The Hubbard Model: Introduction and Selected Rigorous Results

The Hubbard model has a special role in Condensed Matter Theory as it is considered as the simplest Hamiltonian model one can write in order to describe anomalous physical properties of some class of real materials. Unfortunately, this model is not exactly solved except for some limits and therefore one should resort to analytical methods, like the Equations of Motion Approach, or to numerical techniques in order to attain a description of its relevant features in the whole r...

The present paper is based on our graduate lectures in condensed-matter physics. We found that the mean-field solution of the Hubbard model is an excellent tool to stimulate students' reflections towards the treatment of realistic magnetic interactions. We show by detailed analytical and numerical calculations how to find the mean-field solution of the model on a square lattice. We then interpret the physical implications of the ground-state magnetic phase diagram in terms of...

High-temperature superconductivity emerges in a host of different quantum materials, often in a region of the phase diagram where the electronic kinetic energy is comparable in magnitude with the electron-electron Coulomb repulsion. Describing such an intermediate-coupling regime has proven challenging, as standard perturbative approaches are inapplicable. Hence, it is of enormous interest to find models that are amenable to be solved using exact methods. While important adva...

We prove analyticity theorems in the coupling constant for the Hubbard model at half-filling. The model in a single renormalization group slice of index $i$ is proved to be analytic in $\lambda$ for $|\lambda| \le c/i$ for some constant $c$, and the skeleton part of the model at temperature $T$ (the sum of all graphs without two point insertions) is proved to be analytic in $\lambda$ for $|\lambda| \le c/|\log T|^{2}$. These theorems are necessary steps towards proving that t...

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...

We study the phase diagram of the Hubbard model in the limit where U, the onsite repulsive interaction, is much smaller than the bandwidth. We present an asymptotically exact expression for T$_c$, the superconducting transition temperature, in terms of the correlation functions of the non-interacting system which is valid for arbitrary densities so long as the interactions are sufficiently small. Our strategy for computing T$_c$ involves first integrating out all degrees of f...

We show that the pre-factors of all terms of the one-dimensional Hubbard model correlation-function asymptotic expansions have an universal form, as the corresponding critical exponents. In addition to calculating such pre-factors, our study clarifies the relation of the low-energy Tomonaga-Luttinger liquid behavior to the scattering mechanisms which control the spectral properties of the model at all energy scales. Our results are of general nature for many integrable intera...

Nagaoka's theorem on ferromagnetism in the Hubbard model with one electron less than half filling is generalized to the case where all possible nearest-neighbor Coulomb interactions (the density-density interaction $V$, bond-charge interaction $X$, exchange interaction $F$, and hopping of double occupancies $F'$) are included. It is shown that for ferromagnetic exchange coupling ($F>0$) ground states with maximum spin are stable already at finite Hubbard interaction $U>U_c$. ...

We present rigorous results for several variants of the Hubbard model in the strong-coupling regime. We establish a mathematically controlled perturbation expansion which shows how previously proposed effective interactions are, in fact, leading-order terms of well defined (volume-independent) unitarily equivalent interactions. In addition, in the very asymmetric (Falicov-Kimball) regime, we are able to apply recently developed phase-diagram technology (quantum Pirogov-Sinai ...

While the exact phase diagram of the Fermi-Hubbard model remains poorly understood despite decades of progress, nearly 60 years ago, Nagaoka proved that a single dopant in an otherwise half-filled Hubbard system can bring about ferromagnetism through kinetic means. The phenomenon was recently observed with ultracold atoms in triangular optical lattices. Here, we explore the kinetic ferromagnetism within the square lattice Hubbard model and its strong-coupling counterpart, the...