Instanton calculus for the self-avoiding manifold model
François David1, Kay Jörg Wiese2
1SPhT-Service de Physique Théorique, CEA/Saclay - 91191 Gif-sur-Yvette Cedex, France
2CNRS-Laboratoire de Physique Théorique de
l'Ecole Normale Supérieure,
24 rue Lhomond, 75005 Paris, France
Abstract
We compute the normalisation factor for the
large order asymptotics of perturbation theory for the self-avoiding
manifold (SAM) model describing flexible tethered (D-dimensional)
membranes in d-dimensional space, and the ε-expansion for this
problem. For that purpose, we develop the methods inspired from
instanton calculus, that we introduced in a previous publication
(Nucl. Phys. B 534 (1998) 555), and we compute the functional
determinant of the fluctuations around the instanton
configuration. This determinant has UV divergences and we show that
the renormalized action used to make perturbation theory finite also
renders the contribution of the instanton UV-finite. To compute this
determinant, we develop a systematic large-d expansion. For the
renormalized theory, we point out problems in the interplay between
the limits ε to 0 and d to infinity, as well as IR
divergences when ε=0. We show that many cancellations between
IR divergences occur, and argue that the remaining IR-singular term is
associated to amenable non-analytic contributions in the large-d limit
when ε= 0. The consistency with the standard
instanton-calculus results for the self-avoiding walk is checked for D
= 1.
cond-mat/0409765 [pdf]
J. Stat. Phys. 120 (2005) 875-1035 [pdf]
Copyright (C) by Kay Wiese. Last edited March 17, 2008.