Generalizing the O(N)-field theory to N-colored manifolds of arbitrary internal dimension D

TL;DR

This paper introduces a geometric generalization of the O(N)-field theory for N-colored membranes of arbitrary dimension D, analyzing its critical properties and phase structure with improved precision and exploring connections to various physical models.

Contribution

It develops a novel N-colored manifold model extending the O(N)-field theory, enabling detailed analysis of critical exponents and phase behavior, including anisotropic and disordered systems.

Findings

Derived a more precise exponent ppa for d=3.

Identified new phases with Ising criticality.

Connected the model to the random bond Ising problem.

Abstract

We introduce a geometric generalization of the O(N)-field theory that describes N-colored membranes with arbitrary dimension D. As the O(N)-model reduces in the limit N->0 to self-avoiding polymers, the N-colored manifold model leads to self-avoiding tethered membranes. In the other limit, for inner dimension D->1, the manifold model reduces to the O(N)-field theory. We analyze the scaling properties of the model at criticality by a one-loop perturbative renormalization group analysis around an upper critical line. The freedom to optimize with respect to the expansion point on this line allows us to obtain the exponent \nu of standard field theory to much better precision that the usual 1-loop calculations. Some other field theoretical techniques, such as the large N limit and Hartree approximation, can also be applied to this model. By comparison of low and high temperature expansions, we arrive at a conjecture for the nature of droplets dominating the 3d-Ising model at criticality, which is satisfied by our numerical results. We can also construct an appropriate generalization that describes cubic anisotropy, by adding an interaction between manifolds of the same color. The two parameter space includes a variety of new phases and fixed points, some with Ising criticality, enabling us to extract a remarkably precise value of 0.6315 for the exponent \nu in d=3. A particular limit of the model with cubic anisotropy corresponds to the random bond Ising problem; unlike the field theory formulation, we find a fixed point describing this system at 1-loop order.