An introduction to numerical methods in low-dimensional quantum systems

Malvezzi, André Luiz

doi:10.1590/s0103-97332003000100004

Cited by 15 publications

(12 citation statements)

References 27 publications

(33 reference statements)

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“…The two curves shown were obtained with different many-body techniques. The full curve was obtained using the densitymatrix renormalization group ͑DMRG͒, 19,20 while the dotted curve was obtained from density-functional theory ͑DFT͒ within the Bethe-Ansatz local-density approximation ͑BA-LDA͒. [21][22][23] In view of the complexity of the problem and the surprising nature of some of our conclusions, we found it advisable to bring two independently developed and implemented many-body methods to bear on the problem.…”

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confidence: 99%

“…DMRG is a well-established numerical technique, whose precision can be improved systematically, at the expense of increased computational effort. 19,20 In our DMRG calculations, truncation errors were kept of the order of 10 −6 or smaller, and increasing the precision beyond this did not affect any of our conclusions. BA-LDA is a more recent development [21][22][23] ͑although the original LDA concept is, of course, widely used in ab initio calculations͒.…”

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Effects of nanoscale spatial inhomogeneity in strongly correlated systems

et al. 2005

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We calculate ground-state energies and density distributions of Hubbard superlattices characterized by periodic modulations of the on-site interaction and the on-site potential. Both density-matrix renormalization group and density-functional methods are employed and compared. We find that small variations in the on-site potential v i can simulate, cancel, or even overcompensate effects due to much larger variations in the on-site interaction U i . Our findings highlight the importance of nanoscale spatial inhomogeneity in strongly correlated systems, and call for a reexamination of model calculations assuming spatial homogeneity. A large part of the complexity of strongly correlated systems arises from the multiple phases that coexist or compete in their phase diagrams. Metallic and insulating phases are separated by metal-insulator transitions, and subject to the formation of various types of long-range order, such as antiferromagnetism, superconductivity, and charge-or spindensity waves. The relative stability of such phases is determined by differences in appropriate thermodynamic potentials, or, at zero temperature, in their ground-state energies. Identification of the appropriate order parameters and calculation of the ground-state energies of the various phases is a complicated problem, and the nature of the phase diagram of many strongly correlated systems is still subject to considerable controversy. It is widely believed, however, that a minimal model containing the essence of strong correlations, and displaying many of the above-mentioned phases, is the homogeneous Hubbard model, which in one dimension and standard notation readsMuch theoretical effort is thus going into the analysis of the homogeneous Hubbard model and the clarification of the nature of its ground state. In a parallel development, nanoscale spatial inhomogeneity has been observed experimentally to be a ubiquitious feature of strongly correlated systems, 1-8 but although its importance is widely recognized, the consequences of such inhomogeneity are still insufficiently understood. The present paper investigates the effects of, and the competition between, two different manifestations of nanoscale inhomogeneity in strongly correlated systems, namely local variations in the on-site potential and in the on-site interaction. We base our analysis on the inhomogeneous Hubbard model

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Effects of nanoscale spatial inhomogeneity in strongly correlated systems

et al. 2005

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“…In view of this, many interesting questions arise, such as how does q * depend on the electronic density and on the Coulomb repulsion U, and if there are sharp 2k F * −4k F * transitions. In order to answer these questions one needs to examine both a wide range of band fillings and large lattices, so here we report on a density matrix renormalization group ͑DMRG͒ approach [23][24][25][26] to this problem.…”

Section: 21mentioning

confidence: 99%

Modulation of charge-density waves by superlattice structures

2006

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We discuss the interplay between electronic correlations and an underlying superlattice structure in determining the period of charge density waves ͑CDW's͒, by considering a one-dimensional Hubbard model with a repeated ͑nonrandom͒ pattern of repulsive ͑U Ͼ 0͒ and free ͑U =0͒ sites. Density matrix renormalization group diagonalization of finite systems ͑up to 120 sites͒ is used to calculate the charge-density correlation function and structure factor in the ground state. The modulation period can still be predicted through effective Fermi wave vectors k F * and densities, and we have found that it is much more sensitive to electron ͑or hole͒ doping, both because of the narrow range of densities needed to go from q * =0 to , but also due to sharp 2k F * −4k F * transitions; these features render CDW's more versatile for actual applications in heterostructures than in homogeneous systems.

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“…In all examined cases we typically keep up to 500 states per block, although in the numerically more difficult cases, where the DMRG results converge slower, we keep up to 1000 states. Truncation errors [20,21], given by the sum of the density matrix eigenvalues of the discarded states, vary from 10 −6 in the worse cases to zero in the best cases. Figure 1: a) The density of zero-momentum excitons n 0 and the total exciton density n T as functions of local hybridization V calculated by DMRG method for three different values of the interband Coulomb interaction U (E f = 0, t f = 0, L = ∞, D = 1).…”

Section: Introductionmentioning

confidence: 99%

Formation and condensation of excitonic bound states in the generalized Falicov-Kimball model

Farkašovský

2017

Phys. Rev. B

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The density-matrix-renormalization-group (DMRG) method and the Hartree-Fock (HF) approximation with the charge-density-wave (CDW) instability are used to study a formation and condensation of excitonic bound states in the generalized Falicov-Kimball model. In particular, we examine effects of various factors, like the f -electron hopping, the local and nonlocal hybridization, as well as the increasing dimension of the system on the excitonic momentum distribution N (q) and especially on the number of zero momentum excitons N 0 = N (q = 0) in the condensate. It is found that the negative values of the f -electron hopping integrals t f support the formation of zero-momentum condensate, while the positive values of t f have the fully opposite effect. The opposite effects on the formation of condensate exhibit also the local and nonlocal hybridization. The first one strongly supports the formation of condensate, while the second one destroys it completely. Moreover, it was shown that the zero-momentum condensate remains robust with increasing dimension of the system.

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An introduction to numerical methods in low-dimensional quantum systems

Cited by 15 publications

References 27 publications

Effects of nanoscale spatial inhomogeneity in strongly correlated systems

Effects of nanoscale spatial inhomogeneity in strongly correlated systems

Modulation of charge-density waves by superlattice structures

Formation and condensation of excitonic bound states in the generalized Falicov-Kimball model

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