Thermal buckling transition of crystalline membranes in a field
TThermal buckling transition of crystalline membranes
Pierre Le Doussal and Leo Radzihovsky Laboratoire de Physique de l’Ecole Normale Sup´erieure, ENS, Universit´e PSL,CNRS, Sorbonne Universit´e, Universit´e de Paris, 75005 Paris, France ∗ Department of Physics, University of Colorado, Boulder, CO 80309 † (Dated: February 19, 2021)Two dimensional crystalline membranes in isotropic embedding space exhibit a flat phase withanomalous elasticity, relevant e.g., for graphene. Here we study their thermal fluctuations in theabsence of exact rotational invariance in the embedding space. An example is provided by a mem-brane 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 elasticityexponents at criticality, while the RG suggests a weakly first order transition. PACS numbers: 64.60Fr,05.40,82.65Dp
Introduction and background.
Experimental realiza-tion of freely suspended graphene [1] and other exfoliatedcrystals, following the 2004 pioneering works of Geim andNovoselov [2], launched extensive research in electronicand mechanical properties of two-dimensional crystallinemembranes[3, 4]. This led to a renaissance in the sta-tistical mechanics of fluctuating elastic membranes, firststudied in the context of soft and biological matter threedecades ago [5–15]. Theoretical interest is also motivatedby the opportunity to explore the nontrivial and rich in-terplay between field theory and geometry [12].The most striking prediction is the existence of a low-temperature stable “flat” phase of a tensionless crys-talline membrane [5], that spontaneously breaks rota-tional symmetry of the embedding space. This is in starkcontrast to canonical two-dimensional field theories forwhich the Hohenberg-Mermin-Wagner theorems[16–18],preclude spontaneous breaking of a continuous symme-try in two dimensions.In such elastic membranes, in a spectacular phe-nomenon of order-from-disorder, thermal fluctuations in-stead stiffen the long-wavelength ( k − ) bending rigidity κ → κ k − η , η >
0, via a universal power-law “cor-rugation” effect, with membrane roughness scaling as h r ms ∼ L ζ , with ζ = (4 − D − η ) / η, ζ and η u = 4 − D − η determined exactly bythe underlying rotational invariance, with a scale depen-dent Young modulus K → K q η u . This was predicted,together with the values of the exponents, by a variety ofcomplementary methods [5, 6, 8, 9, 15]. It was verified innumerical simulations [19] and continues to be exploredexperimentally [20].Most theoretical studies to-date have focused on stress-free fluctuating membranes in an isotropic embeddingenvironment [5, 6, 8–10, 14, 15, 21–25], as appropriatefor e.g., soft matter realizations of a membrane in anisotropic fluid (though see interesting generalizations forspherical shells[26, 27]). However, many experiments on graphene and other solid-state membranes (even somesuspended ones) may be subjected to embedding spaceanisotropy and/or external stresses due to the presence ofa substrate [28–30], clamping[31, 32], or electric and mag-netic fields[33, 34]. Orientational fields could also be im-posed by suspending the membrane in a nematic fluid[35–37]. It is interesting to consider for instance the case ofan uniaxial easy axis field tending to order the mem-brane’s normal, and/or the application of a boundarystress σ . In all previous theoretical descriptions, the ro-tational invariance in the embedding space was assumedand the response found to be controlled by the thermaltensionless membrane fixed point[6]. The case of weakfield or stresses is treated by simply introducing a cut-off for the isotropic critical fluctuations, beyond a largescale ξ = ( κ/σ ) ν , that diverges with a vanishing σ . Suchperturbations then lead to an anomalous response, thatin the context of tension predicts a non-Hookean stress-strain relation ε ∼ σ α with α < FIG. 1. A schematic illustration of a critical membrane tunedto a buckling transition, subjected to an external in-planeisotropic stress σ ij = σδ ij , stabilized and balanced by anexternal field (cid:126)E , which tends to align the normals (blue vec-tors). In this Letter we describe such experimental geome-tries, illustrated in Fig.1, where the imposed stress a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b and anisotropy lead to qualitatively richer and uni-versal buckling phenomenology. Generic buckling is acomplex out-of-plane instability of a sheet subjected tocompression, that results in a strongly distorted, non-perturbative state. Instead, here we focus on a gentler,continuous form of this transition, where the instabilityis controlled by an external field. Specifically, we con-sider an externally oriented membrane tuned to a buck-ling transition by a compressional boundary stress ap-plied within the plane explicitly selected by the orienta-tional field [41]. The compressive stress can be tuned toa critical value, σ c to cancel out at quadratic order the(embedding-space) rotational symmetry breaking fields.Our key observation is that at this new buckling crit-ical point (to which the isotropic flat membrane criti-cal point[6] is unstable), although at harmonic order themembrane appears to be rotationally invariant and stress-free, thus exhibiting strong thermal fluctuations, it ad-mits new important elastic nonlinearities that are not ro-tationally invariant. These lead to a critical membrane,tuned to the buckling point, that is, thus qualitativelydistinct from the conventional tensionless membrane[42]. Results.
Subjecting a crystalline membrane to alateral compressive isotropic boundary stress σ , tunedto a critical tensionless buckling point σ c and stabi-lized by an orienting field, we find a new buckling uni-versality class, distinct from the isotropic tensionlessmembrane[5, 6, 8, 9, 15]. We propose a model basedon symmetry arguments, supported by more detailedconsiderations. We use two complementary approachesto analyze the properties of the resulting critical state.The first is the self-consistent screening approximation(SCSA) which was found to provide an accurate descrip-tion for the isotropic case [9, 15]. Thermal fluctuationsand elastic nonlinearities at the buckling transition leadto a universal anomalous elasticity with exponent η anis = 0 . , (1)characterizing the divergence of the effective length-scaledependent bending rigidity κ ( k ) ∼ k − η . The in-planeelastic moduli remain finite at the critical point, i.e., η anis u = 0 [44]. This is at variance with the tensionless isotropic membrane for which SCSA predicts universalexponents η ≈ . η u ≈ .
358 [9]. The correspondingroughness h rms ∼ L ζ of the critically buckled membraneis characterized by a universal roughness exponent ζ anis = 0 . , (2)and it is thus rougher than a tensionless isotropic mem-brane, with a roughness exponent ζ ≈ .
59 [9].We complement this SCSA calculation by an RG anal-ysis in an expansion in (cid:15) = 4 − D . It confirms the insta-bility of the standard anomalous elasticity fixed pointof the isotropic, tensionless membrane, under break-ing of the embedding space rotational symmetry. Let us recall that for the isotropic membrane the elasticnonlinearities destabilize the harmonic theory (i.e., theGaussian fixed point) beyond the length scale ξ isoNL ∼ ( κ T K ) − D . If the anisotropy perturbation is very weak,e.g., w ∼ µ , , λ , (cid:28) K (see below for definitions ofthese anisotropy parameters), the membrane still expe-riences the standard isotropic anomalous elasticity up toscales ξ isoNL , crossing over to the new anisotropic criticalbehavior beyond the crossover length ξ anis NL = ξ isoNL (cid:18) K w (cid:19) /ρ , ρ = (cid:15)d c d c + 24 + O ( (cid:15) ) , (3)where ρ is the crossover exponent obtained from lin-earization of the RG flow around the isotropic fixed point.If the anisotropy perturbation is stronger, the thermalfluctuations and elastic nonlinearities directly destabilizethe harmonic theory at scales of order ξ isoNL . Beyond thesescales, the RG flows to a new stable buckling criticalpoint, which, within the (cid:15) -expansion, is however accessi-ble only for space codimension d c = d − D > d c = 1, we interpret theresulting runaway flows as a weakly first order transition,as for the standard crumpling transition. We note thatthe SCSA is exact for the large d c limit, and confirmthat the two methods match in their common regime ofvalidity. Model of anisotropic membrane buckling.
The coordi-nates of the atoms in the d -dimensional embedding spaceare denoted (cid:126)r ( x ) ∈ R d , with the atoms labeled by theirposition x ∈ R D in the internal space. For graphene D = 2, and atoms span a triangular lattice, describedhere in the continuum limit. The deformations with re-spect to the flat sheet are described by D phonon fields u α ( x ), and d c = d − D height fields (cid:126)h ∈ R d c (orthogonalto the (cid:126)e α ) as (cid:126)r ( x ) = ( x α + u α ( x )) (cid:126)e α + (cid:126)h ( x ), where the (cid:126)e α are a set of D orthonormal vectors. While the physicalcase corresponds to d = 3 and d c = 1, it is useful to studythe theory for a general d c . The nonlinear strain tensormeasures the deformation of the induced metric relativeto the preferred flat metric, u αβ = ( ∂ α (cid:126)r · ∂ β (cid:126)r − δ αβ ) (cid:39) ( ∂ α u β + ∂ β u α + ∂ α (cid:126)h · ∂ β (cid:126)h ) to the accuracy needed here,with the O (( ∂u ) ) phonon nonlinearities irrelevant andtherefore neglected (see below). The tensor u αβ encodesfull rotational invariance in the embedding space, its ap-proximate form being invariant under infinitesimal rota-tions by θ , i.e., the O ( θ ) term vanishes under the (ap-parent) distortion u = x (cos θ − h = x sin θ , whichcorresponds to a rigid rotation, with the correspondingvanishing of the exact strain tensor.Here we build on the model of a rotationally invari-ant tensionless membrane. Its Hamiltonian is the sum ofcurvature energy and in-plane stretching energy F [ (cid:126)h, u α ] = (cid:90) d D x (cid:20) κ ∂ (cid:126)h ) + τ u αα + µ ( u αβ ) + λ u αα ) (cid:21) (4)where κ is the bending modulus, λ, µ the in-plane Lam´eelastic constants. The parameter τ controls the preferredextension of the membrane in the (cid:126)e α plane.External orientational and boundary stresses introducenew relevant elastic nonlinearities, with five new inde-pendent couplings, that by symmetry lead to a modifiedeffective Hamiltonian F = F + F , where F breaksrotational invariance in the embedding space, F [ (cid:126)h, u α ] = (cid:90) d D x (cid:18) γ ∂ α (cid:126)h ) (5)+ λ ∂ α u α ( ∂ β (cid:126)h ) + λ ∂ α (cid:126)h ) ] + µ ∂ α u β ( ∂ α (cid:126)h · ∂ β (cid:126)h ) + µ ∂ α (cid:126)h · ∂ β (cid:126)h ] (cid:19) , retaining in-plane isotropy and the h → − h invarianceas a feature of our geometry, preserving the equivalencebetween the two sides of the membrane.We now study the membrane with parameters tuned tothe thermal buckling critical point defined by the renor-malized γ R = 0. Integrating over the in-plane phononmodes u α and, rescaling for convenience all elastic con-stants by 1 /d c , we obtain an effective Hamiltonian forthe height field, F [ (cid:126)h ] = (cid:90) d D x [ κ ∂ (cid:126)h ) + γ ∂ α (cid:126)h ) ] + 14 d c (cid:90) d D x d D y × ∂ α (cid:126)h ( x ) · ∂ β (cid:126)h ( x ) R αβ,γδ ( x − y ) ∂ γ (cid:126)h ( y ) · ∂ δ (cid:126)h ( y ) , (6)with a non-local quartic tensorial interaction, which inFourier space is given by[46] R αβ,γδ ( q ) = (cid:88) i =1 w i ( W i ) αβ,γδ ( q ) . (7)The W i are five projectors in the space of four indextensors, equal to bilinear combinations of longitudinal P Lαβ ( q ) = q α q β /q and transverse P T ( q ) = δ αβ − P Lαβ ( q )projectors on the wave vector q . The five ”bare cou-plings” w i are given in the Supplementary Material (SM)[50] in terms of the bare elastic moduli in (4) and (5), to-gether with the basis tensors W i . The important featuresare the following. When rotational symmetry breakingis absent, γ = 0, µ = µ = λ = λ = 0, the couplings w , w , w vanish and w = µ , w = µ + ( D − µλλ + 2 µ , (8)leading to (the q dependence suppressed) R αβ,γδ = ( w − w ) P Tαβ P Tγδ + w
12 ( P Tαγ P Tβδ + P Tαδ P Tβγ ) , (9) which is the usual quartic coupling associated to F .When λ and λ are turned on, while µ = µ = 0, all w i are nonzero except w = 0. Finally, when all couplingsin F are nonzero, all w i are nonzero. SCSA analysis.
