\prime}})}{p^{4}|{\bf
q}-{\bf p}|^{4}},$ (104) $\displaystyle\approx$
$\displaystyle-\frac{Td_{c}}{\kappa^{2}}R_{\alpha\beta,\gamma^{\prime}\delta^{\prime}}({\bf
q})R_{\gamma^{\prime\prime}\delta^{\prime\prime},\gamma\delta}({\bf
q}){\int_{\Lambda_{\ell}e^{-d\ell}}^{\Lambda_{\ell}}{d^{D}p\over(2\pi)^{D}}}\frac{p_{\gamma^{\prime}}p_{\delta^{\prime}}p_{\gamma^{\prime\prime}}p_{\delta^{\prime\prime}}}{p^{8}}$
(105)

where in the second line we have kept only the leading terms in $D=4$.

Similarly, the contribution from the second (vertex correction) diagram is
given by

$\displaystyle\delta R_{\alpha\beta,\gamma\delta}^{(2)}({\bf q},{\bf k}_{3})$
$\displaystyle=$ $\displaystyle-4\frac{T}{\kappa^{2}}{\rm
sym}R_{\alpha\beta,\gamma^{\prime}\delta^{\prime}}({\bf
q}){\int_{\Lambda_{\ell}e^{-d\ell}}^{\Lambda_{\ell}}{d^{D}p\over(2\pi)^{D}}}\frac{p_{\gamma^{\prime}}(q_{\delta^{\prime}}-p_{\delta^{\prime}})p_{\gamma^{\prime\prime}}(q_{\delta^{\prime\prime}}-p_{\delta^{\prime\prime}})R_{\gamma^{\prime\prime}\gamma,\delta^{\prime\prime}\delta}({\bf
p}-{\bf k}_{3})}{p^{4}|{\bf q}-{\bf p}|^{4}},$ (106) $\displaystyle\approx$
$\displaystyle-4\frac{T}{\kappa^{2}}R_{\alpha\beta,\gamma^{\prime}\delta^{\prime}}({\bf
q}){\int_{\Lambda_{\ell}e^{-d\ell}}^{\Lambda_{\ell}}{d^{D}p\over(2\pi)^{D}}}\frac{p_{\gamma^{\prime}}p_{\delta^{\prime}}p_{\gamma^{\prime\prime}}p_{\delta^{\prime\prime}}R_{\gamma^{\prime\prime}\gamma,\delta^{\prime\prime}\delta}({\bf
p})}{p^{8}}$ (107)

where ${\rm sym}$ denotes the symmetrization
$(\alpha,\beta)\leftrightarrow(\gamma,\delta)$. Finally, the contribution from
the third (box) diagram is

$\displaystyle\delta R_{\alpha\beta,\gamma\delta}^{(3)}({\bf q},{\bf k}_{1})$
$\displaystyle=$
$\displaystyle-4\frac{T}{\kappa^{2}}{\int_{\Lambda_{\ell}e^{-d\ell}}^{\Lambda_{\ell}}{d^{D}p\over(2\pi)^{D}}}\frac{p_{\gamma^{\prime}}(q_{\delta^{\prime}}-p_{\delta^{\prime}})p_{\gamma^{\prime\prime}}(q_{\delta^{\prime\prime}}-p_{\delta^{\prime\prime}})R_{\alpha\gamma^{\prime},\gamma\delta^{\prime}}({\bf
k}_{1}-{\bf p})R_{\gamma^{\prime\prime}\beta,\delta^{\prime\prime}\delta}({\bf
k}_{2}+{\bf p})}{p^{4}|{\bf q}-{\bf p}|^{4}},$ (108) $\displaystyle\approx$
$\displaystyle-4\frac{T}{\kappa^{2}}{\int_{\Lambda_{\ell}e^{-d\ell}}^{\Lambda_{\ell}}{d^{D}p\over(2\pi)^{D}}}\frac{p_{\gamma^{\prime}}p_{\delta^{\prime}}p_{\gamma^{\prime\prime}}p_{\delta^{\prime\prime}}R_{\alpha\gamma^{\prime},\gamma\delta^{\prime}}({\bf
p})R_{\gamma^{\prime\prime}\beta,\delta^{\prime\prime}\delta}({\bf
p})}{p^{8}}.$ (109)

where we recall that ${\bf q}={\bf k}_{1}+{\bf k}_{2}$. Note that one should
symmetrize with the crossed diagram but at the level of the last step
exchanging $\gamma^{\prime\prime}$ and $\delta^{\prime\prime}$ does not make a
difference.

Figure 2: Feynman diagrams for one-loop corrections to the quartic vertex
$R_{\alpha\beta,\gamma\delta}({\bf q})$, with (a) ”vacuum polarization”
$\delta R^{(1)}_{\alpha\beta,\gamma\delta}({\bf q})$, (b) ”vertex correction”
$\delta R^{(2)}_{\alpha\beta,\gamma\delta}({\bf q},{\bf k}_{3})$, (c) ”box
diagram” $\delta R^{(3)}_{\alpha\beta,\gamma\delta}({\bf q},{\bf k}_{1})$.

To evaluate these integrals we now insert the decomposition
$R_{\alpha\beta,\gamma\delta}(q)=\sum_{i=1}^{5}w_{i}(W_{i}(q))_{\alpha\beta,\gamma\delta}$
and use the definitions and the properties of the projectors summarized in
Section A. We further use the formula for the angular averages (denoted
$\langle\dots\rangle$, where $\hat{k}=k/|k|$)
$\langle\hat{k}_{\alpha}\hat{k}_{\beta}\rangle=\frac{1}{D}\delta_{\alpha\beta}$
and
$\langle\hat{k}_{\alpha}\hat{k}_{\beta}\hat{k}_{\gamma}\hat{k}_{\delta}\rangle=\frac{1}{D(D+2)}\left(\delta_{\alpha\beta}\delta_{\gamma\delta}+\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\gamma\beta}\right)$LRReview
.

Denoting $\delta w=(\delta w_{1},\delta w_{2},\delta w_{3},\delta w_{4},\delta
w_{5})$ the one loop corrections to the couplings $w_{i}$ from the first
diagram are

$\displaystyle\delta
w=\frac{-d_{c}}{D(D+2)}\bigg{(}2w_{1}^{2},2w_{2}^{2},(D+1)w_{3}^{2}+2\sqrt{D-1}w_{4}w_{3}+3w_{4}^{2},w_{4}(\sqrt{D-1}w_{4}+3w_{5})+w_{3}(Dw_{4}+\sqrt{D-1}w_{5}+w_{4}),$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(D+1)w_{4}^{2}+2\sqrt{D-1}w_{5}w_{4}+3w_{5}^{2}\bigg{)}\times\frac{T}{\kappa^{2}}{\int_{\Lambda_{\ell}e^{-d\ell}}^{\Lambda_{\ell}}{d^{D}p\over(2\pi)^{D}}}\frac{1}{p^{4}}$
(110)

The contribution of the second diagram reads

$\displaystyle\delta
w=\frac{-4}{D(D+2)}\bigg{(}w_{1}(2w_{5}-w_{2}),w_{2}(2w_{5}-w_{2}),\frac{w_{2}}{2}((D^{2}-3)w_{3}+\sqrt{D-1}(D+1)w_{4})+w_{5}(Dw_{3}+\sqrt{D-1}w_{4}+w_{3}),$
(111)
$\displaystyle~{}~{}~{}\frac{w_{2}((D^{2}-1)w_{3}+\sqrt{D-1}(D^{2}+D-4)w_{4}+(D^{2}-1)w_{5})+2w_{5}((D-1)w_{3}+\sqrt{D-1}(D+4)w_{4}+(D-1)w_{5})}{4\sqrt{D-1}},$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}w_{5}(\sqrt{D-1}w_{4}+3w_{5})+\frac{w_{2}}{2}(\sqrt{D-1}(D+1)w_{4}+(D-1)w_{5})\bigg{)}\times\frac{T}{\kappa^{2}}{\int_{\Lambda_{\ell}e^{-d\ell}}^{\Lambda_{\ell}}{d^{D}p\over(2\pi)^{D}}}\frac{1}{p^{4}}$

The contribution of the third diagram reads

$\displaystyle\delta
w=\frac{-1}{D(D+2)}\bigg{(}\left(D^{2}-2\right)w_{2}^{2}+4Dw_{5}w_{2}+8w_{5}^{2},\left(D^{2}-2\right)w_{2}^{2}+4Dw_{5}w_{2}+8w_{5}^{2},$
$\displaystyle(D^{2}+D-3)w_{2}^{2}+4(D+1)w_{5}^{2}+4w_{5}w_{2},\sqrt{D-1}(w_{2}-2w_{5}){}^{2},(D^{2}-1)w_{2}^{2}+4(D-1)w_{5}w_{2}+12w_{5}^{2}\bigg{)}\frac{T}{\kappa^{2}}{\int_{\Lambda_{\ell}e^{-d\ell}}^{\Lambda_{\ell}}{d^{D}p\over(2\pi)^{D}}}\frac{1}{p^{4}}$

where we have kept the explicit factors $D$ in the geometric factors.

