# Density-matrix renormalization group

The Density Matrix Renormalization Group (DMRG) is one of the most powerful numerical techniques for studying many-body systems. It was developed in 1992 by Steven R. White at the University of California in Irvine to overcome the problems arising in the application of the Numerical Renormalization Group (NRG) to quantum lattice many-body systems such as the Hubbard model of strongly correlated electrons. Since then the approach has been extended to a great variety of problems in all fields of physics and to quantum chemistry. More than 1800 scientific papers on the topic *density matrix renormalization* have been published through the end of the year 2012. In 2003 Steve White was awarded the Aneesur Rahman Prize for Computational Physics by the American Physical Society for *his development, application, and dissemination of the numerical density matrix renormalization group (DMRG) method.* In addition, the original paper introducing DMRG has been selected as the milestone Physical Review Letter for the year 1992.

In applications to quantum lattice systems, DMRG consists in a systematic truncation of the system Hilbert space, keeping a small number of important states in a series of subsystems of increasing size to construct wave functions of the full system. In DMRG the states kept to construct a **renormalization group** transformation are the most probable eigenstates of a reduced **density matrix** instead of the lowest energy states kept in a standard NRG calculation. DMRG techniques for strongly correlated systems have been substantially improved and extended since their conception in 1992. They have proved to be both extremely accurate for low-dimensional problems and widely applicable. They enable numerically exact calculations (i.e., as good as exact diagonalizations) on large lattices with up to a few thousand particles and sites (compared to less than a few tens for exact diagonalizations).

Originally, DMRG has been considered as a renormalization group method. Recently, the interpretation of DMRG as a matrix-product state has been emphasized. From this point of view, DMRG is an algorithm for optimizing a variational wavefunction with the structure of a matrix-product state. This formulation of DMRG has revealed the deep connection between the density-matrix renormalization approach and quantum information theory and has lead to significant extensions of DMRG algorithms. In particular, efficient algorithms for simulating the time-evolution of quantum many-body systems have been developed since 2004.

## References

*Computational Many Particle Physics*, H. Fehske, R. Schneider, and A. Weiße (Eds.), Lecture Notes in Physics**739**, Springer-Verlag, Berlin, Heidelberg, 2008, Part IX*The density-matrix renormalization group in the age of matrix product states*by U. Schollwöck, published in Annals of Physics**326**, 96 (2011) or e-print arXiv:1008.3477- U. Schollwöck,
*The density-matrix renormalization group*, Rev. Mod. Phys.**77**, 259 (2005) or e-print cond-mat/0409292