The form (6) is suitable to apply theSCSA method, which is exact in the limit of large d c .The calculation is performed in the SM [50] and par-allels the one in Section IV. A of [15]. Consider thetwo point correlation of the height field in Fourier space, (cid:104) h i ( k ) h j ( k (cid:48) ) (cid:105) = G ( k )(2 π ) d δ d ( k + k (cid:48) ) δ ij . If we neglectthe quartic nonlinearities in (6) we find G ( k ) = G ( k ) =1 / ( γk + κk ). The nonlinearities leads to a nonzeroself-energy σ ( k ) = G ( k ) − − γk − κk . Together withthe renormalized interaction tensor, ˜ R ( q ), it satisfies theSCSA equations σ ( k ) = 2 d c (cid:90) q k α ( k β − q β )( k γ − q γ ) k δ ˜ R αβ,γδ ( q ) G ( k − q )(10)˜ R ( q ) = R ( q ) − R ( q )Π( q ) ˜ R ( q ) (11)where Π( q ) encodes the screening of the in-plane elastic-ity by out of plane fluctuationsΠ αβ,γδ ( q ) = 14 (cid:90) p v αβ ( q , q − p ) v γδ ( q , q − p ) G ( p ) G ( q − p )(12)and v αβ ( p , p (cid:48) ) = p α p (cid:48) β + p (cid:48) α p β . One can decomposeΠ( q ) = (cid:80) i =1 π i ( q ) W i ( q ) and ˜ R ( q ) = (cid:80) i =1 ˜ w i ( q ) W i ( q ),where ˜ w i ( q ) are the momentum dependent renormal-ized couplings. Looking for a solution which behaves atsmall k as G ( k ) (cid:39) Z − κ /k − η , and evaluating the inte-grals π i ( q ) [50] one finds that they diverge at small q as π i ( q ) (cid:39) Z − κ a i q − (4 − D − η ) where a i = a i ( η, D ). From(11) we find that the renormalized couplings are softenedat small q as ˜ w i ( q ) ∝ Z κ c i q η u , with η u = 4 − D − η and c i = 1 /a i for i = 1 , (cid:18) c c c c (cid:19) (cid:39) (cid:18) a a a a (cid:19) − (13)Inserting this into the self-energy equation (10) and per-forming the integrals we find that the factors of Z κ canceland the self-consistent equation, which implicitly deter-mines η as a function of D, d c is given by d c (cid:88) i =1 , b i a i + b a − b a + b a a a − a , (14)where b i = b i ( η, D ) are self-energy integrals, given withthe a i ( η, D ) in the SM. Note that here we have consideredthe case where all bare couplings w i are nonzero. For aphysical membrane, D = 2, (14) reduces to finding theroot of a cubic equation d c = 24( η − (2 η + 1)( η − η (2 η − . (15)For d c = 1 we obtain our main result (1). For large d c we find η = 2 /d c + O (1 /d c ). The roughness of a size L membrane is characterized by h rms = (cid:104) h (cid:105) / (cid:39) L ζ where ζ = (4 − D − η ) /
2. Hence for d c = 1 we find ζ = 0 . q → ˜ w i ( q )˜ w j ( q ) = c i c j (16)for any pair ( i, j ) such that the bare couplings w i , w j arenonzero. Near D = 4 we find that these renormalizedcouplings take values such that the interaction energybecomes v [( ∂ α (cid:126)h ) ] + v ( ∂ α (cid:126)h · ∂ β (cid:126)h ) , i.e., local in thefields ∂ α (cid:126)h . This property however does not hold for D <
4, e.g., one finds c /c = ( D + η − / (2 − η ) instead ofunity for D = 4, η = 0. Thus the critical point requires afully non-local five coupling description. The c i are givenin [50]. In the physical case of D = 2 and d c =1 we find c i = { , . , . , − . , . } , (17)and the universal λ/µ = − .
978 and the Poisson ratio, σ anis = − . , (18)to be contrasted with σ = − / w = w = w = 0. The corresponding renormalizedcouplings also vanish, which amounts to b = b = b = 0in (14), leading to d c b a + b a , (19)which is precisely the SCSA equation for the anomalousflat phase of the isotropic membrane, leading for D = 2 to η = d c + √ − d c + d c , and η (cid:39) . ζ = 0 .
590 for d c = 1[9, 15]. Near D = 4 one recovers η = d c +24 (cid:15) + O ( (cid:15) ) fromthe Aronovitz-Lubensky’s (cid:15) -expansion[6]. Another fixedmanifold is w = 0, corresponding to a choice of barecouplings so that ( µ + µ ) = µ ( µ + µ ), which includesthe choice µ = µ = 0, leading to ˜ w ( q ) = 0 and d c b a + b a − b a + b a a a − a . (20)This leads to yet another fixed point with slightly differ-ent exponents. For D = 2 and d c = 1 we find η = 0 . ζ = 0 . D = 4 we find η = d c +36 (cid:15) + O ( (cid:15) ).Universal amplitude ratios have c = 0. RG analysis.
As a nontrivial check and for further in-sight, we have complemented this SCSA calculation andresults using an RG analysis, controlled by an (cid:15) = 4 − D expansion near D = 4. We have calculated the one-loop corrections to the Hamiltonian (6) and obtained the RG equations for the five dimensionless couplingsˆ w i = w i /κ C Λ − (cid:15)(cid:96) of the form ∂ (cid:96) ˆ w i = (cid:15) ˆ w i + a ijk ˆ w i ˆ w k ,where the a ijk and details of the calculation are given in[50]. The anomalous dimension of the out-of-plane heightfield h defines the exponent η given by η = 112 (10 ˆ w −
18 ˆ w + 5 ˆ w + 3 ˆ w − w ) , (21)with ˆ w = √ w , and evaluated at the fixed point ofinterest w ∗ i (see below). The anomalous dimension of thephonon field is given by η u = 112 ( ˆ w − ˆ w ) . (22)The isotropic membrane corresponds to the space ˆ w =ˆ w = ˆ w = 0, which is preserved by the RG flow andalong which ∂ (cid:96) ˆ w = −
112 ˆ w (( d + 20) ˆ w + 10 ˆ w ) , (23) ∂ (cid:96) ˆ w = −
524 ˆ w (( d + 4) w + 8 ˆ w ) . (24)The isotropic membrane fixed point is ˆ w ∗ = (cid:15)d +24 , ˆ w ∗ = (cid:15) d +24) , corresponding to ˆ µ ∗ = (cid:15) d , ˆ λ ∗ = − (cid:15) d [6]. Di-agonalizing the RG flow for ˆ w i = ˆ w ∗ i + δ ˆ w i around thisfixed point in the larger space of five couplings showsthat, in addition to the two negative eigenvalues − − d c d c +24 within the plane δ ˆ w , of the isotropic membrane,(i) there is a marginal direction mixing δ ˆ w , , (eigen-value 0) (ii) there are two unstable directions with eigen-values d c d c +24 with δw , nonzero (in the large d c limit thiseigenspace is purely along δw , ). Hence, consistent withthe SCSA findings, the isotropic membrane fixed pointis unstable to anisotropy of the orientational field andexternal boundary stress.To determine where the general flow goes we searchedfor attractive fixed points of the RG equations. We foundone such fixed point in the subspace of couplings ˆ w i atwhich, the interaction energy is fully local in the gradi-ents ∂ α (cid:126)h and parameterized by two couplings v , v asdefined above. This subspace is preserved by the RGand also arises in the study of the crumpling transition.In fact the RG flow within this subspace is identical tothe one obtained in [45] with d replaced by d c . It ad-mits a stable FP for d c > D = 4, with the SCSA (which is exact for large d c andany D ), predicting a new fixed point for membrane inanisotropic embedding space. For the physical membrane D = 2 and d c = 1, while the SCSA predicts this new”anisotropic buckling transition” to be continuous, theRG, if extrapolated from D = 4, suggests a weakly firstorder transition, as argued for the crumpling transition[22, 23, 45].To reach the new anisotropic buckling critical pointrequires tuning γ = γ c , so that γ R = 0. Slightly awayfrom criticality the correlation length is long but finite, ξ ∼ | δγ | − ν , diverging with a vanishing δγ = γ − γ c .Linearizing the RG flow around the fixed point yields δγ ( L ) ∼ δγL θ , where θ = − (cid:15)d c (1 − d c + O ( d c )), see theSM [50]. By balancing κ ( ξ ) ξ − ∼ δγ ( ξ ) ξ − and usingthat κ ( ξ ) ∼ ξ η we obtain the correlation length exponentas ν = 1 / (2 + θ − η ). Model development.
Until now we argued for the model(4,5) based on symmetry considerations. Here, as illus-trated in Fig.1, we develop an explicit model of a mem-brane undergoing buckling in the absence of rotationalinvariance in the embedding space. We consider an elas-tic membrane in an external field (cid:126)E (taken along thez-axis) that aligns the membrane’s normal ˆ n along thefield. We thus expect the energy density to be a mono-tonic function of ˆ n · (cid:126)E , namely of the small tilt angle θ , H o rient = α θ + ˜ α θ + . . . , (25)with α >
0, ˜ α >
0. Combining this orienta-tional field energy with the Hamiltonian for an elas-tic membrane[12, 15], subjected to an in-plane compres-sional boundary stress σ >
0, isotropic in the membrane’sxy plane, and using that, to lowest order θ ∼ | ∂ α h | , weobtain, H = κ ∂ h ) + µu αβ + λ u αα + σ∂ α u α + α ∂ α h ) + α ∂ α h ) + . . . . (26)We note that the external stress, σ is an in-planeboundary term, that induces a stress-dependent inwarddisplacement of the membrane’s edges. Observing that σ∂ α u α = σu αα − σ ( ∂ α h ) , the first term can be ac-commodated by simply changing the preferred extensionof the membrane without breaking the embedding spacerotational symmetry (i.e., it amounts to a redefinition ofthe parameter τ in (4), which determines the preferredmembrane’s projected area [51]). The negative in-planestrain ∂ α u α induced by positive σ can be relieved by amembrane tilt, ( ∂ α h ) >
0, stress-free in the actual planeof the membrane. The lowering of the energy associatedwith the membrane tilt is then given by the second term,i.e., H σ = − σ ( ∂ α h ) , which, neglecting bending en-ergy and boundary conditions, is unbounded, since tiltis unconstrained in the absence of the orientational field.Putting these ingredients together and rescaling xy co-ordinate system, we obtain the Hamiltonian governinga buckling transition of a membrane in an orientationalfield, H = κ ∂ h ) + µu αβ + λ u αα + γ ∂ α h ) + α ∂ α h ) + . . . , (27) where γ = α − σ is the critical parameter which can betuned to γ c to reach the buckling transition (with γ c = 0at T = 0), studied in here. Conclusion.
To summarize, in this Letter, in con-trast to previous works on tensionless crystalline mem-branes, we studied a thermal elastic sheet tuned by anexternal boundary stress to a critical point of a buck-ling transition, stabilized by an orientational field. Wefind that this breaking of embedding rotational symmetryhas profound effects, and leads to a new class of anoma-lous elasticity, that we have explored in detail here usingthe SCSA and RG analyses. With much recent interestin elastic sheets, most notably graphene and other vander Waals monolayers, we hope that our predictions willstimulate further experiments to probe the rich universalphenomenology predicted here for an elastic membranetuned to a buckling transition in an anisotropic environ-ment.
Note Added:
We have recently became aware of anongoing work by S. Shankar and D. R. Nelson on a mem-brane subjected to a boundary stress or strain, which, incontrast to our work only breaks embedding rotationalsymmetry at the boundary.
Acknowledgments.
We thank John Toner, David Nel-son and Suraj Shankar for enlightening discussions. LRalso acknowledges support by the NSF grants MR-SEC DMR-1420736, Simons Investigator Fellowship, andthanks ´Ecole Normale Sup´erieure for hospitality. PLDacknowledge support from ANR under the grant ANR-17-CE30-0027-01 RaMaTraF. Both authors thank KITPfor hospitality. This research was supported in part bythe National Science Foundation under Grant No. NSFPHY-1748958. ∗ [email protected] † [email protected][1] The structure of suspended graphene sheets . J. C. Meyer,A. K. Geim, M. I. Katsnelson, K. S. Novoselov, T. J.Booth, and S. Roth,
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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 αβ,γδ ( q ) introduced in the text, which are symmetric in α ↔ β , in γ ↔ δ and in ( α, β ) ↔ ( γ, δ ). The product of such tensors is defined as ( T · T (cid:48) ) αβ,γδ = T αβ,γ (cid:48) δ (cid:48) T (cid:48) γ (cid:48) δ (cid:48) ,γδ , the identitybeing I αβ,γδ = ( δ αγ δ βδ + δ αδ δ βγ ). We recall the definition [15] of the five ”projectors” W i , i = 1 , . . . ,
5, which spanthe space of such four index tensors( W ) αβ,γδ ( q ) = 1 D − P Tαβ P Tγδ , ( W ) αβ,γδ ( q ) = P Lαβ P Lγδ , (28)( W ) αβ,γδ ( q ) = ( W a ) αβ,γδ ( q ) + ( W b ) αβ,γδ ( q ) , (29)( W a ) αβ,γδ ( q ) = 1 √ D − P Tαβ P Lγδ , ( W b ) αβ,γδ ( q ) = 1 √ D − P Lαβ P Tγδ , (30)( W ) αβ,γδ ( q ) = 12 ( P Tαγ P Lβδ + P Tαδ P Lβγ + P Lαγ P Tβδ + P Lαδ P Tβγ ) , (31) W ( q ) = 12 ( δ αγ δ βδ + δ αδ δ βγ ) − W ( q ) − W ( q ) − W ( q ) , (32) W ( q ) + W ( q ) = 12 ( P Tαγ ( q ) P Tβδ ( q ) + P Tαδ ( q ) P Tβγ ( q )) (33)where P Tαβ = δ αβ − q α q β /q and P Lαβ = q α q β /q are the standard transverse and longitudinal projection operatorsassociated to q . The first two projectors W , W are mutually orthogonal and orthogonal to the other three. Notethat while R , being symmetric, can be expressed in terms of the symmetric tensors W i , i = 1 , ..
