arXiv:cond-mat/0404706 · Physical Review Letters

Density Matrix Renormalization Group and Periodic Boundary Conditions: A Quantum Information Perspective

Frank Verstraete, Diego Porras, J. I. Cirac

Published Nov 23, 2004• 580 citations

Abstract

We introduce a picture to analyze the density matrix renormalization group (DMRG) numerical method from a quantum information perspective. This leads to a variational formulation of DMRG which allows for dramatic improvements in the case of problems with periodic boundary conditions. The picture also explains some features of the method in terms of entanglement and teleportation.

DOI: 10.1103/physrevlett.93.227205

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arXiv:cond-mat/0404706 · Physical Review Letters

Density Matrix Renormalization Group and Periodic Boundary Conditions: A Quantum Information Perspective

Frank Verstraete, Diego Porras, J. I. Cirac

Published Nov 23, 2004• 580 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevlett.93.227205

Related papers

Tensor Networks: a quantum-information perspective on numerical renormalization groups

Pietro Silvi

Exact many-body quantum problems are known to be computationally hard due to the exponential scaling of the numerical resources required. Since the advent of the Density Matrix Renormalization Group, it became clear that a successful strategy to work around this obstacle was to develop numerical methods based on the well-known theoretical renormalization group. In recent years, it was realized that quantum states engineered via numerical renormalization allow a variational representation in terms of a tensor network picture. The discovery provided a further boost to the effectiveness of these techniques, not only due to the increased flexibility and manipulability, but also because tensor network states embed a direct interface to the entanglement they carry, so that one can directly address many-body quantum correlations within these variational ansatz states. This lead to the application of several numerical tools, originally developed in the field of quantum-information, to approach condensed matter problems.Exact many-body quantum problems are known to be computationally hard due to the exponential scaling of the numerical resources required. Since the advent of the Density Matrix Renormalization Group, it became clear that a successful strategy to work around this obstacle was to develop numerical methods based on the well-known theoretical renormalization group. In recent years, it was realized that quantum states engineered via numerical renormalization allow a variational representation in terms of a tensor network picture. The discovery provided a further boost to the effectiveness of these techniques, not only due to the increased flexibility and manipulability, but also because tensor network states embed a direct interface to the entanglement they carry, so that one can directly address many-body quantum correlations within these variational ansatz states. This lead to the application of several numerical tools, originally developed in the field of quantum-information, to approach condensed matter problems.

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{\mathrm{sin}}^{2}

-deformed interaction [A. Gendiar, R. Krcmar, and T. Nishino, Prog. Theor. Phys.122, 953 (2009)] keep spatial homogeneity, and are relatively free from the boundary effect. By comparing the numerical results of those various boundary conditions, we show that the open boundary condition sometimes gives unreliable results even after the finite-size scaling. We conclude that the examination of the boundary condition dependence is required besides the usual treatment based on the system size or accuracy dependence in cases where the long-range quantum fluctuation is important.

{m}^{5}

{m}^{3}

, where

m

is the matrix dimension, and $m\ensuremath{\simeq}100--1000$ in typical cases. We test the method on the

S=\frac{1}{2}

and

S=1

Heisenberg chains. It is also applicable to nontranslationally invariant cases. The method makes ground-state calculations with periodic boundary conditions about as efficient as traditional DMRG calculations for systems with open boundaries.

q=3,

4 Potts model, as well.