Low-temperature behavior of the O(N) models below two dimensions.

Abstract

We investigate the critical behavior and the nature of the low-temperature phase of the O(N) models treating the number of field components N and the dimension d as continuous variables with a focus on the d≤2 and N≤2 quadrant of the (d,N) plane. We precisely chart a region of the (d,N) plane where the low-temperature phase is characterized by an algebraic correlation function decay similar to that of the Kosterlitz-Thouless phase but with a temperature-independent anomalous dimension η. We revisit the Cardy-Hamber analysis leading to a prediction concerning the nonanalytic behavior of the O(N) models' critical exponents and emphasize the previously not broadly appreciated consequences of this approach in d<2. In particular, we discuss how this framework leads to destabilization of the long-range order in favor of the quasi-long-range order in systems with d<2 and N<2. Subsequently, within a scheme of the nonperturbative renormalization group we identify the low-temperature fixed points controlling the quasi-long-range ordered phase and demonstrate a collision between the critical and the low-temperature fixed points upon approaching the lower critical dimension. We evaluate the critical exponents η(d,N) and ν^{-1}(d,N) and demonstrate a very good agreement between the predictions of the Cardy-Hamber type analysis and the nonperturbative renormalization group in d<2.