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arXiv:cond-mat/9811170

The Density Matrix Renormalization Group, Quantum Groups and Conformal Field Theory

Germán Sierra, Miguel Angel Martin-Delgado

Published Nov 12, 1998• 9 citations
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Abstract

We present an overview of the Density Matrix Renormalization Group and its connections to Quantum Groups, Matrix Products and Conformal Field Theory. We emphasize some common formal structures in all these theories. We also propose two-dimensional extensions of the variational matrix product ansatzs.
DOI
10.48550/arxiv.cond-mat/9811170

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We introduce a Lagrangian formulation of the Density Matrix Renormalization Group (DMRG). We present Lagrangians which when minimised yield the optimal DMRG wavefunction in a variational sense, both within the general matrix product ansatz, as well as within the canonical form of the matrix product that is constructed within the DMRG sweep algorithm. Some of the results obtained are similar to elementary expressions in Hartree-Fock theory, and we draw attention to such analogies. The Lagrangians introduced here will be useful in developing theories of analytic response and derivatives in the DMRG.
The density matrix renormalization group (DMRG) is a numerical method for studying low dimensional strongly correlated models. The study of these models is one of the most active areas of condensed matter physics today and has been spurred not only by the discovery of high temperature superconductivity but also by experiments on new compounds. In this thesis we study fundamentals of the density matrix renormalization group method as well as applications of the method to Heisenberg spin chains. The introductory text is meant as a brief overview of the DMRG as well as Heisenberg spin systems and provides an introduction to the papers. In papers I and II we investigate the density matrix renormalization group with the aim to better understand the underlying structure of the method. We find that if the renormalization converges to a fixed point in the thermodynamic limit, the resulting basis states can be written as a matrix product form. We also show an explicit relation of the DMRG to variational calculations using the fact that the matrix product states can be rederived through a variational ansatz without any reference to the DMRG calculation. It is then shown how the matrix product states can be used to define a set of Bloch waves for describing elementary excitations of the Heisenberg spin-1 chain. The low lying energy spectrum is calculated. In papers III and IV we investigate the thermodynamics of impurities in a spin-1/2 chain. For this purpose we have shown that a recently developed transfer matrix DMRG method can be used to calculate impurity properties. Both impurity corrections to thermodynamic properties and local properties at finite temperature can be obtained. The method gives quicker and more accurate results than quantum Monte Carlo. The numerical results show good agreement with field theory calculations for a number of impurity configurations and the cross-over function for the magnetic susceptibility from high- to low-temperature behavior shows the expected data collapse. Our results also give quantitative predictions for experiments on doped spin-1/2 chains.
We introduce a Lagrangian formulation of the density matrix renormalisation group (DMRG). We present Lagrangians which, when minimised, yield the optimal DMRG wavefunction in a variational sense, both within the general matrix product ansatz and within the canonical form of the matrix product that is constructed within the DMRG sweep algorithm. Some of the results obtained are similar to elementary expressions in Hartree-Fock theory, and we draw attention to such analogies. The Lagrangians introduced here will be useful in developing theories of analytic response and derivatives in the DMRG.
The density matrix renormalization group (DMRG) has an underlying variational ansatz, the matrix product state (MPS). The accuracy of this ansatz is controlled by its virtual dimension. In most applications for quantum chemistry this parameter is very large, and the high-accuracy or FCI regime is studied. With a good orbital choice and ordering, and by exploiting the symmetry group of the Hamiltonian, DMRG then allows to retrieve highly accurate results in active spaces beyond the reach of FCI [1,2]. However, DMRG can also be of use with smaller virtual dimensions. I will discuss how fundamental ideas from Hartree-Fock (HF) theory are transferable to DMRG. Both methods have a variational ansatz. The time-independent variational principle leads for HF and DMRG to a self-consistent mean-field theory in the particles and lattice sites, respectively. The time-dependent variational principle results in the random-phase approximation (RPA). RPA reveals the elementary excitations: particle and site excitations for HF and DMRG, respectively. Exponentiation of these excitations provides a nonredundant parameterization of the ansatz manifold, formulated by Thouless’ theorem. A Taylor expansion in the nonredundant parameters leads to the configuration interaction expansion [3,4]. References [1] S. Wouters, W. Poelmans, P. W. Ayers and D. Van Neck, Computer Physics Communications 185, 1501-1514 (2014). [2] S. Wouters, T. Bogaerts, P. Van Der Voort, V. Van Speybroeck and D. Van Neck, Journal of Chemical Physics 140, 241103 (2014). [3] S. Wouters, N. Nakatani, D. Van Neck and G. K.-L. Chan, Physical Review B 88, 075122 (2013). [4] N. Nakatani, S. Wouters, D. Van Neck and G. K.-L. Chan, Journal of Chemical Physics 140, 024108 (2014).