Fluctuations and Lower Critical Dimensions of Crystalline Membranes Joseph Aronovitz, Leonardo Golubovic, T.C

Fluctuations and lower critical dimensions of crystalline membranes Joseph Aronovitz, Leonardo Golubovic, T.C. Lubensky

To cite this version:

Joseph Aronovitz, Leonardo Golubovic, T.C. Lubensky. Fluctuations and lower critical dimensions of crystalline membranes. Journal de Physique, 1989, 50 (6), pp.609-631. 10.1051/jphys:01989005006060900. jpa-00210941

HAL Id: jpa-00210941 https://hal.archives-ouvertes.fr/jpa-00210941 Submitted on 1 Jan 1989

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Classification Physics Abstracts

64.60Fr - 68.35Rh - 05.40+J

Fluctuations and lower critical dimensions of crystalline membranes

Joseph Aronovitz (*), Leonardo Golubovi0107 (**) and T. C. Lubensky Department of Physics, University of Pennsylvania, Philadelphia, PA 19104-6396, U.S.A.

(Reçu le 16 août 1988, accepté sous forme définitive le 21 novembre 1988)

Résumé. 2014 Nous présentons l’étude de membranes cristallines de connectivité fixée et dimension D dans un espace à d dimensions. Nous étudions la transition de plissage et les fluctuations autour de la phase plane avec des arguments qualitatifs et une analyse exacte dans la limite où d est grand. En particulier, nous étudions la nature de la transition entre le comportement élastique critique et non linéaire dans la phase plane près de la transition de plissage. Avec deux approches différentes, nous montrons que la dimension critique inférieure Dlc(d), en dessous de laquelle il n’existe que la phase plissée à température finie, peut être exprimée en fonction de d. Une approche détermine Dlc(d) en considérant les effets de désordre sur l’ordre à longue distance des vecteurs tangents à la membrane dans la phase basse température, dus aux ondes capillaires. L’autre approche identifie Dlc(d) à la dimension à laquelle la dimension de Hausdorff de la membrane à la transition de plissage du deuxième ordre est égale à celle de la phase plane. Nous calculons explicitement Dlc(d) dans la limite de d grand et obtenons Dlc(d) 2, au moins pour d suffisamment grand. Tout au long de l’article, nous présentons des arguments qualitatifs qui montrent que, en général, Dlc(d) 2. Nous considérons aussi la dimension critique inférieure Du(d) de l’ordre de position des particules qui constituent la membrane. Nous trouvons Du(~) = 3 et calculons la correction en 1/d de ce résultat. En termes qualitatifs, nous montrons que 2 ~ Du(d) ~ 3 avec Du(d) = 2, quand d = D = Du(d) = 2 et Du(d) = 3 quand d ~ ~.

Abstract. 2014 We study flexible D-dimensional fixed-connectivity crystalline membranes fluctuating in a d-dimensional embedding space. We address both the crumpling transition and fluctuations around the flat phase by means of qualitative arguments and exact analyses in the limit of large d. In particular, we investigate the nature of the crossover between critical and nonlinear elastic behavior in the flat phase near the crumpling transition. Using two different approaches, we argue that the lower critical dimension, Dlc(d), below which only the low-rigidity crumpled phase exists at finite temperatures can be computed as a function of d. One approach determines Dlc(d) by considering disordering effects of capillary waves on the long-range order of the membrane tangent vectors in the low-temperature phase. The other identifies Dlc(d) as the

(*) Current address : Loomis Laboratory of Physics, University of Illinois, 1110 West Green St., Urbana, Il 61801, U.S.A. (**) On leave from Boris Kidric Institute of Nuclear Sciences, Institute of Theoretical Physics, Vinca, P.O. Box 522, Beograd, Yugoslavia.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01989005006060900 610

dimension at which the Hausdorff dimension of the membrane at the second-order crumpling transition is equal to that of the flat phase. We explicitly calculate Dlc(d) in the large-d limit and find that Dlc(d) 2, at least for sufficiently large d. Throughout the paper we present qualitative arguments that Dlc(d) 2 is satisfied in general. We also consider the lower critical dimension Du(d) of the positional order of particles comprising the membrane. We find Du(~) = 3, and calculate the 1/d correction to this result. On qualitative grounds we argue that 2 ~ Du(d) ~ 3, with Du(d) = 2, when d = D = Du(d) = 2, and Du(d) = 3, when d ~ ~.

1. Introduction.

Membranes are, in general, D-dimensional flexible sheets of particles fluctuating in a d- dimensional embedding space, with D : d. Specific examples of physical interest are flexible polymers [1] with D = 1 and crystalline [2], hexatic [2, 3] and liquid [4-6] membranes with D = 2. Recent numerical [7] and theoretical [2, 3, 8-11] studies show that the nature of membrane fluctuations is strongly dependent on details of the intrinsic order of its constituent particles. For example, a 2-dimensional incompressible fluid membrane fluctuating at constant area at any temperature is, like a 1-dimensional polymer, a crumpled object, having no long-range orientational order of normals erected at points on its surface [5, 6]. On the other hand, 2-dimensional crystalline membranes, characterized by a fixed internal connectivity of constituent particles, are believed to exhibit more interesting phase behavior. At high flexibility (i.e., low rigidity) they become, like fluid membranes, crumpled, fractal objects with a Hausdorff dimension dH bigger than 2 [11]. However, numerical simulations [7] indicate that, at sufficiently high rigidities, these membranes exhibit a flat phase, with dH = 2, separated by a, presumably, second order crumpling transition from the low rigidity, crumpled phase. A theoretical argument supporting these simulations has been proposed by Nelson and Peliti [2]. According to them, long-range orientational order occurring at high bare rigidities is due to a long-range interaction between local Gaussian curvatures mediated by transverse phonons of the crystalline membrane. The net effect of this interaction is a strong dependence of the bending rigidity As) on wave number q at small q :

