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Physical review. B, Condensed matter

Hubbard model in one dimension with a general bond-charge interaction: Analytical ground-state solution for the pairing of two particles

Emmanuel Vallejo, Oracio Navarro

Published May 28, 2003• 5 citations
Publisher

Abstract

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

DOI
10.1103/physrevb.67.193105

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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
We have modified an existing real space renormalization group scheme to remove some physically inconsistent features and used the rectified method to study the ground state of the one-dimensional Hubbard model with nearest-neighbour repulsion and the bond-charge interaction for half filling. For the nearest-neighbour interaction we find a phase diagram which agrees nicely with the Monte Carlo results. For the case of bond-charge interaction we have reproduced some available exact results and found some interesting information in the parameter regime where no exact solution is available.
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)
The single-band Hubbard Hamiltonian [1] is the simplest model of interacting electrons in a lattice.It was developed by John Hubbard, who introduced the model to take into account the local electronic correlations in narrow energy bands.The model has a kinetic term that allows hopping of electrons between atomic sites and a term that considers the on-site interaction between electrons.This model is largely used in solid state physics to study magnetic properties of solids, insulator-metal transition (Mott transition) and high temperature superconductors.However, this is a problem that has exact analytical solution in a very few limiting cases and must be treated approximately or numerically.In this work, we propose a new methodology to analytically solve the Hubbard Hamiltonian, mapping it into a two-site model (the Hubbard dimer).To obtain the Green's function for the lattice, we employed the cumulant expansion technique, using as a 'seed' the exact two-site Green's function (the atomic method) [2].From the Green's function, we obtained the density of states [DOS] and the occupation numbers as functions of the external parameters of the model, and compared the results with the Hubbard I approximation [1].
SE RESOLVIO EL HAMILTONIANO DE HUBBARD EXTENDIDO PARA DOS PARTICULAS EN UNA RED UNIDIMENSIONAL, USANDO EL METODO DEL MAPEO EN EL ESPACIO REAL Y LA TECNICA DE LA FUNCION DE GREEN. ESTE HAMILTONIANO CONSIDERA LAS INTERACCIONES INTRA-ATOMICAS U E INTER-ATOMICAS V . EL METODO MAPEA EL PROBLEMA DE MUCHOS CUERPOS CORRELACION DE UN PROBLEMA EQUIVALENTE DE AMARRE FUERTE EN UN ESPACIO DE MAYOR DIMENSION. HACIENDO UN ANALISIS DEL PROBLEMA EN ESTE ESPACIO SE OBTUVO LA SOLUCION ANALITICA PARA LA ENERGIA DE ENLACE DEL ESTADO BASE. LOS RESULTADOS ESTAN EN COMPLETO ACUERDO CON LA SOLUCIONES OBTENIDAS TANTO NUMERICAMENTE [1] COMO EN EL ESPACIO RECIPROCO [2]. Abstract A SOLUTION TO THE EXTENDED HUBBARD HAMILTONIAN FOR THE CASE OF TWO-PARTICLES IN AN INFINITE ONE-DIMENSIONAL LATTICE IS PRESENTED, USING A REAL-SPACE MAPPING METHOD AND THE GREEN FUNCTION TECHNIQUE. THIS HAMILTONIAN CONSIDERS THE ON-SITE (U) AND THE NEAREST-NEIGHBOR (V ) INTERACTIONS. THE METHOD IS BASED ON MAPPING THE CORRELATED MANY-BODY PROBLEM ONTO AN EQUIVALENT SITE-IMPURITY TIGHT-BINDING ONE IN A HIGHER DIMENSIONAL SPACE. IN THIS NEW SPACE WE OBTAINED THE ANALYTICAL SOLUTION FOR THE GROUND STATE BINDING ENERGY. RESULTS ARE IN AGREEMENT WITH THE NUMERICAL SOLUTION OBTAINED PREVIOUSLY [1], AND WITH THOSE OBTAINED IN THE RECIPROCAL SPACE [2].
The Hubbard model is investigated in the framework of lattice density functional theory (LDFT). The single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ with respect to the lattice sites is considered as the basic variable of the many-body problem. A new approximation to the interaction-energy functional $W[\ensuremath{\gamma}]$ is proposed which is based on its scaling properties and which recovers exactly the limit of strong electron correlations at half-band filling. In this way, a more accurate description of W is obtained throughout the domain of representability of ${\ensuremath{\gamma}}_{\mathrm{ij}},$ including the crossover from weak to strong correlations. As examples of applications results are given for the ground-state energy, charge-excitation gap, and charge susceptibility of the Hubbard model in one-, two-, and three-dimensional lattices. The performance of the method is demonstrated by comparison with available exact solutions, with numerical calculations, and with LDFT using a simpler dimer ansatz for W. Goals and limitations of the different approximations are discussed.