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arXiv:0808.0666 · Physical Review Letters

Density Matrix Renormalization Group in the Heisenberg Picture

Michael J. Hartmann, Javier Prior, Stephen R. L. Clark, Martin Bodo Plenio

Published Feb 2, 2009• 85 citations
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Abstract

In some cases the state of a quantum system with a large number of subsystems can be approximated efficiently by the density-matrix renormalization group, which makes use of redundancies in the description of the state. Here we show that the achievable efficiency can be much better when performing density-matrix renormalization group calculations in the Heisenberg picture, as only the observable of interest but not the entire state is considered. In some nontrivial cases, this approach can even be exact for finite bond dimensions.

DOI
10.1103/physrevlett.102.057202

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In some cases the state of a quantum system with a large number of subsystems can be approximated efficiently by the density-matrix renormalization group, which makes use of redundancies in the description of the state. Here we show that the achievable efficiency can be much better when performing density-matrix renormalization group calculations in the Heisenberg picture, as only the observable of interest but not the entire state is considered. In some nontrivial cases, this approach can even be exact for finite bond dimensions.
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 $\left\{|{\ensuremath{\psi}}_{i}\ensuremath{\rangle}\right\}$ 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 ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}}_{ij}=\ensuremath{\langle}{\ensuremath{\psi}}_{i}|\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}|{\ensuremath{\psi}}_{j}\ensuremath{\rangle}$; 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)O(10)O(10) times. Furthermore, for moderate LLL the errors of the DMRG and PT-DMRG both scale linearly with ${L}^{\ensuremath{-}1}$ (with LLL 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 $\sqrt{{L}^{\ensuremath{-}1}}$. 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.
Using the Density Matrix Renormalization Group (DMRG) one can simulate spin systems without the computation time growing as 2 , making it feasible to simulate large systems. In this paper, the impact of using the DMRG in the Schrodinger picture and in the Heisenberg picture is investigated for three different models, a spin1 2 XX chain, a spin1 2 XXZ chain and a Bose-Hubbard model. Simulations in the Heisenberg model will turn out to perform better only in the XX chain for certain operators. For all the other tested models, the Schrodinger picture is to be preferred. We will also try to shed some light on why the simulations in the Heisenberg picture fail so quickly.
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The transfer-matrix density-matrix renormalization-group method for one-dimensional quantum lattice systems has been developed by considering the symmetry property of the transfer matrix and introducing the asymmetric reduced density matrix. We have evaluated a number of thermodynamic quantities of the anisotropic spin-1/2 Heisenberg model using this method and found that the results agree very accurately with the exact ones. The relative errors for the spin susceptibility are less than ${10}^{\ensuremath{-}3}$ down to T=0.01JT=0.01JT=0.01J with 80 states kept.
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.
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