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Theory and Application of the Density Matrix Renormalization Group

Ming-Chiang Chung

Published Jan 1, 2002• 0 citations
Publisher

Abstract

I begin, in chapter 2, by discussing the DMRG algorithms.The infinite-system algorithm and finite-system algorithm will be introduced. In chapter 3, coupled oscillators in one and two dimensions will be discussed in detail. As a bridge between bosonic and fermionic systems, I will consider the coherent- state treatment for solvable bosons in chapter 4. In chapter 5 the solvable fermionic systems in one and two dimension will be the subject. As an application, in chapter 6, the Ising plane with line-like defects will be treated using the transfer-matrix DMRG. Chapter 7, finally, contains a summary of the main results.

DOI
10.17169/refubium-4748

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I begin, in chapter 2, by discussing the DMRG algorithms.The infinite-system algorithm and finite-system algorithm will be introduced. In chapter 3, coupled oscillators in one and two dimensions will be discussed in detail. As a bridge between bosonic and fermionic systems, I will consider the coherent- state treatment for solvable bosons in chapter 4. In chapter 5 the solvable fermionic systems in one and two dimension will be the subject. As an application, in chapter 6, the Ising plane with line-like defects will be treated using the transfer-matrix DMRG. Chapter 7, finally, contains a summary of the main results.
The density matrix renormalization group (DMRG) method and its applications to finite temperatures and two-dimensional systems are reviewed. The basic idea of the original DMRG method, which allows precise study of the ground-state properties and low-energy excitations, is presented for models which include long-range interactions. The DMRG scheme is then applied to the diagonalization of the quantum transfer matrix for one-dimensional systems, and a reliable algorithm at finite temperatures is formulated. Dynamic correlation functions at finite temperatures are calculated from the eigenvectors of the quantum transfer matrix with analytical continuation to the real frequency axis. An application of the DMRG method to two-dimensional quantum systems in a magnetic field is demonstrated and reliable results for quantum Hall systems are presented.
(April 2006) 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, 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 on 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
The density matrix renormalization group (DMRG) was introduced by Steven White in 1992 as a method for accurately describing the properties of one-dimensional quantum lattices. The method, as originally introduced, was based on the iterative inclusion of sites on a real-space lattice. Based on its enormous success in that domain, it was subsequently proposed that the DMRG could be modified for use on finite Fermi systems, through the replacement of real-space lattice sites by an appropriately ordered set of single-particle levels. Since then, there has been an enormous amount of work on the subject, ranging from efforts to clarify the optimal means of implementing the algorithm to extensive applications in a variety of fields. In this article, we review these recent developments. Following a description of the real-space DMRG method, we discuss the key steps that were undertaken to modify it for use on finite Fermi systems and then describe its applications to quantum chemistry, ultrasmall superconducting grains, finite nuclei and two-dimensional electron systems. We also describe a recent development which permits symmetries to be taken into account consistently throughout the DMRG algorithm. We close with a description of the outlook for future applications of the method.
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.
The density matrix renormalization group (DMRG) is one of the most powerful numerical methods for studying two-dimensional quantum lattice systems, despite a perception that it is only suitable for one dimension. Reviewing past applications of DMRG in 2D demonstrates its success in treating a wide variety of problems, although it remains underutilized in this context. We present techniques for performing cutting-edge 2D DMRG studies including methods for ensuring convergence, extrapolating finite-size data, and extracting gaps and excited states. Finally, we consider a selection of recently developed 2D tensor network methods and compare the performance of one of these to 2D DMRG.