Renormalization Theory for Interacting Crumpled Manifolds

TL;DR

This paper develops a renormalization framework for a continuous D-dimensional elastic manifold interacting with an impurity, using intrinsic geometry to handle non-locality and proving perturbative renormalizability beyond traditional local field theory methods.

Contribution

It introduces a novel renormalization approach for non-local models of extended objects, establishing perturbative renormalizability and critical behavior analysis.

Findings

Renormalization proven for D-dimensional manifolds with impurity interaction.

Identification of upper critical dimension d* for the model.

Establishment of scaling laws and critical exponents for the delocalization transition.

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.