New trends in density matrix renormalization

Abstract

The density matrix renormalization group (DMRG) has become a powerful numerical method that can be applied to low-dimensional strongly correlated fermionic and bosonic systems. It allows for a very precise calculation of static, dynamic and thermodynamic properties. Its field of applicability has now extended beyond condensed matter, and is successfully used in quantum chemistry, statistical mechanics, quantum information theory, and nuclear and high-energy physics as well. In this article, we briefly review the main aspects of the method and present some of the most relevant applications so as to give an overview of the scope and possibilities of DMRG. We focus on the most important extensions of the method such as the calculation of dynamical properties, the application to classical systems, finite-temperature simulations, phonons and disorder, field theory, time-dependent properties and the ab initio calculation of electronic states in molecules. The recent quantum information interpretation, the development of highly accurate time-dependent algorithms and the possibility of using the DMRG as the impurity-solver of the dynamical mean field method (DMFT) give new insights into its present and potential uses. We review the numerous very recent applications of these techniques where the DMRG has shown to be one of the most reliable and versatile methods in modern computational physics.

Keywords:

• Density matrix renormalization
• Low-dimensional systems
• Numerical methods
• Quantum information
• Strongly correlated electrons

Acknowledgments

The author is grateful for numerous fruitful discussions on DMRG with most of the people involved with this method, in particular with P. Horsch (who drew her attention to the method in its very early stages), S. White, I. Peschel, X. Wang, T. Xiang, T. Nishino, M.A. Martín-Delgado, G. Sierra, S. Ramasesha, E. Jeckelmann, R. Noack, U. Schollwöck, P. Schmitteckert, D. García and M. Rozenberg.

The author is Staff Researcher at the National Council for Science and Technology, CONICET (Argentina) and Fellow of the Guggenheim Foundation (2005). This project was performed under grants PICT 02 03-12742 and PICT 03-13829 of the ANPCyT (Argentina).