# The Density Matrix Renormalization Group and its time-dependent variants

#### Abstract

Since its creation in 1992, the density matrix renormalization group (DMRG) method [1] has evolved and mutated. From its original formulation in a condensed matter context, it has been adapted to study problems in verious fields, such as nuclear physics and quantum chemistry, to become one of the dominant numerical methods to study strongly correlated systems. The purpose of these lectures is to provide a clear and pedagogical introduction to the DMRG, and its time-dependent variants, using simple examples, and pieces of code. Reviews on the method abound [2, 3, 4, 5, 6, 7]. In terms of nice introductions, I refer the reader to the lectures notes by Noack and Manmanna, also originated after the Vietri School[6]. When planning this set of lectures, I decided that I would try to conceive them as a natural continuation of this work. Therefore, I strongly encourage the reader to look at Ref.[6] first as a warm-up. In these lectures, we shall cover many technical aspects of the DMRG, from a “traditional”, or “conventional” perspective, describing the theoretical fundamentation, as well as the details of the algorithm, briefly touching on some recent developments in connection to matrix product states (MPS). In the second part, I will describe the time-dependent extensions of the method, illustrating its application with several examples. I will accompany the discussion with pseudo code, and code snippets that can be freely downloaded from http://physics.uwyo.edu/~adrian/programs/. Documentation on how to install and run these applications is provided in the appendices. The programs are in C++, but written on top of libraries that implement all the basic objects, such as vectors and matrices. Therefore, the codes read pretty much like Matlab, and hopefully the reader will find little difficulty understanding them. To conclude, I will provide a basic tutorial of the ALPS (Algorithms for Physics Simulations)[9] DMRG code. ALPS (http://alps.comp-phys.org/) provides a framework and collection of libraries and programs for simulating strongly correlated problems. The DMRG application is part of this collection that includes other powerful methods, such as exact diagonalization, quantum and classical Monte Carlo, and dynamical mean-field (DMFT), to mention a few. The ALPS DMRG application puts all the power of the DMRG method in a very friendly and accessible interface that enables anyone with no previous knowledge of the method or coding to run state-of-the-art simulations.

### Cite this paper

@inproceedings{Feiguin2012TheDM, title={The Density Matrix Renormalization Group and its time-dependent variants}, author={Adrian E. Feiguin}, year={2012} }