Localized density matrix minimization and linear-scaling algorithms

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• 作者：Rongjie Lai^a ; ^{lair@rpi.edu" class="auth_mail" title="E-mail the corresponding author} ; Jianfeng Lu^b ; ^{jianfeng@math.duke.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Localized density matrix ; ℓ_{1ℓ1 norm} ; Hamiltonian ; Finite temperature ; Linear-scaling algorithms
• 刊名：Journal of Computational Physics
• 出版年：2016
• 出版时间：15 June 2016
• 年：2016
• 卷：315
• 期：Complete
• 页码：194-210
• 全文大小：1593 K

文摘

We propose a convex variational approach to compute localized density matrices for both zero temperature and finite temperature cases, by adding an entry-wise 1" class="mathmlsrc">1-s2.0-S0021999116300146&_mathId=si1.gif&_user=111111111&_pii=S0021999116300146&_rdoc=1&_issn=00219991&md5=98df40878ca53242ae3c9bdc631d6529" title="Click to view the MathML source">ℓ₁ regularization to the free energy of the quantum system. Based on the fact that the density matrix decays exponentially away from the diagonal for insulating systems or systems at finite temperature, the proposed 1" class="mathmlsrc">1-s2.0-S0021999116300146&_mathId=si1.gif&_user=111111111&_pii=S0021999116300146&_rdoc=1&_issn=00219991&md5=98df40878ca53242ae3c9bdc631d6529" title="Click to view the MathML source">ℓ₁ regularized variational method provides an effective 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 1" class="mathmlsrc">1-s2.0-S0021999116300146&_mathId=si1.gif&_user=111111111&_pii=S0021999116300146&_rdoc=1&_issn=00219991&md5=98df40878ca53242ae3c9bdc631d6529" title="Click to view the MathML source">ℓ₁ 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.