Abstract
The Density Matrix Renormalization Group (DMRG) method is widely used by computational physicists as a high accuracy tool to obtain the ground state of large quantum lattice models. Since the DMRG method has been originally developed for 1-D models, many extended method to a 2-D model have been proposed. However, some of them have issues in term of their accuracy. It is expected that the accuracy of the DMRG method extended directly to 2-D models is excellent. The direct extension DMRG method demands an enormous memory space. Therefore, we parallelize the matrix-vector multiplication in iterative methods for solving the eigenvalue problem, which is the most time- and memory-consuming operation. We find that the parallel efficiency of the direct extension DMRG method shows a good one as the number of states kept increases.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Kennedy, T., Nachtergaele, B.: The Heisenberg Model - a Bibliography (1996), http://math.arizona.edu/~tgk/qs.html
Rasetti, M. (ed.): The Hubbard Model: Recent Results. World Scientific, Singapore (1991)
Montorsi, A. (ed.): The Hubbard Model. World Scientific, Singapore (1992)
Machida, M., Yamada, S., Ohashi, Y., Matsumoto, H.: Novel superfluidity in a trapped gas of Fermi atoms with repulsive interaction loaded on an optical lattice. Phys. Rev. Lett. 93, 200402 (2004)
Machida, M., Yamada, S., Ohashi, Y., Matsumoto, H., Machida, et al.: Reply. Phys. Rev. Lett. 95, 218902 (2005)
Yamada, S., Imamura, T., Machida, M.: 16.447 TFlops and 159-Billion-dimensional Exact-diagonalization for Trapped Fermion-Hubbard Model on the Earth Simulator. In: Proc. of SC 2005 (2005), http://sc05.supercomputing.org/schedule/pdf/pap188.pdf
Yamada, S., Imamura, T., Kano, T., Machida, M.: High-Performance Computing for Exact Numerical Approaches to Quantum Many-Body Problems on the Earth Simulator. In: Proc. of SC 2006 (2006), http://sc06.supercomputing.org/schedule/pdf/gb113.pdf
White, S.R.: Density Matrix Formulation for Quantum Renormalization Groups. Phys. Rev. Lett. 69, 2863–2866 (1992)
White, S.R.: Density-matrix algorithms for quantum renormalization groups. Phys. Rev. B 48, 10345–10355 (1993)
Hager, G., Wellein, G., Jackemann, E., Fehske, H.: Stripe formation in dropped Hubbard ladders. Phys. Rev. B 71, 75108 (2005)
Noack, R.M., Manmana, S.R.: Diagonalization and Numerical Renormalization- Group-Based Methods for Interacting Quantum Systems. In: Proc. of AIP Conf., vol. 789, pp. 91–163 (2005)
Cullum, J.K., Willoughby, R.A.: Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Theory, vol. 1. SIAM, Philadelphia (2002)
Knyazev, A.V.: Preconditioned eigensolvers - An oxymoron? Electronic Transactions on Numerical analysis 7, 104–123 (1998)
Knyazev, A.V.: Toward the optimal eigensolver: Locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput. 23, 517–541 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Yamada, S., Okumura, M., Machida, M. (2008). High Performance Computing for Eigenvalue Solver in Density-Matrix Renormalization Group Method: Parallelization of the Hamiltonian Matrix-Vector Multiplication. In: Palma, J.M.L.M., Amestoy, P.R., Daydé, M., Mattoso, M., Lopes, J.C. (eds) High Performance Computing for Computational Science - VECPAR 2008. VECPAR 2008. Lecture Notes in Computer Science, vol 5336. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92859-1_5
Download citation
DOI: https://doi.org/10.1007/978-3-540-92859-1_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-92858-4
Online ISBN: 978-3-540-92859-1
eBook Packages: Computer ScienceComputer Science (R0)