###  .8 D 2. Correction to the bending rigidity $\kappa$

The correction to the self-energy to first order in perturbation theory $O(R)$
can be read off from (69) as

$\displaystyle\delta\Sigma(k)=\frac{2T}{\kappa}k_{\alpha}k_{\delta}\int_{q}(k_{\beta}+q_{\beta})(k_{\gamma}+q_{\gamma})R_{\alpha\beta,\gamma\delta}({\bf
q})\frac{1}{|{\bf k}+{\bf q}|^{4}}$ (112)

from which we will identify the corrections to $\kappa$ and $\gamma$ from the
small external momentum $k$ expansion

$\displaystyle\delta\Sigma(k)=\delta\gamma\,k^{2}+\delta\kappa\,k^{4}+O(k^{6})$
(113)

The calculation of (112) proceeds by inserting again
$R_{\alpha\beta,\gamma\delta}({\bf q})=\sum_{i=1}^{5}w_{i}(W_{i}({\bf
q}))_{\alpha\beta,\gamma\delta}$, performing the expansion at small $k$ of the
numerator, and the resulting contractions of indices. In the course of the
calculation one needs the leading behavior near $D=4$ and expansion in $k$ of
three integrals. One uses the expansion

$\frac{1}{|{\bf k}+{\bf q}|^{4}}=\frac{1}{q^{4}}(1-4\frac{{\bf k}\cdot{\bf
q}}{q^{2}}-2\frac{k^{2}}{q^{2}}+12\frac{({\bf q}\cdot{\bf
k})^{2}}{q^{4}}+O(k^{3}))$ (114)

The first integral is

$\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{|{\bf q}+{\bf
k}|^{4}}q_{\alpha}=\int\frac{d^{D}q}{(2\pi)^{D}}\frac{q_{\alpha}}{q^{4}}\left(1-4\frac{{\bf
k}\cdot{\bf
q}}{q^{2}}+O(k^{2})\right)=-4k_{\beta}\frac{\delta_{\alpha\beta}}{D}\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{q^{4}}+O(k^{2})$
(115)

It can also be obtained by taking the ratio $\lim_{b\to 0,D\to
4}\frac{I_{\alpha}(a=2,b)}{I(a=2,b)}=-p_{\alpha}$ using Eqs. A34 and A43 in
LRReview .

The second integral is

$\displaystyle\int\frac{d^{D}q}{|{\bf q}+{\bf
k}|^{4}}\frac{q_{\alpha}q_{\beta}q_{\gamma}}{q^{2}}=\int\frac{d^{D}q}{q^{4}}\frac{q_{\alpha}q_{\beta}q_{\gamma}}{q^{2}}\left(1-4\frac{{\bf
k}\cdot{\bf q}}{q^{2}}+O(k^{2})\right)$ (116)
$\displaystyle=-\frac{4}{D(D+2)}(\delta_{\alpha\beta}k_{\gamma}+\delta_{\alpha\gamma}k_{\beta}+k_{\alpha}\delta_{\beta\gamma})\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{q^{4}}+O(k^{2})$
(117)

One can check that this is also the result from A34 and A51 in LRReview , i.e.
$\lim_{b\to 0,D\to 4,D=2a+2b}\frac{I_{\alpha\beta\gamma}(a=2,b+1)}{I(a=2,b)}$,
being careful to obey the constraint $D=2a+2b$ when taking the limits.

The third integral is

$\displaystyle\int\frac{d^{D}q}{|{\bf q}+{\bf
k}|^{4}}q_{\alpha}q_{\beta}=\int\frac{d^{D}q}{q^{4}}q_{\alpha}q_{\beta}\left(1-4\frac{{\bf
k}\cdot{\bf q}}{q^{2}}-2\frac{k^{2}}{q^{2}}+12\frac{({\bf q}\cdot{\bf
k})^{2}}{q^{4}}\right)$ (118)
$\displaystyle=\frac{\delta_{\alpha\beta}}{D}\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{q^{2}}-2k^{2}\frac{1}{D}\delta_{\alpha\beta}\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{q^{4}}+\frac{12}{D(D+2)}k_{\gamma}k_{\delta}(\delta_{\alpha\beta}\delta_{\gamma\delta}+\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma})\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{q^{4}}$
(119)
$\displaystyle=\frac{\delta_{\alpha\beta}}{D}\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{q^{2}}+\left(\frac{1}{D}\delta_{\alpha\beta}(-2+\frac{12}{D+2})k^{2}+\frac{24}{D(D+2)}k_{\alpha}k_{\beta}\right)\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{q^{4}}$
(120)

It can also be obtained from $\lim_{b\to 0,D\to
4}\frac{I_{\alpha\beta}(a=2,b)}{I(a=2,b)}=p_{\alpha}p_{\beta}$ from A34 and
A48 in LRReview .

We finally obtain the corrections $\delta\gamma$ and $\delta\kappa$ as

$\displaystyle\delta\gamma=\frac{2T}{\kappa}\frac{\left((D-1)w_{2}+2w_{5}\right)}{2D}\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{q^{2}}=|_{D\to
4}\frac{2T}{\kappa}\frac{1}{8}\left(3w_{2}+2w_{5}\right)\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{q^{2}}$
(121)

$\displaystyle\delta\kappa=\frac{2T}{\kappa}\frac{-\left(D^{2}+D-2\right)w_{2}+(D-2)(D+1)w_{1}+Dw_{3}-6\sqrt{D-1}w_{4}-2Dw_{5}+w_{3}+11w_{5}}{D(D+2)}\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{q^{4}}$
(122) $\displaystyle=|_{D\to
4}\frac{2T}{\kappa}\frac{1}{24}\left(10w_{1}-18w_{2}+5w_{3}-6\sqrt{3}w_{4}+3w_{5}\right)\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{q^{4}}$
(123)

where we will calculate the remaining integral using momentum shell
$\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{q^{4}}\to{\int_{\Lambda_{\ell}e^{-d\ell}}^{\Lambda_{\ell}}{d^{D}q\over(2\pi)^{D}}}\frac{1}{q^{4}}$.
The correction $\delta\gamma$ (given by a UV divergent integral) obtained
above is analogous to the usual non universal shift in the critical
temperature for $O(N)$ models, and of little interest to us since we will tune
the bare $\gamma$ so that the system is at its critical point $\gamma_{R}=0$
(i.e. $\gamma+\delta\gamma=0$). Said otherwise, the bare term in the model is
$\frac{1}{2}(\gamma-\gamma_{c})(\nabla h)^{2}$.

###  .9 D 3. Final RG equations

We now use that the integral
${\int_{\Lambda_{\ell}e^{-d\ell}}^{\Lambda_{\ell}}{d^{D}p\over(2\pi)^{D}}}\frac{1}{p^{4}}=C_{D}\frac{1}{\epsilon}(e^{\epsilon
d\ell}-1)\Lambda_{\ell}^{-\epsilon}=C_{4}\Lambda_{\ell}^{-\epsilon}d\ell+O(\epsilon)$
with $\epsilon=4-D$ and $C_{4}=\frac{1}{8\pi^{2}}$. We define the scaled
dimensionless coupling

$\tilde{w}_{i}=\frac{T}{\kappa^{2}}w_{i}C_{4}\Lambda_{\ell}^{-\epsilon}$ (124)

To derive the flow equation we calculate $\partial_{\ell}\tilde{w}_{i}$ taking
into account (i) the rescaling (ii) the sum of the three diagrams which
correct $R$ (specifying $D=4$) leading to
$\delta\tilde{w}_{i}=\beta_{i}[\tilde{w}]d\ell=\sum_{j,k}c_{ijk}\tilde{w}_{j}\tilde{w}_{k}d\ell$
(iii) the extra term from the correction
$\delta(\kappa^{-2})=-\frac{2}{\kappa^{3}}\delta\kappa$ which leads to the
$\eta$ function, $\eta[\tilde{w}]$. This leads to the RG equation

