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arXiv:cond-mat/0311407 · Physical review. B, Condensed matter

Plaquette operators used in the rigorous study of ground states of the periodic Anderson model in D=2 dimensions

Zsolt Gulácsi

Published Oct 10, 2002• 18 citations
arXivHTMLPDF

Abstract

The derivation procedure of exact ground states for the periodic Anderson model (PAM) in restricted regions of the parameter space and $D=2$ dimensions using plaquette operators is presented in detail. Using this procedure, we are reporting exact ground states for PAM in two dimensions and finite value of the interaction, whose presence do not require the next-to-nearest-neighbor extension terms in the Hamiltonian. In order to do this, a completely new type of plaquette operator is introduced for PAM, based on which a localized phase is deduced whose physical properties are analyzed in detail. The obtained results provide exact theoretical data that can be used for the understanding of system properties leading to metal-insulator transitions, strongly debated in recent publications in the frame of PAM. In the described case, the loss of the localization character is connected to the breakdown of the long-range density-density correlations rather than Kondo physics.

DOI
10.1103/physrevb.66.165109

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The derivation procedure of exact ground states for the periodic Anderson model (PAM) in restricted regions of the parameter space and D=2D=2D=2 dimensions using plaquette operators is presented in detail. Using this procedure, we are reporting exact ground states for PAM in two dimensions and finite value of the interaction, whose presence do not require the next-to-nearest-neighbor extension terms in the Hamiltonian. In order to do this, a completely new type of plaquette operator is introduced for PAM, based on which a localized phase is deduced whose physical properties are analyzed in detail. The obtained results provide exact theoretical data that can be used for the understanding of system properties leading to metal-insulator transitions, strongly debated in recent publications in the frame of PAM. In the described case, the loss of the localization character is connected to the breakdown of the long-range density-density correlations rather than Kondo physics.
We construct a class of exact ground states of three-dimensional periodic Anderson models (PAMs), including the conventional PAM, on regular Bravais lattices at and above 3∕43∕43∕4 filling, and discuss their physical properties. In general, the fff electrons can have a (weak) dispersion, and the hopping and the nonlocal hybridization of the ddd and fff electrons extend over the unit cell. The construction is performed in two steps. First the Hamiltonian is cast into positive semidefinite form using composite operators in combination with coupled nonlinear matching conditions. This may be achieved in several ways, thus leading to solutions in different regions of the phase diagram. In a second step, a nonlocal product wave function in position space is constructed which allows one to identify various stability regions corresponding to insulating and conducting states. The compressibility of the insulating state is shown to diverge at the boundary of its stability regime. The metallic phase is a non-Fermi-liquid with one dispersing and one flat band. This state is also an exact ground state of the conventional PAM and has the following properties: (i) it is nonmagnetic with spin-spin correlations disappearing in the thermodynamic limit, (ii) density-density correlations are short ranged, and (iii) the momentum distributions of the interacting electrons are analytic functions, i.e., have no discontinuities even in their derivatives. The stability regions of the ground states extend through a large region of parameter space, e.g., from weak to strong on-site interaction UUU. Exact itinerant, ferromagnetic ground states are found at and below 1∕41∕41∕4 filling.
The periodic Anderson model (PAM) is studied within the framework of dynamical mean-field theory, with particular emphasis on the interaction-driven Mott transition it contains, and on resultant Mott insulators of both Mott-Hubbard and charge-transfer type. The form of the PAM phase diagram is first deduced on general grounds using two exact results, over the full range of model parameters and including metallic, Mott, Kondo and band insulator phases. The effective low-energy model which describes the PAM in the vicinity of a Mott transition is then shown to be a one-band Hubbard model, with effective hoppings that are not in general solely nearest neighbour, but decay exponentially with distance. This mapping is shown to have a range of implications for the physics of the problem, from phase boundaries to single-particle dynamics; all of which are confirmed and supplemented by NRG calculations. Finally we consider the locally degenerate, non-Fermi liquid Mott insulator, to describe which requires a two-self-energy description. This is shown to yield a number of exact results for the associated local moment, charge, and interaction-renormalised levels, together with a generalisation of Luttinger's theorem to the Mott insulator.
The Kondo interaction coupling, Heisenberg exchange coupling, and Coulomb interactions within d-sites, were introduced in a one dimensional Periodic Anderson Model Hamiltonian (PAMH) to further investigate the effects of interaction parameters on the ground state energy of systems with heavy fermions (HF) behavior. Periodic Anderson model PAM being one of the most successful model for studying the heavy fermions System (HFS) was used in an extended version (mixed Kondo-Anderson representation) on a system of three-electrons interacting on three-sites cluster. Exact Diagonalization technique (EDT) normally used to solve conventional PAM calculation was considered in this work for a very small cluster. Hamiltonian used to describe this model contains the usual term describing the kinetic energy of the system, on-site coulomb repulsion and a hopping integral. The Hamiltonian is acted on the different Hilbert states of the lattice system and results of the interactions were obtained in terms of hopping integral, coulomb repulsions, exchange couplings and the hybridization term. Graphs of ground state energy Eo plotted agains tthese interaction parameters were presented in a clear format. As these parameters were varied numerically through a finite range of values, the individual effects of these parameters on the system’s ground state energy were observed and discussed. Hence, the results obtained from this work shows theoretically how the tuning of the Columbic interaction within the conduction band provides information that sheds light on the underlying physics of the heavy fermions systems models. Results obtained from this work further demonstrate the reliability of the model Hamiltonians that we have harnessed and the importance of considering electron-lattice interactions as well as interactions that account for magnetic impurities for the proper description of heavy fermions material.