X (q ) determines the spectrum of capillary waves - excitations associated with displacements of the membrane in directions normal to the plane of the flat phase. The self-consistent theory of reference [2] yields 17 h = 1 for 2-dimensional membranes. The theory of fluctuations about the flat phase was extended in reference [8] to general D and d. There it was shown that to first order in e = 4 - D the harmonic fixed point with 17 h = 0 becomes unstable for all D 4 to an attractive anharmonic fixed point with 17h:> 0. At this fixed point, there is a softening relative to harmonic theory of excitations associated with displacements in the plane of the membrane (phonons) in addition to the hardening of capillary waves discussed above. Within an harmonic theory for displacements about the flat phase, the lower critical dimension DI, for the crumpling transition is 2. Anharmonicities, however, increase the bending rigidity, stabilizing the flat phase and presumably depressing De, below 2. A continuum model for the crumpling transition was introduced in references [9] and [10]. Within this model, there is a mean field second order transition for all D > Dc = 4. For D :5 4, there is a second order critical transition for d:> dc (D) = 219 + 0 (4 - D). For d d,, there is a fluctuation induced first order transition [9]. Throughout this 4 presumably paper, we will consider the case of a second-order crumpling transition. In fact, simulations of 611

reference [7] indicate within size limits that in d = D + 1 = 3 the transition is second order. Thus it may be that dc (2) 3. The crumpling transition is analogous to transitions in magnetic systems [12, 13] with O (N ) symmetry with tangent vectors of the membrane the analog of the order parameter of the ferromagnetic phase and the embedding-space dimension d the analog of the number of spin components N. There are, however, two significant features of the present problem that distinguish it from the O (N ) magnetic systems : (1) the lower critical dimension is less than 2 rather than equal to 2 and (2) anharmonicities in the flat phase lead to a breakdown of harmonic elasticity at small wave number for all membrane dimensions D less than the upper critical dimension De = 4. The interplay between them and these properties of the crumpling transition are the subject of this paper. We will use as a basis for our discussion a version of the model introduced in reference [9] in which intra-membrane self-avoidance interactions are ignored, i.e., we will consider only so- called phantom membranes [7]. Though self-avoidance could affect the critical behavior of the crumpling transition [9], its effects on the long-distance scaling behavior of fluctuations in the flact phase are likely to be irrelevant, i.e., the exponent ’q h in equation (1.1) should be the same for both phantom and self-avoiding membranes. Thus, all basic conclusions concerning the lower critical dimension of the flat phase presented here are unlikely to be altered by self- avoidance. We will consider only fixed-connectivity membranes without thermally excited [2, 3] or quenched dislocations or disclinations [14], which destabilize [15] the flat phase of two- dimensional membranes. The model of reference [9] allows a unified treatment of the crumpling transition and of the flat and crumpled phases. In section 2, we review this model in some detail and discuss the crossover from critical fluctuations to non-linear elastic fluctuations in the flat phase. In section 3, we discuss the lower critical dimension Dtc(d) and its determination either by considerations of the Hausdorff dimension of the membrane at the crumpling transition or of the destruction of long-range orientational order in the flat phase. We also present arguments that deviations from harmonic elasticity in the flat phase below some critical D are essential in order that phonons and capillary waves become equivalent at Dlc. Our primary new results are based upon exact evaluations, presented in section 4, of the anomalous critical exponent 17 (D, d ) of the crumpling transition to first order in an expansion in 1/d and of the exponent nh in equation (1.1) to first order in 1/dc where dc = d - D is the codimension of the flat membrane. We employ a calculational scheme similar to that developed by Ma [16, 13] for calculating critical exponents in O (N ) models in an expansion in 1/N. At D = 2, we find q (2, d) = nh (2, d ) = 2/d to leading order in 1/d in agreement with general arguments of section 3 and results obtained by David and Guitter [10] using a totally different approach. Our result for nh also agrees with that of reference [8] near D = 4. Knowledge of tel (D, d ) and nh(D, d ) allows us to show that, in the large d limit, Dtc(d) = 2 - 2/d « 2 using again arguments presented in section 3. We also calculate to order 1/d the lower critical dimension Du (d) = 3 - O(1/d) below which there is no long-range periodic positional order of the membrane. We note that, as observed above, the functions Dtc(d) and Du(d) are likely to be the same for phantom and self- avoiding crystalline membranes.

2. Models for crystalline membranes and basic fluctuation phenomena. Recently models to study fluctuation effects at the crumpling transition [9, 10] and in the flat phase [8] of crystalline membranes have been proposed. In this section we review these models and phases and the phase transitions they predict. We will also discuss some scaling and crossover phenomena not previously treated in the literature. 612

In general, models for membranes characterized by a fixed connectivity of constituent particles are functionals of position vectors R (x ), where R (x ) e R d gives the position in the d- dimensional embedding space of the mass point indexed by the coordinate x in a D- dimensional intemal manifold. The internal coordinates x label mass points in the unstretched and uncompressed flat membrane and in accordance with classical theories of elasticity [17] are considered to be continuous variables. The volume of points in this internal manifold is thus proportional to the mass of the membrane. We take the point of view that x is the continuum limit of a discrete variable labeling lattice sites in a regular D-dimensional crystalline lattice with lattice parameter a. This lattice with long-range periodic positional order is the zero temperature ground state of our model. Paczuski, Kardar and Nelson [9] have recently proposed a continuum Landau-Ginzburg- Wilson model for the crumpling transition in crystalline membranes whose Hamiltonian is (in the absence of self-avoidance)

and where over is assumed. where aa - 8,/ ax, (a = 1, 2, ..., D ) summation repeated indices The first term in (2.1) is the bending energy with K a rigidity constant. The second term in (2.1) is the stretching energy,

This stretching energy is that appropriate to an isotropic intemal manifold of mass points. If x is actually the continuum limit of points on a crystal lattice, there will be additional (except for hexagonal lattice in two dimensions) terms quartic in aaR reflecting the point group symmetry of the lattice. We will ignore these terms in this paper. The stretching term in the H can be expressed in terms [10],of the metric tensor,

relating displacements in the embedding space to the displacements in the internal manifold. The Hamiltonian H of equation (2.1) describes long wavelength fluctuations of the membrane. It is understood that there is an upper wavenumber cut off of order 2 7T 1 a. The elastic constants K, il and A are, therefore, those K(a ), u (a ), and À (a ), for a system with upper cutoff 2 w la. Throughout most of this paper we will assume jj = À + 2 ù ID - il > 0. There are, however, interesting phenomena when il = 0 and B ~ 0 which we will occasionally mention. When expressed as a function of the tangent vectors,

H is similar in form to the phenomenological cp4-Hamiltonian [12, 13]. It has a mean field phase transition at ro = 0 separating a crumpled phase with (ma) = 0 and 0 (d ) symmetry from a flat phase with (ma) # 0 and 0 (D) x 0 (d - D ) symmetry. The flat phase is characterized by nonzero average position vectors, 613 where the vectors ea are any set of D d-dimensional vectors satisfying e. - eb = ab. In mean field theory,

where J5 = À + 2 il ID. More generally, for d sufficiently large [9] and D 4, there is a second order non-classical transition at ro = roc with

where {3 =A 1/2. The correlation functions of R (x ) and the order parameter ma (x ) in Fourier space are and

