Hubbard model
Encyclopedia
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 atom
s trapped in optical lattice
s. 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.
of their constituent atoms, and then 'hopping' between atoms during conduction. Mathematically, this is represented as a 'hopping integral' or 'transfer integral' between neighboring atoms, which can be viewed as the physical principle that creates electron bands in crystal
line materials, due to overlapping between atomic orbitals. Width of the band depends upon the overlapping amplitude. However, the more general band theories do not consider interactions between electrons explicitly. They consider interaction of a single electron with potential of nuclei and other electrons in the average way only. By formulating conduction in terms of the hopping integral, however, the Hubbard model is able to include the so-called 'onsite repulsion', which stems from the Coulomb repulsion between electrons at the same atomic orbitals. This sets up a competition between the hopping integral, which is a function of the distance and angles between neighboring atoms, and the on-site Coulomb repulsion, which is not considered in the usual band theories. The Hubbard model can therefore explain the transition from metal to insulator in certain transition metal oxides
as they are heated by the increase in nearest neighbor spacing, which reduces the 'hopping integral' to the point where the onsite potential is dominant. Similarly, this can explain the transition from conductor to insulator in systems such as rare-earth
pyrochlore
s 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 onsite repulsion.
has only one electron, in the so-called 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 half-full. The 1-D 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 components. The first component is the hopping integral. The hopping integral is typically represented by the letter t because it represents the kinetic energy of electrons hopping between atoms. The second term in the Hubbard model is then the on-site repulsion, typically represented by the letter U because it represents the potential energy arising from the charges on the electrons. Written out in second quantization notation, the Hubbard Hamiltonian
then takes the form:
, where 
represents nearest-neighbor interaction on the lattice.
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 inter-atomic spacing is increased. In the limit where the spacing is infinite (or if we ignore the first term), the chain simply resolves into a set of isolated magnetic moment
s. Additionally, when there are some contributions from the first term, but the material remains an insulator, the overlap integral provides for exchange interactions between neighboring magnetic moments which may lead to a variety of interesting magnetic correlations, such as ferromagnetic, antiferromagnetic, etc. depending on the exact solutions of the model. One dimensional Hubbard model was solved by Lieb
and Wu by means of Bethe ansatz
. During 1990-2000 essential progress has been achieved: a hidden symmetry
was discovered, the scattering matrix, correlation functions
, thermodynamic and quantum entanglement
were evaluated, see.
.
Consider the following description of a Mott-Hubbard insulator:
2 --> Ni3+O2- + Ni1+O2-
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 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.
Exact treatment of the Hubbard model at absolute zero
is possible using the Lanczos algorithm
, which produces static as well dynamic properties of the system. This method requires the storing of three vectors of the size of the number of states, 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 provide an exact solution
. For low temperatures and large lattice sizes convergence problems appear that lead to an exponential growth of computational effort due to the so-called 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, which 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.
Electrical insulation
thumb|250px|[[Coaxial Cable]] with dielectric insulator supporting a central coreThis article refers to electrical insulation. For insulation of heat, see Thermal insulation...
. The Hubbard model, named after John Hubbard
John Hubbard (physicist)
John Hubbard was a British physicist, best known for the Hubbard model for interacting electrons, the Hubbard-Stratonovich transformation, and the Hubbard approximations. He graduated from Imperial College London, receiving a B.Sc. and a Ph.D...
, is the simplest model of interacting particles in a lattice, with only two terms in the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
(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
Bloch wave
A Bloch wave or Bloch state, named after Felix Bloch, is the wavefunction of a particle placed in a periodic potential...
, 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
High-temperature superconductivity
High-temperature superconductors are materials that have a superconducting transition temperature above . From 1960 to 1980, 30 K was thought to be the highest theoretically possible Tc...
. More recently, the Bose-Hubbard model has been used to describe the behavior of ultracold atom
Ultracold atom
Ultracold atoms is a term used to describe atoms that are maintained at temperatures close to 0 kelvins , typically below some tenths of microkelvins , where their quantum-mechanical properties become important...
s trapped in optical lattice
Optical lattice
An optical lattice is formed by the interference of counter-propagating laser beams, creating a spatially periodic polarization pattern. The resulting periodic potential may trap neutral atoms via the Stark shift. Atoms are cooled and congregate in the locations of potential minima...
s. Recent ultracold atom
Ultracold atom
Ultracold atoms is a term used to describe atoms that are maintained at temperatures close to 0 kelvins , typically below some tenths of microkelvins , where their quantum-mechanical properties become important...
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.
