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Physical review. B, Condensed matter

Approximate diagonalization using the density matrix renormalization-group method: A two-dimensional-systems perspective

Shoudan Liang, Hanbin Pang

Published Apr 1, 1994• 108 citations
Publisher

Abstract

In order to extend the density-matrix renormalization-group (DMRG) method to two-dimensional systems, we formulate two alternative methods to prepare the initial states. We find that the number of states that are needed for accurate energy calculations grows exponentially with the linear system size. We also analyze how the states kept in the DMRG method manage to preserve both the intrablock and interblock Hamiltonians, which is the key to the high accuracy of the method. We also prove that the energy calculated on a finite cluster is always a variational upper bound.

DOI
10.1103/physrevb.49.9214

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In order to extend the density-matrix renormalization-group (DMRG) method to two-dimensional systems, we formulate two alternative methods to prepare the initial states. We find that the number of states that are needed for accurate energy calculations grows exponentially with the linear system size. We also analyze how the states kept in the DMRG method manage to preserve both the intrablock and interblock Hamiltonians, which is the key to the high accuracy of the method. We also prove that the energy calculated on a finite cluster is always a variational upper bound.
In order to extend the density-matrix renormalization-group (DMRG) method to two-dimensional systems, we formulate two alternative methods to prepare the initial states. We find that the number of states that is needed for accurate energy calculations grows exponentially with the linear system size. We also analyze how the states kept in the DMRG method manage to preserve both the intrablock and interblock Hamiltonians, which is the key to the high accuracy of the method. We also prove that the energy calculated on a finite cluster is always a variational upper bound.
An extension of the density-matrix renormalization-group (DMRG) method is presented. Besides the two groups or classes of block states considered in White's formulation, the retained m states, and the neglected ones, we introduce an intermediate group of block states having the following p largest eigenvalues ${\ensuremath{\lambda}}_{i}$ of the reduced density matrix: ${\ensuremath{\lambda}}_{1}>~\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}>~{\ensuremath{\lambda}}_{m}>~{\ensuremath{\lambda}}_{m+1}>~\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}>~{\ensuremath{\lambda}}_{m+p}.$ These states are taken into account when they contribute to intrablock transitions but are neglected when they participate in more delocalized interblock fluctuations. Applications to one-dimensional models (Heisenberg, Hubbard, and dimerized tight binding) show that in this way the involved computer resources can be reduced without significant loss of accuracy. The efficiency and accuracy of the method is analyzed by varying m and p and by comparison with standard DMRG calculations. A Hamiltonian-independent scheme for choosing m and p and for extrapolating to the limit where m and p are infinite is provided. Finally, an extension of the 3-classes approach is outlined, which incorporates the fluctuations between the p states of different blocks as a self-consistent dressing of the block interactions among the retained m states.
The density-matrix renormalization-group (DMRG) method can provide approximate ground-state wave functions for many one-dimensional systems with a high degree of accuracy. Conventional DMRG techniques, however, cannot be expected to give accurate excited-state information because the excited states of interest are not explicitly included in the calculations. We discuss techniques for extending DMRG to accurately target and investigate arbitrary excited states. As an example we apply the method to investigate the properties of excitons within the extended Hubbard model. Our results show that excited-state calculations within the standard DMRG procedure give results that are quantitatively incorrect.
Abstract The physical properties of a quantum many-body system can, in principle, be determined by diagonalizing the respective Hamiltonian, but the dimensions of its matrix representation scale exponentially with the number of degrees of freedom. Hence, only small systems that are described through simple models can be tackled via exact diagonalization. To overcome this limitation, numerical methods based on the renormalization group paradigm that restrict the quantum many-body problem to a manageable subspace of the exponentially large full Hilbert space have been put forth. A striking example is the density-matrix renormalization group (DMRG), which has become the reference numerical method to obtain the low-energy properties of one-dimensional quantum systems with short-range interactions. Here, we provide a pedagogical introduction to DMRG, presenting both its original formulation and its modern tensor-network-based version. This colloquium sets itself apart from previous contributions in two ways. First, didactic code implementations are provided to bridge the gap between conceptual and practical understanding. Second, a concise and self-contained introduction to the tensor-network methods employed in the modern version of DMRG is given, thus allowing the reader to effortlessly cross the deep chasm between the two formulations of DMRG without having to explore the broad literature on tensor networks. We expect this pedagogical review to find wide readership among students and researchers who are taking their first steps in numerical simulations via DMRG. Graphic abstract
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.
A density-matrix renormalization-group (DMRG) method for calculating dynamical properties and excited states in low-dimensional lattice quantum many-body systems is presented. The method is based on an exact variational principle for dynamical correlation functions and the excited states contributing to them. This dynamical DMRG is an alternate formulation of the correction vector DMRG but is both simpler and more accurate. The finite-size scaling of spectral functions is discussed and a method for analyzing the scaling of dense spectra is described. The key idea of the method is a size-dependent broadening of the spectrum. The dynamical DMRG and the finite-size scaling analysis are demonstrated on the optical conductivity of the one-dimensional Peierls-Hubbard model. Comparisons with analytical results show that the spectral functions of infinite systems can be reproduced almost exactly with these techniques. The optical conductivity of the Mott-Peierls insulator is investigated and it is shown that its spectrum is qualitatively different from the simple spectra observed in Peierls (band) insulators and one-dimensional Mott-Hubbard insulators.