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Springer eBooks

Applications of Quantum Information in the Density-Matrix Renormalization Group

Örs Legeza, R. M. Noack, Jenő Sólyom, L. Tincani

Published Dec 8, 2007• 64 citations
Publisher
DOI
10.1007/978-3-540-74686-7_24

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The density-matrix renormalization group (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly become the method of choice for numerical studies of such systems. Its applications to the calculation of static, dynamic, and thermodynamic quantities in these systems are reviewed here. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and nonequilibrium statistical physics, and time-dependent phenomena is also discussed. This review additionally considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by the DMRG.
This chapter starts with an introductory survey on the physical background and historical events that lead to the emergence of the density matrix renormalization group (DMRG) and its tensor network generalization. We then briefly overview the major progress on the renormalization group methods of tensor networks and their applications in the past three decades. The tensor network renormalization was initially developed to solve quantum many-body problems, but its application field has grown constantly. It has now become an irreplaceable tool for investigating strongly correlated problems, statistical physics, quantum information, quantum chemistry, and artificial intelligence.