$\displaystyle\partial_{\ell}\tilde{w}_{i}=\epsilon\tilde{w}_{i}+\beta_{i}[\tilde{w}]-2\eta[\tilde{w}]\tilde{w}_{i}\quad,\quad\eta[\tilde{w}]=\frac{\partial_{\ell}\kappa}{\kappa}$
(125)

where (123) leads to (from now on for notational convenience we will suppress
the tilde on $w$)

$\displaystyle\eta[w]=\frac{1}{12}\left(10w_{1}-18w_{2}+5w_{3}+3w_{5}-6w_{44}\right)$
(126)

gives the $\eta$ exponent at the fixed point. Putting all together, the final
RG equations are (with $w_{44}=\sqrt{3}w_{4}$)

$\displaystyle\partial_{\ell}w_{1}=\epsilon
w_{1}+\frac{1}{12}\left(-(d_{c}+20)w_{1}^{2}+2\left(19w_{2}-5w_{3}-5w_{5}+6w_{44}\right)w_{1}-7w_{2}^{2}-4w_{5}^{2}-8w_{2}w_{5}\right)$
(127) $\displaystyle\partial_{\ell}w_{2}=\epsilon
w_{2}+\frac{1}{12}\left(-(d_{c}-31)w_{2}^{2}-20w_{1}w_{2}-2\left(5w_{3}+9w_{5}-6w_{44}\right)w_{2}-4w_{5}^{2}\right)$
$\displaystyle\partial_{\ell}w_{3}=\epsilon
w_{3}+\frac{1}{24}\bigg{(}-5d_{c}w_{3}^{2}-d_{c}w_{44}^{2}-2d_{c}w_{3}w_{44}-17w_{2}^{2}+46w_{3}w_{2}$
$\displaystyle-10w_{44}w_{2}-20w_{3}^{2}-20w_{5}^{2}-40w_{1}w_{3}+24w_{3}w_{44}-4w_{5}\left(w_{2}+8w_{3}+w_{44}\right)\bigg{)}$
$\displaystyle\partial_{\ell}w_{44}=\epsilon
w_{44}+\frac{1}{24}\bigg{(}-w_{44}\left(5(d_{c}+4)w_{3}+(d_{c}-24)w_{44}+40w_{1}\right)-w_{5}\left(3(d_{c}+2)w_{3}+(3d_{c}+28)w_{44}\right)$
$\displaystyle-3w_{2}^{2}+\left(-15w_{3}-3w_{5}+56w_{44}\right)w_{2}-18w_{5}^{2}\bigg{)}$
$\displaystyle\partial_{\ell}w_{5}=\epsilon
w_{5}+\frac{1}{72}\left(-9(d_{c}+12)w_{5}^{2}-6w_{5}\left((d_{c}-10)w_{44}+20w_{1}-27w_{2}+10w_{3}\right)-5\left(d_{c}w_{44}^{2}+9w_{2}^{2}+6w_{44}w_{2}\right)\right)$

Large $d_{c}$ limit. In the above RG equations (127) the couplings $w_{i}$
have not been rescaled by $1/d_{c}$. If one rescales them, and then take the
large $d_{c}$ limit one obtains

$\displaystyle\partial_{\ell}w_{1}=\epsilon w_{1}-\frac{w_{1}^{2}}{12}\
,\quad\partial_{\ell}w_{2}=\epsilon w_{2}-\frac{w_{2}^{2}}{12}\
,\quad\partial_{\ell}w_{3}=\epsilon
w_{3}-\frac{1}{24}\left(5w_{3}^{2}+2w_{44}w_{3}+w_{44}^{2}\right)\ ,$ (128)
$\displaystyle\partial_{\ell}w_{44}=\epsilon
w_{44}-\frac{1}{24}\left(3w_{5}\left(w_{3}+w_{44}\right)+w_{44}\left(5w_{3}+w_{44}\right)\right)\
,\quad\partial_{\ell}w_{5}=\epsilon
w_{5}-\frac{1}{72}\left(9w_{5}^{2}+6w_{44}w_{5}+5w_{44}^{2}\right).$

Recall that $w_{i}$ here is in fact the rescaled coupling $\tilde{w}_{i}$
given in (124). Hence comparing with (92) (the factor $T$ being omitted there)
we see that we can identify $w_{i}\equiv\frac{1}{8\pi^{2}}\hat{w}_{i}$.
Inserting into (128) we obtain a set of RG equations for the $\hat{w}_{i}$
which, as one can check using $\lim_{\epsilon=4-D\to 0}\epsilon
a_{i}(D,0)=\frac{1}{192\pi^{2}}\\{2,2,5,\sqrt{3},3\\}$ and
$w_{44}=\sqrt{D-1}w_{4}$, agree exactly with the RG equations at large $d_{c}$
(93) for $D=4$. Finally note that $\eta[w]=O(1/d_{c})$ at large $d_{c}$
consistent with the SCSA and large $d_{c}$ expansion.

###  .10 D 4. Analysis of the RG equations

Instability of the isotropic membrane fixed point. The case of the
rotationally invariant membrane is obtained setting $w_{2}=w_{4}=w_{5}=0$,
which is a manifold preserved by the RG. The RG flow (127) then reduces within
this subspace $(w_{1},w_{3})$ to

$\displaystyle\partial_{\ell}w_{1}=\epsilon
w_{1}-\frac{1}{12}w_{1}\left((d_{c}+20)w_{1}+10w_{3}\right)\ ,$ (129)
$\displaystyle\partial_{\ell}w_{3}=\epsilon
w_{3}-\frac{5}{24}w_{3}\left((d_{c}+4)w_{3}+8w_{1}\right).$ (130)

We recall that in that subspace $(w_{1},w_{3})$ are related to $(\mu,\lambda)$
via $w_{1}=\mu$ and $w_{3}=\mu+(D-1)\frac{\mu\lambda}{\lambda+2\mu}$ as
obtained from (45), and given in (8) in the text. Using that relation one can
derive RG equations for $\mu$ and $\lambda$ which can be checked to be
identical to the one in Ref. AL  (taking into account a difference by a factor
of $4$ in the definition of $\mu,\lambda$ there). There are four fixed points

$\left\\{w_{1}\to
0,w_{3}\to\frac{24\epsilon}{5(d_{c}+4)}\right\\},\left\\{w_{1}\to\frac{12\epsilon}{d_{c}+24},w_{3}\to\frac{24\epsilon}{5(d_{c}+24)}\right\\},\left\\{w_{1}\to
0,w_{3}\to 0\right\\},\left\\{w_{1}\to\frac{12\epsilon}{d_{c}+20},w_{3}\to
0\right\\}\ ,$ (131)

which correspond to (in the same order footnote3 )

$\displaystyle(\mu,\lambda)=(0,0);(\frac{12\epsilon}{24+d_{c}},\frac{-4\epsilon}{24+d_{c}});(0,\frac{2\epsilon}{d_{c}});(\frac{12\epsilon}{20+d_{c}},\frac{-6\epsilon}{20+d_{c}})\
.$ (132)

The second one is the standard fixed point which describes the isotropic flat
membrane within the $\epsilon$-expansion AL . The third one describes the
fixed connectivity fluid (zero shear modulus), that is a model for nematic
elastomer membranesLRelastomer . The fourth one is located on the line where
the bulk modulus vanishes, i.e.
$2\mu+D\lambda=\frac{2(D-1)w_{1}w_{3}}{Dw_{1}-w_{3}}=0$ which separates the
thermodynamically stable and unstable regions of parameters, and controls the
transition between these regions. The exponent $\eta$ is given by

$\eta=\eta[w]=\frac{5}{12}(2w_{1}+w_{3})=\frac{5\mu(\lambda+\mu)}{2(\lambda+2\mu)}$
(133)

and gives $\eta^{\rm iso}=\frac{12\epsilon}{24+d_{c}}$ for the isotropic
membrane, as in AL .

