  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.
