Hubbard model

The Hubbard model is an approximate model used, especially in solid-state 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 on-site interaction. The particles can either be fermions, as in Hubbard's original work, or bosons, in which case the model is referred to as the "Bose–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 long-range 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 high-temperature superconductivity. 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.

Narrow energy band theory

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's 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.
Example: 1D chain of hydrogen atoms

The hydrogen atom has only one electron, in the so-called s orbital, which can either be spin up ( \( \uparrow \)) or spin down ( \( \downarrow \) ). 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.

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, t. The second term is the on-site interaction of strength U that represents the electron repulsion. Written out in second quantization notation, the Hubbard Hamiltonian then takes the form

\( {\displaystyle {\hat {H}}=-t\sum _{i,\sigma }\left({\hat {c}}_{i,\sigma }^{\dagger }{\hat {c}}_{i+1,\sigma }+{\hat {c}}_{i+1,\sigma }^{\dagger }{\hat {c}}_{i,\sigma }\right)+U\sum _{i}{\hat {n}}_{i\uparrow }{\hat {n}}_{i\downarrow },} \)

where \( {\displaystyle {\hat {n}}_{i\sigma }={\hat {c}}_{i\sigma }^{\dagger }{\hat {c}}_{i\sigma }} \) is the spin-density operator for spin σ {\displaystyle \sigma } \sigma on the i {\displaystyle i} i-th site. The total density operator is \( {\displaystyle {\hat {n}}_{i}={\hat {n}}_{i\uparrow }+{\hat {n}}_{i\downarrow }} \) and occupation of i-th site for the wavefunction \( \Phi \) is\( {\displaystyle n_{i}=\langle \Phi \vert {\hat {n}}_{i}\vert \Phi \rangle } \). 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 interaction to hopping, \( {\displaystyle U/t} \), is varied. This ratio can be modified by, for example, increasing the inter-atomic spacing, which would decrease the magnitude of t {\displaystyle t} t without affecting U. In the limit where \( {\displaystyle U/t\gg 1} \), the chain simply resolves into a set of isolated magnetic moments. If \( {\displaystyle U/t} \) 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 one-dimensional 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.[2]
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 charge-transfer insulators.

Consider the following description of a Mott–Hubbard insulator:

\( {\displaystyle (\mathrm {Ni} ^{2+}\mathrm {O} ^{2-})_{2}\longrightarrow \mathrm {Ni} ^{3+}\mathrm {O} ^{2-}+\mathrm {Ni} ^{1+}\mathrm {O} ^{2-}.} \)

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:

\( {\displaystyle \mathrm {Ni} ^{2+}\mathrm {O} ^{2-}\longrightarrow \mathrm {Ni} ^{1+}\mathrm {O} ^{1-}.} \)

This is known as charge transfer and results in charge-transfer 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.[3][4] One major goal of this research is to determine the low-temperature phase diagram of this model, particularly in two-dimensions. Approximate numerical treatment of the Hubbard model on finite systems is possible via a number of methods.

One such method, the Lanczos algorithm, can produce 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 finite-temperature auxiliary-field 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 so-called fermion sign problem.

The Hubbard model can also be studied within dynamical mean-field theory (DMFT). This scheme maps the Hubbard Hamiltonian onto a single-site 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 U 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

Bloch's theorem
Electronic band structure
Solid-state physics
Bose–Hubbard model
t-J model
Heisenberg model (quantum)
Dynamical mean-field theory
Stoner criterion

References

Altland, A.; Simons, B. (2006). "Interaction effects in the tight-binding system". Condensed Matter Field Theory. Cambridge University Press. pp. 58 ff. ISBN 978-0-521-84508-3.
Essler, F. H. L.; Frahm, H.; Göhmann, F.; Klümper, A.; Korepin, V. E. (2005). The One-Dimensional Hubbard Model. Cambridge University Press. ISBN 978-0-521-80262-8.
Scalapino, D. J. (2006). "Numerical Studies of the 2D Hubbard Model": cond–mat/0610710.arXiv:cond-mat/0610710. Bibcode:2006cond.mat.10710S.

LeBlanc, J. (2015). "Solutions of the Two-Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms". Physical Review X. 5 (4): 041041.arXiv:1505.02290. Bibcode:2015PhRvX...5d1041L. doi:10.1103/PhysRevX.5.041041.