Efficient perturbation theory to improve the density matrix renormalization group

Tirrito, Emanuele, Ran, Shi-Ju, Ferris, Andrew J., McCulloch, Ian P. and Lewenstein, Maciej (2017) Efficient perturbation theory to improve the density matrix renormalization group. Physical Review B: Condensed Matter and Materials Physics, 95 6: . doi:10.1103/PhysRevB.95.064110


Author Tirrito, Emanuele
Ran, Shi-Ju
Ferris, Andrew J.
McCulloch, Ian P.
Lewenstein, Maciej
Title Efficient perturbation theory to improve the density matrix renormalization group
Journal name Physical Review B: Condensed Matter and Materials Physics   Check publisher's open access policy
ISSN 1550-235X
2469-9950
Publication date 2017-02-01
Sub-type Article (original research)
DOI 10.1103/PhysRevB.95.064110
Open Access Status Not yet assessed
Volume 95
Issue 6
Total pages 11
Place of publication College Park, MD, United States
Publisher American Physical Society
Collection year 2018
Language eng
Formatted abstract
The density matrix renormalization group (DMRG) is one of the most powerful numerical methods available for many-body systems. It has been applied to solve many physical problems, including the calculation of ground states and dynamical properties. In this work, we develop a perturbation theory of the DMRG (PT-DMRG) to greatly increase its accuracy in an extremely simple and efficient way. Using the canonical matrix product state (MPS) representation for the ground state of the considered system, a set of orthogonal basis functions {|ψi)} is introduced to describe the perturbations to the ground state obtained by the conventional DMRG. The Schmidt numbers of the MPS that are beyond the bond dimension cutoff are used to define these perturbation terms. The perturbed Hamiltonian is then defined as Hij=(ψi|Ĥ|ψj); its ground state permits us to calculate physical observables with a considerably improved accuracy compared to the original DMRG results. We benchmark the second-order perturbation theory with the help of a one-dimensional Ising chain in a transverse field and the Heisenberg chain, where the precision of the DMRG is shown to be improved O(10) times. Furthermore, for moderate L the errors of the DMRG and PT-DMRG both scale linearly with L-1 (with L being the length of the chain). The linear relation between the dimension cutoff of the DMRG and that of the PT-DMRG at the same precision shows a considerable improvement in efficiency, especially for large dimension cutoffs. In the thermodynamic limit we show that the errors of the PT-DMRG scale with √L-1. Our work suggests an effective way to define the tangent space of the ground-state MPS, which may shed light on the properties beyond the ground state. This second-order PT-DMRG can be readily generalized to higher orders, as well as applied to models in higher dimensions.
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: School of Mathematics and Physics
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