The Hubbard Model: Introduction and Selected Rigorous Results

We investigate two-site electronic correlations within generalized Hubbard model, which incorporates the conventional Hubbard model (parameters: $t$ (hopping between nearest neighbours), $U$ (Coulomb repulsion (attraction)) supplemented by the intersite Coulomb interactions (parameters: $J^{(1)}$(parallel spins), $J^{(2)}$ (antiparellel spins)) and the hopping of the intrasite Cooper pairs (parameter: $V$). As a first step we find the eigenvalues $E_{\alpha} $ and eigenvector...

Significant advances in numerical techniques have enabled recent breakthroughs in the study of various properties of the Hubbard model - a seemingly simple, yet complex model of correlated electrons that has been a focus of study for more than half a century. In particular, it captures the essence of strong correlations, and is believed to possess various emergent, low energy states and collective excitations characteristic of cuprate high-temperature superconducting material...

In 1968 we published the solution of the ground state energy and wave function of the one-dimensional Hubbard model, and we also showed that there is no Mott transition in this model. Details of the analysis have never been published, however. As the Hubbard model has become increasingly important in condensed matter physics, relating to topics such as the theory of high-T_c superconductivity, it is appropriate to revisit the one-dimensional model and to recall here some deta...

The Hubbard model arises naturally when electron-electron interactions are added to the tight-binding descriptions of many condensed matter systems. For instance, the two-dimensional Hubbard model on the honeycomb lattice is central to the ab initio description of the electronic structure of carbon nanomaterials, such as graphene. Such low-dimensional Hubbard models are advantageously studied with Markov chain Monte Carlo methods, such as Hybrid Monte Carlo (HMC). HMC is the ...

As a prototype model of antiferromagnetism, we propose a repulsive Hubbard Hamiltonian defined on a graph $\L={\cal A}\cup{\cal B}$ with ${\cal A}\cap {\cal B}=\emptyset$ and bonds connecting any element of ${\cal A}$ with all the elements of ${\cal B}$. Since all the hopping matrix elements associated with each bond are equal, the model is invariant under an arbitrary permutation of the ${\cal A}$-sites and/or of the ${\cal B}$-sites. This is the Hubbard model defined on the...

In this paper, we study the Hubbard model with intersite Coulomb interaction in the ionic limit (i.e. no kinetic energy). It is shown that this model is isomorphic to the spin-1 Ising model in presence of a crystal field and an external magnetic field. We show that for such models it is possible to find, for any dimension, a finite complete set of eigenoperators and eigenvalues of the Hamiltonian. Then, the hierarchy of the equations of motion closes and analytical expression...

We present analytical results of fundamental properties of one-dimensional (1D) Hubbard model with a repulsive interaction, ranging from fractional excitations to universal thermodynamics, interaction-driven criticality, correlation functions, Contact susceptibilities and quantum cooling. Using the exact solutions of the Bethe Ansatz equations of the Hubbard model, we first rigorously calculate the gapless spin and charge excitations, exhibiting exotic features of fractionali...

A new representation for electrons is introduced, in which the electron operators are written in terms of a spinless fermion and the Pauli operators. This representation is canonical, invertible and constraint-free. Importantly, it simplifies the Hubbard interaction. On a bipartite lattice, the Hubbard model is reduced to a form in which the exchange interaction emerges simply by decoupling the Pauli subsystem from the spinless fermion bath. This exchange correctly reproduces...

The Hubbard model on the kagom\'e lattice has highly degenerate ground states (the flat lowest band) in the corresponding single-electron problem and exhibits the so-called flat-band ferromagnetism in the many-electron ground states as was found by Mielke. Here we study the model obtained by adding extra hopping terms to the above model. The lowest single-electron band becomes dispersive, and there is no band gap between the lowest band and the other band. We prove that, at h...

We uncover a disorder-driven instability in the diffusive Fermi liquid phase of a class of many-fermion systems, indicative of a metal-insulator transition of first order type, which arises solely from the competition between quenched disorder and interparticle interactions. Our result is expected to be relevant for sufficiently strong disorder in d = 3 spatial dimensions. Specifically, we study a class of half-filled, Hubbard-like models for spinless fermions with (complex) ...