So far we have discussed particles moving in space that is a continuum. In solidstate physics it is more useful to employ what is known as a tight-binding picture where an electron is assumed to be tied to a particular atom, occasionally hopping to a neighboring atom. In this picture, the electron’s position is a discrete quantity-it is either on one atom or on its neighbor and not anywhere in between. This tightbinding picture may be derived using appropriate basis functions. For our purposes, we postulate that the kinetic motion of the electron is brought about by hopping. Therefore the kinetic energy is c†iσc jσ, (9.1) where < i j > signifies that sites i and j are nearest neighbors. The negative sign implies that hopping lowers the energy of the system by an amount t. But there is a price to be paid for hopping, in the form of a potential energy. The potential energy is assumed to be short-ranged, namely it is present only if a site has two electrons so that they repel with some energy U . But Pauli’s exclusion principle forbids two electrons with the same spin from residing at the same site. Thus while hopping lowers energy, hopping onto a site already occupied by an electron is either forbidden or has an energy cost U . The potential energy may be written as V =U∑ i ni↑ni↓, (9.2) where niσ = c † iσciσ is the number of electrons in site i. These two put together, form the famous Hubbard model of condensed matter physics. It is mathematically well defined given that short distances have a lower bound, namely the lattice spacing. The phases of this model depend on the ratio U/t and also the number of electrons per site (and temperature, if present). One can have variants of this model as well. The extended Hubbard model includes interaction between the nearest neighbors. Then we have a term such as
So far we have discussed particles moving in space that is a continuum. In solidstate physics it is more useful to employ what is known as a tight-binding picture where an electron is assumed to be tied to a particular atom, occasionally hopping to a neighboring atom. In this picture, the electron’s position is a discrete quantity-it is either on one atom or on its neighbor and not anywhere in between. This tightbinding picture may be derived using appropriate basis functions. For our purposes, we postulate that the kinetic motion of the electron is brought about by hopping. Therefore the kinetic energy is c†iσc jσ, (9.1) where < i j > signifies that sites i and j are nearest neighbors. The negative sign implies that hopping lowers the energy of the system by an amount t. But there is a price to be paid for hopping, in the form of a potential energy. The potential energy is assumed to be short-ranged, namely it is present only if a site has two electrons so that they repel with some energy U . But Pauli’s exclusion principle forbids two electrons with the same spin from residing at the same site. Thus while hopping lowers energy, hopping onto a site already occupied by an electron is either forbidden or has an energy cost U . The potential energy may be written as V =U∑ i ni↑ni↓, (9.2) where niσ = c † iσciσ is the number of electrons in site i. These two put together, form the famous Hubbard model of condensed matter physics. It is mathematically well defined given that short distances have a lower bound, namely the lattice spacing. The phases of this model depend on the ratio U/t and also the number of electrons per site (and temperature, if present). One can have variants of this model as well. The extended Hubbard model includes interaction between the nearest neighbors. Then we have a term such as
The Hubbard model is one of the most important models of many-body physics. It is a simplified model for how electrons move and interact in a crystal. It consists of a hopping term, where electrons can move between neighboring sites (sometimes more distant neighbors are allowed). When two electrons (of opposite spin) occupy the same lattice site, they feel a Coulomb repulsion. This model can be exactly solved only in one dimension, in the limit of infinite dimensions, or for small systems via exact diagonalization. It has been studied by thousands of physicists over the years. While much is known about the model, even more remains unknown, so it is still a great frontier to examine further. In this chapter, we focus on the known solutions and provide concrete examples and rigorous theorems about the behavior of the model. This includes the symmetries of spin, particle number, and pseudospin, the Nagaoka theorem for ferromagnetism, spin-reflection positivity, and spin and pseudospin towers.
