Large N Expansion for 4-Epsilon Dimensional Oriented Manifolds in Random Media

TL;DR

This paper develops a large N expansion approach to analyze the behavior of oriented manifolds in random media near four dimensions, revealing non-analytic corrections and scaling properties for different disorder types.

Contribution

It generalizes the functional renormalization group method to large N and epsilon expansions, providing new insights into the phase behavior of manifolds in random media.

Findings

Interface width scales as a power law with a calculated roughness exponent.

Non-analytic corrections in 1/N are identified in the double limit of small epsilon and large N.

Crossover behavior between short-range and long-range disorder regimes is characterized.

Abstract

The equilibrium statistical mechanics of a d dimensional ``oriented'' manifold in an N+d dimensional random medium are analyzed in d=4-epsilon dimensions. For N=1, this problem describes an interface pinned by impurities. For d=1, the model becomes identical to the directed polymer in a random medium. Here, we generalize the functional renormalization group method used previously to study the interface problem, and extract the behavior in the double limit epsilon small and N large, finding non-analytic corrections in 1/N. For short-range disorder, the interface width scales as a power law of the width. We calculate the roughness exponent characterizing this power law for small epsilon and large N, as well as other properties of the phase. We also analyze the behavior for disorder with long-range correlations, as is appropriate for interfaces in random field systems, and study the crossover between the two regimes.