Let us now discuss the stability of the isotropic membrane to the non-
rotationally invariant terms (due to an external orienting field $\vec{E}$) in
the model. For this we calculate the eigenvalues and associated eigenvectors
(represented as columns) of the Hessian around the isotropic fixed point,
which are given by

$\displaystyle\left\\{0,-\frac{\epsilon d_{c}}{d_{c}+24},\frac{\epsilon
d_{c}}{d_{c}+24},\frac{\epsilon d_{c}}{d_{c}+24},-\epsilon\right\\}$ (134)
$\displaystyle\left(\begin{array}[]{ccccc}\frac{16}{d_{c}+24}&-\frac{1}{2}&-\frac{19d_{c}+78}{5(d_{c}+12)}&-\frac{19d_{c}-6}{3(d_{c}+12)}&\frac{5}{2}\\\
0&0&\frac{1}{5}(-d_{c}-2)&-\frac{d_{c}}{3}&0\\\
-\frac{2(d_{c}+8)}{5(d_{c}+24)}&1&-\frac{23d_{c}-24}{25(d_{c}+12)}&-\frac{26(d_{c}-6)}{15(d_{c}+12)}&1\\\
1&0&0&1&0\\\ 0&0&1&0&0\\\ \end{array}\right)\ .$ (140)

The second and last columns are the two stable directions which are also
obtained if one diagonalises the flow inside the isotropic subspace. In the
full space of five couplings however, we see that the isotropic fixed point is
unstable in two directions, with eigenvalue $\rho=\frac{\epsilon
d_{c}}{d_{c}+24}$, and marginal in a third direction.

Crossover for small anisotropy. To discuss the effect of a small anisotropy
let us first recall the analysis of the length scales in the isotropic
membrane. The dimensionless couplings $\tilde{w}_{1},\tilde{w}_{3}$ (we
temporarily restore the tilde) at scale $L$ are of order

$\displaystyle\tilde{w}_{1,3}\sim\frac{TK_{0}}{\kappa^{2}}L^{4-D}\quad,\quad
L<L_{\rm anh}\sim(\frac{\kappa^{2}}{TK_{0}})^{1/(4-D)}\ ,$ (141)
$\displaystyle\tilde{w}_{1,3}\simeq\tilde{w}^{*}_{1,3}\sim\frac{TK_{0}(L)}{\kappa(L)^{2}}L^{4-D}\quad,\quad
L>L_{\rm anh}\ ,$ (142)

where $L_{\rm anh}$ is the length scale below which the harmonic theory holds
(and the elastic moduli and bending rigidity equal their bare values). For
$L>L_{\rm anh}$ these are corrected and one has $\kappa(L)\sim\kappa(L/L_{\rm
anh})^{\eta}$ and $K_{0}(L)\sim K_{0}(L/L_{\rm anh})^{-(4-D-2\eta)}$. The
length $L_{\rm anh}$ is itself determined when $\tilde{w}_{1,3}$ reach numbers
of order unity, of order their value at the fixed point.

Consider now the model in presence of very small bare symmetry breaking
couplings $\mu_{1},\mu_{2},\lambda_{1},\lambda_{2}$ assumed to be of the same
order. Then, from (45) the bare $w^{0}_{2},w^{0}_{5},w^{0}_{44}$ are linear
combinations of those, hence small and of the same order. These couplings are
relevant and grow as

$\displaystyle\tilde{w}_{i}\sim\frac{Tw_{i}^{0}}{\kappa^{2}}L^{4-D}\quad,\quad
L<L_{\rm anh}\ ,$ (143)
$\displaystyle\tilde{w}_{i}\sim\frac{Tw_{i}^{0}}{\kappa^{2}}L_{\rm
anh}^{4-D}(\frac{L}{L_{\rm anh}})^{\rho}\quad,\quad L>L_{\rm anh}\ ,$ (144)

where $w_{i}^{0}$ denote any linear combination of the bare symmetry breaking
couplings ($i=2,4,5$) and $\rho$ was calculated above in the $\epsilon$
expansion. The length scale $L_{\rm anis}$ beyond which anisotropy will change
the property of the system is obtained when $\tilde{w}_{i}$ becomes of order
unity, hence

$L_{\rm anis}\sim L_{\rm
anh}(\frac{K_{0}}{w_{i}^{0}})^{1/\rho}\quad,\quad\rho=\frac{\epsilon
d_{c}}{d_{c}+24}+O(\epsilon^{2})\ ,$ (145)

whenever $w_{i}^{0}\sim\mu_{1,2},\lambda_{1,2}\ll K_{0}$.

Search for new fixed points

We now study the RG flow (127) in the five parameter space, for general
codimension $d_{c}$.

For the physical case, $d_{c}=1$, we find 12 real fixed points. However all of
them are repulsive, one with two unstable directions, the others with even
more. Hence around $D=4$ there is no perturbative fixed point and we have a
runaway RG flow.

We find that an attractive fixed point exists only for high enough $d_{c}$.
The situation is very similar to the one for the crumpling transition, with
$d$ replaced by $d_{c}$. For instance, for $d_{c}=220$ we find one, and only
one, fully attractive fixed point

$w_{i}=\\{0.05063,0.05063,0.01912,\ -0.03150,0.04012\\},$ (146)

with eigenvalues ${-1.,-0.86129,-0.86129,-0.46296,-0.08355}$ One can check
that this fixed point lies in the manifold

$\displaystyle w_{1}=w_{2}=\mu_{0}\quad,\quad
w_{3}=\frac{1}{2}(D-1)\lambda_{0}+\mu_{0}\quad,\quad
w_{4}=\frac{1}{2}\sqrt{D-1}\lambda_{0}\quad,\quad
w_{5}=\frac{1}{2}\lambda_{0}+\mu_{0}\ ,$ (147)

with $\mu_{0}=0.050628$ and $\lambda_{0}=-0.021002$ This is the manifold
mentioned in the text which leads to a purely local interaction between the
tangent fields, i.e.

$R_{\alpha\beta,\gamma\delta}(q)=\frac{\mu_{0}}{2}(\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma})+\frac{\lambda_{0}}{2}\delta_{\alpha\beta}\delta_{\gamma\delta}\
.$ (148)

One can check by inserting (147) into (127) that this manifold is preserved by
the RG. Furthermore, inside this manifold one can check inserting (147) into
(126) that $\eta[w]=0$ to the order $O(\epsilon)$, and that the RG flow can be
written as

$\displaystyle\partial_{\ell}\mu_{0}=\epsilon\mu_{0}+\frac{1}{12}\left(-(d_{c}+21)\mu_{0}^{2}-\lambda_{0}^{2}-10\lambda_{0}\mu_{0}\right)\
,$ (149)
$\displaystyle\partial_{\ell}\lambda_{0}=\epsilon\lambda_{0}+\frac{1}{12}\left(-(6d_{c}+7)\lambda_{0}^{2}-2(3d_{c}+17)\lambda_{0}\mu_{0}-(d_{c}+15)\mu_{0}^{2}\right)\
.$ (150)

Defining $u=\mu$ and $v=\lambda/2+\mu/4$ one can check that these equations
are identical to the Eqs. (5a,b) in Ref. PKN  (for their $u,v$) setting there
$K_{4}=1/4$. Hence they are identical to those of the crumpling transition but
with $d\to d_{c}$. From PKN  we know that this fixed point exists only for
$d>219$. This fixed point, which we interpret here as describing the
anisotropic membrane in its flat phase at the buckling transition found here
within the RG in the $D=4-\epsilon$ expansion is the one found within the SCSA
(and large $d_{c}$) expansion described in the Section D4. While in the RG it
disappears near $D=4$ for $d_{c}<219$, within the SCSA it survives for the
physical dimension $D=2$ and $d_{c}=1$. Hence while the RG suggests a
fluctuation driven first order transition in the physical dimension, the SCSA
suggests a continuous transition. The question of which is the most accurate
description is beyond the scope of the present work and would presumably
require numerical simulations, as was the case for the crumpling transition
(see e.g. MouhannaCrumpling  for discussion and references).

