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

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 $d\leq 2$ and $N\leq 2$ 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 $\eta$. 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 favour 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 $\eta(d,N)$ and $\nu^{-1}(d,N)$ and demonstrate a very good agreement between the predictions of the Cardy-Hamber type analysis and the nonperturbative renormalization group in $d<2$.