Theory (Narrow energy band theory)
The Hubbard model is based on the tight-binding approximation from solid state physics. In the tight-binding approximation, electrons are viewed as occupying the standard orbitalsAtomic orbital
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...
of their constituent atoms, and then 'hopping' between atoms during conduction. Mathematically, this is represented as a 'hopping integral' or 'transfer integral' between neighboring atoms, which can be viewed as the physical principle that creates electron bands in crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
line materials, due to overlapping between atomic orbitals. Width of the band depends upon the overlapping amplitude. However, the more general band theories do not consider interactions between electrons explicitly. They consider interaction of a single electron with potential of nuclei and other electrons in the average way only. By formulating conduction in terms of the hopping integral, however, the Hubbard model is able to include the so-called 'onsite repulsion', which stems from the Coulomb repulsion between electrons at the same atomic orbitals. This sets up a competition between the hopping integral, which is a function of the distance and angles between neighboring atoms, and the on-site Coulomb repulsion, which is not considered in the usual band theories. The Hubbard model can therefore explain the transition from metal to insulator in certain transition metal oxides
Transition metal oxides
Transition metal oxides comprise a class of materials that contain transition elements and oxygen. They include insulators as well as metals. Often the same material may display both types of transport properties, hence a Metal-Insulator transition, obtained by varying either temperature or...
as they are heated by the increase in nearest neighbor spacing, which reduces the 'hopping integral' to the point where the onsite potential is dominant. Similarly, this can explain the transition from conductor to insulator in systems such as rare-earth
Rare earth element
As defined by IUPAC, rare earth elements or rare earth metals are a set of seventeen chemical elements in the periodic table, specifically the fifteen lanthanides plus scandium and yttrium...
pyrochlore
Pyrochlore
Pyrochlore 2Nb2O6 is a solid solution between the niobium end member , and the tantalum end member .-Occurrence:...
s as the atomic number
Atomic number
In chemistry and physics, the atomic number is the number of protons found in the nucleus of an atom and therefore identical to the charge number of the nucleus. It is conventionally represented by the symbol Z. The atomic number uniquely identifies a chemical element...
of the rare-earth metal
Metal
A metal , is an element, compound, or alloy that is a good conductor of both electricity and heat. Metals are usually malleable and shiny, that is they reflect most of incident light...
increases, because the lattice parameter increases (or the angle between atoms can also change — see Crystal structure
Crystal structure
In mineralogy and crystallography, crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry...
) as the rare-earth 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 atomHydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...
has only one electron, in the so-called s orbital, which can either be spin up (
) or spin down (). This orbital can be occupied by at most two electrons, one with spinSpin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
up and one down (see Pauli exclusion principle
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
).
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 1-D 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 components. The first component is the hopping integral. The hopping integral is typically represented by the letter t because it represents the kinetic energy of electrons hopping between atoms. The second term in the Hubbard model is then the on-site repulsion, typically represented by the letter U because it represents the potential energy arising from the charges on the electrons. Written out in second quantization notation, the Hubbard Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
then takes the form:
, where represents nearest-neighbor interaction on the lattice.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 inter-atomic spacing is increased. In the limit where the spacing is infinite (or if we ignore the first term), the chain simply resolves into a set of isolated magnetic moment
Magnetic moment
The magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...
s. Additionally, when there are some contributions from the first term, but the material remains an insulator, the overlap integral provides for exchange interactions between neighboring magnetic moments which may lead to a variety of interesting magnetic correlations, such as ferromagnetic, antiferromagnetic, etc. depending on the exact solutions of the model. One dimensional Hubbard model was solved by Lieb
Elliott H. Lieb
Elliott H. Lieb is an eminent American mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, condensed matter theory, and functional analysis....
and Wu by means of Bethe ansatz
Bethe ansatz
In physics, the Bethe ansatz is a method for finding the exact solutions of certain one-dimensional quantum many-body models. It was invented by Hans Bethe in 1931 to find the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic Heisenberg model Hamiltonian...
. During 1990-2000 essential progress has been achieved: a hidden symmetry
Yangian
Yangian is an important structure in modern representation theory, a type of a quantum group with origins in physics. Yangians first appeared in the work of Ludvig Faddeev and his school concerning the quantum inverse scattering method in the late 1970s and early 1980s. Initially they were...
was discovered, the scattering matrix, correlation functions
Correlation function (statistical mechanics)
In statistical mechanics, the correlation function is a measure of the order in a system, as characterized by a mathematical correlation function, and describes how microscopic variables at different positions are correlated....
