DensityMatrix Renormalization Group (DMRG) web page
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What is DMRG?
The Density Matrix Renormalization Group (DMRG) is one of the most powerful
numerical techniques for studying manybody 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 manybody 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 lowdimensional
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 matrixproduct state
has been emphasized.
From this point of view, DMRG is an algorithm for optimizing
a variational wavefunction with the structure of
a matrixproduct state.
This formulation of DMRG has revealed the deep connection between
the densitymatrix renormalization approach and quantum information theory
and has lead to significant extensions of DMRG algorithms.
In particular, efficient algorithms for simulating the timeevolution of quantum
manybody 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
 DMRG and TEBD codes are included in the
Algorithms and Libraries for Physics Simulations (ALPS)
software package.
 ITensor
is a C++ library for creating efficient and flexible physics simulations
based on tensor product wavefunctions.

A DMRG visualization applet created at the University of Göttingen
(Germany) helps to understand how DMRG works
 A DMRG code in Fortran90
is developed at the Scuola Normale Superiore di Pisa (Italy).
 DMRG++
is a free and open source DMRG code using generic programming (C++ templates)
which is developed at Oak Ridge National Laboratory (USA).
 Snake
is a free and open source DMRG program written in C++ and matlab
which can be used to calculate the ground state, realtime dynamics and finite
temperature properties of quantum 1D or impurity systems.
 BLOCK
implements the DMRG algorithm for quantum chemistry.
 An Open Source MPS code is in
preparation.
The preceding Open Source TEBD code has been included in the
ALPS package
Other web sites about DMRG
References
 Videos

Videos of lectures on DMRG and tensor network states given at the Boulder Summer School 2010
by Steve White, Frank Verstraete and others
 Books

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, SpringerVerlag,
Berlin, Heidelberg, 2008, Part IX.
See
www.springeronline.com/9783540746850.

I. Peschel, X. Wang, M. Kaulke, and K. Hallberg (Eds.),
DensityMatrix Renormalization, A New Numerical Method in Physics,
in the Serie Lecture Notes in Physics , Springer, Berlin, 1999.
 Lecture notes
 Reinhard M. Noack and Salvatore R. Manmana,
Diagonalization and Numerical RenormalizationGroupBased
Methods for Interacting Quantum Systems
in AIP Conf. Proc. 789, 93163 (2005).
Eprint: condmat/0510321.
 E. Jeckelmann and H. Fehske,
Exact numerical methods for electronphonon 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. 247284, IOS Press, Amsterdam, 2006.
Eprint: condmat/0510637.
 Review articles
 The densitymatrix 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
(Eprint: arXiv:1008.3477)
 Naokazu Shibata,
Quantum Hall Systems Studied by the Density Matrix Renormalization Group Method,
Prog. Theor. Phys. Suppl. 176, 182 (2008).
EPrint: arXiv:0906.1036.
 E. Jeckelmann,
Densitymatrix renormalization group methods for momentum and frequencyresolved dynamical
correlation functions,
Progress of Theoretical Physics Supplement 176, pp. 143164 (2008).
Eprint: arXiv:0808.2620 [condmat.strel].
 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 143224 (2008).
 K. Hallberg,
New Trends in Density Matrix Renormalization,
Advances in Physics 55, pp 477526 (2006).
EPrint: condmat/0609039.
 U. Schollwöck and S.R. White,
Methods for Time Dependence in DMRG,
in G. G. Batrouni and D. Poilblanc (eds.),
Effective models for lowdimensional strongly correlated systems,
p. 155, AIP, Melville, New York (2006).
Eprint: condmat/0606018.
 U. Schollwöck, The densitymatrix renormalization group,
Rev. Mod. Phys. 77, 259 (2005) (57 pages).
Eprint: condmat/0409292.
 Conference proceedings

Proceedings
(talk slides) of the international workshop on
New Development of Numerical Simulations in LowDimensional Quantum Systems:
From Density Matrix Renormalization Group to Tensor Network Formulations
at the Yukawa Institute for Theoretical Physics (YITP) of the Kyoto University (2010).
 Proceedings
(talk slides, posters, and pictures) of the
workshop
Density Matrix Renormalization Group and other Advances in Numerical Renormalization
Group Methods in Beijing (2010)
 Proceedings
(pictures, abstracts, talk slides, and posters) of
the workshop Recent Progress and
Prospects in DMRG in Leiden (2004)

The proceedings of the first DMRG conference in Dresden in 1998 were
published in the book
DensityMatrix 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
Past conferences
 International workshop on
New Development of Numerical Simulations in LowDimensional Quantum Systems:
From Density Matrix Renormalization Group to Tensor Network Formulations
at the Yukawa Institute for Theoretical Physics (YITP), Kyoto University, Japan,
October 2729, 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 213, 2004.

International Seminar and Workshop on
Density Matrix Renormalization Group
and other recent Advances in Numerical Renormalization Group Methods,
MPIKS, Dresden, Germany, August 24September 18, 1998.
URL: http://dmrg.info 
Eric Jeckelmann 
Last update: March 30, 2015.