The above simple relation between GRR (q ) and G (q ) is valid for the fixed-connectivity, dislocationless membranes we are considering. In mean field theory, G-1 (q ) = ro [1 + (q e)2] where q = 1 q1 and e = (K/ro)I/2 is the mean field correlation length. In the critical region, G (q ) and GRR (q ) satisfy the scaling equations,

where e - 1’0 - ’0 c 1- JI is the correlation length. The susceptibility G (q = 0 ) diverges 1’0 - ’0 c 1- ’Y with y satisfying the Fisher scaling law y = (2 - q ) v. Similarly the order parameter exponent satisfies 8 = (D - 2 + 11 ) v/Z. As in 04 theories, the exponents /3, y and q are respectively 1/2, 1 and 0 in mean field theory. As usual, crossover from mean field to critical behavior occurs when e >’ eG where is the Ginzburg length [18]. The membrane in the crumpled and flat phases and at the critical point is characterized by a Hausdorff dimension dH relating mass (contained within a radius 1 x1 from the origin of the internal manifold) to the volume in embedding space via where RG (x ) is determined by

(RG(x) is proportional to the radius of gyration [11] of the part of the membrane contained between 1 x1 and the origin). In the flat phase, RG (X) - 1 x1 and dH = D. In the crumpled phase and at the critical point, 614

In the crumpled phase (arising in mean field theory when ro > 0 in the action (2.1)) nonlinearities of the theory (see Eq. (2.2)) as well as bending terms are irrelevant so that at small q, GRR - q- 2. Then by equation (2.11), one finds that for D 2, dH = 2 D/(2 - D), while for D > 2, RG (x) tends to a constant for large x, which is consistent with equation (2.11) with dH = 00 [for D = 2, 7?c(x)- (In 1 x 1 )112, and, consequently, du = 00]. Self-avoidance modifies these results [11]. At the crumpling transition, critical point correlations of position vectors are given by equation (2.9b) so that by equation (2.11) and (2.13), we obtain dH = oo for D > 4 and

for D 4. When D --+ 4 from below, ’T1(D,d)-+O, and dH diverges. We now tum to a more detailed investigation of fluctuations in the flat phase. First we introduce displacement variables U (x ) with respect to the average positions via

The stretching Hamiltonian equation (2.2) is a function of the strain,

which measures distortions with respect to the equilibrium membrane characterized by the background metric tensor g ab 0 = J2 8ab = °a (R) . ab (R). In this equation Ua = ea . U. In terms of U ab’ the stretching Hamiltonian is .

where we have dropped a term independent of Uab(X), Note that there is a term linear in Uaa(x) in HSt. The equation of state for J = 1 (mua) 1 is determined [19] by adjusting the coefficient of this term to give (Uaa(x» = 0. In the harmonic approximation, Ue (x) is linear in U (x ), and (U aa(x» = 0 when the coefficient of the linear term in equation (2.17) is zero, i.e., when the mean field equation of state, equation (2.6), is satisfied. In the anharmonic theory, Uab (x ) is a nonlinear function of U (x ), and there are resultant corrections to the mean field equation of state. The tensor of equation (2.16) is the correct strain tensor for describing deviations from a state characterized by the metric tensor g 0 ab = j2 6e. The coefficients of the linear and nonlinear terms in Uab(x) differ by a factor of J. It is usual in discussions of non-linear elasticity [17] to define the strain so that the coefficients of these two terms are the same. To arrive at this more usual form for the strain, we can either rescale U (x ) via

or rescale lengths via x’ = Jx and define 615

The first rescaling makes U (x ) analogous to the transverse magnetization and u (x ) to an angle in a Heisenberg ferromagnet. The second rescaling converts go to the unit metric 8 ab and is more in the spirit of classical elasticity. Both rescalings lead to

where it is understood that u if a function of x’ and derivatives are with respect to x’ for the second case. In what follows we will use the’transformation of equation (2.18) and absorb factors of J2 into the elastic and bending constants by setting g = j4 il A = j4 X, and K = J2 K.

now = We decompose u (x ) into D phonon coordinates ua (x ) (a 1, ..., D) (represented by u e R D) and (d - D ) height coordinates h perpendicular to the hyper plane defined by the e,,’s :

with h. ea = 0. In terms of these variables the flat phase Hamiltonian is where

and

Ho is harmonic in u and h, whereas Hl contains quartic h field interactions as well as trilinear u - h couplings. H2 contains all terms in H not explicitly included in Ho and Hl. In particular it contains a term linear in eQuQ and terms quartic in daube We observe that when H2 is removed from equation (2.22), the resulting Hamiltonian is harmonic in u, and the u fields can be integrated exactly [2] in the partition function f DhDu exp [- H(u, h)]. The term proportional to K (a,, dauf in Ho is not normally included in the flat phase elastic energy because, at small wavenumber q, it is unimportant compared to the » (8,,Ub )2 and À (oaua)2 terms. Near the critical point, however, this term must be included for a correct treatment of fluctuations at wavenumbers with q) > 1. The nonlinear elastic Hamiltonian describing fluctuations with q) « 1 is identical to equations (2.22) to (2.24) with K(a ) = j2 K(a), A (a ) = J4 À (a ), and w (a ) = J4 il (a ) replaced by their renormalized values, K(g), À (g), and g (e appropriate to systems with a wave number cutoff 2 7T 1 g rather than 2 7T la. Standard scaling arguments yield

in the critical region where e » eG- 616

At the harmonic level (Eq. (2.23)), capillary waves and phonons are decoupled and one obtains

where, as defined in the introduction, dc = d - D is the codimension,

and where ( ) o implies averages with respect to Ho. We see that harmonic phonon and capillary wave fluctuations are qualitatively different. Symmetry arguments can be used to establish that the renormalized correlation functions Ghh and Guu (defined as the averages in Eqs. (2.26) and (2.27) but with respect to the full Hamiltonian Eq. (2.22)) have to satisfy

The nonlinear terms in equation (2.22) lead to significant renormalization of Ghh and Guu for D 4 where the long-wavelength excitation spectrum is controlled by an anharmonic fixed point whose properties were analyzed to first order in e = 4 - D in reference [8]. For wavelengths greater than the nonlinear length [8],

which by equation (2.25) and the discussion preceding it can be written

the elastic constants appearing in equation (2.26) are renormalized at small q and behave as

