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arXiv:cond-mat/9905208 · APS March Meeting Abstracts

Decay properties of the density matrix and Wannier functions for interacting systems

Stefan Goedecker, Erik Koch

Published May 14, 1999• 0 citations
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Abstract

For non-interacting electrons the one-particle density matrix and the related Wannier functions characterize a material as insulating or metallic. Introducing many-body Wannier functions, we show that this characterization can be carried over to interacting systems. In particular, we find that an exponential decay of the density matrix characterizes not only band insulators but also Mott insulators. The properties of the many-body Wannier functions differ, however, from those of the Wannier functions of a non-interacting systems.

DOI
10.48550/arxiv.cond-mat/9905208

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For non-interacting electrons the one-particle density matrix and the related Wannier functions characterize a material as insulating or metallic. Introducing many-body Wannier functions, we show that this characterization can be carried over to interacting systems. In particular, we find that an exponential decay of the density matrix characterizes not only band insulators but also Mott insulators. The properties of the many-body Wannier functions differ, however, from those of the Wannier functions of a non-interacting systems.
In the framework of the band theory, we consider two tight-binding models of insulators. The first one, proposed recently by Taraskin et al, is a translationally invariant system, built out of two independent non-overlapping bands of single-particle orbitals that are coupled by a weak inter-band hybridization. This kind of insulator exhibits unphysical properties: we show, in particular, that the one-body density matrix does not depend on the width of the gap between the bands. Consequently, there is no delocalization effect with increasing metallicity. In the second model there are also two bands. However, they are not imposed by construction but are created from a band of single-particle orbitals due to the breaking of the translational symmetry by a periodic potential. These bands are separated by a gap for all nonzero values of the unique energy parameter of the model. We demonstrate that the one-body density matrix has the same structure as in the first model. As a result, the large distance asymptotic formulae derived by Taraskin et al in dimensions D=1,2,3D=1,2,3D=1,2,3, apply as well, but only for very large gap widths. In D=1 and in the diagonal direction of D=2 cases, we derive a stronger asymptotic formula, valid for all gap widths. The both kinds of asymptotic formulae are composed of a dimension-dependent power-law factor and a gap-dependent exponentially decaying factor. The latter asymptotics implies that the exponential decay rate vanishes linearly with the vanishing gap. In non-diagonal directions, we have found numerically that the linear scaling is replaced by the square root one. Independently of the direction, the exponential decay rate grows logarithmically with the gap width, for sufficiently large gaps.
We present an analytical study of the spatial decay rate $\ensuremath{\gamma}$ of the one-particle density matrix $\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{r},{\stackrel{\ensuremath{\rightarrow}}{r}}^{\ensuremath{'}})\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}\ensuremath{\gamma}|\stackrel{\ensuremath{\rightarrow}}{r}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{r}}^{\ensuremath{'}}|)$ for systems described by single-particle orbitals in periodic potentials in arbitrary dimensions. This decay reflects electronic locality in condensed matter systems and is also crucial for O(N)O(N)O(N) density functional methods. We find that $\ensuremath{\gamma}$ behaves contrary to the conventional wisdom that generically $\ensuremath{\gamma}\ensuremath{\propto}\ensuremath{\surd}\ensuremath{\Delta}$ in insulators and $\ensuremath{\gamma}\ensuremath{\propto}\ensuremath{\surd}T$ in metals, where $\ensuremath{\Delta}$ is the direct band gap and TTT is the temperature. Rather, in semiconductors $\ensuremath{\gamma}\ensuremath{\propto}\ensuremath{\Delta}$, and in metals at low temperature $\ensuremath{\gamma}\ensuremath{\propto}T$.
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.
The one-body picture underlies our understanding of weakly interacting solids but breaks down in strongly correlated systems. We develop a general framework, based on the single-particle Green's function and the one-body reduced density matrix (1RDM), to characterize correlated electronic phases. Applying it to the Hubbard diamond chain, we combine density matrix renormalization group and cellular dynamical mean-field theory to construct symmetry-resolved effective orbitals and track their evolution across its Mott transitions, while the 1RDM purity provides a scalar indicator of the phase boundaries. These tools offer a general route to extend one-body concepts to correlated materials.
Reduced density matrix functional theory for the case of solids is presented and an exchange-correlation functional based on a fractional power of the density matrix is introduced. We show that compared to other functionals, this produces more accurate behavior for total energies as a function of particle number for finite systems. Moreover, it captures the correct band-gap behavior for conventional semiconductors, as well as strongly correlated Mott insulators, where a gap is obtained in the absence of any magnetic ordering.
In this paper a new and more comprehensive characterization of the insulating state of matter is developed. This characterization includes the conventional insulators with energy gap as well as systems discussed by Mott which, in band theory, would be metals. The essential property is this: Every low-lying wave function $\ensuremath{\Phi}$ of an insulating ring breaks up into a sum of functions, $\ensuremath{\Phi}={{\ensuremath{\Sigma}}_{\ensuremath{-}\ensuremath{\infty}}}^{\ensuremath{\infty}}{\ensuremath{\Phi}}_{M}$, which are localized in disconnected regions of the many-particle configuration space and have essentially vanishing overlap. This property is the analog of localization for a single particle and leads directly to the electrical properties characteristic of insulators. An Appendix deals with a soluble model exhibiting a transition between an insulating and a conducting state.