E. Renormalization group for the $u,h$ theory

Here we perform the one loop RG study on the $u,h$ theory given in (35), (36),
i.e. before integration over the phonons. It allows to obtain some extra
information (the renormalization of $\mu$) and provides a useful check on the
RG flow of the previous Section. We can rewrite the model as

${\cal F}[u,\vec{h}]=\int
d^{D}x\,\,\frac{1}{2}(\nabla^{2}h)^{2}+\frac{1}{2}(G^{u})^{-1}_{\alpha\beta}u_{\alpha}u_{\beta}+u_{\alpha}C^{\mu+\mu_{1},\lambda+\lambda_{1}}_{\alpha,\gamma\delta}A_{\gamma\delta}+\frac{1}{2}C^{\mu+\mu_{2},\lambda+\lambda_{2}}_{\alpha\beta,\gamma\delta}A_{\alpha\beta}A_{\gamma\delta}\quad,\quad
C^{\mu,\lambda}_{\alpha,\gamma\delta}=-\partial_{\beta}C^{\mu,\lambda}_{\alpha\beta,\gamma\delta}\
,$ (151)

where we have defined, in Fourier space, the $uhh$ vertex

$C^{1}_{\alpha,\gamma\delta}({\bf
q})=C^{\mu+\mu_{1},\lambda+\lambda_{1}}_{\alpha,\gamma\delta}({\bf
q})=-i\left((\lambda+\lambda_{1})q_{\alpha}\delta_{\gamma\delta}+(\mu+\mu_{1})(\delta_{\alpha\delta}q_{\gamma}+\delta_{\alpha\gamma}q_{\delta})\right)\
,$ (152)

and the bare phonon propagator

$(G^{u})^{-1}_{\alpha\beta}({\bf q})=\mu P_{\alpha\beta}^{T}({\bf
q})+(\lambda+2\mu)P_{\alpha\beta}^{L}({\bf q})\ .$ (153)

Here we calculate the corrections to the vertices, hence we evaluate to lowest
order in the perturbation theory in the nonlinearities, the vertices of the
effective action $\Gamma_{uu},\Gamma_{uhh},\Gamma_{hhhh}$. These vertices will
give us the corrections respectively to $(\mu,\lambda)$,
$(\mu_{1},\lambda_{1})$ and $(\mu_{2},\lambda_{2})$. The corresponding
diagrams are shown in the Fig.3.

Figure 3: Feynman diagrams for one-loop corrections to the renormalized
vertices in the $u-h$ description of the critical buckling membrane.

###  .11 E. 1 Calculation of $\Gamma_{uu}$

Let us calculate the one loop corrections to the phonon propagator, given by
the single diagram in Fig.3. The effective action for the $u^{2}$ term is
given, to one loop, as

$\displaystyle\frac{1}{2}\int_{\bf q}u({\bf q})\cdot\Gamma_{uu}({\bf q})\cdot
u(-{\bf q})=\int_{\bf q}\frac{1}{2}u_{\alpha}({\bf q})u_{\beta}(-{\bf
q})[G^{-1}_{\alpha\beta}({\bf
q})-C^{\mu+\mu_{1},\lambda+\lambda_{1}}_{\alpha,\gamma\delta}({\bf
q})C^{\mu+\mu_{1},\lambda+\lambda_{1}}_{\alpha^{\prime},\gamma^{\prime}\delta^{\prime}}(-{\bf
q})\langle A_{\gamma\delta}({\bf q})A_{\gamma^{\prime}\delta^{\prime}}(-{\bf
q})\rangle_{0}]\ ,$

where here and below $\cdot$ denotes index summations. We have the following
average, performed with the quadratic action

$\displaystyle\langle A_{\gamma\delta}({\bf
q})A_{\gamma^{\prime}\delta^{\prime}}(-{\bf
q})\rangle_{0}=\frac{1}{2}d_{c}\Pi_{\gamma\delta,\gamma^{\prime}\delta^{\prime}}({\bf
q})\quad,\quad\Pi_{\alpha\beta,\gamma\delta}({\bf q})={\rm
sym}\int_{p}p_{\alpha}(q_{\beta}-p_{\beta})p_{\delta}(q_{\gamma}-p_{\gamma})G({\bf
p})G({\bf q}-{\bf p})\ ,$ (155)

which leads to

$\displaystyle(\Gamma_{uu})_{\alpha\beta}({\bf q})=G^{-1}_{\alpha\beta}({\bf
q})-\frac{d_{c}}{2}C^{\mu+\mu_{1},\lambda+\lambda_{1}}_{\alpha,\gamma\delta}({\bf
q})C^{\mu+\mu_{1},\lambda+\lambda_{1}}_{\alpha^{\prime},\gamma^{\prime}\delta^{\prime}}(-{\bf
q})\Pi_{\gamma\delta,\gamma^{\prime}\delta^{\prime}}({\bf q})\ .$ (156)

Within the Wilson RG and to leading order in $\epsilon$ one has

$\displaystyle\Pi_{\alpha\beta,\gamma\delta}({\bf
q})\simeq\frac{1}{\kappa^{2}}\int_{p}\frac{p_{\alpha}p_{\beta}p_{\delta}p_{\gamma}}{p^{8}}=\frac{1}{\kappa^{2}}S_{\alpha\beta,\gamma\delta}\int_{p}\frac{1}{p^{4}}\
,$ (157)

i.e., the dependence in the external momentum $q\ll p$ is subdominant, where
$p$ is the internal momentum in the loop. We have defined

$S^{(4)}_{\alpha\beta,\gamma\delta}=S_{\alpha\beta,\gamma\delta}=\frac{1}{D(D+2)}(\delta_{\alpha\beta}\delta_{\gamma\delta}+\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma})\
.$ (158)

Hence we obtain

$\displaystyle\Gamma_{uu}(q)=G^{-1}(q)-\frac{d_{c}}{2\kappa^{2}}C^{1}({\bf
q})\cdot S\cdot(C^{1}(-{\bf q}))^{T}\int_{p}\frac{1}{p^{4}}\ .$ (159)

Replacing $\int_{p}\frac{1}{p^{4}}\to C_{4}\Lambda_{\ell}^{-\epsilon}$ and
performing the contractions, one obtains the following corrections to $\mu$
and $\lambda$

$\displaystyle\delta\mu=-\frac{d_{c}}{12\kappa^{2}}(\mu+\mu_{1})^{2}C_{4}\Lambda_{\ell}^{-\epsilon}d\ell\
,$ (160)
$\displaystyle\delta\lambda=-\frac{d_{c}}{24\kappa^{2}}[12(\lambda+\lambda_{1})^{2}+2(\mu+\mu_{1})^{2}+12(\lambda+\lambda_{1})(\mu+\mu_{1})]C_{4}\Lambda_{\ell}^{-\epsilon}d\ell\
.$

Exponent $\eta_{u}$. The first equation can be rewritten to obtain the
anomalous dimension of the phonon field, i.e the exponent $\eta_{u}$ defined
by $\mu(L)\sim L^{-\eta_{u}}$,

$\displaystyle\eta_{u}=-\frac{\delta\mu}{\mu
d\ell}=\frac{d_{c}}{12}g_{\mu}=\frac{d_{c}}{12}(\tilde{w}_{1}-\tilde{w}_{2})\
,$ (161)

where we have defined the proper dimensionless coupling
$g_{\mu}=\frac{(\mu+\mu_{1})^{2}}{\mu\kappa^{2}}C_{4}\Lambda_{\ell}^{-\epsilon}$
which we related to the $\tilde{w}_{i}$ using (45) and (124). At the fixed
point this gives the exponent $\eta_{u}$:

\- at the isotropic membrane fixed point $\tilde{w}_{2}=0$ and
$\tilde{w}_{1}=\frac{12\epsilon}{d_{c}+24}$, leading to
$\eta_{u}=\frac{d_{c}\epsilon}{d_{c}+24}$. Since
$\eta=\frac{12\epsilon}{d_{c}+24}$ we check, to first order in $\epsilon$ the
exact relation (to all orders), $\eta_{u}=\epsilon-2\eta$ guaranteed by
rotational invariance AL .

\- at the anisotropic membrane fixed point $\tilde{w}_{1}=\tilde{w}_{2}$ hence
$\eta_{u}=0$ to order $O(\epsilon)$. The relation $\eta_{u}=\epsilon-2\eta$
does not hold (since $\eta=0$ there, to $O(\epsilon)$).

Note that one can also define the screening exponent $\eta_{w}$ for the
coupling constants $w_{i}$ such that $w_{i}(L)\sim L^{-\eta_{w}}$. It is given
by the graphical corrections $\beta_{i}=\frac{\delta w_{i}}{w_{i}}$. Since the
RG equation for the scaled dimensionless coupling $\tilde{w}_{i}$ reads
$\partial_{\ell}\hat{w}_{i}=(\epsilon-2\eta(\hat{w})-\beta_{i}[\tilde{w}])\tilde{w}_{i}$,
at any fixed point one must have $\beta_{i}=\epsilon-2\eta$. In presence of
anisotropy, $\beta_{i}$ becomes different from $\eta_{u}$. The nonlinear
interactions are still screened, since $\eta<\epsilon/2$ at the anisotropic
fixed point, but this screening is not directly related to the renormalization
of $\mu$ and $\lambda$.

