Density-Matrix Renormalization Group (DMRG) web page

• Snake is a free and open source DMRG program written in C++ and matlab which can be used to calculate the ground state, real-time dynamics and finite temperature properties of quantum 1D or impurity systems.


• Statistics of DMRG papers and citations from 1993 to 2012.
  

What is DMRG?

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 1300 scientific papers on the topic density matrix renormalization have been published through the end of the year 2008. 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.

Introductions to DMRG can be found in the books and lecture notes listed below. The numerous applications of DMRG are discussed in the listed review articles and conference proceedings.

Software

1. A DMRG code is included in the Algorithms and Libraries for Physics Simulations (ALPS) software package.


2. ITensor is a C++ library for creating efficient and flexible physics simulations based on tensor product wavefunctions.


3. A DMRG visualization applet created at the University of Göttingen (Germany) helps to understand how DMRG works


4. A DMRG code in Fortran90 is developed at the Scuola Normale Superiore di Pisa (Italy).


5. DMRG++ is a free and open source DMRG code using generic programming (C++ templates) which is developed at Oak Ridge National Laboratory (USA).


6. Snake is a free and open source DMRG program written in C++ and matlab which can be used to calculate the ground state, real-time dynamics and finite temperature properties of quantum 1D or impurity systems.

Other web sites about DMRG

• Nishino's DMRG homepage at the Kobe University (Japan).
• Article about DMRG in Wikipedia.

References

• Videos
  
  
  1. Videos of lectures on DMRG and tensor network states given at the Boulder Summer School 2010 by Steve White, Frank Verstraete and others
  
  
• Books
  
  
  1. Several chapters in the following book are devoted to an introduction to the DMRG method and its extensions: Computational Many Particle Physics, H. Fehske, R. Schneider, and A. Weiße (Eds.), Lecture Notes in Physics 739, Springer-Verlag, Berlin, Heidelberg, 2008, Part IX.
     See www.springeronline.com/978-3-540-74685-0.
  
  
  2. I. Peschel, X. Wang, M. Kaulke, and K. Hallberg (Eds.), Density-Matrix Renormalization, A New Numerical Method in Physics, in the Serie Lecture Notes in Physics , Springer, Berlin, 1999.


• Lecture notes
  
  
  1. Reinhard M. Noack and Salvatore R. Manmana, Diagonalization- and Numerical Renormalization-Group-Based Methods for Interacting Quantum Systems in AIP Conf. Proc. 789, 93-163 (2005).
     E-print: cond-mat/0510321.
  
  
  2. E. Jeckelmann and H. Fehske, Exact numerical methods for electron-phonon problems in Proceedings of the International School of Physics "Enrico Fermi" - Course CLXI on Polarons in Bulk Materials and Systems with Reduced Dimensionality, edited by G. Iadonisi, J. Ranninger, and G. De Filippis, pp. 247-284, IOS Press, Amsterdam, 2006.
     E-print: cond-mat/0510637.


• Recent review articles
  
  
  1. The density-matrix renormalization group in the age of matrix product states by U. Schollwöck, published in Annals of Physics 326, 96 (2011) -- doi:10.1016/j.aop.2010.09.012 (E-print: arXiv:1008.3477)
  
  
  2. Naokazu Shibata, Quantum Hall Systems Studied by the Density Matrix Renormalization Group Method, Prog. Theor. Phys. Suppl. 176, 182 (2008).
     E-Print: arXiv:0906.1036.
  
  
  3. E. Jeckelmann, Density-matrix renormalization group methods for momentum- and frequency-resolved dynamical correlation functions, Progress of Theoretical Physics Supplement 176, pp. 143-164 (2008).
     E-print: arXiv:0808.2620 [cond-mat.str-el].
  
  
  4. F. Verstraete, V. Murg, and J.I. Cirac, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Advances in Physics 57, pp 143-224 (2008).
  
  
  5. K. Hallberg, New Trends in Density Matrix Renormalization, Advances in Physics 55, pp 477-526 (2006).
     E-Print: cond-mat/0609039.
  
  
  6. U. Schollwöck and S.R. White, Methods for Time Dependence in DMRG, in G. G. Batrouni and D. Poilblanc (eds.), Effective models for low-dimensional strongly correlated systems, p. 155, AIP, Melville, New York (2006).
     E-print: cond-mat/0606018.
  
  
  7. U. Schollwöck, The density-matrix renormalization group, Rev. Mod. Phys. 77, 259 (2005) (57 pages).
     E-print: cond-mat/0409292.


• Conference proceedings
  1. Proceedings (talk slides) of the international workshop on New Development of Numerical Simulations in Low-Dimensional Quantum Systems: From Density Matrix Renormalization Group to Tensor Network Formulations at the Yukawa Institute for Theoretical Physics (YITP) of the Kyoto University (2010).
  
  
  2. Proceedings (talk slides, posters, and pictures) of the workshop Density Matrix Renormalization Group and other Advances in Numerical Renormalization Group Methods in Beijing (2010)
  
  
  3. Proceedings (pictures, abstracts, talk slides, and posters) of the workshop Recent Progress and Prospects in DMRG in Leiden (2004)
  
  
  4. The proceedings of the first DMRG conference in Dresden in 1998 were published in the book Density-Matrix Renormalization, A New Numerical Method in Physics, I. Peschel, X. Wang, M. Kaulke, and K. Hallberg (Eds.), in the Serie Lecture Notes in Physics , Springer, Berlin, 1999.


• Preprints
  
  
  • Nishino's list of e-prints related to DMRG.

Past conferences

• International workshop on New Development of Numerical Simulations in Low-Dimensional Quantum Systems: From Density Matrix Renormalization Group to Tensor Network Formulations at the Yukawa Institute for Theoretical Physics (YITP), Kyoto University, Japan, October 27-29, 2010.


• International workshop on Density Matrix Renormalization Group and other Advances in Numerical Renormalization Group Methods, Beijing, China, August 23 to September 3, 2010.


• International Workshop on Recent Progress and Prospects in DMRG held at the Lorentz Center, Leiden University, The Netherlands, August 2-13, 2004.


• International Seminar and Workshop on Density Matrix Renormalization Group and other recent Advances in Numerical Renormalization Group Methods, MPIKS, Dresden, Germany, August 24-September 18, 1998.