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Cambridge University Press eBooks

Real-Space DMRG

Tao Xiang

Published Aug 31, 2023• 0 citations
Publisher

Abstract

This chapter introduces the density matrix renormalization group (DMRG) in real space. The infinite and finite lattice algorithms of DMRG, and the approaches for targeting more than one eigenstate and for implementing DMRG in two dimensions by mapping a two-dimensional lattice onto a one-dimensional one, are discussed. The one-dimensional antiferromagnetic Heisenberg model of both integer and half-integer spins is used to demonstrate the method.

DOI
10.1017/9781009398671.008

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•Jan 1, 2002•0 citations
This chapter introduces the density matrix renormalization group (DMRG) in real space. The infinite and finite lattice algorithms of DMRG, and the approaches for targeting more than one eigenstate and for implementing DMRG in two dimensions by mapping a two-dimensional lattice onto a one-dimensional one, are discussed. The one-dimensional antiferromagnetic Heisenberg model of both integer and half-integer spins is used to demonstrate the method.
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.
We propose an approach to implement the density-matrix renormalization group (DMRG) in two dimensions. With this approach the initial blocks of a $L\ifmmode\times\else\texttimes\fi{}L$ lattice are built up directly from the matrix elements of a $(L\ensuremath{-}1)\ifmmode\times\else\texttimes\fi{}(L\ensuremath{-}1)$ lattice and the topological characteristics of two-dimensional lattices are preserved in the iteration of DMRG. By applying it to the spin-1/2 Heisenberg model on both square and triangular lattices, we find that this approach is significantly more efficient and accurate than other two-dimensional DMRG methods currently in use.
A density matrix renormalization group (DMRG) approach is presented for quantum systems of two spatial dimensions. In particular, it is shown that it is possible to create a multichain-type two-dimensional (2D) DMRG approach that utilizes previously determined system and environment blocks at all points. One first builds up an effective quasi-1D system and environment blocks of width L and these quasi-1D blocks are then used to as the initial building blocks of the 2D infinite-lattice algorithm introduced here. This algorithm is found to be competitive with those results of previous 2D DMRG algorithms and also of the best of other approximate methods. An illustration of this is given for the spin-half and spin-one Heisenberg models on the square lattice. The best results for the ground-state energies per bond of the spin-half and spin-one square-lattice Heisenberg antiferromagnets for the $N=20\ifmmode\times\else\texttimes\fi{}20$ lattice using this treatment are given by ${E}_{g}{/N}_{B}=\ensuremath{-}0.3321$ and ${E}_{g}{/N}_{B}=\ensuremath{-}1.1525,$ respectively.
Abstract. The Density Matrix Renormalization Group (DMRG) was introduced by Steven White in 1992 as a method for accurately describing the properties of onedimensional 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 realspace 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 an outlook for future applications of the method.
We demonstrate how to parallelize the density matrix renormalization group (DMRG) algorithm in real space through a straightforward modification of serial DMRG. This makes it possible to apply at least an order of magnitude more computational power to challenging simulations, greatly accelerating investigations of two-dimensional systems and large parameter spaces. We discuss details of the algorithm and present benchmark results including a study of valence-bond-solid order within the square-lattice Q2{Q}_{2}Q2​ model and N\'eel order within the triangular lattice Heisenberg model. The parallel DMRG algorithm also motivates an alternative canonical form for matrix product states.
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.