Hubbard model

The Hubbard model is an approximate model used, especially in solid-state physics, to describe the transition between conducting and insulating systems. The Hubbard model, named after John Hubbard, is the simplest model of interacting particles in a lattice, with only two terms in the Hamiltonian (see example below): a kinetic term allowing for tunneling ('hopping') of particles between sites of the lattice and a potential term consisting of an on-site interaction. The particles can either be fermions, as in Hubbard's original work, or bosons, when the model is referred to as either the 'Bose–Hubbard model' or the 'boson Hubbard model'. The Hubbard model is an approximate model used, especially in solid-state physics, to describe the transition between conducting and insulating systems. The Hubbard model, named after John Hubbard, is the simplest model of interacting particles in a lattice, with only two terms in the Hamiltonian (see example below): a kinetic term allowing for tunneling ('hopping') of particles between sites of the lattice and a potential term consisting of an on-site interaction. The particles can either be fermions, as in Hubbard's original work, or bosons, when the model is referred to as either the 'Bose–Hubbard model' or the 'boson Hubbard model'. The Hubbard model is a good approximation for particles in a periodic potential at sufficiently low temperatures that all the particles are in the lowest Bloch band, as long as any long-range interactions between the particles can be ignored. If interactions between particles on different sites of the lattice are included, the model is often referred to as the 'extended Hubbard model'. The model was originally proposed (in 1963) to describe electrons in solids and has since been the focus of particular interest as a model for high-temperature superconductivity. More recently, the Bose–Hubbard model has been used to describe the behavior of ultracold atoms trapped in optical lattices. Recent ultracold atom experiments have also realised the original, fermionic Hubbard model in the hope that such experiments could yield its phase diagram. For electrons in a solid, the Hubbard model can be considered as an improvement on the tight-binding model, which includes only the hopping term. For strong interactions, it can give qualitatively different behavior from the tight-binding model and correctly predicts the existence of so-called Mott insulators, which are prevented from becoming conducting by the strong repulsion between the particles. The Hubbard model is based on the tight-binding approximation from solid-state physics, which describes particles moving in a periodic potential, sometimes referred to as a lattice. For real materials, each site of this lattice might correspond with an ionic core, and the particles would be the valence electrons of these ions. In the tight-binding approximation, the Hamiltonian is written in terms of wannier states, which are localized states centered on each lattice site. Wannier states on neighboring lattice sites are coupled, allowing particles on one site to 'hop' to another. Mathematically, the strength of this coupling is given by a 'hopping integral', or 'transfer integral', between nearby sites. The system is said to be in the tight-binding limit when the strength of the hopping integrals falls off rapidly with distance. This coupling allows states associated with each lattice site to hybridize, and the eigenstates of such a crystalline system are Bloch wave functions, with the energy levels divided into separated energy bands. The width of the bands depends upon the value of the hopping integral. The Hubbard model introduces a contact interaction between particles of opposite spin on each site of the lattice. When the Hubbard model is used to describe electron systems, these interactions are expected to be repulsive, stemming from the screened Coulomb interaction. However, attractive interactions have also been frequently considered. The physics of the Hubbard model is determined by competition between the strength of the hopping integral, which characterizes the system's kinetic energy, and the strength of the interaction term. The Hubbard model can therefore explain the transition from metal to insulator in certain interacting systems. For example, it has been used to describe metal oxides as they are heated, where the corresponding increase in nearest-neighbor spacing reduces the hopping integral to the point where the on-site potential is dominant. Similarly, the Hubbard model can explain the transition from conductor to insulator in systems such as rare-earth pyrochlores as the atomic number of the rare-earth metal increases, because the lattice parameter increases (or the angle between atoms can also change — see Crystal structure) as the rare-earth element atomic number increases, thus changing the relative importance of the hopping integral compared to the on-site repulsion. The hydrogen atom has only one electron, in the so-called s orbital, which can either be spin up ( ↑ {displaystyle uparrow } ) or spin down ( ↓ {displaystyle downarrow } ). This orbital can be occupied by at most two electrons, one with spin up and one down (see Pauli exclusion principle). Now, consider a 1D chain of hydrogen atoms. Under band theory, we would expect the 1s orbital to form a continuous band, which would be exactly half-full. The 1D chain of hydrogen atoms is thus predicted to be a conductor under conventional band theory.