###  .12 E 2. Calculation of $\Gamma_{uhh}$

We now calculate the vertex corrections given by the three diagrams in Fig.3.
They are corrections to the term $uhh$ in (151), which we write in the form

$\int_{q}u_{\alpha}(-{\bf q})\,\delta V_{\alpha,\beta\gamma}({\bf
q})\,A_{\beta\gamma}({\bf q})\ .$ (162)

One obtains for the first diagram

$\displaystyle\delta V^{(1)}_{\alpha,\beta\gamma}({\bf
q})=-\frac{d_{c}}{2}(C^{1}({\bf q})\cdot S\cdot
C^{2})_{\alpha,\beta\gamma}\frac{1}{\kappa^{2}}\int_{\bf p}\frac{1}{p^{4}}\ ,$
(163)

where $C^{1}({\bf q})$ is the three index tensor given in (152) (i.e., the
bare $uhh$ vertex) and we denote here and below $C^{2}$ the four index tensor
$C^{\mu+\mu_{2},\lambda+\lambda_{2}}_{\alpha\beta,\gamma\delta}$ (entering the
bare $h^{4}$ vertex) defined in (35). The second diagram gives the correction

$\displaystyle\delta V^{(2)}_{\alpha,\beta\gamma}({\bf
q})=-C^{1}_{\alpha,\alpha^{\prime}\alpha^{\prime\prime}}({\bf
q})S_{\alpha^{\prime}\beta^{\prime}\alpha^{\prime\prime}\gamma^{\prime}}C^{2}_{\beta^{\prime}\beta,\gamma^{\prime}\gamma}\frac{1}{\kappa^{2}}\int_{\bf
p}\frac{1}{p^{4}}\ .$ (164)

Finally, the third diagram gives, using that $C^{1}_{\beta,\gamma\delta}({\bf
q})=-iq_{\alpha}C^{1}_{\alpha\beta,\gamma\delta}$ (where the momentum
independent four index tensor
$C^{\mu+\mu_{1},\lambda+\lambda_{1}}_{\alpha\beta,\gamma\delta}$ is denoted
$C^{1}$)

$\displaystyle\delta V^{(3)}_{\alpha,\beta\gamma}({\bf
q})=C^{1}_{\alpha,\beta^{\prime}\gamma^{\prime}}({\bf
q})\left[S^{(8)}_{\beta^{\prime}\gamma^{\prime}\beta^{\prime\prime}\gamma^{\prime\prime}s^{\prime}s^{\prime\prime}\alpha^{\prime}\alpha^{\prime\prime}}\left(\frac{1}{2\mu+\lambda}-\frac{1}{\mu}\right)+\frac{1}{\mu}S^{(6)}_{\beta^{\prime}\gamma^{\prime}\beta^{\prime\prime}\gamma^{\prime\prime}s^{\prime}s^{\prime\prime}}\delta_{\alpha^{\prime}\alpha^{\prime\prime}}\right]C^{1}_{\alpha^{\prime}s^{\prime},\beta^{\prime\prime}\beta}C^{1}_{\alpha^{\prime\prime}s^{\prime\prime},\gamma^{\prime\prime}\gamma}\frac{1}{\kappa^{2}}\int_{\bf
p}\frac{1}{p^{4}},$

where we defined the 6 and 8 index symmetric tensors, schematically,

$\displaystyle
S^{(6)}_{\beta^{\prime}\gamma^{\prime}\beta^{\prime\prime}\gamma^{\prime\prime}s^{\prime}s^{\prime\prime}}=\frac{1}{D(D+2)(D+4)}(\delta\delta\delta+14\
\text{terms})\ ,$ (166) $\displaystyle
S^{(8)}_{\beta^{\prime}\gamma^{\prime}\beta^{\prime\prime}\gamma^{\prime\prime}s^{\prime}s^{\prime\prime}\alpha^{\prime}\alpha^{\prime\prime}}=\frac{1}{D(2+D)(4+D)(6+D)}(\delta\delta\delta\delta+104\
\text{terms})\ .$ (167)

Performing the contractions we obtain for $i=1,2,3$

$\delta V^{(i)}_{\alpha,\beta\gamma}({\bf
q})=[A_{i}q_{\alpha}\delta_{\beta\gamma}+B_{i}(q_{\beta}\delta_{\alpha\gamma}+q_{\gamma}\delta_{\alpha\beta})]\frac{1}{\kappa^{2}}\int_{\bf
p}\frac{1}{p^{4}}\ .$ (168)

To display the results more compactly we define the new variables

$\Lambda_{i}=\lambda_{i}+\lambda\quad,\quad M_{i}=\mu_{i}+\mu\quad,\quad
i=1,2\ .$ (169)

In terms of these variables the coefficients $A_{i},B_{i}$ read

$\displaystyle
A_{1}=-\frac{d_{c}}{12}\left(3\Lambda_{1}\left(2\Lambda_{2}+M_{2}\right)+M_{1}\left(3\Lambda_{2}+M_{2}\right)\right)\quad,\quad
B_{1}=-\frac{d_{c}}{12}M_{1}M_{2}\ ,$ (170) $\displaystyle
A_{2}=\frac{1}{12}\left(-3\Lambda_{1}\left(\Lambda_{2}+5M_{2}\right)-M_{1}\left(\Lambda_{2}+7M_{2}\right)\right)\quad,\quad
B_{2}=-\frac{1}{12}M_{1}\left(\Lambda_{2}+M_{2}\right)\ ,$ $\displaystyle
A_{3}=\frac{3\Lambda_{1}^{3}\mu+\Lambda_{1}M_{1}^{2}(9\lambda+34\mu)+M_{1}^{3}(5\lambda+14\mu)+13\Lambda_{1}^{2}\mu
M_{1}}{12\mu(\lambda+2\mu)}\quad,\quad
B_{3}=\frac{M_{1}\left(\Lambda_{1}^{2}\mu+M_{1}^{2}(-(\lambda-2\mu))+4\Lambda_{1}\mu
M_{1}\right)}{12\mu(\lambda+2\mu)}\ .$

From these coefficients we directly obtain the corrections

$\delta\Lambda_{1}=(A_{1}+A_{2}+A_{3})\frac{1}{\kappa^{2}}\int_{\bf
p}\frac{1}{p^{4}}\quad,\quad\delta
M_{1}=(B_{1}+B_{2}+B_{3})\frac{1}{\kappa^{2}}\int_{\bf p}\frac{1}{p^{4}}\ .$
(171)

Putting all contributions together and replacing $\int_{\bf
p}\frac{1}{p^{4}}\to C_{4}\Lambda_{\ell}^{-\epsilon}$, we obtain, from the
vertex corrections

$\displaystyle\delta
M_{1}=\frac{1}{12}M_{1}\left(M_{2}\left(-\left(d_{c}+1\right)\right)-\Lambda_{2}+\frac{\Lambda_{1}^{2}\mu+M_{1}^{2}(-(\lambda-2\mu))+4\Lambda_{1}\mu
M_{1}}{\mu(\lambda+2\mu)}\right)\frac{1}{\kappa^{2}}C_{4}\Lambda_{\ell}^{-\epsilon}d\ell\
,$ (172)
$\displaystyle\delta\Lambda_{1}=\frac{1}{12}\bigg{(}-d_{c}\left(3\Lambda_{1}\left(2\Lambda_{2}+M_{2}\right)+M_{1}\left(3\Lambda_{2}+M_{2}\right)\right)$
$\displaystyle+\frac{3\Lambda_{1}^{3}\mu+\Lambda_{1}M_{1}^{2}(9\lambda+34\mu)+M_{1}^{3}(5\lambda+14\mu)+13\Lambda_{1}^{2}\mu
M_{1}}{\mu(\lambda+2\mu)}-3\Lambda_{1}\left(\Lambda_{2}+5M_{2}\right)-M_{1}\left(\Lambda_{2}+7M_{2}\right)\bigg{)}\frac{1}{\kappa^{2}}C_{4}\Lambda_{\ell}^{-\epsilon}d\ell\
.$

To recover the result for the isotropic membrane one sets
$\Lambda_{i}=\lambda$ and $M_{i}=\mu$ and the above corrections reduce to

$\displaystyle\delta\mu=-\frac{d_{c}}{12\kappa^{2}}\mu^{2}C_{4}\Lambda_{\ell}^{-\epsilon}d\ell\
,$ (173)
$\displaystyle\delta\lambda=-\frac{d_{c}}{12\kappa^{2}}\left(6\lambda^{2}+6\lambda\mu+\mu^{2}\right)C_{4}\Lambda_{\ell}^{-\epsilon}d\ell\
.$

This simplification occurs because the second and third diagram exactly cancel
due to rotational invariance. Indeed the corrections (173) coincide with (160)
(setting $\mu_{1}=\lambda_{1}=0$ there).