Thus at small q, equations (2.26) and (2.27) should be replaced by

and

The exponents 17h and 17u are not independent. The requirement that (daUa)2 and daUa(dbh. abh ) must rescale the same way under renormalization in order to preserve rotational invariance implies [8] that

is satisfied for all D and d for which the anharmonic fixed point is stable. 617

The exponents 71 u and 71 h were calculated to first order in e = 4 - D in reference [8]. Within the region of thermodynamic stability with u > 0, B = À + 2 u /D > 0, they are given by and

These results clearly satisfy equation (2.33). Tlh and q. are positive according to equations (2.34) and (2.35) at least for sufficiently small e. They are also positive in the large d limit as we will show in section 4.2. It is likely that these exponents remain positive for all values of s and 1 /d, as will be further discussed in the next section. If so, then for D 4, from equations (2.30) to (2.32), one can see that anharmonicity tends to stabilize orientational order [since KR (q) stiffens for q - 0] and to destabilize positional order [since IL R(q) and k R (q ) vanish as q - 01. The crossover from nonlinear to harmonic behavior occurs at q nI 1 or equivalently at e.,IL - 1 for a sample with internal manifold volume LD. At low temperatures e eG and nl j(2/ Íi. Both K and ji are expressed in reduced units : they are physical elastic constants divided by the temperature. Thus, if physical elastic constants are finite at T = 0, nl will diverge as T-1/(4 - D) as T ---> 0 and the region of applicability of the harmonic theory will grow. In the critical region, efl - e so that there is a direct crossover from fluctuations characteristic of the critical point to those of the nonlinear elastic theory as qe passes through one. We close this section with a brief discussion of extemal forces and stresses. For simplicity, we confine our attention to forces exerted at the boundary of the membrane. In general the total force, F, on a volume element of the membrane can be expressed as an integral over a force density f (x ) via F = dDx f (x ). In the phantom surfaces we are considering, forces can only be transmitted locally with respect to intemal coordinates x. This implies that the force density f (x ) can be expressed as a gradient with respect to x of a d x D component stress tensor with = d : Qa components cr,,j a 1, ..., D, and j = 1,...,

The work done by the system in displacing mass points by 5R(x) is then dDx 5R(x). f (x ), implying that the Hamiltonian due to extemal stresses to be added to equation (2.1) is

Thus the stress tensor a a is the field conjugate the order parameter ma (x) of the crumpling transition. The free energy near the crumpling transition, therefore, satisfies the scaling relation

where à = (D + 2 - q ) v /2 is the gap exponent and u is any component of Qa. Similarly at the critical point, 618 with

In the flat phase, the coupling to the stress is obtained by replacing ma in equation (2.37) by J(aaUb eb + aah). Since, by equation (2.37), ma will align in the plane spanned by the set of D vectors a a’ the only stresses we need to consider in the flat phase are those with components Uab = eb ab. Scaling at the nonlinear low-temperature fixed point [8] then implies that the free density energy obeys the scaling relation

for any component cr,,b - J- 1 cr. This implies a breakdown of the linear Hooke’s law relation between stress and strain to

where

Thus the relative change of the membrane base area AB, defined as the projection of the membrane onto the plane (2.5), due to isotropic stress is

In addition, one can use Ward identities resulting from the rotational invariance of H to show that

in the presence of an extemal stress. As we have already noted q. and Tlh are expected to be positive functions of D and d, implying an enhanced response (with respect to harmonic theory) of strain to extemal stress. In fact, this enhancement due to thermal fluctuations is to be expected on physical grounds. To see this, note that in the presence of thermally excited capillary waves, a free surface of the flat phase occupies at any finite temperature a smaller base area than that corresponding to its zero temperature configuration. This is a crumpling effect due to the thermally induced orientational disorder of the membrane tangents. It is clear that the finite temperature crumpled configuration is easier to stretch by an extemal force then is the zero temperature configuration. In a similar way a flat unstretchable piece of paper becomes stretchable after being mechanically crumpled ; in our case this crumpling is due to thermally excited capillary waves. Thus, crumpling effects are likely to enhance the response of the surface to weak extemal stretching forces relative to that of the linear Hooke’s law, presumably yielding a positive Tl u in (2.44), i. e. , by equation (2.43) positive Tlu [and positive 2 - nh]. In section 3, we will argue, on different grounds, that fluctuations tending to weaken the orientational order of the membrane (i.e., to crumple it), should soften the surface phonons and so yield a positive q..

3. Lower critical dimensions.

In this section we will discuss the various lower critical dimensions associated with the destruction of long-range orders and the disappearance of the crumpling transition. We will 619

focus mostly on the lower critical dimension Dlc(d) below which the crumpling transition ceases to exist. At the critical point, we expect the membrane to occupy a smaller volume than it does in the flat phase, i.e. we expect the Hausdorff dimension of the membrane to be greater than that of the flat phase. Thus, dH > D or, by equation (2.14), 2 - D - Tl (D, d ) 0. As the crumpling critical point is pushed by fluctuations deeply into the low temperature (i.e. high rigidity) region, dH should approach D. When the transition temperature is depressed to zero [i.e. at Dlc(d)], dH and D become equal implying

Note that if q is positive, then Dlc(d) 2. In section 4.1, we will use this equation to generate an expansion of D1c in powers of 1/d. It is the orientational order of the flat phase that distinguishes it from the crumpled phases. The lower critical dimension is thus the dimension at which fluctuations destroy the orientational order of the flat phase. The components of the tangent vectors in directions normal to the hyperplane of ordering, equation (2.5), are proportional to a,h, so that the condition for the existence of orientational order is that

be finite. From this we obtain the equation

relating Dlc(d) and llh. The condition for the existence of long-range positional order of the membrane is that local phonon fluctuations,

be finite. Thus below a lower critical dimension Du (d ) satisfying

there is no long-range positional order. Within the harmonic approximation, n = nh = 0, and by equations (3.3) and (3.5), Dlc(d) = Du(d) = 2. On the other hand positive values of n. and Tlh would imply and

