The Hubbard Model: Introduction and Selected Rigorous Results

We present the first rigorous examples of non-singular Hubbard models which exhibit ferromagnetism at zero temperature. The models are defined in arbitrary dimensions, and are characterized by finite-ranged hoppings, dispersive bands, and finite on-site Coulomb interaction U. The picture, which goes back to Heisenberg, that sufficiently large Coulomb interaction can revert Pauli paramagnetism into ferromagnetism has finally been confirmed in concrete examples.

We have rigorously shown that a strong Hubbard repulsion can cause superconductivity. The model, which has a particular set of local symmetries, manifests the phase diagram of many unconventional superconductors; anti-ferromagnetism paramagnetism, superconductivity, and even ferromagnetism are all observed. The key technique in the analysis is an exact non-linear fermion transformation, which preserves the correlated motion of electrons while removing the strong interactions....

Despite its abundance in nature, predicting the occurrence of ferromagnetism in the ground state is possible only under very limited conditions such as in a flat band system with repulsive interaction or in a band with a single hole under infinitely large Coulomb repulsion, etc. Here, we propose a general condition to achieve saturated ferromagnetism based on the quantum geometry of electronic wave functions in itinerant electron systems. By analyzing multi-band repulsive Hub...

This work deals with the one-dimensional Hubbard model. It has seven chapters: 1. Introduction; 2. The Hubbard Model; 3. Algebraic Representation for the Hilbert Space of the 1D Hubbard Model; 4. Pseudoparticle Perturbation Theory; 5. Excitations and Finite-Size Effects; 6. Zero-Temperature Transport; 7. Conclusions.

We show that a particular class of variational wave functions reproduces the low-energy properties of the Hubbard model in one dimension. Our approach generalizes to finite on-site Coulomb repulsion the fully-projected wave function proposed by Hellberg and Mele [Phys. Rev. Lett. {\bf 67}, 2080 (1991)] for describing the Luttinger-liquid behavior of the doped $t{-}J$ model. Within our approach, the long-range Jastrow factor emerges from a careful minimization of the energy, w...

Whether spin-independent Coulomb interaction in an electron system can be the origin of ferromagnetism has been an open problem for a long time. Recently, a "constructive" approach to this problem has been developed, and the existence of ferromagnetism in the ground states of certain Hubbard models was established rigorously. A special feature of these Hubbard models is that their lowest bands (in the corresponding single-electron problems) are completely flat. Here we study ...

A wide variety of experimental platforms, ranging from semiconductor quantum-dot arrays to moir\'e materials, have recently emerged as powerful quantum simulators for studying the Hubbard model and its variants. Motivated by these developments, here, we investigate a generalization of the Hubbard model which includes the effects of long-range Coulomb interactions. Working on finite-sized two-dimensional square and triangular lattices, we use exact diagonalization and density-...

We propose a new mechanism which can lead to ferromagnetism in Hubbard models containing triangles with different on-site energies. It is based on an effective Hamiltonian that we derive in the strong coupling limit. Considering a one-dimensional realization of the model, we show that in the quarter-filled, insulating case the ground-state is actually ferromagnetic in a very large parameter range going from Tasaki's flat-band limit to the strong coupling limit of the effectiv...

The Hubbard model describes interacting electrons on a lattice,a situation which occurs in various solid state materials and devices. The aim of the present paper is to briefly discuss this model and its applications in the study of transport properties of Q1D and Q2D systems. Several examples are taken from materials and fields that have been intensively studied in recent years.

The Holstein model has been widely accepted as a model comprising electrons interacting with phonons; analysis of this model's ground states was accomplished two decades ago. However, the results were obtained without completely taking repulsive Coulomb interactions into account. Recent progress has made it possible to treat such interactions rigorously; in this paper, we study the Holstein--Hubbard model with repulsive Coulomb interactions. The ground state properties of the...