Interplay of the scaling limit and the renormalization group: Implications for symmetry restoration

Abstract

Symmetry restoration is usually understood as a renormalization-group induced phenomenon. In this context, the issue of whether one-loop renormalization-group equations can be trusted in predicting symmetry restoration has recently been the subject of much debate. Here we advocate a more pragmatic point of view and expand the definition of symmetry restoration to encompass all situations where the physical properties have only a weak dependence upon an anisotropy in the bare couplings. Moreover we concentrate on universal properties, and so take a scaling limit where the physics is well described by a field theory. In this context, we find a large variety of models that exhibit, for all practical purposes, symmetry restoration: even if symmetry is not restored in a strict sense, physical properties are surprisingly insensitive to the remaining anisotropy. Although we have adopted an expanded notion of symmetry restoration, we nonetheless emphasize that the scaling limit also has implications for symmetry restoration as a renormalization-group induced phenomenon. In all the models we considered, the scaling limit turns out only to permit bare couplings which are nearly isotropic and small. Then the one-loop $\ensuremath{\beta}$ function should contain all the physics and higher loop orders can be neglected. We suggest that this feature generalizes to more complex models. We exhibit a large class of theories with current-current perturbations [of which the $\mathrm{SO}(8)$ model of interest in two-leg Hubbard ladders and armchair carbon nanotubes is one] where the one-loop $\ensuremath{\beta}$ functions indicate symmetry restoration; thus we argue that these results can be trusted within the scaling limit.