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Localized density matrix minimization and linear-scaling algorithms. (English) Zbl 1349.82007
Summary: 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 exponentially away from the diagonal for insulating systems or systems at finite temperature, the proposed $$\ell_1$$ 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 $$\ell_1$$ 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.

##### MSC:
 82B10 Quantum equilibrium statistical mechanics (general) 65F99 Numerical linear algebra 65K10 Numerical optimization and variational techniques
SIESTA; UNLocBoX
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