Fixed-point structure and effective fractional dimensionality for O(N) models with long-range interactions.

We study, by renormalization group methods, O(N) models with interactions decaying as power law with exponent d+σ. When only the long-range momentum term p(σ) is considered in the propagator, the critical exponents can be computed from those of the corresponding short-range O(N) models at an effective fractional dimension D(eff). Neglecting wave function renormalization effects the result for the effective dimension is D(eff)=2d/σ, which turns to be exact in the spherical model limit (N→∞).

Renormalization group flow of hexatic membranes.

We investigate hexatic membranes embedded in Euclidean D-dimensional space using a reparametrization invariant formulation combined with exact renormalization group equations. An XY model coupled to a fluid membrane, when integrated out, induces long-range interactions between curvatures described by a Polyakov term in the effective action. We evaluate the contributions of this term to the running surface tension, bending, and Gaussian rigidities in the approximation of vanishing disinclination (vortex) fugacity.

O(N)-universality classes and the Mermin-Wagner theorem.

We study how universality classes of O(N)-symmetric models depend continuously on the dimension d and the number of field components N. We observe, from a renormalization group perspective, how the implications of the Mermin-Wagner-Hohenberg theorem set in as we gradually deform theory space towards d = 2. For a fractal dimension in the range 2 < d < 3, we find, for any N ≥ 1, a finite family of multicritical effective potentials of increasing order.

Fixed points of higher-derivative gravity.

We recalculate the beta functions of higher-derivative gravity in four dimensions using the one-loop approximation to an exact renormalization group equation. We reproduce the beta functions of the dimensionless couplings that were known in the literature, but we find new terms for the beta functions of Newton's constant and of the cosmological constant. As a result, the theory appears to be asymptotically safe at a non-Gaussian fixed point rather than perturbatively renormalizable and asymptotically free.