多体理论中张量网络的对称性约化I。$SU(2)$代数的自动符号求值
Symmetry reduction of tensor networks in many-body theory I. Automated symbolic evaluation of $SU(2)$ algebra
摘要：
随着（核）多体理论的不断发展，用于求解定态薛定谔方程的基本形式的复杂性也在不断增加。在原子核多体从头算方法中，只要有效地利用角动量（即$SU(2)$、量子数），就可以将其简化为解析的工作方程。相应的过程构成了一个繁琐且容易出错的过程，但却是这些多体框架实现的一个组成部分。事实上，这种对称性简化是将现代模拟推进到更高精度的关键一步，因为使用自适应张量可以将计算复杂度降低几个数量级。虽然在过去已经尝试了自动化的（反）交换规则与费米和玻色子代数在发挥作用的工作方程的推导，有没有系统的帐户，以实现相同的目标，他们的对称性减少。在这项工作中，第一个版本的自动化工具进行基于图论的角动量减少。以通用张量网络的无限制表达式作为输入，代码在几秒钟内以错误安全的方式提供其角动量约简形式。几个国家的最先进的多体方法作为例子，以证明该方法的一般性，并强调对多体社区的潜在影响。
英文摘要：
The ongoing progress in (nuclear) many-body theory is accompanied by an ever-rising increase in complexity of the underlying formalisms used to solve the stationary Schrödinger equation. The associated working equations at play in state-of-the-art ab initio nuclear many-body methods can be analytically reduced with respect to angular-momentum, i.e. $SU(2)$, quantum numbers whenever they are effectively employed in a symmetry-restricted context. The corresponding procedure constitutes a tedious and error-prone but yet an integral part of the implementation of those many-body frameworks. Indeed, this symmetry reduction is a key step to advance modern simulations to higher accuracy since the use of symmetry-adapted tensors can decrease the computational complexity by orders of magnitude. While attempts have been made in the past to automate the (anti-) commutation rules linked to Fermionic and Bosonic algebras at play in the derivation of the working equations, there is no systematic account to achieve the same goal for their symmetry reduction. In this work, the first version of an automated tool performing graph-theory-based angular-momentum reduction is presented. Taking the symmetry-unrestricted expressions of a generic tensor network as an input, the code provides their angular-momentum-reduced form in an error-safe way in a matter of seconds. Several state-of-the-art many-body methods serve as examples to demonstrate the generality of the approach and to highlight the potential impact on the many-body community.
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