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

Abstract

We study, by renormalization group methods, $\mathrm{O}(N)$ models with interactions decaying as power law with exponent $d+\ensuremath{\sigma}$. When only the long-range momentum term ${p}^{\ensuremath{\sigma}}$ is considered in the propagator, the critical exponents can be computed from those of the corresponding short-range $\mathrm{O}(N)$ models at an effective fractional dimension ${D}_{\mathrm{eff}}$. Neglecting wave function renormalization effects the result for the effective dimension is ${D}_{\mathrm{eff}}=\frac{2d}{\ensuremath{\sigma}}$, which turns to be exact in the spherical model limit $(N\ensuremath{\rightarrow}\ensuremath{\infty})$. Introducing a running wave function renormalization term the effective dimension becomes instead ${D}_{\mathrm{eff}}=\frac{(2\ensuremath{-}{\ensuremath{\eta}}_{\mathrm{SR}})d}{\ensuremath{\sigma}}$. The latter result coincides with the one found using standard scaling arguments. Explicit results in two and three dimensions are given for the exponent $\ensuremath{\nu}$. We propose an improved method to describe the full theory space of the models where both short- and long-range propagator terms are present and no a priori choice among the two in the renormalization group flow is done. The eigenvalue spectrum of the full theory for all possible fixed points is drawn and a full description of the fixed-point structure is given, including multicritical long-range universality classes. The effective dimension is shown to be only approximate, and the resulting error is estimated.