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Symmetry reduction of tensor networks in many-body theory
The European Physical Journal A ( IF 2.7 ) Pub Date : 2020-10-26 , DOI: 10.1140/epja/s10050-020-00233-6
A. Tichai , R. Wirth , J. Ripoche , T. Duguet

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.



中文翻译:

多体理论中张量网络的对称约简

(核)多体理论的不断发展伴随着用于解决平稳薛定ding方程的基本形式主义复杂性的不断提高。可以相对于角动量(即SU)解析地简化现有的从头算核多体方法中起作用的相关工作方程(2),在对称限制的情况下有效使用量子数。相应的过程构成了繁琐且易于出错的过程,但仍是那些多主体框架的实现中不可或缺的一部分。实际上,这种对称性降低是将现代仿真提高到更高精确度的关键步骤,因为使用自适应对称张量可以将计算复杂度降低几个数量级。尽管过去曾试图自动化与费米离子和玻色子代数相关的(反)换向规则,但在工作方程的推导中却发挥了作用,但还没有系统地实现对称性降低的相同目标。在这项工作中,介绍了执行基于图论的角动量减少的自动化工具的第一版。以通用张量网络的对称性不受限制的表达式作为输入,该代码以错误安全的方式在几秒钟内提供了减小角动量的形式。几种最新的多体方法用作示例,以说明该方法的一般性并突出显示对多体社区的潜在影响。

更新日期:2020-10-30
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