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 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.

Critical behavior atm-axial Lifshitz points: Field-theory analysis and ε-expansion results

“…To clarify this long-standing controversy, it proved useful to work directly in position space and to compute the Laurent expansion of the dimensionally regularized distributions associated with the Feynman diagrams. There are two other classes of difficult problems where this technique has demonstrated its potential: field theories of polymerized ͑teth-ered͒ membranes [44][45][46] and critical behavior in systems with boundaries. 35,47 In the present study an additional complication had to be mastered: The free propagator at the Lifshitz point, which because of anisotropic scale invariance is a generalized homogeneous function rather than a simple power of the distance ͉xϪxЈ͉, involves a complicated scaling function.…”

Renormalization of crumpled manifolds

“…where ρ → 0 + is the dilation factor, and x i (ρ) = x i if i / ∈ P. This transformation has an immediate correspondent in terms of mutual distances: a ij → a ij (ρ), depending on both P and p. Under this transformation, the interaction polynomial P G (a ν ) factorizes into [15]:…”

Interacting crumpled manifolds: Exact results to all orders of perturbation theory

“…In this letter, we pursue this road further, calculating contributions to the partition-function exactly for a manifold of toroidal or spherical shape. We obtain the expansion up to order (2 − D) 4 . This information can then be used to extrapolate away from D = 2.…”

“…-The statistical mechanics of fluctuating lines and surfaces is a subject of great interest, which poses fundamental problems and has remained challenging for more than 20 years. One particular universality class, which has been studied extensively in the past, are polymerized or "tethered" membranes [1,2,3,4,5,6,7,8,9]. These are two-dimensional networks, where the bond-length fluctuates, but never breaks up.…”

“…As a first step to prove renormalizability of the full problem, [3,4] analysed (3) and indeed showed renormalizability to all orders in perturbation theory for all dimensions 0 < D < 2. (3) has essential features in common with SAM: Its critical embedding dimension tends to infinity as the internal dimension approaches D = 2.…”