We consider the zero-dimensional O(N) vector model as a simple example to calculate n-point correlation functions using perturbation theory, the large-N expansion, and the functional renormalization group (FRG). Comparing our findings with exact results, we show that perturbation theory breaks down for moderate interactions for all N, as one should expect. While the interaction-induced shift of the free energy and the self-energy are well described by the large-N expansion even for small N, this is not the case for higher-order correlation functions. However, using the FRG in its one-particle irreducible formalism, we see that very few running couplings suffice to get accurate results for arbitrary N in the strong coupling regime, outperforming the large-N expansion for small N. We further remark on how the derivative expansion, a well-known approximation strategy for the FRG, reduces to an exact method for the zero-dimensional O(N) vector model.
O(N)-invariant Hierarchical Renormalization Group Fixed Points by Algebraic Numerical Computation and ε-Expansion
推广Pinn，Portt和Wieczerkowski的方法，得到了一个分量（N=1），维数d在2到4之间的分层模型，我们计算了N和d（0 < d < 4，-2 <= N <= 20）的分层重整化群方程的O（N）对称不动点.计算了线性化RG方程在不动点处的谱，并从谱中提取了临界指数ν，并与Borel-Pade-Schummed ε-展开式进行了比较.
Operator product expansion coefficients from the nonperturbative functional renormalization group
利用Blazot-Ménén-Galain-Wschebor近似下的非微扰泛函重整化群（FRG），计算了三维$\mathrm{O}(N)$普适类和$2 \leq d \leq 4$维Ising普适类中与算子$\mathcal{O}_1\simφ$和$\mathcal{O}_2\simφ^2$相关的算子乘积展开（OPE）系数$c_{112}$.当可用时，从共形引导和蒙特卡罗模拟的精确结果和估计比较非常好，我们的结果，而FRG是能够提供的值在整个范围内的$d$和$N$考虑。
Anomalous scaling and large-scale anisotropy in magnetohydrodynamic turbulence: Two-loop renormalization-group analysis of the Kazantsev--Kraichnan kinematic model
