Dependence of Critical Exponents Accuracy on the Number of Fields in the O(N) -Invariant \phi^4 Model

Abstract

The study of large-$N$ cases of the $O(N)$-symmetric model at criticality has attracted significant attention in recent years. In particular, a recent high-precision Monte Carlo study ( Physical Review B 105, 054428 (2022)) reported the most accurate estimates to date for the critical exponents $\nu$, $\eta$, and $\omega$ for $N \ge4$. Within the framework of the renormalization group (RG)---the cornerstone of the modern theory of critical phenomena---powerful computational approaches such as the $\varepsilon$-expansion and non-perturbative RG are available. Since the effective expansion parameter in the $\varepsilon$-expansion, $\sigma = \frac{3}{N+8}$, decreases with increasing N, one expects resummation techniques to become progressively more accurate in the large-N regime. To test this expectation, we apply the entire-hypergeometric resummation algorithm developed by Shalaby et al. to the recently obtained seven-loop divergent $\varepsilon$-series for $N \ge 4$, yielding high-precision estimates for $\nu$, $\eta$, and $\omega$ . Our analysis confirms the anticipated improvement in accuracy with increasing N. To assess the significance of these results, we note that at the same seven-loop order the $O(2)$ case exhibits errors that are an order of magnitude larger than those obtained from experiment, Monte Carlo simulations, and conformal-field-theory analysis. In contrast, for sufficiently large N, the errors in the present work are of the same order of magnitude as those from Monte Carlo and non-perturbative RG methods, demonstrating the strong predictive power of our resummation approach in the large-N regime.