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arXiv:cond-mat/0309057 · Physical review. B, Condensed matter

Density matrix renormalization group calculations for two-dimensional lattices: Application to the spin-half and spin-one square-lattice Heisenberg models

DAMIAN J. J. FARNELL

Published Oct 13, 2003• 15 citations

Abstract

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.

DOI: 10.1103/physrevb.68.134419

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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.

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.

It has proved difficult to extend the density-matrix renormalization-group technique to large two-dimensional systems. I here present an approach where the calculation is done directly in two dimensions. This makes it possible to use an infinite-system method, and the fixed point in two dimensions is studied. By applying several related blocking schemes to the two-dimensional Heisenberg model I find that there exists an algorithm for which the energy per site decreases monotonically as the system size increases, thereby showing the potential feasibility of this method.

An anisotropic Heisenberg spin S = 1 model on a two-dimensional hexagonal lattice (formed from interacting chains) with antiferromagnetic nearest-neighbor interactions of two types is analyzed by the real space renormalization group method. The problem of the influence of interchain pairing on the critical properties of the model is studied, and the phase diagram of the model is constructed. The two-dimensional density matrix renormalization group algorithm is used to calculate the ground state energy for the isotropic case as a function of the ratio between interchain and intrachain interactions.

The application of real space renormalization group methods to quantum lattice models has become a topic of great interest following the development of the density matrix renormalization group by White.This method has been used to find the ground and low-lying excited state energies and wave functions of quantum spin models in which the form of the ground state is not clear, for instance because the interactions are frustrated.It has also been applied to fermion problems where the tendency for localization due to the strong Coulomb repulsion is opposed by the lowering of the kinetic energy which occurs as a result of electron transfer.The approach is particularly suitable for one-, or quasi-one-dimensional problems.The method involves truncating the Hilbert space in a systematic and optimised manner.Results for the ground state energy are thus variational bounds.The results for low-lying energies and correlation functions for one-dimensional systems have unprecedented accuracy and the method has become the method of choice for solving one-dimensional quantum spin problems.We review the method and results obtained for the spin-1 chain with biquadratic exchange as well as the spin-1/2 model with competing nearest and next nearest neighbour exchange will be described.More recently, the density matrix renormalization group has been applied to reformulate the coupling constant renormalization group approach which is appropriate for the study of critical properties.This approach has been applied to the anisotropic spin-1/2 Heisenberg chain.Finally, we discuss recent work which has borne promising applications in two dimensions -the Ising model and the two-dimensional Hubbard model.