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

Liang, Shoudan; Pang, Hanbin

doi:10.1103/physrevb.49.9214

Cited by 102 publications

(101 citation statements)

References 10 publications

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“…7, is through a bond dimension χ MPS that grows exponentially in the linear size of the lattice. Once more, however, using an MPS with very large χ MPS (which can again be afforded due to the relatively mild scaling of computational costs with χ MPS ) and finite size scaling arguments, an MPS has been successfully used to study ground states of two-dimensional lattice models [103][104][105][106][107] . The present analysis has also reminded us of an important limitation of PEPS and MERA in D > 1 dimensions.…”

Section: Discussionmentioning

confidence: 99%

Tensor Network States and Geometry

2011

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Tensor network states are used to approximate ground states of local Hamiltonians on a lattice in D spatial dimensions. Different types of tensor network states can be seen to generate different geometries. Matrix product states (MPS) in D = 1 dimensions, as well as projected entangled pair states (PEPS) in D > 1 dimensions, reproduce the D-dimensional physical geometry of the lattice model; in contrast, the multi-scale entanglement renormalization ansatz (MERA) generates a (D+1)-dimensional holographic geometry. Here we focus on homogeneous tensor networks, where all the tensors in the network are copies of the same tensor, and argue that certain structural properties of the resulting many-body states are preconditioned by the geometry of the tensor network and are therefore largely independent of the choice of variational parameters. Indeed, the asymptotic decay of correlations in homogeneous MPS and MERA for D = 1 systems is seen to be determined by the structure of geodesics in the physical and holographic geometries, respectively; whereas the asymptotic scaling of entanglement entropy is seen to always obey a simple boundary lawthat is, again in the relevant geometry. This geometrical interpretation offers a simple and unifying framework to understand the structural properties of, and helps clarify the relation between, different tensor network states. In addition, it has recently motivated the branching MERA, a generalization of the MERA capable of reproducing violations of the entropic boundary law in D > 1 dimensions.

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Section: Discussionmentioning

confidence: 99%

Tensor Network States and Geometry

2011

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“…In practice the extension of the DMRG to more than 1D is to map a higher dimensional lattice onto a 1D one, namely to choose a path to order all lattice sites [10]. The mapping breaks the lattice symmetry and introduces long range interactions among lattice sites.…”

Section: Introductionmentioning

confidence: 99%

Two-dimensional algorithm of the density-matrix renormalization group

Xiang

Lou

2001

Phys. Rev. B

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We propose a new approach to implement the density matrix renormalization group (DMRG) in two dimensions. With this approach the initial blocks of a L×L lattice are built up directly from the matrix elements of a (L − 1) × (L − 1) lattice and the topological characteristics of two dimensional lattices is preserved in the iteration of DMRG. By applying it to the spin-1/2 Heisenberg model on both square and triangle lattices, we find that this approach is significantly more efficient and accurate than other two-dimensional DMRG methods currently in use.

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“…The analysis made use of the Liang and Pang approach to 2D lattices [49], which unrolls the 2D lattice onto a 1D lattice with long-range interactions. The calculations included up to m = 1200 states per block.…”

Section: Applications Of the Non-abelian Dmrg Formalismmentioning

confidence: 99%

The density matrix renormalization group for finite fermi systems

Dukelsky

P̧ittel

2004

Rep. Prog. Phys.

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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.

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

Cited by 102 publications

References 10 publications

Tensor Network States and Geometry

Tensor Network States and Geometry

Two-dimensional algorithm of the density-matrix renormalization group

The density matrix renormalization group for finite fermi systems

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