Hubbard model
The Hubbard model is an approximate model used, especially in solidstate physics, to describe the transition between conducting and insulating systems.[1] 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 onsite interaction. The particles can either be fermions, as in Hubbard's original work, or bosons, in which case 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, where all the particles may be assumed to be in the lowest Bloch band, and longrange interactions between the particles can be ignored. If interactions between particles at 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 a focus of particular interest as a model for hightemperature 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 tried to realise the original, fermionic model, in the hope that such experiments could yield its phase diagram.[2]
For electrons in a solid, the Hubbard model can be considered as an improvement on the tightbinding model, which includes only the hopping term. For strong interactions, it can give qualitatively different behavior from the tightbinding model, and correctly predicts the existence of socalled Mott insulators, which are prevented from becoming conducting by the strong repulsion between the particles.
Narrow energy band theory
The Hubbard model is based on the tightbinding approximation from solidstate 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 tightbinding 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 tightbinding 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 nearestneighbor spacing reduces the hopping integral to the point where the onsite potential is dominant. Similarly, the Hubbard model can explain the transition from conductor to insulator in systems such as rareearth pyrochlores as the atomic number of the rareearth metal increases, because the lattice parameter increases (or the angle between atoms can also change – see Crystal structure) as the rareearth element atomic number increases, thus changing the relative importance of the hopping integral compared to the onsite repulsion.
Example: 1D chain of hydrogen atoms
The hydrogen atom has only one electron, in the socalled s orbital, which can either be spin up () or spin down (). 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 halffull. The 1D chain of hydrogen atoms is thus predicted to be a conductor under conventional band theory.
But now consider the case where the spacing between the hydrogen atoms is gradually increased. At some point we expect that the chain must become an insulator.
Expressed in terms of the Hubbard model, on the other hand, the Hamiltonian is now made up of two terms. The first term describes the kinetic energy of the system, parameterized by the hopping integral, . The second term is the onsite interaction of strength that represents the electron repulsion. Written out in second quantization notation, the Hubbard Hamiltonian then takes the form
where is the spindensity operator for spin on the th site. The total density operator is and occupation of th site for the wavefunction is . Typically t is taken to be positive, and U may be either positive or negative in general, but is assumed to be positive when considering electronic systems as we are here.
If we consider the Hamiltonian without the contribution of the second term, we are simply left with the tight binding formula from regular band theory.
When the second term is included, however, we end up with a more realistic model that also predicts a transition from conductor to insulator as the ratio of interation to hopping, , is varied. This ratio can be modified by, for example, increasing the interatomic spacing, which would decrease the magnitude of without affecting . In the limit where , the chain simply resolves into a set of isolated magnetic moments. If is not too large, the overlap integral provides for superexchange interactions between neighboring magnetic moments, which may lead to a variety of interesting magnetic correlations, such as ferromagnetic, antiferromagnetic, etc. depending on the parameters of the model. The onedimensional Hubbard model was solved by Lieb and Wu using the Bethe ansatz. Essential progress has been achieved in the 1990s: a hidden symmetry was discovered, and the scattering matrix, correlation functions, thermodynamic and quantum entanglement were evaluated.[3]
More complex systems
Although the Hubbard model is useful in describing systems such as a 1D chain of hydrogen atoms, it is important to note that in more complex systems there may be other effects that the Hubbard model does not consider. In general, insulators can be divided into Mott–Hubbard type insulators (see Mott insulator) and chargetransfer insulators.
Consider the following description of a Mott–Hubbard insulator:
This can be seen as analogous to the Hubbard model for hydrogen chains, where conduction between unit cells can be described by a transfer integral.
However, it is possible for the electrons to exhibit another kind of behavior:
This is known as charge transfer and results in chargetransfer insulators. Note that this is quite different from the Mott–Hubbard insulator model because there is no electron transfer between unit cells, only within a unit cell.
Both of these effects may be present and competing in complex ionic systems.
Numerical treatment
The fact that the Hubbard model has not been solved analytically in arbitrary dimensions has led to intense research into numerical methods for these strongly correlated electron systems.[4] One major goal of this research is determine the lowtemperature phase diagram of this model, particularly in twodimensions.
Exact treatment of the Hubbard model on finite systems is possible using the Lanczos algorithm, which produces static and dynamic properties of the system. Ground state calculations using this method require the storage of three vectors of the size of the number of states. The number of states scales exponentially with the size of the system, which limits the number of sites in the lattice to about 20 on currently available hardware. With projector and finitetemperature auxiliaryfield Monte Carlo, two statistical methods exist that also can obtain certain properties of the system. For low temperatures, convergence problems appear that lead to an exponential growth of computational effort with decreasing temperature due to the socalled fermion sign problem.
The Hubbard model can also be studied within dynamical meanfield theory (DMFT). This scheme maps the Hubbard Hamiltonian onto a singlesite impurity model, a mapping that is formally exact only in infinite dimensions and in finite dimensions corresponds to the exact treatment of all purely local correlations only. DMFT allows one to compute the local Green's function of the Hubbard model for a given and a given temperature. Within DMFT, one can compute the evolution of the spectral function and observe the appearance of the upper and lower Hubbard bands as correlations increase.
See also
References
 Altland, A.; Simons, B. (2006). "Interaction effects in the tightbinding system". Condensed Matter Field Theory. Cambridge University Press. pp. 58 ff. ISBN 9780521845083.
 Quintanilla, J.; Hooley, C. (2009). "The strongcorrelations puzzle". Physics World. 22 (6): 32–37. Bibcode:2009PhyW...22f..32Q. doi:10.1088/20587058/22/06/38.
 Essler, F. H. L.; Frahm, H.; Göhmann, F.; Klümper, A.; Korepin, V. E. (2005). The OneDimensional Hubbard Model. Cambridge University Press. ISBN 9780521802628.

Scalapino, D. J. (2006). "Numerical Studies of the 2D Hubbard Model": cond–mat/0610710. arXiv:condmat/0610710. Bibcode:2006cond.mat.10710S. Cite journal requires
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Further reading
 Hubbard, J. (1963). "Electron Correlations in Narrow Energy Bands". Proceedings of the Royal Society of London. 276 (1365): 238–257. Bibcode:1963RSPSA.276..238H. doi:10.1098/rspa.1963.0204. JSTOR 2414761.
 Bach, V.; Lieb, E. H.; Solovej, J. P. (1994). "Generalized Hartree–Fock Theory and the Hubbard Model". Journal of Statistical Physics. 76 (1–2): 3. arXiv:condmat/9312044. Bibcode:1994JSP....76....3B. doi:10.1007/BF02188656.
 Lieb, E. H. (1995). "The Hubbard Model: Some Rigorous Results and Open Problems". Xi Int. Cong. Mp, Int. Press (?). 1995: cond–mat/9311033. arXiv:condmat/9311033. Bibcode:1993cond.mat.11033L.
 Gebhard, F. (1997). "Metal–Insulator Transition". The Mott Metal–Insulator Transition: Models and Methods. Springer Tracts in Modern Physics. 137. Springer. pp. 1–48. ISBN 9783540614814.
 Lieb, E. H.; Wu, F. Y. (2003). "The onedimensional Hubbard model: A reminiscence". Physica A. 321 (1): 1–27. arXiv:condmat/0207529. Bibcode:2003PhyA..321....1L. doi:10.1016/S03784371(02)017855.