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International Journal of Quantum Chemistry

Asymptotic analysis of the localization spread and polarizability of 1‐D noninteracting electrons

Gian Luigi Bendazzoli, Stefano Evangelisti, Antonio Monari

Published Apr 4, 2011• 26 citations
Publisher

Abstract

Abstract According to the modern Theory of the Insulating State [Resta, J Chem Phys 2006, 124, 104104], the metallic behavior of a N‐electron system with open boundary conditions is characterized by a localization spread λβγ diverging in the thermodynamic limit. This quantity, which is the second‐moment cumulant of the position operator (per electron), cannot in general be evaluated in closed form but for simple model systems. In this article, we perform an asymptotic analysis of λβγ for a gas of N non‐interacting electrons in a 1‐Dimensional box and a Hückel chain of N equivalent sites. The asymptotic behavior of the closely related polarizability tensor is also investigated for these exactly solvable models. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

DOI
10.1002/qua.23036

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Abstract According to the modern Theory of the Insulating State [Resta, J Chem Phys 2006, 124, 104104], the metallic behavior of a N‐electron system with open boundary conditions is characterized by a localization spread λβγ diverging in the thermodynamic limit. This quantity, which is the second‐moment cumulant of the position operator (per electron), cannot in general be evaluated in closed form but for simple model systems. In this article, we perform an asymptotic analysis of λβγ for a gas of N non‐interacting electrons in a 1‐Dimensional box and a Hückel chain of N equivalent sites. The asymptotic behavior of the closely related polarizability tensor is also investigated for these exactly solvable models. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011
The localization spread gives a criterion to decide between metallic and insulating behavior of a material. It is defined as the second moment cumulant of the many-body position operator, divided by the number of electrons. Different operators are used for systems treated with open or periodic boundary conditions. In particular, in the case of periodic systems, we use the complex position definition, which was already used in similar contexts for the treatment of both classical and quantum situations. In this study, we show that the localization spread evaluated on a finite ring system of radius R with open boundary conditions leads, in the large R limit, to the same formula derived by Resta and co-workers [C. Sgiarovello, M. Peressi, and R. Resta, Phys. Rev. B 64, 115202 (2001)] for 1D systems with periodic Born-von Kármán boundary conditions. A second formula, alternative to Resta's, is also given based on the sum-over-state formalism, allowing for an interesting generalization to polarizability and other similar quantities.
Abstract The perturbation expansion coefficients for the ground‐state energy of the half‐filled one‐dimensional Hubbard model with N = 4ν + 2, (ν = 1, 2,…) sites and satisfying cyclic boundary conditions were calculated in the Hückel limit (U/β → 0), where β designates the one‐electron hopping or resonance integral, and U, the two‐electron on‐site Coulomb integral. This was achieved by solving—order by order—the Lieb–Wu equations, a system of transcendental equations that determines the set of quasi‐momenta {ki} and spin variables {τα} for this model. The exact values for these quantities were found for the N = 6 member ring up to the 20th order in terms of the coupling constant B = U/2β, as well as numerically for 10 ⩽ N ⩽ 50, and the N = 6 Lieb–Wu system was reduced to a system of sextic algebraic equations. These results are compared with those of the infinite system derived analytically by Misurkin and Ovchinnikov [Teor. Mat. Fiz. 11, 127 (1972)]. It is further shown how the results of this article can be used for the interpolation by the root of a polynomial. It is demonstrated that a polynomial of relatively small degree provides a very good approximation for the energy in the intermediate coupling region. © 1995 John Wiley & Sons, Inc.
Electronic polarizability of finite chains is accurately calculated from the total energy variation of the system produced by small but finite static electric fields applied along the chain direction. Normalized polarizability, that is, polarizability divided by chain length, diverges as the second power of length for metallic systems but approaches a constant value for insulating systems. This behaviour provides a very convenient way to characterize the wave-function malleability of finite systems as it avoids the need of attaching infinite contacts to the chain ends. Hubbard model calculations at half filling show that the method works for a small U = 1 interaction value that corresponds to a really small spectral gap of 0.005 (hopping t = -1 is assumed). Once successfully checked, the method has been applied to the long-range hopping model of Gebhard and Ruckenstein showing 1/r hopping decay (Gebhard and Ruckenstein 1992 Phys. Rev. Lett. 68 244; Gebhard et al 1994 Phys. Rev. B 49 10926). Metallicity for U values below the reported metal-insulator transition is obtained but the surprise comes for U values larger than the critical one (when a gap appears in the spectral density of states) because a steady increase of the normalized polarizability with size is obtained. This critical size-scaling behaviour can be understood as corresponding to a molecule which polarizability is unbounded. We have checked that a real transfer of charge from one chain end to the opposite occurs as a response to very small electric fields in spite of the existence of a large gap of the order of U for one-particle excitations. Finally, ab initio quantum chemistry calculations of realistic poly-acetylene chains prove that the occurrence of such critical behaviour in real systems is unlikely.
The insulating state of matter is characterized by the excitation spectrum, but also by qualitative features of the electronic ground state. The insulating ground wavefunction in fact: (i) displays vanishing dc conductivity; (ii) sustains macroscopic polarization; and (iii) is localized. The idea that the insulating state of matter is a consequence of electron localization was first proposed in 1964 by W. Kohn. I discuss here a novel definition of electron localization, rather different from Kohn's, and deeply rooted in the modern theory of polarization. In fact the present approach links the two features (ii) and (iii) above, by means of essentially the same formalism. In the special case of an uncorrelated crystalline solid, the localization of the many-body insulating wavefunction is measured - according to our definition - by the spread of the Wannier orbitals; this spread diverges in the metallic limit. In the correlated case, the novel approach to localization is demonstrated by means of a two-band Hubbard model in one dimension, undergoing a transition from band insulator to Mott insulator.
Abstract The metal‐insulator transition induced by electron correlation (Mott transition) is studied with ab initio full CI methods in model systems. These are linear chains of equally spaced hydrogen atoms. Different indicators of the metal‐insulator transition (maximum of the localization tensor or of the polarizability) areused and discussed. It is shown that the different indicators give concordant results, and that the transition is associated to a change in electron correlation, measured by the “particle‐hole” symmetric correlation entropy. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem 111:3416–3423, 2011
The insulating state of matter is characterized by the excitation spectrum, but also by qualitative features of the electronic ground state. The insulating ground wave function in fact (i) sustains macroscopic polarization, and (ii) is localized. We give a sharp definition of the latter concept and we show how the two basic features stem from essentially the same formalism. Our approach to localization is exemplified by means of a two-band Hubbard model in one dimension. In the noninteracting limit, the wave function localization is measured by the spread of the Wannier orbitals.