High-accuracy critical exponents of $O(N)$ hierarchical sigma models

We perform high-accuracy calculations of the critical exponent $\gamma$ and its subleading exponent for the $3D$ $O(N)$ Dyson’s hierarchical model for $N$ up to 20. We calculate the critical temperatures for the nonlinear sigma model measure $\delta (\overrightarrow{\varphi }.\overrightarrow{\varphi }-1)$. We discuss the possibility of extracting the first coefficients of the $1/N$ expansion from our numerical data. We show that the leading and subleading exponents agree with the Polchinski equation and the equivalent Litim equation, in the local potential approximation, with at least 4 significant digits.

Figure 1

${log}_{10}\frac{|\beta (l_{\max })-{\beta }_c|}{{\beta }_c}$ calculated for $l_{\max }=40$ to $l_{\max }=60$ for $N=10$ (filled circles), $N=11$ (empty circles), $N=12$ (empty triangles) up to $N=30$ (empty squares).

Figure 2

Minimal value of $l_{\max }$ in order to have ${log}_{10}\frac{|{\beta }_c(l_{\max })-{\beta }_c(\infty ))|}{{\beta }_c(\infty )}=-20$ versus $N$.