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European Journal of Physics

Complete set of commuting observables for a two-site Hubbard model

Nunzia Cerrato, Canio Noce

Published May 8, 2019• 1 citation
Publisher

Abstract

Abstract The aim of this paper is to present an exact solution of the Hubbard model in a simplified case. Considering the simplest problem of a two-site entity, a solution of the model is obtained by using a suitable complete set of commuting observables. Owing to this approach, we show that some features of the physics of insulating and metallic systems may be understood within a simple unified picture. Furthermore, we are confident that this purely quantum mechanical approach turns out to be of particular interest for a general reader and undergraduate or graduate physics students.

DOI
10.1088/1361-6404/ab2004

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Abstract The aim of this paper is to present an exact solution of the Hubbard model in a simplified case. Considering the simplest problem of a two-site entity, a solution of the model is obtained by using a suitable complete set of commuting observables. Owing to this approach, we show that some features of the physics of insulating and metallic systems may be understood within a simple unified picture. Furthermore, we are confident that this purely quantum mechanical approach turns out to be of particular interest for a general reader and undergraduate or graduate physics students.
The description of solids at a microscopic level is complex, involving the interaction of a huge number of its constituents, such as ions or electrons. It is impossible to solve the corresponding many-body problems analytically or numerically, although much insight can be gained from the analysis of simplified models. An important example is the Hubbard model, which describes interacting electrons in narrow energy bands, and which has been applied to problems as diverse as high-Tc superconductivity, band magnetism, and the metal-insulator transition. This 2005 book presents a coherent, self-contained account of the exact solution of the Hubbard model in one dimension. The early chapters will be accessible to beginning graduate students with a basic knowledge of quantum mechanics and statistical mechanics. The later chapters address more advanced topics, and are intended as a guide for researchers to some of the more topical results in the field of integrable models.
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].
We present the exact ground-state wave function and energy of the generalized Hubbard model, subjected to the condition that the number of double occupied sites is conserved, for a wide, physically relevant range of parameters. For one hole and one double occupied site the existence of the ferromagnetic ground-state is proved which allow one to determine the critical value of the on-site repulsion corresponding to the point of metal-insulator transition. For the one dimensional model the exact solution for special values of the parameters is obtained. 1
We present the exact ground-state wave function and energy of the generalized Hubbard model, subject to the condition that the number of doubly occupied sites is conserved. For a wide, physically relevant range of parameters. For one hole and one doubly occupied site the existence of the ferromagnetic ground state is proved which allows one to determine the critical value of the on-site repulsion corresponding to the point of metal-insulator transition. For the one-dimensional model the exact solution for special values of the parameters is obtained.
We present an experimental quantum simulation of the two-site Hubbard Hamiltonian, using the NMR technique. By varying the ratio between the Coulomb interaction U and the kinetic energy Jh (U/Jh), it was possible to observe changes in the behavior of the system, showing that the simulation carries the physics of the Hubbard model, such as the occurrence of an analog of the crossover from a metallic to a Mott-insulator regime in the limit U ​≫Jh. The experimental results were compared with the theoretic predictions, showing good fidelity. The problem is relevant to the areas of Quantum Computation and Solid State Physics, and establishes a possible new approach to treat the model.
The aim of this paper is to present a self contained introduction to the Hubbard model and some of its applications.The paper consists of two parts: the first will introduce the basic notions of the Hubbard model starting from the motivation for its development to the formulation of the Hamiltonian,and some methods of calculation within the model. The second part will discuss some applications of the model to 1D and 2D systems,based on a combination of the author's results with those from the literature.