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Physica status solidi. C, Conferences and critical reviews/Physica status solidi. C, Current topics in solid state physics

Two particles correlation in a one‐dimensional Fibonacci lattice within the generalized Hubbard model

A. Quiroz, José Eduardo Espinosa

Published Jul 27, 2005• 0 citations
Publisher

Abstract

In this work we analyzed the ground-state symmetry for the cases of two particles in a linear Fibonacci chain, using the generalized Hubbard Hamiltonian and a previously developed real-space mapping method. The method is based on mapping the correlated many-body problem onto an equivalent site-and bond-impurity one-body problem in a higher dimensional space, where the problem can be solved exactly. For the case of two particles, it is obtained a ground-state when the correlated hopping interactions are included. The two particle problem is analyzed by looking at the phase diagram for the bound state. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

DOI
10.1002/pssc.200461768

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Abstract During the last two decades, two of the most important discoveries in condensed matter physics have been the discovery of quasicrystals and the discovery of high‐Tc ceramic superconductors. These topics have generated a large number of experimental and theoretical studies in the physics of low dimension. They have also modified some of the concepts in solid state physics. For instance, it was believed that the five‐fold symmetry was incompatible with a long‐range order and it was not expected that ceramic materials with a high‐Tc and a short coherence length exhibit superconductivity. Therefore, it is important to revise both the spatial symmetry and the electronic correlation to identify how they affect the physical properties of materials. The study of these subjects is complex since we cannot use the reciprocal space to study quasicrystals and the electronic correlation in many‐body systems has not entirely been solved. Even in one‐dimensional quasiperiodic structures, the interactions between electrons have often been neglected and only few results have been obtained. In this work, we solved the cases of two and three interacting particles in a Fibonacci lattice using a real‐space method, the Green function technique, the renormalized perturbation expansion method and the Hubbard model. For the case of two interacting particles an analytical solution for the pairing phase diagram was obtained using the extended Hubbard Hamiltonian. For the case of three interacting particles the binding energy was numerically calculated. The results present here are compared with those obtained for the periodic and binary lattices. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

