Hubbard model

The Hubbard model is an

The Hubbard model introduces short-range interactions between electrons to the tight-binding model, which only includes kinetic energy (a "hopping" term) and interactions with the atoms of the lattice (an "atomic" potential). When the interaction between electrons is strong, the behavior of the Hubbard model can be qualitatively different from a tight-binding model. For example, the Hubbard model correctly predicts the existence of Mott insulators: materials that are insulating due to the strong repulsion between electrons, even though they satisfy the usual criteria for conductors, such as having an odd number of electrons per unit cell.

History

The model was originally proposed in 1963 to describe electrons in solids.^[4] Hubbard, Martin Gutzwiller and Junjiro Kanamori each independently proposed it.^[2]

Since then, it has been applied to the study of high-temperature superconductivity, quantum magnetism, and charge density waves.^[5]

Narrow energy band theory

The Hubbard model is based on the

The hydrogen atom has 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).

Under

Expressed using the Hubbard model, the Hamiltonian is made up of two terms. The first term describes the kinetic energy of the system, parameterized by the hopping integral, ${\displaystyle t}$. The second term is the on-site interaction of strength ${\displaystyle U}$ 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 ${\displaystyle \sigma }$ on the ${\displaystyle i}$-th site. The density operator is ${\displaystyle {\hat {n}}_{i}={\hat {n}}_{i\uparrow }+{\hat {n}}_{i\downarrow }}$ and occupation of ${\displaystyle i}$-th site for the wavefunction ${\displaystyle \Phi }$ 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, but is assumed to be positive when considering electronic systems.

Without the contribution of the second term, the Hamiltonian resolves to the tight binding formula from regular band theory.

Including the second term yields a 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 ${\displaystyle t}$ without affecting ${\displaystyle U}$. 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

Although Hubbard is useful in describing systems such as a 1D chain of hydrogen atoms, it is important to note that more complex systems may experience other effects that the Hubbard model does not consider. In general, insulators can be divided into Mott–Hubbard insulators and charge-transfer insulators.

A Mott–Hubbard insulator can be described as

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

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. Unlike Mott–Hubbard insulators electron transfer happens only within a unit cell.

Both of these effects may be present and compete 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.^[7][8] 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 various methods.

One such method, the

The Hubbard model can 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 ${\displaystyle U}$ and a given temperature. Within DMFT, the evolution of the

Simulator

Stacks of heterogeneous 2-dimensional

• Anderson impurity model
• 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