5, we will need at someintermediate stages of the calculations some products (such as Π ∗ R see below), which are not symmetric. Hencewe introduced W a and W b , which together with W i , i = 1 , , W make the representation complete undertensor multiplication. The rules for the tensor multiplication T (cid:48)(cid:48) = T (cid:48) ∗ T of the tensors T = (cid:80) i =1 w i W i + w a W a + w b W b + w W and T (cid:48) = (cid:80) i =1 w (cid:48) i W i + w (cid:48) a W a + w (cid:48) b W b + w (cid:48) W are w (cid:48)(cid:48) = w (cid:48) w , w (cid:48)(cid:48) = w (cid:48) w , (cid:18) w (cid:48)(cid:48) w (cid:48)(cid:48) a w (cid:48)(cid:48) b w (cid:48)(cid:48) (cid:19) = (cid:18) w (cid:48) w (cid:48) a w (cid:48) b w (cid:48) (cid:19) (cid:18) w w a w b w (cid:19) , (34)with T (cid:48)(cid:48) = (cid:80) i =1 w (cid:48)(cid:48) i W i + w (cid:48)(cid:48) a W a + w (cid:48)(cid:48) b W b + w (cid:48)(cid:48) W . B. Integration over in-plane deformations
The integration over the phonon fields u α ( x ) of the Gibbs measure ∼ e −F [ (cid:126)h,u α ] /T , with F = F + F given by (4)and (5) leads to the Gibbs measure ∼ e −F [ (cid:126)h ] /T for the height fields with an effective Hamiltonian of the form (6) inthe text (we set τ = 0). To perform it we use a method slightly different from the one in e.g. [15] Section III B. Letus introduce the elastic matrix C µ,λαβ,γδ = λδ αβ δ γδ + µ ( δ αγ δ βδ + δ αδ δ βγ ) (35)and denote ˜ u αβ = ( ∂ α u β + ∂ β u α ) and A αβ = ∂ α (cid:126)h · ∂ β (cid:126)h . We then rewrite the model F = F + F as F [ u,(cid:126)h ] = (cid:90) d D x (cid:20) κ ∇ h ) + 12 C µ,λαβ,γδ ˜ u αβ ˜ u γδ + ˜ u αβ C µ + µ ,λ + λ αβ,γδ A γδ + 12 C µ + µ ,λ + λ αβ,γδ A αβ A γδ (cid:21) (36)We must treat separately the contributions of the in plane strains which are uniform (i.e. with zero momentum), andthose with nonzero wavevector, i.e. the phonons. B.1 Phonon integration: nonzero wavevector .We recall the phonon field propagator (cid:104) u α ( q ) u β ( q (cid:48) ) (cid:105) = T (2 π ) D δ D ( q + q (cid:48) ) (cid:32) P Tαβ ( q ) µq + P Lαβ ( q )(2 µ + λ ) q (cid:33) (37)from which the in-plane strain correlator at nonzero wavevector is obtained as (cid:104) ˜ u αβ ( q )˜ u γδ ( q (cid:48) ) (cid:105) = T (2 π ) D δ D ( q + q (cid:48) ) D αβ,γδ ( q ) (38)with, for q (cid:54) = 0, D αβ,γδ ( q ) = 14 (cid:34) ˆ q α ˆ q γ (cid:32) P Tβδ ( q ) µ + P Lβδ ( q )2 µ + λ (cid:33) + 3 permutations (cid:35) (39)This tensor has a simple expression in terms of the projectors (suppressing the indices and the q dependence) D = 12 µ W + 12 µ + λ W (40)The integration over the phonon field in (36) using (37) is then a simple quadratic Gaussian integral leading to theform given in Eq. (6) in the main text F eff ( (cid:126)h ) = 1 d c (cid:90) q (cid:54) =0 R αβ,γδ ( q ) A αβ ( − q ) A γδ ( q ) (41)where the interaction tensor is R = 12 C µ + µ ,λ + λ − C µ + µ ,λ + λ · D · C µ + µ ,λ + λ (42)Thanks to the projectors its explicit calculation is easy. One decomposes C µ,λ = 2 µ ( W + W + W + W ) + λ [( D − W + √ D − W + W ] (43)and uses the above multiplication rules for the W i ’s. One obtains R αβ,γδ ( q ) = (cid:88) i =1 w i W i ( q ) (44)in terms of the five ”elastic constants” w i w = µ + µ (45) w = µ + µ − ( µ + µ ) µw = µ + µ + ( D − λ + λ − ( λ + λ ) λ + 2 µ ) w = 12 ( D − (cid:18) λ + λ − ( λ + λ )( λ + λ + 2 µ + 2 µ ) λ + 2 µ (cid:19) , w = √ D − w w = 12 (cid:18) λ + λ + 2 µ + 2 µ − ( λ + λ + 2 µ + 2 µ ) λ + 2 µ (cid:19) Note that this is true under the condition that the phonon propagator is positive definite i.e. µ > , µ + λ > R αβ,γδ ( q ) given above is understood to explicitly exclude the zero-mode q = 0, whichwe address below. The stability of the zero-mode requires µ > µ + Dλ >
0, which is a more stringent condition.Note that when µ = µ = 0 one has w = 0. When in addition λ = λ = 0 one has w = w = w = 0 and w = µ , w = µ + ( D − µλλ + 2 µ (47)as given in the text. Note that in general there are 5 couplings w i and 6 original couplings. Inversion thus determinesonly the following five ratio as µ + µ = w (48)( µ + µ ) µ = w − w ( λ + λ ) µ = 4 ( w − w ) ( w − w + w ) ( D ( w − w ) − w + w + 2 w ) λ + λ = 2 (cid:0) w − ( D −
1) ( w − w ) ( w − w ) (cid:1) ( D −
1) ( D ( w − w ) − w + w + 2 w ) λµ = 2( D −
1) ( w − w ) D ( w − w ) − w + w + 2 w − µ + µ and λ + λ are the two h vertex couplings in the original u, h theory (before integrating phonons) andand ( λ + λ ) µ and ( µ + µ ) µ are the natural uhh vertex couplings combination appearing in perturbation theory. Finally λ/µ is a ratio of elastic constants. Hence the overall elastic constant scale, µ , remains undetermined and must becalculated separately from the u, h theory. Note that combining the above equations, one also obtains the followingratio λ + λ µ + µ = 2( w − w − w ) D ( w − w ) − w + w + 2 w (49)0Finally, in the case µ = µ = 0 one has w = 0 and one can invert the above relations for all remaining couplings µ = w , λ = − w ( − Dw + w − w + w + 2 w ) D ( w − w ) − w + w + 2 w (50) λ = 2 w ( − Dw + w + w ) D ( w − w ) − w + w + 2 w , λ = 2 w (2( D − w + w ) − D − w (( D − w + w )( D −
1) ( D ( w − w ) − w + w + 2 w ) (51)consistent with the above result.One can also ask about necessary conditions for the quartic form in the effective stretching energy (6) to be positivedefinite. Positivity of the quartic form k α ( q − k ) α R αβ,γδ ( q ) k β ( q − k ) δ (52)for any choice of k , k , q implies for instance: (i) choosing all k i aligned with qw > k = k and considering various limits we also find w > , ( D − w + w > w ≥ w for the above equations (48) to make sense. B.2 zero-mode
We must treat separately the uniform part of the nonlinear strain tensor, u αβ ( q = 0). It is the sum of the uniformpart of the in-plane strain tensor, which we denote ˜ u αβ and of A αβ = [( ∂ α h )( ∂ β h )]( q = 0). The energy per unitvolume associated to this zero-mode is f (˜ u , A ) = µ (˜ u αβ + A αβ ) + λ u αα + A αα ) + λ ˜ u αα A αα + 2 µ ˜ u αβ A αβ + µ ( A αβ ) + λ A αα ) , (55)which can be rewritten as f (˜ u , A ) = 12 C µ,λαβ,γδ ˜ u αβ ˜ u γδ + ˜ u αβ C µ + µ ,λ + λ αβ,γδ A γδ + ( µ + µ )( A αβ ) + λ + λ A αα ) . (56)Minimizing the energy over the D ( D + 1) / u αβ (or integratingthe Gibbs measure, which is equivalent since the energy is quadratic in the ˜ u αβ ) we obtain the minimum[˜ u ] αβ = − [ C µ,λ ] − αβ,γ (cid:48) δ (cid:48) C µ + µ ,λ + λ γ (cid:48) δ (cid:48) ,γ,δ A γ,δ = − µ + µ µ A αβ + λµ − λ µµ (2 µ + Dλ ) δ αβ A γγ , (57)where we have used that [ C µ,λ ] − αβ,γδ = − λ µ (2 µ + Dλ ) δ αβ δ γδ + 14 µ ( δ αγ δ βδ + δ αδ δ βγ ) . (58)Plugging back this minimum into the energy we find f eff [ h ] = f ( u , A ) = 12 ¯ C αβ,γδ A αβ A γδ , (59)where ¯ C = C µ + µ ,λ + λ αβ,γδ − C µ + µ ,λ + λ · [ C µ,λ ] − · C µ + µ ,λ + λ . (60)Upon explicit calculation the final result is f eff [ h ] = ( µ − µ − µ µ )( A αβ ) + 12 ( λ − Dλ (2 λ + λ ) µ − λµ + 4 λ µ ( µ + µ ) µ ( Dλ + 2 µ ) )( A αα ) . (61)Note that it vanishes when the new terms breaking rotational symmetry are absent i.e. when µ = µ = λ = λ = 0.These zero-mode terms are thus generated only by the bulk anisotropy since we are working in the fixed stress settingand freely integrate over the zero-mode of the in-plane strain. We leave their study for the future [57].1 B.3 Stability
Here we note that we can rewrite F + F = (cid:90) d D x (cid:104) κ ∂ (cid:126)h ) + τ u αα + γ ∂ α (cid:126)h ) + f el (cid:105) . (62)Using the traceless tensors and the traces as f el = µ (cid:18) ˜ u αβ − D δ αβ ˜ u γγ + µ + µ µ ( A αβ − D δ αβ A γγ ) (cid:19) + 2 µ + Dλ D (cid:18) ˜ u αα + 2( µ + µ ) + D ( λ + λ )2 µ + Dλ A αα (cid:19) (63)+ˆ µ (cid:18) A αβ − D δ αβ A γγ (cid:19) + B A αα , (64)with ˆ µ = µ + µ − ( µ + µ ) µ (65) B = 12 D (cid:18) µ + µ ) + D ( λ + λ ) − (2( µ + µ ) + D ( λ + λ )) µ + Dλ (cid:19) , (66)where we recall that A αβ = ∂ α (cid:126)h · ∂ β (cid:126)h . Let us set τ = 0. Note that ˆ µ = w as defined in (45). Hence we see that,since the traceless part and the trace are independent, for w > B > A αβ = 0, and, in turn from the two first squares, ˜ u αβ = 0.Hence in that case u α = 0, (cid:126)h = 0 is indeed the stable ground state. We note that, in contrast, in the rotationallyinvariant case (i.e., setting µ = µ = λ = λ ), the same reasoning leads to the zero energy minimum condition, u αβ = ˜ u αβ + A αβ = 0, instead of the above anisotropic condition of ˜ u αβ and A αβ vanishing separately. This isexpected 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 calculationin Ref.15.
