论文标题
严格的重新归一化小组的温和介绍：一个工作的典范
Gentle introduction to rigorous Renormalization Group: a worked fermionic example
论文作者
论文摘要
我们对关键现象的大部分理解是基于重新规范化组（RG）的概念，但是其固定点的实际确定通常基于近似值和截断，并且物理量的预测通常具有有限的精度。但是，可以给予RG固定点完全严格且非扰动的表征，这就是在此处介绍的符号式费米的模型，该模型具有非局部（“远程”）动力学项，这取决于参数$ \ varepsilon $和Quartic相互作用。我们确定了固定点所属的Banach相互作用空间，并通过收敛近似方案确定它。 Banach空间不仅限于相关的相互作用，但它包含与短距离内核的所有可能无关的术语，就像在大距离处像伸展指数一样衰减。由于该模型具有与$ ϕ^4 $或ISING模型共同的许多特征，因此结果可以用作测试RG研究中截断和近似值的有效性的基准。该分析基于来自建设性RG的结果，我们提供了教程和独立的介绍。此外，我们证明了固定点在$ \ varepsilon $中是分析性的，这是一个令人惊讶的事实，依赖于问题的费用性质。
Much of our understanding of critical phenomena is based on the notion of Renormalization Group (RG), but the actual determination of its fixed points is usually based on approximations and truncations, and predictions of physical quantities are often of limited accuracy. The RG fixed points can be however given a fully rigorous and non-perturbative characterization, and this is what is presented here in a model of symplectic fermions with a nonlocal ("long-range") kinetic term depending on a parameter $\varepsilon$ and a quartic interaction. We identify the Banach space of interactions, which the fixed point belongs to, and we determine it via a convergent approximation scheme. The Banach space is not limited to relevant interactions, but it contains all possible irrelevant terms with short-ranged kernels, decaying like a stretched exponential at large distances. As the model shares a number of features in common with $ϕ^4$ or Ising models, the result can be used as a benchmark to test the validity of truncations and approximations in RG studies. The analysis is based on results coming from Constructive RG to which we provide a tutorial and self-contained introduction. In addition, we prove that the fixed point is analytic in $\varepsilon$, a somewhat surprising fact relying on the fermionic nature of the problem.
