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arXiv:cond-mat/9902279 · Physical review. B, Condensed matter

Two-dimensional infinite-system density-matrix renormalization-group algorithm

Patrik Henelius

Published Oct 1, 1999• 19 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.60.9561

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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.
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.
A variant of White's density-matrix renormalization-group scheme, which is designed to compute low-lying energies of one-dimensional quantum lattice models with a large number of degrees of freedom per site is described. The method is tested on two exactly solvable models---the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg chain and a dimerized XY\mathrm{XY}XY spin chain. To illustrate the potential of the method, it is applied to a model of spins interacting with quantum phonons. It is shown that the method accurately resolves a number of energy gaps on periodic rings that are sufficiently large to afford an accurate investigation of critical properties via the use of finite-size scaling theory.
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.
I revisit the infinite-size variant of the Density Matrix Renormalization Group (iDMRG) algorithm for obtaining a fixed-point translationally invariant matrix product wavefunction in the context of one-dimensional quantum systems. A crucial ingredient of this algorithm is an efficient transformation for obtaining the matrix elements of the wavefunction as the lattice size is increased, and I introduce here a versatile transformation that is demonstrated to be much more effective than previous versions. The resulting algorithm has a surprisingly close relationship to Vidal's Time Evolving Block Decimation for infinite systems, but allows much faster convergence. Access to a translationally invariant matrix product state allows the calculation of correlation functions based on the transfer matrix, which directly gives the spectrum of all correlation lengths. I also show some advantages of the Matrix Product Operator (MPO) technique for constructing expectation values of higher moments, such as the exact variance $$.
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.