Renormalization Theory for Interacting Crumpled Manifolds

Abstract

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 delta-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 d<d* where d*=2D/(2-D) is the upper critical dimension, the perturbative expansion is UV finite, while UV divergences occur as poles at d=d*. The standard proof of perturbative renormalizability for local field theories (the BPH theorem) does not apply to this model. We prove perturbative renormalizability to all orders by constructing a subtraction operator 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 epsilon-expansion 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 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.

126 pages (+ 24 figures not included available upon request), harvmac, SPhT/92/124

Cited by in corpus (33)

• Functional Renormalization Group and the Field Theory of Disordered Elastic Systems
• Classical Particle in a Box with Random Potential: exploiting rotational symmetry of replicated Hamiltonian
• $C_T$ for Non-unitary CFTs in Higher Dimensions
• Self-avoiding Tethered Membranes at the Tricritical Point
• Characteristic polynomials of random Hermitian matrices and Duistermaat-Heckman localisation on non-compact Kaehler manifolds
• Renormalization and Hyperscaling for Self-Avoiding Manifold Models
• New Renormalization Group Results for Scaling of Self-Avoiding Tethered Membranes
• Hessian spectrum at the global minimum of high-dimensional random landscapes
• Renormalization of Crumpled Manifolds
• Rotational Symmetry Breaking in Multi-Matrix Models
• Scale and Conformal Invariance in Higher Derivative Shift Symmetric Theories
• Three-dimensional Folding of the Triangular Lattice
• A spin glass model for reconstructing nonlinearly encrypted signals corrupted by noise
• Comment on "Superinstantons and the Reliability of Perturbation Theory in Non-Abelian Models"
• Functional renormalization-group approach to decaying turbulence
• Field Theory of Disordered Elastic Interfaces at 3-Loop Order: The $\beta$-Function
• Super-rough phase of the random-phase sine-Gordon model: Two-loop results
• The tubular phase of self-avoiding anisotropic crystalline membranes
• Classification of Perturbations for Membranes with Bending Rigidity
• Hyperscaling for polymer rings
• Anomalous dimensions of partially-conserved higher-spin currents from conformal field theory: bosonic ${\varphi }^{2n}$ theories
• Towards a fully automated computation of RG-functions for the 3-$d$ O(N) vector model: Parametrizing amplitudes
• Generalizing the O(N)-field theory to N-colored manifolds of arbitrary internal dimension D
• Interacting crumpled manifolds
• Dynamical Selection of Critical Exponents
• Interacting Crumpled Manifolds: Exact Results to all Orders of Perturbation Theory
• Renormalization Group Approach to Interacting Crumpled Surfaces: The hierarchical recursion
• Field Theory of the RNA Freezing Transition
• Anomalous dimensions from conformal field theory: Generalized ${\varphi }^{2n+1}$ theories
• Scaling behavior of tethered crumpled manifolds with inner dimension close to D=2: Resumming the perturbation theory
• Theory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles
• Existence of a Phase with Finite Localization Length in the Double Scaling Limit of N-Orbital Models
• Principles of non-local field theories and their application to polymerized membranes