, thermodynamic and quantum entanglement
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
were evaluated, see.
More Complex Systems
Although the Hubbard model is useful in describing systems such as a 1-D 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 and charge transfer insulatorsCharge transfer insulators
Charge transfer insulators are a class of materials predicted to be conductors following conventional band theory, but which are in fact insulators due to a charge transfer process. Unlike Mott insulators, where the insulating properties arise from electrons hopping between unit cells, the...
.
Consider the following description of a Mott-Hubbard insulator:
2 --> Ni3+O2- + Ni1+O2-
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:
- Ni2+O2- --> Ni1+O1-
This is known as charge transfer, and results in charge transfer insulators
Charge transfer insulators
Charge transfer insulators are a class of materials predicted to be conductors following conventional band theory, but which are in fact insulators due to a charge transfer process. Unlike Mott insulators, where the insulating properties arise from electrons hopping between unit cells, the...
. 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 cannot be solved analytically in arbitrary dimensions has led to intense research into numerical methods for these strongly correlated electron systems.Exact treatment of the Hubbard model at absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....
is possible using the Lanczos algorithm
Lanczos algorithm
The Lanczos algorithm is an iterative algorithm invented by Cornelius Lanczos that is an adaptation of power methods to find eigenvalues and eigenvectors of a square matrix or the singular value decomposition of a rectangular matrix. It is particularly useful for finding decompositions of very...
, which produces static as well dynamic properties of the system. This method requires the storing of three vectors of the size of the number of states, 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
Auxiliary field Monte Carlo
Auxiliary field Monte Carlo is a method that allows the calculation, by use of Monte Carlo techniques, of averages of operators in many-body quantum mechanical or classical problems .-Reweighting procedure and numerical sign problem:The distinctive ingredient of "auxiliary field Monte Carlo" is...
two statistical methods exist that also can provide an exact solution
Exact solution
In mathematics and especially physics, an exact solution is a solution to a problem that encapsulates the whole mathematics or physics of the problem without using an approximation.-References:*...
. For low temperatures and large lattice sizes convergence problems appear that lead to an exponential growth of computational effort due to the so-called sign problem.
The Hubbard model can also be studied within dynamical mean field theory
Dynamical mean field theory
Dynamical Mean Field Theory is a method to determine the electronic structure of strongly correlated materials. In such materials, the approximation of independent electrons, which is used in Density Functional Theory and usual band structure calculations, breaks down...
(DMFT). This scheme maps the Hubbard Hamiltonian onto a single site impurity model, which allows one to compute the local Green's function
Green's function
In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...
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
- Bloch waveBloch waveA Bloch wave or Bloch state, named after Felix Bloch, is the wavefunction of a particle placed in a periodic potential...
- Electronic band structureElectronic band structureIn solid-state physics, the electronic band structure of a solid describes those ranges of energy an electron is "forbidden" or "allowed" to have. Band structure derives from the diffraction of the quantum mechanical electron waves in a periodic crystal lattice with a specific crystal system and...
- Solid state physics
- Bose–Hubbard model
- t-J modelT-J modelThe t-J model was first derived in 1977 from the Hubbard model by Józef Spałek. The model describes strongly correlated electron systems. It is used to calculate high temperature superconductivity states in doped antiferromagnets.The t-J Hamiltonian is:...
- Dynamical Mean Field TheoryDynamical mean field theoryDynamical Mean Field Theory is a method to determine the electronic structure of strongly correlated materials. In such materials, the approximation of independent electrons, which is used in Density Functional Theory and usual band structure calculations, breaks down...
External links
- Electron Correlations in Narrow Energy Bands, J. Hubbard, Proc. Roy. Soc. A 276 238 (1963)
- The one-dimensional Hubbard model - A reminiscence by E.H. Lieb and F.Y. Wu
- Generalized Hartree-Fock Theory and the Hubbard Model by V. Bach, E. H. Lieb, and J. Solovej
- The Hubbard Model: Some Rigorous Results and Open Problems by E.H. Lieb
- The One-Dimensional Hubbard Model by Fabian H. L. Essler , Holger Frahm, Frank Göhmann, Andreas Klümper and Vladimir KorepinVladimir KorepinVladimir Korepin is a Russian-American physicist and mathematician. He is a professor at the C. N. Yang Institute of Theoretical Physics of the State University of New York at Stony Brook...
- Some Results and Publication of Vladimir Korepin
- The strong-correlations puzzle by J. Quintanilla and C. Hooley
- The Mott metal insulator transitions by Florian Gebhard