Abstract
The density-matrix renormalization group (DMRG) method is widely used by computational physicists as a high accuracy tool to explore the ground state in large quantum lattice models, e.g., Heisenberg and Hubbard models, which are well-known standard models describing interacting spins and electrons, respectively, in solid states. After the DMRG method was originally developed for 1-D lattice/chain models, some specific extensions toward 2-D lattice (n-leg ladder) models have been proposed. However, high accuracy as obtained in 1-D models is not always guaranteed in their extended versions because the original exquisite advantage of the algorithm is partly lost. Thus, we choose an alternative way. It is a direct 2-D extension of DMRG method which instead demands an enormously large memory space, but the memory explosion is resolved by parallelizing the DMRG code with performance tuning. The parallelized direct extended DMRG shows a good accuracy like 1-D models and an excellent parallel efficiency as the number of states kept increases. This success promises accurate analysis on large 2-D (n-leg ladder) quantum lattice models in the near future when peta-flops parallel supercomputers are available.
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References
Cullum, J.K., Willoughby, R.A.: Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Vol. 1: Theory. SIAM, Philadelphia (2002)
Hager G., Wellein G., Jackemann E., Fehske H.: Stripe formation in dropped Hubbard ladders. Phys. Rev. B 71, 075108 (2005)
Hallberg K.A.: New trends in density matrix renormalization. Adv. Phys. 55, 477–526 (2006)
Kennedy, T., Nachtergaele, B.: The Heisenberg Model—a Bibliography (1996). http://www.math.ucdavis.edu/~bxn/qs.html
Knyazev A.V.: Preconditioned eigensolvers—an oxymoron?. Electron. Trans. Numer. Anal. 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)
Machida M., Yamada S., Ohashi Y., Matsumoto H.: Novel superfluidity on 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., Matsumoto H. et al.: Reply. Phys. Rev. Lett. 95, 218902 (2005)
Machida M., Yamada S., Ohashi Y., Matsumoto H.: On-site pairing and microscopic inhomogeneneity in confined lattice fermion systems. Phys. Rev. A 74, 053621 (2006)
Machida M., Okumura M., Yamada S.: Stripe formation in fermionic atoms on a two-dimensional optical lattice inside a box trap: Density-matrix renormalization-group studies for the repulsive Hubbard model with open boundary conditions. Phys. Rev. A 77, 033619 (2008)
Montorsi, A. (eds): The Hubbard Model. World Scientific, Singapore (1992)
Noack R.M., Manmana S.R.: Diagonalization and numerical renormalization—group-based methods for interacting quantum systems. Proc. AIP Conf. 789, 91–163 (2005)
Rasetti, M. (eds): The Hubbard Model: Recent Results. World Scientific, Singapore (1991)
Schollwoeck U.: The density-matrix renormalization group. Rev. Mod. Phys. 77, 259–315 (2005)
Schrieffer, J.R., Brooks, J.S. (eds): Hand Book of High-Temperature Superconductivity. Springer, New York (2007)
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)
White S.R., Scalapino D.J.: Stripes on a 6-Leg Hubbard Ladder. Phys. Rev. Lett. 91, 136403 (2003)
Yamada, S., Imamura, T., Machida, M.: 16.477 TFLOPS and 159-Billion-dimensional Exact-diagonalization for Trapped Fermion-Hubbard Model on the Earth Simulator. In: Proc. of SC2005 (2005)
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 SC2006 (2006)
Yamada S., Okumura M., Machida M.: Direct extension of density-matrix renormalization group toward two-dimensional quantum lattice systems: studies for parallel algorithm, accuracy, and performance. J. Phys. Soc. Jpn. 78, 094004 (2009)
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Yamada, S., Okumura, M., Imamura, T. et al. Direct extension of the density-matrix renormalization group method toward two-dimensional large quantum lattices and related high-performance computing. Japan J. Indust. Appl. Math. 28, 141–151 (2011). https://doi.org/10.1007/s13160-011-0027-z
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DOI: https://doi.org/10.1007/s13160-011-0027-z