Phy5670/HubbardModel: Difference between revisions
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Utility<br> | Utility<br> | ||
The model referred to as the Hubbard model appeared in the literature for the first time in 1963, in two subsequent independent papers -- the first by Gutzwiller, and the second by Hubbard -- as an attempt to describe in a simplified way the effect of correlations for <math>d</math>-electrons in a transition metals, In deed in the latter case the feature that electrons form a (narrow) delocalized <math>d</math>-band is in competition with the quasi-atomic behavior originated from correlations, which would make plausible an atomic description of the problem. The model hamiltonian <math>H</math> consists of two contributions,<br> | :The model referred to as the Hubbard model appeared in the literature for the first time in 1963, in two subsequent independent papers -- the first by Gutzwiller, and the second by Hubbard -- as an attempt to describe in a simplified way the effect of correlations for <math>d</math>-electrons in a transition metals, In deed in the latter case the feature that electrons form a (narrow) delocalized <math>d</math>-band is in competition with the quasi-atomic behavior originated from correlations, which would make plausible an atomic description of the problem. The model hamiltonian <math>H</math> consists of two contributions,<br> | ||
:<math>H = \sum_{\mathbf{i},\mathbf{j}}\sum_{\sigma}t_{\mathbf{i},\mathbf{j}}\left(a_{\mathbf{i},\sigma}^{\dagger}a_{\mathbf{j},\sigma} + h.c.\right) + U\sum_{\mathbf{i}}n_{\mathbf{i},\uparrow}n_{\mathbf{i},\downarrow}</math> , (1.1.1) | :<math>H = \sum_{\mathbf{i},\mathbf{j}}\sum_{\sigma}t_{\mathbf{i},\mathbf{j}}\left(a_{\mathbf{i},\sigma}^{\dagger}a_{\mathbf{j},\sigma} + h.c.\right) + U\sum_{\mathbf{i}}n_{\mathbf{i},\uparrow}n_{\mathbf{i},\downarrow}</math> , (1.1.1) | ||
a kinetic term describing the motion of electrons between neighboring sites (the hopping integral <math>t_{\mathbf{i},\mathbf{j}}</math> is usually restricted to nearest-neighbors, and is assumed translationally invariant, namely <math>t_{\mathbf{i},\mathbf{j}} = -t, t > 0</math>), and an on-site term, which approximates the interactions among electrons, whose strength is given by the parameter <math>U</math>. <math>U > 0</math> corresponds to repulsive Coulomb interaction, whereas <math>U < 0</math> could eventually describe an effective attractive interaction mediated by the ions. <math>\mathbf{i},\mathbf{j}</math> label the sites of a <math>D</math>-dimensional lattice <math>\mathbf{\Lambda}</math>, <math>\sigma = \uparrow,\downarrow</math> denotes the spin, and <math>a_{\mathbf{i},\sigma}^{\dagger},a_{\mathbf{j},\sigma}</math> are the electrons creation and annihilation operators, with <math>n_{\mathbf{i},\sigma} = a_{\mathbf{i},\sigma}^{\dagger}a_{\mathbf{i},\sigma}</math>. | a kinetic term describing the motion of electrons between neighboring sites (the hopping integral <math>t_{\mathbf{i},\mathbf{j}}</math> is usually restricted to nearest-neighbors, and is assumed translationally invariant, namely <math>t_{\mathbf{i},\mathbf{j}} = -t, t > 0</math>), and an on-site term, which approximates the interactions among electrons, whose strength is given by the parameter <math>U</math>. <math>U > 0</math> corresponds to repulsive Coulomb interaction, whereas <math>U < 0</math> could eventually describe an effective attractive interaction mediated by the ions. <math>\mathbf{i},\mathbf{j}</math> label the sites of a <math>D</math>-dimensional lattice <math>\mathbf{\Lambda}</math>, <math>\sigma = \uparrow,\downarrow</math> denotes the spin, and <math>a_{\mathbf{i},\sigma}^{\dagger},a_{\mathbf{j},\sigma}</math> are the electrons creation and annihilation operators, with <math>n_{\mathbf{i},\sigma} = a_{\mathbf{i},\sigma}^{\dagger}a_{\mathbf{i},\sigma}</math>. | ||
Revision as of 17:26, 8 December 2010
The Hubbard Model
Introduction
What it is
History of it
Utility
- The model referred to as the Hubbard model appeared in the literature for the first time in 1963, in two subsequent independent papers -- the first by Gutzwiller, and the second by Hubbard -- as an attempt to describe in a simplified way the effect of correlations for -electrons in a transition metals, In deed in the latter case the feature that electrons form a (narrow) delocalized -band is in competition with the quasi-atomic behavior originated from correlations, which would make plausible an atomic description of the problem. The model hamiltonian consists of two contributions,
- , (1.1.1)
a kinetic term describing the motion of electrons between neighboring sites (the hopping integral is usually restricted to nearest-neighbors, and is assumed translationally invariant, namely ), and an on-site term, which approximates the interactions among electrons, whose strength is given by the parameter . corresponds to repulsive Coulomb interaction, whereas could eventually describe an effective attractive interaction mediated by the ions. label the sites of a -dimensional lattice , denotes the spin, and are the electrons creation and annihilation operators, with .
The One-Dimensional Hubbard Model
Main focus, since cannot be solved exactly in Higher D.
Solution by Bethe Ansatz
Thermodynamic Properties
Higher Dimensions
Short discussion of usefulness of numerical results/methods
Related Physical Systems
Mott Insulators
Ultra-Cold atoms.