###  .13 E. 3 Calculation of $\Gamma_{hhhh}$

We now calculate the corrections to the $h^{4}$ vertex in (151). They are
given by the six diagrams in Fig.3. We recall that we denote $C^{2}$ the four
index tensor $C^{\mu+\mu_{2},\lambda+\lambda_{2}}_{\alpha\beta,\gamma\delta}$
which appears in the bare $h^{4}$ vertex.

The first diagram gives the following correction to $C^{2}$

$\displaystyle\delta C^{2}=-\frac{d_{c}}{2}C^{2}\cdot S\cdot
C^{2}\frac{1}{\kappa^{2}}\int_{\bf p}\frac{1}{p^{4}}\ .$ (174)

The second and third diagram give respectively

$\delta C^{2}_{\alpha\beta,\gamma\delta}=-2{\rm
sym}\,C^{2}_{\alpha\alpha^{\prime},\gamma\gamma^{\prime}}C^{2}_{\beta\beta^{\prime},\delta\delta^{\prime}}S_{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\delta^{\prime}}\frac{1}{\kappa^{2}}\int_{\bf
p}\frac{1}{p^{4}}\quad,\quad\delta
C^{2}_{\alpha\beta,\gamma\delta}=-2\,C^{2}_{\alpha\beta,\alpha^{\prime}\beta^{\prime}}S_{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\delta^{\prime}}C^{2}_{\gamma\gamma^{\prime},\delta\delta^{\prime}}\frac{1}{\kappa^{2}}\int_{\bf
p}\frac{1}{p^{4}}\ .$ (175)

The fourth diagram is more complicated

$\displaystyle\delta
C^{2}_{\alpha\beta,\gamma\delta}=-2C^{1}_{r_{1}s_{1},\alpha\alpha^{\prime}}C^{1}_{r_{2}s_{2},\gamma\gamma^{\prime}}C^{1}_{r_{3}s_{3},\beta\beta^{\prime}}C^{1}_{r_{4}s_{4},\delta\delta^{\prime}}\bigg{[}\langle\hat{p}_{\alpha^{\prime}}\hat{p}_{\beta^{\prime}}\hat{p}_{\gamma^{\prime}}\hat{p}_{\delta^{\prime}}\hat{p}_{s_{1}}\hat{p}_{s_{2}}\hat{p}_{s_{3}}\hat{p}_{s_{4}}\hat{p}_{r_{1}}\hat{p}_{r_{2}}\hat{p}_{r_{3}}\hat{p}_{r_{4}}\rangle\left(\frac{1}{\lambda+2\mu}-\frac{1}{\mu}\right)^{2}$
(176)
$\displaystyle+\left(\langle\hat{p}_{\alpha^{\prime}}\hat{p}_{\beta^{\prime}}\hat{p}_{\gamma^{\prime}}\hat{p}_{\delta^{\prime}}\hat{p}_{s_{1}}\hat{p}_{s_{2}}\hat{p}_{s_{3}}\hat{p}_{s_{4}}\hat{p}_{r_{3}}\hat{p}_{r_{4}}\rangle\delta_{r_{1}r_{2}}+\langle\hat{p}_{\alpha^{\prime}}\hat{p}_{\beta^{\prime}}\hat{p}_{\gamma^{\prime}}\hat{p}_{\delta^{\prime}}\hat{p}_{s_{1}}\hat{p}_{s_{2}}\hat{p}_{s_{3}}\hat{p}_{s_{4}}\hat{p}_{r_{1}}\hat{p}_{r_{2}}\rangle\delta_{r_{3}r_{4}}\right)\frac{1}{\mu}\left(\frac{1}{\lambda+2\mu}-\frac{1}{\mu}\right)$
$\displaystyle+\frac{1}{\mu^{2}}S^{(8)}_{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\delta^{\prime}s_{1}s_{2}s_{3}s_{4}}\delta_{r_{1}r_{2}}\delta_{r_{3}r_{4}}\bigg{]}\frac{1}{\kappa^{2}}\int_{\bf
q}\frac{1}{q^{4}}\ ,$

where $\langle\dots\rangle$ denote angular averages and $\hat{\bf p}={\bf
p}/p$ a unit vector. It was convenient to use that notational trick, rather
than the symmetric tensors of order 10 and 12, as it allows the contractions
to be taken more easily. This is equal to

$\displaystyle\delta
C^{2}_{\alpha\beta,\gamma\delta}=\bigg{(}-2C^{1}_{r_{1}s_{1},\alpha\alpha^{\prime}}C^{1}_{r_{2}s_{2},\gamma\gamma^{\prime}}C^{1}_{r_{3}s_{3},\beta\beta^{\prime}}C^{1}_{r_{4}s_{4},\delta\delta^{\prime}}\frac{1}{\mu^{2}}S^{(8)}_{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\delta^{\prime}s_{1}s_{2}s_{3}s_{4}}\delta_{r_{1}r_{2}}\delta_{r_{3}r_{4}}$
(177)
$\displaystyle-2\left(\frac{1}{\lambda+2\mu}-\frac{1}{\mu}\right)^{2}(\Lambda_{1}+2M_{1})^{4}S_{\alpha\beta\gamma\delta}$
$\displaystyle-2\left(\frac{1}{\lambda+2\mu}-\frac{1}{\mu}\right)\frac{1}{\mu}(\Lambda_{1}+2M_{1})^{2}[2((\Lambda_{1}+2M_{1})^{2}-M_{1}^{2})S_{\alpha\beta\gamma\delta}+\frac{M_{1}^{2}}{D}(\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma})]\bigg{)}\frac{1}{\kappa^{2}}\int_{\bf
p}\frac{1}{p^{4}}\ .$

The fifth diagram leads to the correction

$\displaystyle\delta C^{2}_{\alpha\beta,\gamma\delta}={\rm
sym}\,4C^{1}_{r_{1}s_{1},\alpha\alpha^{\prime}}C^{1}_{r_{2}s_{2},\gamma\gamma^{\prime}}C^{2}_{\beta^{\prime}\beta,\delta^{\prime}\delta}\bigg{[}\left(\frac{1}{\lambda+2\mu}-\frac{1}{\mu}\right)S^{(8)}_{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\delta^{\prime}s_{1}s_{2}r_{1}r_{2}}+\frac{1}{\mu}S_{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\delta^{\prime}s_{1}s_{2}}\delta_{r_{1}r_{2}}\bigg{]}\frac{1}{\kappa^{2}}\int_{\bf
q}\frac{1}{q^{4}}\ ,$ (178)

and finally, the sixth diagram, to

$\displaystyle\delta C^{2}_{\alpha\beta,\gamma\delta}={\rm
sym}\,2C^{2}_{\alpha\beta,\alpha^{\prime}\beta^{\prime}}C^{1}_{r_{1}s_{1},\delta\delta^{\prime}}C^{1}_{r_{2}s_{2},\gamma\gamma^{\prime}}\bigg{[}\left(\frac{1}{\lambda+2\mu}-\frac{1}{\mu}\right)S^{(8)}_{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\delta^{\prime}s_{1}s_{2}r_{1}r_{2}}+\frac{1}{\mu}S_{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\delta^{\prime}s_{1}s_{2}}\delta_{r_{1}r_{2}}\bigg{]}\frac{1}{\kappa^{2}}\int_{\bf
q}\frac{1}{q^{4}}\ .$ (179)

Performing the contractions, in total we find for the corrections to the
$h^{4}$ vertex

