标题： 局域化密度矩阵最小化和线性标度算法 显示英文标题

标题： Localized density matrix minimization and linear scaling algorithms

摘要： 我们提出了一种凸变分方法，通过在量子系统的自由能中添加逐元素的$\ell_1$正则化项，来计算零温度和有限温度情况下的局域密度矩阵。 基于绝缘体系统或有限温度系统中密度矩阵远离对角线处呈指数衰减的事实，所提出的$\ell_1$正则化变分方法提供了一种很好的近似原始量子系统的方法。 我们提供了近似行为的理论分析，并基于 Bregman 迭代设计了保证收敛的数值算法。 更重要的是，$\ell_1$正则化系统自然导致具有带状结构的局域密度矩阵，这使我们能够开发出计算成本与问题规模成线性关系的近似算法来寻找局域密度矩阵。 显示英文摘要

摘要： We propose a convex variational approach to compute localized density matrices for both zero temperature and finite temperature cases, by adding an entry-wise $\ell_1$ regularization to the free energy of the quantum system. Based on the fact that the density matrix decays exponential away from the diagonal for insulating system or system at finite temperature, the proposed $\ell_1$ regularized variational method provides a nice way to approximate the original quantum system. We provide theoretical analysis of the approximation behavior and also design convergence guaranteed numerical algorithms based on Bregman iteration. More importantly, the $\ell_1$ regularized system naturally leads to localized density matrices with banded structure, which enables us to develop approximating algorithms to find the localized density matrices with computation cost linearly dependent on the problem size.