基于O（N）格林函数的多费米子经典自由度耦合Monte Carlo方法的数值化研究
A Study of Parallelizing O(N) Green-Function-Based Monte Carlo Method for Many Fermions Coupled with Classical Degrees of Freedom
摘要：
在凝聚态物理中，费米子与经典自由度相互作用的模型被应用于各种各样的系统。对于这类模型，Weiße [Phys. Rev. Lett. { \bf 102}，150604（2009）]最近提出了一种非常有效的数值方法，称为O（$N$）基于格林函数的蒙特卡罗（GFMC）方法，其中使用核多项式展开技术来避免大小为$N$的费米子哈密顿矩阵的完全数值对角化，其通常花费O（$N^3$）计算复杂度。基于这一背景，本文将GFMC方法应用于三维空间的双交换模型。我们主要集中在GFMC方法的实现使用MPI在基于CPU的集群和Nvidia的计算统一设备架构（CUDA）编程技术在基于GPU的（基于图形处理单元）集群。讨论了该算法的时间复杂度和在机群上的并行实现细节。我们还显示了性能缩放的哈密顿矩阵的大小和节点的数量增加，分别。性能评估表明，对于一个$32^3$哈密顿函数，单个GPU表现出更高的性能，相当于使用MPI并行化的30多个CPU核心。
英文摘要：
Models of fermions interacting with classical degrees of freedom are applied to a large variety of systems in condensed matter physics. For this class of models, Weiße [Phys. Rev. Lett. {\bf 102}, 150604 (2009)] has recently proposed a very efficient numerical method, called O($N$) Green-Function-Based Monte Carlo (GFMC) method, where a kernel polynomial expansion technique is used to avoid the full numerical diagonalization of the fermion Hamiltonian matrix of size $N$, which usually costs O($N^3$) computational complexity. Motivated by this background, in this paper we apply the GFMC method to the double exchange model in three spatial dimensions. We mainly focus on the implementation of GFMC method using both MPI on a CPU-based cluster and Nvidia's Compute Unified Device Architecture (CUDA) programming techniques on a GPU-based (Graphics Processing Unit based) cluster. The time complexity of the algorithm and the parallel implementation details on the clusters are discussed. We also show the performance scaling for increasing Hamiltonian matrix size and increasing number of nodes, respectively. The performance evaluation indicates that for a $32^3$ Hamiltonian a single GPU shows higher performance equivalent to more than 30 CPU cores parallelized using MPI.
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