C.1. SCSA equations
The SCSA is given by the pair of coupled equations (10) and (11) given in the text for the self-energy σ ( k ) = G ( k ) − − κk and for the renormalized interaction ˜ R αβ,γδ ( q ). The equation (11) involves tensor multiplication. Wecan thus decompose Π( q ) = (cid:80) i =1 π i ( q ) W i ( q ) and ˜ R ( q ) = (cid:80) i =1 ˜ w i ( q ) W i ( q ), as indicated in the text, where ˜ w i ( q ) arethe momentum dependent renormalized couplings and π i ( q ) are polarization integrals calculated below. The rules forthe tensor multiplication were given in the previous section. Since the tensors R αβ,γδ ( q ), ˜ R αβ,γδ ( q ) and Π αβ,γδ ( q )are symmetric in α ↔ β , in γ ↔ δ and in ( α, β ) ↔ ( γ, δ ), they can be parameterized in terms of five couplings (i.e.,with w a = w b = w ).We can now solve the equation (11) and find the renormalized couplings ˜ w i ( q ) as˜ w ( q ) = w w π ( q ) , ˜ w ( q ) = w w π ( q ) , (67) (cid:18) ˜ w ( q ) ˜ w ( q )˜ w ( q ) ˜ w ( q ) (cid:19) = (cid:18) w w w w (cid:19) (cid:18)(cid:18) (cid:19) + (cid:18) π ( q ) π ( q ) π ( q ) π ( q ) (cid:19) (cid:18) w w w w (cid:19)(cid:19) − . (68)These can be substituted into (10) to express the self-energy as σ ( k ) = 2 d c (cid:88) i =1 , (cid:90) q ˜ w i ( q ) G ( k − q ) k α ( k β − q β )( W i ) αβ,γδ ( q ) k γ ( k δ − q δ ) , (69)The above equations form a closed set of SCSA equations for the five renormalized elastic coupling constants ˜ w i ( q ),together with the self energy σ ( k ). The complete Dyson equation for the self-energy contains an additional UV2divergent ”tadpole” diagram contribution, which scales as k . The integral in (69) also contains a component thatscales as k at small k . Both contributions have been substracted by tuning the bare coefficient γ in order to sit atthe critical point.To solve the above SCSA equations at the critical point, we look for a solution with the long wavelength form G ( k ) (cid:39) Z − κ /k − η . The π i ( q ) integrals have been calculated in the Appendix B of [15]. They diverge for small q as: π i ( q ) (cid:39) Z − κ a i ( η, D ) q − (4 − D − η ) . (70)For the amplitudes a i ( η, D ) we find a = 2 A , a = A − η ) D + η − , a = A ( D + 1) , a = A √ D − D + 2 η − , (71) a = AD − η ( −
22 + 31 D − D + D + 43 η − Dη + 5 D η − η + 8 Dη + 4 η ) , with A = A ( η, D ) = Γ(2 − η − D/ D/ η/ D/ η/ π ) D/ Γ(2 − η/ − η/ D + η ) . (72)To compute the self-energy we define the amplitudes b i ( η, D ) through: (cid:90) q q − D − η | k − q | − (4 − η ) k α ( k β − q β )( W i ) αβ,γδ ( q ) k γ ( k δ − q δ ) = b i ( η, D ) k − η . (73)The explicit calculation in the Appendix B of [15] gives b = B ( D − D + 1) , b = − B ( D − D − η ) D − η , b = B ( D + 1) , (74) b = 2 B √ D − η − , b = BD − η ( −
22 + 15 D − D + 43 η − Dη − η + 4 Dη + 4 η ) , where B = B ( η, D ) = Γ( η/ D/ η/ − η )4(4 π ) D/ Γ(2 − η/ D/ η )Γ( D/ − η/ . (75) 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 correspondsto the ”anisotropic fixed point” discussed in the text. In the limit q → w ( q ) (cid:39) π ( q ) , ˜ w ( q ) (cid:39) π ( q ) , , (cid:18) ˜ w ( q ) ˜ w ( q ) , ˜ w ( q ) ˜ w ( q ) (cid:19) (cid:39) (cid:18) π ( q ) π ( q ) π ( q ) π ( q ) (cid:19) − . (76)independent of the bare values, as long as they are nonzero. Substituting Eqs.(68),(72),(75) into (69) we see thatfactors of Z κ cancel and we find the self-consistent equation: d c (cid:88) i =1 , b i ( η, D ) a i ( η, D ) + b ( η, D ) a ( η, D ) − b ( η, D ) a ( η, D ) + b ( η, D ) a ( η, D ) a ( η, D ) a ( η, D ) − a ( η, D ) . (77)Putting everything together, after considerable simplifications, the equation determining the exponent η = η anis ( D, d c ) is found to be: d c = D ( D + 1)( D − η )( D − η )(2 D − η )Γ[ η ]Γ[2 − η ]Γ[ η + D ]Γ[2 − η ]2(2 − η )(5 − D − η )( D + η − D + η ]Γ[2 − η − D ]Γ[ η + D ]Γ[ D + 2 − η ] , (78)which in D = 2 reduces to d c = 24( η − (2 η + 1)( η − η (2 η − . (79)3as given in the main text, leading to η anis ( D = 2 , d c = 1) = 0 . .. . In the limit of large d c , the solution of (78)behaves as η anis ( D, d c ) (cid:39) C ( D ) d c + O (1 /d c ) , C ( D ) = ( D − (2 D − D + 2)2(5 − D )( D − (cid:0) − D (cid:1) Γ (cid:0) D + 2 (cid:1) Γ (cid:0) D (cid:1) , with C (2) = 2. As discussed in [15] the leading coefficient C ( D ) in the 1 /d c expansion is an exact result, while thehigher 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 inembedding space dimension d = 1 instead of d = 3. Not surprisingly then, the leading term in the large d c expansionabove then coincides with the one in the 1 /d expansion for the crumpling transition of Ref. 6. We can also expandour SCSA prediction in (cid:15) = 4 − D , finding η anis ( D, d c ) (cid:39) d c (4 − D ) + O ((4 − D ) ) , (80)consistent with the vanishing of the leading order O ( (cid:15) ) of η 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 discussedin the text, from (76) we obtain ˜ w i ( q ) (cid:39) Z κ c i q − D − η /A (81) c i = 1 /a i for i = 1 , , (cid:18) c c c c (cid:19) (cid:39) (cid:18) a a a a (cid:19) − Inserting the ˜ w i ( q ) into (48), we obtain the renormalized couplings of the u, h theory. More precisely we obtain the h couplings ˜ µ ( q ) + ˜ µ ( q ), ˜ λ ( q ) + ˜ λ ( q ), and the uh couplings (˜ µ ( q ) + ˜ µ ( q )) / ˜ µ ( q ) and (˜ λ ( q ) + ˜ λ ( q )) / ˜ λ ( q ). Thesefour couplings thus vanish as q − D − η at small q . In addition we obtain the ratio ˜ λ ( q ) / ˜ µ ( q ) which has a finite limitat small q . The determination of ˜ µ ( q ) however requires an additional calculation (see below), with the result that˜ µ ( q ) ∼ q η u where η u is now an independent exponent (at variance with the rotationally invariant case where one has η u = 4 − D − η ). For this anisotropic fixed point, η u = 0, i.e. ˜ µ (0) is finite. Hence we find that ˜ µ ( q ) → − ˜ µ (0) atsmall q , so that the h coupling can vanish at small q as ˜ µ ( q ) + ˜ µ ( q ) ∼ q − D − η , and similarly for ˜ λ ( q ). A similarproperty holds for the uh couplings.Let us now determine the amplitude ratio, which are universal at the fixed point. From (81) we obtain the amplituderatio in the long wavelength limit as lim q → ˜ w i ( q )˜ w j ( q ) = c i c j (82)for any pair ( i, j ), with, using (71) and (81) we obtain c = 12 , c = − D + η − η − , c = 14 (cid:18) − D ( D + 3)( D + η −
1) + Dη − − D + 3)( D + 2 η −
5) + 2 (cid:19) (83) c = − √ D − D + η − D + 2 η − η − D + η − D + 2 η − , c = ( D + 1)( D + η − η − D + η − D + 2 η −
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 fixedpoint. Inserting the value of η for D = 2 and d c =1 we find, at the anisotropic fixed point c i ( D = 2 , d c = 1) = (cid:26) , . , . , − . , . (cid:27) (85)From this, using (48), we find lim q → ˜ λ ( q ) / ˜ µ ( q ) = − . σ R ( q ) = ˜ λ ( q )2˜ µ ( q )+( D − λ ( q ) = − . D = 2 and large d c we find, up to O (1 /d c ) terms c i = (cid:26) , d c ,
13 + 136 d c , d c , d c (cid:27) (86)4Hence for D = 2, the anisotropic membrane fixed point converges as d c → + ∞ to the one of the isotropic membranesince ˜ w , ˜ w , ˜ w are parametrically smaller in that limit than ˜ w and ˜ w (which span the couplings of the isotropicmembrane). However, from (83) we can state that these two fixed points are different at infinite d c for D >
2. In thislimit one can simply set η → d c → + ∞ c i = (cid:26) , D − , D − D + 2240 − D , ( D − D − D − √ D − , − D + D + 28 ( D − D + 5) (cid:27) (87)while for the isotropic one c i = { , , D +1 , , } , see below. Hence, for d c = + ∞ , the anisotropic fixed point leavesthe 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 localinteraction between the gradient fields ∂ α (cid:126)hR αβ,γδ = µ δ αγ δ βδ + δ αδ δ βγ ) + λ δ αβ δ γδ . (88)for some constants denoted µ and λ (these are denoted 4 v and 4 v respectively in the main text). It is realized bythe choice w = w = µ , w = 12 ( D − λ + µ , w = 12 √ D − λ , w = 12 λ + µ . (89)Note that the two eigenvalues of the matrix formed by the w i , i = 3 , ,
5, are then µ , and µ + Dλ . Replacing d c by d this is also the subspace corresponding to the bare action of the Landau theory for the crumpling transition [45].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 preservedby the SCSA). In D = 4 at large d c it is indeed preserved (consistent with the RG), since in that case one haslim d c → + ∞ c i = (cid:26) , , , − √ , (cid:27) (90)which indeed belongs to the subspace (89). However, from the above discussion, we expect the two-loop correctionsin the RG to fail to preserve this subspace. This indicates that the study of the RG of the crumpling transition tohigher order in (cid:15) will be qualitatively different from the one given in [45], a subject we leave for future investigation. 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 ∂ (cid:96) = − q∂ q on both sides of equations (67) we obtain, ∂ (cid:96) ˜ w i ( q ) = − w i (1 + w i π ( q )) ( − q∂ q ) π i ( q ) = − ˜ w i ( q ) ( − q∂ q π i ( q )) (cid:39) − ˜ w i ( q ) κ − (cid:15)q − (cid:15) a i (0 , D ) (91)where we have used (70) setting η →
0, i.e., using the bare propagator with Z κ = κ . The natural dimensionlesscoupling for the RG is ˆ w i := ˜ w i ( q ) κ − q − (cid:15) (92)In terms of these couplings we obtain the RG equation for d c = + ∞ , exact for any (cid:15) = 4 − D , ∂ (cid:96) ˆ w i = (cid:15) ˆ w i − (cid:15)a i (0 , D ) ˆ w i , i = 1 , ∂ (cid:96) (cid:18) ˆ w ˆ w ˆ w ˆ w (cid:19) = (cid:15) (cid:18) ˆ w ˆ w ˆ w ˆ w (cid:19) − (cid:15) (cid:18) ˆ w ˆ w ˆ w ˆ w (cid:19) (cid:18) a (0 , D ) a (0 , D ) a (0 , D ) a (0 , D ) (cid:19) (cid:18) ˆ w ˆ w ˆ w ˆ w (cid:19) (94)The fixed point of these RG equations which describes the anisotropic membrane for d c = + ∞ is then ˆ w i = ˆ w ∗ i withˆ w ∗ i = 1 a i (0 , D ) , i = 1 , , (cid:18) ˆ w ∗ ˆ w ∗ ˆ w ∗ ˆ w ∗ (cid:19) = (cid:18) a (0 , D ) a (0 , D ) a (0 , D ) a (0 , D ) (cid:19) − (95)5is consistent with the above analysis (81). The calculation of the exponent η to leading order O (1 /d c ) is then asfollows. If one calculates − k∂ k ( σ ( k ) /k ) from (69) one obtains a convergent integral. Replacing ˜ w i ( q ) = κ q (cid:15) ˆ w i in(69) and using (73) we can write the RG function η = η ( ˆ w ) as η = − k∂ k ( σ ( k ) /k ) = 2 d c (cid:88) i =1 , ˜ b i ( D ) ˆ w i , ˜ b i ( D ) = lim η → ηb i ( η, D ) , (96)where in the r.h.s we used η as a regulator to obtain the needed (finite) integral. One can then easily check that atthe fixed point (95) the exponent η = η ( ˆ w ∗ ) recovers the result η (cid:39) C ( D ) /d c predicted by the self-consistent equation(77).In the above RG equations (93) we have neglected the renormalization of κ which is subdominant in 1 /d c . We cannow take it into account and define accordingly the running RG couplings as ˆ w i := ˜ w i ( q ) κ − q η − (cid:15) = ˜ w i ( q )˜ κ ( q ) − q − (cid:15) .This allows to write the SCSA equations as RG flow equations for any d c as follows ∂ (cid:96) ˆ w i = ( (cid:15) − η ) ˆ w i − ( (cid:15) − η ) a i ( D, η ) ˆ w i , i = 1 , ∂ (cid:96) (cid:18) ˆ w ˆ w ˆ w ˆ w (cid:19) = ( (cid:15) − η ) (cid:18) ˆ w ˆ w ˆ w ˆ w (cid:19) − ( (cid:15) − η ) (cid:18) ˆ w ˆ w ˆ w ˆ w (cid:19) (cid:18) a ( D, η ) a ( D, η ) a ( D, η ) a ( D, η ) (cid:19) (cid:18) ˆ w ˆ w ˆ w ˆ w (cid:19) , (98)where the η RG function, η = η ( ˆ w ), is defined as η := − k∂ k ˜ κ ( k ) = − k∂ k ( σ ( k ) /k ) = 2 d c η (cid:88) i =1 , b i ( η, D ) ˆ w i . (99)The fixed point of these RG equations, corresponding to all bare w i being nonzero, i.e., the anisotropic membranefixed point, is given byˆ w ∗ i = 1 a i ( D, η ∗ ) , i = 1 , , (cid:18) ˆ w ∗ ˆ w ∗ ˆ w ∗ ˆ w ∗ (cid:19) = (cid:18) a ( D, η ∗ ) a ( D, η ∗ ) a ( D, η ∗ ) a ( D, η ∗ ) (cid:19) − , (100)where η ∗ is determined by (99) at the fixed point. Equivalence with the full SCSA equation (77) is then immediatelyfollows. Other fixed points