Richard T. Scalettar Physics Department, University of California, Davis, California 95616, U.S.A. George G. Batrouni INLN, Universite´ de Nice - Sophia Antipolis, CNRS; 1361 route des Lucioles, 06560 Valbonne, France Originally introduced almost fifty years ago, the Hubbard Hamiltonian, H = −t ∑ 〈ij〉 ∑ i (ni↑− 12)(ni↓− 1 2 )−μ ∑ i (ni↑+ni↓), (11.1) has been widely, and very successfully, used to model many-body effects and quantum phase transitions (QPTs) in condensed matter, despite the simplifications it embodies [1-4]; see also Chap. 1. The Hamiltonian, given by Eq. (11.1), contains a kinetic energy term t which describes the hopping of two sets of fermionic particles, labeled by σ, on near-neighbor sites 〈ij〉 of the lattice.1 These fermions feel an interaction energy U if they occupy the same site, a term written here in a symmetric form which makes chemical potential μ = 0 correspond to half-filling ρ = 〈∑σ niσ〉 = 1 at any value of t, U or temperature T , as long as the lattice is bipartite. Many different geometries have been studied, including 1D chains, 2D square, triangular, and honeycomb structures, and 3D cubic lattices.
We consider an extended Hubbard model of interacting fermions on a lattice. The fermion kinetic energy corresponds to a tight-binding Hamiltonian with nearest-neighbor (-t) and next-nearest-neighbor (t∣IH) hopping matrix elements. In addition to the on-site Hubbard interaction (U) we also consider a nearest-neighbor repulsion (V). We obtain the zero-temperature phase diagram of our model within the Hartree-Fock approximation. We consider ground states having charge- and spin-density-wave ordering as well as states with orbital antiferromagnetism or spin nematic order. The latter two states correspond to particle-hole binding with ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ symmetry in the charge and spin channels, respectively. For t∣IH=0, only the charge-density-wave and spin-density-wave states are energetically stable. For nonzero t∣IH, we find that orbital antiferromagnetism (or spin nematic) order is stable over a finite portion of the phase diagram at weak coupling. This region of stability is seen to grow with increasing values of t∣IH.
The Hubbard model with occupation-dependent hopping rate exhibits superconductivity in a wide range of parameters within mean-field (BCS) theory. Here we study pair binding energies in this model for small clusters by exact diagonalization of the Hamiltonian. The model is defined by on-site and nearest-neighbor repulsions U and V, and occupation-dependent hopping rate t(n)=t+n\ensuremath{\Delta}t. We present results for one-dimensional chains and for two-dimensional square lattices of sizes 4\ifmmode\times\else\texttimes\fi{}4, 6\ifmmode\times\else\texttimes\fi{}6, and 8\ifmmode\times\else\texttimes\fi{}8. As a function of carrier density n the pair binding energy first increases and then decreases, and vanishes beyond a critical density. BCS results for the pair binding energy are found to be in remarkably good agreement with the exact results. In particular, BCS theory accurately reproduces the range of interaction parameters where pair binding exists in the exact solution. The model is found to exhibit no tendency to phase separation or clustering of more than two particles even for parameters giving rise to strong pair binding, in contrast to other tight-binding models where pairing occurs.
Originally, the Hubbard model has been derived for describing the behaviour of strongly-correlated electrons in solids. However, since over a decade now, variations of it are also routinely being imple-mented with ultracold atoms in optical lattices, allowing their study in a clean, essentially defect-free environment. Here, we review some of the rich literature on this subject, with a focus on more recent non-standard forms of the Hubbard model. After an introduction to standard (fermionic and bosonic) Hubbard models, we discuss briefly common models for mixtures, as well as the so called extended Bose-Hubbard models, that include interactions between neighboring sites, next-neighboring sites, and so on. The main part of the review discusses the importance of additional terms appearing when refining the tight-binding approximation on the original physical Hamiltonian. Even when restricting the models to the lowest Bloch band is justified, the standard approach neglects the density-induced tunneling (which has the same origin as the usual on-site interaction). The importance of these contri-
It is well known that an electron in a single-band tight-binding lattice coherently driven by an external dc electric field undergoes Bloch oscillations. In the framework of the extended Hubbard model, here it is shown theoretically that two strongly interacting electrons with on-site and nearest-neighbor site repulsion undergo a sequence of Bloch oscillations and Zener tunneling, the so-called Bloch-Zener oscillations (BZOs), among different narrow bands formed by bound particle states. Such a kind of BZOs arise because of nearest-neighbor site particle interaction and are thus rather distinct than BZOs observed for a single electron in biperiodic lattices supporting two single-particle minibands.