Precision calculation of critical exponents in the $O(N)$ universality classes with the nonperturbative renormalization group

We compute the critical exponents $\nu$, $\eta$ and $\omega$ of $O(N)$ models for various values of $N$ by implementing the derivative expansion of the nonperturbative renormalization group up to next-to-next-to-leading order [usually denoted $\mathcal{O}({\mathrm{\partial }}^4)$]. We analyze the behavior of this approximation scheme at successive orders and observe an apparent convergence with a small parameter – typically between $1\mathrm{/}9$ and $1\mathrm{/}4$ – compatible with previous studies in the Ising case. This allows us to give well-grounded error bars. We obtain a determination of critical exponents with a precision which is similar or better than those obtained by most field theoretical techniques. We also reach a better precision than Monte-Carlo simulations in some physically relevant situations. In the $O(2)$ case, where there is a longstanding controversy between Monte-Carlo estimates and experiments for the specific heat exponent $\alpha$, our results are compatible with those of Monte-Carlo but clearly exclude experimental values.