$\displaystyle\delta
M_{2}=\frac{1}{12}\bigg{[}\frac{-1}{\mu^{2}(\lambda+2\mu)^{2}}\bigg{(}\mu^{2}M_{2}^{2}\left(d_{c}+21\right)(\lambda+2\mu)^{2}+\Lambda_{1}^{4}\mu^{2}+M_{1}^{4}\left(7\lambda^{2}+44\lambda\mu+76\mu^{2}\right)$
(180) $\displaystyle+8\Lambda_{1}\mu M_{1}\left(2M_{1}^{2}(\lambda+4\mu)-5\mu
M_{2}(\lambda+2\mu)\right)+2\Lambda_{1}^{2}\mu\left(2M_{1}^{2}(\lambda+8\mu)-5\mu
M_{2}(\lambda+2\mu)\right)\frac{1}{\kappa^{2}}C_{4}\Lambda_{\ell}^{-\epsilon}$
$\displaystyle-20\mu
M_{2}M_{1}^{2}(\lambda+2\mu)(\lambda+4\mu)+8\Lambda_{1}^{3}\mu^{2}M_{1}\bigg{)}-\Lambda_{2}^{2}+2\Lambda_{2}\left(\frac{\Lambda_{1}^{2}\mu+2M_{1}^{2}(\lambda+4\mu)+4\Lambda_{1}\mu
M_{1}}{\mu(\lambda+2\mu)}-5M_{2}\right)\bigg{]}\frac{1}{\kappa^{2}}C_{4}\Lambda_{\ell}^{-\epsilon}d\ell\
,$

and

$\displaystyle\delta\Lambda_{2}=\frac{1}{12}\bigg{[}]-\Lambda_{2}^{2}\left(6d_{c}+7\right)-2\Lambda_{2}M_{2}\left(3d_{c}+17\right)+M_{2}^{2}\left(-\left(d_{c}+15\right)\right)-\frac{\left(\Lambda_{1}^{2}\mu+M_{1}^{2}(-(\lambda-2\mu))+4\Lambda_{1}\mu
M_{1}\right){}^{2}}{\mu^{2}(\lambda+2\mu)^{2}}$
$\displaystyle+\frac{4M_{2}\left(\Lambda_{1}^{2}\mu+2M_{1}^{2}(\lambda+4\mu)+4\Lambda_{1}\mu
M_{1}\right)}{\mu(\lambda+2\mu)}+\frac{8\Lambda_{2}\left(\Lambda_{1}^{2}\mu+2M_{1}^{2}(\lambda+4\mu)+4\Lambda_{1}\mu
M_{1}\right)}{\mu(\lambda+2\mu)}\bigg{]}\frac{1}{\kappa^{2}}C_{4}\Lambda_{\ell}^{-\epsilon}d\ell\
.$ (181)

To recover the result for the isotropic membrane one sets
$\Lambda_{i}=\lambda$ and $M_{i}=\mu$ and the above corrections reduce
exactly, once again, to (173). Here the simplification arises from the last
five diagram cancelling due to rotational invariance.

###  .14 E. 4 Final RG equations

We can now put together $\delta\mu$, $\delta\lambda$ from (160), $\delta
M_{1}=\delta\mu+\delta\mu_{1}$,
$\delta\Lambda_{1}=\delta\lambda+\delta\lambda_{1}$ from (172), and $\delta
M_{2}=\delta\mu+\delta\mu_{2}$,
$\delta\Lambda_{2}=\delta\lambda+\delta\lambda_{2}$ from (180). This leads to
the complete set of corrections to the six couplings, which is bulky and which
we will not display here in full (see below). Let us denote $m_{i}$,
$i=1,\dots,6$ these couplings. These corrections read schematically $\delta
m_{i}=d_{ijk}m_{j}m_{k}$. To obtain the final RG flow one defines scaled
dimensionless couplings $\tilde{m}_{i}$, as in (124), and take into account
the corrections to $\kappa$ as we did in (125), leading to
$\partial_{\ell}\tilde{m}_{i}=\epsilon\tilde{m}_{i}+d_{ijk}\tilde{m}_{j}\tilde{m}_{k}-2\eta[w[m]]\tilde{m}_{i}$.
Here we denote $w[m]$ the $w_{i}$ expressed as functions of the $m_{i}$ via
the Eq. (45), and we have used the same formula (126) for the $\eta[w]$
function.

To check that these are consistent with the RG equations obtained via the
quartic theory in Section D, we simply need to compare the above corrections
$\delta m$ and $\delta w$, i.e summing (.7), (111) and (.7) which can be
written as $\delta w_{i}=\beta_{i}[w]d\ell=c_{ijk}w_{j}w_{k}d\ell$. We have
performed the check as follows. We have evaluated in two ways

$\displaystyle\delta w_{i}[m]=\frac{\partial w_{i}[m]}{\partial m_{j}}\delta
m_{j}=\frac{\partial w_{i}[m]}{\partial m_{j}}d_{ijk}m_{j}m_{k}d\ell\ ,$ (182)
$\displaystyle\delta w_{i}[m]=c_{ijk}w_{j}[m]w_{k}[m]d\ell\ ,$ (183)

and using $w[m]$ from Eq. (45) we have shown using Mathematica that the two
lines above are identical functions of the $m_{i}$. This provides a quite non
trivial check of these two lengthy calculations. Hence the RG equation for the
5 couplings $w_{i}$ can be deduced from the one for the 6 couplings $m_{i}$.
The reverse is not true however, there is, in the general case, additional
information in the 6 coupling flow, as we discussed above in Section E. 1 it
allows to obtain $\delta\mu$ and from it we obtained there the exponent
$\eta_{u}$, related to the anomalous dimension of the phonon field. Let us
indicate for completeness the combination of couplings which enters the
exponent $\eta$

$\eta=\frac{1}{12}\left(10w_{1}-18w_{2}+5w_{3}+3w_{5}-6w_{44}\right)=\frac{M_{1}\left(2\Lambda_{1}\mu+3\lambda
M_{1}+5\mu M_{1}\right)}{2\mu(\lambda+2\mu)}\ .$ (184)

To express the RG flow it is natural to define the dimensionless ratio
$r=\lambda/\mu$ and the four dimensionless coupling constants associated to
the nonlinear terms in the action

$\tilde{M}_{1}^{2}=\frac{M_{1}^{2}}{\mu\kappa^{2}}C_{4}\Lambda_{\ell}^{-\epsilon}\quad,\quad\tilde{\Lambda}_{1}^{2}=\frac{\Lambda_{1}^{2}}{\mu\kappa^{2}}C_{4}\Lambda_{\ell}^{-\epsilon}\quad,\quad\tilde{M}_{2}=\frac{M_{2}}{\kappa^{2}}C_{4}\Lambda_{\ell}^{-\epsilon}\quad,\quad\tilde{\Lambda}_{2}=\frac{\Lambda_{2}}{\kappa^{2}}C_{4}\Lambda_{\ell}^{-\epsilon}\
,$ (185)

and then $\mu$ can still flow with eigenvalue $\eta_{u}$. Since the RG
equations for these couplings are bulky let us only display them here to
leading order in large $d_{c}$, and we have dropped the tilde for notational
convenience

$\displaystyle\partial_{\ell}r=\frac{1}{12}d_{c}\left(-6\Lambda_{1}^{2}+rM_{1}^{2}-6\Lambda_{1}M_{1}-M_{1}^{2}\right)\quad,\quad\partial_{\ell}M_{1}=\frac{\epsilon}{2}M_{1}+\frac{1}{24}M_{1}\left(M_{1}^{2}-2M_{2}\right)d_{c}\
,$ (186)
$\displaystyle\partial_{\ell}\Lambda_{1}=\frac{\epsilon}{2}\Lambda_{1}+\frac{1}{24}d_{c}\left(-12\Lambda_{1}\Lambda_{2}+\Lambda_{1}M_{1}^{2}-6\Lambda_{2}M_{1}-6\Lambda_{1}M_{2}-2M_{2}M_{1}\right)\
,$ (187) $\displaystyle\partial_{\ell}M_{2}=\epsilon
M_{2}-\frac{1}{12}M_{2}^{2}d_{c}\quad,\quad\partial_{\ell}\Lambda_{2}=\epsilon\Lambda_{2}+\frac{1}{12}d_{c}\left(-6\Lambda_{2}^{2}-6\Lambda_{2}M_{2}-M_{2}^{2}\right)\
.$ (188)

It is easy to see that the only attractive fixed point of these equations (and
of the complete equations for any $d_{c}>219$) is such that

$M_{1}=0\quad,\quad\Lambda_{1}=0\quad,\quad
M_{2}=\frac{12}{d_{c}}+O(\frac{1}{d_{c}^{2}})\quad,\quad\Lambda_{2}=-\frac{4}{d_{c}}+O(\frac{1}{d_{c}^{2}})\
.$ (189)

This is in agreement with the RG analysis using the $h^{4}$ theory presented
above. Indeed this anisotropic fixed point lies in the manifold (147) in the
$w_{i}$ variables, which in the current variables imply the constraints
$\mu+\mu_{1}=0$, $\mu_{0}=\mu+\mu_{2}$,
$\lambda_{0}=\lambda+\lambda_{2}-\frac{(\lambda+\lambda_{1})^{2}}{\lambda+2\mu}$.
The fixed point (189) obeys these constraints and one can check that the