As discussed in the main text, there are a number of other subspaces which are preserved by renormalization withinthe SCSA method (hence also at large d c ). These can be labeled as S i ,...i n , with 1 ≤ i < i < · · · < i n ≤
5, wherethe only nonzero bare couplings w i are w i , . . . w i n . For those with w = 0, i.e., i , . . . , i n ∈ { , , , } , there arefour with n = 1, five with n = 2 (that is S , S , S , S , S ) together with S and S (note that w = 0 is notpreserved unless one has also w = 0 or w = 0). Then, one has S , S , S , S with w (cid:54) = 0. In each ofthese subspaces there is a fixed point denoted by P i ,...i n . It is obtained from (100) by setting to zero the ˆ w ∗ i not inthe set { i , . . . , i n } (disregarding their corresponding equation, except a which must be set to zero when ˆ w = 0).Their associated SCSA equation is obtained as d c = (cid:80) i =1 b i ( η, D ) ˆ w ∗ i . Let us give some examples.1. The fixed point P describes the isotropic flat phase. Its exponent η is determined by d c = b a + b a i.e., Eq. (19)in the main text, leading to the well known value η = 4 / (1 + √
15) = 0 . .. , ζ = (8 − √
15) = 0 . .. for the exponents describing out-of-plane fluctuations of the physical membrane D = 2 , d c = 1. The amplitudesare c = A/a and c = A/a which gives c i = { , , D +1 , , } , leading to lim q → ˜ λ ( q ) / ˜ µ ( q ) = − D +2 and tothe universal Poisson ratio, σ R = − / S has ˆ w ∗ = 0 and describes the case where µ = µ = 0. The exponent η is determined by(20) in the text, which for D = 2 gives d c η −
4) + 6 η − η + 1615 − η + 8 . (101)For d c = 1 one finds η = 0 . ζ = 0 . c i are then given by (83), where one sets c = 0. For D = 2, d c = 1 inserting the above value of η one finds c i = (cid:26) , , . , − . , . (cid:27) . (102)Note that the manifold ˆ w = 0 is however not preserved within the (cid:15) -expansion (see analysis in section below).6 Remark . One bonus of these RG equations, as compared to the original self-consistent equations, is that one candetermine the direction of the RG flow, the Hessian around each fixed point, and the various crossovers in the flow. Forinstance, to determine the Hessian around a fixed point ˆ w ∗ i , with associated exponent η ∗ , we need the variation of η .Variation of (99) around the fixed point gives δη = − (cid:80) j =1 b j δ ˆ w j (cid:80) k =1 b (cid:48) k ˆ w ∗ k , where we have denoted a i ≡ a i ( D, η ∗ ), b i ≡ b i ( D, η ∗ ), a (cid:48) i ≡ ∂ η b i ( D, η ) | η = η ∗ , b (cid:48) i ≡ ∂ η b i ( D, η ) | η = η ∗ . Using this, one can obtain the Hessian, and the flow around the fixedpoint. We defer this study to the future[57]. D. Renormalization group calculation for the h theory Here we present the details of the one-loop RG calculation for the quartic model h defined in Eq. (6) of the maintext. The power-counting is the same as in the standard φ O ( N ) model with quartic nonlinearities which are relevantfor D < D uc = 4. This allows us to control the RG analysis by an expansion in (cid:15) = 4 − D around D = 4 [52–54] Herewe will simply display the calculation using the momentum shell RG, i.e introducing a running UV cutoff Λ (cid:96) = Λ e − (cid:96) and integrating the internal momentum in the shell Λ (cid:96) e − d(cid:96) < q < Λ (cid:96) . However, we have checked all of our formulaalso using dimensional regularization for D < h vertex δR andthe correction to the bending rigidity δκ . Away from criticality one also needs to calculate the correction to γ . D 1. Correction to the quartic interaction
Having constructed the generic vertex R αβ,γδ ( q ), the analysis of the diagrams is then quite similar to that of the O ( N ) model[52–54]. There are three distinct channels contributing to the renormalization of R αβ,γδ ( q ), with onlyone of them taken into account in the large d c and SCSA analysis. The corrections to the quartic coupling can bewritten as the sum δR = δR (1) + δR (2) + δR (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 δR (1) αβ,γδ ( q ) = − T d c κ R αβ,γ (cid:48) δ (cid:48) ( q ) R γ (cid:48)(cid:48) δ (cid:48)(cid:48) ,γδ ( q ) (cid:90) Λ (cid:96) Λ (cid:96) e − d(cid:96) d D p (2 π ) D p γ (cid:48) ( q δ (cid:48) − p δ (cid:48) ) p γ (cid:48)(cid:48) ( q δ (cid:48)(cid:48) − p δ (cid:48)(cid:48) ) p | q − p | , (104) ≈ − T d c κ R αβ,γ (cid:48) δ (cid:48) ( q ) R γ (cid:48)(cid:48) δ (cid:48)(cid:48) ,γδ ( q ) (cid:90) Λ (cid:96) Λ (cid:96) e − d(cid:96) d D p (2 π ) D p γ (cid:48) p δ (cid:48) p γ (cid:48)(cid:48) p δ (cid:48)(cid:48) p (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 δR (2) αβ,γδ ( q , k ) = − Tκ sym R αβ,γ (cid:48) δ (cid:48) ( q ) (cid:90) Λ (cid:96) Λ (cid:96) e − d(cid:96) d D p (2 π ) D p γ (cid:48) ( q δ (cid:48) − p δ (cid:48) ) p γ (cid:48)(cid:48) ( q δ (cid:48)(cid:48) − p δ (cid:48)(cid:48) ) R γ (cid:48)(cid:48) γ,δ (cid:48)(cid:48) δ ( p − k ) p | q − p | , (106) ≈ − Tκ R αβ,γ (cid:48) δ (cid:48) ( q ) (cid:90) Λ (cid:96) Λ (cid:96) e − d(cid:96) d D p (2 π ) D p γ (cid:48) p δ (cid:48) p γ (cid:48)(cid:48) p δ (cid:48)(cid:48) R γ (cid:48)(cid:48) γ,δ (cid:48)(cid:48) δ ( p ) p (107)where sym denotes the symmetrization ( α, β ) ↔ ( γ, δ ). Finally, the contribution from the third (box) diagram is δR (3) αβ,γδ ( q , k ) = − Tκ (cid:90) Λ (cid:96) Λ (cid:96) e − d(cid:96) d D p (2 π ) D p γ (cid:48) ( q δ (cid:48) − p δ (cid:48) ) p γ (cid:48)(cid:48) ( q δ (cid:48)(cid:48) − p δ (cid:48)(cid:48) ) R αγ (cid:48) ,γδ (cid:48) ( k − p ) R γ (cid:48)(cid:48) β,δ (cid:48)(cid:48) δ ( k + p ) p | q − p | , (108) ≈ − Tκ (cid:90) Λ (cid:96) Λ (cid:96) e − d(cid:96) d D p (2 π ) D p γ (cid:48) p δ (cid:48) p γ (cid:48)(cid:48) p δ (cid:48)(cid:48) R αγ (cid:48) ,γδ (cid:48) ( p ) R γ (cid:48)(cid:48) β,δ (cid:48)(cid:48) δ ( p ) p . (109)where we recall that q = k + k . Note that one should symmetrize with the crossed diagram but at the level of thelast step exchanging γ (cid:48)(cid:48) and δ (cid:48)(cid:48) does not make a difference.To evaluate these integrals we now insert the decomposition R αβ,γδ ( q ) = (cid:80) i =1 w i ( W i ( q )) αβ,γδ and use the defi-nitions and the properties of the projectors summarized in Section A. We further use the formula for the angularaverages (denoted (cid:104) . . . (cid:105) , where ˆ k = k/ | k | ) (cid:104) ˆ k α ˆ k β (cid:105) = D δ αβ and (cid:104) ˆ k α ˆ k β ˆ k γ ˆ k δ (cid:105) = D ( D +2) ( δ αβ δ γδ + δ αγ δ βδ + δ αδ δ γβ )[15].7 FIG. 2. Feynman diagrams for one-loop corrections to the quartic vertex R αβ,γδ ( q ), with (a) ”vacuum polarization” δR (1) αβ,γδ ( q ),(b) ”vertex correction” δR (2) αβ,γδ ( q , k ), (c) ”box diagram” δR (3) αβ,γδ ( q , k ). Denoting δw = ( δw , δw , δw , δw , δw ) the one loop corrections to the couplings w i from the first diagram are δw = − d c D ( D + 2) (cid:18) w , w , ( D + 1) w + 2 √ D − w w + 3 w , w ( √ D − w + 3 w ) + w ( Dw + √ D − w + w ) , ( D + 1) w + 2 √ D − w w + 3 w (cid:19) × Tκ (cid:90) Λ (cid:96) Λ (cid:96) e − d(cid:96) d D p (2 π ) D p (110)The contribution of the second diagram reads δw = − D ( D + 2) (cid:18) w (2 w − w ) , w (2 w − w ) , w D − w + √ D − D + 1) w ) + w ( Dw + √ D − w + w ) ,w (( D − w + √ D − D + D − w + ( D − w ) + 2 w (( D − w + √ D − D + 4) w + ( D − w )4 √ D − ,w ( √ D − w + 3 w ) + w √ D − D + 1) w + ( D − w ) (cid:19) × Tκ (cid:90) Λ (cid:96) Λ (cid:96) e − d(cid:96) d D p (2 π ) D p (111)8The contribution of the third diagram reads δw = − D ( D + 2) (cid:18) (cid:0) D − (cid:1) w + 4 Dw w + 8 w , (cid:0) D − (cid:1) w + 4 Dw w + 8 w , ( D + D − w + 4( D + 1) w + 4 w w , √ D − w − w ) , ( D − w + 4( D − w w + 12 w (cid:19) Tκ (cid:90) Λ (cid:96) Λ (cid:96) e − d(cid:96) d D p (2 π ) D p where we have kept the explicit factors D in the geometric factors. D 2. Correction to the bending rigidity κ The correction to the self-energy to first order in perturbation theory O ( R ) can be read off from (69) as δσ ( k ) = 2 Tκ k α k δ (cid:90) q ( k β + q β )( k γ + q γ ) R αβ,γδ ( q ) 1 | k + q | (112)from which we will identify the corrections to κ and γ from the small external momentum k expansion δσ ( k ) = δγ k + δκ k + O ( k ) (113)The calculation of (112) proceeds by inserting again R αβ,γδ ( q ) = (cid:80) i =1 w i ( W i ( q )) αβ,γδ , performing the expansionat small k of the numerator, and the resulting contractions of indices. In the course of the calculation one needs theleading behavior near D = 4 and expansion in k of three integrals. One uses the expansion1 | k + q | = 1 q (1 − k · q q − k q + 12 ( q · k ) q + O ( k )) (114)The first integral is (cid:90) d D q (2 π ) D | q + k | q α = (cid:90) d D q (2 π ) D q α q (cid:18) − k · q q + O ( k ) (cid:19) = − k β δ αβ D (cid:90) d D q (2 π ) D q + O ( k ) (115)It can also be obtained by taking the ratio lim b → ,D → I α ( a =2 ,b ) I ( a =2 ,b ) = − p α using Eqs. A34 and A43 in [15].The second integral is (cid:90) d D q | q + k | q α q β q γ q = (cid:90) d D qq q α q β q γ q (cid:18) − k · q q + O ( k ) (cid:19) (116)= − D ( D + 2) ( δ αβ k γ + δ αγ k β + k α δ βγ ) (cid:90) d D q (2 π ) D q + O ( k ) (117)One can check that this is also the result from A34 and A51 in [15], i.e. lim b → ,D → ,D =2 a +2 b I αβγ ( a =2 ,b +1) I ( a =2 ,b ) , beingcareful to obey the constraint D = 2 a + 2 b when taking the limits.The third integral is (cid:90) d D q | q + k | q α q β = (cid:90) d D qq q α q β (cid:18) − k · q q − k q + 12 ( q · k ) q (cid:19) (118)= δ αβ D (cid:90) d D q (2 π ) D q − k D δ αβ (cid:90) d D q (2 π ) D q + 12 D ( D + 2) k γ k δ ( δ αβ δ γδ + δ αγ δ βδ + δ αδ δ βγ ) (cid:90) d D q (2 π ) D q (119)= δ αβ D (cid:90) d D q (2 π ) D q + (cid:18) D δ αβ ( − D + 2 ) k + 24 D ( D + 2) k α k β (cid:19) (cid:90) d D q (2 π ) D q (120)It can also be obtained from lim b → ,D → I αβ ( a =2 ,b ) I ( a =2 ,b ) = p α p β from A34 and A48 in [15].We finally obtain the corrections δγ and δκ as δγ = 2 Tκ (( D − w + 2 w )2 D (cid:90) d D q (2 π ) D q = | D → Tκ
