J. Phys. Soc. Jpn. 77, 114713 (2008) [8 Pages]

Shifted Conjugate-Orthogonal–Conjugate-Gradient Method and Its Application to Double Orbital Extended Hubbard Model

+ Affiliations
1Center for Research and Development of Higher Education, The University of Tokyo, Tokyo 113-00332Core Research for Evolutional Science and Technology, Japan Science and Technology Agency (CREST-JST), Sanbancho Bldg., 5 Sanbancho, Chiyoda-ku, Tokyo 102-00753Department of Computational Science and Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-86034Department of Applied Mathematics and Physics, Tottori University, 4-101 Koyama-Minami, Tottori 680-8550

We discuss the shifted conjugate-orthogonal–conjugate-gradient (COCG) method, which can be used to solve a series of linear equations generated by a number of scalar shifts, without performing time-consuming matrix–vector operations, except at a single reference energy. This is a type of CG method and is robust and has an expression of the accuracy of the calculated result that requires no additional calculation cost. The shifted COCG is useful for calculating the Green's function of a many-electron Hamiltonian, which has a very large dimension. We applied the shifted COCG method to the double orbital extended Hubbard model with twelve electrons on a periodic \(\sqrt{8} \times \sqrt{8}\) site system, with the dimension of the Hamiltonian equal to 64,128,064, and we found that the ground state is an insulator. We also discuss the main points of reducing the amount of memory required to apply COCG algorithm.

©2008 The Physical Society of Japan


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