A report on Hubbard model

Approximate model used, especially in solid-state physics, to describe the transition between conducting and insulating systems.

- Hubbard model

9 related topics with Alpha

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Mott insulator

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Mott insulators are a class of materials that are expected to conduct electricity according to conventional band theories, but turn out to be insulators (particularly at low temperatures).

Mott insulators are a class of materials that are expected to conduct electricity according to conventional band theories, but turn out to be insulators (particularly at low temperatures).

One of the simplest models that can capture Mott transition is the Hubbard model.

A hypothetical example of a large number of carbon atoms being brought together to form a diamond crystal, demonstrating formation of the electronic band structure. The right graph shows the energy levels as a function of the spacing between atoms. When far apart (right side of graph) all the atoms have discrete valence orbitals p and s with the same energies. However, when the atoms come closer (left side), their electron orbitals begin to spatially overlap. The orbitals hybridize, and each atomic level splits into N levels with different energies, where N is the number of atoms. Since N is a very large number in a macroscopic sized crystal, the adjacent levels are energetically close together, effectively forming a continuous energy band. At the actual diamond crystal cell size (denoted by a), two bands are formed, called the valence and conduction bands, separated by a 5.5 eV band gap. Decreasing the inter-atomic spacing even more (e.g., under a high pressure) further modifies the band structure.

Electronic band structure

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In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called band gaps or forbidden bands).

In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called band gaps or forbidden bands).

A hypothetical example of a large number of carbon atoms being brought together to form a diamond crystal, demonstrating formation of the electronic band structure. The right graph shows the energy levels as a function of the spacing between atoms. When far apart (right side of graph) all the atoms have discrete valence orbitals p and s with the same energies. However, when the atoms come closer (left side), their electron orbitals begin to spatially overlap. The orbitals hybridize, and each atomic level splits into N levels with different energies, where N is the number of atoms. Since N is a very large number in a macroscopic sized crystal, the adjacent levels are energetically close together, effectively forming a continuous energy band. At the actual diamond crystal cell size (denoted by a), two bands are formed, called the valence and conduction bands, separated by a 5.5 eV band gap. Decreasing the inter-atomic spacing even more (e.g., under a high pressure) further modifies the band structure.
Fig 1. Brillouin zone of a face-centered cubic lattice showing labels for special symmetry points.
Fig 2. Band structure plot for Si, Ge, GaAs and InAs generated with tight binding model. Note that Si and Ge are indirect band gap materials, while GaAs and InAs are direct.

The Hubbard model is an approximate theory that can include these interactions.

Dynamical mean-field theory

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Method to determine the electronic structure of strongly correlated materials.

Method to determine the electronic structure of strongly correlated materials.

Likewise, DMFT maps a lattice problem (e.g. the Hubbard model) onto a single-site problem.

Bose–Hubbard model

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The Bose–Hubbard model gives a description of the physics of interacting spinless bosons on a lattice.

The Bose–Hubbard model gives a description of the physics of interacting spinless bosons on a lattice.

It is closely related to the Hubbard model which originated in solid-state physics as an approximate description of superconducting systems and the motion of electrons between the atoms of a crystalline solid.

John Hubbard (physicist)

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John Hubbard (27 October 1931 – 27 November 1980) was a British physicist, best known for the Hubbard model for interacting electrons, the Hubbard–Stratonovich transformation, and the Hubbard approximations.

Tight binding

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Approach to the calculation of electronic band structure using an approximate set of wave functions based upon superposition of wave functions for isolated atoms located at each atomic site.

Approach to the calculation of electronic band structure using an approximate set of wave functions based upon superposition of wave functions for isolated atoms located at each atomic site.

Modern explanations of electronic structure like t-J model and Hubbard model are based on tight binding model.

Elliott H. Lieb

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American mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, condensed matter theory, and functional analysis.

American mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, condensed matter theory, and functional analysis.

In particular, his scientific works pertain to quantum and classical many-body problem, atomic structure, the stability of matter, functional inequalities, the theory of magnetism, and the Hubbard model.

Bethe ansatz

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Ansatz method for finding the exact wavefunctions of certain one-dimensional quantum many-body models.

Ansatz method for finding the exact wavefunctions of certain one-dimensional quantum many-body models.

Since then the method has been extended to other models in one dimension: the (anisotropic) Heisenberg chain (XXZ model), the Lieb-Liniger interacting Bose gas, the Hubbard model, the Kondo model, the Anderson impurity model, the Richardson model etc.

Numerical sign problem

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Problem of numerically evaluating the integral of a highly oscillatory function of a large number of variables.

Problem of numerically evaluating the integral of a highly oscillatory function of a large number of variables.

Condensed matter physics — It prevents the numerical solution of systems with a high density of strongly correlated electrons, such as the Hubbard model.