consistent with the previous observation that nonlinear effects tend to stabilize orientational and to destabilize positional order. We present now an alternative derivation of the equation (3.3) for Dlc. It sheds more light on basic trends of n h and TI u as functions of D and d and gives some idea of why the phenomenon of the breakdown of classical elasticity for D less than some critical dimensionality (which actually is 4, Ref. [8]) should exist at all. On physical grounds one can expect that, upon lowering D (at fixed d), the tendency of thermal fluctuations to destabilize the orientational order should increase. On the other hand, the most prominent consequence 620 of this order is the existence of two rather different kinds of elementary excitations of the membrane : phonons and capillary waves. Thus if upon lowering D, thermal fluctuations have an increasing tendency to destabilize long-range orientational order, one should expect that the difference between phonons and capillary waves should tend to disappear with decreasing D. Within the harmonic theory, these two kinds of excitations are qualitatively different for any D (see Eqs. (2.26) and (2.27)). Actually, this is somewhat paradoxical since even within harmonic theory alone there exists a critical dimension for the disappearance of long-range orientational order, D1c = 2. Thus, the existence of another critical dimension D, below which harmonic theory breaks down, in such a way that the difference between the spectra of phonons and capillary waves is lessened, is not so surprising. It tums out that De = 4 (Ref. [8]). From equations (2.31) to (2.35) one can see that for D 4 the difference between these two kinds of excitations still exists, but, because of the positive signs of nh and nu, it is somewhat lessened ; phonons are softer, while capillary waves are harder than in harmonic theory. We expect that inequalities ’qh::. 0 and nu > 0 should be satisfied beyond the e-expansion [8] or the large d limit results of section 4. They are consequences of the simple fact that, upon lowering D, thermal fluctuations tend to weaken the orientational order and to diminish the difference between the two distinct elementary excitations (whose distinction hinges on the existence of the orientational order) of the flat phase by softening phonons (nu > 0) and stiffening capillary waves (nh > 0). This, if correct, immediately implies inequalities (3.6) and (3.7) for D1c and Du. As long as long range orientational order exists, the distinction between phonons and capillary waves should persist : phonons should be harder than the capillary waves, or, by (2.31) and (2.32), for D:> D1c(d). Precisely at D = D1c(d), the distinction between these two kinds of excitations should disappear. Thus we arrive at

This expression is identical to equation (3.3) when 17u is expressed in terms of 17h via equation (2.33). The reasoning leading to equation (3.9) was, however, quite different from that leading to equation (3.3). Precisely at D1c the spectra of phonons and capillary waves scale identically :

The nature of the physics embedded in the equation (3.10) becomes more clear when one observes that by equations (3.1) and (3.3)

Thus, at D = Dlc(d) there is equality between two rather different exponents : 17h, associated with the low temperature fixed point governing scaling of phonons and capillary waves, and n, which is associated with the crumpling transition. By (3.11) the scaling of position variables correlations at the transition (see Eq. (2.9b)) is identical to that of phonons and capillary waves (see Eq. (3.10)). Also, the Hausdorff dimension of the membrane at the transition becomes equal to that of the flat phase at Dlc. Thus, the two distinct fixed points governing scaling behavior at the transition and in the flat phase become identical at D = Dlc. Most likely they merge when D tends to D1c from above, leaving a single unstable fixed point for D Dlc. This scenario is consistent with disappearance of the flat phase for D Dl,. 621

These conclusions are in agreement with the results of the 1/d expansion of section 4. For example, since both n and TJh are positive and of order 1/d, one obtains from equation (3.11)

and by inserting (3.12) back into (3.11) and expanding in powers of 1/d, one gets

at most. Calculations presented in section 4 indeed confirm that for D = 2, n and nh are equal to order 1 /d. However, q (D = 2, d ) and 17h(D = 2, d ) should differ at higher order in 1/d since n (D, d ) and 17h(D, d ) are only equal at D = D1c and Dlc 2. Further insight into physics at the lower critical dimension can be obtained from the nonlinear Hooke’s law of equations (2.42) and (2.44). At D = D1c, q, = 1 by equations (2.43) and (3.9). Thus, by equation (2.44), at the lower critical dimension 8AB/AB 0’0 = 1. In other words, even in the absence of external stress, the membrane has a tendency to deform spontaneously at D = Dlc as would be expected when the membrane is crumpled at any finite temperature. Finally, we recall that the breakdown of harmonic elasticity leads to Du > 2 and the absence of long-range positional order in D = 2 dimensional membranes. In bulk crystals with D = d, Du = 2. Because capillary waves, present when the codimension dc is positive, enhance the disordering effects of phonons, Du (dc) is presumably an increasing function of dc with Du (dc) 2. We note that equation (3.5) determining Du can be rewritten with the aid of equation (2.33) as

In the large dc limit, nh = o (l/dc) (see Sect. 4.2), and Du(dc) = 3 - o (1/dc). So in conclusion, Du(dc) is most likely an increasing function satisfying 2 : Du(dc) : 3 with Du(dc = 0) = 2 and Du(dc = oo) = 3.

4. Crystalline membranes in large-d limit.

In this section we consider fluctuations of crystalline membranes in limit of large dimensionality d of the embedding space. The crumpling transition is studied in section 4.1, while the breakdown of harmonic elasticity theory of the flat phase is considered in section 4.2.

4.1 CRUMPLING TRANSITION IN THE LARGE-d LIMIT. - We first study the phase transition of the model (2.1) in the large-d limit. Our main goal will be to calculate the exponent and so, using (3.1), to calculate Dlc(d). The essential features of the calculation are similar to those of the large-N limit of standard N-vector models [16, 12, 13], with N (the number of spin components) analogous to d (the number of components of R(x)). For example, a well defined large-d limit exists provided one presumes, in analogy with spin systems [16, 12, 13], that the quartic couplings À and il of the stretching energy (2.2) are O(l/d). Thé bending constant k of (2.1), being similar to the spin stiffness, is kept at 0 (1 ). With these assumptions, in the crumpled phase (ro> rc) the leading diagram contributing to the self energy

1 defined is given in figure 1(b). In equation , while GRR (q ) is 622 by (2.9b). The dashed interaction lines in figure 1 represent the bare quartic interaction, while full lines are renormalized propagators, given to leading order by

From the diagram in figure 1(b) we obtain the equation,

relating the bare ro to its renormalized counterpart rR. From (4.3), it is easy to show that, for D > 2, rR goes to zero at the second order transition occurring at ro = rc, with

For ro r,,, the flat phase will be thermodynamically stable as long as both a and B are positive, as will be assumed in the following. When D - 2, rc diverges, and for D 2 the flat phase is completely suppressed by thermal fluctuations. Thus, to leading order in 1/d, one has that D1c = 2. In this leading order derivation, one should notice that the integral in (4.4a) can be rewritten as

which, to the leading order in 1/d (Eq. (4.2)), diverges in the infrared when D 2. At higher orders in 1/d, the relevant integral will become

