ε-Expansion for Self-Avoiding Tethered Surfaces of Fractional Dimension

Aronovitz, Joseph A.; Lubensky, T. C.

doi:10.1209/0295-5075/4/4/003

Cited by 75 publications

(51 citation statements)

References 10 publications

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“…The radius of gyration that characterizes the crumpled phase is predicted to scale as a power of the internal size of the membrane, R G (L) ∼ L ν [48][49][50]. Flory type arguments, which are based on dimensional analysis predict ν = (D + 2)/(d + 2).…”

Section: Crumpling Transition and The Crumpled Phasementioning

confidence: 99%

Anomalous elasticity, fluctuations and disorder in elastic membranes

2018

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Motivated by freely suspended graphene and polymerized membranes in soft and biological matter we present a detailed study of a tensionless elastic sheet in the presence of thermal fluctuations and quenched disorder. The manuscript is based on an extensive draft dating back to 1993, that was circulated privately. It presents the general theoretical framework and calculational details of numerous results, partial forms of which have been published in brief Letters [1,2]. The experimental realization atom-thin graphene sheets [3] has driven a resurgence in this fascinating subject, making our dated predictions and their detailed derivations timely. To this end we analyze the statistical mechanics of a generalized D-dimensional elastic "membrane" embedded in d dimensions using a self-consistent screening approximation (SCSA), that has proved to be unprecedentedly accurate in this system, exact in three complementary limits: (i) d → ∞, (ii) D → 4, and (iii) D = d. Focusing on the critical "flat" phase, for a homogeneous two-dimensional (D = 2) membrane embedded in three dimensions (d = 3), we predict its universal roughness exponent ζ = 0.590, length-scale dependent elastic moduli exponents η = 0.821 and ηu = 0.358, and an anomalous Poisson ratio, σ = −1/3. In the presence of random uncorrelated heterogeneity the membrane exhibits a glassy wrinkled ground state, characterized by ζ = 0.775, η = 0.449, η u = 1.101 and a Poisson ratio σ = −1/3. Motivated by a number of physical realizations (charged impurities, disclinations and dislocations) we also study power-law correlated quenched disorder that leads to a variety of distinct glassy wrinkled phases. Finally, neglecting self-avoiding interaction we demonstrate that at high temperature a "phantom" sheet undergoes a continuous crumpling transition, characterized by a radius of gyration exponent, ν = 0.732 and η = 0.535. Many of these universal predictions have received considerable support from simulations. We hope that this detailed presentation of the SCSA theory will be useful to further theoretical developments and corresponding experimental investigations on freely suspended graphene. Contents

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Section: Crumpling Transition and The Crumpled Phasementioning

confidence: 99%

Anomalous elasticity, fluctuations and disorder in elastic membranes

2018

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“…This extension can be measured, for instance, by the radius of gyration R G of the noninteracting manifold, defined as 17) where Tr ′ means the sum over the non-zero eigenvalues of the Laplacian ∆ on the closed manifold. Consequently we have…”

Section: ⋄ Existence Of a Wilson Functionmentioning

confidence: 99%

“…and we can now let ρ → 0 and get 17) which means that, in this limit, the tree has been disconnected into several components on which its determinant is exactly factorized.…”

Section: Appendix B Factorization Of the Measurementioning

confidence: 99%

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Renormalization theory for interacting crumpled manifolds

David¹,

Duplantier²,

Guitter³

1993

Nuclear Physics B

100

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We consider a continuous model of D-dimensional elastic (polymerized) manifold fluctuating in d-dimensional Euclidean space, interacting with a single impurity via an attractive or repulsive δ-potential (but without self-avoidance interactions). Except for D = 1 (the polymer case), this model cannot be mapped onto a local field theory. We show that the use of intrinsic distance geometry allows for a rigorous construction of the high-temperature perturbative expansion and for analytic continuation in the manifold dimension D. We study the renormalization properties of the model for 0 < D < 2, and show that for bulk space dimension d smaller that the upper critical, the perturbative expansion is ultra-violet finite, while ultraviolet divergences occur as poles at d = d ⋆ . The standard proof of perturbative renormalizability for local field theories (the Bogoliubov Parasiuk Hepp theorem) does not apply to this model. We prove perturbative renormalizability to all orders by constructing a subtraction operator R based on a generalization of the Zimmermann forests formalism, and which makes the theory finite at d = d ⋆ . This subtraction operation corresponds to a renormalization of the coupling constant of the model (strength of the interaction with the impurity). The existence of a Wilson function, of an ǫ-expansionà la Wilson Fisher around the critical dimension, of scaling laws for d < d ⋆ in the repulsive case, and of non-trivial critical exponents of the delocalization transition for d > d ⋆ in the attractive case is thus established. To our knowledge, this study provides the first proof of renormalizability for a model of extended objects, and should be applicable to the study of self-avoidance interactions for random manifolds.

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“…Hausdorff dimension d H = 2D/(2 − D); and the finiteness of the upper critical dimension d ⋆ = 2d H for the SA-interaction allows for an ǫ-expansion about d ⋆ [6][7][8], performed via a direct renormalization method adapted from that of des Cloizeaux in polymer theory [9].…”

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confidence: 99%

Renormalization of crumpled manifolds

1993

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We consider a model of D-dimensional tethered manifold interacting by excluded volume in IR d with a single point. By use of intrinsic distance geometry, we first provide a rigorous definition of the analytic continuation of its perturbative expansion for arbitrary D, 0 < D < 2. We then construct explicitly a renormalization operation R, ensuring renormalizability to all orders. This is the first example of mathematical construction and renormalization for an interacting extended object with continuous internal dimension, encompassing field theory.

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ε-Expansion for Self-Avoiding Tethered Surfaces of Fractional Dimension

Cited by 75 publications

References 10 publications

Anomalous elasticity, fluctuations and disorder in elastic membranes

Anomalous elasticity, fluctuations and disorder in elastic membranes

Renormalization theory for interacting crumpled manifolds

Renormalization of crumpled manifolds

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