18 (3 w + 2 w ) (cid:90) d D q (2 π ) D q (121)9 δκ = 2 Tκ − (cid:0) D + D − (cid:1) w + ( D − D + 1) w + Dw − √ D − w − Dw + w + 11 w D ( D + 2) (cid:90) d D q (2 π ) D q (122)= | D → Tκ (cid:16) w − w + 5 w − √ w + 3 w (cid:17) (cid:90) d D q (2 π ) D q (123)where we will calculate the remaining integral using momentum shell (cid:82) d D q (2 π ) D q → (cid:82) Λ (cid:96) Λ (cid:96) e − d(cid:96) d D q (2 π ) D q . The correction δγ (given by a UV divergent integral) obtained above is analogous to the usual non universal shift in the criticaltemperature for O ( N ) models, and of little interest to us since we will tune the bare γ so that the system is at itscritical point γ R = 0 (i.e. γ + δγ = 0). Said otherwise, the bare term in the model is ( γ − γ c )( ∇ h ) . D 3. Final RG equations
We now use that the integral (cid:82) Λ (cid:96) Λ (cid:96) e − d(cid:96) d D p (2 π ) D p = C D (cid:15) ( e (cid:15)d(cid:96) − − (cid:15)(cid:96) = C Λ − (cid:15)(cid:96) d(cid:96) + O ( (cid:15) ) with (cid:15) = 4 − D and C = π .We define the scaled dimensionless coupling ˜ w i = Tκ w i C Λ − (cid:15)(cid:96) (124)To derive the flow equation we calculate ∂ (cid:96) ˜ w i taking into account (i) the rescaling (ii) the sum of the three diagramswhich correct R (specifying D = 4) leading to δ ˜ w i = β i [ ˜ w ] d(cid:96) = (cid:80) j,k c ijk ˜ w j ˜ w k d(cid:96) (iii) the extra term from the correction δ ( κ − ) = − κ δκ which leads to the η function, η [ ˜ w ]. This leads to the RG equation ∂ (cid:96) ˜ w i = (cid:15) ˜ w i + β i [ ˜ w ] − η [ ˜ w ] ˜ w i , η [ ˜ w ] = ∂ (cid:96) κκ (125)where (123) leads to (from now on for notational convenience we will suppress the tilde on w ) η [ w ] = 112 (10 w − w + 5 w + 3 w − w ) (126)gives the η exponent at the fixed point. Putting all together, the final RG equations are (with w = √ w ) ∂ (cid:96) w = (cid:15)w + 112 (cid:0) − ( d c + 20) w + 2 (19 w − w − w + 6 w ) w − w − w − w w (cid:1) (127) ∂ (cid:96) w = (cid:15)w + 112 (cid:0) − ( d c − w − w w − w + 9 w − w ) w − w (cid:1) ∂ (cid:96) w = (cid:15)w + 124 (cid:18) − d c w − d c w − d c w w − w + 46 w w − w w − w − w − w w + 24 w w − w ( w + 8 w + w ) (cid:19) ∂ (cid:96) w = (cid:15)w + 124 (cid:18) − w (5( d c + 4) w + ( d c − w + 40 w ) − w (3( d c + 2) w + (3 d c + 28) w ) − w + ( − w − w + 56 w ) w − w (cid:19) ∂ (cid:96) w = (cid:15)w + 172 (cid:0) − d c + 12) w − w (( d c − w + 20 w − w + 10 w ) − (cid:0) d c w + 9 w + 6 w w (cid:1)(cid:1) Large d c limit . In the above RG equations (127) the couplings w i have not been rescaled by 1 /d c . If one rescalesthem, and then take the large d c limit one obtains ∂ (cid:96) w = (cid:15)w − w , ∂ (cid:96) w = (cid:15)w − w , ∂ (cid:96) w = (cid:15)w − (cid:0) w + 2 w w + w (cid:1) , (128) ∂ (cid:96) w = (cid:15)w −
124 (3 w ( w + w ) + w (5 w + w )) , ∂ (cid:96) w = (cid:15)w − (cid:0) w + 6 w w + 5 w (cid:1) . w i here is in fact the rescaled coupling ˜ w i given in (124). Hence comparing with (92) (the factor T beingomitted there) we see that we can identify w i ≡ π ˆ w i . Inserting into (128) we obtain a set of RG equations for theˆ w i which, as one can check using lim (cid:15) =4 − D → (cid:15)a i ( D,
0) = π { , , , √ , } and w = √ D − w , agree exactlywith the RG equations at large d c (93) for D = 4. Finally note that η [ w ] = O (1 /d c ) at large d c consistent with theSCSA and large d c expansion. D 4. Analysis of the RG equations
Instability of the isotropic membrane fixed point . The case of the rotationally invariant membrane isobtained setting w = w = w = 0, which is a manifold preserved by the RG. The RG flow (127) then reduces withinthis subspace ( w , w ) to ∂ (cid:96) w = (cid:15)w − w (( d c + 20) w + 10 w ) , (129) ∂ (cid:96) w = (cid:15)w − w (( d c + 4) w + 8 w ) . (130)We recall that in that subspace ( w , w ) are related to ( µ, λ ) via w = µ and w = µ + ( D − µλλ +2 µ as obtained from(45), and given in (8) in the text. Using that relation one can derive RG equations for µ and λ which can be checkedto be identical to the one in Ref. [6] (taking into account a difference by a factor of 4 in the definition of µ, λ there).There are four fixed points (cid:26) w → , w → (cid:15) d c + 4) (cid:27) , (cid:26) w → (cid:15)d c + 24 , w → (cid:15) d c + 24) (cid:27) , { w → , w → } , (cid:26) w → (cid:15)d c + 20 , w → (cid:27) , (131)which correspond to (in the same order [56])( µ, λ ) = (0 , (cid:15)
24 + d c , − (cid:15)
24 + d c ); (0 , (cid:15)d c ); ( 12 (cid:15)
20 + d c , − (cid:15)
20 + d c ) . (132)The second one is the standard fixed point which describes the isotropic flat membrane within the (cid:15) -expansion[6]. The third one describes the fixed connectivity fluid (zero shear modulus), that is a model for nematic elastomermembranes[55]. The fourth one is located on the line where the bulk modulus vanishes, i.e. 2 µ + Dλ = D − w w Dw − w = 0which separates the thermodynamically stable and unstable regions of parameters, and controls the transition betweenthese regions. The exponent η is given by η = η [ w ] = 512 (2 w + w ) = 5 µ ( λ + µ )2( λ + 2 µ ) (133)and gives η iso = (cid:15) d c for the isotropic membrane, as in [6].Let us now discuss the stability of the isotropic membrane to the non-rotationally invariant terms (due to anexternal orienting field (cid:126)E ) in the model. For this we calculate the eigenvalues and associated eigenvectors (representedas columns) of the Hessian around the isotropic fixed point, which are given by (cid:26) , − (cid:15)d c d c + 24 , (cid:15)d c d c + 24 , (cid:15)d c d c + 24 , − (cid:15) (cid:27) (134) d c +24 − − d c +785( d c +12) − d c − d c +12) 52 ( − d c − − d c − d c +8)5( d c +24) − d c − d c +12) − d c − d c +12)
11 0 0 1 00 0 1 0 0 . (135)The second and last columns are the two stable directions which are also obtained if one diagonalises the flow insidethe isotropic subspace. In the full space of five couplings however, we see that the isotropic fixed point is unstable intwo directions, with eigenvalue ρ = (cid:15)d c d c +24 , and marginal in a third direction.1 Crossover for small anisotropy . To discuss the effect of a small anisotropy let us first recall the analysis of thelength scales in the isotropic membrane. The dimensionless couplings ˜ w , ˜ w (we temporarily restore the tilde) atscale L are of order ˜ w , ∼ T K κ L − D , L < L anh ∼ ( κ T K ) / (4 − D ) , (136)˜ w , (cid:39) ˜ w ∗ , ∼ T K ( L ) κ ( L ) L − D , L > L anh , (137)where L anh is the length scale below which the harmonic theory holds (and the elastic moduli and bending rigid-ity equal their bare values). For L > L anh these are corrected and one has κ ( L ) ∼ κ ( L/L anh ) η and K ( L ) ∼ K ( L/L anh ) − (4 − D − η ) . The length L anh is itself determined when ˜ w , reach numbers of order unity, of order theirvalue at the fixed point.Consider now the model in presence of very small bare symmetry breaking couplings µ , µ , λ , λ assumed to be ofthe same order. Then, from (45) the bare w , w , w are linear combinations of those, hence small and of the sameorder. These couplings are relevant and grow as˜ w i ∼ T w i κ L − D , L < L anh , (138)˜ w i ∼ T w i κ L − D anh ( LL anh ) ρ , L > L anh , (139)where w i denote any linear combination of the bare symmetry breaking couplings ( i = 2 , ,
5) and ρ was calculatedabove in the (cid:15) expansion. The length scale L anis beyond which anisotropy will change the property of the system isobtained when ˜ w i becomes of order unity, hence L anis ∼ L anh ( K w i ) /ρ , ρ = (cid:15)d c d c + 24 + O ( (cid:15) ) , (140)whenever w i ∼ µ , , λ , (cid:28) K . 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 unstabledirections, the others with even more. Hence around D = 4 there is no perturbative fixed point and we have arunaway RG flow.We find that an attractive fixed point exists only for high enough d c . The situation is very similar to the one forthe crumpling transition, with d replaced by d c . For instance, for d c = 220 we find one, and only one, fully attractivefixed point w i = { . , . , . , − . , . } , (141)with eigenvalues − ., − . , − . , − . , − . w = w = µ , w = 12 ( D − λ + µ , w = 12 √ D − λ , w = 12 λ + µ , (142)with µ = 0 . λ = − . R αβ,γδ ( q ) = µ δ αγ δ βδ + δ αδ δ βγ ) + λ δ αβ δ γδ . (143)One can check by inserting (142) into (127) that this manifold is preserved by the RG. Furthermore, inside thismanifold one can check inserting (142) into (126) that η [ w ] = 0 to the order O ( (cid:15) ), and that the RG flow can bewritten as ∂ (cid:96) µ = (cid:15)µ + 112 (cid:0) − ( d c + 21) µ − λ − λ µ (cid:1) , (144) ∂ (cid:96) λ = (cid:15)λ + 112 (cid:0) − (6 d c + 7) λ − d c + 17) λ µ − ( d c + 15) µ (cid:1) . (145)2Defining u = µ and v = λ/ µ/ u, v ) setting there K = 1 /
4. Hence they are identical to those of the crumpling transition but with d → d c .From [45] we know that this fixed point exists only for d > D = 4 − (cid:15) expansion is the one found within the SCSA (and large d c ) expansion described in the Section D4. While in the RGit disappears near D = 4 for d c < D = 2 and d c = 1.Hence while the RG suggests a fluctuation driven first order transition in the physical dimension, the SCSA suggestsa continuous transition. The question of which is the most accurate description is beyond the scope of the presentwork and would presumably require numerical simulations, as was the case for the crumpling transition (see e.g. [23]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 thephonons. It allows to obtain some extra information (the renormalization of µ ) and provides a useful check on theRG flow of the previous Section. We can rewrite the model as F [ u,(cid:126)h ] = (cid:90) d D x
12 ( ∇ h ) + 12 ( G u ) − αβ u α u β + u α C µ + µ ,λ + λ α,γδ A γδ + 12 C µ + µ ,λ + λ αβ,γδ A αβ A γδ , C µ,λα,γδ = − ∂ β C µ,λαβ,γδ , (146)where we have defined, in Fourier space, the uhh vertex C α,γδ ( q ) = C µ + µ ,λ + λ α,γδ ( q ) = − i (( λ + λ ) q α δ γδ + ( µ + µ )( δ αδ q γ + δ αγ q δ )) , (147)and the bare phonon propagator ( G u ) − αβ ( q ) = µP Tαβ ( q ) + ( λ + 2 µ ) P Lαβ ( q ) . (148)Here we calculate the corrections to the vertices, hence we evaluate to lowest order in the perturbation theory inthe nonlinearities, the vertices of the effective action Γ uu , Γ uhh , Γ hhhh . These vertices will give us the correctionsrespectively to ( µ, λ ), ( µ , λ ) and ( µ , λ ). The corresponding diagrams are shown in the Fig.3. FIG. 3. Feynman diagrams for one-loop corrections to the renormalized vertices in the u − h description of the critical bucklingmembrane. E. 1 Calculation of Γ uu Let us calculate the one loop corrections to the phonon propagator, given by the single diagram in Fig.3. Theeffective action for the u term is given, to one loop, as12 (cid:90) q u ( q ) · Γ uu ( q ) · u ( − q ) = (cid:90) q u α ( q ) u β ( − q )[ G − αβ ( q ) − C µ + µ ,λ + λ α,γδ ( q ) C µ + µ ,λ + λ α (cid:48) ,γ (cid:48) δ (cid:48) ( − q ) (cid:104) A γδ ( q ) A γ (cid:48) δ (cid:48) ( − q ) (cid:105) ] , (149)where here and below · denotes index summations. We have the following average, performed with the quadraticaction (cid:104) A γδ ( q ) A γ (cid:48) δ (cid:48) ( − q ) (cid:105) = 12 d c Π γδ,γ (cid:48) δ (cid:48) ( q ) , Π αβ,γδ ( q ) = sym (cid:90) p p α ( q β − p β ) p δ ( q γ − p γ ) G ( p ) G ( q − p ) , (150)which leads to (Γ uu ) αβ ( q ) = G − αβ ( q ) − d c C µ + µ ,λ + λ α,γδ ( q ) C µ + µ ,λ + λ α (cid:48) ,γ (cid:48) δ (cid:48) ( − q )Π γδ,γ (cid:48) δ (cid:48) ( q ) . (151)Within the Wilson RG and to leading order in (cid:15) one hasΠ αβ,γδ ( q ) (cid:39) κ (cid:90) p p α p β p δ p γ p = 1 κ S αβ,γδ (cid:90) p p , (152)i.e., the dependence in the external momentum q (cid:28) p is subdominant, where p is the internal momentum in the loop.We have defined S (4) αβ,γδ = S αβ,γδ = 1 D ( D + 2) ( δ αβ δ γδ + δ αγ δ βδ + δ αδ δ βγ ) . (153)Hence we obtain Γ uu ( q ) = G − ( q ) − d c κ C ( q ) · S · ( C ( − q )) T (cid:90) p p . (154)Replacing (cid:82) p p → C Λ − (cid:15)(cid:96) and performing the contractions, one obtains the following corrections to µ and λδµ = − d c κ ( µ + µ ) C Λ − (cid:15)(cid:96) d(cid:96) , (155) δλ = − d c κ [12( λ + λ ) + 2( µ + µ ) + 12( λ + λ )( µ + µ )] C Λ − (cid:15)(cid:96) d(cid:96) . Exponent η u . The first equation can be rewritten to obtain the anomalous dimension of the phonon field, i.e theexponent η u defined by µ ( L ) ∼ L − η u , η u = − δµµd(cid:96) = d c g µ = d c