Generally, I’ will diverge at D1c(d), so that in agreement with (3.1). We also observe that, to the leading order in 1/d, the exponents characterizing the crumpling transition are given by the same formulas as the corresponding exponents in spin systems [16, 12, 13]. This fact was first reported in reference [9]. Thus, for D 4,

as can be easily verified from equations (4.2) to (4.4). Also, .

where a, as in spin systems, gives the specific heat singularity at the transition. We now proceed to calculate 17 to 0 (1/d) following the standard treatment of spin systems [16, 13, 12]. The calculation is based on the observation that since 17 = 0 (1/d), and since GRR (q) scales at criticality according to equation (2.9b), this correlation function can be expanded in powers of 1/d, in the following way 623

Then, as for ordinary spin systems, q can be calculated by extracting singular behavior of the form q4ln q from the self energy diagrams (shown in Figs. l(c) and 1(d)) which give 1/d corrections to X. The thick line in these figures represents an effective quartic interaction U, given by the sum of bubble chains indicated in figure 1(e). Diagrams from figure 1(c) give only O(q2) contributions to X and so only effect the value of r,,. Corrections of the form q4ln q come from the diagram in figure 1(d). This diagram’s contribution to X(q) is given by

where implicit summation over all repeated indices is assumed as usual. The summation of bubble graphs that contribute to the fourth rank, D-dimensional tensor Uab, a’ b’ (P) which enters (4.8) is more complicated then the summation in the magnetic case because of the tensor nature of the surface’s bare quartic interaction in equation (2.2). The bare vertex (shown in Fig. 1(a)), can be represented in the form where and

Thus, the effective quartic interaction of figure 1(e) has the structure

where p = q + q, II (p ) is the polarization loop tensor :

and implicit summations over all indices are assumed (for instance (q x q ) U°(k x k) - (q x lj)ab CJ2"cd(k x k)cd]. The geometric series in (4.10) can be summed by considering U° and IIU° to be linear operators acting on the D (D + 1 )/2 dimensional vector space of symmetric, second rank tensors S or, in the special case il = 0, on the one dimensional subspace spanned by the symmetric tensor 8ab. With this interpretation, we see that

As in the case of spin systems [16, 13, 12], the polarization loop integral (4.11) is ultraviolet convergent for D 4 and can be calculated by taking the momentum cut off to be infinite. Thus, for D 4, (4.11) has the form,

for some tensor f(q) which is only a function of the unit vector 1 = q/ 1 q 1 . For D 4, II diverges in the infrared so that the inverse of (1 + IIU°) in (4.12) is dominated by the polarization loop term. We thus see that 624

Fig. 1. - Diagrams of the perturbation theory of the action (2.1) in the crumpled phase in the large-d limit. (a) The fundamental interaction vertex. Full lines represent renormalized propagators and dashed lines the bare quartic interaction. (b) Self-energy diagram of 0 (1 ). (c) and (d) Self-energy diagrams of 0(1/d). (e) The effective quartic interaction represented by thick lines in (c) and (d). Diagrams of the effective action for capillary waves, equation (4.21), have the same structure without, however, tadpole diagrams of (b) and (c) which are forbidden by symmetry. Thus, any allowed diagram of the flat phase has the structure of (f). where it is important to realize that II is inverted as a linear operator on the appropriate vector space. Then, as in standard critical phenomena, (4.14) is inserted into (4.8), and the index 17 is calculated by extracting the coefficient of the singularity q4ln q. Although the presence of such a singularity is easy to conjecture on homogeneity grounds from (4.8), (4.13) and (4.14), the detailed calculation of n requires the inversion of the polarization tensor. This inversion tums out to be a rather nontrivial mathematical task. Moreover, some deep physics concerning Dtc lies just beyond the solution of this algebraic problem. To see this, consider the polarization tensor (4.11) in the limit D --+ 2. It is easy to show that each individual component IIab,a’b,(q) diverges (for any q) in this limit. At first glance one might conclude that the inverse of II goes to zero, causing & X = 0 in equation (4.8) to vanish and consequently causing n to be zero to O(l/d). Then, by the arguments of section 3, one would conclude that Dtc = 2 even within the corrected (to 0(l/d)) theory. Actually, in the standard N-vector model [16, 13, 12] 17 vanishes when D --> 2 in the large-N limit precisely because of such a divergence of the polarization loop integral. However, in our case the tensorial nature of the polarization loop operator makes an important difference. As a linear operator on S, II can be represented as 625 where Pk is the projection operator to the subspace corresponding to the eigenvalue Àk of the polarization loop operator, i. e. II4rk = Àk lkk, where .pk E S is any member of the À k eigenspace. In this representation, II-1 is easy to compute :

Since 11 has the property nab, a’ b’ = na’ b’, ab’ all k in (4.15a) are real. It is clear now, by (4.15a), that the divergence of components of II, occurring when D - 2, is a consequence of a divergence of at least some, but not necessarily all, eigenvalues Àk(D). Thus, by (4.15b), one can see that q need not vanish to 0(l/d) in the limit D --+ 2. In fact a single eigenvalue does remain finite, and 11 (D = 2, d ) is indeed nonzero. Since inverting or diagonalizing II is of some importance for the physics of Dlc, we describe this procedure in some detail. On the basis of the theory’s underlying O(D ) invariance (i. e. , symmetry of (2.1) with respect to rotations in the D-dimensional internal manifold), all fourth rank tensor operators entering our calculation can be written in the following form (for concreteness, we treat f) :

where fk, k = 0, 1, ..., 4, are pure D-dependent numbers. We now consider (4.16)’s action on S. After introducing the natural scalar pro.duct ( Ip (1)@ 4( (2) >= 4,,,(bl) b2), one can show that the symmetry f ab, a’ b’ = ta’ b’, ab allows S ta be decomposed into mutually orthogonal, invariant (upon action of f ) subspaces. In detail, the direct application of (4.16) shows that S admits a direct sum decomposition into three invariant orthogonal subspaces : S = Sl + S2 + S3. In this decomposition, Si is the two-dimensional space of second-rank tensors of the form cf¡ab = Xl 8ab + x2 qa qb. The space S2 + S3 is orthogonal to Sl. Consequently any w E (S2 + S3 ) satisfies w = 0 and 4a 4b cfab = 0 (these relations come from the aforementioned form of the scalar product). The space S2 is (D - 2 ) (D + 1)/2- dimensional, and any w belonging to S2 satisfiës 4a cfab = 0, as well as being orthogonal to Sl. Finally, the (D - 1 )-dimensional subspace S3 is the orthogonal complement of Sl + S2 in S. From (4.16) it can be shown that S2 and S3 are eigenspaces of degenerate eigenvalues of f. Thus, the problem of inverting f reduces to the trivial inversion of the symmetric 2 x 2 matrix which represents f on Si. An explicit représentation of f-1 is given in the appendix. The above decomposition allows us to invert II, and so (after much algebra) eventually to calculate q. The detailed calculation of the integral (4.8) turns out to be quite lengthy, although straightforward. Here we confine ourselves to the final formula for q:

where B (x, y ) is the standard beta function. By (4.17a), q is continuous and nonvanishing when D -+ 2, in the large-d limit. In fact, in this limit (in which all individual components IIab, a’ b’ diverge), only one eigenvector of II, of the form ’P,,,b - 8ab - qa qb (belonging to Sl) yields a nonvanishing contribution to H- 1. Inserting D = 2 into (4.17a) yields directly that 626

Combining equations (2.16) and (4.17b), we see that the Hausdorff dimension of a D = 2 dimensional membrane at the crumpling transition is

Thus, for large enough d, dH > D = 2. According to the arguments of section 3, this inequality is consistent with the existence of a finite temperature, second-order crumpling transition, and as well as with the existence of a flat phase at low temperatures. As d --+ oo, dH --+ 2. In this limit one may expect that critical temperature of the transition goes to zero, as is also argued by David and Guitter [10]. Dlc can be determined by equation (3.3) as a function of d, in the large-d limit

Thus Dtc(d) is indeed less then two, at least for sufficiently large d. We also observe that when both e = 4 - D and 1/d are small, n = 0 (£3)/d, as opposed to the order 0 (a 2)Id which one might have expected-based on the knowledge that the 0() contribution to n in N-vector model [16], is zero. Thus any nonvanishing 0 ( £2) contribution to q has to be O(1/d2) or smaller in the large-d limit. Finally, we point out that the above techniques can also be used to treat either the case of il = 0, or that of B = Ã + 2 ù ID = 0. We quote here only the results for the simpler case of zero shear modulus (u = 0) membranes (so called fixed connectivity fluids [8]) for which Uo projects onto the one-dimensional subspace of S which is spanned by âQb. II is now easily inverted on this subspace, and we compute that

We note that 71 (D = 2) = 0 + 0 (1/d2) for this case, yielding that Dtc = 2 + 0 (1/ d2) for the fixed connectivity fluid. Other results [8] suggest that De, = 2 exactly for this case.

4.2 FLAT PHASE IN THE LARGE-dc LiMIT. - We now outline the calculation of the exponent 71h, which characterizes the anomalous behavior of capillary waves of the flat phase (see Eq. (2.31)) for D 4. The starting point of this calculation is the phenomenological Hamiltonian (2.22) for fluctuations around the flat phase [8]. In fact, since the capillary wave field h has de components (we recall that dc = d - D is the codimension of the flat phase), the natural expansion parameter in high embedding dimensions is 1/dc. rather than 1/d. However, because 1/dc = 1/d + O (1/d2), to the leading order in l/d, the 1/dc entering our basic formulas in the following can be replaced by 1/d. A well defined large-d,, limit of the theory (2.22) exists if one assumes that k - g - K ~ 0 (dc), or equivalently that À and 1£ are of 0(1/ dc), and that K = 0 (1 ). With either of these sets of assumptions it is easy to count, in powers of 1 /dc, the diagrams of the theory (2.22) which contribute to the capillary wave correlation function Ghh (q ), introduced in section 2. An inspection of the theory’s 1/dc expansion shows that none of the interactions contained in H2 contribute to n h to O(1/dc). Thus H2 can be (at least to this order) removed from (2.22), yielding an action which is harmonic in the u-fields. Then, as already observed in section 2, the phonons can be integrated out of the partition function. This integration results in an effective Hamiltonian 627 for capillary waves, already derived for the special case D = 2 by Nelson and Peliti [2]. In general D this Hamiltonian has the form

with

In (4.22) we introduced the transverse projector

and a composite field Sab (q ) which is the Fourier transform of

For the special case D = 2 it can be shown that mab(q)2 = maa(q)2, so that (4.21) reduces to the effective Hamiltonian of reference [2]. In context of the effective Hamiltonian (4.21), diagrams of the perturbation theory have the same form as those in figure 1, provided one identifies propagators (full lines) and bare interactions (dashed lines) with those entering equation (4.21). It can be shown that the momentum integral in (4.21) has to be considered in the sense of the principal value, with q = 0 being removed. Diagrammatically, this results in the absence of tadpole diagrams such as those in figures 1(b) and 1(c). The leading diagram contributing to Ghh (q ) is thus given by figure 1(d), where the thick line is, once again, an effective interaction, which is calculated as the sum of bubble chains depicted in figure 1(e). As before, l1h can be calculated to O(1/dc) by matching the contribution from the diagram in figure 1(d) to

The basic difference between the present theory and the problem considered in section 3.1 is the absence of the O(q2) term in [Ghh(q)]-1. This absence is the consequence of the theory’s underlying rotational symmetry (see Eq. (2.28)) [8]. In fact, the 0 (q2) term is absent to all orders in the perturbation theory of the effective Hamiltonian (4.21). To see this, note that for theory (4.21), any self-energy diagram which is not a tadpole has, in general, the form represented in figure 1(f). The diagram must contain a pair of propagators with attached interaction lines, each entering a block which may contain any diagrammatic pattern coming from (4.21). Then, by equations (4.21) to (4.24), it is straightforward to show that the potential 0 (q2) contribution of this diagram to [Ghh (q)]-1 vanishes - the transverse projection operators entering equation (4.22) annihilate any such contribution. Since, as mentioned above, tadpoles are forbidden, the effective Hamiltonian (4.21) is, to all orders of perturbation theory, consistent with symmetry constraints (2.28). For zero shear modulus (IL = 0), quartic h self interactions in (4.21) vanish and the effective capillary wave theory is harmonic. Thus, for this special case, one has nh = 0 to O(1/d2). In fact, we expect, on the basis of general arguments of reference [8], that zero shear crystalline membranes have 17 h = 0 to aff orders of perturbation theory, and, by equation (3.3), Dtc = 2. The calculation of 17 h (for positive IL) is similar in spirit but considerably less complex than that of n. As in section 4.1, the most complex part of the calculation is the inversion of the 628 polarization loop tensor operator II(P) (this time on the subspace of S transverse to p). We now give resulting formulas for nh which differ for B > 0 and B = 0 case. For B > 0 we find

which, for D = 2, gives

Thus, by (4.17b) and (4.17)

as was previously argued in section 3 (Eq. (3.13)) on qualitative grounds. Thus, to O(l/d), ’TJ and nh are equal for D = 2-dimensional membranes -the difference between them should appear at O(1/d2). Using equations (4.27) and (3.3) one obtains

for the lower critical dimension of the orientational order, in agreement with equation (4.19). Equations (2.33) and (4.26) can be used to calculate the O(l/d) correction to ,q,. Note that 71u, the exponent which characterizes the anomalous behavior of phonons, is nonzero to 0 (1 ) in the 1 /d expansion

