Phy5670/HubbardModel

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} -electrons in a transition metals, In deed in the latter case the feature that electrons form a (narrow) delocalized Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} -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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} consists of two contributions,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}} , (1.1.1)

a kinetic term describing the motion of electrons between neighboring sites (the hopping integral Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_{\mathbf{i},\mathbf{j}}} is usually restricted to nearest-neighbors, and is assumed translationally invariant, namely Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_{\mathbf{i},\mathbf{j}} = -t, t > 0} ), and an on-site term, which approximates the interactions among electrons, whose strength is given by the parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} . Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U > 0} corresponds to repulsive Coulomb interaction, whereas Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U < 0} could eventually describe an effective attractive interaction mediated by the ions. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{i},\mathbf{j}} label the sites of a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} -dimensional lattice Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{\Lambda}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma = \uparrow,\downarrow} denotes the spin, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{\mathbf{i},\sigma}^{\dagger},a_{\mathbf{j},\sigma}} are the electrons creation and annihilation operators, with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_{\mathbf{i},\sigma} = a_{\mathbf{i},\sigma}^{\dagger}a_{\mathbf{i},\sigma}} .

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.