12 ( ˜ w − ˜ w ) , (156)where we have defined the proper dimensionless coupling g µ = ( µ + µ ) µκ C Λ − (cid:15)(cid:96) which we related to the ˜ w i using (45)and (124). At the fixed point this gives the exponent η u :- at the isotropic membrane fixed point ˜ w = 0 and ˜ w = (cid:15)d c +24 , leading to η u = d c (cid:15)d c +24 . Since η = (cid:15)d c +24 we check,to first order in (cid:15) the exact relation (to all orders), η u = (cid:15) − η guaranteed by rotational invariance [6].- at the anisotropic membrane fixed point ˜ w = ˜ w hence η u = 0 to order O ( (cid:15) ). The relation η u = (cid:15) − η does nothold (since η = 0 there, to O ( (cid:15) )).Note that one can also define the screening exponent η w for the coupling constants w i such that w i ( L ) ∼ L − η w . Itis given by the graphical corrections β i = δw i w i . Since the RG equation for the scaled dimensionless coupling ˜ w i reads ∂ (cid:96) ˆ w i = ( (cid:15) − η ( ˆ w ) − β i [ ˜ w ]) ˜ w i , at any fixed point one must have β i = (cid:15) − η . In presence of anisotropy, β i becomesdifferent from η u . The nonlinear interactions are still screened, since η < (cid:15)/ µ and λ .4 E 2. Calculation of Γ uhh We now calculate the vertex corrections given by the three diagrams in Fig.3. They are corrections to the term uhh in (146), which we write in the form (cid:90) q u α ( − q ) δV α,βγ ( q ) A βγ ( q ) . (157)One obtains for the first diagram δV (1) α,βγ ( q ) = − d c C ( q ) · S · C ) α,βγ κ (cid:90) p p , (158)where C ( q ) is the three index tensor given in (147) (i.e., the bare uhh vertex) and we denote here and below C thefour index tensor C µ + µ ,λ + λ αβ,γδ (entering the bare h vertex) defined in (35). The second diagram gives the correction δV (2) α,βγ ( q ) = − C α,α (cid:48) α (cid:48)(cid:48) ( q ) S α (cid:48) β (cid:48) α (cid:48)(cid:48) γ (cid:48) C β (cid:48) β,γ (cid:48) γ κ (cid:90) p p . (159)Finally, the third diagram gives, using that C β,γδ ( q ) = − iq α C αβ,γδ (where the momentum independent four indextensor C µ + µ ,λ + λ αβ,γδ is denoted C ) δV (3) α,βγ ( q ) = C α,β (cid:48) γ (cid:48) ( q ) (cid:20) S (8) β (cid:48) γ (cid:48) β (cid:48)(cid:48) γ (cid:48)(cid:48) s (cid:48) s (cid:48)(cid:48) α (cid:48) α (cid:48)(cid:48) (cid:18) µ + λ − µ (cid:19) + 1 µ S (6) β (cid:48) γ (cid:48) β (cid:48)(cid:48) γ (cid:48)(cid:48) s (cid:48) s (cid:48)(cid:48) δ α (cid:48) α (cid:48)(cid:48) (cid:21) C α (cid:48) s (cid:48) ,β (cid:48)(cid:48) β C α (cid:48)(cid:48) s (cid:48)(cid:48) ,γ (cid:48)(cid:48) γ κ (cid:90) p p , (160)where we defined the 6 and 8 index symmetric tensors, schematically, S (6) β (cid:48) γ (cid:48) β (cid:48)(cid:48) γ (cid:48)(cid:48) s (cid:48) s (cid:48)(cid:48) = 1 D ( D + 2)( D + 4) ( δδδ + 14 terms) , (161) S (8) β (cid:48) γ (cid:48) β (cid:48)(cid:48) γ (cid:48)(cid:48) s (cid:48) s (cid:48)(cid:48) α (cid:48) α (cid:48)(cid:48) = 1 D (2 + D )(4 + D )(6 + D ) ( δδδδ + 104 terms) . (162)Performing the contractions we obtain for i = 1 , , δV ( i ) α,βγ ( q ) = [ A i q α δ βγ + B i ( q β δ αγ + q γ δ αβ )] 1 κ (cid:90) p p . (163)To display the results more compactly we define the new variablesΛ i = λ i + λ , M i = µ i + µ , i = 1 , . (164)In terms of these variables the coefficients A i , B i read A = − d c
12 (3Λ (2Λ + M ) + M (3Λ + M )) , B = − d c M M , (165) A = 112 ( − (Λ + 5 M ) − M (Λ + 7 M )) , B = − M (Λ + M ) ,A = 3Λ µ + Λ M (9 λ + 34 µ ) + M (5 λ + 14 µ ) + 13Λ µM µ ( λ + 2 µ ) , B = M (cid:0) Λ µ + M ( − ( λ − µ )) + 4Λ µM (cid:1) µ ( λ + 2 µ ) . From these coefficients we directly obtain the corrections δ Λ = ( A + A + A ) 1 κ (cid:90) p p , δM = ( B + B + B ) 1 κ (cid:90) p p . (166)Putting all contributions together and replacing (cid:82) p p → C Λ − (cid:15)(cid:96) , we obtain, from the vertex corrections δM = 112 M (cid:18) M ( − ( d c + 1)) − Λ + Λ µ + M ( − ( λ − µ )) + 4Λ µM µ ( λ + 2 µ ) (cid:19) κ C Λ − (cid:15)(cid:96) d(cid:96) , (167) δ Λ = 112 (cid:18) − d c (3Λ (2Λ + M ) + M (3Λ + M ))+ 3Λ µ + Λ M (9 λ + 34 µ ) + M (5 λ + 14 µ ) + 13Λ µM µ ( λ + 2 µ ) − (Λ + 5 M ) − M (Λ + 7 M ) (cid:19) κ C Λ − (cid:15)(cid:96) d(cid:96) . i = λ and M i = µ and the above corrections reduce to δµ = − d c κ µ C Λ − (cid:15)(cid:96) d(cid:96) , (168) δλ = − d c κ (cid:0) λ + 6 λµ + µ (cid:1) C Λ − (cid:15)(cid:96) d(cid:96) . This simplification occurs because the second and third diagram exactly cancel due to rotational invariance. Indeedthe corrections (168) coincide with (155) (setting µ = λ = 0 there). E. 3 Calculation of Γ hhhh We now calculate the corrections to the h vertex in (146). They are given by the six diagrams in Fig.3. We recallthat we denote C the four index tensor C µ + µ ,λ + λ αβ,γδ which appears in the bare h vertex.The first diagram gives the following correction to C δC = − d c C · S · C κ (cid:90) p p . (169)The second and third diagram give respectively δC αβ,γδ = − C αα (cid:48) ,γγ (cid:48) C ββ (cid:48) ,δδ (cid:48) S α (cid:48) β (cid:48) γ (cid:48) δ (cid:48) κ (cid:90) p p , δC αβ,γδ = − C αβ,α (cid:48) β (cid:48) S α (cid:48) β (cid:48) γ (cid:48) δ (cid:48) C γγ (cid:48) ,δδ (cid:48) κ (cid:90) p p . (170)The fourth diagram is more complicated δC αβ,γδ = − C r s ,αα (cid:48) C r s ,γγ (cid:48) C r s ,ββ (cid:48) C r s ,δδ (cid:48) (cid:20) (cid:104) ˆ p α (cid:48) ˆ p β (cid:48) ˆ p γ (cid:48) ˆ p δ (cid:48) ˆ p s ˆ p s ˆ p s ˆ p s ˆ p r ˆ p r ˆ p r ˆ p r (cid:105) (cid:18) λ + 2 µ − µ (cid:19) (171)+ ( (cid:104) ˆ p α (cid:48) ˆ p β (cid:48) ˆ p γ (cid:48) ˆ p δ (cid:48) ˆ p s ˆ p s ˆ p s ˆ p s ˆ p r ˆ p r (cid:105) δ r r + (cid:104) ˆ p α (cid:48) ˆ p β (cid:48) ˆ p γ (cid:48) ˆ p δ (cid:48) ˆ p s ˆ p s ˆ p s ˆ p s ˆ p r ˆ p r (cid:105) δ r r ) 1 µ (cid:18) λ + 2 µ − µ (cid:19) + 1 µ S (8) α (cid:48) β (cid:48) γ (cid:48) δ (cid:48) s s s s δ r r δ r r (cid:21) κ (cid:90) q q , where (cid:104) . . . (cid:105) denote angular averages and ˆ p = p /p a unit vector. It was convenient to use that notational trick, ratherthan the symmetric tensors of order 10 and 12, as it allows the contractions to be taken more easily. This is equal to δC αβ,γδ = (cid:18) − C r s ,αα (cid:48) C r s ,γγ (cid:48) C r s ,ββ (cid:48) C r s ,δδ (cid:48) µ S (8) α (cid:48) β (cid:48) γ (cid:48) δ (cid:48) s s s s δ r r δ r r (172) − (cid:18) λ + 2 µ − µ (cid:19) (Λ + 2 M ) S αβγδ − (cid:18) λ + 2 µ − µ (cid:19) µ (Λ + 2 M ) [2((Λ + 2 M ) − M ) S αβγδ + M D ( δ αγ δ βδ + δ αδ δ βγ )] (cid:19) κ (cid:90) p p . The fifth diagram leads to the correction δC αβ,γδ = sym 4 C r s ,αα (cid:48) C r s ,γγ (cid:48) C β (cid:48) β,δ (cid:48) δ (cid:20) (cid:18) λ + 2 µ − µ (cid:19) S (8) α (cid:48) β (cid:48) γ (cid:48) δ (cid:48) s s r r + 1 µ S α (cid:48) β (cid:48) γ (cid:48) δ (cid:48) s s δ r r (cid:21) κ (cid:90) q q , (173)and finally, the sixth diagram, to δC αβ,γδ = sym 2 C αβ,α (cid:48) β (cid:48) C r s ,δδ (cid:48) C r s ,γγ (cid:48) (cid:20) (cid:18) λ + 2 µ − µ (cid:19) S (8) α (cid:48) β (cid:48) γ (cid:48) δ (cid:48) s s r r + 1 µ S α (cid:48) β (cid:48) γ (cid:48) δ (cid:48) s s δ r r (cid:21) κ (cid:90) q q . (174)Performing the contractions, in total we find for the corrections to the h vertex δM = 112 (cid:20) − µ ( λ + 2 µ ) (cid:18) µ M ( d c + 21) ( λ + 2 µ ) + Λ µ + M (cid:0) λ + 44 λµ + 76 µ (cid:1) (175)+8Λ µM (cid:0) M ( λ + 4 µ ) − µM ( λ + 2 µ ) (cid:1) + 2Λ µ (cid:0) M ( λ + 8 µ ) − µM ( λ + 2 µ ) (cid:1) κ C Λ − (cid:15)(cid:96) − µM M ( λ + 2 µ )( λ + 4 µ ) + 8Λ µ M (cid:19) − Λ + 2Λ (cid:18) Λ µ + 2 M ( λ + 4 µ ) + 4Λ µM µ ( λ + 2 µ ) − M (cid:19) (cid:21) κ C Λ − (cid:15)(cid:96) d(cid:96) , δ Λ = 112 (cid:20) ] − Λ (6 d c + 7) − M (3 d c + 17) + M ( − ( d c + 15)) − (cid:0) Λ µ + M ( − ( λ − µ )) + 4Λ µM (cid:1) µ ( λ + 2 µ ) + 4 M (cid:0) Λ µ + 2 M ( λ + 4 µ ) + 4Λ µM (cid:1) µ ( λ + 2 µ ) + 8Λ (cid:0) Λ µ + 2 M ( λ + 4 µ ) + 4Λ µM (cid:1) µ ( λ + 2 µ ) (cid:21) κ C Λ − (cid:15)(cid:96) d(cid:96) . (176)To recover the result for the isotropic membrane one sets Λ i = λ and M i = µ and the above corrections reduce exactly,once again, to (168). Here the simplification arises from the last five diagram cancelling due to rotational invariance. E. 4 Final RG equations