physica status solidi (b)•Jun 13, 2005•1 citation
In this work we analyzed the ground-state symmetry for the cases of two particles in a linear Fibonacci chain, using the generalized Hubbard Hamiltonian and a previously developed real-space mapping method. The method is based on mapping the correlated many-body problem onto an equivalent site-and bond-impurity one-body problem in a higher dimensional space, where the problem can be solved exactly. For the case of two particles, it is obtained a ground-state when the correlated hopping interactions are included. The two particle problem is analyzed by looking at the phase diagram for the bound state. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
In this work we solve a general Hubbard Hamiltonian for two interacting particles in a periodic and nonperiodic infinite one-dimensional lattice, using a real-space mapping method, the renormalized perturbation expansion (RPE), and the Green-function technique. This Hamiltonian considers a general bond-charge interaction, the on-site interaction, and the general intersite interaction. The real-space method is based on mapping the correlated many-body problem onto an equivalent site- and bond-impurity tight-binding problem in a higher-dimensional space. Analyzing the periodic and the quasiperiodic lattices in this new space, we obtained the analytical solution for the binding condition at the ground state. Our general results for the periodic chain reproduce completely the limit cases of the numerical solution obtained previously and those obtained in reciprocal space
At non-half-filling, a one-dimensional interacting electronic model (t–U–R) is studied analytically. The particular correlated hopping keeps particle–hole symmetry of the Hubbard model. Physically, the model involved is equivalent to an extended Hubbard chain incorporating a two-body (X) and a three-body (X˜) site-bond interactions with X˜=2X. The analysis of weak-coupling theory leads to the ground-state phase diagram, which consists of three types of metallic phases, characterized by the charge- (CDW) and spin-density-wave (SDW) and singlet-superconductivity (SS) instabilities.
This thesis addresses the question of a strongly interacting many-body problem: the Hubbard model on one- and two-dimensional lattices. The technique employed is the small-cluster method wherein the full many-body Hamiltonian for finite clusters with periodic boundary conditions (PBC's) is solved exactly. We first apply this technique to an extended one-dimensional chain and a two-dimensional square-lattice Hubbard model. We then construct a two-dimensional overlayer/substrate Hubbard model with first- and second-nearest-neighbor hopping, on-site and nearest-neighbor interactions, and overlayer-site orbital energy. The ground-state properties are investigated and discussed in a many-body picture.
The ground-state instabilities for a one-dimensional lattice system of electrons with onsite (Hubbard) and bond-site (hopping) interactions are analyzed in a perturbative approach. The zero temperature phase diagram at different band fillings is drawn; an attractive (repulsive) bond-site interaction favors the appearance of a superconductor state at low concentrations of electrons (holes). A comparison with the exact results for the Hubbard model and previous works for particular cases is also discussed.
In this Thesis, we develop and apply an accurate method to determine the ground-state properties of strongly-correlated electrons in the framework of lattice model Hamiltonians. In lattice density-functional theory (LDFT) the basic variable is the one-particle density matrix γ. From a generalized HK theorem, the ground state energy Egs[γgs] = minγ E[γ] is obtained by minimizing the energy-functional E[γ] over all the physical or representable γ. The energy functional can be divided into two contributions: the kinetic-energy functional T [γ], which linear dependence on γ is exactly known, and the correlation-energy functional W [γ], which dependence on γ is not explicitly known. Finding accurate approximations for W [γ] constitutes the actual challenge of this thesis. Part of this work is based on previous studies which derive an approximation of W [γ] for the Hubbard Hamiltonian based on scaling hypothesis and an exact analytical results for the Hubbard dimer. However, this approach is limited to spin independent and homogeneous systems. In order to extend the scope of LDFT, we have developed three different approaches to derive W [γ] in order to study broken symmetry problems. First we extend the previous scaling functional to system having charge transfer. By a systematic study of the dependence of W [γ] on the charge distribution we find similar scaling properties as for the homogeneous case. An extension to the Hubbard model on bipartite lattices is then derived and applied to finite and infinite systems with repulsive and attractive interactions. The high accuracy of this functional is demonstrated. Nevertheless, this approach is difficult to transfer to more complex systems, since the calculation of W [γ] applies to the system as a whole. To overcome this problem, we derive another approximation based on local scaling properties. This functional is formulated on-site so that it can be applied to any ordered or disordered Hamiltonian with on-site interaction. As applications we study the metal-insulator transition in the ionic Hubbard model in oneand two-dimensions, as well as the one-dimensional Hubbard chain with first and second nearest neighbor hoppings. Finally, we develop a numerical approach of W [γ], on the basis on exact diagonalizations of an effective many-body Hamiltonian corresponding to a cluster surrounded by an effective medium. This effective Hamiltonian depends on the density matrix γ and allows to derive approximation to W [γ] which improve systematically with increasing cluster size. The formulation is spin-dependent and allows a straightforward generalization to multi-orbital correlated systems such spd-Hamiltonians. In addition it take into account the effects of short-range charge and spin fluctuations in the functional. The accuracy of the method is demonstrated for the Hubbard model by comparison with the Bethe-Ansatz solution (1D) and quantum Monte Carlo simulations (2D). Finally, an outlook or relevant future developments of the theory is provided.
Abstract During the last two decades, two of the most important discoveries in condensed matter physics have been the discovery of quasicrystals and the discovery of high‐Tc ceramic superconductors. These topics have generated a large number of experimental and theoretical studies in the physics of low dimension. They have also modified some of the concepts in solid state physics. For instance, it was believed that the five‐fold symmetry was incompatible with a long‐range order and it was not expected that ceramic materials with a high‐Tc and a short coherence length exhibit superconductivity. Therefore, it is important to revise both the spatial symmetry and the electronic correlation to identify how they affect the physical properties of materials. The study of these subjects is complex since we cannot use the reciprocal space to study quasicrystals and the electronic correlation in many‐body systems has not entirely been solved. Even in one‐dimensional quasiperiodic structures, the interactions between electrons have often been neglected and only few results have been obtained. In this work, we solved the cases of two and three interacting particles in a Fibonacci lattice using a real‐space method, the Green function technique, the renormalized perturbation expansion method and the Hubbard model. For the case of two interacting particles an analytical solution for the pairing phase diagram was obtained using the extended Hubbard Hamiltonian. For the case of three interacting particles the binding energy was numerically calculated. The results present here are compared with those obtained for the periodic and binary lattices. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)