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arXiv:cond-mat/9403069

Can Density-matrix renormalization group be Applied to two dimensional systems?

Shoudan Liang, Hanbin Pang

Published Mar 17, 1994• 0 citations
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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 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.

DOI
10.48550/arxiv.cond-mat/9403069

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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 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.
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.
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.
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 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 highly anisotropic two-dimensional (2D) systems is presented. The method consists of applying the usual DMRG in two steps. In the first step, a pure one-dimensional calculation along the longitudinal direction is made in order to generate a low-energy Hamiltonian. In the second step, the anisotropic 2D lattice is obtained by coupling in the transverse direction the 1D Hamiltonians. The method is applied to the anisotropic quantum spin-half Heisenberg model on a square lattice.
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.