We can now put together δµ , δλ from (155), δM = δµ + δµ , δ Λ = δλ + δλ from (167), and δM = δµ + δµ , δ Λ = δλ + δλ from (175). This leads to the complete set of corrections to the six couplings, which is bulky andwhich we will not display here in full (see below). Let us denote m i , i = 1 , . . . , δm i = d ijk m j m k . To obtain the final RG flow one defines scaled dimensionless couplings ˜ m i , as in(124), and take into account the corrections to κ as we did in (125), leading to ∂ (cid:96) ˜ m i = (cid:15) ˜ m i + d ijk ˜ m j ˜ m k − η [ w [ m ]] ˜ 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 η [ w ] function.To check that these are consistent with the RG equations obtained via the quartic theory in Section D, we simplyneed to compare the above corrections δm and δw , i.e summing (110), (111) and (112) which can be written as δw i = β i [ w ] d(cid:96) = c ijk w j w k d(cid:96) . We have performed the check as follows. We have evaluated in two ways δw i [ m ] = ∂w i [ m ] ∂m j δm j = ∂w i [ m ] ∂m j d ijk m j m k d(cid:96) , (177) δw i [ m ] = c ijk w j [ m ] w k [ m ] d(cid:96) , (178)and using w [ m ] from Eq. (45) we have shown using Mathematica that the two lines above are identical functions ofthe m i . This provides a quite non trivial check of these two lengthy calculations. Hence the RG equation for the 5couplings w i can be deduced from the one for the 6 couplings m i . The reverse is not true however, there is, in thegeneral case, additional information in the 6 coupling flow, as we discussed above in Section E. 1 it allows to obtain δµ and from it we obtained there the exponent η u , related to the anomalous dimension of the phonon field. Let usindicate for completeness the combination of couplings which enters the exponent ηη = 112 (10 w − w + 5 w + 3 w − w ) = M (2Λ µ + 3 λM + 5 µM )2 µ ( λ + 2 µ ) . (179)To express the RG flow it is natural to define the dimensionless ratio r = λ/µ and the four dimensionless couplingconstants associated to the nonlinear terms in the action˜ M = M µκ C Λ − (cid:15)(cid:96) , ˜Λ = Λ µκ C Λ − (cid:15)(cid:96) , ˜ M = M κ C Λ − (cid:15)(cid:96) , ˜Λ = Λ κ C Λ − (cid:15)(cid:96) , (180)and then µ can still flow with eigenvalue η u . Since the RG equations for these couplings are bulky let us only displaythem here to leading order in large d c , and we have dropped the tilde for notational convenience ∂ (cid:96) r = 112 d c (cid:0) − + rM − M − M (cid:1) , ∂ (cid:96) M = (cid:15) M + 124 M (cid:0) M − M (cid:1) d c , (181) ∂ (cid:96) Λ = (cid:15) + 124 d c (cid:0) − Λ + Λ M − M − M − M M (cid:1) , (182) ∂ (cid:96) M = (cid:15)M − M d c , ∂ (cid:96) Λ = (cid:15) Λ + 112 d c (cid:0) − − M − M (cid:1) . (183)It is easy to see that the only attractive fixed point of these equations (and of the complete equations for any d c > M = 0 , Λ = 0 , M = 12 d c + O ( 1 d c ) , Λ = − d c + O ( 1 d c ) . (184)7This is in agreement with the RG analysis using the h theory presented above. Indeed this anisotropic fixed point liesin the manifold (142) in the w i variables, which in the current variables imply the constraints µ + µ = 0, µ = µ + µ , λ = λ + λ − ( λ + λ ) λ +2 µ . The fixed point (184) obeys these constraints and one can check that the values for M andΛ are consistent with those for the fixed point of (144) at large d c (and in fact, as one can check, for any d c > M and Λ flow to zero exponentially with (cid:96) , at the anisotropic fixed point we see that the flowof r = λ/µ and the flow of µ , which is given (exactly) by1 µ ∂ (cid:96) µ = − M d c − M (2Λ + (3 r + 5) M ) r + 2 (185)lead to finite, but non-universal values for λ and µ . This is consistent with the exponent η u = 0 as claimed above. F. Renormalization group flow of γ Until now we have assumed γ (and τ ) to be tuned so that the system is at the critical point (the buckling transition),i.e. γ R = 0. Now we assume a small deviations away and calculate the RG flow of γ , and the associated (independent)critical exponent ν . To check consistency, we perform the calculation both in the h theory and in the uh + h theory. F.1. Flow of γ in quartic h theory To obtain the flow of γ to linear order in γ , we expand the height field propagator at small γ as G ( k ) = 1 κk + γk = 1 κk − γκ k + O ( γ ) . (186)Let us call here δσ ( k ) = δγk + O ( k ) the part of the self-energy proportional to O ( γ ) at small γ (there is also a O (1)part calculated in Section D.2 which determines the shift in the critical point γ c (see discussion there) but which isof no interest to us here. To lowest order in perturbation theory the self-energy is given by two diagrams, the sunsetdiagram in (112), leading to δγ s , and the tadpole diagram δγ t , with δγ = δγ s + δγ t . From the sunset diagram onehas from (112) δσ s ( k ) = − γκ k α k γ d c (cid:88) i =1 , w i (cid:90) q k − q ) ( k β − q β )( W i ) αβ,γδ ( q )( k δ − q δ ) = δγ s k + O ( k ) . (187)Within Wilson RG, to lowest order in (cid:15) one can write δσ s ( k ) = − γκ k α k γ d c (cid:88) i =1 , w i (cid:90) q q β q δ q ( W i ) αβ,γδ ( q ) = − γκ k d c (cid:20)(cid:18) − D (cid:19) w + 1 D w (cid:21) (cid:90) q q . (188)In addition there is the tadpole contribution, leading to the O ( γ ) correction δγ t σ t ( k ) = k α k γ R αβ,γδ (cid:90) q q γ q δ G ( q ) ⇒ δγ t k = − γk α k γ R αβ,γδ (cid:104) q γ q δ (cid:105) κ (cid:90) q q , (189)where R is the k = 0 component of the vertex. From (59) it is equal to R = ¯ C where ¯ C given in (60), and moreexplicitly, from (61) R αβ,γδ = 12 (cid:18) M − M µ (cid:19) ( δ αγ δ βδ + δ αδ δ βγ ) + Dλ Λ µ − D Λ µ + 2Λ µ + 2 λM − µM µ ( Dλ + 2 µ ) δ αβ δ γδ . (190)Using (cid:104) q γ q δ (cid:105) = D δ γδ , performing the contractions, one finds, for D = 4 δγ t = − γ (cid:18) + M − (2Λ + M ) (2 λ + µ ) (cid:19) κ (cid:90) q q . (191)We can express the following combination using the w i + M − (2Λ + M ) (2 λ + µ ) = 3 w (3 w + w + 2 w ) + 4 (cid:0) w − w w (cid:1) w − w − w + 6 w . (192)8Hence we obtain the flow for γ in terms of the rescaled couplings defined in (124), dropping the tilde ( ˜ w i → w i ) forsimplicity ∂ (cid:96) γ = − γ (cid:20) d c (cid:18) w + 12 w (cid:19) + 14 (cid:32) w (3 w + w + 2 w ) + 4 (cid:0) w − w w (cid:1) w − w − w + 6 w (cid:33) (cid:21) . (193)One can immediately check that for the isotropic membrane the right hand side vanishes exactly. This arises fromrotational invariance, there are no corrections to γ . Here the bare γ is tuned to the critical point γ c and the flowequation (193) is, more properly, the RG equation for the deviations to criticality γ → γ − γ c .If one now inserts the values for the couplings at the anisotropic fixed point, or more generally of any couplingssatisfying the constraints (142), one finds that the ratio appearing in (193) is of the form 0 divided by 0, i.e. it isundetermined. We resolve this ambiguity in the next section by studying the u − h theory. To this end we study thecorrelation length exponent related to the eigenvalue of γ . Correlation length exponent ν From the propagator (186) the bare correlation length is ξ = (cid:112) κ/γ . Let us write (193) as ∂ (cid:96) γ = θγ . At the fixedpoint γ ( L ) = γ L θ , where γ is the bare value. The correlation length ξ is defined by balancing κ ( ξ ) ξ − ∼ γ ( ξ ) ξ − .Taking into account that κ ( ξ ) ∼ ξ η , we obtain ξ = γ − ν , ν = 12 + θ − η . (194) F.2. Flow of γ in quartic h, u theory We now calculate the corrections to γ within the model described in (36),(35), and also in (146), whose RG wasstudied in Section E. The nonlinear terms are12 C αβγδ ∂ α u β ∂ γ (cid:126)h · ∂ δ (cid:126)h + 18 C αβγδ ( ∂ α (cid:126)h · ∂ β (cid:126)h )( ∂ γ (cid:126)h · ∂ δ (cid:126)h ) , where we recall that C , αβγδ = Λ , δ αβ δ γδ + M , ( δ αβ δ γδ + δ αβ δ γδ ) (195)in terms of the coupling defined in (164). In Fourier space, we recall that the propagator of the phonon field u α isgiven by (37) and the propagator of the height field h field by (186).The contribution to δγ = δγ u + δγ h is given by (i) two sunset diagrams, giving δγ s u and δγ s h : they correspondrespectively to expansion to second order in the cubic phonon vertex and to first order expansion in the quartic vertex(ii) two tadpole diagrams δγ t u and δγ t h .The ”sunset” diagram involving phonons gives the following correction, evaluated to lowest order in (cid:15)δγ s u = − (cid:18) (cid:19) ˆ k γ ˆ k γ (cid:48) C αβγδ C α (cid:48) β (cid:48) γ (cid:48) δ (cid:48) (cid:90) > q ( k − q ) δ ( k − q ) δ (cid:48) q α q α (cid:48) [ κ ( k − q ) + γ ( k − q ) ] q (cid:34) P Tββ (cid:48) ( q ) µ + P Lββ (cid:48) ( q )2 µ + λ (cid:35) , = − ˆ k γ ˆ k γ (cid:48) C αβγδ C α (cid:48) β (cid:48) γ (cid:48) δ (cid:48) (cid:90) > q q δ q δ (cid:48) q α q α (cid:48) [ κq + γ ] q (cid:34) P Tββ (cid:48) ( q ) µ + P Lββ (cid:48) ( q )2 µ + λ (cid:35) , = − ˆ k γ ˆ k γ (cid:48) C αβγδ C α (cid:48) β (cid:48) γ (cid:48) δ (cid:48) (cid:20) Λ C d(cid:96)κµ (cid:18) δ ββ (cid:48) (cid:104) q δ q δ (cid:48) q α q α (cid:48) (cid:105) − µ + λ µ + λ (cid:104) q δ q δ (cid:48) q α q α (cid:48) q β q β (cid:48) (cid:105) (cid:19) − γκ µ (cid:18) δ ββ (cid:48) (cid:104) q δ q δ (cid:48) q α q α (cid:48) (cid:105) − µ + λ µ + λ (cid:104) q δ q δ (cid:48) q α q α (cid:48) q β q β (cid:48) (cid:105) (cid:19)(cid:21) C Λ − (cid:15)(cid:96) d(cid:96). (196)Using Mathematica and our spherical averages of product of q α ’s, we find, δγ s u = − (cid:18) Λ κ − γκ (cid:19) µ (Λ + 4Λ M + 10 M ) + 3 λM µ (2 µ + λ ) C Λ − (cid:15)(cid:96) d(cid:96). (197)9The total correction δγ s involving the ( ∂h ) vertex is given by the sum of the sunset and tadpole diagram as δγ h = δγ s h + δγ t h = 2 × k α ˆ k γ C αβγδ (cid:90) > q q β q δ κq + γq + 2 × d c k α ˆ k β C αβγδ (cid:90) > q q γ q δ κq + γq , = 14 C αβγδ (cid:18) D δ βδ ˆ k α ˆ k γ + 2 d c D δ γδ ˆ k α ˆ k β (cid:19) (cid:90) > q κq + γ . In D = 4 we find, δγ h = 14 (cid:18) Λ κ − γκ (cid:19) [Λ + 5 M + d c (2Λ + M )] C Λ − (cid:15)(cid:96) d(cid:96). (198)We need to calculate the tadpole diagram involving the phonons. It arises from the term at zero momentum A αβ C αβ,γδ (cid:104) ˜ u γδ (cid:105) in the energy (56). The expectation value (cid:104) ˜ u γδ (cid:105) of the in-plane strain field is given in (57) as (cid:104) ˜ u (cid:105) = − [ C µ,λ ] − C (cid:104) A (cid:105) . Hence we find δγ t u k = γ d c k α k β [ C · [ C µ,λ ] − · C ] αβ,γδ (cid:104) q γ q δ (cid:105) κ (cid:90) q q (199)leading to δγ t u = γ + M ) λ + µ κ (cid:90) q q . (200)Putting all four contributions together we obtain the O ( γ ) total correction as δγ = γ κ (cid:20) µ (Λ + 4Λ M + 10 M ) + 3 λM µ (2 µ + λ ) − (Λ + 5 M ) + d c (cid:18) (2Λ + M ) λ + µ − (2Λ + M ) (cid:19) (cid:21) C Λ − (cid:15)(cid:96) d(cid:96) , (201)which leads to the RG flow equation by defining the dimensionless scaled couplings. One can check using (48), (49)and (50) that the RG flow obtained here is formally identical to the one obtained in (193).However, now one can check that the indeterminacy mentioned in the previous section is resolved. Indeed in theexpression (193) there is a factor M both in numerator and denominator, and since M = 0 at the anisotropic fixedpoint this led to an ambiguous expression. However, above these factors cancel and the exponent θ at the fixed pointcan be unambiguously determined from (201). One finds, setting M = Λ = 0 θ = −
14 ( d c (2Λ + M ) + Λ + 5 M ) . (202)We can insert M = (cid:15)d c (12 − d c + O ( d c )) and Λ = (cid:15)d c ( − − d c + O ( d c )), which can be obtained from the RG inthe previous section, and obtain θ = − (cid:15)d c (1 − d c + O ( 1 d c )) . (203) G. EFFECT OF THE PARAMETER τ As indicated in the text, the parameter τ simply changes ζ , such that ζ is the ratio of the projected area of themembrane on its preferred plane (here xy ) to its internal size L . To see this, we rewrite the energy density in F interms of trace and traceless parts of the nonlinear stress tensor µ ( u αβ − D δ αβ u γγ ) + Bu αα + τ u αα , (204)where B = µ + Dλ D . Completing the square and defining u α = ˜ u α − BD τ x α , the energy density becomes µ (˜ u αβ − D δ αβ ˜ u γγ ) + B (˜ u αα ) − τ B . (205)0Here ˜ u α is the ”centered” phonon field and ˜ u αβ = ( ∂ α u β + ∂ β ˜ u α + ∂ α (cid:126)h · ∂ β (cid:126)h ) its associated nonlinear strain. Thenew parameterization for the positions in the embedding space is thus (cid:126)r α = [ ζx α + ˜ u α ] (cid:126)e α + (cid:126)h , ζ = 1 − BD τ . (206)In fact ζ is also the order parameter of the crumpling transition, and the term τ u αα is identical to the term t ( ∂ α (cid:126)r )2