Both 17h and 17u are positive for 4 > D > Dlc(d) to the order we calculated them in the 1 /d-expansion. The same is true for the O(4 - D ) values of these exponents [8]. On the basis of the physical arguments presented in section 3, we expect that the positivity of 17 h and 17 u is generally satisfied. The most striking effects of this positivity are the inequalities DI, 2 (Eq. (3.6)) and Du > 2 (Eq. (3.7)). Equation (3.14) can be used to derive the lower critical dimensionality of the positional order Du(dc) in terms of an expansion in 1/dc

so that, by equation (4.26)

Thus, at least for sufficiently large d, Du (dc) 3. One has that Du(dc = 0) = 2 (see Sect. 3), and that Du(dc = oo ) = 3. Du(dc) is, presumably, an increasing function of dc, satisfying 629

Finally, the limit B = 0, with IL > 0, is special. In this limit, U0 projects onto a smaller subspace of the transverse symmetric tensors. This leads to a value of TJh different from the B > 0, g > 0, case (see Ref. [8]). In the large-d limit we obtain for B = 0 case

while the corresponding 11u can be obtained by equation (2.33). Thus in a crystalline membrane having small but positive bulk modulus B there should exist a crossover from the regime where 11 h is given, in the large-d limit by (4.33) to another regime at longer length scales, where 11h is given, in the large-d limit, by (4.26), or by (2.34) for small e = 4 - D, reference [8].

5. Discussion and comparison with related work.

In this section we summarize the consequences of our results on D = 2-dimensional membranes. The qualitative arguments presented in this work suggest that the lower critical dimension the flat phase De, is, for any d, less than two. The explicit calculations presented in section 4 support this conclusion, at least for sufficiently large d (see Eqs. (4.19) and (4.29)). Thus, D = 2-dimensional membranes have a second order crumpling transition at least for large enough d. At the associated critical point, these membranes have a Hausdorff dimension dH (Eq. (4.18)) larger than that of the flat phase (fôr which dH = 2). However, equation (4.18) implies that when d - oo, dH tends to two from above. These results imply the existence of a flat phase below some finite critical temperature, which tends to zero when d - oo and D = 2. In the same limit, the exponent nh, which characterizes the anomalous behavior of capillary waves (Eq. (4.27)), tends to zero. This implies that anharmonicities stabilizing the low-temperature phase vanish in the large-d limit, and once again we conclude that the crumpling transition temperature is depressed to zero when d - oo . Our large-d results were obtained using standard methods for the study of critical, phenomena [16, 12, 13]. Recently, David and Guitter [10] considered the spécial. case D = 2 in the large-d limit by a renormalization group method. Their study also indicatès the existence of the crumpling transition critical point at a finite temperature which tends to zero as d ---> oo. This behavior, as well as other quantitative details of reference [10] (the values of q, nh, and dH at the transition) are in agreement with our results discussed in the previous paragraph. Our general-D treatment also allows us to determine that in the limit of large d the lower critical dimension Du(dc) for the disappearance of long-range positional order is greater than two. We believe that Du(dc) is an increasing function of dc with D(dc = 0) = 2 and Du(dc = (0) = 3. The arguments and calculations presented in the paper favor the conclusion that realistic D = 2-dimensional fixed-connectivity membranes embedded in a space with d = 3 have a low-temperature flat phase that is orientationally but not positionally ordered. This phase is separated from a high-temperature, orientationally-disordered crumpled phase by a finite transition. The mechanism to this is the temperature crumpling physical leading conclusion- breakdown of harmonic elasticity in the flat phase. It would be interesting to study this mechanism by numerical simulations (which have already given some evidence of the existence of the flat phase [7]), i.e. to measure the exponents ’qh, n. and 11 cr introduced in section 2. One potential difficulty in such a simulation is finite size effects, which we will briefly discuss for membranes with fi = B. For these membranes the discussion of section 2 implies that in order for anharmonic effects to be 630 important (hence observable), the linear size of the system must be large compared to the nonlinearity length Çnl. For D = 2, this length behaves as

outside the critical region of the crumpling transition (i.e., when e eG), and behaves as

within the critical region. Thus, within the critical region e., diverges in the same way as does the correlation length e, while it remains roughly constant in a temperature range which is close to the critical point but still not in the critical region. These remarks presume a second order crumpling transition. If the transition is driven first order by fluctuations, the critical region corresponding to equation (5.2) might be absent. Finally, deep in the low temperature phase, the behavior of e., can be model dependent. Thus, if the bare physical elastic constants are finite at T = 0 (implying that k - A - 1 / T at low T), gnl will diverge, by equation (5.1) as T-1/2. Thus there is a minimum in ew (T) which presumably lies just outside the critical region of the crumpling transition. Nonlinear effects are most easily observed for the temperature corresponding to this minimum.

Acknowledgments.

This work was supported in part the National Science Foundation under Grant No. DMR85- 20272.

Appendix.

In this appendix, we give the explicit formula needed to invert the operator f of equation (4.16), when considered as a linear operator on the space of symmetric tensors S. It is first useful to introduce the normalized transverse and longitudinal tensors

We next point out that the unit operator on S can be represented in the form

where (10 ),,b = 1 ab, a’ 6 0,,,’b’ for any 0 E S. One can then show that the operators P3 and P2 defined respectively as and where 631 for any 0 E S, are projection operators onto eigenspaces of f. Using P2, P3 and the notation of (A3c), we can represent f in the form

where F a{3 is the 2 x 2 matrix

Then, finally

where F -1 is the matrix inverse of F.

References

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